Series in High Energy Physics, Cosmology, and Grayitytion
The
Standard Model
and Beyond
**'"' \\
Paul Langacker
0:
,. ^ CRC Press
' Taylor & Francis Group
A TAYLOR & FRANCIS BOOK

The Standard Model and Beyond

Series in High Energy Physics, Cosmology, and Gravitation
Series Editors: Brian Foster, Oxford University, UK
Edward W Kolb, Fermi National Accelerator Laboratory, USA
This series of books covers all aspects of theoretical and experimental high energy physics,
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Particle and Astroparticle Physics
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Joint Evolution of Black Holes and Galaxies
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Gravitation: From the Hubble Length to the Planck Length
I Ciufolini, E Coccia, V Gorini, R Peron, and N Vittorio (Eds.)
Neutrino Physics
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The Galactic Black Hole: Lectures on General Relativity and Astrophysics
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The Mathematical Theory of Cosmic Strings: Cosmic Strings in the Wire Approximation
M R Anderson
Geometry and Physics of Branes
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Modern Cosmology
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Gravitation and Gauge Symmetries
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P Fre, V Gorini, G Magli, and U Moschella (Eds.)
Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics
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Series in High Energy Physics, Cosmology, and Gravitation
The Standard Model
and Beyond
Paul Langacker
Institute for Advanced Study
Princeton, New Jersey, USA
@
CRC Press
Taylor & Francis Group
Boca Raton London New York
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Library of Congress Cataloging-in-Publication Data
Langacker, P.
The standard model and beyond / Paul Langacker.
p. cm. — (Series in high energy physics, cosmology, and gravitation)
Includes bibliographical references and index.
ISBN 978-1-4200-7906-7 (hardcover : alk. paper)
1. Standard model (Nuclear physics) I. Title.
QC794.6.S75L36 2010
539.7'2-dc22 2009040058
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Contents
Preface xi
1 Notation and Conventions 1
1.1 Problems 5
2 Review of Perturbative Field Theory 7
2.1 Creation and Annihilation Operators 7
2.2 Lagrangian Field Theory 9
2.3 The Hermitian Scalar Field 10
2.3.1 The Lagrangian and Equations of Motion 10
2.3.2 The Free Hermitian Scalar Field 12
2.3.3 The Feynman Rules 12
2.3.4 Kinematics and the Mandelstam Variables 15
2.3.5 The Cross Section and Decay Rate Formulae 17
2.3.6 Loop Effects 24
2.4 The Complex Scalar Field 25
2.4.1 £7(1) Phase Symmetry and the Noether Theorem ... 27
2.5 Electromagnetic and Vector Fields 30
2.5.1 Massive Neutral Vector Field 32
2.6 Electromagnetic Interaction of Charged Pions 33
2.7 The Dirac Field 39
2.7.1 The Free Dirac Field 40
2.7.2 Dirac Matrices and Spinors 41
2.8 QED for Electrons and Positrons 50
2.9 Spin Effects and Spinor Calculations 56
2.10 The Discrete Symmetries P, C, CP, T, and CPT 58
2.11 Two-Component Notation and Independent Fields 71
2.12 Quantum Electrodynamics (QED) 74
2.12.1 Higher-Order Effects 75
2.12.2 The Running Coupling 78
2.12.3 Tests of QED 82
2.12.4 The Role of the Strong Interactions 87
2.13 Mass and Kinetic Mixing 94
2.14 Problems 97
v

vi The Standard Model and Beyond
3 Lie Groups, Lie Algebras, and Symmetries 103
3.1 Basic Concepts 103
3.1.1 Groups and Representations 103
3.1.2 Examples of Lie Groups 105
3.1.3 More on Representations and Groups 106
3.2 Global Symmetries in Field Theory 112
3.2.1 Transformation of Fields and States 112
3.2.2 Invariance (Symmetry) and the Noether Theorem ... 114
3.2.3 Isospin and SU{3) Symmetries 119
3.2.4 Chiral Symmetries 130
3.2.5 Discrete Symmetries 132
3.3 Symmetry Breaking and Realization 133
3.3.1 A Single Hermitian Scalar 133
3.3.2 A Digression on Topological Defects 136
3.3.3 A Complex Scalar: Explicit and Spontaneous
Symmetry Breaking 137
3.3.4 Spontaneously Broken Chiral Symmetry 140
3.3.5 Field Redefinition 141
3.3.6 The Nambu-Goldstone Theorem 143
3.3.7 Boundedness of the Potential 145
3.3.8 Example: Two Complex Scalars 146
3.4 Problems 149
4 Gauge Theories 157
4.1 The Abelian Case 158
4.2 Non-Abelian Gauge Theories 160
4.3 The Higgs Mechanism 164
4.4 The Rz Gauges 169
4.5 Anomalies 177
4.6 Problems 179
5 The Strong Interactions and QCD 183
5.1 The QCD Lagrangian 186
5.2 Evidence for QCD 188
5.3 Simple QCD Processes 193
5.4 The Running Coupling in Non-Abelian Theories 198
5.4.1 The RGE Equations for an Arbitrary Gauge Theory . 199
5.5 Deep Inelastic Scattering 202
5.5.1 Deep Inelastic Kinematics 203
5.5.2 The Cross Section and Structure Functions 204
5.5.3 The Simple Quark Parton Model (SPM) 209
5.5.4 Corrections to the Simple Parton Model 213
5.6 Other Short Distance Processes 217
5.7 The Strong Interactions at Long Distances 220
5.8 The Symmetries of QCD 221

Table of Contents vii
5.8.1 Continuous Flavor Symmetries 221
5.8.2 The (3*, 3) + (3,3*) Model 223
5.8.3 The Axial £7(1) Problem 229
5.8.4 The Linear a Model 232
5.8.5 The Nonlinear a Model 235
5.9 Other Topics 236
5.10 Problems 237
6 The Weak Interactions 239
6.1 Origins of the Weak Interactions 239
6.2 The Fermi Theory of Charged Current Weak Interactions . . 245
6.2.1 ii Decay 250
6.2.2 i/ce~ -> i/ce~ 256
6.2.3 7T and K Decays 258
6.2.4 Nonrenormalization of Charge and the Ademollo-Gatto
Theorem 266
6.2.5 P Decay 268
6.2.6 Hyperon Decays 273
6.2.7 Heavy Quark and Lepton Decays 274
6.3 Problems 278
7 The Standard Electroweak Theory 281
7.1 The Standard Model Lagrangian 281
7.2 Spontaneous Symmetry Breaking 287
7.2.1 The Higgs Mechanism 287
7.2.2 The Lagrangian in Unitary Gauge after SSB 289
7.2.3 Effective Theories 304
7.2.4 The fy Gauges 306
7.3 The Z, the W, and the Weak Neutral Current 308
7.3.1 Purely Weak Processes 309
7.3.2 Weak-Electromagnetic Interference 321
7.3.3 Implications of the WNC Experiments 328
7.3.4 Precision Tests of the Standard Model 330
7.3.5 The Z-Pole and Above 340
7.3.6 Implications of the Precision Program 350
7.4 Gauge Self-Interactions 358
7.5 The Higgs 362
7.5.1 Theoretical Constraints 362
7.5.2 Experimental Constraints and Prospects 368
7.6 The CKM Matrix and CP Violation 371
7.6.1 The CKM Matrix 372
7.6.2 CP Violation and the Unitarity Triangle 376
7.6.3 The Neutral Kaon System 378
7.6.4 Mixing and CP Violation in the B System 391
7.6.5 Time Reversal Violation and Electric Dipole Moments 397

Vlll
The Standard Model and Beyond
7.6.6 Flavor Changing Neutral Currents (FCNC) 401
7.7 Neutrino Mass and Mixing 404
7.7.1 Basic Concepts for Neutrino Mass 405
7.7.2 The Propagators for Majorana Fermions 417
7.7.3 Experiments and Observations 418
7.7.4 Neutrino Oscillations 425
7.7.5 The Spectrum 438
7.7.6 Models of Neutrino Mass 440
7.7.7 Implications of Neutrino Mass 446
7.8 Problems 447
8 Beyond the Standard Model 453
8.1 Problems with the Standard Model 453
8.2 Supersymmetry 463
8.2.1 Implications of Supersymmetry 463
8.2.2 Formalism 469
8.2.3 Supersymmetric Interactions 482
8.2.4 Supersymmetry Breaking and Mediation 490
8.2.5 The Minimal Supersymmetric Standard Model (MSSM)
493
8.2.6 Further Aspects of Supersymmetry 505
8.3 Extended Gauge Groups 508
8.3.1 SU(2) x 17(1) x U(l)f Models 5l6
8.3.2 SU(2)L x SU(2)R x 17(1) Models 519
8.4 Grand Unified Theories (GUTs) 525
8.4.1 The SU(5) Model 527
8.4.2 Beyond the Minimal SU(5) Model 534
8.5 Problems 537
A Canonical Commutation Rules 541
B Derivation of a Simple Feynman Diagram 545
C Unitarity, the Partial Wave Expansion and the Optical
Theorem 547
D Two, Three, and n-Body Phase Space 549
E Calculation of the Anomalous Magnetic Moment of the
Electron 555
F Breit-Wigner Resonances 559
G Implications of P, C, T, and G-parity for Nucleon
Matrix Elements 563

Table of Contents ix
H Collider Kinematics 567
I Quantum Mechanical Analogs of Symmetry Breaking 573
References 577
1.1 Field Theory 577
1.2 The Standard Model and Particle Physics 578
1.3 The Strong Interactions, QCD, and Collider Physics 579
1.4 The Electroweak Interactions 580
1.5 CP Violation 581
1.6 Neutrinos 581
1.7 Supersymmetry, Strings, and Grand Unification 582
1.8 Astrophysics and Cosmology 583
1.9 Groups and Symmetries 584
1.10 Articles 584
Web Sites 631
Index 635

Preface
In the last few decades there has been a tremendous advance in our
understanding of the elementary particles and their interactions. We now have a
mathematically consistent theory of the strong, electromagnetic, and weak
interactions—the standard model—most aspects of which have been
successfully tested in detail at colliders, accelerators, and non-accelerator
experiments. It also provides a successful framework and has been strongly
constrained by many observations in cosmology and astrophysics. The standard
model is almost certainly an approximately correct description of Nature down
to a distance scale l/1000£/l the size of the atomic nucleus.
However, nobody believes that the standard model is the ultimate theory:
it is too complicated and arbitrary, does not provide an understanding of the
patterns of fermion masses and mixings, does not incorporate quantum
gravity, and it involves several severe fine-tunings. Furthermore, the origins of
electroweak symmetry breaking, whether by the Higgs mechanism or
something else, are uncertain. The recent discovery of non-zero neutrino mass can
be incorporated, but in more than one way, with different implications for
physics at very short distance scales. Finally, the observations of dark matter
and energy suggest new particle physics beyond the standard model.
Most current activity is directed towards discovering the new physics which
must underlie the standard model. Much of the theoretical effort involves
constructing models of possible new physics at the TeV scale, such as super-
symmetry or alternative models of spontaneous symmetry breaking. Others
are examining the extremely promising ideas of superstring theory, which
offer the hope of an ultimate unification of all interactions including
gravity. There is a lively debate about the implications of a landscape of possible
string vacua, and serious efforts are being made to explore the consequences
of string theory for the TeV scale. It is likely that a combination of such
bottom-up and top-down ideas will be necessary for progress. In any case,
new experimental data are urgently needed. At the time of this writing the
particle physics community is eagerly awaiting the results of the Large Hadron
Collider (LHC) and is optimistic about a possible future International Linear
Collider. Future experiments to elucidate the properties of neutrinos and to
explore aspects of flavor, and more detailed probes of the dark energy and
dark matter, are also anticipated.
The purpose of this volume is to provide an advanced introduction to the
physics and formalism of the standard model and other non-abelian gauge
theories, and thus to provide a thorough background for topics such as super-
XI

Xll
The Standard Model and Beyond
symmetry, string theory, extra dimensions, dynamical symmetry breaking,
and cosmology. It is intended to provide the tools for a researcher to
understand the structure and phenomenological consequences of the standard
model, construct extensions, and to carry out calculations at tree level. Some
"old-fashioned" topics which may still be useful are included. This is not a
text on field theory, and does not substitute for the excellent texts that
already exist. Ideally, the reader will have completed a standard field theory
course. Nevertheless, Chapter 2 of this book presents a largely self-contained
treatment of the complicated technology needed for tree-level calculations
involving spin-0, spin-|, and spin-1 particles, and should be useful for those
who have not studied field theory recently, or whose exposure has been more
formal than calculational*. It does not attempt to deal systematically with
the subtleties of renormalization, gauge issues, or higher-order corrections.
An introductory-level background in the ideas of particle physics is assumed,
with occasional reference to topics such as gluons or supersymmetry before
they are formally introduced. Similarly, occasional reference is made to
applications to and constraints from astrophysics and cosmology. The necessary
background material may be found in the sources listed in the bibliography.
Chapter 1 is a short summary of notations and conventions and of some
basic mathematical machinery. Chapter 2 contains a review of calculational
techniques in field theory and the status of quantum electrodynamics.
Chapters 3 and 4 are concerned with global and local symmetries and the
construction of non-abelian gauge theories. Chapter 5 examines the strong
interactions and the structure and tests of Quantum Chromodynamics (QCD)*
Chapters 6 and 7 examine the electroweak interactions and theory,
including neutrino masses. Chapter 8 considers the motivations for extending the
standard model, and examines supersymmetry, extended gauge groups, and
grand unification. There are short appendices on additional topics. The
bibliographies list many useful reference books, review articles, research papers,
and Web sites. No attempt has been made to list all relevant original articles,
with preference given instead to later articles and books that can be used to
track down the original ones. Supplementary materials and corrections are
available at http://www.sns.ias.edu/~pgl/SMB/. Comments, corrections,
and typographical errors can also be sent through that site.
I would like to thank Mirjam Cvetic, Jens Erler, Hye-Sung Lee, Gil Paz,
Liantao Wang, and Itay Yavin for reading and commenting on parts of the
manuscript, Lisa Fleischer for help in the preparation of the manuscript, and
my wife Irmgard for her extreme patience during the writing.
Paul Langacker
July 4, 2009
*Most calculations, especially at the tree-level, are now carried out by specialized computer
programs, many of which are included in the list of Web sites, but it is still important to
understand the techniques that go into them.

1
Notation and Conventions
In this chapter we briefly survey our notation and conventions.
Conventions
We generally follow the conventions used in (Langacker, 1981). In
particular, (/j,,v,p,a) are Lorentz indices; (z,j, k = 1---3) are three-vector indices;
(z, j, /c = 1 • • • N) are also used to label group generators or elements of the
adjoint representation; (a, 6, c) run over the elements of a representation, while
(a,/?,7) and (r,s,t) refer to the special cases of color and flavor,
respectively. (a,/3) are also occasionally used for Dirac indices. (ra,n) are used
as horizontal (family) indices, labeling repeated fermions, scalars, and
representations. The summation convention applies to all repeated indices except
where indicated. Operators are represented by capital letters (Tl, Q, Y), their
eigenvalues by lower case letters (tl,q,y), and their matrix representations
by (L1,Lq,Ly)- In Feynman diagrams, ordinary fermions are represented by
solid lines, spin-0 particles by dashed lines, gluons by curly lines, other gauge
bosons by wavy lines, and gluinos, neutralinos, and charginos by single or
double lines. Experimental errors are usually quoted as a single number, with
statistical, systematic, and theoretical uncertainties combined in quadrature
and asymmetric errors symmetrized.
Units and Physical Constants
We take h = c = 1, implying that E, p, m, ^, j have "energy units." One
can restore units at the end of a calculation using the values of ft, c, and he
listed in Table 1.1.
Operators and Matrices
The commutator and anti-commutator of two operators or matrices are
[A, B] = AB- BA, {A, B} = AB + BA. (1.1)
The transpose, adjoint, and trace of an n x n matrix M are
transpose: MT (Mj6 = M6a), adjoint: Mf = MT* (1.2)
1

2
The Standard Model and Beyond
TABLE 1.1
Conversions and physical constants. For more precise values, see (Amsler
et al., 2008). The fine structure constant is a = e2/4ir, where e > 0 is the
— 1/2
charge of the positron. The Planck constant is Mp = GN .
h ~ 6.6 x 10"22 MeV-s c-3.0x 1010 cm/s he ~ 197 MeV-fm
a-1 ~ 137.04 a~l(M2) ~ 128.9 sin2 6W(M2) ~ 0.2312
ap(M2) - 0.034 dg\M\) ~ 0.010 as(M2) ~ 0.118
GF ~ 1.17 x 10"5 GeV"2 Mw ~ 80.40 GeV Mz ~ 91.19 GeV
rae - 0.511 MeV raM ~ 105.7 MeV mr ~ 1.78 GeV
mp ~ 938 MeV mn± ~ 140 MeV mK± ~ 494 MeV
MP - 1.22 x 1019 GeV k ~ 1.16 x 104 °K/eV
trace: TrM = ^Maa, Tr(M1M2) = Tr(M2M1), TrM = TrMT.
(1.3)
a=l
Vectors, Metric, and Relativity
Three-vectors and unit vectors are denoted by x and x = x/\x\, respectively.
We do not distinguish between upper and lower indices for three-vectors; e.g.,
the inner (dot) product x • y may be written as xlyl,x'lyi, or Xiyi. The Levi-
Civita tensor is e^fc, where i,j,k = 1 • • • 3, is totally antisymmetric, witji
^123 = 1. Its contractions are !
^ijk^ijk = u, ^ijk^ijm = ^km , .,
^ijk^imn = ^jm^kn ^jn^kmi
where the Kronecker delta function is
^-{ki7r (15)
Cijk is useful for vector cross products and their identities. For example,
(AxB)i = €ijkAjBk (1.6)
(lxB).(Cx5) = e^e^-ftG^ = (I-C)(B.D)-(1-5)(C.5). (1.7)
Notations for four-vectors and the metric are given in Table 1.2.
The four-momentum of a particle with mass m is / = (E,p) with p2 =
E2 —p2 = m2. (The symbol p is occasionally used to represent \p\ rather
than a four-vector, but the meaning should always be clear from the context.)
The velocity (3 and energy are given by

Notation and Conventions
Under a Lorentz boost by velocity (3l (the relativistic addition of (3l to /?,
which is equivalent to going to a new Lorentz frame moving with — (3l)
where
with
E' = 1L(E + 0L • P), p' =P±+ 7l(P\\ + PlE),
P\\=PlPl'P, P±=P-P\\, 1L = -^==5=.
V1 ~/*L
(1.9)
(1.10)
(1.11)
TABLE 1.2
Notations and conventions for four-vectors and the metric
contravariant four-vector A^ = (A°,A), xM = (t,x)
covariant four-vector A^ = g^uAy = (A0, — A), xM =
(*,-£)
metric
Lorentz invariant
derivatives
antisymmetric tensor
contractions
^=^ = diag(l, -1,-1,-1)
{!» M = v
A • B = A^B» = g^A^B" = A°B° -A-3
d = a^ = & - v2
a4°
d-A = d^A» = ^ + V • if
a*dH = ad^fc - (<9^a)6
e^a, with e0i23 = +1 and e0123 = -1
eLivpae
'-fjti/par
= -24 ewe
-\xvpr
-6<
^%,ru, = -2 (<??<£ -<£<£)
Translation Invariance
Let PM be the momentum operator, \i) and |/) momentum eigenstates,
P"|t)=pt'|i>, P"|/)=p?|/>, (1.12)
and let O(x) be an operator defined at spacetime point x, so that
0(x)
AP>x
O(0)e
-iPx
(1.13)

4 The Standard Model and Beyond
Then the x dependence of the matrix element (f\0(x)\i) is given by
</|O(x)|i>=e^/-«)-*</|O(0)|i). (1.14)
The combination of Lorentz and translation invariance is Poincare invariance.
The Pauli Matrices
The 2x2 Pauli matrices a = (cti,0-2,0-3) (also denoted by r, especially for
internal symmetries) are Hermitian, &i = aj, and defined by
[<Ti,<Tj] = 2ieijkak. (1.15)
A convenient representation is
*-(! J). -("-;)• -(;.!)■ ("«>
There is no distinction between Oi and a1. Some useful identities include
2
Tr <7i = ^ aiaa = 0, TV (a^) = 2Sij
a=l \ ' '
{(7i, Gj] = 2(^7 => O^i = I, OiOj = (Jyi + iCijkCTk.
The last identity implies
(A- ff)(B • a) = A- BI + i(Ax B) • ff, (1.1$)
where A and J5 are any three-vectors (including operators) and A • a is a 2 x 2
matrix. Thus, (A• a)2 = A2/ for an ordinary real vector A with A= \A\, and
e*x* = (cosA)I + i(sinA)A-a. (1.19)
Any 2x2 matrix M can be expressed in terms of a and the identity by
M = ^Tr (M)/ + ^Tr (Ma) • a. (1.20)
The SU(2) Fierz identity is given in Problem 1.1.
The Delta and Step Functions
The Dirac delta function S(x) is defined (for our purposes) by
/
+oo
S(x - a)g(x)dx = g(a) (1-21)
for sufficiently well-behaved g(x). Useful representations of S(x) include
1 /*+0° 1
S(x-a) = — / eik^x-a^dk = — li
V ' 2-Kj-oc 7T7-
lim
o
7
(x — a)2 +72J
(1.22)

Notation and Conventions
The derivative of S(x) is defined by integration by parts,
r+°° r+°°d6(x-a) ... dg
/+oo r-\
8'{x - a)g(x)dx = I
-oo «/-c
-g(x)dx = —
dx *y } dx
Suppose a well-behaved function f(x) has zeroes at #oi- Then
(1.23)
'<**»-E^- <->
The step function, 0(#), is defined by
0, x < xl
0(x-*') = {£ l^, (1.25)
from which S(x) = dO/dx.
1.1 Problems
1.1 Let Xn, n = 1 • • • 4 be arbitrary Pauli spinors (i.e., two-component
complex column vectors). Then the bilinear form xlnaiXn is an ordinary number.
Prove the Fierz identity
(xl^Xs) • (X2<?Xi) = 2r/F(X4Xi)(X2X3) - (xlxsXx^Xi)*
where rjj? = + 1. (The identity also holds for anticommuting two-component
fields if one sets rjp = —1.) Hint: expand the 2x2 matrix X1X2 m (X4X1XX2X3)
using (1.20).
1.2 Justify the result (1.24) for 8(f(x)).
1.3 Calculate the surface area f dfln of a unit sphere in n-dimensional
Euclidean space, so that f dnk = f dQ,n J0°° kn~ldk. Show that the general
formula yields
J dn± = 2, J dQ2 = 2tt, J dCt3 = 4tt, J dQ4 = 2tt2.
Hint: Use the Gaussian integral formula
/+00 r~
e-ax2dx= J- for Re a >0
to integrate f dnke~ak in both Euclidean and spherical coordinates.

6 The Standard Model and Beyond
1.4 Show that the Lorentz boost in (1.10) can be written as
\p\\ J Vsinh2/L ^shyLJ \p\\ J '
where
is the rapidity of the boost.

2
Review of Perturbative Field Theory
Field Theory is the basic language of particle physics (i.e., of point particles).
It combines quantum mechanics, relativistic kinematics, and the notion of
particle creation and annihilation. The basic framework is remarkably
successful and well-tested. In this book we will work mainly with perturbative
field theory, characterized by weak coupling.
The Lagrangian of a field theory contains the interaction vertices.
Combined with the propagators for virtual or unstable particles one can compute
scattering and decay amplitudes using Feynman diagrams. Here we will review
the rules (but not the derivations) for carrying out field theory calculations
of amplitudes and the associated kinematics for processes involving spin-0,
spin-^, and spin-1 particles in four dimensions of space and time, and give
examples, mainly at tree level. Much more detail may be found in such field
theory texts as (Bjorken and Drell, 1964, 1965; Weinberg, 1995; Peskin and
Schroeder, 1995).
2.1 Creation and Annihilation Operators
Let |0) represent the ground state or vacuum, which we define as the no
particle state (we are ignoring for now the complications of spontaneous symmetry
breaking). The vacuum is normalized (0|0) = 1. We will use a covariant
normalization convention*. For a spin-0 particle, define the creation and
annihilation operators for a state of momentum pas a\p) and a(p), respectively,
i.e.,
a(j?)|0)=0, a\p)\0) = \p), (2.1)
*The alternative non-covariant convention, in which some formulas are simpler, is
Wntifyyahd)')] = 63(p-p'), where an(p) is related to a(p) by a(p) = yj,(2n)32Epan(p).
The corresponding non-covariantly normalized single-particle state is \p)n = an(p) |0), and
the integration over physical states is simply / d3p. Yet another possibility is box
normalization in a volume V = L3, leading to discrete three-momenta with ith component
Pi =raj^,Tij = 1,2, ••• , and commutators [ob(p),o^(p')] =$£$'•
7

8 The Standard Model and Beyond
where \p) describes a single-particle state with three-momentum p, energy
Ep = yfp2 + m2, and velocity (3 = p/Ep. We assume the commutation rules
(for Bose-Einstein statistics)
[a(p),a\p')} = (2^)32Ep63(p-p'), [a(p),a(p')} = [at(p),a+(p')] = 0,
(2.2)
which correspond to the state normalization
(p\p')^(2n)32EpS3(p-p'). (2.3)
This is Lorentz invariant because of the Ep factor. This can be seen from the
fact that the integration over physical momenta is
d3p d4p
(2tt)32£p (2tt)3
6(p2-m2)G(p% (2.4)
where 9(x) is the step function and we have used (1.24). The right-hand side
of (2.4) is manifestly invariant. The additional 2(27r)3 factor is for convenience.
The interpretation of (2.2) is that each momentum p of a non-interacting
particle can be described by a simple harmonic oscillator. The number
operator Af(p), which counts the number of particles with momentum p in a state,
and the total number operator TV, which counts the total number of particles,
are given by
N{p) = a\p)a{p), N = J *f2E N(p). (2J5)
(2.1)-(2.5) actually hold for any real or complex bosons, provided one adds
appropriate labels for particle type and (in the case of spin-1, 2, • • •) for spin.
For real (i.e., describing particles that are their own antiparticles) spin-0 fields
of particle types a and 6, for example,
[aa(p), al(p')} = Sab(2ir)32Ep83(p-p'), \aa(p), ab(p')] = [a£ (p), a+(p')] = 0.
(2.6)
Similarly, for a complex scalar, describing a spin-0 particle with a distinct
antiparticle, it is conventional to use the symbols a* and tf for the particle
and antiparticle creation operators, respectively. (It is a convention which
state is called the particle and which the antiparticle.) E.g., for the 7r+ state,
a+(p)|0) = |7r+(p)>, &V)|0) = \n~(p)), (2.7)
with
\b(p),b\p')} = (2ir)32Ep83(p-p'), [a(p),b^(p')} - [a(p),b(p')} = 0.
(2.8)
The creation and annihilation operators for fermions are similar, except that
they obey the anti-commutation rules appropriate to Fermi-Dirac statistics.

Review of Perturbative Field Theory 9
The creation operator for a spin-| particle is a^(p, s), where s refers to the
particle's spin orientation, which may be taken with respect to a fixed z axis
or with respect to p (helicity). Then,
(2.9)
{a(p, s), aT(p', s')} = a(p, s)aT(p', s') + aT(p', s')a(p, 5)
= (2n)32Ep83(p-p')8aa>.
Similarly, for the antiparticle,
{b(p,s),b\p',s')} = (27r)32Ep63(p-p')6ss,, (2.10)
while
{a, a} = {6,6} = {a, 6} = {a, 6f} - 0 (2.11)
for all values of p,p',s, and s'. Fermion and boson operators commute with
each other, e.g., [aboson, ^fermion] = 0.
Non-interacting multi-particle states are constructed similarly. For
example, the state for two identical bosons is
\P1P2) = \P2P1) = at(pi)at(ft)|0> (2.12)
with
(piP2\P3P4) = (27r)32El(27r)32E2 [S3^ -p3)S3(p2 -ft)
+ 53(Pl-ft)53(P2-P3)].
Similarly, for two identical fermions,
\p1SuP2S2) = -\P2S2\P1S1) = at(pi3i)at(ft32)|0), (2.14)
with
<pi«i;ft«2|ft«3;P4«4> = (27r)32E1(27r)32E2 [<*3(pi -P3)^lS3(53(p2 -p4)5S2S4
- <*3(Pi -p4)(5SlS4(53(P2 -ft)^2S3] • (2-15)
2.2 Lagrangian Field Theory
Consider a real or complex field </>(#), where x = (t,x). The (Hermitian)
Lagrangian density
C(<t>(x),d^(x),<t>\x),d^(x)) (2.16)

10
The Standard Model and Beyond
contains information about the kinetic energy, mass, and interactions of </>.
We will generally use the simpler notation £(</>, dM</>), or just £(#), with the
understanding that for a complex field C can depend on both </> and its
Hermitian conjugate </>*. (2.16) is trivially generalized to the case in which there
are more than one field. It is useful to also introduce the Lagrangian L(t) and
the action S by integrating C over space and over space-time, respectively,
//*+oo /»
d3x £(</>,<9M</>), S = / dt L(t) = / d4x £(</>,<9M</>). (2.17)
The Euler-Lagrange equationss of motion for </> are obtained by minimizing
the action with respect to </>(#) and </>*(#),
%-*m-* (218)
and similarly for </>*. The fields </> are interpreted as operators in the Heisenberg
picture, i.e., they are time-dependent while the states are time independent.
Other quantities, such as the conjugate momentum, the Hamiltonian, and the
canonical commutation rules, are summarized in Appendix A.
2.3 The Hermitian Scalar Field
A real (or more accurately, Hermitian) spin-0 (scalar) field, which satisfies
</>(#) = </>*(#), is suitable for describing a particle such as the 7r° which has no
internal quantum numbers and is therefore the same as its antiparticle.
2.3.1 The Lagrangian and Equations of Motion
The Lagrangian density for a Hermitian scalar is
C{4>,d^) = l- [(dM4>)(d»<t>) - mV] - TO)
^[(^)2-mV]-TO).
(2.19)
The first two terms correspond respectively to canonical kinetic energy and
mass (the \ is special to Hermitian fields), while the last describes interactions.
(dM</>)2 is a shorthand for (dM</>)(dM</>). The interaction potential is
<t>3 <t>4 <t>b
V/(</>) = k-^- + A— + c + di<j) + db^- + • • • + non - perturbative, (2.20)

Review of Perturbative Field Theory
11
where the k\ factors are for later convenience in cancelling combinatoric
factors^, and "non-perturbative" allows for the possibility of non-polynomial
interactions. The constant c is irrelevant unless gravity is included. A
nonzero d\ will necessarily induce a non-zero vacuum expectation value (VEV),
(O|0|O) 7^ 0, suggesting that one is working in the wrong vacuum. The di
term can be eliminated by a redefinition of <j> —» </>' = constant + </>, as will
be described in Chapter 3. C and <f> have dimensions of 4 and 1, respectively,
in mass units, so the coefficient of </>fc has the mass dimension 4 — k. The
dk terms with k > 5 are known as non-renormalizable or higher-dimensional
operators. They lead to new divergences in each order of perturbation theory,
with dk typically of the form dk = Ck/Mk~4, where c^ is dimensionless and
M is a large scale with dimensions of mass. Such terms would be absent in
a renormalizable theory, but may occur in an effective theory at low energy,
where they describe the effects of the exchange of heavy particles (or other
degrees of freedom) of mass A4 that are not explicitly taken into account in
the field theory. (In Chapters 6 and 7 we will see that an example of this is
the four-fermi operators that are relevant to describing the weak interactions
at low energy.) Keeping just the renormalizable terms (and c = d\ = 0), one
has 34
W) = 4r+ Atr (2-21)
where k (dimensions of mass) and A (dimensionless) describe three- and four-
point interactions, respectively, as illustrated in Figure 2.1. Prom the Euler-
FIGURE 2.1
Three- and four-point interactions of a Hermitian scalar field </>. The factor is
the coefficient of </>n/n! in z£, as described in Appendix B.
Langrange equation (2.18), one obtains the field equation
(□ + m2) </> + ^ = (□ + m2) </> + Ky + A^- = 0, (2.22)
where □ +ra2 = d^d^ +m2 is the Klein-Gordon operator. The expression for
the Hamiltonian density is given in Appendix A.
* Conventions for such factors may change, depending on the context.

12
The Standard Model and Beyond
2.3.2 The Free Hermitian Scalar Field
Let </>o = </>o be the solution of (2.22) in the free (or non-interacting) limit
k — A = 0, i.e.,
(n + m2)</>o(x)=0. (2.23)
(2.23) can be solved exactly, and then small values of the interaction
parameters k and A can be treated perturbatively (as Feynman diagrams). The
general solution is
M*) = 4>\{x) = J {J*f2Ep [a(p)e-^ + aHp)e+i>-*] , (2.24)
where x = (t,x) and p = (Ep,p) with Ep = yjp2 + m2, i.e., the four-
momentum p in the Fourier transform is an on-shell momentum for a particle
of mass m. The canonical commutation rules for </>o and its conjugate
momentum given in Appendix A will be satisfied if the Fourier coefficients a(p)
satisfy the creation-annihilation operator rules in (2.2).
It is useful to define the Feynman propagator for </>o,
iAF(x-x') = <O|T[0o(s),0o(s')]|O>, (2.25)
where
T[<h{x), M*')} = ©(* " t')<t>o(x)<t>o(x') + Q(t' - t)(t>0{x')(t>o{x) (2.26)
represents the time-ordered product of (j>o(x) and (f)o(xf). In (2.26) &(t — t') is
the step function, defined in (1.25). Aj?(x — x') is just the Green's function
of the Klein-Gordon operator, i.e.,
(Ux + m2) AF(x - x') = -S4(x - x'), (2.27)
where Ux refers to derivatives w.r.t. x. The momentum space propagator
AF(k) is
AF(k)= f d4xe+ikxAF(x)
_ ! _ ! (2.28)
~ k2 - m2 + ie ~ kl - k2 - m2 + ie
k is an arbitrary four-momentum, i.e., it need not be on shell. The on-shell
limit is correctly handled by the ie factor in the denominator, where e is a
small positive quantity that can be taken to 0 at the end of the calculation.
2.3.3 The Feynman Rules
The Feynman rules allow a systematic diagrammatic representation of the
terms in the perturbative expansion (in k and A) of the transition amplitude

Review of Perturbative Field Theory
Mfi between an initial state i and a final state /. The derivation is beyond
the scope of this book, but can be found in any standard field theory text.
(The derivation of a simple example is sketched in Appendix B.) Heuristic
derivations may also be found, e.g., in (Bjorken and Drell, 1964; Renton,
1990). For the Hermitian scalar field with the potential (2.21), the rules are
• Draw each connected topologically-distinct diagram in momentum space
corresponding to initial (final) states i (/), with internal lines
corresponding to virtual (intermediate) particles. The internal and external
lines are joined at three- and four-point vertices corresponding to the
interactions in Vj. Each external and internal line has an associated
four-momentum, which is off-shell for the virtual particles. It is
convenient to put an arrow on the line to indicate the direction of momentum
flow. This direction is only a convention, and for the Hermitian scalar
field (with no internal quantum numbers) there is no restriction on how
many arrows flow into or out of a vertex.
• There is a factor of —in at every three-point vertex and a factor — iX
at each four-point vertex, as in Figure 2.1. These correspond to the
coefficients of </>3/3! and </>4/4!, respectively, in iC, with the 1/3! for k
cancelling against 3! ways to associate the three lines with the three
fields in </>3, and similarly for the A term.
• There is a factor of iAjr(k) = ki_^,ie for each internal line with four-
momentum k.
• Four-momentum is conserved at each vertex, implying that the overall
four-momentum is conserved (i.e., Mfi is only defined for T,pi = £/?/).
• Integrate over each unconstrained internal momentum (there will be one
for each internal loop in the diagram), with a factor f /f^yr-
• There may be additional combinatoric factors* associated with the
interchange of internal lines for fixed vertices, e.g., a factor of 1/n! if n
internal lines connect the same pair of vertices, as in Figure 2.2.
The ordering and arrangement of the external lines in a Feynman diagram is
irrelevant (although the relative ordering between two diagrams does matter
for fermions). In this book we we usually, but not always, place the initial
state particles at the bottom and the final particles at the top.
As a simple example, the tree-level diagrams for the 2 —> 2 scattering
amplitude Mfi = (p3P4\M\piP2) are shown in Figure 2.3. Applying the Feynman
*In general, there may be subtleties involving combinatorial factors (or, signs when fermions
are involved), especially in higher-order diagrams, which are best resolved by returning to
the original derivation.

14
The Standard Model and Beyond
\
\ .-- /
/
/ x
/
/
/
/
/
y > * -V-
\
\
FIGURE 2.2
Diagrams with additional factors of 1/2! (left) and 1/3! (right), required
because not all of the (4!)2 ways of associating the four fields at each vertex with
four lines lead to distinct diagrams.
PS P4 PS P4 PS P4
n y n y ^^ "^ ^~
Pi
+
IIP
P2
PS
= -i\
pi
P4
J
— IK *---►--* -IK
^/iAF(pi3)X>y
n. y
+
V
+ A *af(p1 + P2);
P2 Pi P2
P4 PS
— IK *---►--* -IK
/iAF(p14)\
Pi P2 Pi P2
FIGURE 2.3
Tree level diagrams for Mji = {pzP4\M\piP2) for a Hermit ian scalar field. The
arrows label the directions of momentum flow, and pij = Pi — Pj. Note that
the arrangement of lines in the last diagram is modified to allow the diagram
to be drawn without crossing lines.

Review of Perturbative Field Theory
15
rules, these diagrams correspond to the expression
Mfi = -zA + (-m)2
+ -; o +
(2.29)
t — m2 u — m2
where s, t, and u are the Mandelstam variables
s = (pi + p2)2 = (j>3+J>4)2
t = (pi-ps)2 = (P4-P2)2 (2.30)
U=(pi- P4)2 = (P3 ~ J>2)2.
For the present (equal mass) case, s = E^,M > 4m2, t < 0, and u < 0,
where £?cm is the total energy in the center of mass. The internal lines are
never on-shell (s,t, u ^ m2) for physical (on-shell) external momenta, so one
can drop the +ie. It is implicit that the external momenta satisfy the four-
momentum conservation p\ + P2 = p% +7)4. The second, third, and fourth
diagrams in Figure 2.3 are said to have s-channel, t-channel, and u-channel
poles, respectively.
2.3.4 Kinematics and the Mandelstam Variables
Let us digress to generalize to the case of a 2 —» 2 scattering process 1 + 2 —»
3 + 4, where we allow for the possibility of inelastic scattering with unequal
masses for 1, 2, 3, and 4. In the absence of spin, the scattering amplitude can
be expressed in terms of the Lorentz invariant Mandelstam variables defined
in (2.30). 5, t, and u are not independent, but are related by
s + t + u = m\+ m\ + m2 + m2. (2.31)
Of course, s = m\ + m^ + 2pi • j>2> etc.
The kinematics is simplest in the center of mass (CM) frame, which is more
accurately the center of momentum, in which the total three-momentum of
the initial and final state vanishes:
Pi = (£7i,Pi), P2 = {E2, -pi)
p3 = (E3,pf), p4 = (£4, -p/),
where pi and pj are respectively the initial and final three-momenta; the
energy and velocity of particle 1 are A
El=pi2+m\, & = |?-, (2.33)
and similarly for 2, 3, and 4; and the CM scattering angle 8 is related by
__Pi'Pf =PiPf cos 6, (2.34)

16
The Standard Model and Beyond
Pi /9
1 //-pi 2
i
2 \
' o3
\4
(iii)
FIGURE 2.4
Scattering kinematics in (i) the center of mass, (ii) the lab, and (iii) the Breit
frames.
as shown in Figure 2.4. In the CM frame, s = (Ex + E2)2 = (E3 + E4)2
is just the square of the total energy. In the physical scattering region, s >
(mi + mi)2 and s > (ra3 + 7714)2. Using P2 = p\ + P2 — Pi one finds
El
s + m\ — m2
2yfs
V~s
7721=7722 2
(2.35)
so that
Pi
f— - [s - (mi + m2)2] [s - (mi - m2)2]
* = \lE\-m\ = I L-^= 1
[s -4m2]1/2
1/2
(2.36)
7721=7722 2
with similar expressions for the other particles. This is sometimes written as
where
Pi
2
2^/5
A(x, y,z) =x + y + z — 2xy — 2xz — 2yz.
(2.37)
(2.38)
The t and u variables, which describe the momentum transfer between
particles 1 and 3 and between 1 and 4, respectively, are given in the CM by
t =m\ + ra2 — 2EiE3 + 2piPf cos 8
> -2p2(l - cos 6) = -Ap2 shT - < 0
mi=m3 ^ v y ^ 2 ~~
7722=7724
w =ra2 + ra2 — 2E1E4 — 2piPf cos 6
Q
-2p2(l + cos6) = -4p2 cos2 - < 0,
(2.39)
7721=7724
7722=7723
where the last expressions are for elastic scattering (mi = 7713, 7712 = 7714 or
mi = 7714, 7712 = 7713), for which Pi = pf = p. Note that t and u are negative

Review of Perturbative Field Theory 17
at high energies (e.g., the last expressions in (2.39) hold with p ~ y/s/2 when
the masses can be neglected), but may be positive at low energies if the masses
are not all equal.
Fixed target experiments are carried out in the laboratory frame, in which
2 is at rest,
p1 = (Eup1), j>2 = (m2,0), (2.40)
with
= s-m -m2
2ra2 y J
A sometimes useful relation between the CM and laboratory variables is
Pi = \Pi\m2/y/s, (2.42)
where pi is the CM momentum in (2.36). Prom energy and momentum
conservation,
E3 + E4 = E± + ra2
\P4\2 = |pi|2 + \p3\2 ~ 2|pi||p3| cosfls,
so that E3 and £4 can be expressed in terms of s and the laboratory scattering
angle #3 for particle 3. The relations between the laboratory variables and t
and u are straightforward. For example,
t = ml + m\- 2E1E3 + 2|p1||p3| cos<93
u = m\ + m\ — 2ra2£,3.
These formulae become especially simple in the special case mi = 7713 = 0
and ra2 = 7714, e.g.,
E3 1
2,2.™ (2'44)
£l 1 + |L(1-COS03)'
t = -2£7i£73(l - cos(93), (2.45)
with \pi\ = Ei and |p3| = E3.
Yet another frame, especially useful for theoretical purposes, is the Breit or
brick wall frame (e.g., Hagedorn, 1964; Renton, 1990), in which the scattered
particle 3 simply reverses the direction of 1. For m\ = 7713, the momenta are
pi = (£?i,p), p3 = (^i,-p), (2.46)
so that
t = -4|p |2. (2.47)
We will see an example in discussing the simple parton model in Section 5.5.
2.3.5 The Cross Section and Decay Rate Formulae
In this section we sketch the derivation of the relation of the transition
amplitude Mfi to the cross section or decay rate. The results apply to any field
theory, not just Hermitian scalars.

18
The Standard Model and Beyond
Two-Body Scattering
Let us first consider the cross section for the 2 —» n process i —» /, where
\i) = W1P2) and |/) = \pjl • • -p/n), as shown in Figure 2.5. (Particle-type and
spin labels are suppressed.)
Pfn Pfl
-X
FIGURE 2.5
The 2 —» n scattering process P1P2 —> P/i "'Pfn (left); 1 —> ^ scattering
Pi —> Pfi" 'Pfn fr°m a potential, represented by a cross (middle); and the
1 —» n decay process pi —> p/2 • • -p/n (right).
As described in Appendix B, the transition matrix element Ufi and
transition (scattering) amplitude Mji are related by (B.l), so the transition
probability is
\ufi
(2n)iSiC£Ph-Pi-P^)Mfi
(2.48)
To interpret the square of the delta function, it is convenient to temporarily
assume a finite volume V for space and a total transition time T, which can
be taken to oo at the end. Then
(2I)V(^pA-p,-K)
1/ '
\JV,T
d4x e*(£fcP/fc-pi-P2)-*
VT(2n)464C£pfk-Pl-p2),
(2.49)
where we have used one delta function to replace the integrand of the other
integral by unity.

Review of Perturbative Field Theory
19
The differential cross section to scatter into Ylk d3pjk is then
HcT = i^i2 y ( * \ ( i ^
T \31-32\\2E1VJ\2E2VJ
fi *Ph v
^2EfkV (2tt)3J
{27r)4S4(Y,kPh -Pi-P2) yr d3pfk
(2.50)
2
In the first line, \Ufi\2/T is the transition rate, |/?i —/?2|/V is the relative flux,
the 1/2EV factors correct for the normalization of the covariant states^, and
the n factors of V/(2tt)3 represent the density of momentum states of the n
final particles. Note that the factors of V and T cancel in the final expression
in (2.50), and that all the particles are on-shell (E2 = p2 + m2). The flux
factor \Pi — /?21 is evaluated using (3j = Pj/Ej.
It should be intuitively clear that the differential cross section is the same
in the lab frame and in collinear frames related to the lab by boosts along the
pi direction, such as the CM frame, but is not the same in arbitrary frames.
This can be seen by the fact that the factor EiE2\/3i — fol in the denominator
is equal to the Lorentz invariant quantity [(pi • p2)2 — m\m%\^l2 in collinear
frames. In fact, the cross section formula is often written in terms of that
quantity, though that is only strictly valid in the collinear frames^. The S4
function and the scattering amplitude Mfi in (2.50) are manifestly Lorentz
invariant. The final state phase space factors are also invariant, as can be
seen in (2.4). The differential cross section can be integrated over the ranges
of momenta of interest. The total cross section for the specific process is
obtained by integrating over all final momenta,
i J
da, (2.51)
where the statistical factor Si = \/l\ must be included for any set of / identical
final particles to avoid multiple counting of the same final state.
Now, consider the example of 2 —» 2 scattering in the CM, as illustrated
in Figure 2.4(i). For a given CM energy y/s = Ecm, the initial and final
momenta Pi, p/, and the energies Ea, a = 1---4 are fixed by (2.35) and
(2.36). It is convenient to introduce spherical coordinates, with the z axis
along pi, so that pj has polar angle 6 and azimuthal angle ip. For spin-0
particles (or unpolarized initial particles with non-zero spin) the scattering
amplitude Mji is independent of if by rotational invariance.
§The inner product (p \p') = (2n)32EpS3(p-pf) = 2EP J (Pxe-^-p'^ goes to 2EpV8ppf
in a finite volume V.
^In applications to astrophysics one is usually interested in the thermal average of cr\Pi —p2\>
so the flux factor cancels (e.g., Kolb and Turner, 1990).

20 The Standard Model and Beyond
The differential cross section is given by
(27T)4<54(p3 +P4-P1- P2) d3p3
da =
4E1E2\Pi-P2\
d3P4
(2tt)32£3 (27r)32£4
\Mfi\*. (2.52)
In the CM,
EiE2\01-02\
E\E<1
J_ J_
E\ E2
oi(E1 + E2) = piy/i. (2.53)
The implicit P4 integral can be done using the four-momentum conservation,
^ (P3 +P4~Pi -P2)
d3p3 d3p4
S(E3 + E4 - y/si)
E3 E4
= 2?r dcos6 S(E3 + E4 - y/s)
d3ps
E3E4
PJdp3
E3E4
(2.54)
where E3 and £4 represent \Jp%2 + m\ and \Jpz2 + 7714, respectively. In
the last expression the azimuthal angle has been (optionally) integrated over,
i.e., d3ps = d(p dcos 8 p2dps —» 2tt dcos 8 p\dp%. The remaining delta function
determines p3. Using p3 dp3 = E3 dE3 = E4 dE4, d(E3+E4)/dEz = I+E3/E4,
and the identity (1.24), the quantity (2.54) is equal to
27r d cos 8
The differential cross section is therefore
Pf_
da
1
dcos8 32ns p
¥ i^i2
\Mfi
mi=m3 327T5
7712=7714
(2.55)
(2.56)
The initial momentum pi is given in (2.36), while pf is of the same form with
mi —»7713 and 7712 —> 7714. The last expression in (2.56) is for elastic scattering,
for which pi = p/. Closely related quantities are
(2.57)
da
dt
1 da
2piPf dcos#'
da
~dn ~
1 da
2n dcos#'
where t is the Mandelstam invariant given by (2.39), and dfl = dip dcos8 is
the solid angle element. The second form is useful if dip is not integrated over.
In our example of the Hermitian scalar field with Vj given by (2.21), the
differential cross section at tree level is obtained from (2.29) and (2.56),
da
1
d cos 8 32ns
A + k2
1
m*
t — m'
+
1
u — m*
(2.58)
where
s = E2CM =4(p2+ra2), t = -2p2(l-cos8), u = -2p2(l+cos<9), (2.59)

Review of Perturbative Field Theory 21
and p = Pi = Pf- The total cross section is then
a=o / , G n dcos6= / —°—- dcos<9, (2.60)
2 J_x dcosv J0 dcosd
where the 1/2 is because the final particles are identical.
One can also use (2.52) to calculate the cross section in the lab frame, using
ExEhlPi -fo\ =E1m2\0i\ =Pim2. (2.61)
The phase space integral can be carried out explicitly in the lab frame, or
can be obtained by Lorentz transforming the CM result. In analogy with the
derivation of (2.55)
o (P3+P4 -Pi ~P2)-^ =— = i 7^TT> (2-62)
^3 ^4 E4
i _i_ iM±.
1 + dE3
where 83 is the laboratory scattering angle of particle 3. dE^/dE^ can be
calculated using the second equation in (2.43), yielding
da = \Mfi\2 ps
d cos 03 327rpira2 Ex + m2 - 2l^- cos 03
(2.63)
Potential Scattering
Consider the 1 —» n scattering of a single particle from a static source (i.e.,
potential scattering), as illustrated in Figure 2.5. This may be an
approximation to scattering from a heavy target particle. In particular, suppose C
contains an interaction term
CI(x) = Cp(x)$(0,g), (2.64)
where $(0, x) is the static source and Cp involves ordinary quantum fields.
Then, in analogy with Eq. B.4 from Appendix B, the tree-level transition
amplitude for i —» /, with \i) = \pi) and |/) = \pfx • ■ -Pfn), is
Ufi~ fd4x (f\i£p{x)\i)$(0,x)
J (2.65)
(2
where
^— Jd4x d3q(f\iCp(0)\i)^(q)e^f-P^>x,
$(«f) = fd3x'e-^£'^(0,x') (2.66)

22 The Standard Model and Beyond
is the Fourier transform of 3>, q = (0,q), and we have used translation invari-
ance for the matrix element. Then, carrying out the x integral,
Ufi = 2ir8(Ef-E1)Mfi, (2.67)
where
M/iH/l^pWIWz-Pi) (2-68)
and pf = YlkPfk- Proceeding as in (2.50), the differential cross section is
\ufi\2 v
da =
T (ft) V2£iV
11 r^so- \Mf
(2.69)
zeM tJi (2n)32Ef«
l\ »
so that energy but not 3-momentum is conserved, as expected. For the
important special case of elastic scattering, i.e., n = 1 with 7712 = m\ and p2 = Pfx,
\PiM = 0Elt2=P, with (3 = |/3i|. (2.70)
But,
5 (E2 - Ei) d3p2 = p2E2dcos0d<p, (2.71)
where 8 and if are the polar and azimuthal scattering angles. Therefore,
The last form holds when Mfi depends only on
\q\2 = \Pi ~Pi\2 = 2p2(l - cos0) = Vsin2 °-, (2.73)
as in the case of spinless or unpolarized particles and a radially symmetric
source. As a simple example, consider
Cj = -^2*(r), (2.74)
where 3>(r) is the spherically symmetric Yukawa potential
*(0,f) = «-_ —» *(*) = __ (2.75)
with r = |x|. 3> can be thought of as arising from the exchange of a heavy
scalar of mass \x between a nucleus and the scattered particle described by
<f>. (This could be an approximate model for the contribution to pion-nucleon
scattering from the exchange of a heavy scalar resonance, such as the a or the
/o(980).) From (2.68) and (2.72) we obtain Mfi = $(q) and
da _ 2ttk2

Review of Perturbative Field Theory
23
Decays
One can also consider the 1 —» n decay process with \i) = \pi) and |/) =
Wfi •••?/»> (Figure 2.5). Similar to (2.48) and (2.50), one has
\Ufi\2 = VT(2n)4S4 \Tpfk -Pi) \Mfi
(2.77)
and differential decay rate
TT d*Pf>
V
LA 2EfkV (2*)*
rest frame
(2«)4S4(ZkPfk-Pi)fi d3pfk
(2.78)
2m i
11 (2*)32£?A
\Mfi
where T is the decay rate of particle 1 and r is its lifetime, and the second
expression is specialized to the rest frame of the decaying particle. The total
decay rate is obtained by integrating over the final particle phase space,
]]Sl jdT,
i J
(2.79)
where Si = 1//! is a statistical factor for / identical particles, analogous to
(2.51).
As an example, consider three distinct Hermitian scalar fields fa, i = 1 • • • 3,
with masses mi. If mi > 7712 + ms it is possible for particle 1 to decay into
2 + 3 as shown in Figure 2.6, provided there is an interaction term to drive
the decay. The simplest appropriate Lagrangian is
V — ZK,
1
i
1
1
1
1
FIGURE 2.6
The two-body decay of the spin-0 scalar 1 into 2 + 3, and the associated
Feynman diagram corresponding to (2.80).

24
The Standard Model and Beyond
3 1
C = E 9 [(Wrf ~ m^<] " **ifc*3, (2.80)
where the kinetic energy and mass terms generalize (2.19). No counting factor
is needed in the interaction term because the fields are distinct. The tree-level
decay amplitude is therefore
Mfi = -in, (2.81)
i.e., the coefficient of 4>i4>24>z m ^£. The decay rate is
2 f (27r)4S4(p3 + P2 - Pi) d3p2 d3p3
*J
2mi (2tt)32£2 (2n)32E3
k,,h /^«* ♦*--«> -£§. <->
= [m2-(m2 + m3)2]l/2[ 2_(m2_m3)2]l/2
2m?
where £2,3 = y P2 2 + mi,3 m tne second line, and the expression for P2 is
analogous to (2.36).
More general techniques for evaluating phase space integrals are discussed
in Appendix D and in (Barger and Phillips, 1997).
2.3.6 Loop Effects
In addition to the tree diagrams shown in Figure 2.3 for the potential in (2.21),
there are a large number of loop diagrams. A sample of the many one-loop
diagrams is shown in Figure 2.7. Of course, the relative strength of these and
higher-loop diagrams depends on the magnitudes of A and k2 .
The first diagram is shown in more detail in Figure 2.8, with the momenta
labelled. The external lines are on-shell, while the two internal lines are not.
Four-momentum is conserved at each vertex, so only the single momentum k
is integrated over. From the Feynman rules, this diagram contributes
_ (-iX)2 f d4k i i
2 J (2tt)4 k2-m2+ie (k - Pl - p2)2 - m2 + ie ( }
to the scattering amplitude, where the \ is due to the identical particles.
The techniques for evaluating such integrals will be illustrated in Appendix
E and are developed in detail in field theory texts. Here we just note that
the integral is logarithmically divergent for large k (the integrand goes as
d4k/k4). Fortunately, the theory is renormahzable", so the divergences can all
II In a renormalizable theory all divergences can be absorbed in a finite number of
quantities, which can either be measured or drop out of final expressions for observables. Non-
renormalizable theories, on the other hand, encounter new divergences at each order in
perturbation theory.

Review of Perturbative Field Theory
25
FIGURE 2.7
Examples of one-loop diagrams contributing to the process P1P2 —> P3P4-
be absorbed in the observed ("renormalized") values of m, A, and the "wave
functions," leading to finite and calculable quantities. The pure (f>3 theory
(i.e., A = 0, k ^ 0) is super-renormalizable: the only divergent diagrams
are the two one-loop diagrams shown on the right in Figure 2.8. (These
may appear as components of larger diagrams.) It should be noted that
the modern view of divergences is that the integrals are not truly infinite.
Rather, they reflect the view that a low energy field theory is an approximate
description valid below some energy scale A, such as the Planck (gravity) scale
MP = G~N1/2 ~ 1.2 x 1019 GeV, above which there is a more complete theory.
A logarithmically divergent integral such as (2.83) is therefore proportional
to A2ln(A/ra). As long as this quantity is small, the sensitivity to the new
physics scale is not great. In addition, the leading logarithmic contributions
can be summed by the renormalization group equations. This will be further
discussed in Section 2.12.2.
2.4 The Complex Scalar Field
Now let us consider the complex, i.e., non-Hermitian, scalar field (j) ^ <ft,
which describes a spin-0 particle which is not the same as its own antiparticle.
For example, </>n+ annihilates a 7r+ or creates 7r~, while </>n- = <f>+ is the
corresponding tt~ field.
The Lagrangian density for a complex scalar field is
C = £t = (d^(d^) - m2tf<t> - VH0, <£f), (2.84)

26
The Standard Model and Beyond
P3
P4
v-*V
^ *
,*v
kk jfe-pi-
P2
%*+~''
^-iAX
Pi
P2
FIGURE 2.8
Left: a typical one-loop diagram, with the momenta flowing through each
line labelled. Right: divergent (sub)diagrams in the super-renormalizable (j)3
theory. The second (tadpole) diagram contributes to the di term in (2.20) and
must be included in the field redefinition that eliminates it.
where the (Hermitian) potential Vi is given by**
V/(0, $) = —((^(j))2 + non-renormalizable.
(2.85)
There is no \ in the kinetic energy and mass terms for a complex field, and
the 1/4 in Vj is for later convenience. Keeping only the quartic term in V/,
the field equation is !
SC
dt
SC
8tf »Sd^
0 =» (□ +m2) 0 +-0(0*0) = 0. (2.86)
A complex scalar field <j> can always be expressed in terms of two real fields
<t>i = <t>l i = l,2,bytt
</)= -7=(<t>l +*02).
$ = ~7k (^i ~ i(h)'
In terms of these, C in (2.84) can be expressed
£ = E5[(Wa-^]-^ + ^
(2.87)
(2.88)
2=1
**There are other possible renormalizable terms in Vj, such as an[((p)n + ((/>* )n], n = 1 • • • 4,
or pm[{(f>)Tn + (^)m]^0, m = 1,2. These would not allow a conserved charge, and would
lead to processes such as 7r+7r+ —► 7r-7r-. Equivalently, they would not exhibit the C/(l)
phase symmetry discussed in Section 2.4.1.
ttlt is a convention whether <f> = (<f>i +i<f>2)/y/2 is identified as cpn+, as is done here, or with
<f>n-. In later chapters we will often take the opposite choice, especially to be consistent
with isospin conventions.

Review of Perturbative Field Theory
27
This is not the most general renormalizable Lagrangian for two real scalars,
because the masses are the same and the potential is of a special form which
leads to a conserved quantum number.
The solution for the free complex field (i.e., A = 0) is
Mx) = J wis, laWe~ip'x + bHp)e+ip-x]
(2.89)
where at(p) and tf(p) create ir+ and tt~ states, respectively, as in (2.7). They
satisfy the commutation rules in (2.2) and (2.8). a and b can be expressed in
terms of the creation operators for the real fields </>io and (foo by
a{p) = -7=[a1(p)+ia2(p)\, b{p) = -/=[ai(p) - ia2(p)\
f i (2-90)
aHp) = ^W\(P) ~ i4tf)}> bHp) = ^k(p) + i4(p)]•
The Feynman propagator for the complex field is
<0|T[^o(x),0j(x')]|O) = iAF(x -x') = ij ^pTrf + u' (2-91)
so that an internal line carrying momentum fcina Feynman diagram is
associated with the factor iApik) defined in (2.28).
In Feynman diagrams, the arrows represent the direction of flow of positive
charge. For example, an external line with an arrow going into the diagram
may represent either an initial state 7r+ or a final state 7r~, and conversely
for an arrow leaving the diagram. Each vertex is associated with a factor
—iX and must involve the same number of entering and exiting lines (charge
conservation). It is conventional to label internal momenta so that the arrow
also coincides with the direction of momentum flow, while the momenta for
external lines are the physical momenta, which are entering the diagram for
initial particles and exiting for final particles. Thus, the physical momentum
flows opposite to the arrow for a 7r~, as illustrated in Figure 2.9. We also
introduce the symbol p for the momentum flowing in the direction of the
arrow, i.e., pn± = ±pn±.
2.4.1 U(l) Phase Symmetry and the Noether Theorem
The Lagrangian density in (2.84) is invariant under the phase transformations
4>{x) - 4?{x) = e^(x) -> 4>{x) + H{x), (2.92)
p small
where 5(f)(x) = i/3(j)(x), which means that
£(0,tf>t) = £(0',4>'t). (2.93)

28
The Standard Model and Beyond
7T+(p3) 7T (p4) K+(P3) , x W (P4)
\iAF(fc + Pi — P3)/
—i\ V — »A V V — iA
iAF(fe)
7T+(pi) 7r_(p2) ^+(Pl) ^ (P2)
FIGURE 2.9
Left: tree-level diagram for 7r+(pi)7r~(p2) —» 7r+(p3)7r~(p4), wherepi^ (£3,4)
are the physical initial (final) momenta. Right: an example of a one-loop
diagram. Ajp(k) is defined in (2.28).
The invariance holds for all /?, i.e., C has a E/(l) symmetry group. U(l) is the
group of 1 x 1 unitary matrices, which is a fancy expression for phase factors.
The symmetry is global **, i.e., (3 is a constant, independent of x.
According to the Noether theorem, any continuous symmetry of the action
leads to a conserved current and conserved charge. For example, translation
invariance of the action under xM —» x'^ = xM — aM leads to energy and
momentum conservation, with the four-momentum operator PM = (H, P) in
(A.5) conserved (see, e.g., Bjorken and Drell, 1965). In the present case,
invariance of C automatically implies that the action is invariant also. As will
be shown below, the Noether theorem then implies that
SC = -(3d^, (2.94)
where
SC = C(<t>', 0;t) - C(<t>, $) = 0 (2.95)
is the change in the Lagrangian density under (2.92) for infinitesimal /?. In
(2.94), JM is the conserved Noether current
J» = -i [(&*</>)*4 - tfd^cf)] = itftyfr (2.96)
In the last expression, we have introduced the symbol 9M, defined by
alfn = aff*b - (d»a)b (2.97)
for any a and b.
The Noether charge
fd3xJ°(x
Q= d6x Ju(x) (2.98)
**The derivative terms in C would not be invariant for 0 = (3{x). Later, we will discuss
how to modify C to yield local (gauge) symmetries, under which <f>(x) —► el^^(j)(x).

Review of Perturbative Field Theory 29
is then conserved, assuming there is no current flow at infinity, because
dQ= f
dt J
d6x
dJ°(x)
dt
j**
dj°(x)
dt
+ V • J(x)
integral =0j
(2.99)
/■
= / d3x d^(x) = 0.
This is most easily interpreted by explicitly calculating Q for non-interacting
fields </>o, for which one has easily that
Q = J J2$2E~ ft&W) ~ 6t(P)*(p)] = ^r+ - *«-• (2-100)
That is, the conserved Noether charge is just the total number of particles
minus the number of antipartides, which coincides with electric charge for
the specific example. It is straightforward to show that the conservation law
continues to hold in every order of perturbation theory provided that the
interaction term respects the symmetry (commutes with Q).
Derivation of the Noether Current for the Complex Scalar Field
Under the infinitesimal phase transformation (2.92), the change in the La-
grangian density is
m = s-£w + Jfi-AW) + li^ + Tn^mtf,
6<j> * 6{d^)
6ft
s(d„w
where
6<t>(x) = i(3<p(x), 8<j>{x)^ = -ip<p(x)i
8{d»4>) = i0d^(x), Sidpttf = -i(3 (aM<A(x))t.
Using the Euler-Lagrange equations (2.18), this is
8£ = i(3
5C
S(d^)
<P -0„
sc
S(d»4>)
T*tJ
where the Noether current is
-6-
8C
rf
= +i^d^<f>.
(2.101)
(2.102)
(2.103)
(2.104)
Finally, for a symmetry*, one has in addition that the Lagrangian density is
unchanged, i.e., SC = 0, so that 9M JM = 0.
*The condition for a symmetry is actually weaker. It is sufficient for the action in (2.17)
to be invariant. We will see examples of this for discrete symmetries in Section 2.10 and
Chapter 6. For a continuous symmetry, it is sufficient to have 8C = —pd^a^, where a1*
transforms as a Lorentz four-vector. In that case, the current J^ — a^ is conserved.

30
The Standard Model and Beyond
2.5 Electromagnetic and Vector Fields
Define the electromagnetic vector potential A^(x) and the field strength tensor
F»V = _pvn = Q»AV _ QVJ^^ (2.105)
related to the electric and magnetic fields E and B by
J?ILV _
(°
Fx
Ey
\EZ
—Ex —Ey — Ez\
0 -Bz By
Bz 0 -Bx
—By Bx 0 J
(2.106)
In the absence of sources, one has the free-field Lagrangian density
C0(A») = -\f^F^ = \{E2- B2) , (2.107)
leading to the free-field equation of motion DAM — d^ (duAu) = 0. (We have
omitted the free-field subscript 0 on F and A for simplicity.)
Co is gauge invariant, i.e., F^ = F'»v = d»A'u - 0M'", and therefore
C0(A») = C0{A^), where
A^ = A^--d^P(x), (2.108)
e •
and (3(x) is an arbitrary differentiable function of (£,#). (The 1/e factor,
where e > 0 is the electric charge of the positron, is inserted for convenience.)
This is known as a U(l) invariance because there is only one function (3(x).
The U(l) gauge invariance will continue to hold for the full Lagrangian
including interactions. It is convenient to work in the Lorentz gauges, dvAv = 0.
This still leaves the freedom of making a further gauge transformation
provided □/? = 0, and one can exploit this freedom to simultaneously choose
A0 = 0 and V • A = 0, the Coulomb or radiation gauge. The radiation gauge
does not have manifest Lorentz or gauge invariance, but is attractive in that
is doesn't involve any unphysical degrees of freedom. The two non-zero
components of A^ are transverse to the photon momentum and correspond to the
two polarization directions. In the radiation gauge, the free-field expression
for the vector potential is
A{x) = j\2^f2E £ [m^)a(p,X)e-ip-x + r(p,X)a\p,X)e^-]
d3p
"P A=l,2
(2.109)
where Ep = \p\, a and a^ are bosonic annihilation and creation operators
satisfying
[a(p,X),a\p',X')]=(27r)32EpS3(p-p')Sxy, (2.110)

Review of Perturbative Field Theory
31
^ = 1,2 refers to the two possible polarization states, and e(p, A) is the photon
polarization in the direction of the photon electric field. The radiation gauge
condition V • A = 0 implies
p-c(p,A) = 0, (2.111)
and the normalization is
e*(p,\)-e(p,\')=6xy. (2.112)
For a linear polarization basis, the e(p, A) are real and can be chosen
c(p,l)xc(p,2)=p. (2.113)
Thus, one can choose e*(p, 1) = p± where p± is an arbitrary unit vector
orthogonal to p, and e(p, 2) = pxp±; e.g., for p in the z direction, e(p, 1) = (1,0,0)
and e*(p,2) = (0,1,0). One can also use the basis of left- and right-handed
circular polarization vectors
c(p,l)=Fic(p,2)
CL,/i(p) = -/| , (2.114)
which correspond to photon helicity (spin measured with respect to the
momentum direction) =pl, respectively. It is useful to introduce the polarization
four-vectors eM(p, A) = (0,e(p, A)), where
P-*(fcA) = 0 ,2115,
e*(p,A)-£(p,A') = -«5AA'. V" J
Thus, the free-field expression for A^1 is obtained from (2.109) by replacing
?(£A)-c"(p,A).
From (2.109) one sees that the amplitude for a physical process is of the
form eM(p, A) • MM for an initial photon, or eM*(p, A) • MM for a final photon,
where the amplitude MM depends on pM and the other momenta and spins in
the process. (The rules for constructing MM will be described in later sections.)
Often, the initial photon beam is unpolarized, or a final photon polarization is
not measured. In that case, one averages (sums) over the initial (final) photon
polarization directions, yielding a rate proportional to
£ c"(p,A)c"*(p,A)
A=l,2
MMJIC, (2.116)
with an extra \ for an average. This is easily evaluated using
T e"(p,A)e-(p,A) = -<T + P^+P^, (2.117)

32
The Standard Model and Beyond
where pr = (Ep, —p) so that pr • p = 2E%. One can use gauge invariance to
show that pMMM = 0, so that the second term in (2.117) does not contribute*,
i.e., (2.116) is just -M^M**.
The Feynman propagator for the free photon field is
/d^k i
e-ifc.(x-x )DF{k) + gauge terms>
(27r)
(2.118)
where
Df^ = -*h- (2-119)
The additional terms in (2.118) are gauge-dependent (e.g., for the radiation
gauge they involve the generalization of the second term in (2.117) to off-
shell momenta). A full treatment of gauge issues is beyond the scope of this
treatment (but see the discussion of the R^ gauges in Chapter 4). However, the
additional terms do not contribute to physical processes by gauge invariance,
so in practice one can simply assign a factor ig^Dpik) = —ig^/^ + ie) for
an internal photon line carrying momentum k in a Feynman diagram.
2.5.1 Massive Neutral Vector Field
The results for the free electromagnetic field are easily generalized to the case
of a massive Hermit ian vector (spin-1) boson V^. The free-field Lagrangian
density is \
C0 = -^G^G^ + ira2V^, with G^ = d»Vu - dvV». (2.120)
It should be emphasized that Co is not gauge invariant under
VV^V + WP (2.121)
because of the mass term and does not lead to a renormalizable theory.
However, it is useful to display the relevant formulae useful for phenomenological
tree-level calculations.
The free field is given by
vw = J {2^f2E E [*"#A)a(£A)e_ipx+eM*(p>A)flt(p>A)e+ipi
tThe analogous situation for QCD and other non-abelian theories is trickier; one can only
drop the second term if ghost contributions are included (e.g., Peskin and Schroeder, 1995).
Another subtlety will be encounted in the example of Compton scattering in Section 2.6.

Review of Perturbative Field Theory
33
with Ep = y/p2 +m2> A massive vector has three polarization states. These
include the two transverse states e(p, 1) and e(p,2) (or cl,r(p)) similar to
the massless case, and a third longitudinal (helicity 0) state
c(p,3) = ^(|p|,0,0,£?p), (2.123)
where we have taken p along the z direction for definiteness. Note that the
magnitudes of the components of e(p, 3) become large at high energy, with
eM(p, 3) ~ p^/m. The three polarization vectors satisfy (2.115), and the
polarization sum is
3 pV
2 e"(p, A)e"*(p, A) = -<T + P-\. (2.124)
iD^(k) = i
y m2
A;2 — m2 + ie
A=l
The momentum space propagator associated with an internal line in a Feyn-
man diagram is
(2.125)
The second term in Dy does n<9£ drop out of calculations and leads to bad
ultraviolent (large k) behavior and thus to non-renormalizability. One also
sees that the limit m —» 0 is not smooth.
A renormalizable gauge invariant theory of massive spin-1 fields, in which
the mass is obtained by the Higgs mechanism* rather than as an elementary
term in £, will be discussed in Chapter 4. In that case, (2.125) will still
correspond to the unitary gauge, in which the physical degrees of freedom are
manifest, but there are other gauges in which the m —» 0 limit is smooth.
2.6 Electromagnetic Interaction of Charged Pions
We are now ready to combine the results of Sections 2.4 and 2.5. Consider
the Lagrangian density
C(</>, A) = (d^d^ - mV</> " \^,F^ - VH0,0f) + £*a(& A) (2.126)
for a complex scalar field <\> and the electromagnetic field AM. The scalar
self-interaction Vi is defined in (2.85), and C^a describes the electromagnetic
*It is also possible to construct a C/(l) gauge invariant theory for a massive vector without
the Higgs or other spontaneous symmetry breaking mechanism by the Stiickelberg
mechanism (Stueckelberg, 1938; Cianfrani and Lecian, 2007).

34
The Standard Model and Beyond
interaction. Its form is dictated by the requirement of gauge invariance under
(2.108). The only way to do this (without introducing non-renormalizable
interactions) is the minimal electromagnetic substitution, familiar from classical
and quantum mechanics. One replaces
id* <* jf ^> jf - qA» (2.127)
in the Lagrangian density in (2.84), where q = e > 0 is the charge of the 7r+,
and identifies the additional terms with C^a- £ will then be invariant under
the generalized gauge (or local) transformation
e HK J (2.128)
(j)^^' = eiq^x^e(j) = ei^x)(j),
i.e., £(</>, A) = £(<//, A'). (2.128) generalizes the global symmetry of Section
2.4.1, i.e., </> —» elP(j) where (3 = constant.
Using the minimal substitution,
C = [(0M + iqA^)4>]\d* + iqA»)4> - m2^c\> - ]f^F^ - V>(0, 0f)
N v ' 4
{d^-iqA^tf (2.129)
= [D^[D»<t>] - m2^cj>- \F^F^ - W,^),
where !
D^ = d^ + iqA" (2.130)
is known as the gauge covariant derivative.
The gauge invariance of (2.129) is obvious except for the first term. Under
(2.128),
^ _ £>'/y = \w + iqAv _ ild"p(xj\ (ei0(x)<t>) = etf(x>D"^
That is, the shift in A* compensates for the change in d*<\> due to the derivative
of /?, leaving £ gauge invariant. Thus, gauge invariance for the charged scalar
requires the existence of a spin-1 field, dictates the form of the interaction with
A*, forbids an elementary mass term (i.e., ^m2AA^A* is not gauge invariant),
and restricts the form of the scalar self-interactions. It turns out that the
theory is then renormalizable.
Prom (2.129) one can read off i£<$>A'-
i£<t>A = -qidvtfMv + q{$ &><!>) Ap + if A^tf 4>
= q{(t>^(j>)A^ + itfApA^h

Review of Perturbative Field Theory 35
where i<fl d^(p is the Noether current for the free complex scalar field § in
(2.104). The Feynman vertex rules can be found from (2.132), inserting the
free-field expressions for <j> and <ft in (2.89) and recalling that a(p) and b(p)
axe the annihilation operators for 7r+ and tt~ , respectively. For this purpose,
it is convenient to explicitly display
d"Mx) = J {2*)£ep H^fl(P )C"*P"X + ipV>Hp)e+ip'x]
d^{x) = J {2n/2Ep [-^b^e~iP'X + ipP"*(p)e+ip'x] >
(2.133)
where p = (Ep,p) with Ep = yjp2 + m2 as usual.
The vertices are displayed in Figure 2.10 for q = e. The three-point vertices
include a factor — iep^+ for each 7r+ and a factor +iep^_ for each ir~. These
are always the physical momenta, whether initial or final. They can both
be written as — iep^±, where pn± = ±pn± is the momentum in the direction
of the arrow. There is also a four-point (seagull) vertex 2ie2g^u, which is
needed for gauge invariance. The Lorentz indices are contracted with eM(fc, A)
for an initial photon of momentum /c, with e*(fc,A) for a final photon, and
with ig^Dpik) = —ig^u/{^2 + ie) for an internal virtual photon line. Each
internal pion has a propagator iAp(k) = i/(k2 — m2 + ie), and there is an
integral f d4k/(27r)4 over each unconstrained internal momentum.
nK and nn Scattering
As a first example, consider the electromagnetic scattering n~ K+ —» n~ K+,
where K± is a complex field with the same electromagnetic couplings (but
different mass) as ir±. They are introduced here to avoid identical particle
effects. We ignore strong interactions of other non-electromagnetic couplings.
There is a single tree-level diagram, as shown in Figure 2.11. The Feynman
rules lead to the transition amplitude
Mfi = ie(pi + p3)
M
-WlAV
(-ie)(P2+P4Y
L(Pi -Ps)2.
-4^a(pl+P3)'(p?+P4) = -4*ta
(2.134)
(Pi -P3)2
where a = e2/47r is the fine structure constant. The Mandelstam variables s,
t, and u are defined in (2.30) and related to the CM scattering angle in (2.39).
§ Interaction terms usually do not modify the form of the Noether currents. Gauge
interactions of complex scalars are an exception because the interaction terms involve derivatives.
For example, the conserved Noether current for C in (2.129) is J^ = i<ft d^cj) — 2q<f>^ A^0,
which appears in the field equation for A in the familiar Maxwell form dpF**" = qJu.

36 The Standard Model and Beyond
AN_ie(PT+ _ Pit-)*1
-*+(p„+)
FIGURE 2.10
Vertices involving charged pions and one or two photons. In the one-photon
diagrams the initial pions enter from the bottom and the final leave from the
top. Antiparticle (tt~) vertices are obtained from particle ones by twisting
the lines and replacing p by p = —p, where p is the physical four-momentum
and p follows the direction of the arrow. The wavy lines represent photons.
Prom (2.56) the differential cross section in the CM is
^a _ * \m |2 _ na2 {s/2p2 + 1 + cos#
d cos 6 32ns 2s I 1 — cos 6
2
7rar
3 + cos (9\2
1-coso) '
(2.135)
y/s^>mK 25
where p ~ (s — m2K)/2yfs since ttik ^> mn. The total cross section is^
-dcos6. (2.136)
.! a cos 9
For 7r+ 7r+ —> 7r+ 7r+ there are two Feynman diagrams due to the different
ways of associating the fields in iC^A with the identical external particles, as
^The integral is divergent for forward scattering due to the massless photon propagator
pole, i.e., the long range Coulomb force. This divergence is already present for classical
Coulomb scattering, and disappears for realistic situations in which screening by other
charges or the finite resolution of the detector are taken into account. Similar comments
apply to 7r+7r+ for both forward and backward scattering and to 7r-7r+.

Review of Perturbative Field Theory
37
it~ (Pi) K+(p2)
it+(p3) w+(p*4) it+(p4) w+(P3)
\ — / \ — /
/Pi"P»\
Pi — Pa
^r+(Pi)
*+(Ps)
-*+(Pi)'
7T+(p2) W + (pi)
*-+(P*2)
TT (p4)
/ Pi — p3 *
W'(ft) -W
T < P1+P2
TT (p2)
FIGURE 2.11
Feynman diagram for 7r~ K+ —» 7r~ if+ (top), 7r+ 7r+ —> 7r+ 7r+ (middle),
and7r+7r- —> 7r+7T~ (bottom).
shown in Figure 2.11. The transition amplitude is
Mfi = ie2
(Pi + P3) * (P2 + P4) (Pi + P4) • (P2 + P3)
Ania
(P3-P1)2
s — u s — t
+—
(P4-Pl)
t
(2.137)
The differential cross section is again given by (2.56), but in this case there
is an extra factor | in the total cross section because the final particles are
identical. At high energies, yfs ^> mn, one finds
da
2na2 /3 + cos2fl
/3 +cos2 fl\2 _ 1 r+1 da
V sin2fl / ' a~2J_l dcosi
dcosO s V sin2fl J ' ~ 2 ./_-, dcosO
There are similarly two diagrams for 7r+ tt~ —> 7r+ 7r~, yielding
M/j = ze2
dcosfl. (2.138)
-(Pi + P3) • (P2 + P4) (Pi - P2) * (P3 - P4)
= 4nia
(P3-Pl)2
-s + u u — t
+
t
Airia
(P1+P2)2
3 + cos2 fl
1 — cos fl
(2.139)
In this case, however, the final particles are not identical. Note that the
amplitude for 7r+ ir~ can be obtained from that for 7r+ 7r+ by the formal

38 The Standard Model and Beyond
substitutions p± —» — P2 and P2 —» — £>4, an example of crossing symmetry.
That is, the amplitude for an outgoing 7r+ of momentum p is the same as that
for an incoming tt~ with momentum — p, as is apparent from the Feynman
rules. Of course, the physical values of p are different for the two cases, since
P° (—P°) must be positive for the first (second) one.
Pion Compton Scattering
Now consider pion compton scattering, 7(fci, Ai)7r+(pi) —> 7(^2, ^2) tt+(P2),
with s = (ki +P1)2, t = (k2 - /ci)2, and u = (p2 - ki)2. The amplitude
corresponding to the diagrams in Figure 2.12 is
Mfi = e2Mel^
-ie(k2 + 2p2r^^(-ie)(k1 + 2Ply
-ie(2pl-k2Y—^{-ie){2p2-kly + 2ie2g^
(2.140)
where €2 = e(A;2,A2) and ei = e(/ci,Ai). It is straightforward to show that
-y(fc2,A2) ?r+(P2) -y(fc2,A2) tt+(P2) T(fc2,A2) *"+(P2)
-y(£i,Ai) -k+(pi) ^ (Pi) ^(fci,Ai) -y(fci,A!) 7r+(ft)
FIGURE 2.12
Diagrams for pion Compton scattering. The third (seagull) diagram is
required by gauge invariance.
Mfi vanishes if t\v is replaced by k\v or e£M by &2M, which is a manifestation
of gauge invariance. This would not occur without the 2ie2g^u.
Recall that our radiation gauge formalism is not manifestly Lorentz
invariant. It is convenient to choose A0 = 0 and V • A = 0 to hold in the
laboratory frame, in which p\ = (m, 0). Then, p\ • e\ = p\ • e£ = 0, as well as
k\ - €1 = /c2 • €% = 0, so that Mfi takes the extremely simple form
Mfi = S7riae2 - c±. (2.141)
The kinematics simplifies greatly compared to the general mass case. From
(2.43) one has (after an appropriate change in labelling)
k2 = 7—^ , (2.142)

Review of Perturbative Field Theory 39
where 6l IS tne laboratory angle of the final photon momentum £2 and ki is
the energy of photon i in the lab. Then, from (2.62)
** ^OVl'i-212- (2-143)
dcos#L m2 \ki
For an unpolarized initial photon and unobserved final polarization one
must average (sum) over the polarization states Ai (A2), yielding
da _ 1 y^ da
dcos6L = 2 x^f dcos6L' ^ ' '
This can be evaluated using (2.117) for both the Ai and A2 sums", with
A£ = fci(l,0,0,1), k$ = k2(l,sin0L,O,cos0L) (2 14g)
fcfr = fci(l,0,0,-l), A£r = k2(l,-sin0L,O,-cos0L),
yielding
^El£i-e2l2 = ^(l+cos2^). (2.146)
A1A2
In this case, however, it is simpler to evaluate the sum using the explicit forms
ex(l) = (0,1,0,0), e2(l) = (O,cos0L,O,-sin0L)
£1(2) = c2(2) = (0,0,l,0).
for the transverse polarization vectors.
2.7 The Dirac Field
The Dirac field i/>a(x), where a = 1 • • • 4 is the spinor index, describes a four-
component spin-1 particle, i.e., ip annihilates the two possible spin states for
a particle and creates the two spin states for the antiparticle. In the absence
of interactions, the Lagrangian density is
Co = $(x)a (ifl- m)a(3 t/>(x)0 = $(x) (ifl-m) ^(x). (2.148)
The sum over a and (3 is written in matrix notation in the second form, in
which ip is a 4-component column vector; the Dirac adjoint
tp(x)=^(x)j° (2.149)
" Since we have already evaluated Mfi in a specific gauge, we cannot drop the second term
on the right of (2.117).

40
The Standard Model and Beyond
is a four-component row vector; and
where 7M, \x = 0 • • • 3 are the 4x4 Dirac matrices, defined by
{y,7"} = 2(r. (2.151)
(There is an implicit 4x4 identity matrix on the right side of (2.151) and
after m in (2.148).) The 7M also must satisfy
(7M)t=7V7° = 7M. (2-152)
so that Co = £q. It is useful to define
75 = 75 = i7V7273, a^=l-[r,Y\- (2-153)
75 enters for spin and chirality projections, for coupling fermions to pseu-
doscalars, and for axial vector currents in the weak interactions. Prom (2.151),
75 = 75t, 757M = ~7M75, and (7s)2 = I. a^v is useful, e.g., in the description
of electric and magnetic dipole moments. An arbitrary 4x4 matrix can
be written as a linear combination of 7,75,7M,7M75, and o^v. E.g., a^^5 is
given in Problem 2.3.
2.7.1 The Free Dirac Field
Prom (2.148), one obtains the free-field (Dirac) equation
(i0-m)<ip(x) = O. (2.154)
The solution to (2.154) is
^{X) = / (2^4? 2 K£ *) *& s^^ + «(£ s)b\p, s)e^"] ,
(2.155)
where a^(p,s) and tf(p,s) are the creation operators for e~ and e+ states,
respectively**,
aUp,s)\0) = |e~(p,s))
6t(p,«)|0) = |c+(p,«)).
**By convention, we are taking ip = tpe- to be the e~ field. The e+ field ipc = ipe+ is of the
same form as (2.155) except a and b are reversed. Charge conjugation and space reflection
will be discussed in more detail in Section 2.10.

Review of Perturbative Field Theory
41
p refers to the physical momentum for both \eT(p, s)), and s runs over the
two independent spin states, a and b and their adjoints satisfy the anticom-
mutation rules in (2.9)-(2.11), e.g., {a(p, s), a* (p', s')} = 63(p- p')5ss>.
In (2.155), u(p,s) and v(p,s) are 4-component Dirac spinors (complex
column vectors). Prom (2.154) they are the solutions to the momentum space
Dirac equation, i.e.,
(ft - m)u(p, s) = (p^ - m)u(p, s) = 0
(lf + m)v(p,s) =0,
where there is an implicit 4x4 identity matrix in the mass term. Taking the
adjoint of (2.157) and using (2.152),
[(^- m)ti]f = u\p^ -m)= u\p^°7^° - (7°)2m)
= u($ — 771)7° = 0,
(2.158)
where u = 1^7° and v = 1^7° are the Dirac adjoints. Thus,
u(p, s)(tf -m) = v(p, s)(tf + m) = 0. (2.159)
Before proceeding to the electrodynamics of fermions, let us consider some
properties of the Dirac matrices and spinors.
2.7.2 Dirac Matrices and Spinors
Explicit Forms for the Dirac Matrices
The Dirac matrices are defined by (2.151) and (2.152). For most calculations
one does not need their explicit form, but it is occasionally useful to have one.
For example, the Pauli-Dirac representation is useful for studying the non-
relativistic limit of an interaction, while the chiral representation is useful at
high energy and for Weyl or Majorana fields, encounted in neutrino physics
and supersymmetry. In the Pauli-Dirac representation
(2.160)
where J is the 2x2 identity matrix and a1 are the Pauli matrices in (1.16).
Similarly, in the chiral representation
.o_ /» M -<_ I ° a'\ -5 (-1 0
0 A
/ 0)
7 \ t n) 7 I _ai q I 7 i q j
*°i = i -°* °\ „« I ** °

42 The Standard Model and Beyond
It is sometimes convenient to rewrite the chiral representation matrices as
V = ( I a;) , (2-162)
where
o» = (/,<?), *» = (/,-<?) = <xM. (2.163)
Traces and Products of Dirac Matrices
Most calculations can be carried out without using the specific forms of the
Dirac matrices by using various trace identities, where the trace is Tr A =
YlctAococ for any square matrix A. Note that Tr A = Tr AT = (Tr A*)* and
Tr (AB) = Tr(BA). For any representation, (2.151) and (2.153) imply
Tr/ = 4, Tr7^ = Tr75 = 0. (2.164)
Define the 4 x 4 matrix
i = a^ (2.165)
for an arbitrary four-vector aM. Then, from (2.151)
tll=- m + 2a-bl =» flfl = a2I. (2.166)
One has immediately that
Tr(^#) = 4 a-6. (2.167)
Other useful trace identities include (see, e.g., Bjorken and Drell, 1964),
Tr(0l ft 0 ft) = 4(a-b c-d + a-d b-c — a-c b • d)
Tr(75 ^$=0 (2.168)
TV (75 i V t $) = W9* a^Kcpdv,
where e^up<T is the totally antisymmetric tensor with e0i23 = +1 and e0123 =
— 1. Also,
Tr(^--- ^n)=Tr(75 ^ • • • fln) = 0 (for n odd)
Tr(fii ••• ^n) = Q>i • a2Tr(^'" fin) - a\ • a3Tr(^2 ^4 • • • fin)
h ai • anTr (^2 • • • ^n-i) (for n even)
Tr(fti fa-- in) =Tr (fin-- fa fa).
Related useful identities are
(2.169)
(2.170)

Review of Perturbative Field Theory 43
Finally, we record the identities
iv [7m iiv H1 + V)] T* [r w t(i± a75)]
-{
64a-cfc-dfor + (2.171)
64 a • d 6 • c for —
where A = ±1. These results can be derived using the identities in (2.168)
and Table 1.2, as will be shown in detail in Section 6.2.1. They are extremely
useful for calculating four-fermion polarization effects, as well as for weak
interaction decay and scattering processes.
Spinor Normalization and Projections
The Dirac spinors satisfy the normalization and orthogonality relations
u(p,s)u(p,sf) = -v(p,s)v(p,sf) = 2m8ss>
u\p, s)u(p, s') = v\p, s)v{p, s') = 2EP 6SS, (2.172)
u(p,s)v(p,s') = u^(p,s)v(-p,sf) = v(p,s)u(p,s') = v^(p,s)u(-p,sf) = 0,
which are just numbers (e.g., Bjorken and Drell, 1965). The projections
V^ u(p, s) u(p, s) =tf + m
' (2-173)
2_, w(p. s) w(p. s) =i> - m>
s
which are useful in summing over spin orientations in physical rates, are 4x4
matrices. The form of (2.173) follows from the Dirac equation, while the
normalization follows by taking the trace and using the first equation in (2.172).
(The second equation (2.172) for s = sf then follows by right-multiplying
(2.173) by 70 and then taking the trace.)
The projections
u(p, s) u(p, s) = (]f + m)
v(p, s) v(p, s) = (jl — m)
1+7*
-0
2 ' (2.174)
(^)
are useful when one does not want to sum over spins. Here, sM is the spin
four-vector. In the particle rest frame, p = (m, 0), it is just a unit vector
s = (0,5) (2.175)
in the spin direction, so that s2 = — 1 and p • s = 0. Boosting to an arbitrary
frame in which p = (3Ep = 7(3m,
s = (7/3 -S, 7*|| +*l), (2.176)

44 The Standard Model and Beyond
where S|| = S-/3/3 and s± = s — s\\ are respectively the components of s parallel
and perpendicular to J3. Thus, s = (0, s) for s-/3 = 0, while 5 = s± = ±7(/3, /3)
for s-/3 = ±1. The latter are known as the positive and negative helicity states,
respectively, or alternatively as right- and left-handed states. The projections
in (2.174) simplify greatly for the helicity states in the relativistic limit:
, * { K (2-177)
where the chiral projection operators Pr,l will be discussed below.
The Propagator
Prom (2.155) and the spinor sums (2.173), one obtains the Feynman
propagator for the free Dirac field**
(0\T\^(xU(x')}\0) =i J -0^e-ik<x-^SF(k), (2.178)
where the momentum space propagator Sf(A;), which is a 4 x 4 matrix, is
SF(k) = 1—^ = 72 * + ™ , . (2.179)
v J }k-m + ic h2 -m2 +ie v y
The last equality follows from (2.166).
Explicit Spinor Forms
Explicit forms for the Dirac spinors are occasionally useful, e.g., for
considering non-relativistic limits. In the Pauli-Dirac representation
ti(p, s) = ^Ep+m I _^ ) , (2.180)
where </>s is a two-component Pauli spinor describing the spin orientation.
Thus,
describe spin orientations in the ±z directions, respectively. <j>±, defined by
h<t>± = -a ■ p<t>± = ±-\p\4>±, (2.182)
ttThe time-ordered product of two fermion fields is defined as in (2.26) except there is a
minus sign before the second term.

Review of Perturbative Field Theory
45
describe positive (negative) helicity states. Similarly, the v spinors are
V(p, s) = y/Ep+TTl I Ep+rnX* \ ? (g^gg)
where
Xs = -ia2Ps. (2.184)
Thus,
Xl = (i)' X2=("o1) (2'185)
represent spins in the ±z directions. The positive (negative) helicity spinors,
which continue to mean that the physical spin is parallel (antiparallel) to the
physical momentum, satisfy
-a ■ pX± = T^\p\x±- (2.186)
The counter-intuitive results in (2.185) and (2.186) are considered in Problem
2.7 and in the discussion of charge conjugation in Section 2.10. Explicit forms
for the helicity spinors are given in Table 2.1, using a phase convention for
which
ia24>*± = T<t>T, ™2X*± = TXt- (2-187)
The explicit u and v spinors in the chiral representation are
~+^" ((* ~ £JT+^) <l>»\ _ (y/p~~t<t>s\
(2.188)
U(P'S) u o \ / ** \ I I / = ,
(I + -E^)xs) \-VP-^Xs
where crM and <rM are defined in (2.163). The second form is easily verified
using (1.20). In the helicity basis, these are
where
X±(p) = lT-E^—- (2-190)
Ej + 771
The chiral representation is especially useful in the case of massless or
relativists fermions, for which the upper (lower) two components of the positive

46
The Standard Model and Beyond
TABLE 2.1
Explicit forms and properties of the helicity spinors corresponding to
spherical angles (0, tp) for p. They are constructed using the rotation
*±(p) = -R(0» ^)5±(^)? where $ = </> or \- The spherical angles of — p are
(n — 0, 7r + y?), so that — z = (7r,7r). The orthonormality and completeness
relations also apply to the fixed spin axis basis.
rotation:
*+(*) =-*-(*) = (£)
*w=-*-<«-(.,$,)
M-fl = -*-<-*)» (.^
<MP) = ie^-M-P)
X+(p) = -i°2<i>+(p)* = 4>-{p)
orthonormality:
completeness:
*)
i?(0,¥?) = e-i<T3^e-i<T25ei<r3^
*_(*)= X+(*)=(?)
"»-*♦«-riff)
<t>- (-P ) = X+ -P) = 1 .„*
V sin 5 /
X±(P) = ±e*ivX?(-P)
X-(p) = -ia24>-(p)* = -0+(p)
4(P)<t>j{P) = % = Xi(p)Xj(p)
Ei^(p)^(p) = E*Xi(p)xJ(p) = -f
(negative) helicity w spinors vanish, and oppositely for v,
u(p, +)^V2E (;+) , u(p, ~)^V2E (+-
v® +) ^T ^ (Xo+) • "<p, -) ^^ ^ (-°
(2.191)
Chiral Fields
For a fermion field -0, one can define left (L) and right (R)-chiral projections
^Pl = Pl1> = X—Y~^, 1>R = W = ^y^, (2-192)
where P^ = PLffl, PlPr = PrPl = 0, P[ R = PLf*, and Pl + Pr = /- ^l
and ^ can be viewed as independent degrees of freedom, with -0 = ipL + Vte-
For a massless fermion the L and #-chiral components correspond to particles
with negative and positive helicity, respectively, i.e., ipL and tpR annihilate
fermions with helicity h = =f|. For antifermions it is just the reverse, ipL and
tpR create antifermion states with h = ±|. For mass m ^ 0 the chiral states
of energy E associated with tpL,R have admixtures of 0(m/E) of the "wrong"
helicity (Problem 2.9). The free-field Lagrangian density in (2.148) can be

Review of Perturbative Field Theory
47
rewritten in terms of the chiral projections as
C = $Li ftipL + $Ri 0ipR - m ($LipR + $riPl) , (2.193)
where ipL,R are defined by
$l = W>l)V = ^Pl7° = i>PR
$r = WrW = ^Pr7° = $Pl
(2.194)
The chiral projections ipL,R are also known as Weyl spinors or Weyl two-
component fields. They can be described as above in four-component notation,
i.e., as two-dimensional projections of four-component fields tp, but it is often
convenient to discard the superfluous components and work in two-component
notation. As the name "chiral" suggests, this is most conveniently displayed
in the chiral representation, for which
*-($■ Mo")' (2195)
Thus
*=(Si)*™=(*L)' ^-U0*)' (2-i96)
where ^l,r are the Weyl two-component fields.
The Lagrangian density (2.148) for the free Dirac field can be written in
terms of the Weyl fields as
C0 = ^Lia^d^L + ^Ria^d^R - m (tf £** + *^*l) , (2.197)
where the four-vectors crM and <rM are the 2 x 2 matrices defined in (2.163). The
Dirac mass term couples the L and R components, while the kinetic energy
terms are diagonal. The free-field Dirac equation becomes
io^d^L - m^R = 0, io-^d^R - mVL = 0. (2.198)
Above, we introduced the chiral fields as projections of a four-component
Dirac field. Alternatively, one can simply define chiral fields as those satisfying
ipL = Pl^l or ipR = PriPr, i.e., not necessarily as projections of another
field ip, and in fact this was done for the L-chiral neutrinos in the original
formulation of the standard model. Equivalent ly, Weyl fields \&l or ^r can
be introduced independently of each other. For example, a single Weyl L field
with Co = WLia^d^L would describe a massless negative helicity particle
and a positive helicity antiparticle.
One can also define the chiral projections of the u and v spinors,
ulMp* s) = Pl,ru(p, s), vl,r(p> s) = pL,Rv(Pi s)- (2.199)

48
If one writes
then
u =
vll,r satisfy the
ft)*
uL = f
Dirac equation
p • auL =
P•vur =
(£p/ +
(EPI-
0
a •
a •
Tfce Si
J , ur
p)ul =
p)uR =
,andar
-t
mufl
rauL.
(2.200)
(2.201)
The chiral components of v = (x>l ®r)T satisfy similar equations, only with
m —> —m. It follows easily that the solutions for u and v are given by (2.188).
For m —» 0 the equations for ul,j? decouple. The u and v spinors of definite
helicity given in (2.191) then coincide with the left and right chiral projections,
PRu(p, +) = u(p, +), PLu(p, -) = ti(p, -)
Prv(p,-) = v(p,-), Plv(p,+) = v(p,+),
showing the flip between chirality and helicity for the v spinors, consistent
with (2.177). The free-field expressions for ipL,R or ^l,r are especially
simple in this case. Prom (2.155) and (2.191)
9(x)LtR = J ^ v^[*=f(p) a(p,=F)c-'p-x±X±(p)6t(p.±)c+ip-x] ,
(2.203)
which is similar to the free Dirac field except there is no sum on spins.
We will further develop the Weyl two-component formalism in Section 2.11
and in Chapters 7 and 8.
Bilinear Forms
Consider the bilinear form w2Mwi, where wi,2 are any two Dirac u or v
spinors. They may even correspond to particles with different masses, which
is relevant, e.g., for weak interaction transitions. M is an arbitrary 4x4
matrix. Then,
(w2MwiY = w{Mw2, (2.204)
where M = 7°M^7° is the Dirac adjoint of M. One finds
M = M for M = 7,7/i,7/i75,^M,/
M = -M for M = 75, a^75 (2.205)
MiM2 = M2 M1 =» ^1 ^2 • • • ^n =^n • • • #2 £i-
There is an equivalent relation,
(^M^j^^M^, (2.206)

Review of Perturbative Field Theory 49
for two Dirac fields, which may be the same or different. Then, for example,
^ \U2Mui\2 = ^^ U2MU1U1MU2
Si,S2 Si,S2
= ^u2a [M(^1+m1)M]a/3w2/3
S2
Tr
M(jfi +mi)M I ^2u2ui J
(2.207)
= Tr [M(fa + mi)M(|fe + m2)] ,
which allow one to express a physical rate in terms of a trace.
For chiral spinors
wL,R = (wLtR)* 7° = wipL,R-y° = w^°PRiL = wPr,l- (2.208)
Therefore, for any two spinors Wi and W2,
w1lTw2l = 0 = w1RTw2R (2.209)
for T = 7,75,a^, or a^75, while
w1lTw2r = 0 = w1RTw2L (2.210)
for r = 7M or 7M75. Equivalent relations for the chiral fields ipL,R were used in
(2.193). Thus, scalar, pseudoscalar, and tensor transitions reverse the chirality
between an initial and final fermion, while vector and axial vector transitions
maintain it.
The Fierz Identities
The Fierz identities are
('01L7M'02L) {^3L1^4l) = ~VF (^1L7M^4l) {$3L1^2l)
{^ir1^^2r) ($zRl^m) = -Vf (/0ii?7M'04i?) ($3r1^2r)
{$lR7^1p2R) (^3L7/x^4l) = %VF $lRlp4L) {^3L^2r) ^ ^^
— — T)J? — — *
{^IR^2l) {^3R^4l) = -y {*PlRlp4L) {^3R^2l)
+ y {^IRV^iPil) {^3RO-^1p2L) ,
where i^u, and tpjR are anticommuting chiral fields and rjj? = — 1. There are
analogous relations for u and v spinors, but with rjj? = +1, e.g.,
(wil^w2l) (^3L7m™4l) = - (wiz/7mW4l) (^3L7m™2l) • (2.212)
The Fierz identities are easily derived by expressing the 4x4 matrices such
35 ^p2L^>3L in terms of the complete set 7,75,7M,7M75, and <rMI/. A related
useful identity is
(W>2l) (^3L<W4*) = 0. (2.213)

50
The Standard Model and Beyond
The Fierz identities are frequently very useful in computations, and are often
used in conjunction with those for charge conjugation (Section 2.10).
2.8 QED for Electrons and Positrons
Just as for pions, one can obtain the Lagrangian density for quantum
electrodynamics (QED), i.e., for the gauge theory of electrons and positrons interacting
with photons, by combining the free field Lagrangians in (2.107) and (2.148)
and applying the minimal electromagnetic substitution, zdM «=> pM —> p^—qA^1,
where q = — e < 0 for the electron field. Thus,
C = $(x) (i& - m) ?P(x) - \f^uF^
= ${x) {ifi + efi-m)^{x) - \f^vF^ (2.214)
= $(x) (i0-m) 1>{x) - eA^x)J^{x) - \f»vF^',
where
J/> = 7M£>M> D» = 5M " ieAV (2-215)
C is clearly invariant under the gauge transformation
A^^A"- -d»(3(x), ip -> eiqf3{x)/etP = e"i/3(xV, (2.216)
e
since D^) —> e~lf3^D^tp, in analogy with (2.131). In the last form of C
J%(x) = -etp(x)^tp(x) (2.217)
is the electromagnetic current.
In Feynman diagrams electrons and positrons are represented by solid lines,
with an arrow indicating the direction of flow of negative electric charge (or
opposite the flow of positive charge). The interaction term in (2.214) implies
three-point vertices involving one photon and two charged fermions, with a
factor ze7M. Charge conservation implies that there is always one arrow
entering (i.e., entering e~ or exiting e+) and one leaving (exiting e~ or entering
e+) the vertex. A final e~ has a factor u(pf,Sf), while an initial e~ has a
factor u(pi, Si). The corresponding factors for a final or initial e+ are v(pf, Sf)
and v(pi,Si), respectively. These are always the physical momenta and spin.
An internal fermion line carrying momentum k corresponds to the propagator
iSp(k) = i kiL^+iz • The sPmor indices always arrange themselves so that
each fermion line running through the diagram is a bilinear form starting with
u or v and ending with u or v. These rules are illustrated in Figure 2.13. As

Review of Perturbative Field Theory 51
U(Pf, Sf) V(P/» 3f)
%&~iix
u(pi, Si)
v(p+,s+)
u(p-,s-)
FIGURE 2.13
Vertices involving the interactions of e± with a photon. The initial fermions
enter from the bottom and the final leave from the top. ± refer to a positron
or electron, respectively. Note the crossing symmetry, i.e., up to an overall
sign the amplitude for an initial positron can be obtained from that for a final
electron except u —» v, always using the physical momentum and spin, and
similarly for the relation of a final positron and initial electron.
in Section 2.6 there is also a factor e* (eM) for a final (initial) photon, a
propagator ig^Drik) = —ig^/(k2 + ie) for an internal virtual photon, and an
integral f d4k/(27r)4 over each unconstrained internal momentum.
Since e± are fermions, the overall sign of a diagram depends on the ordering
of particles in the states. The overall sign is rarely needed, but there may be
crucial relative signs between two diagrams due to the anti-commutation rules
for the creation and annihilation operators. In particular, there is a relative
minus sign between diagrams involving the exchange of two external lines,
and a factor of —1 for a closed fermion loop, as illustrated in Figure 2.14.
More complicated sign ambiguities are best resolved by going directly to the
expression for the transition amplitude.
en Scattering
First consider e~(fci) 7r+(pi) —» e~(k2) tt+(P2)- The lowest order diagram,
shown in Figure 2.15, yields the transition amplitude
Mfi = ieu(k2,s2)'j^u{kus1) ( _ 2 ) [-ie(p2 +Pi)*}, (2.218)

52
The Standard Model and Beyond
U(P3, 83)
v=y
™(P4, S4) U(P4, 34)
JP\—p2lK
u(pi,ai) it(p2,s2) it(pi,si)
™(P3, S3)
lt(p2, 82)
v/
A f
/*\
FIGURE 2.14
Top: relative minus signs between diagrams involving exchanged fermion lines.
Bottom: a closed loop diagram, with an extra factor of —1.
u(k2, s2)
* (P2)
U>f(P3, S3) V/(P4, S4)
T< P1+P2
u(Sl'Sl) W+(fi) ue_(ft, Sl) *e+(p2, *2)
FIGURE 2.15
Upper left: Lowest order diagram for e~(ki) 7r+(pi) —> e-^) 7r+(P2)- Upper
right: Diagram for e"(pi) e+(p2) —► /(P3) /(A), where / ^ e*.

Review of Perturbative Field Theory
53
where the Feynman rules for the pion vertex are taken from Figure 2.10. Using
(2.56), the unpolarized differential cross section in the CM is
Si,«2 \ Sl,S2 /
where s = (pi + ki)2 and we have averaged (summed) over si (52). The
squared matrix element is evaluated using (2.205), (2.207) and the trace
identities in (2.167) and (2.168),
le4
\Mfi\2 = -^(p2+Pi)ll(P2+Pi)v ^(u2^ui)(ui<iuu2)
si,s2
le4
= 2^2(P2+Pi)M(P2+Pi)I/,n:[7/i(^i+me)7,/(fe + me)] (2.220)
= 2^(p2 + pi)M(p2 + Pi)„ [Kh% + k?k$ - g^fa • k2 - ml)] ,
where Ui = w(/ci, $$) and £ = (fci — k2)2. This is easily evaluated using the
kinematic relations in Section 2.3.4. For example, ignoring the pion and electron
masses, (2.220) becomes 16ira2([(s - u)/t}2 - 1).
e~e+ Annihilation
Now, consider e~(pi) e+ (p2) —> /(pb) /(pi), where / ^ e1^1 is a fermion with
electric charge Q/e. There is a single tree-level diagram, shown in Figure 2.15,
which implies
Mfi = {-iQ/e u37m^4) ( J (+^e v2-ypui)
= f6 u3^v4 v2^uu (2.221)
where s = (pi + P2)2; ^3 and V4 are spinors for a mass ray; and ^2 and u\
correspond to mass rae. Averaging over both s± and 52, the unpolarized cross
section is
J?_ _^_2L\mfi\\ (2.222)

54
where
The Standard Model and Beyond
i*„i».i Y. m>? = T
Si, 2=l--4
me'^->0
x TY (7^ + me)7^2 - me))
4Q2fe4
= ^— [P4i/P3/X + P4/xP3i/ - 0/xi/(P3 ' P4 + m^)]
x [pffi + p^2 " <^(Pi ' P2 + ™2)] (2.223)
SQ}e4 2
^— [Pi -P4P2'P3 + Pi' P3P2 ' P4 + mfpi • p2J ,
where me is neglected in the last line. In that limit,
pif2 = (£,0,0,±£), p3,4 = (£,±p/sin0,O,±p/cos0),
in the CM, where
*-«-^.
A/s — 4m?
Then,
Pi • P4 = P2 • P3 = E2{\ + /?/ cos6)
Pi * P3 = P2 • P4 = £2(1 - /?/ COS (9)
P!-p2 = 2£?2, m2 = (l-(32)E2,
implying
da _ KQ2fa2Pf
dcos 0 2s
Integrating over cos#, the total cross section is
(2-$+$cos20).
AnQ}a2Pf (3-0}
2^2
35
47rQ2a
> —
0/->l 35
(2.224)
(2.225)
(2.226)
(2.227)
(2.228)
Bhabha Scattering
The diagrams for Bhabha (e~e+ —» e~e+) scattering are shown in Figure
2.16. The amplitude is
M = ie2
^37^4^27^1 U3'JflUiV2'JfJ'V4
(Pi +P2)2
(Pi -P3)
(2.229)

Review of Perturbative Field Theory
55
e+(P4)
e (Pa)
e+(P4)
e"(pi) e+(p2)
FIGURE 2.16
Feynman diagrams for e~(pi)e+(p2) —> e~(p3) e+QJi). The relative minus
sign between the two diagrams is independent of state conventions.
Ignoring the e mass, the unpolarized cross section is then da/dcos 8
\Mfi\2/327rs, where
\Mfi\2 = \52\Mf
spins
e
~4
^ (7m 2*47^3)^(7^7^2) , TV(7m ^7,^3)^(7^ ^47^2)
+
t2
Tr (7m fa^ ^27m J^n" fa) 1* (7m ^i7*/ #27m ^47^ fa)
. (2.230)
s£ st
The traces are easily evaluated using the identities in Section 2.7.2, yielding
da is of
d cos 6 s
2 r+2
tz + uz
+ ■
s2 + u2 2u21
t2
+
St
(2.231)
Electron Compton Scattering
The Feynman diagrams for e~ Compton scattering j(ki)e~ (pi) —> 7(^2)6" (^2)
are the same as the first two diagrams in Figure 2.12 except that the pions
are replaced by electrons. The corresponding amplitude is
M = ^eiAie)2u2
r
i>x+ #1 -ra(
V 1 V
-7+7
fa- #2 - m(
-r
ux. (2.232)
It is straightforward to show that the differential laboratory cross section
averaged (summed) over the initial (final) electron spins is given by the Klein-
Nishina formula
da
-kg? (ki
dcos0l 2m2 \ki
ki+%+i^-\
(2.233)
/C2//C1 has the same form as in pion Compton scattering, Equation (2.142),
except m —» me. If one also averages and sums over the initial and final

56 The Standard Model and Beyond
photon spins, one obtains (using (2.146))
da na2 (k
d cos 8l m2 \ k\
*2 . *1 .2/1
— + sinz 6L
ki #2
(2.234)
which yields the classical Thomson cross section atot = S7ra2/3ml in the limit
k2 ~ ki < me.
2.9 Spin Effects and Spinor Calculations
We have so far mainly focussed on spin-averaged calculations. However, many
experiments involve polarized initial particles or measure the final
polarization, e.g., by their decay distributions. The calculation of fermion
polarization effects can always be carried out using the standard trace techniques,
which yield the absolute squares of amplitudes, provided one uses the spin
projections in (2.174) and (2.177). However, it is often simpler to
calculate the amplitudes directly using the explicit forms for the 7 matrices and
Dirac spinors (e.g., Hagiwara and Zeppenfeld, 1986). Further simplifications
are achieved because of space reflection invariance, which relates different
spin amplitudes. (Even for the weak interactions, which violate reflection
invariance, different amplitudes may be related up to an overall coefficient.)
However, such calculations are carried out in a specific Lorentz frame and are
therefore not manifestly invariant. Both techniques can be tedious for
nonzero masses, so in this section we will illustrate calculations assuming that
the masses are negligible. An example of a calculation for a massive fermion
is given in Problem 2.19.
The spin-averaged amplitude-squared for e~(pi)e+(p2) —» f(Pz)f(P4),
given in (2.223), reduces to
|M|2 = Q2e4(1 + cos2 0) (2.235)
for me = rrif = 0. Now consider the amplitude for definite spins in the helicity
basis, which we denote by M(hsh4,hih2), where hi = ±^ is the ith particle
helicity. For massless particles, by (2.210) the only nonzero amplitudes are
for h\ = — /&2 and hs = —/i4, since vector and axial vector interactions do not
reverse chirality. We will see in Section 2.10 that reflection invariance implies
that there are only two independent amplitudes, and that
M (+-,+-) = M (-+,-+), M (-+,+-) = M (+-,-+), (2.236)
where we have simplified the notation by only writing the sign of hi. Momen-

Review of Perturbative Field Theory
tarily keeping the masses, the absolute square for arbitrary spins is
57
\M\
sz
l±p±)lM + mi)(l±±A}
x Tr
[7"(A + me) (I±£* ) y (A -™e) (I±£*
(2.237)
where ^ is the spin vector given in (2.176). Taking the masses to zero and
using (2.177), this reduces to
|M(+-,+-)|2 = |M(-+,-+)|2
o2
-^-Tr (7„ jrf47/1 jfePi)Tr (7" 2*17" jM>l)
o2
(2.238)
-^—(Pi -P4 P2-Pz) = <22e4(l + cos<9)2,
where the calculation of the traces and contraction of the Lorentz indices was
carried out using (2.171). Similarly,
|M(+-,-+)|2 = |M(-+,+-)|2
Qje4 2 (2.239)
= —2~- (pi *P3 P2' P4">= Q2/e4 (! -cos of •
(2.238)-(2.239) reproduce the spin-averaged result in (2.235).
Now let us repeat the calculation using the explicit gamma matrices and
spinors in the chiral representation, given in (2.161) and (2.188), noting that
o o r o i o I ~ °~l 0 \
7 7=7, 77= -7 7t = ( Q +aM '
Then (using s = (V2E)4),
M (+-,+-) = Z?5zf-ti+(3)7Mt;_(4)t;_(2)7'4fi+(l)
(2.240)
(2.241)
-zQ/e2 [0+(3)tx_(4)X-(2)t0+(l) -0+(3)VX-(4)x-(2)V0+(l)]
and
M(-+,+-)= + fQ/e2^_(3)tx+(4)x-(2)V+(l)
+ 0_(3)ViX+(4)xt-(2VV+(l)] .
Prom (2.224) and Table 2.1 the helicity spinors are
(2.242)
md-(i), x-(2)=(;
/ - sin f \
<^+(3) = x+(4) =
COS
sin
(2.243)

58 The Standard Model and Beyond
The first terms in (2.241) and (2.242) vanish. The second terms are most
easily evaluated using the SU(2) Fierz identities in Problem 1.20,
<K3) VX(4)x(2) V</>(1) = 20(3)t0(l)X(2)tx(4) - 0(3)tx(4)X(2)t0(l),
(2.244)
yielding
M (+-, +-) = M (-+, -+) = 2iQfe2 cos2 9- = iQfe2 (1 + cos0) (2.245)
M (-+, +-) = M (+-, -+) = -2iQfe2 sin2 °- = -iQfe2 (1 - cos(9).
2.10 The Discrete Symmetries P, C, CP, T, and CPT
Space reflection and charge conjugation are symmetries of the strong and
electromagnetic interactions, but are violated by the weak interactions because
of their chiral nature. The product CP is also violated in the weak sector,
though more weakly, due ultimately to complex phases in the Yukawa
interactions between the fermion and Higgs fields. Such phases may be large, but
their effects are small because they require mixing between all three fermion
families to be observable. Time reversal is presumably also violated, because
of the CPT theorem, which states that any local, Lorentz-invariant, unitary
field theory must be invariant under the product CPT (Streater and Wight-
man, 1980). In this section we introduce the basic formalism for the discrete
symmetries. Physical consequences and tests will be described in Chapters 6
and 7. More detailed descriptions of the discrete symmetries may be found,
in (e.g., Bjorken and Drell, 1965; Gasiorowicz, 1966; Weinberg, 1995; Sozzi,
2008).
Space Reflection
Under space reflection (P) a classical vector changes sign, e.g.,
£—►-£, p —> -p. (2.246)
However, an axial vector such as orbital angular momentum L = x x p is left
invariant, and it is reasonable to define spin and total angular momentum to
have the same property
J^J. (2.247)
Both vectors and axial vectors transform as vectors under rotations. Scalars
are rotational scalars which do not change sign under P. Examples are t,

Review of Perturbative Field Theory 59
E, and \p |2. Pseudoscalars, such as J • p, do change sign. In particular,
helicity, /i = 5 • p, is a pseudoscalar and reverses under P. Vector and axial**
four-vectors transform, respectively, as
*"* t* v»> A" r_A"' (2-248>
so that x = (t,x) and p = (E,p) are vectors. The fermion spin vector sM in
(2.176) is an axial four-vector. The angular momentum J is part of a second
rank antisymmetric tensor, U = \<eij\zUk^ where
LTV = xV - xV 7» V- (2-249)
A single particle state is assumed to transform similarly,
P\ps) = Vp\ - ps), P\ph) =T)P\-p- h), (2.250)
where (in this section) s represents spin with respect to a fixed axis and h
represents helicity*. The phase rjp is the intrinsic parity, which depends on
the particle type. One must have rjp = ±1 to ensure P2 = I. Invariance of
the action in (2.17) requires invariance of the Lagrangian, and therefore that
PC (t, x) P~l = C (t, -x). (2.251)
For the example of the free complex scalar,
C = (<9M</>)t(<9^</>) - m2</>+</>, (2.252)
this can be accomplished for
P</>(£, x)P~l = T)P(t)(t, -x). (2.253)
The mass term is obviously invariant. For the kinetic term,
8^(x) -+ VPd^{x') = VPQ^ttx') = VP-faTtK*') = Vpd'»<t>(x'), (2.254)
where x' = (*,-£). Thus, \d^{x)\2 -> \d'^(j){x')\2. (2.253) follows from the
free field expression in (2.89) provided
Pa\p)P~l = rypat(-p), Pb\p)P~l = r)Pb\-p), (2.255)
where a^(p) and tf(p) are the particle and antiparticle creation operators.
The transformation of a Hermitian scalar field is the same as in (2.253) and
(for a*(p)) in (2.255).
use the symbol A^ both for gauge fields and for axial vectors. The meaning should
always be clear from the context.
*In the helicity form, there may be additional (p, ^-dependent phases, depending on the
phase conventions for the helicity states.

60 The Standard Model and Beyond
For the free Dirac field, with
£ = tp(t,x)[i0-m]ip(t,x), (2.256)
we must choose Pip (£, x) P_1 in such a way that
P$ (t, x) ip{t,x)P = ip (t, -x) ip (t, -x) (2.257)
and
Pip (t, x) ^ip (t, x) P~l = ip (t, -x) 7m^ (t, -£). (2.258)
The lowering of the index on 7M compensates <9M = <9'M, analogous to (2.254),
i.e., ip(x) ftip(x) = $(x') ftip(x'). These conditions can be satisfied if
PiP (t, x) P-1 = 7V (*, -x) (2.259)
=» P^ (t, f) P"1 = (7V (*, ~x))] 7° = ^ (*, -£)f 7°7° = ^ (*, -*) 7°,
using 7°7^7° = 7^ from (2.151). (We have taken 77P.0 = +1 for simplicity.)
For chiral fields,
^t 0 -r .- ^ n (2-260)
PH>l,r (t, 2) P-1 = (7Vr,l (t, -x)Y 7° = $r,l (t, -x) 7°
Using the expression (2.155) for the free Dirac field and the relations*
7% (p, s)=u (-p, s), 7% (p, s) = -v (-p, s), (2.2,61)
which follow from the explicit spinor forms in (2.7.2), we see that (2.259) is
equivalent to
Pa\p,s)P~l = a\-p,s), Pb\p,s)p-1 = -b\-p,s). (2.262)
Thus, a fermion and its antifermion must have opposite intrinsic parity.
An arbitrary fermion bilinear form transforms as
PtPa (t, x) TiPh (t, x) P"1 = VPaVPb^a («, -x) 7^7°^ (*, -£) • (2.263)
The values of Tp = 7^7° are listed in Table 2.2. In particular, F = 1,75,7M,
and 7M75 transform as a scalar, pseudoscalar, vector, and axial vector,
respectively, for T]pa7]pb = 1.
As will be discussed in Chapters 3 and 5, the Yukawa interaction between
a neutral pion n° and nucleons can be phenomenologically described by
CnN = ig* {vibP ~ nihn) 7T°, (2.264)
^For the phase conventions in Table 2.1, the transformations in the helicity basis are
j°u(py<f>±(p)) = ±e±i(*u (-p, (pzf(-p)) and 7°v(p,x±(p)) = TeTi(fiv (-p,Xq=(-p)), where
(f is the azimuthal angle of p.

Review of Perturbative Field Theory
61
TABLE 2.2
Transformation of the complete set of Dirac matrices and of their related
chiral forms under Hermitian conjugation (the Dirac adjoint T) in
(2.206)), space reflection (rp) in (2.263), charge conjugation (rc) in
(2.291), and time reversal (I\) in (2.320). The pseudotensor a^^5 is not
independent (Problem 2.3), but is included for completeness.
scalar
pseudoscalar
vector
axial vector
tensor
pseudotensor
r
~T
75
7"
7M75
o»v
CTM"75
Pl,r
1»Pl,r
°»uPl,r
r
7ort7o
1
-75
7M
7M75
a^
-a^j5
Pr,l
-fPh,R
<t^Pr,l
Tp
7°r7°
1
-75
7m
-7^75
&HV
-<V75
Pr,l
1^Pr,l
<?HvPr,L
Tc
CYTC-X
1
75
-r
7M75
-a'1"
-a^j5
Pl,r
-7»Pr,l
-<t^Pl,r
rt
Tr»T-l
1
75
7m
7m75
—O'tiv
-a^vl5
Pl,r
1»Pl,r
-&hvPl,r
where gn is a real coupling, and where p and n respectively represent the
proton and neutron fields. Since the nucleon term is a pseudoscalar, reflection
invariance requires that the 7r° must be a pseudoscalar, i.e., rjn = —1. Of
course, this was originally ascertained by experiment (see, e.g., Gasiorowicz,
1966), i.e., by the observation of the reaction tt~D —> nn, where the deuteron
D has Jp = 1+ and the n~ was established to be in an S-wave. The neutrons
had to be in an odd-parity 3P\ state, since it is the only antisymmetric J = 1
state available. Similarly, the 7r° was shown to have odd parity by the angular
distribution in n° -> e+e~ e+e~ (Abouzaid et al., 2008). The i in (2.264) is
required by Hermiticity.
The interaction terms in (2.129) and (2.214) indicate that the
electromagnetic field Aq couples to vector currents, where in this section, we use the
symbol Aq to distinguish it from an axial vector. Reflection invariance of the
electromagnetic interactions therefore requires that Aq transforms as a vector
PA%(t, x)P~l = AQ^{t, -£), (2.265)
which is consistent with the classical expectation E —» — E and B —» +B.
Using the free field results in Section 2.5, this is equivalent to
Pat(fl \h)p-1 = a\-p, -Afc), (2.266)
where the polarization label A^ represents the photon helicity. This can be
seen from the free field expression (2.109), using the relation
c/i(SAfc) = cM(-p,-Afc), (2.267)

62
The Standard Model and Beyond
which follows from (2.113) and (2.114) under the convention
e"(jf, A) = (-l)A+1eM(-p, A), A = 1,2 (2.268)
for the linear polarization vectors. All of these results also apply to massive
vectors, with three helicity states.
The explicit transformations of the fields in (2.253), (2.259), and (2.265)
were justified for free fields. However, the same transformation laws will
continue to hold in the presence of the strong and electromagnetic interactions,
or any others which respect reflection invariance. This is apparent in the
interaction picture (see, e.g., Peskin and Schroeder, 1995, and Appendix B)
in which the interacting fields can be formally expressed in terms of the non-
interacting ones and the interaction Hamiltonian Hi, and using [P, Hj] = 0.
To illustrate the importance of this, consider the matrix elements of the vector
and axial currents, ^al^b and ^al^^b, respectively, where ipa,b represent
strongly interacting fields such as nucleon or quark fields, and a can be the
same as or different from b.
(a{pa, Sa)\$a7n1pb\b{pb, Sh)) = u{pa, Sa)r^u(p6, Sb)
_ A (2.259)
(a(Po,*o)|^o7/i7 ^b\b(Pb,sb)) =u(pa,sa)r^u(pb,sb).
The fields are evaluated at x = 0 (otherwise, by translation invariance, there
would be a factor el^Va~Vb^'x on the right). In the absence of strong or
electromagnetic interactions, one would have T^ = 7M and T^ = 7M75. However,
such corrections can yield more complicated matrix elements. Prom Lorentz
invariance, T^A can involve linear combinations of 7M, cr^uqv', gM, 7^75, cr^uqu^,
and #M75, where q=pa—Pb- Each of these can be multiplied by an arbitrary
form factor that can depend on the invariant q2. Other four-vectors, such as
those involving pa^ + pb^ are not independent (see Problem (2.4)). However,
reflection invariance of the strong and electromagnetic interactions restricts
the possibilities. Assuming the same intrinsic parities for a and 6,
(a(pa,sa)\^a7^b\b(Pb,sb)) = (a(pa, sa)\P~lP^al^bP'1 P\b(Pb, sb))
= (a(-pa, Sa)\$a"f07^01pb\b(-pb, Sb))
= u(-pa, sa)Tv^{q')u{-pb, sb)
= u(pa, sah°rv»(qf)7°u(pb, sh), (2.270)
where we have used (2.261). The notation Tv^{q') indicates that it is
evaluated using q'^ = p'a^ — p'h = q^. Reflection invariance therefore requires
T» = 70r^(q')70- (2-271)
Comparison with Table 2.2 shows that T^ can only contain 7^, a^uqu, and q^.
Similarly, T^ can only contain 7M75,crMI/gI/75, and gM75. (Higher-order terms
in the weak interactions, which violate reflection invariance, could induce

Review of Perturbative Field Theory
63
the wrong terms. However, one can always treat such effects explicitly in
perturbation theory, using P invariance for calculating the matrix elements.)
As another application, consider M(a,i(pi,hi)), the matrix element of a
scattering or decay process involving k external particles with particle types
a momenta pi, and helicities hi, i = 1- - - k, some of which are in the initial
and some in the final state. If the relevant interactions are reflection invariant,
then it follows from the interaction picture expression that the amplitude is
the same as the amplitude for the reflection-reversed process,
M{ai{pi, hi)) = r)M(ai(-pi, -hi)), (2.272)
in which rj = ±1 is associated with the intrinsic parities. This result is
especially useful for k = 3 or 4, i.e., decays into 2 or 3 particles or 2 —» 2
scattering, because in that case one can use ordinary rotational invariance to
show that M(ai(—pi,—hi)) = M(ai(pi,—hi)). This is obvious in the center
of mass frame, or in a frame in which one particle is at rest, because the
remaining momenta then lie in a plane and can be reversed by a rotation by
7r around an axis perpendicular to the plane*. Therefore, for k < 4,
M{ai{pi, hi)) = r)M(ai(pi, -hi)), (2.273)
which can be very useful in simplifying calculations.
Charge Conjugation
Charge conjugation changes particles into antiparticles, without affecting their
momenta or spin, e.g.,
C\a(p,s))=VCa\ac(p,s)), (2.274)
where ac is the antiparticle to a and rjca is a phase factor. (It is basically a
convention as to which is called the particle and which the antiparticle.) For
a complex scalar field, this implies
CtfC'1 = r,c*4>, C^C-1 = rfc<t>4>\ (2.275)
since <ft is the field for the antiparticle to </>. (2.275) implies
Cat(p)C-x = nc+tfip), C6t(p)C-1 = v*c^(P) (2-276)
for the creation operators in (2.89). One can always make a phase
transformation </>' = y/±rjc<j>(t> so that the new field has r]c<j>' = ±1. For a Hermit ian
scalar, one must have r)c<$> = ±1,
C(f>C-1 = ±</>, Ca\p)C-1 = ±a\p). (2.277)
see this for an arbitrary frame, a violation would require terms of the form p\ • (p2 x ps)
in M. However, this would have to arise from a Lorentz invariant e^upaPa^PbuPcpPda >
which would vanish for k < 4 by momentum conservation. Such terms could arise for k > 4.

64
The Standard Model and Beyond
If one expresses a complex scalar <j> with 77c 0 = ±1 in terms of two Hermitian
scalars </>ij2, as in (2.87), then rjc4>1 = —Tlcfo = ±1- Results similar to (2.275)-
(2.277) hold for the transformation of a vector field A^. For QED the gauge
field couples to a C-odd vector current, so one must choose Aq^ —> —Aq^. (C
is not conserved for the chiral theories that will be introduced in Chapter 4,
which involve both vector and axial currents with opposite C transformations.
It is nevertheless useful to define the transformations of the gauge fields in
this way.) The intrinsic C phases can be determined experimentally for (C-
invariant) strong and electromagnetic processes. For example, 7r° decays to
two photons, so rjcn° = ric1 = +1-
For QED, the electron state is transformed into a positron,
C\e-(p,s))=r1ce\e+(p,s)), (2.278)
and similarly for other fermions. One requires
&PC-1 = rfCi,r, CrpC-1 = nc^c = vci, (Vc)f 70, (2.279)
where ip is the e~ (or fermion) field, and ipc is the e+ (or antifermion) field.
Only the relative phases of different fermion fields in bilinears are usually
relevant^; in the following we will set rjc^ = 1 for simplicity. From (2.155),
the free fermion fields are given by
Tp{*) = J {2^2E E Kp> «) a& s)e~ip'x+v& s)6t(p> s)e+ipi
r(x) = J {2^f2E E Hp> «) b& s)e~ip'x+v& s)atw s)e+ip'x].:
(2.280)
so one must choose
Cat(^«)C-1=6t(S*), Cb^(p,s)C-1 = a^(p,s). (2.281)
Prom (2.280) it is apparent that tpc is related to the Hermitian conjugate
ip^. The precise relation, which is extremely useful even if a theory is not
charge conjugation invariant, is
^ =C$T = C (^t7°)T = C7°VT, r = -^C-1 (2.282)
(in indices, tp^ = Capijip = {^7°T)ctl3 ipa)- C is a 4 x 4 Dirac matrix, which can
be determined by the requirement that the free Lagrangian density in (2.148)
is C-invariant, i.e.,
CC0C~l = i>c(x) (iff-m) ipc(x) = C0. (2.283)
§Majorana masses, which we will encounter in connection with neutrinos in Section 7.7,
and supersymmetry in Section 8.2 are an important exception.

Review of Perturbative Field Theory 65
(2.283) is satisfied if and only if C satisfies
C_17MC = -7J, (2.284)
where one must use the anticommuting nature of the fermion field, and the
fields are considered to be normal-ordered so that c-numbers can be ignored^.
Consistency between (2.280) and (2.282) (or between (2.157) and (2.159))
further requires that the u and v spinors are related by
v(p, s) = Cu(p, s)T = (C70T) u(p, *)*, u(p, s) = Cv(p, s)T, (2.285)
which implies
u(p, s) = -v{p, s)TC\ v(p, s) = -u(p, s)TCf. (2.286)
These relations hold both for the case of a fixed spin axis or for helicity spinors.
The explicit form of C depends on the representation used for the 7 matrices
and is seldom needed. However, C takes the simple form
C = -C~l = -Cf = -CT = ±z727° (2.287)
in the Pauli-Dirac (+^727°) and chiral (—Z727°) representations. The off-
diagonal nature of a2 accounts for the extra minus sign in (2.186).
For arbitrary fermion fields tpa^ (with the same intrinsic phases),
(2.288)
In particular, the electromagnetic current
J% = -^e-V^e- (2-289)
in QED transforms as
CJ&C-1 = -^e+7^e+ = +&-We- = ~J%, (2-290)
so that the interaction —cAq^Jq in (2.214) is invariant for Aq^ —» —Aq^.
An arbitrary fermion bilinear transforms as
C^aT^C-1 = ram = ^rc^a, (2.291)
where
rc = (c_1rc)T = cttc~1. (2.292)
^From (2.282) and (2.284) we have that ip and ipc have opposite intrinsic parities under
space reflection, consistent with the comment following (2.262) and ensuring that C and P
commute.

66 The Standard Model and Beyond
The Tc are listed in Table 2.2. The spinor identity corresponding to (2.291)
differs by a sign, i.e.,
waTwb = -wcbTcwca, (2.293)
where each w can be either & u or v spinor, and
uc{p, s) = v{p, s), vc(p, s) = u{p, s) (2.294)
is the charge conjugate spinor.
The chiral fermion fields transform as
CiPLC-1 =?Pcl = Pl*I>c, OPrC-1 =rR = PriPc, (2.295)
where ipli'ip'fi) is the field that annihilates a left (right)-chiral antiparticle or
creates a right (left)-chiral particle. Prom (2.282) and (2.295)
rL = pLr = cVr, rR = pRr = c$z
where", e.g., ^TR = [{ipR)h°]T = 7°TW>fl)tT- Thus, $CL ~ ^R up to Dirac
indices. Similar results hold for the transformations (2.285) and (2.286) of
the chiral spinors. The flip of chirality in (2.296) is at first confusing. It is
associated with the fact that C does not change the spin or helicity, but the
relation between chirality and helicity is reversed for antiparticles, as can be
seen in (2.191). E.g., in the massless limit tpcL and ipR both create positive
helicity particles or annihilate negative helicity antiparticles. Note that 7M and
7M75 transform with opposite signs, so that a left-chiral current is mapped
onto a right-chiral one,
c^Li^bLC-1 = raLrrbL = -^7^*. (2.297)
Thus, the charged current weak interactions, which only involve left-chiral
currents, violate both space reflection and charge conjugation invariance.
Similarly,
C$aLlpbRC-1 = $CaL1pCbR = +$bL1paR
The relations (2.291), (2.293), and (2.297) are frequently used in conjunction
with the Fierz identities in (2.211). For example, combining (2.211) and
(2.297),
= - WbLl^^dL) WcLI^Il) •
"Some authors use the notation ipR or (iPr) rather than ip^ to emphasize that the Dirac
adjoint operation acts on ipn rather than on ip.

Review of Perturbative Field Theory
67
The transformation laws for boson and fermion fields continue to hold in
the presence of the strong and electromagnetic interactions, which are C-
invariant. Even including the weak interactions, which violate C, the same
relations hold for the unperturbed fields when the weak effects are treated
perturbatively, analogous to the discussion of reflection non-invariance. The
spinor identities in (2.285) and (2.286) are valid independent of whether C is
violated.
The implications of C-invariance and the related concept of G-parity for
strong transitions and for matrix elements of operators will be discussed in
Section 3.2.5 and Appendix G.
CP Transformations
It is straightforward to write Lagrangians that violate P or C, but more
difficult to find ones that violate CP invariance, at least for a small number
of fields, due to the combination of Hermiticity, Lorentz invariance, and the
fact that the overall phases of physical fields or states are not observable in
quantum mechanics. In particular, one can always choose the phases of the
fields so that any given term in a Lagrangian is manifestly CP invariant. CP
violation can therefore only emerge when two or more terms clash and cannot
be invariant simultaneously. Note that C and P commute.
To make this more explicit, the P and C transformations of spin-0 fields <j>
and generic spin-1 fields W^ defined in the previous sections always imply
4>{x) -^ ^(a/), W»(x) -^ VwWl(x'), (2.300)
where x' = (£, —x), rj^ and rjw are products of C and P phases, and the f are
omitted for Hermitian fields. Similarly,
<P(x) —»i^W) = -v;C^T(x')
(2 301)
*l,r{*) -^ ^7°V£,l(*') - -v;CtP[Tr(x>).
Thus, CP maps ipL,R onto its own Hermitian conjugate. In contrast, P and C
each maps a chiral fermion field onto a different one, tpL,R —► ipR,L, ipL,R —►
vr L- The elementary fermion bilinear forms /0a(x)r,0b(x) for the T's in Table
2.2 all satisfy
^a{x)T^h{x) —> 7£a6^(* Wa(*') = Vhb [M^TM^)}+, (2-302)
where r/pa6 = VaVb'nr^ with rjr = +1 for T = 1 or 7s, and rjr = — 1 for
T = 7M,7M75,crMI/, and a^j5. It is understood that Lorentz indices in T are
lowered in the second and third expressions. (2.302) can be easily verified
from the transformations in (2.206), (2.263), and (2.291), along with Table
2.2. The upshot of (2.300) and (2.302) is that any Poincare invariant term in

68
The Standard Model and Beyond
Ci{x), such as
\<t>l<t>b\2, 4l, ^^- Mb<f>c talntoW", (2.303)
as well as a mass or kinetic energy term, is mapped onto its Hermitian
conjugate evaluated at x' up to an overall phase Ci(x) > r)lC\(x'). Furthermore,
the 77 phases can always be chosen (or the fields redefined) so that rji = 15
and therefore &(x) + C\(x) ► C\(x') + Ci(x') (or just &(x) ► &{x') if
it is Hermitian). CP invariance of L results if this can be done for all of the
terms simultaneously. Let us illustrate by an example involving two complex
scalars </>a and </>*>, with
£ = E (l5^l2 - ™2 W2) - v(^ </>»)■ (2-304)
r=a,b
For the special case
V = gabb4>a4>l+9lbb4>l4>l\ (2-305)
one has
V -^ gahhfllvftitt2 + 9l»t?k&4>a4%- (2-306)
The model is CP invariant even \£gabb = \9abbW°t is complex because we have
the freedom to choose the CP phases r)a^. For example, C(x) —» C(x') for the
choice rja = exp(2za) and 7/6 = 1. This can be cast in a simpler form by a field
redefinition, in which one rewrites C in terms of a new field </>a = </>a exp(za),
so that •
V = \gabb\(4>a4>l + 4>l4>l2), (2-307)
with the other terms unchanged in form. This is invariant for rja = rjt, = 1 (for
which the transformations of </>a and </>a are consistent). On the other hand,
the potential
V = gabb<t>a<t>l + 9aab4>l4>b + 9aaa<t>l + ^- (2.308)
with complex coefficients is not CP invariant because the two CP phases rja^
are not sufficient to make all three terms invariant (except for special cases
such as gaaa = 0 or all phases = 0). Equivalently, one can perform two phase
redefinitions of the fields, but that is not enough to make three coefficients
real in general. This example illustrates the origin of CP violation in the
standard model, in which there are not enough quark field redefinitions to
remove all of the phases from both their mass terms and the charged current
weak couplings simultaneously.
Time Reversal and CPT
Under a time reversal (T) transformation, classical trajectories are replaced
by their time-reversed ones,
x(t)^x(-t), p(t)^-p(-t), J(t)^-J(-t). (2.309)

Review of Perturbative Field Theory
69
The electromagnetic current and gauge potential satisfy
JqmW V J£(x"}' Aq»{x) V AQ{x'^ (2'310)
where x" = (-«,£), so that £(x) -> £(x") and B(x) -> -JB(x"). Time
reversal is more complicated in quantum mechanics. In analogy with (2.309)
one defines a time reversal operator T so that
TxiT-1 = xu TpiT-1 = -Pi, (2.311)
where Xi and Pi are the position and momentum operators. However,
consistency with the canonical commutation rules [#i,/>j] = iSij (or with the
Schrodinger equation) requires that T is antiunitary, i.e., TcT~l = c* for any
c-number c. This extra complex conjugation significantly complicates many
calculations. 'The antiunitarity is also required in field theory for the
consistency of the canonical commutation relations for the fields. A consequence of
antiunitarity is that
(a|0|6) = (Ta\TOT-l\Tb)\ (2.312)
where O is an operator and \Tb) and (Ta\ are time reversed states. For
example,
T\a(p, 8)) = \Ta(p, s)) = ±\a(-p, -*)>, (2.313)
for a single particle state with spin orientation s measured with respect to a
fixed axis. For multiparticle states, T also interchanges initial (in) states with
final (out) states.
Invariance of the action under time reversal requires TC(x)T~l = C(x").
This is ensured for free spin-0 fields for
T4>{x)T-l=r)T4>ct>{x"), (2.314)
where <j> can be either Hermitian or complex, and tjt^ is a phase. (In the
complex case </> = (fa + ifa)/y/2 this requires tjti = —rjT2 for the Hermitian
components fa and fa.) For an electrically charged complex field, (2.314)
implies that (2.310) is satisfied for the contribution
J+f{x) = iqcj)\x)t^(j)(x) -> -iq(f>\x"fd»(t>(x") = J^(x") (2.315)
since 9M = —d'^. Similarly,
TW^(x)T~l = t]twW^(x") (2.316)
for an arbitrary Hermitian or complex spin-1 field.
For a Dirac field, invariance of the free field action or the requirement
(2.310) for JQ implies
T^(x)T~l=r]Tll)T^(x"),
(2.317)

70
The Standard Model and Beyond
where T is a Dirac matrix satisfying
T^T-1 = 7"T = %■ (2.318)
In the Pauli-Dirac and chiral representations
T = T* =T~l = -T* = i7173. (2.319)
A fermion bilinear therefore transforms as
Ti>a{x)T^h{x)T-1 = i>a{x")Tti>h(x"), Tt = TT*T~\ (2.320)
assuming the same T-phases for ipa,b' The values of Ft are listed in Table 2.2.
Implications of T-invariance for matrix elements will be discussed in Appendix
G. The u and v spinors transform as
Tufa s) = u{-p, -s)*eia, Tvfa, s) = v{-p, -s)*ei/3, (2.321)
where a and (3 are phases that depend on s. For the conventions in (2.180)
and (2.188), exp(za) = exp(i/3) = =fi for s in the ±z direction.
The CPT theorem, i.e., {CPT)C{x){CPT)~l = C{-x) for a Hermitian,
local, Poincare-invariant £, can be demonstrated heuristically by an extension
of the discussion following (2.303). There it was argued that CP mapped each
term onto its Hermitian conjugate, up to a phase depending on the phases
of the complex coefficients and the intrinsic CP phases of the fields in that
term. The same occurs for CPT, except that the phase can always be chosen
to be +1. To see this, by combining the transformations for T, P, and C,;we
find that
<j>(x)
- 5„ ,*T (2-322)
where the upper (lower) sign is for the Pauli-Dirac (chiral) representation,
and r\ = t\tt\pt\'q. Analogous to (2.302), the fermion bilinear forms transform
as
$a(x)Tth{x) ——* fjrab [^a(-x)TM-x)]*, (2.323)
where ffrab = VaVbflr, with 7yr = 1 for T = l,75,aMI/, or a^^5, and r)r = —1
for T = 7^ or 7M75. CPT also complex conjugates any coefficients, and of
course -^ = -gA^. Therefore, if one chooses the phases so that fj^ = +1
for all </>, i)y = — 1 for all V, and f}^ = —1 for all tp every Poincare invariant
term in C(x) will be mapped onto its Hermitian conjugate evaluated at —x,
leaving the action invariant. (Any universal value for fj^ would suffice for a
fermion number conserving theory, but fj^ = real is required for manifest CPT
invariance in the presence of Majorana mass terms.) More rigorous derivations
of the CPT theorem may be found in (e.g., Streater and Wightman, 1980;
Weinberg, 1995).

Review of Perturbative Field Theory
71
The CPT theorem implies that the mass, intrinsic properties, and total
lifetime of a particle and its antiparticle must be equal (see, e.g., Sozzi, 2008).
However, CPT does allow partial rate asymmetries (Okubo, 1958) if CP is
violated, i.e., the decays rates T(a —» 61,2) for a decay of a into 61 or 62 can
differ from the corresponding antiparticle decay rates T(ac —» b^ 2), provided
that the sums of the partial rates are the same. This is important for many
models of baryogenesis (Chapter 8). Experimental searches for CPT and
Lorentz invariance violation are reviewed in (Kostelecky and Russell, 2008;
Amsler et al., 2008).
2.11 Two-Component Notation and Independent Fields
Weyl two-component fields (or Weyl spinors) were briefly introduced in (2.196)
and the subsequent discussion. The description of the discrete symmetries
for fermions is somewhat simpler when re-expressed in the two-component
language. First, consider the case of a four-component Dirac field -0, written in
terms of two-components Weyl L and R fields as in (2.196), i.e., tp = '
Using the explicit chiral forms
^
R
«-(-r£). ^-(yry --(:::>)■ <->
the transformations under P and C in (2.259) and (2.282) become
*l,r{x) y **,£(*'), **,*(*) "^ *£,*(*), (2-325)
where x' = (£, —x) and we have set rjp^ = rjc^ = +1. The charge conjugate
Weyl fields are defined as
*r = ^2*L n = -«r2*«> (2-326)
where * is a shorthand for fT, i.e.,
n,R = *ITr (2-327)
is a column vector with components that are the adjoints of those of ^l,r-
(2.325) and (2.326) require that 9cLjR(x) —> -^cR,L(x'), i.e., the fields and
their charge conjugates have the opposite intrinsic parity. Both P and C map
one Weyl field onto another, e.g., \I>l is mapped unto ^r or \I>^. On the other
hand, CP maps a Weyl field onto its own adjoint:
*l(x)—> *cR(x') =+ia2n(x')
, o (2-328)
9R{x) -^ 91&) = -i^R^'l

72 The Standard Model and Beyond
with the ia2 acting like a raising/lowering operator on the helicity indices
(cf. Eq. 2.187). Thus, the CP transformation always exists, even in a theory
involving a single Weyl field, while P and C are only defined when one has
both ^l and VR.
Under time reversal,
9L,R{x) -^ -<72*l,*(x"), (2.329)
where x" = (—t,x) and we have used (2.317) with tjt^ = 1. Combining
(2.328) and (2.329)
*L,fl(x) -—> =R*l,«(-*)*, (2.330)
consistent with (2.322).
Fermion mass and kinetic energy terms are expressed in two-component
notation in (2.197). Some important fermion bilinear forms become
i>a15lpb = *L*6R - *1r *6L (2-331)
: aM and ctm are defined in (2.1
a"" = - {a^o" - <7"ct") ,
J* = -** = -*£,
63), and
4
,« = +,« = +l-tijkak.
(2.332)
(2.333)
The transformations of the bilinear forms under the discrete symmetries can
easily be rewritten from those given in Section 2.10 and Table 2.2. Especially
useful are the charge conjugation identities such as
K{%R = *Jt*««. K{^%L = -*!**"*««, (2.334)
analogous to (2.297) and (2.298). The Fierz identities for the two-component
fields can be read off from (2.211) or derived directly from Problem 1.1. For
example,
(*It<r"*6t) h\Lo^dL) = hlLa^dL) falL&^bi)
/ W ( / N / N (2-335)
(*1R^*6R) (*lLa^dL) = -2 (*lR*dL) (*lL*bR) ■

Review of Perturbative Field Theory
73
Combining (2.334) and (2.335) one finds identities such as
(vIl^^l) (*L^*<*l) = 2 (n^dL) (vLKr) ■ (2-336)
As a very simple example, let us repeat the calculation of the amplitude
for e"'(pi) ^(ft) —> /(ft)/(ft) given in Section 2.9 one more time, using
two-component notation. The QED interaction for the e~ is
C = -eA»J% = +eA^ (tf£*"*L + *^M*fl) , (2.337)
and similarly for the other fermions, so that
Af (+-,+-) = =^</+(3)/_(4)|^a"*fl|0>(0|*^*fl|e+(l)ei(2))
M (-+,+-) = ^^(/-(3)/+(4)|*^*L|0)(0|*JiaM*R|e+(l)ec_(2))
(2.338)
in an obvious notation. Using the free field expression in (2.203), the
amplitudes become
M (+-,+-) = -iQ/e2 [^+(3)ta"(-x-(4)) (-X-(2))V^+(1)]
M(-+,+-) = -iQ/e2 [^_(3)ta"(+x+(4))(-x-(2))t<r^+(l)] ,
which reproduces 2.241 and 2.242.
We will mainly utilize the two-component notation in connection with
neutrino masses and for displaying the interaction terms in a supersymmetric
theory. Much more extensive expositions, including such topics as Feynman
rules and propagators in two-component form are described in the books on
supersymmetry and in (Dreiner et al., 2008). A more compact version of the
two-component notation will be introduced in Chapter 8.
Independent Fermion Fields
It was emphasized in (2.192) that the left- and right-chiral projections ipL,R =
Pl,riP or their associated Weyl fields ^l,r could be considered as
independent degrees of freedom. However, from (2.282) and (2.296) it is clear that the
conjugate fields tpc and ip^ R are not independent, but are related by ipc ~ ip^
and iPlR~ ipR L. In many cases it is convenient to work in terms of the ipL,R
(or ^l,r) fields, as is conventionally done for QED, quantum chromodynam-
ics, and most treatments of the standard electroweak model. However, it is
sometimes easier to work in terms of the L chiral fields for both the particles
and antiparticles, i.e., ipL and i\)cL (or \I>l, \I>£) instead, regarding the ipR and
tycR as dependent. This is typically done when discussing grand unification
or supersymmetry, for example. To illustrate this, the QED electromagnetic

74 The Standard Model and Beyond
current Jq can be written in a number of equivalent ways, including
(2.340)
where ipe and tpec = i\)ce are respectively the e and e+ fields. In two-
component notation
JQ = -*eL^"*eL - *Ifl<r"*e* = -*\L^eL + *\cL<J^e<L- (2.341)
Similarly, a fermion mass term can be reexpressed as
-C = m^ip = m ($LipR + $r$l) = m ($LC^f + ipfCtpL)
(2.342)
where we have used (2.296) and (2.287). The two forms in the second line
emphasize two different interpretations of a Dirac mass. The ty*R ^l
expression, for example, can be viewed as annihilating an L chiral field and creating
an R chiral one, while the equivalent —^c^ia2^L form can be interpreted as
the annihilation of two distinct L chiral fields.
2.12 Quantum Electrodynamics (QED)
Quantum electrodynamics represents the merger of three great ideas of
modern physics: classical electrodynamics as synthesized by Maxwell, quantum
mechanics, and special relativity. The basic formulation of QED was
completed by 1930; it combined the Dirac theory of the electron (with its correct
predictions of the lowest order electron magnetic dipole moment and the
existence of positrons) with the quantization of the electromagnetic field into
individual photons. A workable prescription for handling the divergent
integrals by renormalization of the electron mass, charge, and wave function
was completed by the early 1950s by Bet he, Feynman, Tomonaga, Schwinger,
Dyson and others (Schwinger, 1958). QED therefore became a
mathematically consistent and well-defined theory, which was subsequently tested to
incredible precision.
In order to consider QED seriously, one must include higher-order (loop)
effects in perturbation theory, both because of the precision of many of the tests
and because some effects (such as light by light scattering via the interaction
of photons with virtual charged particle loops) only occur at loop level. The
calculation of higher order effects is greatly complicated by divergences and
the need to renormalize (while maintaining gauge invariance). A systematic

Review of Perturbative Field Theory
75
study is beyond the scope of this book, and only brief comments are made.
The subject is studied in detail in standard texts on field theory. Another
complication is that strong interaction effects are very important and cannot
be ignored in studying the electromagnetic interactions of strongly interacting
particles (hadrons), such as protons, neutrons, and pions. They even enter at
higher orders (through loops involving virtual hadrons) in the
electrodynamics of electrons and muons. Fortunately, a great deal can be said about these
strong interaction corrections using symmetry principles.
2.12.1 Higher-Order Effects
Analogous to the Hermitian scalar case in Section 2.3.6, QED has
logarithmically divergent diagrams such as those shown in Figure 2.17. To take this
FIGURE 2.17
Electron-photon vertex (top left), and one-loop corrections, corresponding to
the vertex correction (top middle), the vacuum polarization (or photon self-
energy) correction (top right), and electron self-energy corrections (bottom).
into account, let us modify the notation in the QED Lagrangian density in
(2.214) to
C = $(x) (ifi + e04 ~ m0)ip(x) - -AF^UF^
4 (2.343)
= $(x) (ifl- mo) t/>(x) - e0J^ - -F^F"",

76
The Standard Model and Beyond
where Jq = —tp'y^ip is the electromagnetic current operator defined in (2.289)
and ip is the electron field, eo and mo are the bare positron charge and mass,
respectively, i.e., the parameters that appear in C. We redefine e and m as the
physical parameters, i.e., the ones actually measured. The renormalizability
of QED implies that the divergences enter only in the relations between the
bare and physical parameters and in the unobservable wave function renor-
malization of the fields**, and that observable quantities are finite to all orders
in perturbation theory when expressed in terms of e and m.
The one and higher-loop vertex corrections imply that the lowest order
electron-photon vertex zeo7M is replaced by a function ieoT^(p2,Pi)/Zi that
can depend on the external momenta. Z\ is the (divergent) vertex renormal-
ization constant, defined by the requirement that TM = 7M at the on-shell
point v\ — v{ — ™>2 and q2 = (p2 — Pi)2 = 0. Similarly, the electron self-
energy and vacuum polarization diagrams modify the electron and photon
propagators so that on shell they take the same form as the free ones except
they are multiplied by divergent wave function renormalization factors £2,3,
Sf^ "> u Z\ ■ » i9^DF(q) - -igv-^, (2.344)
p — m + ze qz +ie
where the position of the pole in Sp{p) defines the physical electron mass.
It is convenient to associate a factor of Z2 or Zj from each electron or
photon line with the vertex. The other Z1/2 is associated with the vertex or
external state at the other end of the line. The overall vertex factor therefore
becomes !
Z Zl/2
ieoY - ieo-V-r^Pi). (2.345)
The renormalization factors ^1,2,3 can be combined with eo to define the
physical charge e. The Ward-Takahashi identity ensures that Z\ = Z2 to all
orders, so we can identify
e-co^fa^coZ,1'2. (2.346)
It can then be shown that rM(j)2,J>i) is finite to all orders when expressed
in terms of e and m, and in particular, Y^{p2,P\) = 7M + 0(a) where
a = e2/47r ~ 1/137 is the fine structure constant The charge renormalization
depends only on the photon vacuum polarization diagrams and is therefore
the same for all particles, e.g., for electrons with charge —eo —» —e, and for
**In addition to the ultraviolet divergences (associated with large momenta in the integrals)
discussed here, there are also infrared divergences associated with low momentum virtual
photons. These can be regulated by introducing a fictitious photon mass; they cancel
against similar terms associated with the emission of soft real photons at energies below
the threshold of the detector.

Review of Perturbative Field Theory
77
^-quarks with charge +2e0/3 —» 2e/3. Similarly, after removing the Z2j3
factors, the electron and photon propagators behave like the free-field ones near
the physical p2 = m2 or q2 = 0 poles, but can have momentum dependent
corrections away from the physical masses. These can also be shown to be
finite to all orders when expressed in terms of e and m. The vacuum
polarization corrections will be discussed in Section 2.12.2 in connection with running
couplings.
The relation of such renormalized quantities to physical on-shell amplitudes
and matrix elements is considered in field theory texts, but it should be
plausible from the above discussion that the on-shell matrix element of —Jq(x)
between physical electron states is just
{P2S2\^{x)1^{x)\pl 8l) = u2r^p2,p1)u1ei<1'x, (2.347)
where q = p2 — Pi and the x dependence follows from translation invariance,
Eq. 1.14. The form of T^(p2,pi) is strongly restricted by symmetry
considerations (which continue to hold in the presence of the strong interactions). In
particular, Lorentz invariance implies that the r.h.s of (2.347) must be a four-
vector, which can only be constructed from p^, p%, and the Dirac matrices.
The Gordon identities in Problem 2.4 indicate that it is sufficient to consider
pf2 in the combination q^ only. Furthermore, QED (and the strong
interactions) are reflection invariant. Therefore, using (2.271) the most general
allowed form is
u2TV-(p2,pi)ui =u2
V^ifa2) + -^q„F2(q2) + q»F3(q2)
ui, (2.348)
where the form factors F\j2js(q2) are Lorentz invariant functions of q2. Charge
conservation, d^Jn = 0, which can be derived to all orders from the
equations of motion or from the Noether theorem of Section 3.2.2, requires that
q2Fs(q2) = 0. One does not expect a S function to develop to any order, so
Fs(q2) must vanish. Also, the Hermiticity of Jq implies that
r"(P2,Pi) = rM(p1)P2), (2.349)
where T is the Dirac adjoint. Prom Table 2.2, F\^2 must therefore be real.
Finally, we have normalized so that F\(0) = 1.
We have seen that eF\ (0) is just the physical electric charge. To interpret
^2(0), consider the interaction of an electron with a static classical gauge
potential A^(x) (cf (2.64) and Problem 2.17). The matrix element of the
interaction Hamiltonian
H = -e0 f d3x tp(x)^ip(x) A^(x) (2.350)
between one-electron states is
(p2s2\H\pY Sl) = -eu2Y^Ul f A^l(x)e-i^d3x = -eu^m A^q).
(2.351)

78
The Standard Model and Beyond
Expanding the fermion bilinear in (2.351) to linear order in the non-relativistic
limit \pi\ <C ra, this reduces (Problem 2.22) to
4>l2 (-2meA0(q) + e(ft + ft) • A(q) + e [1 + F2(0)] a • 5(g)) </>Sl, (2.352)
where <j>8l 2 are the two-component Pauli spinors and B is the Fourier
transform of the magnetic field B = V x A The conventional non-relativistic
interaction Hamiltonian for ane" in an external field is
Hi = -eA0(x) + ^ (ft I(x) + A{x) • p) - /Zc • J3(£) + 0(e2), (2.353)
where /2e = —g^BS is the electron magnetic dipole moment operator, \xb =
e/2m is the Bohr magneton, S = a/2 is the spin operator, and g is the
electron ^-factor, which is not predicted in the non-relativistic theory. The
three terms correspond to the Coulomb interaction, and the orbital and spin
magnetic moment interactions, respectively. The momentum space matrix
element of Hi (corrected to agree with our covariant state normalization) is
{p2S2\H\plSl) =2mjd3xe-i^<t>l2Hi<t>Sl, (2.354)
with the p operators in Hi replaced by the ft,i eigenvalues. This coincides
with (2.352) provided that one identifies the ^-factor as g = 2 [1 + ^(0)],
where the 2 is the relativistic Dirac contribution (from 7MFi(0)), and F2(0) =
(g — 2)/2 is the anomalous QED term, usually denoted by ae. The leading
contribution, from the vertex diagram in Figure 2.17, is F2(0) = a/2n, as
first calculated by Schwinger. The calculation is sketched in Appendix E as
an illustration of the techniques for calculating Feynman loop integrals. The
relation of field theory matrix elements to non-relativistic potentials is further
discussed in Problem 2.24.
2.12.2 The Running Coupling
We saw in (2.346) that the renormahzation of the electric charge is due to the
vacuum polarization diagram of Figure 2.17, with the divergent parts of the
electron self-energy and vertex diagrams cancelling by the Ward-Takahashi
identity. Now, let us consider the vacuum polarization as a function of q2.
The one-loop vacuum polarization diagram in Figure 2.18, can be added to
to the tree level photon propagator to yield (see, e.g., Peskin and Schroeder,
1995)
^g^iv 2 IQilv 2
qz Q
e,
2
A2
1"i&taS» +*'<•'>
(2.355)
where we have also included a factor of e\ from the outside vertices. A is
an ultraviolet cutoff of the divergent momentum integral. In practice, the

Review of Perturbative Field Theory
79
FIGURE 2.18
One photon exchange and one-loop vacuum polarization "bubble."
cutoff must be introduced in a way that respects gauge invariance, such as
in the Pauli-Villars or dimensional regularization schemes. II(q2) is a finite
function that vanishes at q2 = 0, so we can identify the photon wave function
renormalization factor as
1-
p2
eo
12tt2
1 A2
(2.356)
which diverges for A —» oo. The r.h.s. of (2.355) can be rewritten
1*9 \1V 2
i + -iW)
-W»v
q2 l-e2U(q2)'
(2.357)
where the corrections to the last form are higher order in e2 or e2. One can
show that this form holds and is finite to all orders, provided II(g2) is expressed
in terms of e2 and m2. The e2nU.n term in the expansion (1 — e2I\)~l =
1 + e2II + e4II2 + • • • corresponds to the diagram with n separate vacuum
polarization bubbles along the line.
This illustrates how the divergences disappear from the expressions for
physical observables when they are written in terms of the renormalized quantities.
However, one may still have a nagging doubt about the underlying
divergences. In fact, the modern view is that the "divergent" momentum integrals
are actually cut off physically at the scale at which the theory is replaced by
a more complete one, and that A should be associated with that scale and
not taken to infinity. One can then interpret the renormalizations as finite
quantities describing, e.g., the difference between a coupling as measured at a
scale much smaller than A and the value it would have at A. A logarithmically
"divergent" term in a weak coupling theory such as QED is then actually a
small effect, e.g., ^ In £ ~ 0.08 for eg ~ e2 and A ~ MP ~ 1019 GeV
(the Planck scale). Most of the divergences in renormalizable theories are of
this logarithmic nature, and results are therefore insensitive to the details of

80
The Standard Model and Beyond
the new physics above A^. Non-renormalizable theories, on the other hand,
typically encounter new divergences of order A2n at n-loop level, and are very
sensitive.
The function U(q2) in (2.357) is
£*L'
U{q)=2^ I dzz^~z)^
m2 - q2z{\ - z)
(2.358)
which vanishes as U(q2) —> —q2/607r2m2 for q2 —> 0. Is is well behaved for
q2 < 0 (i.e., ^-channel exchange, as in Figure 2.18), but has a branch point at
q2 = 4ra2 associated with the e+e~ threshold in s-channel processes. In the
limit Q2 = —q2 ^> m2 (positive for ^-channel exchange),
The replacement in (2.357) is universal for all photon exchanges. It is
useful to introduce a running or effective fine structure constant ae// (Q2) ~
e2(Q2)/47r = a/(l — 47raII(—Q2)). Such considerations are most useful for
large Q2, so we will actually define the running quantity as
where Q2 is an arbitrary reference scale, such as m2. This coincides with
aeff for Q2 ^> Q2 ~ m2, and approaches the bare coupling ao = e2/47r for
Q2 —> A2 (up to higher-order corrections). (2.360) is equivalent to
1 i O2
6;ln^2, (2.361)
a(Q2) a(Q2) Q2
where b' = 4nb = 1/3tt. Thus, a(Q2)~l runs linearly with InQ2 at large Q2.
(Higher-order corrections to the vacuum polarization bubble lead to small
nonlinear effects). The running a(Q2) therefore increases logarithmically from its
value ~ 1/137 for small Q2, so the QED interaction strength should be larger
for high energy processes. This was motivated here for spacelike momentum
transfers q2 < 0, but continues to hold even in the timelike region, where
higher-order corrections are minimized if one uses a(|g2|). Equivalently, the
effective Coulomb interaction scales as a(Q ~ l/r)/r, and therefore the
effective a increases at smaller separations. There is a simple interpretation
of this: an electron charge in a dielectric medium is screened for larger
separations, leading to an interaction strength falling faster than 1/r. At small
t^The quadratically divergent (but renormalizable) corrections to scalar self-energies, such
as the Higgs mass-square in the standard model, are a critical exception. This will be
discussed in Chapter 8.

Review of Perturbative Field Theory
81
separations, however, the screening is less effective and a test charge feels the
full electric charge. The same mechanism applies here, except the dielectric is
really due to the quantum fluctuations of the vacuum, as represented by the
virtual e+e~ loop. We will see in Chapter 5 that the gluon self-interactions
in quantum chromodynamics have the opposite effect of antiscreening (for
which there is no simple classical analog), leading to a decrease in the strong
coupling at large momenta or short distance (asymptotic freedom).
The running of a(Q2) is described by the renormalization group equation
(RGE)
^^=/3(92) = ^a2(Q2) + 0(a3)> (2.362)
where b' = 1/3tt is due to the one-loop diagram in Figure 2.18, and the
other terms are higher-loop contributions to the vacuum polarization bubbles.
(2.361) is the solution in one-loop approximation. The running in (2.361) is
valid for Q2 » ra2; there is little effect for Q2 < ra2. However, for Q2 > ra2,
where raM ~ 200 rae is the muon mass and rae = ra is the e~ mass, one
should also include the effect of the muon loop in the photon propagator, i.e.,
b' -> 2/3tt, and
1 1 2 O2
-77^ = ~r^\ - t" ln ^2 (2-363)
a(Q2) a (raj) 3tt m2
for Q2 > ra2. Thus, b' changes discontinuously and \/a has a kink near the
particle threshold. (The exact form of the threshold, including constant terms,
whether it occurs at raM or 2raM, etc, depends on the details of the
renormalization scheme.) Quark loops also contribute to the vacuum polarization and
the running. If one could ignore the strong interactions, then a quark of flavor
r (e.g., u, d, s) would contribute a term 3q2/3n to b' for Q2 > ra2, where qr
is the quark electric charge in units of e, rar is its mass, and the 3 is because
there are 3 quark colors. Thus
b' = ^ £ Crq2r, (2.364)
mr<Q
where the sum includes both quarks and charged leptons, with Cr = 1 (lep-
tons) and Cr = 3 (quarks). Unfortunately, this is a poor approximation for
the strongly interacting particles, which cannot really be treated as free quarks
at low energies. Multiple gluon exchanges between the quarks in the vacuum
polarization diagram, or hadronic bound state effects, invalidate the quark
part of (2.364). A more reliable approximation can be obtained by replacing
the perturbative loop by a dispersion relation
3£i 91
3* Q2o
/•OO
/ F(Q2,s)R(s)ds (2.365)
in the 1/a equation. R(s) = &e+e- (s)/(Aira2/3s) is the ratio of the total cross
section for e+e~ —» hadrons at CM energy yfs divided by the cross section

82
The Standard Model and Beyond
for e+e~ —» ^+//~, which can be taken from experiment, and F(Q2,s) is
a known function (see, e.g., Eidelman and Jegerlehner, 1995). One therefore
finds that 1/a decreases from ~ 137 for Q ~ 0 to ~ 129 at the the Z mass. The
extrapolation can be done quite reliably, but the small (0(0.02%)) uncertainty
is still the largest theoretical uncertainty in the precision electroweak program.
Closely related hadronic uncertainties are also significant in the interpretation
of the measured anomalous magnetic moment of the muon.
The running a(Q2) effect was sketched here in an on-shell renormalization
scheme, i.e., e was defined in terms of the electron-photon vertex on shell, and
m as the location of the pole in the propagator. In the minimal subtraction
schemes ('t Hooft, 1973; Bardeen et al., 1978) one defines renormalized
couplings and masses at an arbitrary renormalization scale //, at which the poles
in dimensional regularization are subtracted. Both the masses and charges
run as a function of /j,. Higher-order corrections to physical processes
involving a single large scale Q are generally minimized by evaluating the couplings
and masses at fj, = Q.
2.12.3 Tests of QED
Quantum electrodynamics is the most successful theory in physics when judged
in terms of the theoretical and experimental precision of its tests. A detailed
review is given in (Kinoshita, 1990). The classical atomic tests of QED, such
as the Lamb shift, atomic hyperfine splittings, muonium (/x+e~ bound states),
and positronium (e+e~ bound states) are reviewed in (Karshenboim, 2005).
The most precise determinations of a and the other physical constants are
surveyed in (Mohr et al., 2008). The currently most precise measurements of
a are compared in Table 2.3.
TABLE 2.3
Most precise determinations of the fine structure constant a = e2/47r.
Ae is defined as [a~l — a~l(ae)] x 106. Detailed descriptions and
references are given in (Mohr et al., 2008).
Experiment
ae = {ge - 2)/2
h/m (Rb, Cs)
Quantum Hall
h/m (neutron)
7p,3He (J- J.)
H+e~ hyperfine
Value of a *
137.035 999 683 (94)
137.035 999 35 (69)
137.036 003 0 (25)
137.036 007 7 (28)
137.035 987 5 (43)
137.036 001 7 (80)
Precision
[6.9 x 10"iU]
[5.0 x 10"9]
[1.8 x 10"8]
[2.1 x 10"8]
[3.1 x 10"8]
[5.8 x 10"8]
Ae
-
0.33 ±0.69
-3.3 ±2.5
-8.0 ±2.8
12.2 ±4.3
-2.0 ±8.0

Review of Perturbative Field Theory 83
7 7
FIGURE 2.19
One-loop and typical two-loop diagrams contributing to the anomalous
magnetic moment of the electron.
The most precise measurement of a is from ae = 1.159 652 181 11(74) x
10~~3, the anomalous magnetic moment of the electron and positron (which are
the same by CPT), which was measured using eT confined in Penning traps,
at the University of Washington and more recently at Harvard University, a
is extracted using the theoretical prediction
ae = ^-- 0.328 478 444 00 (-) + 1.181 234 017 (-Y
a
2^
1.7283(35) (-) +fle
1.7283(35) (-) +a?w + a*ad,
(2.366)
where the first four terms are the pure-QED contributions, involving photons,
electrons, muons, and taus. A few representative diagrams are shown in Figure
2.19. The one-loop term is calculated in Appendix E. The two- and three-
loop contributions have been calculated fully analytically, while the four-loop
term (which involves well over 800 Feynman diagrams) requires numerical
integration of the Feynman parametric integrals, resulting in the quoted error
in the coefficient. The electroweak and hadronic contributions involving W, Z,
Higgs, and strongly interacting particles are estimated to be ~ 0.02973(52) x
10~12 and ~ 1.682(20) x 10~12, respectively (for reviews of the theory, see,
e.g., Czarnecki and Marciano, 2001; Mohr et al., 2008). In interpreting very
precise measurements, especially in extracting parameters from them, one
must always be concerned that possible effects from (unknown) physics beyond
the standard model might lead to an error. This possibility can never be
totally eliminated. However, in the case of ae the effects of physics at scale
M are usually of 0(me/M)2, which is < 10~12 for M ~ 1 TeV, and therefore
irrelevant at the present precision of~6.6xl0-10.
It can be seen in Table 2.3 that there are a variety of determinations of a at
the few x 10~8 level from atomic transitions, the quantum hall effect, neutron
scattering from a silicon crystal, p and 3He gyromagnetic ratio measurements
involving the Josephson effect, and muonium hyperfine structure. The overall
agreement of these measurements and a variety of others from atomic physics

84
The Standard Model and Beyond
with that from ae is impressive. The current average, strongly dominated by
ac, is a"1 = 137.035 999 679(94).
The agreement, which involves the calculation of ae to high order, validates
not only QED but the entire formalism of gauge invariance and renormaliza-
tion theory. Other basic predictions of gauge invariance (assuming it is not
spontaneously broken, which would lead to electric charge nonconservation -
see Chapter 4), are that the photon mass ra7 and its charge q1 (in units of e)
should vanish. The current upper bounds are extremely impressive (Amsler
et al., 2008)
ra7 < 1 x 10-18 eV , qi < 5 x 10-30, (2.367)
based on astrophysical effects (the survival of the Solar magnetic field and
limits on the dispersion of light from pulsars).
QED has also been tested at high energies, especially at the e+e~ colliders
PEP (at SLAC), PETRA (DESY), and TRISTAN (KEK), which operated
below the Z pole where the s-channel photon diagram dominates (Wu, 1984;
Kiesling, 1988). While not as precise as the low energy tests, these results
all confirmed the QED predictions. QED was also tested indirectly in the Z
pole experiments at LEP (CERN) and SLC (SLAC), and above the Z pole
at LEP 2, where it was an important ingredient for calibrations and entered
interferences and radiative corrections. Finally, the running of a with Q2 has
been confirmed experimentally (for a review, see Mele, 2006).
The Anomalous Magnetic Moment of the Muon
The anomalous magnetic moment of the muon aM has been measured to high
precision in the Brookhaven 821 experiment (Bennett et al., 2006), in which
the precession of the \x magnetic moment relative to its momentum in a
storage ring was monitored using the parity-violating correlation between the
momentum of the decay e*1 and the ^ spin direction (Section 6.2). The
value obtained was a™p = 11 659 208.0(5.4)(3.3) x 10-10, where the two
errors are respectively statistical and systematic. aM is especially important
because it is expected to be more sensitive to new physics than most of the
other probes, with deviations typically of 0(m^/M)2, i.e., 0(m^/me)2 larger
than those for ae. The SM expectation is (for reviews, see Jegerlehner, 2007,
2008; Miller et al., 2007; Mohr et al., 2008) a™ = a%ED + afw + a*ad. The
pure QED part, from diagrams analogous to Figure 2.19, has been calculated
to four loops (three analytically) and the leading contributions to the five-loop
diagrams estimated, leading to (Hocker and Marciano article in Amsler et al.,
2008)
aQED =£- + 0.765857410(27) (-) + 24.05050964(87) (-)
+ 130.8055(80) (-)4 + 663(20) (-)* (2'368)
=116 584 718.10(0.16) x 10-11,

Review of Perturbative Field Theory
85
using a (slightly older) value of a from ae. The electroweak diagrams include
the contributions of the W, Z and Higgs (H) bosons, shown in Figure 2.20.
The first two diagrams dominate, yielding
H H MM
FIGURE 2.20
One-loop electroweak contributions to aM = (g^ — 2)/2.
4W ~ ^| (| + \(l -4sin20Ho) ~ 194.8 x KT11 (2.369)
at one loop, where Gp and sin2 6w are respectively the Fermi constant and
weak angle. Including the two-loop diagrams, which are enhanced by large
logarithms,
a£w = 154(2) x HT11, (2.370)
with little dependence on the Higgs mass. An original goal of the BNL 821
experiment was to achieve sensitivity to a^w. However, there is a
significant contribution and uncertainty from the hadronic contributions a^ad, such
as shown in Figure 2.21. In particular, the hadronic vacuum polarization
diagram yields a large contribution which cannot be reliably calculated in
perturbation theory. It is closely related to the hadronic contributions to
the running of a mentioned in Section 2.12.2. It can be estimated from the
measured cross section for e+e~ —» hadrons using a dispersion relation
similar to (2.365), but with a slightly different weighting function (Eidelman and
Jegerlehner, 1995). The integral is dominated by the low energy region (such
as the p resonance), yielding a!^d ~ 6 894(46) x 10~n. Although this is
quite precise, the uncertainty is comparable to the experimental uncertainty
in aM. The situation is confused by the fact that one can also obtain a^ad from
hadronic r decays, which are related by isospin. Applying isospin breaking
corrections leads to a value 7 103(58) x 10~n, which differs by ~ 2.8a. The
origin of the discrepancy is unclear. There are also small but nonnegligible
three-loop vacuum polarization diagrams and hadronic light by light diagrams

86
The Standard Model and Beyond
7
had II
FIGURE 2.21
Two-loop hadronic vacuum polarization (left) and three-loop hadronic light
by light (right) diagrams. The shaded blobs represent hadronic states.
(Figure 2.21). The latter cannot be directly related to experimental data and
relies on model calculations. The sum of the three-loop diagrams is around
22(35) x 10-11. Using the e+e~ value for a^ad, one finds the standard model
expectation (Amsler et al., 2008)
< = 116 591 788(58) x 10
"n =» AaM = ae*V-alM = 292(86) x 10"11,
(2.371)
a 3.4cr discrepancy. Using the r decay determination of a|jad, on the other
hand, reduces the discrepancy to only 0.9cr.
The situation is therefore uncertain. If one accepts the value of a^ in
(2.371) then there is a strong suggestion of new physics contributions to aM.
For example, in the supersymmetric extension of the standard model the
diagrams in Figure 2.22 would give an additional contribution that would be
FIGURE 2.22
New contributions to aM in the supersymmetric extension of the SM. jl and
v are the superpartners of the \x and v, while x±,Q are superpartners of the
electroweak gauge bosons and Higgs fields.

Review of Perturbative Field Theory
87
significant for relatively low masses for the supersymmetric partners and/or
relatively large tan (3 (the ratio of the expectation values of the two Higgs
doublets in the theory). The central value of the discrepancy in (2.371) would
be accounted for (Czarnecki and Marciano, 2001) if
msusY ~ 67>/tan/3 GeV, (2.372)
where msusY is the typical mass of the new particles in Figure 2.22.
2.12.4 The Role of the Strong Interactions
Many QED processes involve strongly interacting particles. Strong interaction
effects complicate the calculations, but in simple enough cases one can still
say a great deal using symmetry principles.
The Pion Electromagnetic Form Factor
Let us start by revisiting the e~7r+ —» e~7r+ scattering amplitude in
Figure 2.15 and Eq. 2.218. From (2.129) and (2.214) the relevant Hamiltonian
interaction density is
Hi = -£/ = eJ£A„ - e2A2(j^(j)
<—► (2 373)
J£ =-^7^ + ^*3^
where ip and <f> are respectively the e~ and 7r+ fields. The amplitude is
schematically
Mfi ~ (e~n+\T[-i fnI(x)d4x,-i jHi{x')(£x'] |e~7r+)
~{-ie)2(e-(k2,s2)\J%\e-{k1,s1))m
-e2u{k2,s2)^u{kus1) (Z^) (^+(ft)|J5k+(ft)), (2-374)
where q = p2 — Pi and the details of the Fourier transforms and momentum-
conserving S functions are not displayed. In the absence of strong interactions,
,<—> ,
one can replace Jq in the last expression by z</>T d^cj) where <j> and cjy are free
fields, yielding
{■«+{P2)\rQ{x)\*+{Pi)) = (P2 +PiYei"-x. (2.375)
In the presence of the strong interactions Jq is still defined by the expression
in (2.373). However, the fields are no longer free, but rather are Heisenberg or
interaction picture fields w.r.t. the strong interactions. For example, consider
the interaction terms

88 The Standard Model and Beyond
where ipp,ipn, and </>n+ are respectively the proton, neutron, and 7r+ fields.
CnN corresponds to the interaction vertices shown in Figure 2.23, and
represents a subset of those in the isospin conserving model described in Section
3.2.3. (The tt~ field is (f)n- = </>* + , while the 7r° and such states as the p
resonance can easily be added.) An alternative and more fundamental de-
FIGURE 2.23
Top: interaction vertices described by (2.376). Middle: representative strong
interaction diagrams contributing to (7r+| Jq|7t+) involving nucleons, pions,
and the p° resonance. Bottom left: diagrams in the quark model description
in which the 7r+ is a ud bound state. Bottom right: parametrization of the
strong interaction effects by a form factor, represented by a shaded circle.
scription is to regard the 7r+ as a bound state of aw (charge +2/3) and d
(charge +1/3) quark and antiquark. In practice, one cannot treat the strong
interactions perturbatively using either description, because the couplings gn
and A, or the vertex relating the bound state 7r+ to the quarks, are too large.
However, one can parametrize our uncertainty by writing matrix elements such
as (7r+| JqI71""1") in a wav which reflects the symmetries of the theory and which
involves one or more form factors, analogous to those introduced for higher-
order QED effects in Section 2.12.1, for the residual unknowns. These form
factors can be measured experimentally, computed using non-perturbative
lattice techniques, or estimated in other theoretical models.
Consider (7r+(p2)|^Q(#)|7r+(Pi))- By an argument similar to the one in

Review of Perturbative Field Theory
89
geCtion 2.12.1, the most general form is
(7r+(p2)|J5(x)|7r+(p1)) = [(pa +Pirf?(q2) + (pa - ptffStf)
(2.377)
The x dependence follows from translation invariance, while the only available
Lorentz four-vectors are (p2 ±P\Y- The form factors f® can depend on q2,
which is the only Lorentz invariant available since the pions are assumed to
be on shell (they could also depend on p\2 if we extended the discussion to
off-shell pions). The electric charge, in units of e, is /+(0). In the absence
of strong interactions we would have /+(0) = 1 and fS(q2) = 0. Since the
strong interactions conserve electric charge, we expect that /+(0) = 1 to be
maintained, i.e., that the electric charge is not renormalized. If this were not
the case, the 7r+ or proton electric charges would differ from those of the e+
due to the strong interactions. The nonrenormalization will be demonstrated
in Section 6.2.4. Furthermore, current conservation**, O^Jq = 0, requires
fS(q2) = 0, analogous to F3 = 0 in (2.348).
/+ (q2) is known as the electromagnetic form factor of the pion. Although
/+ (0) = 1, the strong interactions (and higher-order QED effects) are
expected to induce a nontrivial q2 dependence. For small q2 one can expand
f2(q2)~l + ±Rlq\ (2.378)
where Rn is known as the pion charge radius. The term is motivated by non-
relativistic quantum mechanics. Suppose the 7r+ were a bound state of a non-
relativistic constituent with charge -he bound in a potential with wave function
ip(x) (this roughly but not exactly mimics the effects of ud constituents).
Then the non-relativist ic analog of f+(q2), which plays a similar role in pion
scattering from a static field, is
F(q)= f tfie^^ix)?. (2.379)
It is obvious in the non-relativist ic case that ^(0) = 1 by the normalization of
the wave function, i.e., charge is not renormalized by the bound state effects.
For small \q\2, and assuming a spherically symmetric wave function (which is
reasonable since the pion spin is 0), it is straightforward to show that
F(q) ~ 1 - ±Rl\q\2 with B* = Jd3x\x\2\^(x)\\ (2.380)
so Rn is the RMS charge radius. The non-relativist ic approximation is not
really valid, but the motivation for the term remains.
^As mentioned in Section 2.6, the conserved current is actually i<f>^ d^cp — 2e<ft A^cp. This
is irrelevant here as we are working to lowest order in e.

90 The Standard Model and Beyond
/+ (q2) can be measured in the spacelike region q2 < 0 by scattering of
pions from an atomic target, or in the reaction e~p —» e~7r+n, in which the
e~ scatters from a virtual 7r+ emitted by the proton, working in a kinematic
region where the exchanged pion is as close as possible to being on shell, f®
can be approximated for small \q2\ by
with y/R2 = 0.672 ± 0.008 fm (Amsler et al., 2008). It can be measured in
the timelike region for q2 > 4m% by e+e~ —> 7r+7r~. For large q2 it falls off
rapidly as 1/q2, while for lower q2 it is dominated by resonances, especially
the p°, which is a spin-1 resonance with mp ~ 770 MeV and a width Tp ~ 150
MeV. Near the p the form factor assumes a classic Breit-Wigner resonance
form (Appendix F), which to first approximation is
^4
fQ(J>
2 rri
f+(<f)\ 1 • (2-382)
+ ' (Q2-rn2)2 + m2T2 }
Corrections to this formula are given in (Gounaris and Sakurai, 1968) and a
plot in (e.g., Quenzer et al., 1978).
The Proton and Neutron Form Factors
Now consider elastic scattering process e~p —» e~p. Similar to (2.373) and
(2.374), the electromagnetic interaction is
CI = -eJ£A^l with J£ = -ipe^ipe + ipp^ipp, (2.383)
where ipe and ipp represent the electron and proton fields, respectively, and
the lowest order amplitude can be written
Mfi = ieuihh^ik!) {^A {-ie)(p{p2)\rQWi)), (2.384)
where q = P2 — Pi and the spin labels are suppressed. Let us first treat the
proton as a pointlike elementary particle. Then
<P(P2)\JqWi)) = fi(ft)7v«(ft). (2-385)
The cross section in the lab frame, i.e., the proton rest frame, is
(2.386)
where we have neglected the electron mass, the proton mass is rap, 6l is
the angle of the scattered electron, dQ,L = dtpd cos 8l —» 2nd cos 8l, the final
electron energy is given by (2.45),
~~ ~ STi _/) x = i , 2fc, - 2^' (2.387)
da
dth,
a2 k2
~ 4k2 sin4 ^h
2 0l
cos
2
q2 . 20l]
"^|sin tJ
fci 1 + ^:(1-C08(?L) l + i^sin2
2

Review of Perturbative Field Theory
91
and?2 = -4/^2 sin2 ^.
When we turn on the strong interactions, we must include the full matrix
element of Jq in (2.384). This is denoted by the shaded circle in Figure
o 24 which also contains some representative strong interaction corrections
to the vertex, both in the approximation of treating the virtual states as
physical hadrons, or in the quark model, for which the proton is a uud bound
state. Just as for the pion, these effects cannot be summed perturbatively.
They must be expressed in terms of form factors that parametrize the matrix
element in a way which takes into account the symmetries of the theory, and
which can be determined by experiment, by model calculations, or by lattice
or other techniques. After making use of the Gordon identities in Problem
e-(k2)
P(P2)
w
e"(fci) P(Pi)
V eT
V e"
V e"
FIGURE 2.24
Top: tree-level amplitude for e~p —» e~p for a point proton and with
proton form factors (shaded circle). Bottom: representative strong interaction
corrections in terms of virtual hadrons (3 left diagrams) or in terms of quark
constituents (right).
2.4 the most general Lorentz and translation invariant matrix element is
<p(fc)| J%(x)\Pm) = u(p2)^Q(q)u(p1)ei^, (2.388)
where
W =
ia
[IV
7M W) + -—qvF%{q2) + q^F^q2)
2m^
+ 7"7V(<?2) + ^^2V) + ^75S3V)
(2.389)
where the form factors Ff and gf depend on q2. We have so far not imposed
additional symmetries such as reflection invariance, so the discussion would be

92
The Standard Model and Beyond
valid even when weak and higher-order electromagnetic effects are included.
Similar to the discussion in Section 2.12.1, the proton electric charge is eFf (0)7
so the nonrenormalization of charge (Section 6.2.4) implies Ff (0) = 1. kv ^
/^(O) is the anomalous magnetic moment of the proton, i.e.,
MP = +9p9nSp, gp = 2(1 + «p), (2.390)
where /j,n = e/2mp is the nuclear magneton. The experimental value kv ~
1.79 is very much larger than the electromagnetic one-loop contribution of
a/2ir, indicating the importance and nonperturbative nature of the strong
interaction corrections. The value, and also the ratio of the proton and neutron
magnetic moments, can be approximately understood in the non-relativistic
quark model (see, e.g., Georgi, 1999). Electromagnetic current conservation
further implies q^u{p2)T^{q)u{pi) = 0, and therefore
F3V) = 0, 2mpg\{q2) + q2gp3(q2) = 0. (2.391)
Prom the discussion in Section 2.10 and Appendix G, gf(q2) ^ 0 requires
the violation of space reflection invariance since Jq transforms as a vector.
^3 7^0 would require the violation of charge conjugation invariance as well.
Since the weak interactions violate P and C they are expected to generate a
nonzero g±3 subject to (2.391), known as an anapole moment, at some level,
e.g., due to W or Z exchange across the photon vertex or a parity-violating
scalar correction to the to pion-nucleon coupling. The anapole moment has
been observed in parity violating transitions in the l33Cs atom (Haxton and
Wieman, 2001; Haxton et al., 2002) at the expected order of magnitude,
though the calculations are difficult and model comparisons with other forms
of hadronic parity violation are not in perfect agreement. A nonzero value
for g% 7^ 0 would require both space reflection and time reversal violation,
and would correspond to an electric dipole moment (EDM) for the proton
(see Problem 2.22). P is violated by the weak interactions, but T violation is
much weaker, especially for the light fermions, so EDMs are predicted to be
extremely small in the standard model, as discussed in Section 7.6. However,
many types of physics beyond the standard model predict EDMs much larger
than in the SM, so experimental searches for the neutron, electron, atomic,
or other EDMs are extremely important.
Let us now restrict our attention to strong interaction corrections to Tq, so
that
r&(«) =
(2.392)
with Ff (0) = 1 and F^O) = kp, analogous to the higher-order QED
corrections to the e~ vertex in (2.348). For q2 < 0 it is conventional to define

Review of Perturbative Field Theory
93
q2 = —q2 > 0, and to define the Sachs, or electric and magnetic*, form factors
GE(q2) = F?(q2)-TF*(q2)
GM(q2) = F?(q2) + F2(q2),
(2.393)
where r = Q2/4ra2 = -g2/4ra2, so that GE(0) = 1 and GM(0) = l + «p. One
can then generalize (2.386) to the Rosenbluth cross section formula for elastic
e-p __> e~p scattering
^ k2 ,^
1 + t 2
dcr
d^L 4fc?sin4^*i
+ 2rGMsin —
(2.394)
The neutron also has electromagnetic properties due to strong interaction
and bound state effects, just as a neutral atom has electromagnetic moments
due to its internal charge distribution. This is most obvious in the quark
picture, where it is a udd bound state, but can also be seen in the virtual
hadron language, e.g., induced by the diagrams analogous to Figure 2.24 in
which the neutron turns into a virtual pn~. The neutron electromagnetic
matrix element is
<n(p2)|JrQ(a?)|n(pi)> = M(p2)
10
[IV
7^W) + —^FJV)
ti(pi), (2.395)
where the neutron total charge is F™ (0) = 0 and F^O) = Kn ~ —1.91 is the
neutron anomalous magnetic moment (gn = 2nn). It is sometimes useful to
define the isovector and isoscalar form factors as
K2 = \ (F?t2 - F?t2), Fft2 = \ (F?i2 + F»2),
(2.396)
and similarly for G^ »
EM'
One can extract the form factors GPE M(Q2) or Ff2(Q2) by measuring the
cross section for e~p —» e~p while varying k\ and 8l for fixed Q2, and also
by scattering from polarized protons (for reviews, see Manohar, 1992; Hyde-
Wright and de Jager, 2004; Arrington et al., 2007; Alberico et al., 2009). They
are described by an approximate dipole form
GpE(Q2)
GpM(Q2)
1 + ««
[i + Q7Q§]2'
(2.397)
where Q2 - 0.71 GeV2 ~ 18.2 fm-2, which holds to -10% for Q2 < 10 GeV2.
The neutron form factors are obtained from scattering from nuclear
targets, such as deuterium (D) or polarized 3He, and are approximated by
*The terms are motivated by the form of the matrix element in the Breit frame (Problem
2.23), which generalizes the discussion of the non-relativistic limit of the e~ vertex at the
end of Section 2.12.1.

94
The Standard Model and Beyond
GnM(Q2)/Kn ~ GPE(Q2) for Q2 < few GeV2, with \GnE(Q2)\ much smaller
(a few %).
The approximate l/Q2 and l/Q4 behaviors of the pion and proton form
factors for large l/Q2 is consistent with the expectation (see, e.g., Lepage and
Brodsky, 1980) of (Q2)~^n~^ for a bound state of n constituent quarks based
on dimension counting rules derivable from QCD. The more rapid falloff for
larger n may be thought of as due to the greater difficulty for the system to
hold together in the scattering.
2.13 Mass and Kinetic Mixing
In general, the Lagrangian density for a theory with more than one field of the
same type (i.e., Hermitian or complex scalar, massive vector, or fermion) may
include non-canonical kinetic energy terms and off-diagonal mass terms. For
example, the Lagrangian density in (2.19) can be generalized for n Hermitian
fields 4>a, cl = 1 • • • n to
C{4>) = X- [dM0TO^ - 0Tm20] - V7(0), (2.398)
where 4> = (0102 • • • <t>n)T is an n-component column vector, /C is a real
symmetric n x n matrix with positive eigenvalues, and rh2 is a real
symmetric matrix*. A non-canonical kinetic energy term (i.e., /C ^ 1), which may
arise if the theory descends from a more fundamental underlying theory, must
be put into canonical form prior to quantization to maintain the classical
relation E2 = p2 + m2. This can be accomplished by defining new fields
</>° = S1/20]c(t), where Ok. is the orthogonal matrix that diagonalizes /C and
S is the (non-orthogonal) diagonal matrix consisting of the eigenvalues of /C.
Then
d^TKd^ = d^s-^olJCOzS-^d^0 = d^0Td^°. (2.399)
The mass and interaction terms must then be rewritten in terms of these
rotated and rescaled fields,
£(4>°) - \ [d»<t>0Td»<t>° - 4>0Tm*<t>0} - V,V), (2.400)
where m2 = S^^Ol^OicS-1^ and V?{4>°) = V^OkS-1'2^). We will
see examples of the effects of this kinetic mixing in Chapter 8.
tThe eigenvalues must be non-negative to avoid spontaneous symmetry breaking.

Review of Perturbative Field Theory
95
The standard way to interpret (2.400) is to perform a second orthogonal
transformation (or a first one if /C was equal to / initially) to diagonalize
m2^ j#e#j (j) = 0^</>0, where 0^lm207n = m2D and ra^ is the diagonal matrix
diag(mi m2'"mn)- Then
£(</>) = ^ [(^0a)2 - mltl] - F7(</>), (2.401)
a=l
where </>a and ma are respectively the ath mass eigenstate field and mass
eigenvalue, and Vi{(j>) = V}°(Om0). The free-field propagator in (2.28) generalizes
trivially in this mass eigenstate basis to
in an obvious notation, so that one can calculate scattering and decay
processes in a straightforward way, with the index a (possibly) changing at the
vertices but not along internal lines.
The same considerations apply to massive vectors fields. For complex
scalars the only difference is that the analogs of /C and m2 must be Her-
mitian, and those of Ok, and Om must be unitary (with T replaced by \).
Such unitary transformations also apply to Dirac fields in the special case of
Hermitian kinetic and mass matrices. Extensions involving 75's (i.e., non-
Hermitian matrices) and Majorana fermions are discussed in Problem 3.28
and in Chapter 7.
In some cases, however, it is easier to treat the small elements (usually
the off-diagonal ones) of m2 by mass insertions, in which they are ignored
in the calculation of the propagators but are instead treated as if they were
interaction terms. We will illustrate this for the case of two Hermitian scalars
<t>\ 2 interacting with a single fermion ip,
C = £^+£ic^o -^?0;2-im^2-m?20?^+^ [Mi + M2] ^, (2.403)
where C^ is the free field density for ip and Cke 0 1S tne canonical kinetic
energy for <j>\2-
The mass eigenstates are
*)-°r(3)- °■(-£.'£!)• (2-404)
where the expressions for 8 and the mass eigenvalues m2 b are elementary and
will not be displayed. The interactions of (j)a^ are given by -0 [ha(j)a + /i&</>&] ip,
where
ha\ _ nr ( hi
hb) \h<
:)=0T T • (2.405)

96
The Standard Model and Beyond
The amplitude for tpiip2 —»ip3ip4 is then given in terms of the mass eigenstates
by
M =i U4U2U3U1
hi
+
t — m* ' t — ml
+ (3 <-> 4).
(2.406)
Now, let us go to the limit in which m\2 is a perturbation, i.e., \m\2\ <C mf,
m!> lmi ~~ mil- Then,
m2a~m\, ml~m\,
m
12
77io — rat
(2.407)
and the bracketed quantity in (2.406) reduces to
hi
+
t — m\ t
-^ + 20hlh2
t — mi t —ru-
ii
(2.408)
Alternatively, one can treat the m\2 term as a perturbation, and work in
terms of the propagators for <j>\ 2 exchange. Then
M = i U4U2U3U1
h\
+
h22
+
277l12^1^2
t — mf t — m\ (t — ml)(t — m2)
+ (3<->4), (2.409)
where the mf2 term is from the mass insertion diagrams in Figure 2.25.
Expressing m\2 in terms of 8 from (2.407) one recovers the result in (2.406) and
(2.408), but (at least in more complicated examples) with considerably less
effort. In addition to the computational simplicity, mass insertions allow one
to keep track of small effects such as symmetry breaking systematically.
V>3
V>4 V>3
V>4 V>3
1 * 2
V>4 V>3
....*....
V>4
lj>l
tp2 lj>l
tp2 V>1
Xj)2 fa
fa
FIGURE 2.25
^-channel diagrams for tpifa —» ip3ip4, treating the mf2 term in (2.403) as a
perturbative mass insertion. There are additional u channel diagrams with
(3 <- 4).

Review of Perturbative Field Theory
97
2.14 Problems
2.1 Prom (A.7), the Hamiltonian for a free complex scalar field is
Show that
H = j (2nf2E Ep ^^a^ + bi^b^
Assume that H is normal ordered.
2.2 Derive (2.63) by Lorentz transforming (2.56). Hint: use the fact that a,
5, and t are invariant.
2.3 Prove directly from the defining relations that
<^75 = -\^"aopa.
2.4 Prove the Gordon decomposition formulas
2m(u2^ui) = u2(p2 +pi)M^i + iu2a^(p2 - Pi)vUi
2ra (^27M75^i) = v>2(p2 — Pi)M75^i + iu2a^(p2 + Pi)i/75^i
0 = u2(p2 -PiTui + iu2a^\p2 +pi)vui
0 = U2(P2 +Pl)M75^l + iU2<Tt"'\P2 -p\)ulbUU
where Ui^ are two Dirac u spinors for a particle of mass m.
2.5 Prove the identity
^YY = i*gvp + 7 V - 7"^ + i^^piaih-
2.6 Show by explicit construction that the Pauli-Dirac and chiral
representations are related by a unitary transformation, i.e., that there exists a unitary
matrix U such that
U1pdu^ = 7ch» UuPD{p, s) = uch(p, s), UvPD(p, s) = -vch(p, s).
(The extra sign in the v-spinor transformation is due to a sign convention.)

98 The Standard Model and Beyond
2.7 The angular momentum operator for a Dirac field can be written as
f = f d3x^(x)^-eijkajk>ip(x) + orbital,
where normal ordering is implied. Show, in the free field limit, that J3 has
the expected behavior
J3\tl>(p,S1,2)) = ±±\l/>&S1,2)), J3\1PC(P,S1,2)) = ±\\1PC(P,S1,2)),
where ip and ipc represent particle and antiparticle states, p is in the z
direction (so that the orbital terms do not enter), and s = S\ or 52 represent spins
in the ±z direction.
2.8 Derive the results in (2.177) for the helicity projections in the massless
limit.
2.9 Weak charged current transitions involve the chiral spinors ul(p, s) and
vl(p,s) defined in (2.199). Show that in the relativistic limit such transitions
mainly involve negative helicity particles or positive helicity antipartides, and
estimate the suppression factor for transitions involving the "wrong" helicity.
2.10 Show in two ways that \u(p2, +)w(pi, —)|2 = 2pi -p2, where u(p, ±) are
the helicity spinors for a massless fermion: (a) directly from the form of the
spinors in the chiral representation, (b) using trace techniques. !
2.11 Prove the Fierz identity in (2.212).
2.12 Suppose an electrically fermion tp of mass m interacts with a Hermitian
scalar </> of mass \x with
Ci = h'tptpcj),
where h is small.
(a) Calculate the spin-averaged differential cross section for ip(pi)ip(p2) —>
ip(P3)ip(p4) in the CM in terms of the invariants 5, t, and u.
(b) Specialize to m = /j, = 0. Show that the scattering is isotropic in that
limit and calculate the total cross section.
2.13 Consider e~(fci)7r+(pi) —» e~ (k2) n+(fo) elastic scattering. Show that
the spin-averaged differential cross section in the pion rest frame is
da 7ra2 cos2 ^
dcos0L ~ 2fc2sin4^[l + 2^sin2^l'
where 6l is the electron scattering angle and we have neglected the electron
mass.

Review of Perturbative Field Theory 99
2.14 Verify the expressions for Bhabha scattering in (2.230) and (2.231).
Rewrite the final result in terms of s and cos 8.
2.15 Calculate the differential cross section for unpolarized M0ller
scattering, e~~e~~ —» e~e~, both in terms of the invariants and 8.
2.16 Calculate the spin-average differential cross section da/dcos8 in the
center of mass for e~(pi)e+ (^2) —> 7r~(P3)7r+(P4)> and the total cross section
a. Neglect the electron mass but not the pion mass. Ignore strong interaction
effects. The angular distribution should be proportional to sin2 8. Interpret
this result.
2.17 (a) Consider the Mott scattering process in which an electron of
momentum p = (3E scatters from a static Coulomb potential of charge Ze,
>H*) = 4^(1.0,0,0).
Show that the unpolarized (Mott) cross section for scattering angle 8 is
da _ (Za)27r(l-/?2sin2f) Z2a2<K
dcos8~ 2/3 V sin4 § ~J^ 2/3 V sin4 § '
The last formula is the the Rutherford cross section.
(b) Suppose the Coulomb potential for an electron in a nuclear field
transformed as a scalar rather than as the time component of a four-vector, i.e.,
Hi = -e^){x)^{x)(j){x), <j>{x) =
47r|£|
Calculate the unpolarized differential cross section, and compare it with the
Mott formula.
2.18 The interaction of the Z (a massive neutral vector boson in the elec-
troweak theory) with a fermion / is
£ = -Gj>(x)<y» (gv - gAl5) ^{x) Z^x),
where G,gv, and g^ are real constants. Calculate the width for Z —» //. Let
Mz and m be the Z and / masses, and set G = 1.
2.19 The A is a heavy spin-| hyperon which decays into p-K~ via the non-
leptonic weak interactions. The decay interaction can be modeled by
£/ = $P(9s ~ #p75)^a</>7t+ + h.c,
where gs and gp are complex constants.
(a) Calculate the width T and the differential width dT/d cos 8 in the A rest

100
The Standard Model and Beyond
frame for a polarized A, where 6 is the angle between 5a and the proton
momentum pp. Use trace techniques.
(b) Show that dT/dcos6 is not reflection invariant for ^ie(gpgg) ^ 0, i.e.,
that it is not invariant under pp —» —pp,SA —> 5a.
(c) Repeat (a), but use explicit expressions for the A and p spinors in the
Pauli-Dirac representation.
2.20 A vector resonance V^ of mass My and width IV couples to massless
fermions a and b with the interaction in (F.12) of Appendix F. Calculate the
total spin-averaged cross section for ad —» bb. Assume that the propagator in
(2.125) is modified to the Breit-Wigner form
iD^(k) = i
4 i ~m~
k2-M* + iMyTv
and express the result in a form similar to (F.ll).
2.21 Consider the interaction
Cl=9 {$aLlpbR<t> + ^bRlpaL^)
between distinct fermions tpa and ^, where </> is a complex scalar and g is
real. Show that the Lagrangian violates P and C, but is CP invariant.
2.22 Consider the non-relativistic limit of the matrix element (2.351) for an
e" ina static external field.
(a) Compute the limit to linear order in the momenta. Hint: use the explicit
forms for the spinors in the Pauli-Dirac representation. It simplifies the
calculation to rewrite U2Ttiu\ using the Gordon decomposition.
(b) Suppose that T^(p2,Pi) in (2.347) contained a term i-^qvlhG<1(q2). This
violates P and T but in principle could be generated by a new interaction.
Show how G2(0) is related to the electric dipole moment de of the electron,
which is defined by the non-relativist ic interaction Hedm = —de-E(x), where
E is an external electric field. Note that the Hermiticity condition (2.349)
requires that Gi is pure imaginary.
2.23 Consider the proton matrix element u(p2)Tq (q)u(pi) of the
electromagnetic current in (2.388), with Tq given by (2.392). Calculate this explicitly in
the Breit frame, in which q° = 0, i.e.,
q = (0,0,0, y^), Pi = (£,0,0,->/gV2), p2 = (E,0,0,+\/Q*/2).
Express the time and space components in terms of the electric and magnetic
form factors defined in (2.393) and interpret the results.

Review of Perturbative Field Theory
101
2.24 Let V(xi — X2) be the potential between two non-identical spin-^
particles in non-relativistic quantum mechanics (NRQM). (V may also depend
on their spin and momentum operators). One shows in time-dependent
perturbation theory that the transition amplitude Ufi from \i) = |piSi,p2S2) to
|/) = |p3*3jP4*4) w^h mi = m3,m2 = m4 is
Ufi = -i(27r)4S4 (p3 + p4 - Pi - P2) 44 (/ dPfe-^Vif)) 0x02,
where <\>% is a two-component Pauli spinor, and V contains appropriate spin
matrices. Note that these states are in our covariant normalization convention,
which has an extra factor y/27r)32Ei ~ y/27r)32rrii for the ith external particle
compared to the usual conventions of NRQM. The corresponding formula in
field theory is
Ufi = (2tt)454 (p3 + p4 - Pi - P2) M,
where M is the scattering amplitude with the phase convention of Appendix
B. Comparing these results, we can read off the equivalent non-relativistic
potential corresponding to a given scattering amplitude. Specifically, for
- Q - - - . Q
Pi = -P2 = P - « > ^3 = -P4 = P + « '
the non-relativistic limit p —> 0, |<f |2 <C m? yields
M -> -z(2m1)(2m2)Vr(?) = -z(2m1)(2m2) /^"^(r),
where we have not displayed the spinors. Non-leading terms in p can be
interpreted as the non-relativistic momentum operator.
(a) Calculate the potential corresponding to the effective four-fermi Hamilto-
nian density Hi = X'ipiipi tyityi-
(b) Consider the interaction
£/ = [gi^iipi +92M2} 4>
between two fermions and a Hermitian scalar of mass m^. Calculate the
potential between tpi and ip2 generated by ^-channel <j> exchange, and show
that it is attractive for g\g2 > 0.
(c) Calculate the potential generated by
Ci = [91^17^1 +92$27*ip2] V^
where V^ is a spin-1 particle of mass My> Show that it is repulsive for gig2 >
0.
(d) Repeat parts (b) and (c) for the case gig2 > 0, but for the potential
between antiparticle 1 and particle 2 and interpret the results. Hint: it is
slightly easier to use the charge conjugation formalism of Section 2.10.

102
The Standard Model and Beyond
(e) Consider the interaction in (2.264) of a 7r° with protons and neutrons.
Calculate the tree-level amplitude for p(pi)n(p2) —» p(P3)n(p4) by ^-channel
7T° exchange, and show that it leads to the non-relativistic potential
"1^ _ e~x 5/1 3 3 \ J
?*""-T +3U + ^ + ^Je J'
where mp ~ mn, ap(an) are the Pauli matrices acting on the p(n) spin,
x = mnr, and S is the tensor operator
S = 3ap • ran • f — 5P • <rn.
2.25 Suppose that a new lepton-flavor violating interaction leads to the
effective interaction
Ceff = -i^F^ea^ [A + B75] p + h.c,
where A and B are respectively magnetic and electric dipole transition
moments. Calculate the decay rate for \x —» cy.
y ' 167rm2

3
Lie Groups, Lie Algebras, and Symmetries
Lie groups and algebras are used to describe continuous global and gauge
symmetries in classical and quantum mechanics and in field theory. A familiar
example is the description of rotational invariance in quantum mechanics. In
particle physics Lie groups are useful not only for space-time symmetries such
as translations, rotations, and Lorentz transformations, but also for internal
symmetries such as isospin. It is assumed that the reader is familiar with such
basic notions as irreducible representations (IRREPs), direct products, and
irreducible tensor operators. Excellent introductions more detailed than the
treatment here include (Slansky, 1981; Georgi, 1999; Gilmore, 2005; Yndurain,
2007).
3.1 Basic Concepts
3.1.1 Groups and Representations
A group G is a set of elements #i,#2 • • • (which may be finite in number,
countably infinite, or continuous) which has
• an associative multiplication law, under which g\g2 = #3 for each gi^ G
G, with gs G G (closure) and (^1^2)^3 = 91(9293) (associative).
• an identity element I G G with Ig = gl = g for all g G G.
• a unique inverse element g_1 for each g G G, such that gg_1 = g_1g = I.
An abelian (commutative) group is a special case, defined by g±g2 = 9291 for
all #1,2 € G. Otherwise, G is non-abelian. A subset of the elements which
itself forms a group under the same multiplication law is a subgroup of G.
(a) An example of an abelian group with a countable number of elements
is the set of integers n with the operation of ordinary addition. The identity
element is 0 (i.e., 0 + n = n + 0 = n) and the inverse is —n. (b) The
set of rational numbers r other than 0 under ordinary multiplication forms
another countable abelian group. The identity element is 1 and the inverse
is 1/r. (c) The cyclic group Zn consists of the nth roots of unity, i.e., G =
{l,u;n,u;£---a;™-1} where uon = e2nl^n. Zn is abelian and finite, (d) The
103

104
The Standard Model and Beyond
group of 2 x 2 matrices of the form C±{I,a1}, where / is the 2x2 identity;
a1, i = 1,2,3, are the Pauli matrices; and C4 = 1, — l,i, or — i (i.e., C4 G Z4)
is finite and non-abelian. Multiplication is defined by the ordinary matrix
product in (1.17), so, e.g., (<t1)-1 = crl. The commutator is [a1, a-7] = 2ieijkcrk.
The subset {/, a3} forms an abelian subgroup, (e) The set of non-singular
mxm matrices A form a non-abelian continuous group under ordinary matrix
multiplication. The identity is the mxm identity matrix and the inverse is
the matrix inverse A~l.
A Lie group G is a continuous group for which the multiplication law
involves differentiable functions of the parameters which label the group
elements. Most of the Lie groups of interest in particle physics are compact,
which means that the parameters form a compact manifold (the Lorentz
group, described in Section 8.2.2, is a notable exception). A Lie group and its
multiplication law can be defined, at least for elements close to the identity
{infinitesimal transformations), in terms of its associated Lie algebra, which
consists of N generators (operators) Tl,i = l,2-N, and their commutation
rules
[Ti,T*] = icijkTk, (3.1)
where a summation on k is implied and the c^ = —Cjik are the structure
constants of G. Without loss of generality, one can choose the Tl to be
Hermitian, in which case the structure constants are real. If all of the c^fc = 0
then G is abelian; otherwise it is non-abelian. An element of a compact G
can be represented in terms of the generators by the unitary operators
UG0) = exp[-i5>r] ^ e-** ee £ ( *.' > , (3!2)
2=1 fc=0
where (31 • • • (3N are N continuous real parameters and the T% are Hermitian.
In particular, the identity element is Ug(0) = /, and the inverse of Ug(0) is
UG0 r1 = UG(-0) = e*f = UG0)*. (3.3)
For small \(3 \ it is sufficient to keep just the linear term in (3.2),
UG0) * I-0-f+ (>(&&), (3.4)
i.e., the generators of the Lie algebra describe the group elements close to
the identity. The Lie algebra also defines the group multiplication law for
arbitrary (3. That is,
Ug(5)Ug0) = e-i3fe-^f = ^(7). (3.5)
7(a, (3) can in principle be expressed in terms of the Lie algebra (the Baker-
Campbell-Hausdorff construction), although there is no closed form
expression. However, for small \a\ and \(3 \
7(a,/3)-f =(a + /3)-f-^[a-f,/3-f]+h.o.t. (3.6)

Lie Groups, Lie Algebras, and Symmetries
105
Now consider a set of n x n dimensional matrices L2, z = 1,2 • • • N. If the
£,* satisfy the same algebra as the generators of a Lie algebra,
[L\U]=icijkL\ (3.7)
then the IS (sometimes written LSn) are said to form a representation of the
algebra, and are Hermitian for the choice of Hermitian T\ That is, we are
considering the T% to be abstract operators, while the IS are a specific matrix
realization. Similarly, the n x n matrices e~l(3'L form a representation of the
group elements Uq{0 ) and have the same multiplication law. For the compact
groups one can choose Hermitian generators normalized such that
Tr (ISIS) = T(L)8ij. (3.8)
The Dynkin index T(L) > 0 depends on which representation is being
considered and on an overall normalization convention, but is independent of i and
j. With these conventions, one can show that c^ is totally antisymmetric in
all three indices (Problem 3.2).
3.1.2 Examples of Lie Groups
The simplest example of a Lie group is the abelian G = E/(l), with a single
generator T, so that
UG((3) = e-W, UG(a)UG(P) = e-«Q+">T. (3.9)
There is an n = 1 dimensional representation, with T = 1 and group
elements Ug((3) —> e~l@. U(\) is named for this representation, i.e., the lxl
dimensional unitary matrices (phase factors).
G = SU(2) is a non-abelian group with N = 3 generators and Cijk = Cijk-
SU(2) is named for its defining representation, the 2x2 unitary matrices
(U(2)) with the extra constraint that they are special, i.e., their determinant
is unity (SU(2)). The generators of the defining representation are IS = ^,
where r% = o% are the Pauli matrices in (1.16). They are Hermitian, so the
group representation elements
U0)
= e-^-f
= COS-J-
>s. Furthermore, Trr1
e-^f] =
:e-iTr(/3.f)
• • ^/5 -
zsin —p • r
2
= 0, so they
= ei0 = 1.
are
special
(3.10)
5
(3.11)
det
The adjoint representation of SU(2) is the 3x3 representation constructed
from the structure constants, [Lladj) = —ie^. There are additional
representations for n = 4,5---00. SU(2) is useful in nature for describing
rotational invariance, the approximate isospin invariance of the strong
interactions, and the weak isospin gauge symmetry of the electroweak interactions.

106
The Standard Model and Beyond
The group SU(3) plays two major roles in the standard model: as a gauge
symmetry associated with color for the strong interactions (QCD), and as
an approximate global flavor symmetry of the strong interactions (the
eightfold way). SU(3) can be defined in terms of its defining representation, the
3x3 unitary matrices with determinant one. There are N = 8 generators,
with (Hermitian) matrices in the defining relation given by L\ = Al/2, where
the Gell-Mann matrices \l,i = 1 • • -8, are given in Table 3.1. The structure
constants Cijk = fijk, h3,k = 1 • • • 8, are listed in Table 3.2. The Gell-Mann
matrices satisfy
T (f f) = \ ^> 1*A« = 0 -> det (>i¥) = 1- (3-12)
There are two diagonal matrices, A3 and A8, i.e., SU(3) has rank two. The X1
satisfy the commutation (Lie algebra) and anticommutation rules*.
[\\\*] = 2ifijk\\ {A*, A'} = ±8i*I + 2dijk\k. (3.13)
The dijk are symmetric in all 3 indices. The nonzero values are listed in Table
3.2. SU(3) has many other representations, including a second inequivalent
3-dimensional conjugate representation L\^ = — Al*/2 = — AlT/2, and the 8
dimensional adjoint (Lladj)jk = —ifijk- There are several SU(2) and U(l)
subgroups of 5C/(3), such as the SU(2) associated with L1'2'3.
TABLE 3.1
The Gell-Mann matrices. In the first entry i = 1,2,3.
/ i 0\ /00 1\ /0 0-i\
V = [ r 0 ) A4 =000 A5 = I 0 0 0
V000/ \100/ \i 0 0/
/000\ /0 0 0\ /l 0 0\
A6 = [ 0 0 1 ) A7 = [ 0 0 -i) Xs = -L [ 0 1 0 I
\0 10/ \0 i 0/ V3\0 0-2/
3.1.3 More on Representations and Groups
The rank of a Lie group is the number of generators that are simultaneously
diagonalizable. The diagonal generators correspond to conserved quantum
*It is sometimes convenient to define A0 = yflf^I. Then, Tr^A-?) = 26^ and {A*, A-*} =
2dijkXk, for i, j, k = 0,1 • • • 8, with d0jfc = y/t/SSjk.

lie Groups, Lie Algebras, and Symmetries
107
TABLE 3.2
The nonzero (totally antisymmetric) structure constants fijk for 5C/(3),
and the nonzero (totally symmetric) d^ defined by the anticommutators
of the Gell-Mann matrices.
/l23 = 1
/l47 = 2
/l56 = ~2
/ — l
7246 — 2
/ — x
7257 - 2
/345 = 2
/367 = — 2
f _ V3
7458 — y
/678 = y
dl18 = 73
^146 = 2
^157 = 2
d228 = 73
^247 = — 2
^256 = 2
^338 = ^
^344 = 2
^355 = 2
^366 = — 2
^377 = — 2
^448 = ~2^3
d558 = -573
d668 = "575
^778 = -573
^888 = -^
numbers if they commute with the Hamiltonian. £/(l),S£/(2), and SU(3)
have rank 1,1, and 2, respectively.
Two n x n representations Ll and L/l of G are equivalent if all A/" of them
are simultaneously related by a similarity transformation, i.e., if there exists
an n x n unitary matrix £/ such that
L'l = ULlU^ for i = 1 - - - iV. (3.14)
Otherwise they are inequivalent. Many groups have inequivalent
representations of the same dimensionality.
A representation U is reducible if it is equivalent to a representation
LH
(L'X 0 0 0 \
0 L% 0 0
0 0 L% 0
V 0 0 0 '••/
(3.15)
in which each element is simultaneously block diagonal (with the same block
dimensions). Otherwise, it is irreducible (an IRREP). States transforming
according to a reducible representation reduce to separate sectors not related
by the symmetry, while all of the states in an IRREP are related. Simple
Lie groups have an infinite number of IRREPs, and they frequently have
inequivalent IRREPs of the same dimension.
A fundamental representation is, roughly speaking, a representation from
which the others can be generated by direct products, in analogy to the
way that any angular momentum j in quantum mechanics may be
generated by combining 2j angular momenta \. The defining representations
(m) of SU(m), such as the 2 of SU(2) in the example, are fundamentals;
SU(m),m > 2, has a second inequivalent fundamental, the m*.

108
The Standard Model and Beyond
The adjoint or regular representation of a Lie group is the NxN dimensional
representation constructed from the structure constants,
(LU)jk = -icijk- (3.16)
It is straightforward to show that Uad^ satisfy (3.7) (Problem 3.3). The adjoint
is essential for defining the self-interactions of the gauge fields in a non-abelian
gauge theory.
If Un is an n dimensional representation of a Lie algebra, then the conjugate
Lln* = — L™ = —LlJ is also a representation. Ln is real^ if it is equivalent to
Un*, i.e., if there exists a unitary U such that — V£ = ULlnW for i = 1 • • • N.
Otherwise, it is complex. The adjoint representation Uad- is always real, with
U = I. For S77(2), the 2 is real with U = r2, i.e.,
L2.=-T- = t*T-t>. (3.17)
The higher dimensional representations are also real. On the other hand,
the m of SU(m) for m > 2 is not equivalent to the m*, e.g., L3* = — 4^-
in SU(3) is not equivalent to L3 = ^-. This is important, for example, for
the Higgs Yukawa couplings in extensions of the electroweak SU{2) group to
higher symmetries.
The Simple Lie Groups
Two groups G\ and Gi commute if [gi,gj] = 0 for all ^G(?i, (jj G G2. Then,
one can define the direct product group G = G\ x G2 with elements gify, or
direct products of more than two groups, such as the standard model group
SU(3) x SU(2) x C/(l). A simple group is (non-rigorously) a non-abelian group
such as SU(3) that is not a direct product*. A semi-simple group is basically
a direct product of simple groups, i.e., a Lie group with no U(l) factors, such
as SU(3) x SU(2).
Cartan has given a classification of the simple Lie algebras. The
classification as well as the IRREPs and their properties are elegantly derived from
Dynkin diagrams (Slansky, 1981), but here we only give the results. There
are four count ably infinite series of classical Lie algebras, and five exceptional
algebras, as listed in Table 3.3. The four series correspond to simple matrix
conditions for the defining representations of the associated groups:
tMathematics books typically work in terms of iL, motivating the term "real".
*More precisely, a subgroup if of a Lie group G is an invariant subgroup if ghg~l G H for
all g G G, h G H. G is simple if it contains no invariant subgroups (other than the identity
and G itself), and semi-simple if it contains no abelian invariant subgroups. Compact semi-
simple Lie groups are either simple or the direct product of two or more simple groups.

Lie Groups, Lie Algebras, and Symmetries
109
• SU(m),m = 2 • • • oo correspond to the mxm complex unitary matrices
Ug = e~l(3'L with unit determinant,
UGU^ = I, detUG = h (3.18)
which implies that (3L are the traceless Hermitian matrices^. Ug leaves
invariant the inner product of two m dimensional complex vectors, i.e.,
ytx = y'^x', where x' = UqX and similarly for y'.
• SO(m) are the mxm real orthogonal matrices Og with unit
determinant^ (i.e., rotations in an m-dimensional real space)
0GOl = I, detOG = l, (3.19)
so that Og — e~l^'L with i(3-L real and antisymmetric. The inner
product yTx of two real vectors is left invariant under an Og transformation.
• Sp(2m) are the real 2m x 2m symplectic matrices M, defined by
MTSM = S, (3.20)
where S is the skew symmetric matrix S = ( j™ n] > where 0m and
Im are respectively the mxm zero and identity matrices. They
therefore leave invariant the quadratic form yTSx, where x and y are 2m
dimensional real vectors.
The defining representations of SU(m) and Sp(2m) are also fundamental and
can be used to generate the higher dimensional IRREPS as direct
products. For SO(m) one can derive higher tensor representations (including
the adjoint) from the defining or vector (m). However, there are additional
double-valued fundamental spinor representations, similar to the familiar 2 of
SO(3) ~ SU(2) (see, e.g., Li, 1974; Slansky, 1981). All of the IRREPS can
be generated as direct products of the fundamental spinor. SU(m), SO(m),
and some of the exceptional groups have found considerable application in
physics. Recently, Sp(2m) has emerged in connection with string theory.
Casimir Invariants
A Casimir invariant is a function f(T) of the group generators T% which
commutes with them, [/(T),Tl] = 0. By Schur's lemma the corresponding
§The U(m) group (which is not simple) is related by U(m) = SU(m) x U(l), where C/(l)
is defined in (3.9) with T the mxm identity matrix.
"The orthogonal group 0(m) consists of the transformations Og and ROg, where Og G
SO(m) and R, which represents a reflection in an odd number of dimensions, is a diagonal
matrix with elements ±1 and det.R = — 1.

110
The Standard Model and Beyond
TABLE 3.3
The Cart an classification of simple Lie algebras. The groups
SO{6) ~ SU(4), SO{4) ~ SU{2) x SU(2), SO(3) ~ SU{2) ~ Sp(2),
and Sp(4) ~ 50(5) have the same Lie algebras, but may differ for
non-infinitesimal transformations. The subscript in the first column is
the rank.
Cart an label
~Ae
Be
ct
De
G2
Fi
E6
E7
E8
Classical group
SU{£ + 1)
so(2e + i)
Sp(2£)
SO{2l)
N
?(FT2)
*(2£+l)
l{2l+l)
1(21-\)
14
52
78
133
248
range
m~
£>2
e>3
e>i
function f(L) of an n x n IRREP L is a multiple of the identity. The coefficient
may depend on the representation and may be used to label it. The simplest
example is the quadratic Casimir
N
L2 = '$2LiLi = C2(L)L (3.21)
2 = 1 |
A familiar example is J2 = j(j + 1)/ for the angular momentum j
representation of the rotation group. C2(L) is related to the Dynkin index T(L) defined
in (3.8) by
T(L)N = C2(L)n. (3.22)
The quadratic Casimir of the adjoint T(Ladj) = C2(Ladj) is also written as
C2(G). Prom (3.16),
cikiCjki=C2{G)8ij. (3.23)
The quadratic Casimirs and Dynkin indices for the defining and adjoint
representations of the classical Lie algebras are given in Table 3.4 (see also van
Ritbergen et al., 1999). Other useful identities (which can be used to construct
other invariants) are
VUV
C2(L) - \C2{G)
V
(3.24)
ci3kUU = -C2(G)Lk
from which Tr U = 0 for the generators of a simple Lie group.

lie Groups, Lie Algebras, and Symmetries
111
SU(m)
SO(m)
Sp(2m)
m2-\
m(m—1)
2
ra(2ra+l)
m
2(ra-2)
ra + 1
m
m
2m
l
2
2
l
2
TABLE 3.4
Quadratic Casimirs and Dynkin indices for the defining representation
Ln and adjoint representation of the classical Lie algebras.
G N C2(G) n T(Ln) C2{Ln)
m2-l
2m
m — 1
2m+l
4
More on SU(m)
Properties of the SU(m) IRREPs and their direct products can be found
systematically from the Young tableaux (e.g., Amsler et al., 2008) or the more
general Dynkin methods. However, many aspects of SU(m) are simple enough
to "do it yourself." For example, the fundamental Llm = Xl/2 with Tr (XlXj) =
25li can be written as an obvious generalization of the 3x3 matrices in Table
3.1, and the structure constants can be calculated from them. An important
property of SU(m) (that does not generalize to the other simple groups) is
that the Llm along with the identity form a complete set which can be used
to write any m x m matrix,
1 -j m -l
M = -Tr (M)I + - Yl ^ (MA*)A*> (3-25)
i=l
where the coefficients are generally complex. In particular, the anticommuta-
tor {A1, AJ} is a linear combination
{A*, A^} = -Si3I + 2dijkXk. (3.26)
m
where the totally symmetric d^h generalize those in Table 3.2. This allows
one to generalize the SU(2) Fierz identity in Problem 1.1 to SU(m),
(x\ AX3) • (xUxi) = 2Vf(xLxi)(xIx3) ' ^(xlxMxlxi), (3-27)
where rjjr = +1 if the \% are ^-dimensional complex vectors or complex scalar
fields and rjjr = — 1 for anticommuting fermion fields.
SU(m) tensor methods, discussed in Section 3.2.3, are especially useful for
constructing SU(m) singlets from direct products.

112
The Standard Model and Beyond
3.2 Global Symmetries in Field Theory
3.2.1 Transformation of Fields and States
In field theory groups consist of symmetry operations which leave the
equations of motion unchanged in form. These may be discrete symmetries, such
as P, C, T discussed in Section 2.10, or discrete internal symmetries, which will
be considered in Section 3.2.5. Here we are more concerned with continuous
groups. One important class is the space-time symmetries, such as space
rotations, Lorentz boosts, and translations. Another, considered in this section,
is internal symmetries, involving the interchanges of fields with similar
properties, changes in their phase, etc. To formalize this, let 3>a(x),a = 1,2 •• -n
be n fields (which may be spin-0, |, 1, etc.) related by a symmetry.
Furthermore, consider a Lie group G of operators Ug(P) = e~l^'T. Then, define a
set of n transformed fields
&a = e-^f^a(x)e^f
= *a-i[^.f,*a] + ^[^.f,t9.f,*a]] + --., (3.28)
where the last form follows from the operator identity in Problem 3.5. Thus,
the transformation of the fields is determined by their commutators with the
group generators T%. Since we assume that the 3>a are transformed into each
other, one must have that
[Ti,*a(x)]=-L\tb*b(x), (3.29)
where Uah are the components of an n x n matrix Ll, which is easily shown
to form an n x n dimensional representation of the Lie algebra of G. Prom
(3.28) and (3.29) one has that
K = (e+i0'ZU*b = U0)ab$b > $a + 0' Lab$b, (3.30)
|/3 | small
which defines how the n fields corresponding to representation L are
transformed into each other. One usually considers the case that L is irreducible.
There is a frequently useful matrix notation for fields A1 transforming under
the adjoint representation" (3.16), in which the A1 are re-expressed in terms
of the elements of an n x n matrix
N
Tr (AL*)
A^AiLi^Ai = ^^ (3.31)
i=l
"The adjoint representation is real, so there is no distinction between upper and lower
indices, i.e., A1 = Ai.

Lie Groups, Lie Algebras, and Symmetries
113
where Ll is an arbitrary non-trivial IRREP of dimension n (usually taken to
be the fundamental or defining). It is then easy to show (Problem 3.7) that
the transformation of A1 —> An = (eL(3'Lad^)ijAj can be expressed in terms of
representation L by
A^A! = f^A^V = e+^Ae-tf'1, AH = ^^'• (3.32)
As described in Chapter 2, if 3>a corresponds to a particle, then the anti-
particle field is given by or closely related to $£. There are two possibilities
for the transformations of non-Hermitian fields. One is that the fields for the
particle and antiparticle are in the same IRREP, such as the pions
* = tt° , (3.33)
which transform as a triplet under SU(2) isospin. This requires that the
representation is real, such as the adjoint in this example. Alternatively, the
particle and antiparticle fields can be in different IRREPs, such as the kaons
under isospin
•-(£) •'-(£)■ <334»
Then, if $ transforms under the n representation Lln, & transforms under
the conjugate representation Lln* = —L1^ = —L1^, which follow by taking the
adjoint of (3.29) and using that T% is Hermitian. Of course, Lln may be real,
as in the SU(2) example or for the adjoint of SU(3).
Prom the expressions (2.89) or (2.155) one sees that for free fields the single
particle state corresponding to 3>a may be constructed by
|a>=at|0>~*£|0>, (3.35)
where in the second expression it is understood that a Fourier transformation
and appropriate projections of Dirac spinors, etc., are to be performed. Thus,
the states \a) and $£|0) transform the same way under G. This continues to
hold for interacting fields as long as the T% commute with the Hamiltonian
H. Then, the action of the generator on the state is
T*\a) ~ T**t|0) = $i:r|0) + LM|0>. (3.36)
Assume for now that the ground state is invariant, i.e., Tl\0) = 0. Then,
Ti\a) = \b)Lla = \b)(b\Ti\a), (3.37)
so that the representation matrix L\a = (6|Tl|a) is just the matrix element of
Tl in the n-dimensional space of particles.

114 The Standard Model and Beyond
3.2.2 Invariance (Symmetry) and the Noether Theorem
The Lagrangian density C is invariant or symmetric under a group of
transformations Ug((3) if they commute, i.e., if
£ = UG0)CUG0)-1 = C (3.38)
for all (3. (A similar definition applies for invariance under discrete
transformations.) Since
C' = C-i\/3.f,c] (3.39)
for small \(3 |, invariance holds if and only if
[T\C] =0 (3.40)
for all i.
The first part of (3.38) defines the transformation of C whether or not
there is an exact symmetry. Since £ is a function of (3>a>dM3>a), and (for a
non-Hermitian field) of (3>t,dM<I>t) one has that
£ = UoiftCUGiP)-1 = £«,cV<, *?,3M*?), (3.41)
where &a is given in (3.30), with an analogous expression for d^'a (since we
are considering global transformations, (3 = constant). The expressions for
$$ and dM3>'J are similar except that U —» —Ll*. It is frequently useful to
consider explicit symmetry breaking, i.e.,
8C = C'-C^0 (but small). (3.42)
One can also have spontaneous symmetry breaking
C' = C butr|0)^0, (3.43)
i.e., the Lagrangian is invariant but the ground state breaks the symmetry (cf.,
the breaking of rotational invariance in a ferromagnet). Both of these cases
will be considered extensively below, but for now consider an exact symmetry,
[r,£]=0andr|0)=0. (3.44)
This implies degenerate multiplets of particles and definite relations between
their interactions. It also implies conserved currents and charges according to
the Noether theorem, which generalizes that for a single complex scalar field
discussed in Section 2.4.1. The Noether theorem for internal symmetries is
d^=0, 1^=0, (3.45)
where the Noether current and charge are
^-'Jk'^-1^-1^1 (3-46)

Lie Groups, Lie Algebras, and Symmetries
115
Ql
J d3xJl0(t,x). (3.47)
Normal ordering on the fields is implied. The Noether theorem can be derived
from the Euler-Lagrange equations, in analogy with the derivation for the U(l)
case in Section 2.4.1. One can use the canonical commutation rules to show
that
[Q\ Q3} = ici3kQ\ [Q\ *a(x)] = -Vah$h(x). (3.48)
That is, one can identify Ql = Tl as a concrete construction of the generators
in terms of the fields.
The Noether currents are also useful for explicitly broken symmetries. The
Noether charges are no longer time independent, but one can use the canonical
commutation rules to show that the the commutation rules in (3.48) still hold
provided the charges and fields are evaluated at equal times. For example,
suppose C = Co + £i, where only Co is invariant,
[T\ Co] = 0 (all i), [T\ d] ^ 0 (some i). (3.49)
Then, the change in C is related to the divergence of J,
8C = C -C= [-i0.f,£i] =-/3.#iJM. (3.50)
This immediately implies
dtAJl = i[Ti,C] =» f d3xd^Ji = -i[T\H], (3.51)
where the last step assumes that any symmetry breaking is in the mass and
interaction terms (i.e., that kinetic energy terms are canonical). The (non-
conserved) T% and d^J1^ are evaluated at the same t. Eq. 3.51 implies that
(a(pa)\Tl\b(pb)) = i(27ry6*(pa - ph) = ^ . (3.52)
hit — rja
This relation is useful when states a and b are not related by the symmetry
and are not degenerate in the symmetry limit. One then has that the leakage
of Tl\b) into \a) is proportional to the symmetry breaking.
The Complex Scalar
As a first example, consider an IRREP relating n complex scalars </)a,a =
1 • • -n, transforming as [Tl, </>a] = —L^fo under some Lie algebra that will
be determined from the symmetries of the Lagrangian density,
n
C = ^ (dM</>a) (d^<t>a) + non-derivative terms. (3.53)
a=l

116 The Standard Model and Beyond
The Noether currents are
4 = i<&Kb% <t>b, (3.54)
where / d M g = f(d^,g) — (d^f)g. The derivative (kinetic energy) terms are
invariant under the group U(n) = SU(n) x £/(l). Under an SU(n)
transformation
0 - e^f0, 0f - 0^"^, (3.55)
where 0 is the n-component column vector (0i 02 • • • 0n)T, 0* is the row
vector (</4 02 '" 0n)» an(^ ^ is tne fundamental representation matrix L^ of
SU(n). The SU(n) invariance is obvious with this matrix notation
CKE = [d^iP*) - (c^)f e-^Le+i^L {&><(>) = (d^ {&*<!>). (3.56)
Cke is also invariant under U(l) transformations, 0 —> el/3/0. £/(n) is the
maximal possible symmetry group of the system; depending on the mass and
interaction terms the symmetry may be smaller.
Including mass terms
£ = (d^ {&*<!>) - 4>^24>, (3.57)
where 0V20 = 0oMo6^& anc* M2 is an n x n matrix with elements n2ah. The
Hermiticity of C requires that fi2 is Hermitian. The eigenvectors and
eigenvalues of ji2 correspond to states of definite mass and to their mass-squares,
respectively. For now, however, let us assume that /j,2 is already diagonal,
ji2 = diag (/j,2 /j%" ' Mn)- Under a group transformation,
0V20 - tfe-'^^e+'^h (3.58)
so the requirement for invariance is that
e-^'Ve+i*£ = M2 (3.59)
for all /?, which is equivalent to
[/3-L,^2] =0. (3.60)
(3.60) determines what subgroup of U(n) survives. For example, if all of the
masses are the same, /j? = /j,2I, then C is invariant under U(n). If all the
masses \j?a are different, then the symmetry is reduced to
U(l)n = [7(1)! x U(l)2 x • • • x C7(l)n, (3.61)
where only 0a transforms nontrivially under E/(l)a, i.e.,
0a -> ei/3°0a, 06 -> 06 for 6 ^ a. (3.62)

Lie Groups, Lie Algebras, and Symmetries 117
For the intermediate case of
H = diag
/
2 2 2 2
\ rii n2 /
(3.63)
with ni fields of mass \i\ and 712 of mass /j,2, the symmetry group is U(n{) x
1/(^2)» with the first (second) set of fields transforming under U(ni)(U(ri2)).
One can also add quartic interaction terms,
C = (d^ (d»(f>) - </>V2</> - ^ \abcd(t>l<t>b 4</>d, (3.64)
abed
where \ahcd = Xbadc (from Hermiticity) and Xabcd = Kdab from Bose
symmetry. In general, this reduces the symmetry to E/(l)**, but there could be a
higher symmetry for specific A's. For example, full U(n) invarance is restored
for
Xabcd = XSabScd, fi = fal
v2
C^id^iPfi-fiiltt-X {<$<!>) .
(3.65)
The Hermitian Scalar
Hermitian scalars, <f>a — (j>\, transform as
[T,4>a\^-Lib4>b. (3.66)
Consistency requires that Ll = —L1*, i.e., the representation is real. The
Lagrangian density
£ = « (^M^a) (d^<t>a) + non-derivative terms (3.67)
implies Noether currents
Jl = -i(dll<l>a)Kb<l>b (3.68)
(the second term in (3.46) is absent for Hermitian fields). The special case
£ = \ [(d^af ~ f(t>a<t>a] " A {^af (3.69)
is 0(n) invariant (Problem 3.8). However,
c=-(d^a)2--d>an2abd>b (37o)
- ^a6c0a</>6</>c ~ Aa6cd</>a</>6</>c</>d
has no symmetries at all in general.
**Even the C/(l) would be broken in the presence of terms like <f>4 + <f>^4.

118
The Standard Model and Beyond
Complex Scalar in a Hermitian Basis
A complex scalar field <j> can always be written in terms of two Hermitian
scalars as in (2.87), </> = (</># + i</>j)/\/2, where (J)rj are Hermitian. For
complex fields that are in the same representation as their adjoints, such
as in (3.33), it is almost always simpler to rewrite the theory in terms of
Hermitian fields. In the case such as (3.34) that the complex fields and their
adjoints transform separately, it is still sometimes useful to go to a Hermitian
basis (especially for formal manipulations), although the complex basis is
usually simpler for explicit calculations. The relation between the bases is
straightforward, but can be confusing.
Suppose </> is an n-component complex field transforming with
representation matrices L^. It is then convenient to introduce the complex 2n-
component vector $ = I ^ 1, with half of the components redundant. $
transforms under the reducible representation
One can introduce 2n Hermitian fields <t>aR,<l>ai, by </>a = -jx((t>aR + i(t>ai), and
the 2n-component real vector <j>h = ( , I. The two bases are related by the
unitary transformation
*=^« A-i=2{Ii-u)' <3i2>
where / is the n x n identity. Hence, the representation matrices for the
symmetry generators in the Hermitian basis are
L,h -AL*A-2^ _i{^ + L ?) ^ _Li; y (3.73)
which are manifestly imaginary and antisymmetric.
Examples: (a) Consider the U(l) group acting on a single complex <j> —»
exp(-M/?)</>, so that L^ = 1. A and the representation matrices are therefore
^710-0' L*=(o-i)' L^(-io)'
while a finite U(l) transformation is just a rotation
</>iA -ipr2 (<t>R\ _ /cos/3 -sin/3 \ (<j>R
4>i J V $1 J Vsm & cos P J V ^
VI
(3.74)
(3.75)
(b) Consider two complex fields <j> = I 0 J transforming as a doublet under
5C/(2), U — rl/2. (The superscripts look ahead to applications to the Higgs.)

Lie Groups, Lie Algebras, and Symmetries
119
In the (reducible) 4-dimensional Hermitian basis (j>h = (</>i </>3 </>2 </>4)T, where
0+ = [<f)i +z</>2)/\/2 and </>° = (</>3 + z</>4)/\/2, the representation matrices are
L£ = 5 ( _Ti,3 T0 ) . L\h = 2 ( T0 T2 ) • (3-76)
Fermions
Now consider the case of n fermions tpa, with
£ = tpai ftlpa - Tpa,™<ab1pb = $10^- tpmip. (3.77)
In the second form, ip is the n-component column vector (^i tp2 * • • ipn)T and
m = raMs an n x n Hermitian matrix^ which can be taken to be diagonal.
ip and ip^ transform as
[T*,^] = -46^, [T\^l] = +L**6^ (3.78)
under a symmetry transformation, and the corresponding Noether current is
4 = da-y^fo. (3.79)
As in the scalar case, the symmetry group is determined by C. One has
C->£ = $i ^e-^'le^'1^ - #-^£me+^L>. (3.80)
The kinetic term is invariant under U(n) = SU(n) x C/(l), but the invariance
condition for the mass term is
e
-^'lme^1 = m <-► [/3 • L,m] = 0. (3.81)
The full U(n) is maintained for n degenerate masses, m = mil, while the
symmetry is reduced to U(l)n for n distinct masses.
3.2.3 Isospin and SU(3) Symmetries
SU(2) Isospin
Isospin is an approximate symmetry of the strong interactions, broken by
~1%. The breaking is ultimately due to the u — d quark mass differences.
This is usually viewed as intrinsic to the strong interactions when discussing
QCD, though the masses are actually associated with the electroweak sector.
There is a comparable breaking from electromagnetism. We first describe a
^Generalized fermion mass terms involving 75 or, equivalently, non-Hermitian matrices,
are considered in Problem 3.28 and Chapter 7.

120
The Standard Model and Beyond
simple model of isospin in terms of the nucleons and pions. Introduce the
nucleon (proton and neutron) and pion fields
r/>:
UJ' '-IH' ' =^i-' *=7r3> (3-82)
where -K{ — -k\ and 71^ annihilate the states \k±). tp and n transform as a
doublet (fundamental) and triplet (adjoint), respectively**, under 5C/(2),
4 = y. (4)ifc = -««*• (3-83)
The diagonal generator is T3 (L3 is diagonal in the 7r±,() basis). Consider the
Lagrangian density
Co = tp(i@-mI + ign75n • f) ^ + \ [{d^f - /Af2] - A (7?2)2 (3.84)
= ^ (i ^ - m/ + iV2^75n) ^ + ^n [(aMn)2 - M2n2] - a (tvn2)2,
with 7T2 = X^=i ^ and n = ^ = I ^ _ 0 I . The gn term is the Yukawa
interaction between the pion and nucleon*. The second (matrix) form in
(3.84) makes it especially easy to read off the Feynman rules for the ppn°,
7in7r0, np7r~, andpn7r+ vertices, viz, —gn75, +<7tt75, —V^9n75, and — y/2gnj5,
respectively.
We should emphasize that experimentally gn is very large: the experimental
ttN coupling Gn observed from low energy ttN and NN interactions, which
may differ from gn by strong interaction corrections, is ~ 13.05(8) (e.g., Gor-
ringe and Fearing, 2004). Therefore, (3.84) should be viewed as a model to
illustrate symmetry considerations and not as a serious perturbative field
theory for the strong interactions. Co is SU(2) and reflection invariant (the 7s
is because the pions are pseudoscalar); using (3.32),
t/> -> e+i^fy, ^ -> ^e-^'2, n -> e&i Ile"^, (3.85)
from which both ^^Iltp and Tr II2 are invariant under SU(2). The invariance
of the pion self-interaction is further discussed in Problem 3.15.
^The convention for the n^ fields in (3.82) is common in particle physics, and
convenient because n+ = (tt-)*. However, the phase conventions usually employed for
states in rotational multiplets in quantum mechanics would instead require the
convention n± = =F (?ri T "T2) /\/2 (Problem 3.12).
*We use the term Yukawa interaction in a generalized sense, i.e., for any 3 point interaction
between a spin-0 and spin-| particles.

lie Groups, Lie Algebras, and Symmetries
121
Now, introduce SU(2) breaking by C = Co + £1, where
£1 = -e^r3^ = -e (^p^p - ^n^n) • (3.86)
jCx thus represents a splitting between the proton and neutron masses,
nip = 77i + e ran = ra — e, (3.87)
which parametrizes contributions both from the quark masses and from elec-
tromagnetismt. It is straightforward to show that
[T\d] = 0, [T1'2^] ^0. (3.88)
Thus, SU(2) is broken to C/(l)^3, corresponding to a conserved charge T3,
with T3p = — Tsn = 1/2, T^i = ±1,7^0 = 0. Actually, £ is invariant under
U(1)t3 x ^(1)b> where the second U{\) corresponds to a conserved fermion
(or in this case, baryon) number BPiTl = l,Bn = 0. The conservation of
of B and T3 is equivalent to that of B and electric charge Q, where Q =
T3 + B/2 when restricted to the fields in this example. C\ transforms as the
T = 1, T3 = 0 component of an irreducible tensor operator. One can therefore
use the Wigner-Eckart theorem for relations between its matrix elements, in
exact analogy to broken rotational invariance in quantum mechanics.
SU(3) Symmetry
SU(3) is an approximate global symmetry of the strong interactions that
extends the SU(2) isospin subgroup. It is valid at the ~ 25% level for masses,
but works better for relations between couplings and matrix elements. SU(3)
was proposed independently by M. Gell-Mann and Y. Ne'eman in the early
1960s to account for the fact that the low lying mesons and baryons could be
associated in octets (the eight-fold way). As described in Section 3.1.2, SU(3)
has eight generators. The fundamental representation matrices L\ = A1/2
and structure constants are listed in Tables 3.1 and 3.2. SU(3) has rank 2,
so two generators, T3 and T8, can be simultaneously diagonalized with the
Hamiltonian. Empirically, these correspond to the strong interaction quantum
numbers by the Gell-Mann-Nishijima relation
where Q is electric charge, ^3 is the third component of isospin, y = B + S is
strong hypercharge, B is baryon number, and S is strangeness (S = 0 for the
pions and nucleons).
TFor simplicity we ignore isospin breaking terms in the interactions or those involving
fj^± — n20. The latter are easily shown to come only from electromagnet ism to leading
order, and not from the quark mass differences.

122
The Standard Model and Beyond
The low-dimensional representations of SU(3) are the 1,3,3*, 6,6*, 8,10,10*,
and 27, where the 1,8 (adjoint), and 27 are real and the others complex. The
observed hadrons can be assigned to the 1,8,10, and 10*. It is convenient to
display the states on weight diagrams, with the axes corresponding to Is and
y. Then, the other generators of the Lie algebra describe transitions from
one state to another. For example, the lowest lying Jp = \ baryons and
Jp = 0~ mesons (J is the spin and P is the intrinsic parity) both transform
under the adjoint (octet) representation, as shown in Figure 3.1. In the
absence of SU(3) breaking, the states in each octet would be degenerate. Each
consists of two isospin doublets with y = ±1, and one isotriplet and one
isosinglet with y = 0. The baryon (hyperon) fields, which annihilate states
with B = 1 and strangeness S = y — B, are given by
E* = i W>i =F #2), £° = ^3, A = ^8
P =-/= W>4 - #5), n=-j=(il>6-iil)7) (3.90)
E° = ~/2 ^6 + ^ ' E~ = 72 ^4 + ^ '
while the pseudoscalar meson (B = 0) fields are
TT* = -7= (01 =F i<fo) , 7T° = </>3, V = 08
K± = 71(04 T *05)' ^° ^= 72(06 T i0?)'
(3.91)
The 7/, which also has Is = y = 0, is an 5C/(3) singlet. The 7r, if, 77,7/ system
is referred to as a nonet = 8 + 1.
Analogous to (3.82) one can define 8-component vectors ip and </>, so that
ip —> exp (z/3 • Ladj)ip, <j> —> exp (z/3 • Ladj)(f> under an SU(3) transformation.
The extension of (3.84) to the baryon and pseudoscalar meson octets is
Co = ipi(i0-m)ipi + - [(dpfa)2 - m20j
+ igffijkipirfipjfa + gddijki)iilbipj(t>k (3*92)
- a((f)j) - Pdijkdimn^j^k^m^n,
where </>^ = <j>j(j>j and fyk and dijk are the coefficients given in Table 3.2.
There are two independent SU(3) invariant meson-baryon interactions, known
as the 'F' and '£)' couplings, which are respectively antisymmetric and
symmetric in the SU(3) indices. This can be understood as follows. An invariant
coupling is just a singlet component of the direct product of the fields in the
interaction term. For SU(2) the meson-baryon interaction in (3.84) involves
the direct product of two doublets and one triplet, (2 x 2) x 3 = (1 + 3) x 3 =

lie Groups, Lie Algebras, and Symmetries
123
n
-1
. y
.p
-$-
S°,A
=o
-l
1 '3
K y+i .K+
7T
K-
■4
^0 ^
7T ,7/
1 '3
-1
K°
FIGURE 3.1
Weight diagrams for the Jp = ^ baryon and Jp = 0~ meson octets. The
anti-baryons are in a separate octet, while the mesons and their antiparticles
are in the same octet.
3 + [1 + 3 + 5], where the IRREPs are labeled by their dimensionality 21 + 1.
The singlet only occurs once in the decomposition, so there is only one
invariant. For SU(3), however, there are two independent ways to form an invariant
from 8x8x8,
8 x 8 = 1 + 8 + 8 + 10 + 10* + 27
8 x (8 x 8) = 8 x 1 + (8 x 8) + (8 x 8) +8 x [10 + 10* + 27]. (3.93)
Similarly, there are now two invariant meson self-interaction terms, associated
with the singlet and symmetric octet components of 8 x 8.
Cq can be written in matrix notation, using
K+
\ K-
(3.94)
as well as
B
( =2. , A2. y+ „
\ a
-S" , Ai
v^ + \/6
=0
-2AU
y/S I
s-£^p (3-95)

124
The Standard Model and Beyond
yielding
Co = TV [B (i 0 - m) B] + ^T> [(^M)2 - /x2M2]
+ 2L-& (Bh5 [B, M}) + -^IV (Bi75 {B, M}) (3.96)
- (a - l(3)(TrM2)2 - 2/3TVM4,
which is invariant from (3.32).
SU(3) Breaking
The degeneracy of the SU(3) multiplets is only good to around 25%, though
the predictions for couplings and amplitudes are typically much better. The
Gell-Mann-Okubo (GMO) ansatz is that the breaking can be described by an
operator that transforms as an octet, i.e.,
C = Co + e8C8, (3.97)
where Co is a singlet (i.e., invariant), eg is a small coefficient, and C8
transforms as the 8th component of an octet of operators, Ci,i = 1 • • • 8:
UgCoUc1 = C0, UgCsUg1 = U^A C, (3.98)
\ / 8 j
or equivalently,
[r, Co] = o, [r, c8] = - {uadj)%j c, = if*^. (3.99)
When first postulated, the actual form of Cg was not known, only its
transformation properties. The power of such an ansatz is that it allows matrix
elements of Cg to be related by SU(3) in terms of one or more parameters
that can be measured (or calculated in a more detailed theory). To illustrate
this, recall the Wigner-Eckart theorem for 5C/(2), which relates the matrix
elements of an irreducible tensor operator* T*, which carries angular
momentum (or isospin) k and z- component q, between states ol\ and a2 with angular
momenta and z components j\,2 and mi^:
(oc2 h m2\T^\ai j± m{) = (a2 j2 \Tk\\ <*i ji)(J2 m2 \ k qx j± m{). (3.100)
The double-barred quantity is the reduced matrix element, which depends
on the operator and states, but is independent of mi,m2, and q, while the
second quantity is a Clebsch-Gordan (CG) coefficient, which leads to selection
rules, m2 = q + mi and |fc — ji\ < j2 < k + ji, and relations between the
nonzero matrix elements. (An excellent table of CG coefficients can be found
t[T\T*}=qT%, [Tl±iT2,T*] = y/(kTq)(k±q+l)T*±l.

lie Groups, Lie Algebras, and Symmetries
125
in the Review of Particle Properties (Amsler et al., 2008, or their website).)
Similarly, for SU(3) the matrix element of an element of an octet of operators
between two octets can be written in terms of two quantities that depend
on the dynamics and the / and d symbols, which are analogs of the CG
coefficients^. Thus,
-es(i\Cs\j) = aifisj + (3diSj, (3.101)
where a and /3 are proportional to eg. The symmetry-breaking term in the
Hamiltonian is — e8 J d3x £8(x), implying that the shift in the mass rriBr of
baryon Br due to the S't/(3)-breaking is
a™ (#r| ~ e8C8\Br) , .
AmB- ~ 2^ ' (3-102)
where the 2rriBr in the denominator is from our covariant normalization
convention. Therefore,
mBr = mBo + crfma + crdmp, (3.103)
where rriB0 is a common (S't/(3)-invariant) mass, crrd are the "CG"
coefficients obtained from if and d by taking the appropriate linear combinations
of indices, mQ ~ a/(2ra£0), and similarly for mp. Ignoring isospin breaking,
there are four masses, Mn = (Mp + Mn)/2, Ms, M~, and Ma, that can be
expressed in terms of three parameters,
'ffl'Bo.ci.B' The latter cannot be
predicted a priori, but there is one linear relation, the Gell-Mann-Okubo relation
(Problem 3.19)
?k±MK = ?Ml±^+0{4), (3,04)
where the result holds to leading order in e8. Experimentally, the GMO
relation works extremely well. The individual masses in GeV
MN ~ 939, M~ ~ 1318, ME - 1193, MA - 1116, (3.105)
indicate SU(3) breaking at the 20% level, but the left- and right-hand sides
of (3.104) are 1129 and 1135 respectively, equal to better than 1%. Similar
formulae apply to other low-lying hadronic states, such as the pseudoscalar
(Jp = 0~) mass-squares ^,/x^,^ and the lowest vector (Jp = 1~) meson
mass-squares, ^^,/x^*,//?. (One must include the effects of octet-singlet
mixing between the 77 and 7/ and between the <j> and uj. See Section 5.8.3.)
The situation is simpler for the Jp = | states, which transform as a 10
(decuplet). There is only one invariant of the form 10 x 8 x 10*, so there is
only one reduced matrix element in (10|£8|10), leading to a linear spacing in
the masses as one goes to lower y. This works very well, as can be seen in
Figure 3.2. In fact, the y = —2, S = — 3 baryon fl~ was not known at the
time SU(3) was proposed. The prediction of its existence and mass from the
GMO ansatz was a great triumph for SU(3).
§One can generalize the CG coefficients to arbitrary SU(3) representations using isoscalar
factors (de Swart, 1963; Amsler et al., 2008).

126
A°. yu *A+
£*"
^*o
-l
Q-
The Standard Model and Beyond
A++ 1230
-<*+
1385
l-l
♦ -2
3 ^3
2
-*0
1530
1672 (predicted)
FIGURE 3.2
Weight diagram for the Jp
I decuplet.
When SU(3) was proposed it was generally believed that isospin was an
exact symmetry of the strong interactions, and that the observed small (1%)
breaking is due to electromagnetic and weak interactions. We now understand
that there is a small breaking component intrinsic to the strong interactions
as well (in part because estimates of the electromagnetic contribution to the
proton-neutron mass difference predict a heavier proton). This can be
incorporated by writing
C = C0 + e3C3 + e8£8, (3.106)
where e3 = 0(1%) and eg = 0(25%). (A simple model for £3 was given in
(3.86), but we do not assume that specific form here.) Additional predictive
power comes from the assumption that £3 and Cg are both operators from the
same octet. SU(3) is very successful, and is extremely useful for relating
various strong and electroweak matrix elements. It is even more successful when
extended to chiral symmetry. The existence of heavier hadrons associated
with the charm (c) and heavier quarks suggests the possibility of considering
SU(4) or higher. However, SU(4) is so badly broken that such an extension
is not very useful.
The Quark Model and SU(3)
The above description of SU(3) in terms of baryons and mesons becomes
simpler when re-expressed in terms of the three lightest quarks and their

Lie Groups, Lie Algebras, and Symmetries
127
fields, u, d, and s, which transform under isospin^ as a doublet (u,d) and
singlet (5). Under SU(3) they transform as a fundamental (triplet) while the
antiquarks (and the q fields) transform as 3*
e 2 Q, q = (u d s) ^> qe l * ,
which is equivalent to
[r*,9«]
K, M] = ±^.
2 ?»i L >™j 2
The quark electric charges and strong hypercharges are given by
^ A3 A8 1
1
2 2^/3 3 v _j
The weight diagrams are shown in Figure 3.3.
(3.107)
(3.108)
(3.109)
y
u
*rh
1
"2
U
y
i
2
■^r/3
FIGURE 3.3
Weight diagrams for the 3 and 3* of SU(3).
"in addition to isospin, SU(3) has two other (non-commuting) SU(2) subgroups, [/-spin
and V-spin. In quark language, these are associated with the d <-* s and u <-* s transitions,
respectively. These are more badly broken than isospin, but are occasionally useful for
specific applications, e.g., Problem 3.20 and (Commins and Bucksbaum, 1983).

128
The Standard Model and Beyond
A quark and antiquark can combine to form an octet and singlet, i.e.,
3 x 3* = 8 + 1. In an obvious matrix notation,
qxq
du dd ds = M + -7= /
^- v/3
octet v^^/
singlet
/ ^u-dd-ss U£ Us
dU tdd-uu-ss fa
2ss—uu—dd
3
SU
3_
sd
(3.110)
1 00
+ - (uu + dd + ss) 0 10
3 * 00 1
where M is the pseudoscalar octet (3.94) expressed in terms of quarks, and
7/ = (uu + dd + ss)/y/3 is the singlet.
The Jp = \ baryons also transform as an octet, but are constructed from
three quarks qqq:
3x 3 = 3*+6
3x3x3 = 3x3* + 3x6
1 + 8 + 8 + 10.
3 +
(3.111)
Thus, both the octet baryons and the J = | decuplet can be constructed
from three quarks. Of course, one must also include space, spin", and color
indices in the construction; in fact, one of the reasons for the prediction of the
color quantum number is that otherwise the fi~ would be a totally symmetric
(J = 3/2) composite of three s quarks, in violation of the spin and statistics
theorem. \
The free quark Lagrangian density is
£quark = ^ qa (i 0 ~ ™<a) Qa
a=u,d,s
2_\Qa(ift — too) qa — rrts (uu — dd) — rrtg (uu + dd — 2ss) /v3
q(ift — mo) Iq — m^qX^q — m%q\%q. (3.112)
"€3£3
Co
-€8£8
£quark would be invariant for degenerate masses, but the 7713,8 terms
automatically transform as components of an octet, so the GMO ansatz for SU(S)
breaking is automatically realized in the quark model. The u, d, and s masses
are related by
mu = mo + 777,3 +
ra8 ra8
—=, rad = rao-m3 + —=,
ms = mo
2ms/V3.
(3.113)
11 For the non-relativistic quark model it is sometimes useful to invoke a mixed SU(6) spin-
flavor symmetry, which is quite successful in describing the baryon spectrum and properties
such as magnetic moments (e.g. Hey and Kelly, 1983; Georgi, 1999; Capstick and Roberts,
2000; Close et al., 2007; Amsler et al., 2008).

Lie Groups, Lie Algebras, and Symmetries 129
They will be further discussed following the extension of SU(3) to a chiral
symmetry.
SU(rn) Tensor Notation
There is a powerful tensor notation for SU(m) (de Swart, 1963), which
generalizes the matrix notation introduced for the adjoint in (3.31). Denote the
components of a field ip transforming as a fundamental m with a lower index,
<0a, and those of an anti-fundamental ( m*) field \ by an upper index, xa,
[T\^a] = -4Vfc, [T\ Xa] = +XbL\a, (3.114)
where Llab = (Llm)ab = — (L^*)^. Of course, the conjugate ift is antifunda-
mental, (ipaV = {^)a- Higher-dimensional representations can be formed as
products of fundamentals, such as ipab, which can be regarded either as an
elementary field or as a product of two fundamentals ipiaip2b> IRREPs can be
formed by symmetrizing and antisymmetrizing, e.g.,
Ipab = ^ (ipab ~ ll>ba\ + ^($ab + lpba\- (3.115)
mxm m(m-l)/2 m(m+l)/2
Mixed tensors, such as 7y£, can again be either an elementary field or a product
ipaXb' Va can De decomposed into an adjoint and singlet,
r& =ti-%%+%%. (3.116)
^^ m m
m2-l 1
The transformation of a general mixed tensor is
\rpi pbi-bi "I __ _ r ic pbi-bi m m m _ j ic pb\-bi . pc--bi jib\ , pb\-c jibi
(3.117)
It is straightforward to prove that the antisymmetric product of m — 1
fundamentals is an m*, i.e.,
eQ—VQ2...Qm = t?Qi , (3.118)
where eai"am = eai...am is the totally antisymmetric tensor in m indices.
With this machinery, it is simple to construct the SU(m) invariant operators
by contracting upper and lower indices of the operators and the e tensor. For
example,
^ar}\ i>abria\ eai'"amtl>ai...am, eai...arnXai'"am, (3.119)
are all invariant (Problem 3.21). The first two operators are also invariant
under the extension to U(m) = SU{m) x £/(l), with i\)a -> eif3tpa, x° -► X°e~i/3,

130
The Standard Model and Beyond
and Pal'.'.'.alk ~^ e^k~l^Pal-..alk' The contractions involving the e tensor are not
U(l) invariant. Using tensor notation, the irN Lagrangian density in (3.84)
becomes
Co = $a(ifi- m) ipa + iV2g^aj37r^b + \ [(dM7r)2 - /aV] - A (tt2)2 ,
(3.120)
where n^ = Uab and n2 = 7r^7rfea.
3.2.4 Chiral Symmetries
We saw in Section 2.7.2 that one can view the left (L) and right (R) chiral
projections tpL,R = Pl,r?P of a fermion field tp as independent degrees of
freedom, with tp = ipL + ipR. The significance of the chiral projections is
that they can have different transformation properties under a global or local
symmetry group G. Suppose
[TSVol] = -VLab^bL, \r,^aR] = -Ur^R. (3-121)
If LlL ^ LlR the transformation is chiral; otherwise it is non-chiral. For
example, the weak interactions are associated with a chiral gauge symmetry,
implying parity violation. The strong interactions obey a non-chiral gauge
symmetry, but have approximate chiral global symmetries. Even for a chiral
symmetry, the fermion representation matrices may be reducible, with some of
the fermions chiral (i.e., their L and R components transform differently), and
others non-chiral or vector (i.e., they transform the same way). The fermion
Noether current for a chiral symmetry is
4 = ^l7m {Ll)ab ipbL + $aR7» {VR)ab *phR, (3.122)
which reduces to (3.79) for the non-chiral case UL = UR = U. In matrix
notation with tp = (ipi ■ ■ ■ ipn)T,
^^[UlPl + UrPr]*
1 - . (3.123)
= 2^ ^ + L«) - ^ ~ L«)7 ] *>
where L1lPl + L1rPr contains both Dirac and internal indices. Under a finite
transformation,
As an example, consider the free fermion Lagrangian density
C = tpai ftlpa - $a™<ab1pb
= $aLl ftlpaL + ^aR^ fi^aR ~ $aLmah1pbR - $aRmab1pbL,
(3.124)
(3.125)

lie Groups, Lie Algebras, and Symmetries
131
for a, 6 = 1 • ■" n- "i is an n x n mass matrix which we assume to be Hermitian
(one can generalize to a non-Hermitian mass matrix, or, equivalently, one
involving Pl,r, as m Problem 3.28). The kinetic part of £ is invariant under the
chiral flavor symmetry U(n)L x U(n)R = SU(n)L x SU(n)R x U(l)v x U(1)A-
The subscripts L and R have no group theoretical significance. Rather, they
indicate that the L-chiral fermions transform as an n under SU(ti)l, but as
a singlet under SU(n)R, and the reverse for the i^-chiral fermions,
SU(n)L : ^aL - (e^M ^feL, ^aR - ^aR
V / ab "> v '
% * ' 1
/ - - x (3-126)
5t/(n)R : ^aL - VaL, V'aR -» U0rL" V6R,
* v ' V / ab
1 V
n
where the /?£, and f3R are independent parameters. The non-chiral (vector)
U(X)v generator is the sum of the U(1)l and U(l)R ones. It corresponds to
fermion number, i.e., tpaL,R —> ^^PaL,R- The axial U(1)a generator is the
difference of the U(\)l,r ones. In QCD the U(1)a generator is broken by
non-perturbative effects, as will be discussed in Chapter 5.
The mass term tyhm^R + h.c. is only invariant for
e-i0L-Lnmei0R-Zn = m (3.127)
First consider the case that m is diagonal with degenerate entries, m = mil:
(a) For mi ^ 0 the mass term breaks the chiral symmetry to the subgroup
with Pl = (3R. This corresponds to the non-chiral SU(n) with UL =
ji _ ji
LR ~ ^n'
(b) For m = 0 the full SU(ti)l x SU(n)R x U(l)v chiral symmetry is
preserved, i.e., chiral symmetries involve vanishing fermion masses.
Of course, one can have hybrid situations, e.g., in which m is diagonal with
blocks of degenerate eigenvalues. Each block with k equal but nonzero masses
will correspond to a non-chiral SU(k) x U(l)v subgroup of the original
symmetry, while a block with / massless fermions will have a chiral SU(1)l x
SU(l)R x U(l)v symmetry.
It turns out that the quark masses are mu < rrid ~ a few MeV, which is
much less than typical hadronic mass scales. Thus the strong interactions have
an approximate SU(2)l x SU(2)r chiral flavor symmetry, explicity broken
by mUjd. This is usefully extended to SU(3)l x SU(3)r, though the latter
is broken by the much larger ms = (9(100) MeV. The precise meaning of
these current (Lagrangian) quark masses, and of the constituent quark masses
~ MPin/3 ~ 300 MeV, will be discussed in Chapter 5.

132
The Standard Model and Beyond
3.2.5 Discrete Symmetries
In addition to the continuous symmetries, many theories have discrete
symmetries characterized by a group with discrete elements. As an example, consider
a single Hermitian field </>, with
C=\(d**)2-\M+«I>-*£-*£■ (3.128)
For general couplings C has no internal symmetries. However, for the
special case a = k = 0, C in invariant under the discrete two-element group
Z2 = {/, R}, where <j> —> —<j> under R. The Z2 symmetry has the obvious
consequence that the number of particles is conserved modulo 2 in any reaction.
As another example, the Lagrangian density for two Hermitian fields
C X (Pi rk^jJ" (Pi rk\^ 1.2/JL2 , jl2x AH</)1 A22</>2 A12</>1 </>2 /o iq0n
is 0(2) invariant for An = A22 = A12. For An = A22 7^ A12 the symmetry
is broken to a discrete subgroup consisting of rotations by {0,7r/2,7r,37r/2}
and of reflections times the same rotations, i.e., to the interchange between </>x
and </>2 as well as possible sign changes for one or both fields. For An^ A22
only the sign changes survive. Discrete symmetries are not associated with
conserved charges and often (as in the Z2 and the An/ A22 examples above)
do not imply that particles fall into degenerate multiplets. Their major
consequence is usually to relate couplings or to remove certain couplings that would
otherwise be possible, and they are often introduced ad hoc in model building
for that purpose. They may also come about as a low energy remnant of a
continuous symmetry in an underlying theory that is valid at high energies.
A more realistic example is G-parity, which is an approximate discrete
symmetry of the strong interactions:
G = CeinT\ (3.130)
where C is the charge conjugation operator and T2 is the second generator of
the SU(2) isospin group. The mesons consisting of u and d quarks and anti-
quarks are eigenstates of G-parity. G-invariance, which follows from QCD if
isospin breaking is neglected, leads to important constraints on their
interactions. For example, the pions have G = — 1 while the 77 has G = +1. G-parity
therefore forbids the decays 77 —» 3n as well as any transitions between an
odd number of pions by the isospin-conserving strong interactions. (77 —» 3n
does proceed much more slowly through isospin breaking effects, especially
the u — d quark mass difference.) Consequences for nucleon matrix elements
are considered in Appendix G.
The G parity assignments can be justified experimentally. For the pions,
Cir°C~l = 7T° since 7r° —» 27 is observed, and C7T+C-1 = +7r~ in our phase
conventions. Similarly, the observed 77 —» 27 decay implies Cr]C~l = +77. The

Lie Groups, Lie Algebras, and Symmetries
133
G-parity then follows because the n and 77 are respectively an isotriplet and
isosinglet. The assignments also follow from the quark model. It is shown
in Problem 3.23 that any color-singlet meson \q1q2) with g12 = u or d has
q = (—l)L+5+/, where L, S, and / are respectively the total orbital angular
momentum, spin, and isospin. The tt and 77 have L = S = 0 so Gv = —Gn = 1.
Similarly, the vector mesons p° and uj are mainly -j=\uu^f dd), respectively,
(Section 5.8.3) in an L = 0, S = 1 state, so Gp = —G^ = 1 and the decays
n —» 27r and u) —» 3tt are allowed.
3.3 Symmetry Breaking and Realization
Symmetries of the equations of motion may be broken explicitly by small
terms in the equations themselves, or spontaneously in the solutions.
Simple quantum-mechanical analogs of many of the possibilities, including no
breaking, explicit breaking, spontaneous breaking, and combined explicit-
spontaneous breaking for discrete and continuous symmetries, are described
in Appendix I. More extensive discussions than the one here can be found
in (e.g., Lee, 1972; Pagels, 1975; Coleman, 1985).
3.3.1 A Single Hermitian Scalar
Consider the Lagrangian density
C=l-{d^f-V{4>) (3.131)
for a single Hermitian scalar, where the potential is
vw4 + T (3-132)
C has no continuous internal symmetries, but does have a discrete Z2
symmetry under </> —» —<\> (this would not be the case for the more general potential
in (3.128)). The equation of motion for <j> is
First consider the solutions to (3.133) for a classical field (j)ciass- The lowest
energy classical solution can be interpreted as the vacuum expectation value
(VEV) or the ground state value of </>, (0|</>|0) = (</>). This can be thought
of as a coherent state or Bose condensation effect (cf., classical solutions to
Maxwell's equations). Prom the expression for the Hamiltonian density in

134
The Standard Model and Beyond
(A.8), the lowest energy solution is for </>(t,x) = constant**, with a value (</>)
that minimizes the potential
dV_
dcj)
0,
d2V
(<t>)
d(f>2
>0,
(<t>)
(3.134)
where the first condition guarantees an extremum, and the second that the
solution is stable (a minimum). One must choose A > 0 so that V is bounded
from below. However, the sign of ji2 is arbitrary. The shape of V{<j>) is shown
in Figure 3.4 for both signs, /j2 > 0 is the familiar case. From (3.133) or
4>{x)
\ v(<t>)
-J- v*
FIGURE 3.4
Left: Potential in (3.132) for /i2 > 0 (dashed), fi2 < 0 (solid), or /i2 < 0
with an explicit symmetry breaking term —acj) (dotted). Right: Domain wall
solution of the classical field equation for </>.
Figure 3.4 the minimum occurs for (</>) = 0, i.e., the vacuum is just "empty
space." One can then quantize </> as in Chapter 2, /j, is the mass of the scalar
particle, and the Z<i symmetry is unbroken.
For /j2 < 0 there are three extrema, at <j> = 0 and at ±.v = ±^/—/j,2/X. The
extremum <j> = 0 is a maximum; it is unstable, since
d?$
dx2
-fi2(f) > 0.
The other two are minima, as can be seen by rewriting
A
4
v{<t>) = -A<t>2-v2)2-
XiA
4
(3.135)
(3.136)
**Any x dependence of the ground state would also violate translation invariance, while
any VEV for higher-spin fields would violate Lorentz invariance. However, as we will see
in the next section, higher energy classical solutions are merely excitations on the ground
state and can involve x dependence or higher spin bosons.

lie Groups, Lie Algebras, and Symmetries
135
Thus, there are two possible degenerate ground states, with (</>) = ±v. One
can quantize around whichever of them is (randomly) chosen, and the Z<i
symmetry (</> —> —<j>) is spontaneously broken. For the + solution, one can
write
</> = */ + </>', (3.137)
where (j)' is a normal quantum field with (</>') = 0. With this substitution,
£ (4>) = C {v + 0') = I (c^)2 - V (0') (3.138)
with
V (<£') = =£- - mV2 + Ai/0'3 + j./.'4. (3.139)
The first term is a constant, which is irrelevant until gravity is considered
(then it becomes a contribution to the cosmological constant). The second
term shows that the </>' field corresponds to a particle with (zeroth-order)
mass-squared ji^, = — 2/j,2 > 0. The third is a cubic self-interaction for <//
induced by the spontaneous symmetry breaking. It is due to the quartic <j>
interaction, with one of the legs replaced by (</>) = v, as seen in Figure 3.5. The
A X A X
v
FIGURE 3.5
Induced cubic and quartic self-interactions for <//.
induced cubic interaction is a manifestation that the discrete Z<i symmetry of
C is spontaneously broken by the ground state (classical) solution, and has
the effect that particle number modulo 2 is no longer conserved. The last term
is a quartic self-interaction that is unaffected by the symmetry breaking.
For the intermediate case /j,2 = 0 it is not sufficient to consider the theory
classically. One must extend the discussion to consider the effective potential,
which reduces to the potential at tree-level in an expansion in the number of
loops. The model considered here is difficult to study; the apparent minimum
is at nonzero v but occurs outside of the range of validity of the expansion.
However, in more realistic theories, such as the standard model, the symmetry
is spontaneously broken for /j? = 0 (Coleman and Weinberg, 1973). The
effective potential is extremely useful for incorporating higher-order effects

136
The Standard Model and Beyond
in the study of symmetry breaking and for considering field theory at finite
temperature (see, e.g., Weinberg, 1995; Quiros, 1999).
One can perturb the potential in (3.132) by linear or cubic terms that
explicitly break the Z2 symmetry, as in (3.128). For definiteness, we will
consider the linear tadpole operator — a</> with a > 0,
V(*) = ^-«*+^. (3.140)
An immediate consequence is that the Z2 symmetry must be violated in the
ground state as well, i.e., (</>) ^ 0. For /j2 > 0 the VEV is induced by the
explicit breaking, v = (</>) = a/fj2 + 0(a3). The most important consequence
is the presence of a small cubic term for the shifted field <// = </> — v,
W) = y<*>'2 + A^'3 + ^>'4, (3.141)
similar to the (larger) one in (3.139). For /j2 < 0 the potential is shifted, as
seen in the dotted curve in Figure 3.4. The deepest (global or true) minimum
is at
^ = ^0 + ^2 +0(a2), (3.142)
where vq = y/—/j,2/\ is the unperturbed minimum. For small enough a there
is another metastable local or false minimum for v < 0. Such metastable local
minima are frequently encountered in field theories, and in some cases it makes
sense to quantize around them rather than around the global minimum (e.g.,
Kusenko et al., 1996; Intriligator et al., 2006). The relevant issues for a realistic
theory with a metastable vacuum are: (a) Is the lifetime of the metastable
vacuum due to tunneling (Linde, 1983) long compared to the 1010 yr age of
the observed Universe? (a) Which vacuum would have been occupied initially
as the Universe cooled?
3.3.2 A Digression on Topological Defects
There are also more energetic classical solutions to (3.133). For example,
the static domain wall solution is an infinite wall perpendicular, e.g., to the
x direction, with </>(#) varying from </>(—oo) = — v to </>(+oo) = +v. This is
illustrated in Figure 3.4, where the center of the wall is at x = 0 and the wall is
parallel to the y and z directions. Energy is stored in the wall in the transition
region near x = 0. The thickness d of the transition region can be estimated
by minimizing the sum of the kinetic energy density ~ (v/d)2 = \/j,\2/\d2 and
the potential energy density (with respect to the minimum) ~ fi>4/^, leading
to d ~ 1/|mI- Thus, the energy density per unit area is ~ d/j,4/X ~ |m|3/^-
Since the wall is infinite in extent, it would take infinite energy to tunnel to
one of the ground states </>(#) = ±z/, so the wall is stable. (Such objects are
known as topological defects.)

Lie Groups, Lie Algebras, and Symmetries
137
This simple model illustrates a generic difficulty with spontaneously broken
discrete symmetries. Such walls would presumably have formed in the early
universe as it cooled from a temperature T much larger than \n\ because
causally disconnected regions would have fallen randomly into either of the
two minima, somewhat like the formation of ferromagnetic domains. Both
walls and anti-walls (making transitions from +v to —v) would have formed.
Most would presumably have been annihilated, but one would expect at least
one to survive in a volume V ~ R3 of the size of our observable universe,
contributing to the energy density and anisotropy of our universe. To get an
idea of the magnitude, let us assume the average energy per unit volume due
to a single domain wall is bounded by the observed average energy density
PtoU
^|~^<^~(3*,0-3e< (3.H3)
Of course, this underestimates the constraint since the observed energy density
is extremely isotropic, unlike a domain wall. Using R ~ 1.4 x 1010yr ~
4x 1017 sec, this yields \/j,\/\1/3 < 30 MeV. Discrete symmetries spontaneously
broken at a larger scale are therefore cosmologically dangerous^.
Spontaneous symmetry breaking in particle physics can lead to other defects
of possible cosmological relevance, which may involve classical values for gauge
fields as well as scalars. These include monopoles, which occur when non-
abelian symmetries are broken down to a subgroup containing a U(l) factor,
and cosmic strings, associated with U(l) symmetries (see Problem 4.1). In
another class are textures, which are not topologically stable but may be long-
lived. These matters are discussed in much more detail in (Coleman, 1985;
Vilenkin, 1985; Kolb and Turner, 1990).
3.3.3 A Complex Scalar: Explicit and Spontaneous
Symmetry Breaking
Consider a complex scalar </> with
Co = (d^ d»4> -V (</>), V (</>) = MV</> + A (<t>U)2 , (3.144)
where A > 0 so that V is bounded from below. As discussed in Section
2.4.1 Co is invariant under the U(l) phase transformations <j> —» el/3</>, with a
conserved charge corresponding to particle number. It is convenient to go to
a Hermitian basis, <f> = (</>x + z</>2)/\/2, as in (2.87), so that
Co = \[
^ (±1 , ±1\ i A / ,2 /2\2
(d^y + {d^2y\ -v(0x,02), v = !j (0? + <&)+- (0? + 4>iy
(3.145)
ftThey can be avoided if the reheating temperature of the universe after a period of inflation
is smaller than fi. Also, the addition of a small explicit symmetry breaking term K<f>3 to
(3.132) would eliminate the problem; the energy difference would lead to an attractive force
between domain walls and anti-walls and therefore more rapid annihilation.

138
The Standard Model and Beyond
As shown in (3.75), the U(l) transformation takes the form of a rotation** in
this basis
(In fact, U{\) and SO(2) are equivalent.) Again, the vacuum corresponds to
a classical solution of the field equations
(fe " ^) *« = - [/*2 + A fa? + ^)] <Pa (3-147)
with minimum energy. This implies a constant z/a = (</>a), with
= 0. (3.148)
dV
d(t>a
1/1,1/2
2
The minimum condition requires that the eigenvalues m2 2 of
/ d2v d2v
( d<j>2 dcpidfo
d2v d2v
d<t>\d<t>2 d<j>2
(3.149)
1/1,1/2
must be non-negative. (These are interpreted as the mass-squares of the
physical mass eigenstate particles when one expands around the minimum.)
For ji2 > 0, the minimum is at v\ = z/2 = 0, as seen in Figure 3.6. One
can quantize around this point, obtaining degenerate </>i?2 with mass-square
ji2 and the relations between the quartic couplings unbroken (or, equivalently,
degenerate <j> and <j>\ with a conserved particle number). Thus, there is an
unbroken symmetry in both the equations of motion and the ground state.
This is known as the Wigner- Weyl realization of the symmetry.
There are two ways in which a Lagrangian symmetry can be broken. One
is to add small explicit breaking terms, as we did for SU(3) in (3.112). For
example, with \j? > 0 and
C = C0- e-<j>l (3.150)
the U(l) ~ SO(2) symmetry would be broken, with nondegenerate masses
m2=fi2, ml=n2 + e, (3.151)
corresponding to the Hermitian mass eigenstates </>i and </>2- In this example,
the quartic relations are not modified at tree-level, though there would be
finite corrections induced at loop-level*. However, there would no longer be a
conserved particle number (Problem 3.25).
^The full symmetry of Co is 0(2). The extra 4>1 —> —<t>2 reflection symmetry corresponds
to cp <-► 0t in the complex basis.
*They are finite because the symmetry breaking is soft, i.e., the coefficient has a positive
power in mass units.

lie Groups, Lie Algebras, and Symmetries 139
v(*>l v(<t>)
s^
02
01
02
01
FIGURE 3.6
Left: The potential in (3.145) for /j,2 > 0. Right: Potential for /j2 < 0.
The other possibility is spontaneous symmetry breaking (SSB), also known
as the Nambu-Goldstone realization of the symmetry (Nambu, 1960; Nambu
and Jona-Lasinio, 1961; Goldstone, 1961). For fj2 < 0 the Mexican hat
potential in (3.145) has its minima away from the origin, as seen in Figure 3.6.
The rotational symmetry leads to a circle of degenerate minima at
-a2
(j)\ + (j)l = ^ = -^->^ (3.152)
A
as illustrated in Figure 3.7. The true ground state must pick a specific point on
this circle, spontaneously breaking the rotational symmetry in much the same
way that the spins in a ferromagnetic domain line up in a specific direction.
Since the classical field carries a U{\) charge, the symmetry is broken and the
charge is not conserved.
Another consequence of the symmetry breaking is the existence of a massless
Nambu-Goldstone boson The Nambu-Goldstone theorem states that for every
spontaneously broken generator of a continuous global symmetry there exists
a massless spin-0 particle, the Goldstone boson.
Let us demonstrate these statements in this example. Without loss of
generality, one can choose axes in the 01,2 plane so that the ground state is at
v\ = v and V2 = 0. (Equivalently, for any choice of ground state one can make
an SO(2) rotation so that this holds for the rotated fields.) Then, define
01=^ + 0'l, 02=02, (3.153)
where (0^) = 0. One can quantize 0^ in the usual way. In terms of 0^,
£ = \ (d„4>\f + \ (d^'2f - v (#, &)
1 4 x (3-154)

The Standard Model and Beyond
</>2
/ 01
FIGURE 3.7
Top view of V{<j>) for /j? < 0. The origin is a local maximum (unstable). The
points on the dashed circle with radius v are degenerate minima.
Similar to the Hermitian field example, the first term is a constant, the second
implies that $[ has mass-squared m\ = —2/j? > 0, the third is an induced
cubic interaction which implies there is no conserved charge, and the last is
the quartic interaction. There is no mass term for </>2, i.e., it is a massless
Goldstone boson. The interpretation is that the potential is flat as one moves
from the minimum in the (j)'2 direction, so that excitations from the ground
state are massless "rolling" modes rather than massive oscillations.
It is possible to combine explicit and spontaneous symmetry breaking.
Suppose there is a small explicit breaking term as in (3.150), with ji2 < 0 and
0 < e <C \/j,\2. This tilts the potential in Figure 3.6 so that there is a minimum
at (3.153) (which is unique except for </>i —» — z/), with
mi
-2M2
ml
e <C m{.
(3.155)
Thus, the Goldstone boson acquires a small mass from the explicit breaking
(it is known as a pseudo-Goldstone boson).
3.3.4 Spontaneously Broken Chiral Symmetry
Let us extend the discussion of spontaneous symmetry breaking to a chiral
symmetry. Consider a single chiral fermion ip = ipL + tpR, and a complex
scalar </>, with
C = ^Li w>l + <M W>r - HlM - h*ML<t>i + (d^y &></> - V (0)
\t
with V (</>) = mV</> + A {^(j))2 ,
(3.156)
where A > 0. Without loss of generality we can take the Yukawa coupling
h to be real and positive (if necessary by absorbing any phase into ipL —►

lie Groups, Lie Algebras, and Symmetries
141
ei(arg/*)^L7 which is not a symmetry). C has a chiral U{\) symmetry t
0_e^, ^L-^Ll tltR^e-Wt/tR, (3.157)
which forbids elementary fermion mass terms. For fj,2 < 0 the symmetry is
spontaneously broken. The scalar part of C can be rewritten in terms of the
shifted fields defined in (3.153), just as in (3.154). The Yukawa terms become
CYuk = ~^W ( 1 + ^ V -^75M, (3-158)
hv - ( #\ h ,T
—7=^ 1 + — p^
where we have used ipLipR + V^l = ipip and Vl^j? - V^l = ipj5tp' Thus,
-0 acquires an effective mass
hv
m^ = —= (3.159)
V2
from the spontaneous breaking of the chiral symmetry. The Hermitian scalars
(/)[ 2 couple as a scalar and pseudoscalar, respectively, to ^, with a Yukawa
coupling h/y/2 = m^jv.
3.3.5 Field Redefinition
In field redefinition one introduces new fields as functions of the original ones.
We saw simple examples in Section 2.10 involving the change in the overall
phases of complex scalar or fermion fields. In some cases, such phase rotations
correspond to U{\) symmetries. However, they are also useful in the absence
of a symmetry for removing or changing the phases of the constants
appearing in C. Field redefinitions were also encountered in Section 2.13 where they
were used to put kinetic energy and mass terms in canonical form. Yet
another example was seen in the model of a spontaneously broken chiral U(l)
symmetry described in Section 3.3.4, which can be described in terms of the
shifted fields </>[ 2 with interactions given in (3.154) and (3.158).
It is sometimes convenient to rewrite the theory in terms of new fields which
are more complicated functions of the original ones. Such field redefinitions
can lead to theories which look very different from the original ones, though for
a wide class of cases they are equivalent in the sense that they have the same
on-shell S (transition) matrices (Haag, 1958; Coleman et al., 1969; Callan
et al., 1969). In the case of SSB, instead of the rectangular Hermitian basis
in (3.145), one can define the fields in a polar basis, i.e.,
0=(^)c*/", (3.160)
V2
trThe individual chiral transformations of the tpL and ^>R are somewhat arbitrary. All that
matters is that 'ipL'ipR —> e~t(3'ipL'ipR- Changing the prescription is equivalent to adding
terms proportional to the unbroken fermion number generator to the chiral charges.

142
The Standard Model and Beyond
where 77 and £ are Hermitian fields. 77 is invariant under a E/(l) transformation
while £ is shifted
V^V, £^£ + /^- (3.161)
The <j> part of C becomes
i.„ ,2 1
£*=o(W+o(0mO2 1 + 7 -^)
2rM" 2
*?
i/
^) = ^-^V + A^3 + ^4,
(3.162)
i.e, £ drops out of the potential and is therefore massless, illustrating that
the Goldstone boson is just the phase of </>. The self-interaction terms in V
for 77 are the same as those for <\)'x in (3.154), and m^ = — 2/j,2. 77 and £ both
have canonical kinetic energy terms (the \/v normalization of the phase in
(3.160) was chosen for that purpose), but there are now derivative couplings
of (dM£)2 to T] and r]2 with coupling strength 1/z/, which enter through the
</> kinetic energy terms. These replace the (j)'2 interactions in (3.154). The
Yukawa interactions become
CYuk = ~^Mr (l + I) e^/v + h.c, (3.163)
i.e., there are interactions with £ to all orders. However, this too can be
simplified by a redefinition
Vr = e^Vfl. (3-164)
£ disappears from £yufc, but reemerges as a derivative coupling to the right-
chiral fermions that enters from ipni ftipR. Altogether, £ = £/ + £</>, where
Cf = $Li W>l + $Ri W>R + ^f^Rl^R ~ ^Mr (l + I) • (3.165)
The prime on ^ has been dropped. It is straightforward to show that the
derivative couplings oid^/v in (3.162) and (3.165) are to the Noether current
associated with the U(l).
The reformulation of the model in the polar basis is not manifestly renor-
malizable, due to the derivative couplings and/or the higher-order couplings
in (3.163), and for considering high energy or higher-order calculations the
original rectangular basis is more useful. However, the new variables define
an effective theory that is very useful at low energies. The physical
particle content and nature of the Goldstone boson are transparent. Also, all of
the £ interactions involve derivatives, which translate into factors of the £
four-momentum p^ in physical amplitudes. Therefore, such amplitudes must
vanish for p^ —» 0. This finds application in the soft pion theorems in the
more realistic case of QCD. Even for p% ^ 0, the formulation is useful for
small external momenta compared with z/, where the interactions are small.
A similar field redefinition is useful for displaying the physical particle content
in spontaneously broken gauge theories (the Higgs mechanism). Calculations
in the various formulations are compared in Problem 3.26.

Lie Groups, Lie Algebras, and Symmetries
143
3.3.6 The Nambu-Goldstone Theorem
As we saw in an example in Section 3.3.3, there are two possible realizations
of an exact continuous symmetry of the Lagrangian (the Goldstone
alternative). In the Wigner-Weyl realization, the ground state also respects the
symmetry. Particles fall into degenerate multiplets with relations between their
couplings, and for a chiral symmetry the chiral fermions are massless. In
the Nambu-Goldstone realization, the symmetry is not respected by the
vacuum, massless Nambu-Goldstone bosons appear, and chiral fermions acquire
effective masses. The converse statement is Coleman's theorem (Coleman,
1966), i.e., that a continuous symmetry of the vacuum is also a symmetry of
the Lagrangian. The various possibilities for the realization and breaking of
continuous symmetries are summarized in Table 3.5.
TABLE 3.5
Possibilities for the realization and breaking of a continuous symmetry.
Exact Lagrangian Symmetry ([Ug, £] = 0)
~UG\0) = \0)
exact symmetry
(Wigner-Weyl)
degenerate multiplets
conserved charges
relations between interactions
chiral: massless fermions
gauge: massless gauge bosons
Ug\0) jt |0>
spontaneous symmetry breaking
(Nambu-Goldstone)
chiral: fermions acquire mass
global: Goldstone bosons
gauge: gauge bosons acquire mass
by Higgs or dynamical mechanism
Explicit breaking ([Ug, £\ j^ 0) (global onlyj
multiplet splitting, etc.
chiral: fermions acquire mass
multiplet splitting, etc.
Goldstone bosons acquire mass
Let us extend the proof of the Nambu-Goldstone theorem at tree level to
an arbitrary scalar sector, which can always be represented by n Hermitian
scalars </>a, a = 1 • • -n. One can write the n fields as a column vector 0 =
(0i </>2 • •-</>n)T, and their possible VEVs by a column vector v = (0) =
{y\ vi • • ■vn)T. Some or all of the va may be zero. The Lagrangian density
consists of the kinetic energy terms for 0, a potential V(</>), and possible terms
involving fermions. The most general renormalizable potential is the quartic
polynomial
V(<j>) = Vq - Ga 0a + -\lah(\>a <t>b + Kabc(t>a 06 0c + Kbcd<f>a 06 0c 0d- (3.166)

144
The Standard Model and Beyond
The coefficients cra, n2ah, Kabc, and Xabcd are real and symmetric in the indices.
v is determined by minimizing the potential, i.e., {dVId<i>a)\v = 0, as in
(3.148). Define the shifted fields <//a = </>a — va with (<//a) = 0 and rewrite V
in terms of <//,
V((j)') = Vb + XAod^o 06 + «o6c0o 06 0c + ^abcdfia 06 0c 0d> (3.167)
where
1 / d3V
.2 = / d2F
1 / d4V
Aabcd
Kabc —
4! \d(j)ad(j)bd(j)cd(t)d
3! \dcj)ad(t>bd(t>c
^abcd-
(3.168)
There is no linear term in V(<//) since we are at a minimum. The eigenvalues
of the mass-squared matrix ji2ah are the physical mass-squares, and the
eigenvectors are in the directions of the mass eigenstate fields. The eigenvalues are
guaranteed to be non-negative by the definition of a minimum.
Now consider the role of a continuous internal symmetry, with generator
representation matrices L\ Under an infinitesimal transformation (and
returning to the original fields)
V(4,)^V(<t,) + 6V(4>), (3.169)
where
^ = I^° = ^(^)>- (3-17°)
Invariance under the transformation requires SV((/>) = 0, and therefore
|^a606 = O. (3.171)
Differentiating (3.171) with respect to </>c and evaluating at </> = z/, one has
A*6(LV)6 = 0, i = l---N. (3.172)
Let us label the generators so that
LV = 0, i = l...M, iV^O, z = M + l---7V. (3.173)
The subgroup G' G G generated by T1 • • • TM leaves the vacuum invariant,
Tl\0) = 0, so G' is unbroken*. However, symmetries associated with the
remaining N—M generators are spontaneously broken, Tl\0) ^ 0, as can easily
*The generators in (3.173) may be linear combinations of the original ones. The precise
meaning is that there are M linearly independent generators for which LV = 0 and N — M
for which Lxv ^ 0.

Lie Groups, Lie Algebras, and Symmetries
145
be seen from (3.29). (M = 0 [G completely broken], and M = N [no breaking]
are special cases.) Prom (3.172) and (3.173) the scalar mass-square matrix
has N - M linearly independent eigenvectors LV with eigenvalue zero. Thus,
there are N—M massless Goldstone bosons, one for each spontaneously broken
generator. \j?ah also has p = n - (N - M) (generally) non-zero eigenvalues,
corresponding to p (generally) massive scalar particles.
The Nambu-Goldstone theorem holds quite generally§. It can be proved to
all orders using the effective potential. It even holds non-perturbatively (Gold-
stone et al., 1962; Weinberg, 1995), as will be shown in an example in Section
5.8. Goldstone bosons do not appear to exist in nature. However, pseudo-
Goldstone bosons, which acquire a small mass from explicit breaking, are
relevant to the chiral flavor symmetries of the strong interactions. Furthermore,
as will be seen in Chapter 4, for a gauge symmetry the Goldstone bosons are
reinterpreted as the longitudinal modes of massive gauge bosons (the Higgs
mechanism).
3.3.7 Boundedness of the Potential
In order to have a sensible theory it is necessary that there be a lowest energy
state, i.e., that the vacuum is stable. This requires that the full effective
potential, including loop effects, running couplings, and possible non-renormalizable
contributions, is bounded from below. However, let us consider the simpler
question of whether the renormalizable tree-level potential V in (3.166) is
bounded from below. V is well-behaved for finite </>, so all we have to do is
make sure that it is bounded from below as </> —» oo for all orientations of the
n-dimensional vector </>. It suffices to consider the case <j>(x) = constant, and
since we are concerned with the large <j> behavior we can ignore spontaneous
symmetry breaking. An arbitrary (constant) <j> can be written^
(f>a = rea, (3.174)
where r = (%2a 0a)1/2 is a radius vector and e is an n-dimensional unit vector
with components ea. One requires
lim V(r,e) =c(e), (3.175)
where the limit c(e) is either a finite number or +oo for all orientations e. If
V —» —oo for any e there is no well-defined ground state and the theory is
unstable.
In most cases the quartic terms in V determine the asymptotic behavior.
If C(e) = Xabcd eaebeced > 0 for all unit vectors e then the theory is bounded,
while if C(e) < 0 for any e the potential is unbounded. If C(e) = 0 for
§An exception, involving the axial C/(l) generator, will be described in Section 5.8.3.
^This form is also often useful for finding the minimum of V.

146
The Standard Model and Beyond
some directions e then one must investigate the other terms in V along those
directions.
It sometimes happens that there are some directions for which V(r, e) is
independent of r, i.e., the quadratic, cubic, and quartic terms vanish so that
V(r, e) = Vq. This is especially common in supersymmetric theories (which
have Vb = 0). If such flat directions correspond to the minimum of V the
ground state is not uniquely defined at the renormalizable tree-level. In some
cases, loop corrections to the effective potential, higher-dimensional (non-
renormalizable) terms, or soft supersymmetry breaking terms lift the flatness
to yield a unique minimum.
As an example, consider the potential
V(4>i, 4>2) = 2^1 4>1 + 2^2 <t>2 + A2 <t>l (3.176)
with A2 > 0. One has
+={£)=r&> (3-m)
where r = (4>\ + cfy)1/2 and e = (cos# sin#)T is a two-dimensional unit
vector. The quartic coefficient is thus C(e) = A2 sin4 0, which is positive for
all directions except sin# = 0. In that case (</>i = ±r, </>2 = 0) the potential
is bounded below, flat, or unbounded for n\ > 0,0, or < 0, respectively.
3.3.8 Example: Two Complex Scalars
As a major example of some of these considerations, let
= —7/2—' ~ —7/2— (0.I/8)
be two complex scalars with Hermitian components 0^, i = 1 • • • 4, and La-
grangian density
C = \dM2 + \d»<t>n\2 -V(<t>i,<t>n)
V = 3\<t>i\2 + ^/|0//|2 + A/I0/I4 + A//I0//I4 + \ni\<t>i\2\<t>ii\2 (3.179)
-Afatn-A'tWir
For V to be bounded below it is sufficient to require that the quartic terms
are positive for all values of </>j, J = /, //, which holds provided
A/,// > 0, A/// > -2A)/2A)f. (3.180)
(The limiting case A7 = A77 = —A777/2, which occurs in supersymmetry,
is considered in Problem 3.31.) \j?j are real, and A is an arbitrary complex
number. C has an internal U{\) symmetry, </>j —» eiqj(3(j)j with qj = —qjj = 1
(which is promoted to U(l) x U(l) for A = 0). W.l.o.g. one can redefine the

Tie Groups, Lie Algebras, and Symmetries
147
phases of </>/,// to make A real and non-negative. Then, if the parameters are
such that the </>j both acquire VEVs, the minimum will occur for (<£/)(</>//) real
and positive. We will assume that the individual VEVs are real and positive,
i.e, ^1,3 = (0i,3) > 0 and ^2,4 = (02,4) = 0. The U(l) symmetry implies that
there are degenerate minima that differ from this by U{\) transformations.
In the Hermitian basis with A real and nonnegative, the last term in V is
Va = -A(</>i</>3 - 0204),
(3.181)
while the other terms depend only on <j>\ + </>% or §\ + <j>\. The nontrivial
extremum conditions for z/2,4 = 0 are
dv
dfa
dv
d<j)3
V
V
l4i + A//1/3 + ■zXiii^l
v\ - Avz = 0
z/3 — Az/i = 0.
(3.182)
These have the trivial solution v\ = z/3 = 0, and possible nontrivial solutions
with v\ ^ 0, ^3 ^ 0. The mass-square matrix for </>i,3 (there is no mixing with
02 4 for either case) corresponding to the trivial solution is
d2v
d2v
„2 = Wf
Vr - I d2v
d2v
^1,3=0
A
A M2J-
(3.183)
This is a minimum provided the two eigenvalues \j?a b are both non-negative.
The trace and determinant are preserved by the unitary transformation which
diagonalizes //^,
Ma + Mb = M/ + M//,
2 2
2 2
(3.184)
so the origin is a minimum provided both of these are positive, in which
case there is no spontaneous symmetry breaking". Otherwise, the origin is a
saddlepoint with one negative eigenvalue (for M/M// ~ ^ < 0), or a maximum
with two negative eigenvalues (for //f + M// < 0 and /J^fijj — A2 > 0). In these
cases, there will be SSB. v\$ can be found numerically from (3.182), and the
mass-square matrix for </>i53 can be written
_ (2\Ivl+Avz/vl
X111V1V3 - A
2\IIv$+Av1/v3J>
(3.185)
where \x\ 7/ have been eliminated using (3.182).
More interesting is the mass-square matrix [x\ for 02,4- For ^1,3 = 0, \s?j is
the same as (3.183) except A —» —A. This has the same eigenvalues as \j?r,
There are no other minima provided (3.180) is satisfied.

148
The Standard Model and Beyond
and the symmetry is unbroken. For the SSB case, one can again use (3.182)
to obtain
/ d2v d2v
Pi
d<j>2d<j>4
= A
^1,3 =
"2. 1
1 ^
^3
(3.186)
This has a zero eigenvalue, corresponding to the Goldstone boson of the broken
U(l). The unnormalized eigenvector is (vi —v^)T. This is in the Lv direction
as expected from (3.173), since in the Hermitian basis,
L =
0
(3.187)
where the 0 in L is the 2 x 2 zero matrix, in analogy with (3.146). The other
eigenvalue is A(jf- + j^), with (unnormalized) eigenvector (i/3 v\)T• For
A —» 0 this state also becomes massless, i.e., both generators of the enhanced
U(l) x U(l) are broken and there are two Goldstone bosons.
An alternative derivation of these results utilizes a polar basis, analogous
to (3.160), i.e.,
0J=(iO+^)c^/,Jf (3J88)
v2
where vi = vi, vjj = z/3, and where 77J and £j are Hermitian. The
normalization of the exponents is chosen so that the kinetic energy terms
Cke = £ \e^j\2 = \Y, (W)2 + (°»h)2
(3.189)
are canonical for real fields, although there are new three- and four-point
interactions involving derivatives that we will not be concerned with here.
Under a U(l) transformation the £j are shifted while the rjj are unchanged,
£/,// -> iiji ± vijiP-
The £j only enter the Va part of the potential
VA = -Airj! + i//)(ry// + 1///) cos ( — + —
\Vl VlI J
(3.190)
(3.191)
which has minima for €f- + &- = 0. We will consider the minimum at £j = 0,
with the other values related by U(l) transformations. The calculation of the
VEVs and the mass matrix for the rjj proceeds as before. The mass matrix
for the £j is obtained by expanding Va to second order,
VA
Avxvz
k + ki
V
Yl + Yl
(y& + vita)
"1 + vi
(3.192)

Lie Groups, Lie Algebras, and Symmetries 149
This reproduces our previous result, with one Goldstone boson and one
massive state. Approximating </>j ~ (t7j+^+^j), the eigenvectors and eigenvalues
are the same as found previously.
3.4 Problems
3.1 Verify the Lie group multiplication rule in (3.6).
3.2 Show that with the normalization (3.8) the structure constants c^fc are
antisymmetric in all three indices.
3.3 Prove the Jacobi identity
[IS, [LP, Lk}} + \V, [Lk, IS]] + [Lk, [IS, LP]] = 0
for an arbitrary representation and use it to prove that the adjoint
representation matrices in (3.16) satisfy the commutation rules in (3.7).
3.4 Use (3.10) to find an explicit expression for 7(5,/?) defined by the
multiplication in (3.5) for the group SU(2).
3.5 Prove the formal power series identity
eABe~A = B + [A,B] + ± [A, [A, B]] + ^ [A, [A, [A, B]]] + ■■■,
where A and B are any two operators or square matrices of the same
dimension.
3.6 Prove that det[eaA] = J2k(aA)k/kl = eaTr A for any Hermitian matrix
A. This implies that the condition for a unitary or orthogonal matrix to be
special (unit determinant) is that the generators be traceless.
3.7 Prove the transformation law (3.32) for a field transforming according
to the adjoint representation.
(W-M2^]-A(<^)2,
3.8 Show that
with </>a = </>t is SO(m) invariant, with </>a, a = 1 • • -m transforming as the
vector representation.
3.9 Derive the Noether currents in (3.54) and (3.79) for complex scalars
and fermions, respectively. In each case, use the canonical commutation rules
in Appendix A to show that the associated Noether charges satisfy the Lie
algebra commutation rules. Assume there are no interaction terms involving
field derivatives.

150
The Standard Model and Beyond
3.10 In the free-field limit the isospin generators for the pion-nucleon fields
in (3.82) can be written as
Ti = J (Wpk E [b± (* -) (f)bb <*s)+d< <*s) ("f)db <* «>
/13 -♦
J-^a){p){-ieijk)ak{p)
where ba and da are respectively the annihilation operators for nucleon a =
porn and for its antiparticle, and ai is the annihilation operator for 7r\
Show that the T% satisfy the SU(2) Lie algebra and that they have the right
commutation rules with the fields.
3.11 Show that the Yukawa interaction ipi^rip • 7? in (3.84) is SU(2)
invariant. Use the representation matrices in (3.83).
3.12 Show that the phase convention (analogous to that used for rotations)
(L1 ± iL2)\Im) = y/{I^m){I±m + l)\Im± 1)
for the states in an isospin / multiplet with 1% eigenvalue m would require
7r± = =p (tti =F i7r2) /V2 instead of (3.82).
3.13 The pion and nucleon transform as isospin triplets and doublets,
respectively. The approximate isospin invariance of the strong interactions implies
that all of the amplitudes for 7TiNa —» ttjN^ can be expressed in terms of
two amplitudes M/, corresponding to total isospin 1 = 1/2 and 3/2. Write
the amplitudes for n+p —» 7r+p, n~p —> n~p, n~p —> 7r°n, and 7r°n —> 7r°n
in terms of the M/. (Note that at low energy the M3/2 amplitude, which is
dominated by the A resonance, strongly dominates. Such calculations can
be extended to SU(3) by means of isoscalar factors (de Swart, 1963; Amsler
et al., 2008).)
3.14 It is sometimes useful to introduce a spurion, which is a fictitious or
composite field with the right quantum numbers to parametrize the effects of
symmetry breaking, and which can therefore be used as a bookkeeping device.
For example, it is known that the effective operator for strangeness-changing
nonleptonic weak transitions, which could in principle involve both AI = \
and AI = § components, is strongly dominated by AI = \. Therefore, the
transitions X4" —» 7r°p, X4" —» 7r+n, and X~ —» 7r~n should be related in the
same way as the isospin-conserving amplitudes for £+S —»7r°p, X+S —» 7r+n,
and YrS —> 7r~n, where 5 is a spin-0 spurion carrying / = 1/2 and ^3 = —1/2.
Show that the amplitudes should be related by
M(X+ -> 7r+n) - M(X" -> 7r"n) = \/2M(X+ -> 7r°p).

Tie Groups, Lie Algebras, and Symmetries
151
This relation, which holds separately for the S and P wave amplitudes
(Problem 2.19), works reasonably well. (This simple example could be handled
easily using the Wigner-Eckart theorem, but the spurion formalism is more
convenient in more complicated cases.) Hint: be careful of the SU(2) phase
conventions as discussed below (3.82).
3.15 Generalize the pion self-interaction term in (3.84) to the case of four
distinct types of pions, ttai = A^-kbx = D^-Kci = C^-Kui = A, each
transforming as an adjoint under SU(2). In general, there are cubic and quartic
interactions such as
Ci = -CijkiAiBjCkDi - dijkAiBjCk-
(There are also other possible terms, such as A2BC or ABD.) However, £j
is only SU(2) invariant for the combinations
d = - caTr (AB)Tr (CD) - C(3Tr (AC)Tr (BD)
- c7Tr (AD)Tr (BC) - dTr (ABC)
where A = A • f/>/2, etc., so that Tr (AB) = A- B. (If A, J5, and C are all
pseudoscalars then reflection invariance would require d = 0.)
(a) Show explicitly that invariants like Tr (ABCD) are not independent.
(b) The existence of only 3 quartic invariants can be justified as follows:
A,B,C, and D are all isospin-1 operators. Thus, the product AB is a
direct sum of operators O1^ with isospin Iab = 0? 1? or 2, and similarly for
CD. However, the only overall singlets involve the pairing with Iab = Icd-
Applying similar reasoning to the cubic term,
2
Li = - Y. vOabOcd ~ di<D\BC,
1=0
where the coefficients c\ and d\ are arbitrary. Construct these invariants
explicitly, and relate the coefficients to ca^j7 and d.
(c) Show why there is only a single invariant in (3.84).
3.16 The effective (non-renormalizable) interaction of a hypothetical doublet
of mesons A with the nucleons is assumed to take the isospin-invariant form
Can = \($t^)(A]tA),
where
and t1 are the Pauli matrices. Calculate the tree-level spin-averaged CM
cross sections for A+p —» A+p, A°p —» A°p, and A+n —> A°p. Assume that
mP = ™>n and mA+ = mAo.

152
The Standard Model and Beyond
3.17 By comparing the SU(2) and SU(3) expressions in (3.84) and (3.92),
show that gn = \{gf + go).
3.18 Let Aj and Bk transform as SU{3) octets. Prove that Fi = ifijkAjBk
and Di = dijkAjBk also transform as octets. Note that the result for F{
generalizes to any simple Lie algebra, while that for Di generalizes to SU(m).
3.19 Starting from the ansatz in (3.97), derive the Gell-Mann-Okubo mass
relation (3.104) for baryons.
3.20 Prom (3.89), the electromagnetic current can be written as Jq^ =
J3 + -4? J^, where J3 and J® are elements of an SU(3) octet of currents.
Jq^ is a singlet under the £/-spin subgroup of 5C/(3), which has generators
U3 = -§T3 + yj\T8, U± = T6 ± iT7. (a) Use 17-spin to show that in the
SU(3) limit the hyperon magnetic moments are related by
1 3
ME+ = Mp> Mn = ME0 = -^ME0 + 2^A
ME" = ME" , MEA = -y (ME° - MA) ,
where /xea is the transition moment observed in E° —> A7.
(b) Use isospin to show that fi^o = \ (me+ + Me- )•
(c) Use SU(3) to show that /x^o = —Ma, allowing all of the magnetic moments
to be expressed in terms of \xv and jin. Hint: use the analog of (3.101).
3.21 Prove that the operator eoci'"OCrnipctl...ctrn in (3.119) is invariant under
SU(m) but not under U(m).
3.22 Let </>a6, a, b = 1 • • • n be a complex scalar transforming as (n, n*) under
SU(n)L x Stf(n)fl, i.e.,
[Tl,<f>ab] = -(Lln)ac(t>cb, [TR,4>ab] = +</>ac(£n)c6>
where TlL and T^ are respectively the generators of SU(ti)l and SU(ti)r.
(a) Calculate a finite transformation of </>.
(b) Show that the Yukawa interaction
CY = h^aL(t)ab^bR + A*^6fl(0o6)VoL = ^L<\>^R + h* $ R<fJ l/l L
and the mass term
An = -M2(0a6)f0a6 = V^ (</>f</>)
are invariant under U{u)l x U(ti)r. In the second forms, </> represents an
n x n matrix with components </>ab, and (<ft)ba = (<l>ab)^-

lie Groups, Lie Algebras, and Symmetries
153
(c) Construct the most general U(ti)l x U(ti)r invariant renormalizable
Lagrangian density for ip and </>.
(d) Display the U(ti)l x U(ti)r Noether currents for the Lagrangian density
in (c). Hint: both <j> and <ft must be included in the sum over fields in (3.46).
3.23 Prove that the color-singlet states \ud), \du), and \uu±dd) are eigen-
states of G parity with eigenvalue (—l)L+5+/, where L, S, and / are
respectively the total orbital angular momentum, spin, and isospin.
3.24 Find the exact domain wall solution to (3.133) corresponding to
Figure 3.4 and use this to justify that the wall thickness is 0(|/x|_1). Use the
Hamiltonian density in (A.8) to compute the energy density (with respect to
the ground state) at a distance x from the wall, and compute the energy per
unit area in the wall. Hint: look for a solution of the form <f>(x) = atanh(6x).
3.25 The 0(2) invariance of (3.145) is explicitly broken by the mass term in
(3.150).
(a) Identify any unbroken symmetries and describe their consequences.
(b) Show that charge is not conserved and draw a Feynman diagram that
demonstrates this, e.g., by allowing the reaction </>+</>~ —» </>+</>+</>+</>~, where
0+ = (</>~)t is the particle created by <ft. Treat the symmetry breaking term
as a mass insertion on an internal line. (There can also be mixing associated
with the external legs, leading to processes such as </>+</>~ —» </>+</>+. However,
the degeneracy of the states for e —» 0 makes this case more complicated to
treat. It is very much like K° — K° mixing treated in Chapter 7.) Note that
this is a spin-0 analog of a Majorana neutrino mass term.
3.26 Consider the spontaneously broken theory of a complex scalar field in
Section 3.3.3, which is described after symmetry breaking by the Lagrangian
density in (3.154) for a rectangular basis, or by (3.162) in a polar basis.
(a) Show that the on-shell amplitudes for <t>i(pi)<t>2(P2) —> <l>i(P3)<l>2(P^) and
l(Pi)€(P2) —> v(P3)^(P4) are the same at tree-level.
(b) Now consider pseudoscalar-fermion scattering ^2(^1)^(^2) —> ^(P^iPiP*)-
The Yukawa interactions are given for three parametrizations in (3.158),
(3.163), and (3.165), and the corresponding scalar interactions in (3.154)
and (3.162). Show that the tree-level amplitudes are the same in all three
parametrizations.
3.27 In theories involving scalars and fermions it is sometimes the case that
each mass eigenstate scalar field couples either as a scalar or a pseudoscalar
to fermions, i.e., that there is a conserved parity. The SM and the MSSM at
tree level are examples of this. As a simpler example, consider a single Dirac
fermion i\) and a single complex scalar </>, with
C =$i ft) + {d^(j))] (<9M</>) - m^L^R - ttC^r^l

154
The Standard Model and Beyond
where fj? is real, while m, h, and k are complex. We ignore 0 self-interactions
for simplicity, and choose /j,2 > 2\k\ > 0 to ensure that 0 does not acquire
a VEV. Consider the cases (a) m = k = 0, (b) m = 0, k ^ 0, (c) and
m/0,K/0, always with h ^ 0, \j? ^ 0. In each case, find the spectrum,
interpret it in terms of the symmetries of £, and determine whether P, C
and CP are conserved. Hint: identify which of the phases in
^l - e^L, 1>R - e^R, 0 - e*Q0
are symmetries and which are field redefinitions.
3.28 The most general fermion number conserving Lagrangian density for a
free Dirac field is
C = tpi fl[a + fry5]^ - tp[c + rfd}ip,
where Hermiticity requires a,b,c, and d to be real. The canonical form in
(2.148), which corresponds to a = 1, c = m, and b = d = 0, is needed
to maintain the classical relation E2 = p2 + m2. However, the more general
form is acceptable (and may emerge from an underlying theory) provided that
C can be put into canonical form by a field redefinition prior to quantization.
Show that for a > \b\ > 0 (and all of the parameters real), one can find a
multiplicative redefinition, ip' = Tip, with
T(a,b,c,d) = Ti(a,b,c,d) + Ts(a,b, c, GQ75,
such that C{tp') is canonical, i.e., C = i//(z 0 — ra)^7, with m > 0. Construct
.T7 explicitly, and find m(a,b, c, d). Specialize your results to the "wrong sign
mass" case, a = 1, c = — |ra|, 6 = d = 0, and interpret the result. Hint: write
tp in terms of it chiral components, ip = ipL+ ipR = PliP + Pr^-, where it will
be seen that d corresponds to a complex mass.
3.29 The potential corresponding to (3.129) can be written for An = A22 as
which is clearly 0(2) invariant for A7 = 0, but only invariant under sign
changes and 0i,2 interchange for A7 ^ 0. Assume that /j2 < 0 and A > 0. Find
the minimum, the physical particle masses, and the unbroken symmetries for
the three cases A7 > 0, A7 = 0, and A7 < 0. In the last case impose the
necessary boundedness condition.
3.30 Consider the potential
V(01, 02,<fe) = y(0? + 02+03) +Ml2 ^2+^13^3
+ M23 0203 + A12 (0? - 0l)2 + A3 <j>\.
Find the conditions for boundedness (assume A12 ^ 0, A3 ^ 0), and the special
cases for which there are flat directions.

Lie Groups, Lie Algebras, and Symmetries
155
3.31 Specialize the example in Section 3.3.8 to the case Xj = \n = —Xm/2 =
X > 0, for which the boundedness condition (3.180) does not hold. This is a
simplified version of the two Higgs doublets in the MSSM, in which the U(l)
is a gauge symmetry.
(a) Find the condition on /x2/7, A, and A for the existence of a minimum of
V. Assume that A is real and non-negative.
(b) Show that A > 0 is a necessary condition to have both i/i^0 and z/3 ^ 0.
(c) For A > 0, what is the condition that the origin (^1,3 = 0) is not a
minimum? If that is the case, is the origin a maximum or a saddlepoint?
(d) Find explicit expressions for v2 =v\ + v\ and 7 = tan_1(z/3/z/i) in terms
of///,//, A and A-
(e) Calculate the nonzero eigenvalue ji\ of \x\ in terms of A and 7.
(f) Express the mass-square matrix /j?r in terms of \j?a ,7 and M| = 2Xu2
(motivated by the MSSM prototype).
(g) Prove the sum rule Ma + Mb = Ma + ^I> where n2a h are the n2R eigenvalues.
(h) Prove that y?a < M§ cos2 27 and /j,2 > /j,\+Mz sin2 27, where the labelling
is such that \j?a < fj,2,
(i) Show that these inequalities are saturated in the large ji2A limit, and
interpret this limit.
3.32 This problem involves a chiral fermion and complex scalar, with an
internal global symmetry that may be both spontaneously and explicitly broken.
Consider the Lagrangian density
L = $i W> + d^d^cj) - h$L<W>R + $R<f>*tl>L]
V^-A(^)2 + ^ + A
where ip = tpL + ipR 1S a fermion, </> is a complex scalar, A > 0, h > 0, and
a>0.
(a) Show that L has a global chiral symmetry U(l) x U(l) for the case a = 0.
(b) Calculate the spectrum of the model (i.e., the masses) for the cases (i)
\li2 > 0, a = 0) and (ii) (/j2 < 0, a = 0).
(c) Calculate the spectrum for the cases (i) (/j2 > 0,a > 0) and (ii) (/j,2 <
0,a > 0). In each case assume that y/\a/\/j,\3 <C 1, and keep only the leading
nonzero term in that parameter.
(d) Interpret the spectrum in each of the above cases in terms of the
symmetries and symmetry breaking.

4
Gauge Theories
In Chapter 3 we considered continuous global symmetries, parametrized by
real constants (3l. In local or gauge symmetries the (31 are promoted to
arbitrary differentiable functions (3l(x) of space and time. Gauge invariance is
sometimes motivated on esthetic grounds (e.g., why should the phase of the
electron field on the Earth be correlated with its phase on Mars?). However,
we will take the more pragmatic view that gauge invariance is a powerful tool
for constructing well-behaved field theories and are the unique renormalizable
field theories for spin-1 particles. In particular, each generator of a gauge
invariant theory must correspond to an (apparently) massless spin-1 gauge
boson, which mediates an (apparently) long-range force, and the diagonal
generators of an unbroken gauge theory correspond to conserved charges (Weyl,
1929). The gauge interactions are uniquely prescribed once one specifies the
gauge group, the representations of the matter fields, and a gauge coupling
constant g for each group factor. This approach is opposite to historical
development: Maxwell's equations of classical electrodynamics were first derived
from observation and consistency, and then it was noticed that they were
invariant under gauge transformations, i.e., that the vector and scalar potential
involved redundant degrees of freedom. In this chapter, we outline the
construction of gauge invariant Lagrangian densities. More detailed treatments
include (Abers and Lee, 1973; Weinberg, 1973c,d; Langacker, 1981).
Let 3>a, a = 1 • • • n, represent n spin-0 or \ fields, which transform as
$a (x) - (eiAlV£)o6*6 (*) = [U (0(x))]ab$b (x) (4.1)
under a gauge transformation (generalizing (3.30)). L\ i = 1 • • • TV, are n x n
representation matrices of the Lie algebra of the gauge group G. (4.1) is
equivalent to
$(x) -> &{x) = e^^Or) = U (8(xj\ $(x), (4.2)
where <£(#) is an n component column vector with components $a(x). If a
theory involves n^ fermion fields ipa in a column vector ip and n^ real or
complex scalars (j)c in a vector 0, then
1>{x) -> e^^Or) = U* (0{xj) ^{x)
/ x (4-3)
0(X) _+ eWx>L*<l>{x) = U* (/?(*)) 4>{x),
157

158 The Standard Model and Beyond
where L^ and Lq are respectively the fermion and scalar representation
matrices. In the case of a chiral symmetry,
L% = ULPL + LRPR with L\ ± LR, (4.4)
so that
1>L{x) - ^x>l^L{x) = UL (p{xj) rl>L{x)
(4-5)
iI>r(x) -> eWM'^nix) = UR (/3(x)) ^R{x).
The transformations of the necessary gauge fields will be detailed below.
4.1 The Abelian Case
As a first example, we take G = 17(1). We already considered the
electromagnetic interactions of charged pions and electrons in Sections 2.6 and 2.8, but
repeat the key results. Under a (non-chiral) gauge transformation
^ _> e-W*)^ fa± _> e^^U^, A^ -> AM - ±3M/?, (4.6)
where ^ is the electron field, <j>^ and <j>n- = <^ + are respectively the 7T+ and
7r~ fields, A is the photon (gauge) field 7, and e > 0. The gauge covariant
derivatives transform as
Thus,
D^ = (&* - ieA») ip -> e-l^x)D^
D»(t>„± = {&> ± ieA») </>n± -> e^^i.
(4.7)
(4.8)
where FMI/ = c^A, - c^A^ is gauge invariant. However, an explicit photon
mass term -^ApA* is no* gauge invariant, so the 7 must be massless. The
Feynman rules for the gauge vertices are given in Figures 2.10 and 2.13. They
are unique except for e and the charge assignments. However, the mass
parameters are arbitrary. Only one pion self-interaction is consistent with 17(1),
but the coefficient A is arbitrary.
These considerations are easily generalized to an arbitrary non-chiral 17(1),
with Tty fermions ipa and n<t> complex scalars <f>b, with charges qa and q'h in units

Gauge Theories
159
of the gauge coupling g. (The simple QED example had qe- = -1, q' + = +1,
and g = e.) The fields transform as
1)a - ei9fl^(xVa, 06 - ei9^(x)^6, AM -+ A^ - -d^ix), (4.9)
where A is the gauge boson. One can think of the charges as the elements
of the (reducible) diagonal fermion and scalar representation matrices L^ =
diag (qi ) and Lj, = diag (q[ q'2 ■ ■ ■ q'n^j. The gauge covariant
derivatives are
D^a = (0" + igqaA») rpa -> e1^^D^a
D»ct>b = (d» + ig4hA») <t>b - e^^D^b,
or in column vector notation,
D^ = (d"J + igLjA") V -» ei0^L*D»ip
D»<j> = (d**J + igL^) 4> -v eil3(-x)L*D^4>,
where / is the n$ x n$ or n^ x n^ identity. The kinetic terms
(4.10)
(4.11)
Co = rpiflrp + (D^ D»<p - ^F^F^ (4.12)
are gauge invariant. These lead to vertices with the massless gauge boson
similar to those in Figures 2.10 and 2.13, except e —> gq'h for fa and — e —> g#a
for ipa. (There are no off-diagonal transitions such as ipi —> fo-)
It is clear that only gqa and #(/£ are physical. One can always rescale g
provided qa^h are rescaled accordingly. For example, this freedom is used in
QED to set the electron charge to —1. In a pure 17(1) theory, the relative
values of the charges are arbitrary. However, if the 17(1) is a subgroup of a
simple group, then the relative charges are fixed by the higher symmetry.
It is straightforward to extend these considerations to a chiral 17(1), i.e.,
in which the L and R charges qaL,R of ipa are different (this is not the case
for QED). Define L^ = LLPL + LrPr, where Ll,r are the diagonal charge
matrices of ipL,R and Pl,r are the chiral projections in (2.192). Then, the
fermion term in (4.12) becomes
JH0tl> = ${iflI-gLi,4)il> = $(i01 - g[LLPL + LRPR]fi)^ (4.13)
and the i/;a vertex in Figure 2.13 is

160
The Standard Model and Beyond
One can add mass and additional non-derivative interaction terms to £0
provided they are invariant under the global 17(1). For example*,
C = C0 ~ Ipamablpb ~ <t>\&d<l>d
+
_ . -. . (4.15)
^a^Cah^b(t>c + ^al^lbl ^b<t>c + h.C. + KbcdK^c^d
would be invariant provided qa = qb for nonzero mab, qc = q'd for nonzero /i^
Qa = Qb + q'c f°r nonzero Tcah or tcah, and q'a + q!h = qc + q'd for nonzero Aa6cd.
4.2 Non-Abelian Gauge Theories
Now consider a non-abelian gauge symmetry (Yang and Mills, 1954), in which
the spin-0 and \ fields transform according to (4.3), where G is a simple
group. Any mass terms, Yukawa interactions, or non-derivative scalar self-
interactions that are invariant under the corresponding global symmetry are
automatically gauge invariant as well. However, the fermion or scalar kinetic
energy terms contain derivatives and are therefore not gauge invariant,
d»<p _> &* \u (p(x)\ $1 = u&*Q + [d*U] $. (4.16)
One must therefore introduce a gauge covariant derivative
where A1^, i — 1 • • • N are real vector gauge fields (one per generator), g is an
arbitrary (real) gauge coupling, A1 L = A%tiL%, and U are the representation
matrices for $, i.e., V = L\,L\, or LlLPL + L^Pr. Of course, d» = d^I,
where / is the n x n identity matrix. In terms of components,
(D^)a = [d»6ab + igA» • Lab] *6. (4.18)
These gauge covariant derivatives specify the interactions of the spin-0 or spin-
\ fields with the gauge bosons in terms of a single gauge coupling g (or one per
group factor for a non-simple group), once the group and representations are
*In (4.15) the matrix /x2 must be Hermitian, and there are constraints on the \abcd fr°m
Hermit icity and Bose statistics. mai is assumed Hermitian, but non-Hermitian fermion
mass matrices can be introduced by rewriting in terms of ipL,R (or allowing 75 terms).
Other types of cubic or quartic scalar self-interact ions could be allowed if they are globally
invariant, such as <j>a4>b4>c + h.c. for q'a + q'h + q'c = 0.

Gauge Theories
161
specified. The Feynman rules for the gauge interactions are shown in Figure
4.1. For example, the fermion kinetic energy term
CKE+ = &V>^ (4.19)
implies that the amplitude for ipb to absorb or emit gauge boson A1 and turn
into ipa is just
-ig^
1"75 , Ti 1+7'
n L_ _i_ n
=LR=L
-ig^L^. (4.20)
This must be sandwiched between appropriate u or v spinors and contracted
with a gauge polarization vector, as described in Chapter 2. Unlike the
abelian case, the non-diagonal generators imply transitions between the
different members of the fermion (or scalar) IRREP that are related by the
symmetry.
Similarly, the kinetic energy term for a complex scalar representation
becomes
(D»4>)' D^ = (0"0)f d^ - igAi (V^LV) + $2i4'Mj 4?VV<S>. (4.21)
The second term on the right yields the three-point vertex in Figure 4.1 since
d^cf) —» —ip*<j> ( d^cft —> +zpM </>*), where p is the momentum entering (exiting)
the vertex for <j> (</>*), i.e., p always flows in the direction of the arrow. The
four-point vertex from iCke^ follows from the last term, taking into account
that there are two ways to contract AlfJ,A^ with the external gauge fields. The
vertices for Hermitian scalars are considered in Problem 4.6.
We must still determine the transformation of A1^. The first requirement is
D»<P -> UD»<P. (4.22)
This implies that the fermion and scalar kinetic energy terms
-> $%V\V*V^ + (£>£</>)+ UlU+D^ (4.23)
are gauge invariant, since WU = I. The necessary transformation is given by
4 • L -> \ • L = UA„ • LU-1 + - (d^U) U~\ (4.24)
where U is any nontrivial IRREP, such as L^ or L^, and U = U(x) = el/3(x)L.
The first term on the right is the expected transformation for an adjoint
representation. It would be present even for a global transformation, as in

162
The Standard Model and Beyond
i,V>
-iflT^U = "^ [LiahPL + L*RabPR]
b
a*, Pa
b>Pb
\
\
/
/
i,/x
-igipa+PbY^ab
a, Pa
b>Pb '
%, /x
3,v
iflV^^j+^i)^
v
3,V
s r
+9Cijk[g^(q-py+g^(p-rr
+9ua{r-qy}^gcijkC^{p,q^
i,/x
3,v
fc, <X
-ig^ijmCkimig^g^-g^gn
-ig^ikmCjimig^g^-g^gn
-ig2cumcjkm(g^gpa-g^gup)
FIGURE 4.1
Feynman rules for the interactions of gauge bosons with fermions, scalars,
and gauge self-interactions. The scalar vertices apply to both complex or
Hermitian scalars. pa and pb in the second vertex are the momenta flowing
in the direction of the arrows, which in some cases are the negative of the
physical momenta. E.g., pa = —pa if the a is twisted downward to represent
an incident antiparticle or Hermitian scalar, while pb = —p^ for an outgoing
antiparticle or Hermitian scalar, as in Figure 2.10. Recall that L\ah = —L\ha
for a Hermitian scalar. In the triple gauge vertex the momenta p, q, and r
all flow into the vertex and satisfy p + q + r = 0. The function C^ua(p, q, r)
is totally antisymmetric if the indices and corresponding momenta are both
interchanged.

Gauge Theories
163
(3.32). The second term generalizes (3.32) to a gauge transformation. One
has
A
,._Tr(VAll.L)
V-
T(L)
(4.25)
where the Dynkin index T(L) is defined in (3.8). It can be shown that Afi is
independent of the representation L. It is then straightforward to establish
that (4.22) holds. For small /3(x), (4.24) reduces to
A^^-CijkPAl-ldpF, (4.26)
where the first term represents the transformation of an adjoint field, and the
second is reminiscent of abelian gauge transformations.
The gauge boson kinetic energy term is
Ckea = --AFluF^ with F%, = d^Al - dvA\ - gcijkA^ . (4.27)
4 N v ' * v '
kinetic energy self-interactions
Unlike the abelian case, the field strength tensor F% contains a quadratic term
proportional to the structure constant, i.e., the non-abelian gauge bosons are
themselves charged. This leads to 3 and 4 point self-interactions, as shown in
Figure 4.1. To see the gauge invariance of Ckea it is convenient to rewrite
^=-^Tr[(iV-L)2] (4.28)
where
F^-L = d^Au • L - dvA^ • L + ig \A^ • L, Av • L~\ . (4.29)
One can then show that
F^ • L -> F^ • L = UF^ LU-\ (4.30)
so that (4.28) is invariant. However, an elementary gauge boson mass term
^MfjAl^Al is not invariant, establishing the statement that there is one
(apparently) massless gauge boson and the associated long range force for each
gauge generator. Altogether, the Langrangian density for a gauge theory of
fermions and complex scalars is
C = -\FlvF^ + faM + (D^D^ + ■■■ (4.31)
Scalar or fermion mass terms, Yukawa couplings, and non-derivative scalar
self-interactions consistent with the global symmetry can be included.

164
The Standard Model and Beyond
Direct Product Groups
The formalism can easily be extended to the case in which the gauge group G
is a direct product of two or more simple or abelian factors, e.g., G = G\ x G2.
The gauge covariant derivatives in (4.18) for scalar or fermion fields become
D»<S> = [d" + i9lA» • U + %92A ■ £2] *, (4.32)
where #m, Am, and Lm, m = 1,2 are respectively the gauge coupling, gauge
bosons, and representation matrices for Gm. Similarly, the gauge boson
kinetic energy terms become a sum over those for the group factors
CKEa = -\Fl„Fr - \FlvFr- (4-33)
The field strength tensors are unchanged,
*m/ii/ = ^M^mi/ — ^An/i — 9mCmijkArntJtArnj/. V^-34)
where cm^fc are the structure constants for Gm. Thus, the fermion and scalar
3 point vertices in Figure 4.1 apply separately to G\ and G2. The same is
true for the gauge self-interactions, which do not connect the gauge bosons of
different group factors because they commute. However, scalar multiplets that
transform under both groups lead to mixed seagull diagrams. The Feynman
rule for the mixed (j^^A1^A^ vertex is
2igig2g^(L\^)ab. (4.35)
4.3 The Higgs Mechanism
We have seen that gauge theories do not allow elementary mass terms for
gauge bosons, because they would break the gauge invariance and spoil the
renormalizability. This appears problematic for the weak interactions, which
are short-ranged and require massive mediators.
Another potential problem, for theories with spontaneous symmetry
breaking, is that they imply massless Goldstone bosons, but there are no known
exactly-massless Goldstone bosons associated with the elementary particle
interactions. Fortunately, when G is a gauge symmetry the two problems of
the unwanted Goldstone bosons and the unwanted massless gauge bosons can
cure each other when the symmetry is spontaneously broken (Schwinger, 1962;
Anderson, 1963). A particularly simple implementation is the Higgs
mechanism, involving elementary spin-0 fields (Higgs, 1964, 1966; Englert and Brout,

Gauge Theories
165
1964; Guralnik et al, 1964). Instead of existing as a massless spin-0
particle, the degree of freedom carried by the Goldstone boson manifests itself as
the longitudinal spin component of a gauge boson, which has in the process
acquired a mass. (One says that the Goldstone boson has been "eaten".)
Remarkably, SSB via the Higgs mechanism preserves the renormalizability of
the theory ('t Hooft, 1971a; 't Hooft and Veltman, 1972; Lee and Zinn-Justin,
1972, 1973).
To illustrate this, consider a 17(1) gauge theory with a single complex scalar
field 0, with
C = -\f^F^ + [(3M + igA^ {&> + igA^4> - faU - K<t>U)2 (4.36)
as in (2.129) or (4.8). We studied the analogous problem of a global 17(1)
symmetry in Section 3.3.3. It was found that for /i2 < 0, <j> acquired a VEV,
<owo> = -^, " = vHr- (4-37)
Expanding <j> = {y + o + %x)I V% around v/\/2, there was one massive physical
scalar 77 and one massless Goldstone boson x-
The minimization of the potential for a gauge symmetry is the same as for
the global case, and (4.37) continues to hold. Rewriting (4.36) in terms of the
new fields,
C = - \F^F^ + ^x? + 9^~A^ + guA^x
+ ^ApA" [2ua + a2 + X2] + gA» (a^x) (4-38)
\{d»°? - £*2 - A^(a2 + X2) ~ ~y + X2)2,
where /i2 = —2/i2 and the constant term in (3.154) is not displayed. We see
that in addition to the kinetic energy terms for the gauge fields and for a and
X there are two new quadratic terms. The third term in (4.38) has the form
of a mass term for the gauge field with mass gv. This mass can be interpreted
as arising from the interaction of the gauge field with the condensate of Higgs
fields. The fourth term in (4.38) is proportional to the gauge field times the
derivative of the Goldstone boson field x- To interpret it, we can recombine
the x and A terms as
^K)2, A'^A^ + te (4.39)
suggesting that d^x/9y ls tne longitudinal component of a now massive vector
field A'^. However, x still enters the cubic and quartic terms in a way that is
hard to interpret.

166
The Standard Model and Beyond
To see what is going on it is useful to use the Kibble reparametrization
(Kibble, 1967), which makes the physical particle content manifest and was already
introduced for the global case in Section 3.3.5. While working at the level of
the classical field theory, one can define new fields 77, £ related to a and \ by
a non-linear field redefinition:
For small a and \ one has approximately
-«/"> (^) „ ^ (4.41)
so that 77 ~ a and £ ~ \- The fields 77 and £ therefore represent the massive
and Goldstone scalars, respectively. In terms of the new fields,
c = - \f%f^' + 9-^fw [i + l
,2
l^vf-^-W-^4,
(4.42)
where
A'lt = All + -^, F'^ = 8^1-8^, (4.43)
which is clearly the Lagrangian density for a massive vector field, including 3
and 4-point gauge and self interactions for the massive scalar 77. The Goldstone
boson has disappeared from the theory.
One can interpret (4.42) as the result of choosing a special gauge. Before
quantizing one can make a 17(1) gauge transformation as in (4.6), choosing
P(x) = -£{x)/v. Then
4>{x) - 4>'(z) = c-*<*>/"0 = (^) , A,(x) - ^ = A, + M.
(4.44)
Eq. 4.42 then follows from (4.36) by gauge invariance, C(A,(j)) = £(A',<//).
This unitary gauge is useful because it makes the physical particle content
of the theory manifest. However, it is not so useful for explicit calculations,
especially in higher orders where it is rather singular.
The number of degrees of freedom was not changed by the Higgs
mechanism. Before SSB there were two massless gauge degrees of freedom and two
Hermitian scalars, while afterwards there are three gauge degrees of freedom
and one massive real scalar.
Now let us consider the non-abelian case with an arbitrary scalar sector.
Just as in the general discussion of the Nambu-Goldstone theorem, it is more
convenient to choose a Hermitian basis for the scalars for formal
manipulations, though a complex basis may be simpler for concrete calculations. We
assume there are n Hermitian scalar fields </>a, which can be arranged in a column

Gauge Theories
167
vector 0 = (0i • • • 0n)T with VEV v = (O|0|O). As discussed in Section 3.3.6,
we assume that M of the generators are not broken, LV = 0, i = 1 • • • M,
while the remaining N — M are broken, i.e., LV ^ 0, z = M + 1 • • • TV. Then,
for a global symmetry we expect that there will be N — M massless Gold-
stone bosons and p = n—(N — M) massive physical scalar particles.
According to (3.172) the Goldstone bosons are linear combinations of the original
Hermitian fields, corresponding to the directions of the massless eigenvectors
iLV, i — M + 1 • • • N. These span an N — M dimensional vector space, but
need not be orthogonal. (The i is because the Ll are imaginary). We can
therefore label the n scalars so that
<t> =
/ v\ + o-i \
Up + ap
Xp+i
\ Xn /
= e[*?:?=M+itiLi]
I v\ + Vi \
Vv +T)V
0
V o J
= e*L
(i/ + f/), (4.45)
in analogy with (4.40). The n — p = N — M fields \% m tne nrst form
are associated with the subspace spanned by the iUv. There are no VEV's
in those directions: we are working in a Hermitian basis, which implies U =
-LlT = —Ll* and therefore that (y\L%v) = 0. The p fields o~i will be associated
with the massive scalars. Not all of the Vi, i = 1 • • • p, are necessarily non-zero.
The second form is a polar reparametrization. Only the broken generators are
included in the exponent, and the £ are the Goldstone boson fields. Unlike
(4.40) we have not attempted to normalize the £l by factors of v, as it is not
needed for a gauge symmetry in unitary gauge*.
The Goldstone fields may then be removed by going to the unitary gauge,
i.e., the Kibble transformation, just as for the U(l) example. Gauge invariance
ensures that C will be unchanged in form under the gauge transformations in
(4.2) and (4.24). In particular, choose /?*(x) = ~C(x), i = M + 1 • • • TV, and
/3*(x) = 0 otherwise. Then, C(A, 0) = C(Af, 0;), where <f)f = v + r). A' is given
generally by (4.24), but for small £l is
< = 4 + ^j4 + ^r
(4.46)
which contains the longitudinal term d^1. The relevant term for our present
considerations is the gauge covariant kinetic term for the Hermitian scalars.
'For a global symmetry the £ fields would have a non-canonical kinetic energy matrix,
and field redefinitions similar to those in Section 2.13 to restore the correct diagonal form
2 SiC^O2 would be required.

168
The Standard Model and Beyond
This becomes, in terms of the new variables,
l-D^D»4> - \{v + v)T(d„ - igA» ■ L)(d„ + igA„ ■ L)(v + rj)
11 (4-47)
= -M?jAi'*A>l + -Dltr,D»T, + g2 {vTUVn) A^A^
where the primes have been dropped for notational simplicity. The Goldstone
fields £ have disappeared from the theory, reemerging as the longitudinal
gauge field components, and the gauge bosons corresponding to the broken
generators have acquired mass, described by the first term in the second line.
The second term is the normal gauge covariant kinetic energy for the physical
7] fields, which include the three and four point gauge interactions, and the
last is an induced cubic gauge interaction.
The gauge boson mass matrix in (4.47) is
M% = Ml = g2vTUUv
= g\v\VU\v) = g\Uv\Uv) = g2 £ (LV)Z(tfi/)a, (4'48)
a=p+l
where the inner product is denned by (x\y) = X^xa2/a- The induced cubic
term in (4.47) may be similarly rewritten as
g2 {yTVUii) A^AJ, = g2{Vv\Lj^ A^AJ, = g2{v\VVv) A^A^
= g2 J2 (L*v):(VV)aA*Ai. (4-49)
a=p+l
It is easy to prove, using the explicit form in (4.48) and the fact that the
representation matrices U are antisymmetric and purely imaginary, that M2
is real, symmetric, and has eigenvalues which are non-negative. From (3.173),
M2 must have the block-diagonal form
M2=('n^2), (4-50)
where the upper diagonal block isMxM dimensional and M2 is (N —
M) x (N — M) dimensional. There are therefore M massless gauge bosons
A1 - • • AM, corresponding to the unbroken generators, and N — M massive
gauge bosons, corresponding to the N — M non-zero eigenvalues of M2. The
N — M Goldstone bosons have been eaten to become the longitudinal modes
of N — M massive gauge bosons AM+1 • • • AN.

Gauge Theories
169
4.4 The R$ Gauges
The unitary gauge for a spontaneously broken theory, introduced in Section
435 is extremely useful for identifying the physical states of the theory, but
it is not very convenient for explicit calculations, especially when higher-
order loop corrections are involved, because it is very singular. It is useful
to work instead in a new class of gauges called the R^ gauges (Fujikawa
et al, 1972; Weinberg, 1973c,d; Lee and Zinn-Justin, 1973), which are less
singular, though the particle content is less obvious. They are therefore better
behaved in higher-order calculations and for proving the renormalizability of
spontaneously broken gauge theories.
Consider a gauge theory with n Hermitian scalars </>a, a = 1 • • • n arranged
in a column vector </>, and m fermion fields represented by a column vector ip.
The most general renormalizable Lagrangian density with a conserved fermion
number is then
C = -^F^ + i(D*»(D^) - V(<j>) + j(iP - mo)* + ^Ta^a, (4.51)
where F^^i = 1 • • • N are the field strength tensors for the gauge group G,
Dp and^ are the covariant derivatives for the scalars and fermions, and V is
the scalar potential containing gauge-invariant terms up to order </>4. There
may be present a fermion bare mass term described by the m x m matrix
mo = tuolPl + worPr, with tuol — mo#' ^ ^ 1S allowed by the symmetries
of the theory. Similarly, there will in general be Yukawa interactions between
the fermions and scalars described bymxm matrices Ta = TJPl + F^Pr*
withr£ = r#.
Just as in Section 4.3, we define a column vector v — (O|0|O) of VEVs, where
va = 0 and some (LV)a ^ 0 for a = p + 1 • • • n, where n — p = N — Mis the
number of broken generators. In an arbitrary gauge one can write (j) = v + <//,
where <// is the shifted field with (0|<//|0) = 0,
(4.52)
We saw in (4.45) that 0^, a — 1 • • p represent physical scalars, while (j)'a^a =
p+l' — n are associated with the Goldstone degrees of freedom. The latter
disappear in the unitary gauge, which can be defined by the condition
(LV|0;) = O, i = l->N. (4.53)
In a general gauge, (j)' will include the unphysical Goldstone bosons. Similar

170
The Standard Model and Beyond
to (4.38) we can rewrite (4.51) in terms of the shifted fields
C = -\FilvFi^ + \{D»4>,){DtlJ>')
- V{y + <t>') + $(i0 - m0 + Taua)il> + ^Ta^a,
where L = L^ = —LT are the representation matrices for the Hermitian
scalars. The terms in the first line represent the gauge and Higgs kinetic
energies and gauge interactions, just as in the non-spontaneously broken case
(Figure 4.1). Both the physical and the Goldstone boson states are included.
The first term in the second line is the induced gauge boson mass term. It is
of the same form as in the unitary gauge, with M2 given in (4.48). The next
will be cancelled by terms added to C to fix the gauge, and the third is the
induced cubic interaction, leading to the cj)'hAl^A^ vertex
ig2g^va{UU + LjL%b (4.55)
shown in Figure 4.2. In the last line, V becomes the scalar potential for
<//, including mass terms for the physical states and scalar self-interactions.
The n x n dimensional mass-squared matrix ft2 is given by (3.168), just as
for the case of a global symmetry. It has p (generally) nonzero eigenvalues
corresponding to the physical scalars, and n—p zero eigenvalues corresponding
to the Goldstone bosons. The next term in (4.54) includes the fermion iriass
matrix
m = m0- rVa = mLPL + mRPR, mL = m^, (4.56)
which may in general have both bare (mo) and spontaneously-generated pieces.
m may involve 75's unless m^ = tur. Techniques for "diagonalizing" m to
obtain the fermion mass eigenvalues and eigenvectors are described in the
standard model context in Section 7.2.2. The Yukawa interactions involving
the shifted fields are given by the last term in (4.54).
The i?£ gauges are denned by the condition
F(A(x),<t>(x),t) = ai(x), (4.57)
where
F = Vl
W,
5M; + |(HiV')
(4.58)
£ is a real parameter which runs from 0 to oo and specifies the gauge, and
al(x) is a number which depends on x only, a1 is averaged with an
exponential weighting factor in the quantization procedure, so its precise value is
unimportant. The limit £ —> 0 corresponds to the unitary gauge.
The quantization procedure in the R^ gauges introduces additional terms
into the effective Lagrangian density which modify the structure of the vector

Gauge Theories
171
and scalar propagators. The Goldstone bosons have not been removed from
the Lagrangian, and occur in internal lines in Feynman diagrams. They do not
occur as external states (except in connection with the equivalence theorem,
introduced in Section 7.5.1). The quantization also introduces terms which
can be represented by a set of N ghost fields 77*, i = 1, • • • , N. These are
fictitious particles that do not correspond to physical states but do circulate
in internal loops. They are needed to ensure unitarity and renormalizability,
and can be treated like complex scalar fields except that they obey Fermi
statistics. The ghost vertices are given by the effective interaction
Cghost = g{d^\)ciikViAl + L-nlnML*^') (4.59)
and are shown in Figure 4.2. There is a factor of —1 for each closed ghost loop.
r\\ and rji are to be viewed as independent anticommuting c-numbers, not
necessarily related by Hermitian or complex conjugation. The notation
emphasizes that there is an effective propagator iDq[x — x') = (0\T[rj(x),rf(x')]\0).
Clear discussions and derivations of the formulae may be found in (Weinberg,
1995; Pokorski, 2000).
i,/x
\
3
fc,/x
9 a
\ /
/ \
y b
b 3,v
ig2g^va{LiLi + LjL%b
FIGURE 4.2
Left: Induced three-point vertex between one scalar and two gauge fields,
derived from a four-point interaction with one external scalar replaced by its
vacuum expectation value. Vertices for ghost interactions with vector (middle)
and scalar (right) fields. The dotted lines represent ghost propagators. Ghost-
ghost-vector vertices involve one outgoing and one incoming momentum, with
the outgoing one appearing in the vertex factor. (Some authors place a dot
near the outgoing line to indicate which momentum appears.)
The momentum space propagator for the gauge bosons in an arbitrary R%
gauge is
iD^(k)
F
k2 - M2/£, J k2 - M2'
1 (4.60)

172
The Standard Model and Beyond
which is an TV x TV matrix in the space of gauge indices*. For practical
calculations it is often convenient to rewrite this (within the subspace of broken
generators) as
iD^(k) = -i[g^-
Wk
~M2
-) —
'■ J k2-
kIMkv
M2 M2k2-M2H
(4.61)
The £ dependent piece, which will ultimately cancel other £ dependent terms
in the Higgs and ghost propagators, is singled out. The ghost propagator,
which is also an N x N matrix, is
iVG(k)
k2-M2H'
(4.62)
The Higgs propagator is an n x n matrix in the space of scalar field indices.
It can be written as
i A* (&) = (/-?)
+ r
k2-n2 k2-M2/^
(4.63)
where V is denned as the projection operator onto the TV — M dimensional
space of Goldstone fields spanned by the vectors zLV, i = M + 1 • • • TV. It is
given explicitly by
V = g2\Liu) (^y {Uu\^Vab = g\Vu)a(^j {Uu)l (4.64)
where
(4.65)
is the inverse of the vector mass-squared matrix in the subspace of massive
vectors (and zero otherwise). The scalar propagator in (4.63) decomposes into
two pieces. The first, proportional to I — V, represents the propagation of
real physical scalars with mass-squared matrix /i2. The second term is the
Goldstone boson contribution, which is present in an arbitrary £ gauge. It is
shorthand for
1
k2-M2/£
g\Vv)a
ab
1
1
M2 k2 - M2H
(Vv)t-
(4.66)
Using Vfi2 = 0, (4.63) can be rewritten
^2
iA^k)
ab
. W (TX \
^)S(^K. <«*>
_/c2-/i2_
where the first term includes both the physical and Goldstone scalars.
*There is an implicit +ie in each propagator, e.g., l/(k2 — M2) —► l/(fc2 — M2 + ie).

Gauge Theories
173
We see that the propagators of gauge, ghost, and Higgs fields can be
decomposed into pieces involving the propagation of physical particles, namely
the first terms in (4.61) and (4.63), as well as pieces which involve unphysical
poles at k2 = *j- which depend upon the gauge. These unphysical poles do
not correspond to physical particles, and are included only in internal lines in
Feynman diagrams. They cancel in the final expressions for physical on-shell
amplitudes, i.e., S matrix elements, which are necessarily gauge invariant.
R,£ gauges for £ ^ 0 are referred to as renormalizable gauges because the
propagators are well-behaved for k2 —> oo, falling as l/k2. In principle it is
best to do calculations with £ arbitrary so that (a) the Feynman diagrams are
well-behaved and manifestly renormalizable, and (b) so that the cancellation
of the £ dependence in the physical S matrix elements can be used as a check
on the correctness of the calculation. In practice, however, it is messy to
carry out calculations for an arbitrary £, and therefore people often make use
of particular gauges for calculations or formal arguments. The £ = 1 gauge
is known as the 't Hooft-Feynman gauge. The gauge propagator takes the
particularly simple form
so the £ = 1 gauge is often convenient for carrying out concrete calculations.
Another useful gauge is £ —> oo, known as the renormalizable or Landau
gauge. The propagators again take a relatively simple form in this limit
iDvW = {k2 _ M2 . <^(*) = F, iA,(k) = w-^, (4.69)
but this gauge still contains ghosts and unphysical scalar fields.
The above gauges are best for calculations of higher orders. However, the
£ —> 0 or unitary gauge is particularly simple for displaying the physical
particle content of the theory. The propagators become
iD^(k)= %_M?^, fl>G(k)=o, i^(k)=(i-p)w-^.
(4.70)
That is, there are no ghost fields and only the physical, non-Goldstone scalar
fields survive. If one is only interested in calculating at tree level, the unitary
gauge is very convenient. However, the gauge boson propagator is badly
behaved as k —> oo; it approaches a constant rather than falling like l/k2.
It therefore induces severe ultraviolet divergences in higher order calculations
which must be handled very carefully.
The ghost fields do not entirely disappear from the theory in the unitary
gauge (Weinberg, 1973c,d). There is an effective multiscalar interaction
Cj = -iS\0)Tr ln(/ + J), (4.71)

174
The Standard Model and Beyond
where
Jij=92(w). WVM- (4-72)
\ / ik
The trace and matrices in (4.71) and (4.72) are restricted to the N — Atf"
dimensional subspace of broken generators of G. Cj is a remnant of the
ghost loops which survives as £ —> 0 because of the factors £_1 in the ghost-
ghost-scalar vertices, which cancel the zeroes in the ghost propagators. Cj
is necessary to cancel divergences due to gauge boson loops in multiscalar
interactions.
Complex Scalars
It is straightforward to re-express the results for the interaction vertices and
propagators in a complex scalar basis, either by "starting from scratch" or by
using the formal results in (3.71)-(3.73). Writing <j> = v + <//, where the fields
and VEVs are now complex, the induced cubic and ghost-Higgs vertices in
(4.54) and (4.59) are replaced by
Caa# = 92 [{v\LlV<t>') + {4>'\UUv)} A\A^
2 (A 7Q\
respectively, while the gauge boson mass matrix in (4.48) becomes
Af£ = g2{v\UV + UU\v). (4.74)
The projection operator V onto the Goldstone subspace is usually obvious,
but is best worked out in the Hermitian basis for complicated cases.
Let us illustrate the R^ constructions for an SU(2) gauge theory involving
a single complex scalar 517(2) doublet. (This is a simplified version of the
standard 517(2) x 17(1) model with the 17(1) gauge coupling, and therefore
electromagnetism, turned off.) The Hermitian 517(2) gauge fields A1^ i =
1 • • • 3, will sometimes be re-expressed as
At = ^4 A* = Al, (4.75)
where A+ = (A~)^ and the labels ± and 0 are suggestive of the electric
charges that will emerge when the model is promoted to 517(2) x 17(1). The
gauge self-interactions for the A fields are derived in Problem 4.7. Similarly,
the scalar doublet can be written
*=(£Y **=(%*)> (4-76)
r
where the conjugate fields are denned as 0 = (0+)t and </>°* = (0°)*. The
most general renormalizable gauge invariant potential for 0,
^) = +mV0 + A(^)2, (4.77)

Gauge Theories
175
is identical to the Higgs potential in the standard model, and will be described
in more detail in Section 7.2. Here we note that A > 0 is needed for vacuum
stability. For /i2 > 0 there is no SSB, i.e., v = (0\<f>\0) = 0, while v + 0 for
//2<0.
In the unbroken case, v = 0, the complex scalar fields </>+ and </>° are
degenerate with mass /i, and their self-interaction terms are
d = -V} = -\{<t>U)2 = -K<t>~<t>+ + 0°>0)2. (4.78)
The scalar gauge interactions can be obtained by writing the general form in
(4.21) in terms of components, using U = rl/2 and r%A% — r3A° + v/2r+A+ +
y/2T-A~:
C* = -if (V*aV " 4>°^4>°) A% - A (f **$»*+) A~
2 (4.79)
- *^= (V >/) A+ + ^(0"0+ + 0°>0)^ • 4,
where A^-A^ = 2j4+M~+A°M°§. The diagonal quantum number associated
with T3 is conserved.
For /i2 < 0 the 517(2) symmetry is completely broken. Similar to the 17(1)
example in (3.152) the minimum occurs for v = v/y/2, where \v\2 = —/i2/A.
Without loss of generality, we can choose v to be real and in the </>° direction:
other orientations of the minimum differ by 517(2) transformations. Thus, we
can write
" = 7i(°)' *=(4»)' <4-80)
where H and z are the Hermitian components of <jp' and w+ = <f>+. Rewriting
the potential in the new variables,
V(</>) = -n2H2 + Xi/H [H2 +z2 + 2w+w~] + ^ [H2 + z2 + 2w+W-}2 ,
(4-81)
where v = y/—y?/\ and we have dropped the additive constant. We therefore
recognize that H is a physical scalar field with mass
while 2, w+, and w~ = (w+y are the Goldstone bosons which should
disappear in the unitary gauge. (The notation is chosen to coincide with the
analogous standard model case.) The second term on the right is an induced
cubic interaction.
§The quartic term is the product of the SU(2) singlets A* • A^ and 4>~<t>+ + </>°*</>° = </>*</>.
This is because the product of two doublets transforms as 0 + 1 under SU(2) while that
for two triplets is 0 + 1 + 2. However, the 1 is antisymmetric for the triplets and vanishes
in this case, so only the singlet components can enter.

176
The Standard Model and Beyond
In the notation following (3.71), <j> can be written in a Hermitian basis as
/ w1 \
4>h =
v + H
w2
\ z )
= Vh + <t>'h, vh =
V
0
W
(4.82)
where w± = (w1 ±iw2)/y/2 and the ordering of the components differs slightly
from the convention in (4.52). Then, using the 517(2) representation matrices
in (3.76),
LW = ~2
0
v
Lh»H = —2
M
0
0
W
LW = +2
0
0
w
, (4.83)
establishing that all three generators are broken and that the vectors iLlhVh
span the Goldstone subspace.
Substituting the representation in (4.80) for 0, the scalar gauge interactions
become
- i9- ([H - iz]*d»w+) A- - i9- (w-*#»[H + iz\) A} (4.§4)
■)
2 2
+ ^vHA» -ifp + ^-l W-W+ +
H2 + z*
A*-A
where we have omitted the irrelevant <f>' — A mixing terms. The three gauge
bosons A1 (or A±,A°) have acquired a common mass
*.-?
(4.85)
by the Higgs mechanism. They are degenerate because the Lagrangian has an
unbroken 0(3) global symmetry after the breaking, as is discussed in Section
7.2.2. The derivative cubic terms always connect H to a Goldstone field (or
two Goldstone fields to each other), so they disappear in the unitary gauge.
(There are important analogs involving physical Higgs fields in models with
extended Higgs sectors, such as in the minimal supersymmetric extension of
the standard model (MSSM).) The induced cubic term vHA2 involves only
the physical scalar field. The projection operator V in (4.63) projects onto
w± and 2, while I — V projects onto H.
As a simple example, let us consider the amplitude for HH —> A+A~ in an
i?£ gauge, with the tree-level diagrams shown in Figure 4.3. Using the gauge
and scalar propagators in (4.61) and (4.63), as well as the interaction vertices

Gauge Theories
in (4-81) and (4.84), the amplitude is
M = (^) ^iDZfa-n) (^) e\v + (Pl ~p2)
+
-w
f3*H(P3 -Pi)n + iPiv\iDw+{P3 -Pi)
+
i#2
Y-jev4*[+i(p2-P4)v+ip2„}
-Je*3-etiDH(pi+P2){-iXp)3\,
+ (Pi <-* Pi)
where the 3! in the last term is combinatoric. The propagators are
k»kv\ 1 _ i k^k"
iD>Z+{k) = -i[gr-
M\ M\ fc2 - M2/£
iDw+(k)
k*-M\/e
iDH(k) =
k2-M2H
177
(4.86)
(4.87)
Using (4.85) as well as ps -63 = p± -ej = 0, the w+ exchange term cancels the £-
dependent part of the A+ term, leaving just the A+ and # exchange diagrams
evaluated in unitary gauge, illustrating the general result that physical on-
shell amplitudes are independent of £.
A+(p3) A"(p4) A+(p3) A-(ft) A+(p3) A"(p4)
„+ \
J*
/ \ / \ ^"
ff('ft) JJ(fi) ff(ft) ff(fi) JJ(ft) ff(fi)
FIGURE 4.3
Tree-level diagrams for HH —> A+A~ in an i?£ gauge. There are two
additional ^-channel diagrams obtained from the first two by pi <-> p2,
4.5 Anomalies
Anomalies refer to quantum effects which break the symmetries associated

178
The Standard Model and Beyond
with the classical equations of motion. In particular, they may occur when
the diverences in a theory cannot be regularized in a way that is consistent
with the original symmetries. The Adler-Bell-Jackiw anomalies (Adler, 1969;
Adler and Bardeen, 1969; Bell and Jackiw, 1969; Bardeen, 1969; Adler, 2004)
are singularities associated with the fermion triangle diagram contributions
to the vertex of three currents, as shown in Figure 4.4. If one or three of
3
k
FIGURE 4.4
Left: The anomalous triangle diagram for the J^J^Jp vertex. Right: The
analogous diagram for the triple gauge boson vertex.
the vertices involve an axial vector (7s) coupling, as can occur for a chiral
symmetry, the diagram diverges linearly, leading to an anomalous divergence
of the currents in perturbation theory that is not revealed by formal
manipulation of the field equations. If one or more of the currents is associated with a
global symmetry of the theory, the anomalous divergence does not cause any
particular problems, and it can even be useful (Adler, 1969; 't Hooft, 1976a),
as we will see in Sections 5.2 and 5.8.3. If the currents are all associated with
gauge symmetries, however, then the diagram contributes to the triple gauge
vertex and cannot be regularized in a way consistent with gauge invariance,
implying that the renormalizability of the theory is lost.
The anomaly coefficient A^ in the vertex of currents i,j, and k is (Georgi
and Glashow, 1972)
Aijk = 2IY Li{L{,LkL} - 2TYUR{VR,LkR}, (4.88)
independent of the fermion masses. We require that each A^k must vanish
for a renormalizable gauge theory.
Another constraint comes from the (universal) interaction of fermions with
gravity (see, e.g., Weinberg, 1995), which leads to a breaking of gauge
invariance in the presence of a gravitational field, proportional to the trace anomaly
T* = TrZ4-TYZ4. (4.89)
We therefore require for this to vanish as well.
n

Gauge Theories
179
We see immediately that there are no triangle anomalies for non-chiral
theories such as QED or QCD, since UL = LlR. The electroweak SM is based
0n a chiral SU{2) x 17(1) group, but as will be seen in Section 7.1 the anomalies
cancel between quarks and leptons or (in non-trivial ways) between L and R.
We discussed in Section 2.10 that instead of working in terms of the L and
i^-chiral particle fields ipL,R, one could just as well express the theory in terms
of the L-chiral particle and antiparticle fields ipL and t/^, where tpR and -0?
are related by (2.296). The 2n fields (ipL xl)cL)T transform under the reducible
2n x 2n-dimensional representation matrices
£* - (Li Si) - (Li -k) ■ {iM}
where the tycL transform as — L1^ since they are basically the adjoints of the
ipR. The anomaly conditions in this basis are
Aijk = 2Tr C{Cj,Ck} = 0, T, = Tr£* = 0, (4.91)
which are equivalent to (4.88) and (4.89) since TrM = TrMT.
4.6 Problems
4.1 The spontaneous breaking of a global or local 17(1) symmetry allows one-
dimensional cosmic string classical solutions, analogous to the two-dimensional
domain walls considered in Section 3.3.2. Consider the complex scalar 0 in a
U(l) gauge theory, with
£ = -\F^F^ + (D^ D»4> - V {</>), V (4>) = /iV</> + A (<^)2 ,
where D^ = (d» + igA»)fa FpV is the field strength tensor, A > 0, and
fj? < 0. It is convenient to rewrite
where we have dropped the irrelevant constant — Ai/*/4. It was shown (Nielsen
and Olesen, 1973) that there is a classical solution for </>(#) and A^{x)
corresponding to a vortex or string running along the z axis from — oo to +oo. In
cylindrical coordinates (r, 0, z) the solution exhibits
i , V a-in0
<P ► ~7=e
so that V —> 0 for r —> oo. Single-valuedness requires that n is an integer.
Find the asymptotic expression for the vector potential AM for the string

180
The Standard Model and Beyond
solution, for which F^v —> 0, D^cj) —> 0, and calculate the magnetic flux
/ B • dS = J (v x A V dS for the solution.
4.2 As mentioned in Section 3.2.5 it often occurs that the spontaneous
breaking of a continuous symmetry leaves a discrete subgroup unbroken. If the
original symmetry was local, the remaining subgroup is known as a discrete
gauge symmetry (e.g., Ibanez and Ross, 1992). As a simple example, consider
a model involving three complex scalars 0*, i = 1 • • • 3, with a U(l) gauge
symmetry. Suppose the potential is of the form
3
^(01,02,03) = $Z K(0i0t) + ycubic,
where Vi(4>\<t>i) are quadratic functions of 0J0* (i.e., quadratic and quartic in
<j>i and 4>\) and
ycubic = ^1010203 + 020102 + ^'C-
(a) Find the 17(1) charges of the 0* for which the theory is 17(1) invariant.
(b) Show that o~\^ can be taken to be real w.l.o.g.
(c) Suppose the V* are such that the minimum of V occurs for (010310) ^ 0
but (0101,210) = 0. Show that a discrete Z3 symmetry remains unbroken.
4.3 Let <£*, i = 1 • • • m2 — 1, be m Hermitian scalars transforming according
to the adjoint representation of SU{m). Define the mxm matrix $ = X^ $?Ll
as in (3.31), where U are the fundamental representation matrices L^. Show
that the gauge covariant derivative of 3> is
DM$ = 3M$ + t0 [;?•£,*]
4.4 From (3.19) the defining (vector) representation of SO(m) consists of the
m(m— l)/2 antisymmetric imaginary mxm matrices. These are conveniently
labelled by two indices, with
(L^)ab = -i(siasi-sisi),
where z,j, a, and b all range from 1 to m. Clearly, L*-7 = —LP% and L%% = 0.
One could restrict the indices so that i < j, but it is convenient not to do so,
provided one is careful about double counting. The trace is
Tr (LijLkl) = 2{SikSjl - 5il5jk).
From the vector representation, one has the Lie algebra
rjiij jikl~\ __ ^ (_fijkj->il _ sjiljijk _|_ zikjijl , £jljiik\

Gauge Theories
181
where the generators TtJ have the same labelling convention as Lij. (It is
instructive to specialize these relations to m = 3.)
(a) Calculate the gauge covariant derivative {D»$)a for a scalar or fermion
field $a, a = l---m, transforming as a vector under SO(m), labelling the
gauge bosons by AlJ = —Aft and the gauge coupling as g.
(b) Calculate the field strength tensor FlJu for AJf.
4.5 The gauge transformation for a non-abelian gauge field is given in (4.24).
(a) Verify that (4.24) reduces to (4.26) for small |/?*|.
(b) Verify the transformation (4.22) for a matter field.
(c) Prove (4.30) for all /? (i.e., not just small |/?*|), which implies that Ckea
is invariant.
4.6 Derive the gauge vertices in Figure 4.1 for Hermitian scalars.
4.7 Specialize the 3 and 4-point gauge vertices in Figure 4.1 to the case of
51/(2). Define the fields A± and A0, as in (4.75). Show that the vertices for
A+(p)Al(q)A-(r), A+A°A° A", and A+A+A~A~ are respectively
igC^cr{p, ^r), -ig2QiiPvcj, ig2Q»»P(j,
and that the others vanish. All of the particles and momenta flow into the
vertices. CMl/a(p, q, r) is defined in Figure 4.1, while
^sifii/pcr — ^g^gpcj g^ipgucr g^o-gi/p'
Note that Q is symmetric in /i <-» v or p <-» a, that Q^up<T = Qpa^iu, and that
4.8 Prove that the gauge boson mass matrix in (4.48) is real, symmetric,
has non-negative eigenvalues, and that N — M eigenvalues are non-zero. Hint:
define an eigenvalue and normalized eigenvector of M2 as A and w, i.e., M2w =
Xw. Use the fact that A = wTM2w = WiM^Wj. Recall that the vectors Uv
span an N — M dimensional space.
4.9 Derive the effective multiscalar interaction in (4.71).

5
The Strong Interactions and QCD
The modern theory of the strong interactions is quantum chromodynamics
(QCD). The basic ingredient is that each of the six flavors or types of quark,
u, d, 5,c, 6, and t, has an additional quantum number color, which takes the
values a = 1,2,3, or red (R), green (G), blue (B), and that there is an
unbroken non-chiral 517(3) gauge symmetry acting on the color index. Thus,
there are 8 massless gauge bosons (gluons), Gl, and a strong gauge coupling
gs and strong fine structure constant as = gl/kn. In this chapter we survey
the properties of the strong interactions and QCD, especially the symmetry
aspects at short and long distances. For more detailed treatments, see, e.g.,
(Brock et al., 1995; Altarelli, 2002; Ellis et al., 2003; Sterman, 2004; Campbell
et al., 2007; Plehn, 2008; Amsler et al., 2008, CTEQ and PDG websites). For
a discussion of the development of QCD, see, e.g., (Gross, 2005).
u d
FIGURE 5.1
QCD interactions.
The quark-quark interaction diagram via one-gluon exchange and the gluon
three- and four-point self interactions are shown schematically in Figure 5.1.
As discussed in Chapter 2, the dominant higher-order vacuum polarization
diagrams in the gluon propagator, shown in Figure 5.2, can be absorbed into an
effective (logarithmically) running coupling #s(/i2) that depends on the renor-
malization scale. Higher-order corrections are minimized if one chooses this
scale comparable to the momentum Q carried by the gluon. The quark-loop
contributions screen the color charge at long distance, making the effective
force weaker. However, the gluon loops anti-screen. The latter dominate for
183

184
The Standard Model and Beyond
six flavors, so the strong force becomes stronger at long distances or low
momentum (infrared slavery), and weaker at short distances or high momentum
(asymptotic freedom). This is in contrast to QED, which has only the charged
particle loops, so that it becomes stronger at short distances.
9s(Q)
9s(Q)
FIGURE 5.2
The running couplings in QCD. The upper left (quark-loop) diagram leads to
a decrease in the effective gs(Q2) at small momentum Q (i.e., long distance r ~
l/Q) due to screening of the charge, analogous to QED. The lower left (gluon-
loop) diagrams have the opposite (anti-screening) behavior. They have no
analog in QED. Ghost diagrams are not shown. On the right are the effective
running QCD coupling, and a sketch of the values of as(Q2) = g2(Q2)/4ir
and the QED coupling a(Q2) = e2(Q2)/A7r. The QCD coupling becomes
small (asymptotically free) for |Q| > 1 GeV, and large (of 0(\)) for |Q| < 1
GeV (infrared slavery).
The Long Distance Regime
The long distance regime, relevant for momenta |Q| < 1 GeV, is characterized
by strong coupling, gs = 0(1). It is therefore non-perturbative. Isolated

The Strong Interactions and QCD
185
quarks and gluons have not been observed; presumably they are confined
and cannot emerge as free particles due to the strong coupling and gluon-self
couplings (infrared slavery (Fritzsch et al., 1973)) that only allow color singlet
asymptotic states (hadrons). These include qq mesons, and qqq baryons. Their
strong interactions are approximately described by phenomenological models,
such as the Yukawa model of pion exchange between nucleons. Within the
underlying QCD theory this is interpreted as an approximation to higher-
order effects, as illustrated in Figure 5.3. Even though perturbation theory
uud
ddu
FIGURE 5.3
Effective Yukawa interaction, due to quark-antiquark exchange with the qq
pair interacting via gluon exchange.
is not useful in this regime, one can still utilize the approximate global flavor
symmetries described in Chapter 3, i.e., 517(2) isospin symmetry relating u
and d at the few % level, and the extension to 517(3) relating u, d, and s at
the 25% level, and their chiral extensions.
The Short Distance Regime
The short distance regime, |Q| » 1 GeV has weak coupling, ys<l
(asymptotic freedom). It can therefore be described in terms of point-like quarks and
gluons with interactions that can be calculated in perturbation theory. For
example, in a deep inelastic scattering process, e~p —> e~X, the final e~ is
observed but the final hadronic states, represented by X, are not observed
and are summed over. (Such processes are known as inclusive, as opposed to
exclusive ones such as e~p —> e~p + 3tt or e~p —> e~ A+ involving a definite
final state.) For Q2 = -q2 > 1 GeV2 the process is described to first
approximation by the photon interacting with a point-like quark, as shown in Figure
5.4. Other parts of the diagram, involving the distribution of quarks within
the proton, and the process of the remaining quarks in the proton and the
scattered quark turning into hadrons (hadronization), are non-perturbative

186
The Standard Model and Beyond
effects. The concept of describing inclusive sums over final states of hadrons
in terms of perturbative calculations involving quarks and gluons is known as
quark-hadron duality.
c-(fc) e-(fc')
FIGURE 5.4
Deep inelastic scattering. The interaction between the lepton e~ and the
quark can be described perturbatively by one photon exchange. However, the
distribution of quarks in the proton, and the hadronization of the remaining
and scattered quark (the blobs) into unobserved hadrons X = X\ + X2 are
non-perturbative.
5.1 The QCD Lagrangian
The quark fields are denoted qra, where a = 1,2,3 or R,G,B is the gauged
color and r = u,d, s, c, 6, t is the (ungauged) flavor index. An alternate
notation is ua = qua, etc. The Dirac indices are suppressed. The quarks
transform under the fundamental (3) representation of 517(3), and are non-chiral,
UL = UR = ^-. The Dirac adjoint field transforms as a 3* and is written q?.
There are 8 Hermitian gauge fields (gluons), Gi = G*f, i = 1 • • • 8. The QCD
Lagrangian density is
CQCD = -\G\VG^ + J2^^<ir0-J2mr^1ra + l^^G^,
r r
(5.1)
where the field strength tensor is
G\v = d„Gt - dvG\ - 9sfijkGiGl (5.2)

The Strong Interactions and QCD
187
The quark gauge covariant derivative is
Dtf = WU = d^ + ^f^- (5-3)
where g
^ = (Grf = EGJ|, (5.4)
with G£ = 0, represents the gluon field in tensor or matrix notation. The mr
in (5-1) are the current quark masses. They are actually generated by
spontaneous symmetry breaking in the full standard model (including the chiral
electroweak part), but can be considered as bare masses when considering
QCD alone. Without loss of generality (for QCD) they can be taken to be
real, nonnegative, and diagonal in flavor.
The interaction terms in (5.2) and (5.3) are the same for all six flavors, so
they are invariant under a global chiral C/(6) x 17(6) flavor symmetry.
However, mc,mb, and mt are large compared to the typical scale of the strong
interactions, A ~ (200 — 300) MeV, so the symmetry is badly broken except
at very high energy. More useful is an approximate 517(3) flavor symmetry in
the limit mu ~ rrid ~ ras, which is valid at the 25% level, and the even better
(1%) 517(2) isospin symmetry, which holds in the limit rrid ~ mu. In fact,
the validity of isospin is more because mu and rrid are both very small
compared to A than because they are degenerate, while that of 517(3) is because
ms ~ 100 MeV (^> rau^) is smaller than A but non-negligible, as described
in Section 3.2.4. These symmetries are enhanced to become chiral for m = 0;
e.g., for mu = rrid = 0 the continuous symmetries of Cqcd become
5l7(3)Coior x 517(2) x 517(2) x 17(1) x 17(1), (5.5)
N v ' s v ' V v '
gauge global global B,Ba
while the 517(2) x 517(2) becomes 517(3) x 517(3) if ms -> 0 as well. The 17(1)
factors are separate L and R chiral baryon numbers. The sum (difference)
of the generators corresponds to baryon ("axial baryon") number B(Ba)-
However, Ba is not a good symmetry of the strong interactions, and its
unsuccessful prediction in the quark model is known as the axial i7(l)^ problem.
Its resolution by non-perturbative effects in QCD will be commented on in
Section 5.8.3.
Another difficulty is the strong CP problem, which refers to the last term
in (5.1). The strong interactions are observed to be reflection invariant (i.e.,
parity P is conserved), as well as invariant under charge conjugation (C), time
reversal (T), and the products CP and CPT. The first three terms in (5.1)
respect these symmetries. However, it is possible to add the final strong CP
term to Cqcd, where 6qcd is a dimensionless constant, and
G> = \^P°Gip° (5.6)

188
The Standard Model and Beyond
is the dual field strength tensor. (G and G are related by exchanging the
analogs of the electric and magnetic fields.) The strong CP term is gauge
invariant and does not spoil the renormalizability of QCD. However, for
Qqcd 7^ 0 it violates P, T, and CP symmetries, and stringent experimental
limits on the electric dipole moment of the neutron require |0qcd| < 10~U.
For pure QCD it is possible to simply impose these symmetries, i.e., to take
Qqcd — 0. However, as will be discussed in Chapter 8, this becomes
problematic in the context of the full standard model, which has other sources of
CP violation.
5.2 Evidence for QCD
QCD is the unique renormalizable field theory consistent with the
observations that existed by ca. 1970, and since that time there has not been any
serious competing theory*. Nevertheless, it is interesting to review some of
the evidence for the ingredients of QCD.
Spin-^ Quarks and Spin-1 Gluons
The first evidence for spin-^ quarks was the success of the constituent quark
model, which successfully classified a large number of hadrons in terms of
three flavors of quarks. The spin-| nature is essential for this classification,
as is evident from the construction of the nucleons out of three quarks or of
the spin-1 p from a qq pair in an S-wave. The very simple description of the
approximate flavor symmetries of the strong interactions and their breaking
in the quark model, i.e., (3.112), also provided evidence.
By 1970, dynamical evidence emerged from the deep-inelastic scattering
process e~p —> e~X with Q2 = —q2 ^> 1 GeV2 shown in Figure 5.4 and to be
described in Section 5.5. The rate observed at the Stanford Linear Accelerator
Center (SLAC) at large Q2 was much larger than would be expected for a "big
fuzzy" proton, calling for the existence of point-like parton constituents of the
proton, reminiscent of the discovery of the atomic nucleus in the Rutherford
experiment. In principle, the partons could be spin-0, spin-|, or higher.
However, the angular distribution of the scattered electron established that they
are spin-1(the Callan-Gross relation), consistent with being quarks. This was
*In fact, prior to the development of QCD many physicists seriously considered
abandoning the ideas of field theory or of any fundamental dynamical equations for the strong
interactions, in favor of the bootstrap, which postulated that there was a unique 5-matrix
consistent with the ideas of unitarity, analyticity, crossing, etc. (see, e.g., Eden et al., 1966;
Collins, 1977).

The Strong Interactions and QCD
189
later confirmed in the distributions observed in deep inelastic i^N and v N
scattering and in their relative strengths.
Additional evidence emerged a few years later in the study of e+e~ —>
hadrons. The observed cross section falls as 1/s, where s is the square of the
CM energy. This is consistent with the behavior expected for the production
0f point-like quarks which later hadronize into hadrons (up to higher-order
QCD corrections), as in (2.228), but not with the more rapid falloff expected
without such constituents. The spin-| nature is established by the observed
1 + cos2 0 angular distribution of (2.227), as opposed to the sin2 0 distribution
predicted for spin-0 (Problem 2.16). On the other hand, the positive evidence
for quarks was apparently contradicted by the non-observation of isolated
quarks. The resolution of that conflict had to await the development of QCD
and the notion of infrared slavery.
The first direct evidence for spin-1 gluons came from the observation of
distinct 3 jet events from e+e~ —> qqG and of the planar broadening of events
in which the third jet could not be resolved, by the TASSO and other
collaborations at PETRA (DESY) in 1979 (see, e.g., Wu, 1984; Bethke, 2007).
Another type of compelling evidence is indirect, i.e., the observed asymptotic
freedom of the strong interactions requires a non-abelian gauge theory. Other
advantages of color octet gluons were especially emphasized in (Fritzsch et al.,
1973).
Evidence for Color
The observed hadrons are all color singlets. Nevertheless, with the benefit of
hindsight, the color quantum number was already needed in the original quark
model. That is because the quark assignments for the baryons and hyperons
(baryons involving an s quark) were totally symmetric in the flavor, spin, and
space indices. In particular, the Q~, which was successfully predicted by the
SU(3) model (Section 3.2.3), was interpreted as
|fi-> = |sVsT), (5.7)
where the arrows all represent spin-up with respect to a reference axis. The
Q~ is therefore symmetric in flavor (all s quarks), spin (all spins in the same
direction), and in space indices (the orbital angular momenta are zero).
However, this violates the spin and statistics theorem, which follows from the union
of relativity and quantum mechanics (see, e.g., Streater and Wightman, 1980),
and which requires that all physical spin-| states should be antisymmetric.
This fundamental difficulty with the quark model is easily resolved by the
introduction of the color quantum number, under the assumption that the
ft~ and other baryon/hyperon states are color singlets, since the projection
of 3 x 3 x 3 onto the singlet is totally antisymmetric,
|jr> = e^l444>.
(5.8)

190
The Standard Model and Beyond
Thus, the color quantum number was needed even before the development of
QCD*, where it played the additional role of a gauge quantum number.
FIGURE 5.5
e+e~ —> qraQr- Left: the blobs represent the quark hadronization. Right:
higher-order QCD corrections.
There are other tests based on counting the number of colors that
contribute to an amplitude or rate. The leading diagrams for e+e~ —> hadrons
are shown in Figure 5.5. At short enough distance, one may regard the
process as first producing quark qra and its antiquark, which may be computed
perturbatively, followed by hadronization, in which the quarks turn into jets
of hadrons such as pions (low momentum (soft) gluons or quarks may be
exchanged between the jets to ensure color neutrality). There are also higher-
order QCD corrections which can be calculated perturbatively. One usually
expresses the theoretical and experimental result in terms of the ratio
R(s) =
o~(e+e
hadrons)
o~(e
+ e-
M+M )
(5.9)
at CM energy yfs, where the denominator is the lowest order theoretical
expression in (2.228) with a running coupling, cr(e+e~ —> /i+/i~) = 47ra2(s)/3s.
R(s) is convenient because the largest energy dependence cancels in the ratio,
and experimentally because the luminosity also cancels. R(s) counts the
number of quark colors and flavors, weighted by the quark electric charge-squared
q%, where qr = | for [u, c, t] and — | for [d, s, 6]. Altogether, one predicts
R
^£<
i+?-+Mil f^y-12.8 r^y
+ ••
(5.10)
^An apparently alternative resolution of the statistics problem, parastatistics (Greenberg,
2008), is in fact equivalent to the existence of the color quantum number.

The Strong Interactions and QCD
191
for massless quarks*, where Nc is the number of colors, and the higher-order
terms use the value of the running as at s. There are additional quark mass
corrections, and only quarks lighter than yfs/2 should be included in the sum.
Thus, the QCD (Nc = 3) prediction (ignoring the higher-order terms) is
R = - for [u,d], - for [uds], — for [udsc], — for [udscb], (5.11)
«j «j O O
in excellent agreement with the experimental result in Figure 5.6. R also
excludes an alternative quark model involving integer electric charges (Nambu
and Han, 1974), at least under the assumption that the quarks may be treated
as pointlike.
FIGURE 5.6
Experimental data on R(s) compared with the lowest order (dashes) and
three-loop (solid) QCD predictions. There are steps at the s, c, and 6
thresholds, given approximately by the locations of the <j> (ss), J/ip (cc), and T (66)
resonances. The perturbative prediction works extremely well above a few
GeV, provided one includes the threshold resonances. At high energies, Z
boson exchange strongly dominates over one photon exchange. Plot courtesy
of the Particle Data Group (Amsler et al., 2008).
Other tests include the ratio of nonleptonic and leptonic decays of the W,
* Similar to QED, there are infrared singularities associated with both virtual and real gluons
as their energy approaches zero, as well as mass singularities that occur in the limit of a
massless quark due to gluons radiated parallel to the quark. These can be shown to cancel
under realistic conditions using dimensional regularization or by introducing a fictitious
gluon mass. (The latter is only possible for diagrams not involving the non-abelian gauge
vertices, since it would violate gauge invariance). See (Field, 1989) for a detailed discussion.

192
The Standard Model and Beyond
since W —> qq, where qq = du, sc counts the number of colors, while the
leptonic decays into £~T>t, £ = e, /i, r do not. In particular, the branching ratio
B(W~
■'■) - r
T(W~ -te-y-)
1
(W~ ^qq) + T (W~ -f t-v) 3 + 2NC Nc=3
> 11%
(5.12)
(up to small corrections from QCD, fermion mass, quark mixing, etc.), in
agreement with the experimental value ~ 10.8%.
Another probe is the Drell- Yan process in which
p P -> t+C + hadrons, £~ = e",/i",r" (5.13)
2 9 —
at large (p£+ +Pe-) ^> lGeV . This is dominated by the qq annihilation
through a 7 or Z, as shown in Figure 5.7, and can be thought of as the
inverse to e+e~ —> qq. The qra and q@ can each be in one of three color states,
FIGURE 5.7
Left: Drell-Yan process, pp —> 7, Z —> ^~^+. Right: The 7r° —> 27 decay.
but only the combination in which a = (3 can contribute, leading to a cross
section 1/NC compared to what would be expected without color, in agreement
with observation. To see this, consider pp scattering in the approximation of
considering the valence quarks only. The initial proton state is
\p)
valence
-7=ta(3/y\qiaqj(3qki),
(5.14)
where i,j, and k refer collectively to the flavor, spin, and space indices. Then
2 / -. \ 2
acolor = 3x2x2x(-L) x (-L) a,
no-color — q ^no-color •> yo.io)
where the 3 represents the three colors which can annihilate, and 22 represent
the color assignments of the remaining quarks, all of which add incoherently.

The Strong Interactions and QCD
193
One can use SU(2)xSU(2) chiral symmetry to show that the chiral anomaly
associated with the triangle diagram in Figure 5.7 dominates the 7r° —> 27
decay amplitude in the m^ —> 0 limit (Adler, 1969; Cheng and Li, 1984).
This again counts the number of quark colors, so that
r (vr0 - 27) ~ (f )2 gS| + h.o.t. = 7.76(2) (f)' eV, (5.16)
where /^ = 130.4(2) MeV is the pion decay constant, which is associated
with the spontaneous breaking of the chiral symmetry and is measured in
7T+ —> y+v decay. The experimental value, 7.8(5) eV therefore yields Nc = 3.
5.3 Simple QCD Processes
In this section we sketch the derivation of some simple QCD processes at tree
level. Some of the calculations are similar to the QED calculations in Chapter
2, except that one has to properly take the color factors into account. Others
involve the gluon self-interactions, which have no QED analog. Processes
involving non-abelian vertices are extremely tedious to carry out by hand,
and are best handled by specialized computer algebra programs (see the list of
websites in the bibliography). However, we will illustrate one relatively simple
example. Higher-order calculations involve all of the subtleties of gauges and
ghost loops (the latter may even appear in tree-level calculations involving
external gluons), which are treated in standard field theory texts.
Color Identities
The calculations are greatly simplified by the use of certain color identities
listed in Table 5.1 for the fundamental representation matrices L\ — Xl/2
(denoted in this section by Ll) and the 517(3) structure constants fak defined
in Tables 3.1 and 3.2. They follow easily from or are special cases of the
identities given in Sections 3.1.2 and 3.1.3 and in the Problems in Chapter 3.
qq and qq Scattering
qq and qq scattering via gluon exchange are very much like the simple QED
processes described in Section 2.8. Here we will neglect the quark masses for
simplicity, but it is straightforward to include them (as would be necessary
for the production of a heavy quark, such as uu —> tt). The process qr/3qra ->
Qs-yilsS, where a,/?,7, and S are color indices and r ^ s are flavor indices,
proceeds through an s channel gluon, as shown in the first diagram in Figure
5.8. The only differences compared with the QED process e~e+ —> // in

194
The Standard Model and Beyond
TABLE 5.1
517(3) color identities for U = Xl/2 and fijk. VV = J2{ UL\ I is the
3x3 identity matrix, and the indices run from 1 to 8. There is no
distinction between upper and lower indices.
UU = \I
Jijkjijm = dvkm
Tr (lMJLk) = \ (dijk + ifijk)
Tx{LiULk)ifijm = -\8km
Tr {IMJVL") = -±6jk
Tr {iStitilS) = *§■
Jilmjjmk ~ Jimkjjlm ~r Jijmjmlk
JilTndjTnk ~ Jimkjjlm ~r Jijmjmlk
= 0
= 0
Tr {ISV) = \5ij
Jijkjijk = ^4
Tr {pULk) ihjk = -6
Tr (UVUU) = -§
Tr (UU) Tr {iSti) = 2
(Jacobi identity)
(2.221) and Figure 2.15 are that -e2Qf —> g* and that there are color factors
on the vertices and gluon propagator, as shown in Figure 4.1, yielding
QyiPs) Q6(P4)
k
Q<y(Ps) Qs(P4)
Sik
n fn \ L^* (n \ 9/3(Pl) Qcc{P2)
qpvPi) Qc*(P2)
FIGURE 5.8
Diagrams for qr/3qra —> Qs-y<is6, where r and 5 are flavor indices. Only the
diagram on the left contributes for r ^ s.
Mfi = {-igs us^LlQ0v4) f —±- J {-igs v27PL«su1)
i9s i
(5.17)
The calculation of the spin-average cross section proceeds as in e~e+ —> //,
Equation (2.223), except it is now convenient to do a color average as well,
in which one averages (sums) over initial (final) quark colors a = 1 • • • 3.
Similarly, in processes involving external gluons one averages (sums) over
initial (final) gluon color indices i = 1 • • • 8. Neglecting the quark masses, the

The Strong Interactions and QCD 195
expression 2Q2fe4(t2 + u2)/s2, derivable from (2.223), is replaced by
(5.18)
Similarly, for annihilation into the same flavor, qrpqrOL —> qrlqr5, the
calculation is similar to Bhabha scattering in (2.229)-(2.231). The first two terms
are obtained by the replacement e4 —> 2#4/9, just as in (5.18). However, the
third (interference) term now has the color factor
9
so that
^WW = l^ {UtiVti) = -±, (5.19)
\Mfl\2 = \A
t2 + u2 s2 + u2 2u21
+
t2 3 st
(5.20)
The spin and color-aver aged squared matrix elements for a number of 2 —> 2
QCD processes, neglecting masses, are listed in Table 5.2, which is expanded
from (Combridge et al., 1977; Barger and Phillips, 1997). More extensive
listings, including mass effects and extensions to supersymmetry, may be found
in (Amsler et al., 2008).
GG -► q0qo
To illustrate a non-trivial non-abelian vertex, consider the process GG —>
qoqo, where qo is a hypothetical spin-0 color triplet, such as a scalar quark in
supersymmetry. There are four tree-level diagrams, as shown in Figure 5.9
Using the vertex factors from Figure 4.1, the amplitude in Feynman gauge is
M =
^\^1v
WsL^(P3 - P4)a f j 9sfijk
x [<r (pa - piY + <r (2pi+p2r - <r>i + ^y]
+ ifi<r{L\V}ae (5.21)
+ (-igs)2 {Vti)ap (^2) (2?3 -PiTiPs -Pi -PaY
+(-igs)2 (VU)a(5 (^2) (Pi " 2p4)^(Pi +P3 -PaY\ ,
where mo is the #o mass, and eiM = eM(pi,Ai) and 62^ = ^(^2,^2) are the
gluon polarization vectors. The straightforward way to calculate \M\2 would
be to first take the absolute square and then use (2.117) for the gluon
polarization sums. However, this would be extremely tedious. The calculation

196
The Standard Model and Beyond
TABLE 5.2
Spin and color-averaged squared amplitudes \M\2/g* for various QCD
subprocesses, characterized by kinematic subprocess invariants s, t, and u.
r ^ s when flavor indices are given. The last two processes involve the
production of a hypothetical spin-0 color-triplet qo, such as one encounters
in supersymmetry. Masses are neglected. The last column is the numerical
value for CM scattering angle 90°, where t = u = —s/2 (neglecting masses).
Expanded from (Combridge et al., 1977; Barger and Phillips, 1997).
\M\2/9J ~90^
qrqs -> qrqs \ 4 (s2+u2\ TT~
qrqs^qrqsJ 9 V <2 J ZZ
qrqr -> qsqs § (^^) 0.2
nn _w nn 4 ( t2+u2 ■ s2+u2 2 u2 \ 0a
qq^qq g[~p~+ ~~^—3^fJ 2-6
nn —> nn 4 /s2+n2 _l s2+*2 2 s2 \ o o
GG - W^ [l (t^ _ 3 (V±u^l x ( 1 \ (0.1
qq^GGJ [e\ tu J 8 \ s* )\X\6A/9J \l.O
qG^qG (^)_|(V^) 6J
GG^GG | (£$£ + ^ + £$£ + 3) 30.4
qq^qo% $% o.i
GG^qoqo i(V)2 + ^ 0-07
P3,<* P4,/3 P3,<* p4,/3
Pi,*, M P2,J, ^ Pi,*, M P2,j, *>
P3, <* p4, /3 P4, /3
\ /
?P3 — Pr
Pi,*, M P2,J, V Pi,*, M P2,J, ^
FIGURE 5.9
Diagrams for Gj,(pi)G£(p2) -> 9oa(P3)<7oMP4), where 90 is a hypothetical
spin-0 color triplet.

The Strong Interactions and QCD
197
w0uld be simplified if the second term in (2.117) did not contribute, but this
requires the calculation to be done in a gauge invariant way (cf., the
discussion of Compton scattering in Section 2.8). In particular, for a non-abelian
theory one must include the negative contribution of fictitious ghost pairs in
the final state (Peskin and Schroeder, 1995, Section 17.4). Although this is
straightforward, it can be avoided by introducing explicit expressions for the
polarization vectors, just as we did in (2.147).
For the qo% final state it is simpler to calculate the amplitudes rather
than their absolute squares, analogous to the fermion helicity calculations in
Section 2.9. The four-momenta in the CM frame are
pif2 = fc(l, 0,0, ±1), p3,4 = £/(l, ±0f sin0,0, ±/3f cos 0), (5.22)
where k = y/s/2, kf = y/s — 4?tiq/2 = /?/£/ = /3/fe, and 6 is the CM
scattering angle. The gluon polarization vectors are
cif2(l) = (0, ±1,0,0), ci,2(2) = (0,0,1,0)
(5.23)
in a linear basis, where en(An) = en(pn, An), n = 1, 2. (The sign for e2(l) was
chosen to be consistent with the space reflection convention in (2.267).) Thus,
6i(Ai)-e2(A2) = (-l)Al+1«5A'A2
Pi -en(A) =p2 -en(A) =0
p3-ei(l) = -p4-ei(l) = ~P3-e2(l) = P* ■ e2(l) = -kfsin6
P3 • e„(2) = p4 • e„(2) = 0,
(5.24)
which greatly simplify the calculation. Denoting the amplitude by M(Ai, A2),
one has M(l, 2) = M(2,1) = 0, and only the g^ terms contribute to M(2,2),
M(2,2) = -^s2
-ifijkL%(^^{L\V}^
(5.25)
t and u are related to 6 by
t-ml = -2fc2(l-/?/cos0),
so that
u — ml
-2/c2(l+/?/Cos0), (5.26)
•-¥)
k
cos6>. (5.27)
M(l, 1) has the same g^teivas as M(2,2), as well as contributions from the
t and u-channel pole terms. These are easily calculated, yielding
M(1,1) = M(2,2)
M(2,2)
-1 +
2fc^ssin26>
(t-ml)(u-ml)
l + 2m5
1 1 Y
2 + 2
- TTIq U — TUq )
(5.28)
= M(2,2)[1 + X],

198
The Standard Model and Beyond
where (5.26) was used to obtain the second form. The spin and color-average
amplitude squared is
1 1
4 64
W2 = ^EE EMAi.Aa)!3
i,j a,/3 Ai,A2
(5.29)
where the factors of 1/4 and 1/64 are respectively due to the averages over
the gluon spins and colors. There is no interference between the two terms in
M(2,2) because they are respectively antisymmetric and symmetric in i and
j. From Table 5.1, the color sums for the squares of these terms are 12 and
28/3 respectively, so that
and
|M|2 =
da
64
1
u — t
kf I »>|2
7
+ 3
[1 + (1 + *)2],
w
da
~d~i
1
r|M|2.
(5.30)
(5.31)
dcosO 327T5 k r"' dt 167rs2
One can obtain the amplitudes for left and right circularly polarized gluons
using (2.114),
M(L, L) = M{R, R) = \ [M(l, 1) - M(2,2)]
-M(2,2)
A^ssin2 6
(t-m§)(u-m§)
1,
M{L, R) = M(R, L) = - [M(l, 1) + M(2,2)]
(5.32)
= M(2,2)
k2fssm26
(t - ml){u - ml)
M(L,R) and M(R,L) vanish in the forward and backward directions
because of angular momentum conservation, i.e, the gluon spins are in the
same direction and cannot be compensated by orbital angular momentum^.
M(L,L) = M(R,R) and M(L,R) = M(R,L) follow from reflection invari-
ance.
5.4 The Running Coupling in Non-Abelian Theories
It was already described in the introduction and (for QED) in Section 2.12.2
that many of the higher-order corrections to QCD (and other field theories)
§The sin2 9 factor is cancelled by the singularities from the t and w-channel poles for ttiq —► 0,
as can be seen from (5.28) and (5.32).

The Strong Interactions and QCD
199
can be absorbed into an effective gs{fi>2) or as(/i2) = #s(/i2)2/47r, where \i
is the renormalization scale. Higher-order corrections are usually minimized
when \i is taken to be a typical momentum scale of the process, such as
^2 __ g2 = |g|2? where q might be the four-momentum carried by an exchanged
gluon. The gluon self-interactions in QCD imply asymptotic freedom, i.e.,
that as(/i2) becomes small at large /i > 0(1 GeV) (short distance), so that
one can treat the quarks and gluons as weakly coupled, and processes such
as deep inelastic scattering can be calculated in perturbation theory. For
small /i (large distance), as(/i2) becomes large and perturbation theory no
longer holds. The strong coupling and gluon self-interactions presumably
lead to the confinement of quarks, gluons, and any colored states, so that
only color singlet hadron states can emerge. In the real world, both regimes
may be relevant to different aspects of a process, e.g., a quark may scatter
in the short distance regime, but its initial distribution in the proton and
its subsequent hadronization are low momentum processes. Fortunately, for
many processes the short and long distance effects can be factorized, with the
former calculable. The latter must be taken from experiment, although their
logarithmnic Q2 dependence is predicted by QCD.
The deep inelastic scattering experiments at SLAC could be understood
in the simple parton model of point-like constituents of the nucleon, which
had been developed somewhat earlier to understand hadronic processes (see,
e.g., Field and Feynman, 1977). However, it was quickly understood that the
interactions of the partons had to be asymptotically free, so that they could
appear point-like at short distances and still be confined in the nucleon. The
breakthrough came in 1973, when it was shown that non-abelian gauge
theories could be asymptotically free (if there are not too many matter fields), and
furthermore that they are the unique asymptotically free renormalizable
theories in four dimensions (Gross and Wilczek, 1973; Politzer, 1973). Combined
with the evidence for three colors, QCD emerged as the unique candidate
theory.
5.4.1 The RGE Equations for an Arbitrary Gauge Theory
The running of the effective gauge coupling g in a gauge theory with a single
group factor is described by the renormalization group equation (RGE)
dg2 = 4n(3(g2) = bg4 + 0(g6) + ■■■, (5.33)
dlnQ
1 loop 2 loop
where the coefficient of the one-loop term in the /? function is
1
(4tt)2
-C2(G)--TF--T0c--T^
(5.34)
where C2{G) is the quadratic Casimir defined after (3.22), with C2{SU{rn)) =
m and C2{U(1)) = 0. 7>, X^, and T$h are respectively the fermion, complex

200
The Standard Model and Beyond
scalar, and Hermitian scalar Dynkin indices
T0c% = IV (L;eJ4) , T^Sn = Tr (L%hL%h) .
T(Lm) = \ in SU(m) for a fundamental representation; for 17(1) a set of
fields with 17(1) charges qi yields T = X^#?- One must sum over all of the
IRREPs in which the particle masses are smaller than Q, so that there are
discontinuities in the slope at the particle thresholds. (5.33) is easily solved
analytically in one-loop approximation,
where a = g2/^7r and M is an arbitrary reference scale. Thus, l/a(Q2) varies
linearly with InQ2. Asymptotic freedom occurs for b < 0. Since C2(G) and
the Dynkin indices are nonnegative, asymptotic freedom always occurs in a
pure non-abelian gauge theory, but not in U(l)^. In the presence of fermion
and scalar fields, asymptotic freedom may or may not hold depending on their
contribution relative to the gauge terms.
The two-loop contributions to (3 are also known (see, e.g., Martin and
Vaughn, 1994). (They are known to four loops for pure QCD (van Ritbergen
et al., 1997).) However, at this order one must also include diagrams involving
other interactions, such as Yukawa interactions or other gauge factors, which
lead to a coupling between their RGEs. More careful treatment of thresholds
and of the renormalization scheme are also needed at this order.
For QCD, one has C2 = 3, Tp = ^f, and T^ch = 0, where nq is the number
of quark flavors lighter than Q, e.g., nq = 3 for ms < Q < mc (perturbative
results are not expected to be valid below ras), nq = 4 for mc < Q < m^, and
nq = 5 for nib < Q < mt. Thus, at one loop
1 1 33-2n, Q2 _ 1 w Q2
+ 10- ln772 = „. /^2n +62,l"77o- (5-37)
<*s{Q2) as(M2) 12tt M2 as(M2) 2 M2'
This can be rewritten
^ 1
a*{Q) = ^T^i' (5-38)
where the scale A" is defined by
A=M^[w^m) (M9)
* Other renormalizable interactions in four dimensions also satisfy RGE for their running
couplings, but these are never asymptotically free (Coleman and Gross, 1973; Zee, 1973).
"One can define A(nQ> as the value relevant to the region with nq light flavors, with the
discontinuities at the thresholds fixed so that as is continuous.

The Strong Interactions and QCD
201
(see Problem 5.3 for a generalization to two loops). In addition to as{Q2)
becoming small for large Q, it also goes to oo at the scale AQCd = A, at least
in one-loop approximation. Of course, the one-loop approximation and
perturbation theory break down in this limit, but one may nevertheless loosely
interpret A as the scale at which as becomes large. This treatment is easily
extended to higher orders, inclusion of quark thresholds, etc. as has been
determined in many ways at different scales, often from the corrections to simple
parton model results. These include hadronic r decays, T spectroscopy and
decays, e+e~ annihilation event shapes above and below the Z, deep inelastic
scattering, and the width for Z —► hadrons (see the QCD review in Amsler
et al., 2008; Bethke, 2007). The running predicted by QCD is clearly
confirmed, as illustrated in Figure 5.10. It is convenient to quote the value of as
0.3
a-0-2
0.1
«uj
-J I i I I M i I
hadronic Jets
e*e rates
I Photo-production
^Fragmentation
! Z width
ep event shaped
PolarizedDI§?
Deep Inelastic Scattering (DIS)
• ' t decays
.—P—
Spectroscopy (Lattice)
Y decay
10
jiGeV
10
0.1
A I L
0.12 0.14
<*s<Mz)
FIGURE 5.10
Left: running of the QCD coupling as a function of the scale /i ~ Q. The
data points are various experimental determinations and the band is the best
fit QCD prediction. Right: measurements of as(/i2) extrapolated to M^
assuming QCD. Plots courtesy of the Particle Data Group (Amsler et al.,
2008).
at the mass of the Z because that is the scale at which the electroweak
couplings are best determined. There is a precise direct measurement of as(Mz)
from the hadronic Z width, and measurements at other scales can be
extrapolated to that scale using the higher order QCD running and the value of A
obtained in a global fit. One recent analysis (Bethke, 2007) yields the
precise value as = 0.1189 ± 0.0010. It should be cautioned, however, that some
of the inputs to the global fit have nontrivial theoretical uncertainties from

202
The Standard Model and Beyond
higher-order effects.
The observed running corresponds to A ~ (200 — 300) MeV, depending
on the exact definition. This sets the scale at which the strong interactions
become strong and determines the approximate scale of such strong interaction
quantities as the nucleon and p masses. In fact, the physical hadrons not
involving the c or b quark receive relatively little contribution from the bare
quark masses (this is especially true of the non-strange ones), and have masses
dominated by the dynamical or const it utent quark mass Mdyn ~ A, which
(along with the pion decay constant /^(A)) is associated with the spontaneous
breaking of the chiral 517(2) x 517(2) or 517(3) x 517(3) chiral symmetry. The
one exception is the pseudoscalar octet, which are pseudo-Golds tone bosons
with masses generated by the explicit chiral breaking from the bare quark
masses (see Sections 3.3.4 and 5.8).
If one ignores the bare masses, there are no dimensionful parameters in the
QCD Lagrangian. However, A emerges by dimensional transmutation, i.e., it
is the scale at which the dimensionless coupling becomes large. This implies
that only dimensionless ratios like mp/A or mp/A are meaningful, at least in
the mr = 0 limit. One could choose to simply define units so that A = 1,
analogous to c = h = 1. However, it is more convenient to keep the "historical
units" for masses, or to scale everything in terms of easily measured quantities
such as mp. Of course, when one considers other interactions, new scales, such
as the Z mass of the Planck scale, also become relevant.
5.5 Deep Inelastic Scattering
Let us consider deep inelastic lepton nucleon scattering (DIS) in more
detail. In this chapter we focus on e~p —> e~X, where the momentum transfer
Q2 = —q2 ^> 1 GeV2, as shown in Figure 5.11. This limit is in the short
distance regime, so to first approximation one can describe the underlying
process as the scattering of the virtual photon with a free quark, as in Figure 5.4
(the simple parton model or SPM). Only the final electron is observed in the
scattering, and the unobserved hadrons X are summed over. These typically
involve complicated multihadron states. Other deep inelastic reactions
substitute e+,p±, and v M for the e~, with neutrino scattering proceeding through
either charged or neutral current processes through the exchange of W± or
Z. (The HERA e±p scatterings involved all of these processes (Abramowicz
and Caldwell, 1999; Klein and Yoshida, 2008).) The initial proton may be
replaced by a nuclear target, which either gives a weighted sum of p and n
cross sections, or, for bubble chamber experiments, sometimes allows one to
identify whether the scattering was from a p or n.
It is difficult to overestimate the importance of deep inelastic scattering.

The Strong Interactions and QCD
203
DIS via 7 or W± exchange was especially important for establishing the
existence of point-like quarks, while the test of the higher-order corrections to the
SPM helped establish QCD. Similarly, the neutral current processes probed
in v M TV —> v M X were extremely important early tests of the standard
electroweak theory**. More detailed descriptions may be found in (Drell and
Walecka, 1964; Renton, 1990; Barger and Phillips, 1997; Abramowicz and
Caldwell, 1999; Ellis et al., 2003; Campbell et al., 2007; Amsler et al., 2008;
Klein and Yoshida, 2008).
5.5.1 Deep Inelastic Kinematics
The kinematics of DIS are indicated in Figure 5.11. Since the final hadrons
are not directly observed, the independent variables are the four-momenta p,
&, and k' of the proton and of the initial and final electron. Except for the
HERA e±p collider, all of the experiments have been performed in the proton
or nuclear rest frame (the lab frame), where the observables are the energies
Ek,k' of the initial and final electrons, and the electron scattering angle 0.
e-(fc) e-(fc') Ek,
FIGURE 5.11
Left: Deep inelastic scattering e~p —> e~X, where X represents unobserved
hadrons. q = k — k' is the four-momentum of the virtual photon. Right:
Kinematics in the proton rest frame. Ek and Ey are the energies of the
initial and final electrons, and 6 is the laboratory scattering angle.
There are a number of useful related kinematic variables, which we express
** Additional critical tests of the electroweak theory and detailed studies of the nucleon (e.g.,
the role of the strange quarks (Beise et al., 2005)) involved elastic and inelastic electron or
muon asymmetries with polarized lepton beams at SLAC, Jefferson Lab, Bates, and Mainz.
Experiments involving both polarized leptons and polarized nucleons at DESY, CERN,
SLAC and Jefferson Lab and involving polarized pp scattering at RHIC have probed the
spin distribution of the constituents of the nucleon (Manohar, 1992; Bass, 2005).

204
The Standard Model and Beyond
both in Lorentz invariant form and in terms of the lab frame observables. The
total CM energy squared is
s = (k + p)2 —>m2e+M2 + 2EkM ~ M2 + 2fcM, (5.40)
lab '
where M is the nucleon mass, and k ~ Ek is the magnitude of the e~ three-
momentum. In the remainder, we neglect the electron mass me. The
momentum transfer-square and energy transfer to the hadrons are
Q2 = -q2 = -(*' - k)2 —> 2kk'{l - cos6)
lab
(5.41)
(5.42)
. _ >Ek-Ek,~k-k'.
M lab
The invariant mass-square of the unobserved final hadrons is
W2 = P£ = (p + g)2 = M2 + 2Mi/ - Q2
—► M2 + 2M(A; - jfc') - 2fcfc'(l - cos0).
lab
Elastic scattering X = p is a special case with W2 = M2 and Q2 = 2Mv.
The next hadronic threshold is for a nucleon plus one pion, i.e., X = p + 7r°
or n + 7T+, corresponding to W2 > (M + mn)2. Still larger W2 can involve
more complicated many-particle states, which are summed over. The three
independent kinematic variables are (s,<32,i/), or equivalently (fc, fc',0) (only
two are independent in the special case of elastic scattering). However, the
hadronic part of the process can only depend on the two Lorentz invariants
Q2 and v. For a fixed initial energy /c, the variables Q2 and v (and therefore
W2) can be varied and determined from k' and 0. The relation of x, W2, and
6 to v and Q2 is shown in Figure 5.12.
It is convenient to define dimensionless variables
x = -^- y- V Ek~Ek' k~k' (543)
X~2Mu' V~Ek- Ek k ■ (5'43)
x is defined kinematically. However, in the SPM x will be interpreted as the
fraction of the proton momentum carried by the scattered parton^. y is the
fraction of the e~ energy in the lab frame that is transferred to the hadrons.
Their ranges are
0 < x < 1, 0<y < w > 1. (5.44)
5.5.2 The Cross Section and Structure Functions
First consider elastic scattering e~p —> e~p for a hypothetical point-like
proton. The spin-averaged cross section from the first diagram in Figure 5.13
tt Early papers often used the variable uj = 1/x.

The Strong Interactions and QCD
205
x = 1
x = 0.5
x = 0.25
2Mv
FIGURE 5.12
Kinematic variables for deep inelastic scattering for fixed initial energy Ek ~
k > rae. The horizontal (vertical) axes are 2Mv = 2My/k (Q2 = 2Mxy/k),
each running from 0 to 2Mk. The sloping solid lines are for fixed 0 < x =
Q2/2Mv < 1. The dashed lines are for fixed hadronic invariant mass-square
W2 = M2 + 2Mv - Q2, where W2 = M2 along the x = 1 line, increasing as
one moves down and to the right. The dotted lines are for fixed laboratory
scattering angle 0, with 03 > 02 > 0\.
FIGURE 5.13
Left: Elastic scattering e~p —> e~p. The shaded circle represents the effects
of the strong interactions. Right: Deep inelastic scattering e~p —> e~X.

206 The Standard Model and Beyond
(with a point vertex) is
_ {2v)H\k> +p'-k-p) d3k' &p> e4
d°~ AMk (2itf2Ek,{2*)*2Evlqie Wl" (5'45)
where q = k — k' = p' — p, and the traces are collected in the leptonic and
hadronic tensors
L? = \Tr WOt + meWUt' + me)} =2
21
L^u = ^ M* + Mh»W + Af)] = 2
(5.46)
The cross section can be rewritten
da a2 k
2 u
dk'dtl q4 k
where dVt = d cos 0dip, and
1 1
= --L^W^ (5.47)
W^ =
(5.48)
The current in (5.48) is the proton part of the electromagnetic current
J% = $p-f1>p = J$, (5.i9)
with
<pV| Jq(O)Ijw) = «(p', a')7"«(p, *). (5.50)
Of course
H^V*') = (PV|J£|ps>*. (5.51)
Jq is actually Hermitian, but it is useful to write (5.48) and (5.50) in a more
general form for a later extension to the weak interactions. The tensor W^v
contains all of the information about the hadrons.
The cross section expression in (5.47) can be immediately extended to elastic
scattering from a physical (strongly-interacting) proton, provided one replaces
(5.50) by
(p'sVm\Ps) "> u(Pf,s')T»Qu(p,s), (5.52)
where rg(p',p) is the vertex function which includes strong corrections. As
discussed in Section 2.12A, the combination of Lorentz covariance,
electromagnetic current conservation, and the observed reflection invariance of the
strong interactions restricts Tq(p',p) to the form
Tq = 7-W) + ^m\ (5-53)

The Strong Interactions and QCD
207
where Fl^iQ2) are f°rm factors that can depend on q2 with Ff(0) = 1 and
fP(Q) = /cp, where kp ~ 1.79 is the anomalous magnetic moment of the
proton. For elastic scattering k' and cos 6 are not independent, but are related
by
k: i
(5.54)
k!_
k
1+A(i_cos0)'
(enforced by an energy-conserving delta function in W^). From (5.47) and
(5.53) one obtains the Rosenbluth cross section formula for elastic scattering
given in (2.394), with the correspondence of notation (fci, k2) —> (A:, kf), 6l —>
0, and mp —> M.
Expression (5.47) continues to hold for the inelastic case provided the
hadronic tensor is redefined as
WL
fiiy
AnM
s / L N n=l v ' prt s' \ n
x(ps\A (0)\XN)(XN\JQ„(0)\ps)
(5.55)
where X^ is an TV particle state which may contain both fermions and bosons.
In the inelastic case k' and cos# are independent variables related to the
invariant mass W2 by (5.42). For unpolarized protons the tensor W^v can
only depend on the four-vectors p and q (it can also depend on the spin
vector s in the polarized case). The only tensors one can construct are
symmetric:
antisymmetric:
PtiQv ~ Q/jlPu,
e^ivpaP Q )
(5.56)
each of which can be multiplied by a function of the Lorentz invariants Q2
and v. One can use the reflection invariance of the strong interactions to
show that the e^pa term is absent. In any case, the leptonic tensor L£u is
symmetric**, so one can keep just the symmetric part of W^. Furthermore,
the electromagnetic current is conserved, O^Jq^ = 0, which implies
<Z"WW = 0. Q'W^ = 0. (5.57)
Thus, only two linear combinations survive, and the most general form is
W^ =
+
^i(qV)
1
M2
p»
p-q
Pu~
p-q
-Qv
W2(Q2,v),
(5.58)
**Both the leptonic and hadronic tensors have e^upa terms for parity-violating weak
processes such as vN —> vX and vN —► [iX due to the interference between the vector and
axial currents.

208
The Standard Model and Beyond
with Q2 = —q2 > 0. The real Lorentz invariant functions Wi,2(Q2,v) are
known as the proton structure functions. They generalize the form factors
F\,2(q2) of elastic scattering, and contain all of the information about the
strong interactions effects. One can combine (5.47) and (5.58) to obtain
da
dk'dtl
a"
^4 0
2
W2 (Q\ u) cos2? + 2W1 (Q\ u) sin2?
(5.59)
for the deep inelastic cross section in the proton rest frame. The solid angle
element is usually integrated over the azimuthal angle, dQ = dipd cos 6 —>
2nd cos 0, and the kinematic variables can be rewritten
dk'd cos 6 =
-dQ2dv = f f dxdy
2kk
k'
(5.60)
to yield
do
dxdy
= 2Mkzy
da
dQ2dv
nMa2y
2/cA/sin4f
W2 (QV) cos2 e- + 2W1(Q2,v) sin2 °-
(5.61)
W\ and W2 can be determined separately from the data by varying 0, /c, and
kl for fixed Q2 and v.
An alternative notation is to use the variables x and Q2 and to define
Fi (*, Q2) = MW1 (Q2, v), F2 (x, Q2) = vW2 (Q2, v) .
(5.62)
This is useful because the simple parton model (QCD) predicts that Fi is
independent of (slowly varying with) Q2 for Q2 large. Then,
d2o _ Sna2Mk
dxdy ~ Q*
_ Sna2Mk
2
F2(x,Q2)(l-2/-^)+2xF1(x,Q2)
2xFl (1 + (12~^)2) + (F2 - 2xF1) (1 - y)
|2"|
(5.63)
MxyF2
2k
In the second form, the last term vanishes for k ^> M, while the middle term
vanishes in the simple quark parton model.
The deep inelastic limit is denned as Q2\v —> 00 with x = Q2/2Mv fixed.
If the proton were an extended fuzzy object, one would expect i^(x, Q2) —> 0
in this limit. However, the MIT-SLAC experiments (Friedman and Kendall,
1972; Mishra and Sciulli, 1989) circa 1970 showed instead that
Fi(x,Q2)
Q2
Fi{x) ^ 0,
(5.64)
a property known as Bjorken scaling (Bjorken, 1969). This scaling can be
understood in the Feynman parton model (e.g., Field and Feynman, 1977;
Drell et al., 1969), in which the proton is made up of hard point-like parton

The Strong Interactions and QCD
209
constituents. The observed y distribution showed that the partons have spin-
I, consistent with QCD and asymptotic freedom. Subsequent experiments
established that the scaling is only approximate, and in fact the slow
(logarithmic) variation of the Fi with Q2 is what one expects from the higher-order
corrections in QCD*.
5.5.3 The Simple Quark Parton Model (SPM)
The cross section for elastic scattering e~p —> e~p from a point proton is
given in (2.386), with the kinematic constraint for k' given in (5.54) (see the
subsequent comment on notation). It is convenient to rewrite (2.386) as
da a2 7T [ 2 6 Q2 . 2 6]
W = ^nTf W r 2 + 2M* Sm 2J ' (5*65)
where we have used dQ2 — 2k'2dcos6, which follows from (5.41) and (5.54).
For elastic scattering, Q2 — 2Mv from (5.42), so (5.65) can also be written as
da a2 7T r 2° Q2 ■ 2 0] / n2
dQ2dv 4fc2sin4ffcA;'
COS 2 + 2M*Sm 2
\ 2MJ
(5.66)
This is a special case of the general formula (5.61) provided we identify the
structure functions as
K J v y (5.67)
"wv>-«("-£)-i«(i-|y-
Now assume that the proton is a bound state of point-like quarks. Consider
the contribution to the structure functions from an individual quark qi with
electric charge e^ which carries four-momentum a^p, where p is the proton
momentum. It is plausible that (5.67) applies for that quark, provided one
replaces M —> XiM and multiplies by e2, i.e.,
(5.68)
(This result can be better derived in the infinite momentum frame, where the
proton and quark masses and the transverse momentum of the proton relative
* Scaling also breaks down at low Q2, where strong coupling effects are important. In
particular, hadronic resonances in the jp channel for fixed W2 are important. It is
interesting that the scaling behavior of the structure functions smoothly interpolates these
resonances (Bloom and Gilman, 1971), as can be understood from finite energy sum rules
(FESR), derivable from analyticity.

210
The Standard Model and Beyond
to the electron direction are negligible.) The contribution of that quark to
the structure functions is
F?(*,Q2) = MW{{Q\v) = \e\?> L - J^)
Fl{x,Q2) = vW\{Q\v) = fas (x, - ^L_J .
To find Fi^Oe, Q2) one must sum over the quark types and integrate over their
possible momenta. This is done in the cross section (i.e., in F\^) because
the different i and Xi lead to different (incoherent) final states. Introduce
the parton distribution function (PDF) qi(Xi) as the probability density (the
absolute square of the momentum space wave function) for finding quark qi
in the proton with momentum fraction #*. Then from (5.69) the predicted
structure functions are
F2(x,Q2)=2xF1(x,Q2)
= Ylj dXiqi(xi)e2Xiti(xi-^^) = ^e2xqi(x).
The simple parton model therefore predicts that the Fi are independent of
Q2 for large Q2 (Bjorken scaling). The quantity xqi{x) is interpreted as the
momentum distribution for parton qit
The predicted relation F2 = 2xFi, the Callan-Gross relation (Callan and
Gross, 1969), is a signature of spin-^ constituents. Scattering from spin-0
constituents would lead to F\ — 0, F2 ^ 0 and therefore a different angular
distribution. An interpretation of this result is that one can show (see, e.g.,
Renton, 1990) that
ii(x'Q)"^~-^-' (571)
where o^ and gt are respectively the total cross sections for 7^ Tp, where
72 T is a virtual photon with momentum </, and L and T refer to longitudinal
(photon helicity 0) and transverse (photon helicities ±1) polarizations. In
the Breit frame (discussed in Section 2.3.4), the virtual photon has four-
momentum q = (0,0,0, —Q), while the incident [final] parton has momentum
^(Q,0,0,Q) [^(Q,0,0, —£?)], as in Figure 2.4. For spin-0 partons one would
have gt = 0 (R = 00) by angular momentum conservation, since there is
no orbital angular momentum along the direction of the photon and parton
momenta. For spin-^ partons, on the other hand, gl = R = 0 using the fact
that helicity is conserved for vector transitions of massless spin-^ particles,
e.g., (2.210).
Even to the extent that the SPM is valid, a real proton is expected to
consist of not only the three valence quarks uud of the quark model, but also
a sea of qq pairs and of gluons produced by soft (low momentum) processes
such as illustrated in Figure 5.14. The anitiquarks contribute to the Fi with

The Strong Interactions and QCD
211
FIGURE 5.14
higher-order soft processes that produce a sea of gluons and qq pairs.
the same formula, except <&(x) —> <&(x),
F2 (x,Q2) = 2xF1 (x,Q2) = ^e2 xfe(x) + £(*)).
(5.72)
The gluons do not contribute directly to electromagnetic processes (they do
contribute to purely hadronic short distance processes). It is convenient to
write
qx{x) = qVi{x) + qsi(x), qi{x) = qSi{x), (5.73)
where qvi and qsi represent the valence and sea quarks, respectively. The
quark distribution functions are determined by long distance effects, and at
present there is no way to reliably calculate them. However, there are a
number of plausible constraints and consistency conditions. One expects
QSi(x) ~qsi(x),
(5.74)
as is suggested by the diagrams in Figure 5.14. However, this is not a rigorous
result and there could be small deviations. A more precise result is
/'
Jo
dx[qSi(x) -qsi{x)] = 0.
(5.75)
This follows from the meaning of valence and sea quarks, and should hold
to the extent that the SPM is valid. Similarly, the proton should have two
valence u quarks and one valence d,
/ uy{x)dx = 2, / dy{x)dx = 1,
Jo Jo
which implies
/ [u(x) - u(x)] dx = 2, / [d(x) - d(x)] dx = 1
Jo Jo
/ [s(x) - s(x)] dx = 0, / [c(x) - c(x)] dx = 0.
Jo Jo
(5.76)
(5.77)

212
The Standard Model and Beyond
These predictions are difficult to measure, but existing data is consistent.
Because of quark mass effects one expects b < c < s < u,d, and for the
kinematic region relevant to the MIT-SLAC experiments it is reasonable to
neglect the 6,6, c, and c. Since the u and d quark masses are so small compared
to A it is a reasonable first approximation to expect u(x) ~ d(x). This is not a
rigorous consequence of isospin since the proton is not an isosinglet, but should
hold approximately because of the isospin invariance of the gluon coupling.
Another constraint is that
Y^ / x [qi(x) + qi(x)] dx+ xG(x)dx = 1, (5.78)
i Jo Jo
where G(x) is the gluon probability distribution, which can be probed in
hadronic processes. (5.78) states that the total momenta of all of the
constituents must add up to the proton momentum.
From (5.72), one predicts
F2(x) ~ -x [u(x) + u(x)] + -x [d(x) + d(x)] + -x [s(x) + s(x)]. (5.79)
y y y
One cannot distinguish the various distribution functions using e~p —> e~X
data alone, but separation between the quarks and antiquarks and between
flavors can be accomplished by considering scattering from neutrons and other
deep inelastic processes, such as
v // V —► MT^ and e±p —> v e X, (5.80)
as will be described in Chapter 7. One finds that the momentum fractions of
the proton constituents are of order
/ xu{x)dx ~ 0.30, / xd(x)dx ~ 0.15, / xG{x)dx ~ 0.50. (5.81)
The momenta carried by the u + d is about 10% that of the u and d, while
the 5 + 5 momentum fraction is about half of that. These rough estimates
apply either to SPM analyses of relatively low Q2 data (e.g., Field and Feyn-
man, 1977), or as the values at a low initial Q2 (e.g., 5 GeV2) in the more
sophisticated QCD improved model described below. The actual momentum
distributions obtained in a recent analysis are shown in Figure 5.15.
The quark distributions and structure functions defined above referred to
the proton, and it is sometimes useful to denote them as W[,Ff,q? to
emphasize that fact. One can define analogous quantities for a neutron target,
such as q™. The SPM prediction is
F2" (x, Q2) = 2xF? (x, Q2) = £ e2 x (tf (*) + #(*)). (5.82)

The Strong Interactions and QCD
213
The neutron distribution functions are not independent, but are related to
those of the proton by isospin, i.e.,
un{x) = dp{x), dn{x) = up{x), sn{x) = sp{x) (5.83)
up to possible small isospin-breaking corrections. Similar results hold for the
antiquarks. It is also useful to define isospin-averaged distribution functions
us(x) = \ W{x) + un(x)} = \ K(z) + dP{x)}
1 i (5-84)
ds{x) = - [<P{x) + d"^)} = - [<F(x) + up(x)} = us(x),
analogous to (2.396). These are the distribution functions per nucleon for
nuclei with equal numbers of protons and neutrons such as 12C or (up to a
correction) 56Fe. For such targets,
Fi (x, Q2) = 2xF? (x, Q2) = J2 <%* [flf (*) + flf (*)] • (5-85)
i
Most experiments are done on heavier nuclei because higher statistics can be
achieved, even though information on the isospin structure is lost.
The simple quark parton model was important because it approximately
described the observed properties of deep inelastic scattering at moderate Q2.
The approximate scaling established the existence of point-like (i.e.,
asymptotically free) constituents of the nucleon, and the relation ^(x) ~ 2xFi(x)
indicated that the constitutents have spin-^. The measured distribution
functions q(x) give more detailed information about the distribution of the quarks
within the nucleon. Further implications of deep inelastic scattering processes
involving W± and Z exchange will be described in Chapter 7. We now turn,
however, to the corrections to the SPM as predicted by QCD.
5.5.4 Corrections to the Simple Parton Model
The scattering from non-interacting point-like quarks in the SPM leads to
(22-independent structure functions Fi^{x^Q2) —> Fi^{x). However, the
asymptotic freedom of QCD predicts that as(Q2) is small but nonzero at
the observed scales, so one expects higher-order corrections to scaling and
the SPM. It is straightforward to show (see, e.g., Renton, 1990; Barger and
Phillips, 1997; Ellis et al., 2003) that diagrams such as those in Figure 5.16
lead to logarithmic Q2 dependence of the structure functions. It is possible
to interpret these corrections as leading to an effective Q2 dependence of the
quark and gluon distribution functions, i.e., the QCD improved parton model
A heuristic picture is that at moderate Q2 (e.g., 20 GeV2) there are
relatively few gluons or sea quarks, at least in the region x > 0.1. At higher
Q2, on the other hand, the virtual photon or other probe can resolve more qq
pairs and gluons associated with the splittings in Figures 5.14 and 5.16. One

214
The Standard Model and Beyond
FIGURE 5.15
Momentum distributions of the gluon (#), valence quarks (uv,dv), and sea
quarks (w, J, s,c) from a recent global analysis (Martin et al., 2007), at \j?
(i.e., Q2) = 20 GeV2 (left), and//2 = 104 GeV2 (right). The heavy quarks
become more important for large (J2 and small x, where mass effects are less
important. Plot courtesy of the Particle Data Group (Amsler et al., 2008).
e"(fc) e-(fc') e"(fc) eT{ti)
FIGURE 5.16
Lowest order QCD corrections to the simple parton model.

The Strong Interactions and QCD
215
therefore expects a somewhat reduced momentum distribution for the valence
quarks, and enhanced sea quark and gluon distributions at low x. This
indeed is the case, as can be seen in Figure 5.15. The most dramatic effect is
for x < 0.05, which has especially been studied at the HERA ep collider at
DESY. The observed dependence of F2(x, Q2) for fixed x and varying Q2 is
shown in Figure 5.17.
The distribution functions fi(x,Q2) where fi = q^q^G therefore depend
both on x and Q2 in a complicated way. Fortunately, one can parametrize
them and test QCD by a two step process. First, fi{x,Ql) can be measured
at some convenient reference scale Qq, thus determining the long distance
effects that cannot be calculated perturbatively or by other presently available
techniques. Then, the InQ2 evolution of the fi(x,Q2) to larger Q2 values
can be predicted from QCD and compared with the experimental data. In
practice, one generally assumes (physically motivated) analytic expressions for
the /^(x, Qq) in terms of several unknown parameters, and then determines
those parameters as well as ots{Q2) (i.e., A) by a fit to the data at all Q2. The
evolution actually depends on all of the distribution functions, including the
gluon's. Therefore, modern fits often make use of all relevant data from the
various deep inelastic and other short distance processes.
It is not sufficient to consider only the lowest order diagrams in Figure
5.16. These scale as as(Q2) InQ2, which is not small. It is possible to sum the
leading order (LO) contributions of 0((as lnQ2)n) for all n by integrating the
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP), or just Altarelli-Parisi,
equations (see, e.g., Altarelli and Parisi, 1977)
dG{x,Q2) _ as{Q2) f1 dw
d\nQ2 ~ 2tt
/SeMsW-^+Ms) «<••<«
dgj(x,Q2) = as(Q2) f1 dw
d\nQ2 2tt
(5.86)
where q^ includes both qi and q^ In (5.86), the Pf^^z) are the splitting
functions which describe the probability for parton fi to emit parton fj
carrying a fraction z < 1 of the fi momentum. Their explicit forms are given
in (e.g., Ellis et al., 2003). The splitting functions are perturbative and can
be expanded in as. The zeroth order term (one factor as is already extracted
in (5.86)) yields the LO approximation. The next to leading order (NLO)
terms, of order 0(a1^(\nQ2)n~1), involve the 0(as) corrections to the
splitting functions. The splitting functions have been calculated to NNLO (for
the original references, see Amsler et al., 2008), which are used in some of the
recent analyses*.
tOther quantities, such as as(/x2) and the parton-parton cross sections for short-distance
hadronic processes must be calculated to the same order.

216
The Standard Model and Beyond
.-* 10
10
10'
10
10'
10
10
10
10
10
10
10
-1
p
I8
k7
1
I6
I5
I4
I3
\2
)
-1
-2
-3
-
1:
x=0.000063
#*** x=0.000102
^ • ** x=0.000162
„•*** A. x=0.000253
^•~# x=0.0004
..• "*"\^? x=0.0005
****** x=0.000632
**.-•••" . x=0.0008
•" ++***** x=0.00102
*****...• x=0.0013
°° ^ .**■***""" x=0.00161
****...• x=0.0021
o ° • ..♦*••
°o *****^ x=0.00253
^1
Proton 1
• HI
• ZEUS
O BCDMS
O E665
□ NMC
A SLAC
°°° •***.. A x=0.0032
o0°^Q- **\.^ **"
ootf"o0*o «*"*^ #**—• x=0.008
ncPOB,0.*^ ###.,t* ****** ****• X = 0-021
DoDooo^«'*^1*0' #.# ,**•*.** *•**•* x=0.032
nnaR,«A-— ' " ,##^#t*..#. X = 0.05
ip******^ dd ^^^y**
aAa nj&Bti&t&G B o esBWW^
^>ft/ft&A AAQSh4Q/S4s n tJ<W&ft^8Mtf<
MA . r . =, „
#* • • • ♦ «*•*»* *•**# ft *
tox* * ****** ****** *
|^&** •♦••• • ******* *
A
A
* x=0.08
* * x=
* ♦
t • *
i t ♦ ,
tM'
0 13 ,
x=0.18
1
r x=0.25
x=0.4
x=0.65
x=0.75
x=0.85 (i„=1)
10
10'
10'
10
10'
10
Q2(GeV")
FIGURE 5.17
F2(x, Q2) for the proton for various fixed x. Plot courtesy of the Particle Data
Group (Amsler et al., 2008).

The Strong Interactions and QCD
217
It is possible to integrate the DGLAP equations numerically. One can also
consider the moments (Mellin transformations) of the distribution functions,
fi(N,Q2)= [ xN-lh{x,Q2)dx. (5.87)
Jo
The moments of the flavor non-singlet (ns) distribution functions (such as
q—qi or qi~Qj) satisfy simple first order equations, which are easily integrated.
In LO the non-singlet moments are predicted to scale as
fns(N,Q2) = fns(N,Ql) (^ij)7" , (5-88)
where the anomalous dimension 7^ is a known function of the number of
light flavors, calculable from Pqq and as. Similarly, the moments of the G and
singlet (i.e., X^fe + Qi)) distribution functions satisfy coupled differential
equations. After solving, the distribution functions can be recovered from the
inverse Mellin transformation. The non-singlet moment evolutions in principle
provide very clean tests of QCD and determinations of as, since they are
independent of the gluon distribution, but in practice the tests are limited by
uncertainties in the distribution functions.
There are many complications to the description of deep inelastic scattering.
For very small rr, as studied at HERA, there are important ln(l/:r)
contributions to the splitting functions, that must be summed. Also, at moderate Q2
values there are mass and higher twist effects, due to diagrams such as gluon
exchange between the scattered and unseattered quarks, which are of order
(1/Q2)n,n > 1. These are difficult to calculate but can be parametrized.
Another complication is that R(x,Q2), defined in (5.71), does not really
vanish, due to QCD corrections and effects neglected in the SPM, such as quark
mass and transverse momentum within the proton. R is difficult to measure
precisely, but is consistent with the expected values < 0.1.
Detailed global analyses* of the parton distribution functions (PDFs) at
NLO and NNLO have been carried out by the CTEQ (Nadolsky et al., 2008)
and MRST (Martin et al., 2007, 2009) collaborations and others (e.g., Alekhin
et al., 2006). Impressive agreement between the data and the QCD predictions
was obtained.
5.6 Other Short Distance Processes
There are many processes observed at hadron colliders that involve short
distance scatterings of the constituent quarks and gluons. These are essential
* Early studies include (Buras and Gaemers, 1978) and (Eichten et al., 1984).

218
The Standard Model and Beyond
for testing QCD and for the cross sections needed both for searches for new
physics and the standard model backgrounds at the Tevatron and LHC.
Standard model processes include pp —> F + X or pp —> F + X, where F is a
specific observed final state such as F = 2 jets or F = 7, Z, or W, which
can be virtual and subsequently decays to leptons (Drell-Yan). In the short
distance regime (e.g., large invariant mass s = p2F, which typically leads to
large momenta pr of the decay products transverse to the beam direction),
the cross section can be factorized into long distance terms that describe the
parton distributions in the initial hadrons and a short distance term
describing the hard scattering of the partons to produce F (Drell and Yan, 1971).
The parton distribution functions are the same ones that are relevant to deep
inelastic scattering. Thus,
a {HAHB -> F + X) = ^ f dxAdxBftA{xA,Q2)f?{xB,Q2)°ij^F{xA,xB),
ij J
(5.89)
where HA^B are the initial hadrons, //* and ff are their PDFs for fa =
#i, #i, G, etc., xAiB are their momentum fractions^, and Q2 is the renormaliza-
tion scale, which can be taken to be, e.g., sorp^. The PDFs evolve according
to the DGLAP equation. aij^jp(xA, xB) is the spin and color-averaged parton
level hard scattering cross section for the relevant subprocess, such as
qq -> qq, qq -> qq, qq -> GG
Gq^Gq, GG^qq, GG -> GG (5.90)
qq -> W; Z, 7 - //, qq - (W9 Z, 7)G.
cfij-+F can be calculated in perturbation theory, to the same order as the
PDFs. Many processes are known to NLO and some to NNLO.
The leading order expressions for massless partons are listed in Table 5.2
(one must replace 5, t, and u by the invariants s, £, and u relevant to the
subprocess). More extensive listings, including mass effects and extensions
to beyond the standard model processes may be found in (e.g., Barger and
Phillips, 1997; Amsler et al., 2008). The leading qq diagrams for pp —> qqX
are shown in Figure 5.18. The separation into short and long distance terms
in (5.89) can be shown to hold to all orders in perturbation theory, and is
expressed in the factorization theorems (Collins and Soper, 1987). (5.89) has
been successfully tested and applied in many processes by the CDF and DO
collaborations at the Tevatron, including Drell-Yan, heavy quark production,
high pT jet production, and W+ jets (Campbell et al., 2007).
Equation (5.89) can adequately describe the production of quarks and glu-
ons. However, they will subsequently fragment (hadronize) into mesons and
§They are related to the rapidity y of F by xa,b = y/s/se±y. The kinematic variables for
a collider process are discussed in more detail in Appendix H.

The Strong Interactions and QCD
219
FIGURE 5.18
Diagrams for pp —> qqX followed by the hadronization of the q and q jets.
There are additional diagrams corresponding to GG —> qq.
baryons (many of which are clustered in jets), which is again a long distance
process. This can be described by a phenomenological fragmentation function
D%(Eh/Ei) s), which describes the probability for parton fa to emit hadron h
carrying fraction Eh/Ei of the parton energy; s is the square of the relevant
renormalization or energy scale (Ellis et al., 2003; Amsler et al., 2008; Ellis
et al., 2008). One can think of the fragmentation as a two step process. First,
the initial partons branch or fragment into a number of lower energy ones.
This can be described by the DGLAP equations for the evolution in 5, and
has the effect that more fragments are produced at lower Eh/Ei for large s.
Eventually the partons form hadrons. There are various phenomenological
models for the actual hadronization, which can be tuned with experimental
data. One popular model involves string fragmentation, in which a classical
color flux tube is formed between two colored partons. When enough energy
is stored in the tube it creates a qq pair and breaks, forming two color
singlet objects. Another popular model is cluster hadronization. Following the
parton branching, including non-perturbative branching of the gluons into qq
pairs, hadrons emerge from neighboring qq color singlet pairs.
There are other powerful techniques that allow separation of perturbative
and non-perturbative effects, e.g., in the decays of mesons involving a single
heavy quark, such as the B+ = ub. In particular, heavy quark effective
theory (HQET) (Isgur and Wise, 1989; Bigi et al., 1997; Neubert, 1994) and
soft collinear effective theory (SCET) (Neubert, 2005) are respectively based
on systematic expansions in Aqcd/^q and Aqcd/E, where tuq and E are
large quark masses and large energies of decay products. Non-relativistic QCD
(NRQCD) (Bodwin et al., 1995) is useful for describing states with two heavy
quarks, such as upsilonium (bb).

220
The Standard Model and Beyond
5.7 The Strong Interactions at Long Distances
At low energies or long distances as —> 0(1) so one cannot use perturbative
techniques. Quarks and gluons are confined, and hadrons become the basic
degrees of freedom. Nevertheless, there are a number of tools available for
understanding the strong interactions. One class involves phenomenological models,
which typically are useful in limited domains of kinematics and parameters.
These include S-matrix theory and models, such as dispersion relations, the
Veneziano (dual resonance) model, Regge theory, and the pomeron (Collins,
1977); the MIT bag model (Chodos et al., 1974); and one boson exchange
potentials in nuclear physics. Another approach, which we touched on in
Chapter 3 and will elaborate on in the next section, involves the chiral
flavor symmetries of the strong interactions, including related techniques such
as current algebra (Adler and Dashen, 1968; D'Alfaro et al., 1973); chiral
perturbation theory (e.g., Pich, 1995; Bernard and Meissner, 2007; Bernard,
2008); QCD sum rules (Colangelo and Khodjamirian, 2000); the 1/NC
expansion ('t Hooft, 1974); the Skyrme model (Skyrme, 1962; Adkins et al., 1983);
and the OZI (Zweig) rule (Lipkin, 1984).
Lattice QCD (e.g., Gupta, 1997; Capitani, 2003; DeGrand and DeTar, 2006)
is a non-perturbative definition of QCD. It is based on approximating space
and time by a discrete lattice of points, allowing the QCD equations to be
solved numerically, typically on a supercomputer. Lattice QCD has had
considerable success in computing the hadron (Davies et al., 2004; Durr et al.,
2008) and gluonium spectra, demonstrating quark confinement, calculating
form factors necessary for the weak interactions (Aubin et al., 2005), and in
computing finite temperature effects.
Confinement
It is believed that under ordinary conditions color is confined, preventing
the production of isolated quarks and gluons. First consider QED. A well-
separated e~ and e+ act as sources of a classical electromagnetic field. The
photons have no self-interactions, and the non-linear effects due to QED are
small. The electromagnetic field therefore spreads out in space, leading to the
familiar V(r) ~ a/r potential between the charged particles. Given sufficient
energy they can get arbitrarily far apart, so there is no confinement. The
qualitative picture for QCD, which is supported by lattice calculations, is
very different. When a quark and antiquark get far enough apart the classical
gluon field they produce forms into a long narrow flux tube due to the strong
coupling and gluon self-interactions. The energy stored in such a flux tube
is proportional to its length, so that there is an effective potential between
the q and q which grows linearly with their separation, V(r) ~ asr. It would
take infinite energy to separate them, so they are confined. Actually, once

The Strong Interactions and QCD
221
the q and q are far enough apart so that asr > 2mn the tube can create a
qq pair and break, leading to two mesons rather than isolated quarks. This
is analogous to cutting a bar magnet, which creates two smaller bar magnets
rather than a monopole-antimonopole pair.
5.8 The Symmetries of QCD
QCD contains a number of accidental global symmetries in addition to the
gauged color St/(3)c (Weinberg, 1973a,b), where the subscript c identifies
SU(3)c as tne grouP associated with color. They are referred to as accidental
because no terms consistent with gauge invariance and renormalizability can
be written to violate them without introducing extra fields. For example, for
6qcd — 0 the Lagrangian density in (5.1) is automatically invariant under
the discrete space reflection (P), charge conjugation (C), and time reversal
(T) transformations1.
5.8.1 Continuous Flavor Symmetries
The gauge interaction terms in Cqcd for nq quark flavors are invariant under
an SU{nq)i x SU{nq)R x U{1)b x U(1)a global chiral flavor symmetry which
commutes with SU(3)C. The quark mass terms break the chiral symmetries,
and the non-chiral subgroup if they are not degenerate. mc^t are all large
compared to the QCD scale A so the flavor symmetry associated with them is
too badly broken to be relevant. Therefore, we will focus on the approximate
SU(3)l x SU(3)r x U(1)b x U(1)a symmetry of the u, d, and s quarks, and,
since ms < A, on its SU(2)L x SU(2)R x U(l)B x U(l)A subgroup, extending
the discussion in Chapter 3. The relevant quark part of the Lagrangian density
is
= qLifiqL + qRifiqii ~ {qLmqR + qRmqL),
where q — (u d s)T represent the quark fields, m = diag(rau rrid ms) is the
diagonal quark mass matrix, and the color indices are suppressed, m can be
written as
m = m0I + rn3A3 + ra8A8, (5.92)
^Parity violation in strong processes can be induced by weak interaction perturbations
(V^ exchange). In a general field theory these could lead to relatively large 0(a) effects
in addition to the safer 0(a/M^) ones. However, for QCD and similar theories, these only
affect quark masses and can be absorbed into the renormalization (Weinberg, 1973b). They
could in principle be problematic for scalar-mediated theories, including supersymmetric
QCD, but there they are suppressed by large superpartner masses.

222
The Standard Model and Beyond
using the notation of (3.112) and (3.113).
The two 17(1) factors have generators
B 3
i J Jg{x)<ex, B* = \f •#<*)<<**> (5.93)
where JB,A are the associated Noether currents
Q
Jt = Yl ^^rc, J£ = Y1 #7M75<?ra, (5.94)
r rex.
with r and a the flavor and color indices, respectively. B is just baryon number
(| for quarks and — | for antiquarks), while Ba is the analogous axial baryon
number. They are related to the chiral 17(1) generators Bl,r (i.e., BL = ±1
for qL(qL) and 0 for <?#(</#), and the reverse for BR) by B(BA) = Qr± Ql-
The baryon number is an accidental symmetry that is conserved to all orders
in perturbation theory in the standard model, though there may be a small
non-perturbative breaking in the electroweak sector due to vacuum tunneling
(instanton) effects. Many extensions of the SM predict B violation at some
very small level, which would lead to proton decay. However, there is no sign
of the axial baryon number symmetry in nature, and in fact the pseudoscalax
spectrum implies that it must be absent, as will be discussed in Section 5.8.3.
It is now believed to be broken by non-perturbative effects associated with
the triangle anomaly in the B^-gluon-gluon vertex and with instantons.
Consider SU(3)L x SU(3)R or its SU(2)L x SU{2)R subgroup. Define the
generators of SU(3)l,r as FlL R, respectively. Their Lie algebra is
FlFi
= ifijkFl [Fk,F>R] =
= ifijkFR,
'fl,fr]
and their commutators with the chiral quark fields are
[Fl,qaL,] = —YqbL'
[FlR,qaL}=0,
[Fl,qaR}=0
[FR,qaR}=-
Aab„
~^qbR
(5.96)
i.e., qL transforms as (3,1), a triplet under SU(3)l and a singlet under
SU(3)r. Similarly, qR transforms as (1,3), and the antiquarks as (3*,1) and
(1,3*). FlL R are associated with the Noether currents
JU = QL-y^QL = q^L^-^QbLa, Jr» = qmn-^QR- (5-97)
It is useful to also define the vector and axial vector generators
Fl = FR + F[ = J V:(x)d3x, Fi5 = FR-F[ = J Ai(x)d3x (5.98)

The Strong Interactions and QCD
223
and their associated currents
K = 4M + JL = TtJ^q* 4 = 4M - J^ = q^^q. (5.99)
The V* are associated with the weak and electromagnetic currents, while the
A1 enter the weak interactions. They do not commute, but rather satisfy
[F\ F>] = ifijkF\ [F\ F'5] = ifijkFk\ [F*5, F*} = ifijkFk.
(5.100)
The vector F% generate the diagonal 517(3) subgroup of 517(3) x 517(3).
The vector and axial currents V* and A1^ are conserved for massless quarks.
Turning on m = diag(rau rrid ms) ^ 0, one obtains from (3.51) (or directly
from the equations of motion) that
d»Vl = -iq
T'm
q, ^Aj, = t9Jy,m}7V (5-101)
As expected, the axial symmetries are broken by the quark masses, even if
they are degenerate, while the vector symmetries are only broken by mass
splittings.
5.8.2 The (3*, 3) + (3, 3*) Model
The (3*, 3) + (3,3*) model of 5£7(3)L x SU(3)R symmetry realization and
breaking (Gell-Mann et al., 1968; Glashow and Weinberg, 1968) was inferred
from the data prior to QCD or even the general acceptance of the quark
model (Pagels, 1975), but emerges naturally in that framework. The basic
idea is the assumption that the strong interactions have an SU(3)l x SU(3)r
invariant part as well as a small breaking term. The observed hadron spectrum
does not exhibit the full SU(3)l x SU(3)r symmetry", so the symmetries
associated with the axial generators are assumed to be spontaneously broken.
The explicit breaking term transforms as a singlet and octet under ordinary
517(3), and as (3*, 3)+ (3, 3*) under 5£7(3)L x SU(3)R. That is manifestly the
case for the quark mass term in (5.91) and (5.92), which constitute a concrete
realization of the (3*, 3) + (3, 3*) model.
The qualitative picture, which is supported by lattice calculations, is
therefore that in the limit mu = rrtd = ms = 0 the symmetries associated with the
8 axial generators Flb are spontaneously broken. According to the Nambu-
Goldstone theorem of Section 3.3.6 this implies the existence of 8 massless
pseudoscalar Goldstone bosons, which are identified with the pseudoscalar
octet 7r±'°,ii:±,ii:0,K0,77. The SE7(3)-singlet rf is not a Goldstone boson
because of the non-perturbative breaking of the associated Ba symmetry, as
"One would expect massless chiral fermions, and each meson isomultiplet would need to
have at least one degenerate partner with the opposite parity, contrary to observations.

224
The Standard Model and Beyond
will be discussed in the next section. Unlike the examples in Chapter 3 the
SSB of the Fl5 is not due to the VEV of an elementary scalar, but rather is
associated with the VEV or vacuum condensate of a composite qq operator
i.e., in the massless limit
(0|t2u|0) = <0|dd|0) = (0\ss\0) = v0= 0{A3) (5.102)
due to the non-perturbative long-distance dynamics. Since the chiral
symmetry is broken, the quarks must acquire a dynamical or constituent common
mass Mdyn ~ Mp/3 ~ 300 MeV proportional to A from the strong dynamics,
i.e., from the interaction with the quarks and gluons in the condensate.
To better justify these statements, let us define the composite pseudoscalar
and scalar field operators
A* A* A*
*"* = -iq^rlbq = -m—qR + iqRirqL
. Z . (5.103)
A2 X1 X1
o-1 = q-^q = qL — qR + toytfL,
for i = 0---8, with A0 = J\l and d0jk = J\sjk. a0 and tt° are SU(S)
singlets and the others are octets. In particular a0 = (uu + dd + ss)/y/6. It
is straightforward to show from (5.96) that
[F*5,^] = idijk<rk, [Fi5,al] = -idijk7rk. (5.104)
Now assume the qq condensate in (5.102), which implies that (0|cr°|0) =
yf^o- Taking the expectation value of the first equation in (5.104),
(O\[Fi5,7ri]\0)=iSijis0. (5.105)
This implies that Fl5\0) ^ 0, i.e., the Fl5 symmetry is spontaneously broken.
It also implies that there must be some angular momentum-0 state in the
Hilbert space with a non-zero inner product with Flb |0), which we assume to
be a singlet particle state that we denote |7rl(<f)) under the anticipation that
it is a bound state with the quantum numbers of the 7rl field. Using (5.98),
<0|Fi5|7rfe(«f)) = J d3x(0\K(x)\nk(q))- (5.106)
But (0|^(x)|7rfc(q)) must be of the form
<0|4 (*)!**($•)) = i^=q»6ike-*x. (5.107)
The q^ follows from Lorentz invariance because q is the only four-vector
available, the Sik is from the unbroken 517(3), and the exp(—iq • x) follows from

The Strong Interactions and QCD
225
Pf
q <r \
&&&*
U&QQ2>
G
Pi
FIGURE 5.19
Left: The pseudoscalar decay constant. Middle: Pion pole contribution to the
induced pseudoscalar form factor in the chiral limit. The cross indicates the
axial current and q = Pf — Pi- Right: Two gluon intermediate state coupling
to the rf.
translation invariance. fn, known as the pseudoscalar decay constant, is
illustrated diagramatically in Figure 5.19. It is determined from the strong
dynamics that lead to the bound state, and is expected to be of O(A). We
will see in Chapter 6 that fn ~ 130 MeV can be measured from pion decay.
Inserting (5.107) into (5.106)
<0|F*V(<f)> = (27r)3q0iSikd3(q)^ (5.108)
where we can take t = 0 since Flb is a symmetry of the equations of motion.
One can insert a complete set of states between Fl5 and 7T7 in (5.105).
Assuming that only the single particle states enter, or at least that they dominate,
we find
/
d3? {{0\Fi5\*k(q)){nk(q)\^\0) - {0\^\nk(q)){nk(q)\Fi5\0)}
(27T)32<?o
= ibii-j^Z = i^ijvo,
(5.109)
where Z1/2, which has dimensions of mass2, is associated with the bound state
wave function of the 7rfe by
(0|7rV(<7)> = Z1/2Sjk. (5.110)
Defining v = i/q/Z1/2, which has dimensions of mass, one has that

226
The Standard Model and Beyond
which is nonzero if the symmetry is spontaneously broken. The Nambu-
Goldstone theorem then follows from d^A1^ = 0 and (5.107)
<0|3 • A*]**) = 0 = ^qHike-1"* =► q2fv = <h = 0- (5.112)
This summarizes the Goldstone alternative: either fn=0 (no SSB) or raj = 0
(there exists a massless spin-0 Goldstone boson).
To justify the generation of a dynamical mass Mdyn ^ 0 let us first derive
the Goldberger-Treiman (GT) relation. To simplify the notation, we work with
the SU(2)l x SU(2)R subgroup. Since the quarks do not appear as physical
states, consider the nucleon isospin doublet ip = (ipp ipn)T\ as in Section 3.2.3.
The matrix elements of the axial currents between the nucleon states will
be considered in more detail in Chapter 6, but for our present purposes all
we need is that the most general form consistent with the strong interaction
symmetries is
(rPf\Ai(xM) = uf [7M75<7i(<?2) + <?m75<?3(<?2)] ytn e^x, (5.113)
where q = Pf — Pi and #1,3 are form factors. g\ is a slowly varying
function, with experimental value #i(0) ~ 1.27 from (3 decay, while the induced
pseudoscalar g$ has a pole at q2 = 0 from the massless Goldstone boson, as
indicated in Figure 5.19. Using (5.107) it is given by
g3(q2)~-^G„, (5.114)
where Gn ~ 13.1 is the effective pion-nucleon coupling denned following
(3.84).
Using d^A1^ and evaluating the divergence of (5.113) at q2 = 0 one obtains
the GT relation
9l(0)Mp = ^. (5.115)
(5.115) should hold in the chiral limit, and experimentally it is satisfied to
better than 2% (Gorringe and Fearing, 2004). Our present purpose, however,
is the observation that even in the chiral limit one has that Mp oc /^ ^ 0 is
induced by the chiral breaking, and that presumably the quarks also acquire
dynamical masses ~ Mp/3.
The picture is perturbed when non-zero current quark masses mq are turned
on in Cqcd- For mu = rrid = ms ^ 0 the axial generators Fz5 are explicitly
broken and the pseudoscalars acquire small mass-squares m2PS oc rag, i.e.,
they become pseudo-Goldstone bosons, analogous to the example in Section
3.3.3. Also, the constituent quark masses are shifted to effective values
Mq = Mdyn + rUq.
(5.116)

The Strong Interactions and QCD
227
The pseudoscalar masses may be obtained by taking the matrix element of
d • A{ = imqX^q = -Imtf (5.117)
from (5.101) between the vacuum and one Goldstone boson state. Using
(5.107) and (5.110),
^=ml = -2mZ1/2 = 6, (5.118)
where the explicit symmetry breaking parameter e has dimensions of mass3.
Finally, it is convenient to define a renormalized field tt1 = tt^Z1/2 so that
(0\^\7rk(q)) = Sjk. (5.119)
In terms of tt1 (5.117) becomes
d'A{ = ^mlv\ (5.120)
v2
which is known as the partially conserved axial current (PCAC) relation. It
again displays the Goldstone alternative in the limit d- A1 = 0, and can be used
as the basis soft pion theorems, which can be used to calculate amplitudes for
physical pions at low energy.
Breaking the degeneracy of the quark masses leads to breaking of the
vector generators Fl. For ms > mu = rrid the vector 517(3) is broken to 517(2)
(isospin), while rrid ^ mu breaks the isospin symmetry, as described in
Section 3.2.3. The diagonal generators F3 and F8, which are related to
electric charge and strong hypercharge by (3.89), are unbroken. The ratios of
the current masses cannot be determined directly because of confinement
and because of Mdyn- However, they can be inferred from the observed
SU(Z)l x SU(3)r symmetry breaking effects (Weinberg, 1977; Langacker and
Pagels, 1979; Gasser and Leutwyler, 1982). To lowest order in symmetry
breaking,
mu _ 2mlo - ml+ + m2K+ - m2K0
ma rn2K0 - m2K+ + m^ +
ms _ m2Ko + m2K+ - m^ +
(5.121)
rrid rn2K0 — m2K+ + m^ +
or ignoring isospin breaking,
o
TUts 771c
ml ~ md + vriu
(5.122)
In practice, higher-order corrections are needed, and additional information
on isospin breaking is obtained from the proton-neutron and K+ — K° mass
differences (which also have electromagnetic components that must be
estimated), p-uj mixing, the rj —> 3n and \j)' —> ip-ir0 decays, and isospin breaking

228
The Standard Model and Beyond
effects in nuclear physics (e.g., Langacker, 1979, 1980). Recent estimates of
the ratios (see the review on Quark Masses in Amsler et al., 2008) are
^-17-22, ^~ 0.3 -0.6. (5.123)
rrid rrid
The larger value of rrid compared to mu explains why mn > rap, since the
electromagnetic effect would make the proton heavier (the two contributions
are of the same order of magnitude). Similar statements apply to m2K+ —
m2Ko < 0, while m^+ — m^0 > 0 is almost entirely due to electromagnetism
(both m^+ and m^0 are proportional to mu + rrid to leading order).
The absolute scales are more difficult, since they depend on strong
interaction matrix elements of the operators in (3.112), and also on the renormaliza-
tion scheme and scale. Current estimates are
mu - 1.5 - 3 MeV, md ~ 3 - 7 MeV, ms ~ 70 - 120 MeV. (5.124)
These are actually running masses, evaluated at a scale // ~ 2 GeV. The u
and d masses are much smaller than other strong interaction scales, such as
the nucleon mass or A, i.e., both 517(2) and SU(2)l x SU(2)r hold at the
< few % level. The near exactness of the chiral symmetry is most evident
from the fact that ra2 is so tiny compared to other hadronic mass-squares.
The success of the vector isospin symmetry is because mu and rrid are both
extremely small even though their ratio is not close to unity, and also because
a is small. ms is small compared to most hadronic masses, but not negligible.
That is why the 517(3) breaking effects are typically of 0(25)%. The one
counterexample is ra^/raj, which is large because of ms ^> mu,d and the
special role of the K and tt as pseudo-Goldstone bosons.
We also record recent determinations of the heavy quark masses
rac(rac) - 1.2 - 1.3 GeV => mpcole ~ 1.5 - 1.8 GeV
mb{mb) ~ 4.1 - 4.4 GeV => mpbole - 4.7 - 5.0 GeV (5.125)
mt{mt) - 161.3 ± 1.8 GeV => mP°le - 170.9 ± 1.8 GeV.
The first values are the running (MS) masses, while the second are the
propagator pole masses, which correspond most closely to kinematic masses. They
are related by
4as(ra2)"1
mpoLe - ra(ra)
1 +
3 7T
(5.126)
The relation is actually known through (9(as(ra2)/7r)3(Melnikov and Ritber-
gen, 2000). The series converges very slowly, especially for rrib and rac, and
it is best to work in terms of the running masses to minimize the associated
uncertainties.

The Strong Interactions and QCD
229
5.8.3 The Axial U(l) Problem
One aspect of QCD and other quark-gluon models that is not present in the
general (3*, 3) + (3,3*) model is that the symmetry is U(3)L x U(3)R rather
than just SU(3)L x SU(3)R x U(l)B as the quark masses go to zero. Assuming
that the dynamics of the extra axial U(l) generator F05 = BA are similar to
those of the Fl5, one might expect that it is spontaneously broken, F05|0) ^ 0,
so that there would be a ninth pseudo-Golds tone boson in addition to the 7r's,
#'s, and 77. Although the 7/(958) has the appropriate quantum numbers,
it turns out that either m^, m^/, or both are just too large to be consistent
with the 7r and K masses, independent of the assumptions on the spontaneous
breaking. Even if the F05 symmetry is not spontaneously broken, the axial
U{1) from U(2)L x U(2)R will be, with m^ predicted to be too light when
the quark masses are turned on. To be more explicit, using standard current
algebra techniques and ignoring SU(2) breaking for simplicity (i.e., 7713 = 0),
one can show (Weinberg, 1975) that the two mass-squared eigenvalues of the
mixed 77 — 7/ system are
3mi HU+^). <^)
1 + 2z2' 3
up to corrections smaller by 0{m2x/rn2K), where z = fo/fn, with /o the decay
constant for F05 defined analogously to fn in (5.107). For z = 1 the two mass
eigenvalues are predicted to be m^ = m^ ~ 140 MeV and m^ = y/2m^ ~
700 MeV, respectively, compared with the observed 77 and 7/ masses of 548
and 958 MeV. For 2: —> 0 (no spontaneous F05 breaking) one state becomes
very heavy (not a Goldstone boson), but the lighter eigenvalue is y/Sm^ ~ 243
MeV. The latter value is the upper limit on the lighter mass. This conflict
with observation was known as the U(1)a problem or the 77 problem. There
were also difficulties associated with the 77 decay.
To resolve the U(1)a problem there needs to be a contribution to the explicit
breaking that is independent of the quark masses. The «S£/(3)-singlet 7/ is
distinguished from the octet states 7r, K, and 77 by the third diagram in
Figure 5.19. The two-gluon intermediate state is a singlet under flavor SU(Z)
and can therefore couple to the 77' but not to the octet states in the 517(3)
limit. This is not by itself sufficient to solve the problem, however. Another
necessary ingredient is the triangle anomaly in the B^-gluon-gluon vertex,
which violates the symmetry and leads to a divergence (Adler, 1969; Bell and
Jackiw, 1969)
&>Al = 2iY^mrqrlbqr + ng-|^G^G^, (5.128)
r=l
where A0 = Yl7=i Qrl^Qr- We have included the explicit breaking from the
quark masses, nq = 2 or 3 is the number of flavors being considered, and G
is the gluon dual field strength tensor defined in (5.6). The GG term (which

230
The Standard Model and Beyond
incidentally has the same structure as the Oqcd term in (5.1)) appears to
do the job, but there is a further complication in that it can be written as a
four-divergence,
G^G\V = d» \e^p<7Giv[Gipa + ?±fijkG"Gk°)] = d»K». (5.129)
The current
AX-n^K^ (5-13°)
is therefore conserved for mr —> 0, apparently reintroducing the problem.
However, K^ is not gauge invariant, and it was shown ('t Hooft, 1976b, 1986)
that there are non-perturbative vacuum tunneling gauge field configurations
(instantons) which fall off sufficiently slowly that they contribute to surface
terms when the divergence is integrated over space-time. The U(1)a anomaly
and instantons therefore generate an 7/ mass which survives when the quark
masses vanish, solving the U(1)a problem and evading the Nambu-Goldstone
theorem.
There is another consequence of these considerations: since the current A®
is formally conserved in the chiral limit, one can use the associated charge
F05 to generate phase rotations on the quark fields. For example, under the
transformation U((3) = exp(—i(3F05) the quarks transform as
q - U(p)qU~l (/?) = ei0^q <=► qL,R - e* V* (5.131)
(cf. (3.28)). However, it can be shown (Callan et al., 1976; Jackiw and Rebbi,
1976) that because of the gauge noninvariance of the K component, the Oqcd
term in (5.1) is not chiral invariant, and that 0qcd is shifted to Oqcd — 2nq(3
under the same transformation. An equivalent formulation is that there are
a continua of possible vacua of QCD, labeled by Oqcd- However, the vacua
are not invariant,
U((3)\0qCD) = \0qcd - 2nq0). (5.132)
The implications for the strong CP problem will be mentioned in Section 8.1.
The Pseudoscalar and Vector Nonets
In Section 3.2.3 we described the Gell-Mann-Okubo formula for the nucleon
masses in the presence of 517(3) breaking. The GMO formula (3.104) followed
from the assumption that the explicit breaking term in C transformed as the
8th component of an 517(3) octet, as in the quark model. Similar
considerations apply to the nonets of mesons, such as the pseudoscalar (7r[138], if [496],
7j[548], r/[958]) and the vector (p[770], AT*[892], <£[1020], u[782]). In each
case, the first state refers to the isospin triplet, the second to the two isospin
doublets, and the last two to the isospin singlets, which may be mixtures of
the isosinglet in the octet and an 517(3) singlet. The numbers in brackets

The Strong Interactions and QCD
231
are the masses in MeV. (We ignore isospin breaking.) The vectors are
interpreted as the lowest lying L = 0, S = 1 qq states in the SU(S) quark model.
The pseudoscalars have the dual (and sometimes conflicting) interpretation as
pseudo-Golds tone bosons, or as the L = 0, S = 0 states in the quark model.
The derivation of the mass formula differs in several ways from that in
(3.104). First, as is clear from chiral perturbation theory or the form of the
free Lagrangian, the analogs of (3.101) and (3.103) should apply to the meson
mass-squares. Second, the meson octets are self-conjugate, so the two isodou-
blets are related and degenerate by charge conjugation. The apparent loss of
a parameter is compensated by the fact that Bose symmetry implies that only
the symmetric (3di&j term in (3.101) is present. Finally, the matrix elements
in (3.101) only involve the states within the original octet. In principle, SU(3)
breaking can mix the octet states with single particle states from outside the
octet, or even with multiparticle states. That is not a significant effect for
the baryons, but it is for the meson nonets which involve an S£/(3)-singlet
that can mix significantly. The result is that the analog of (3.104) for the
pseudoscalar and vector octets is
Am2K = 3m28 + m2, 4rr&. = 3m28 + m2, (5.133)
where ra28 and ra2^ are respectively m2PSS and rnf/gg, the 88 entries of the
pseudoscalar and vector octet mass-squared matrices. However, the S77(3)-
breaking can also induce m2p08 = m2pm or rriy0S = m^80 mixing terms
between the octet and singlet, so the 77 — 77' and (j>—uj systems are really described
by 2 x 2 mass matrices** such as
where m2P00 is the singlet mass-square in the absence of mixing. r]g and 770
are the original octet and singlet states, given in the quark model by
778 = -7= {uu + dd — 2ss) , 770 = -7= (uu + dd + ss) . (5.135)
v6 v3
Within this simple two-state approximation, the mass eigenstates are
7] = rigcosOp — ryosinflp, 7/ = 778 sin#p + 770 cos#p, (5.136)
where 6p is the pseudoscalar mixing angle. One finds
m2m = cos2 Opml + sin2 0Pra2,. (5.137)
**Isospin-breaking (es) effects would extend the discussion to include 7r° — rj — rj' and
p° — <j> — uj mixing. Also, mixing with excited octet or singlet qq states or with gluonium is
possible. We will ignore these complications.

232
The Standard Model and Beyond
From the GMO formula in (5.133) as well as confirming (and sign) information
from decays involving the r\ and 7/ one obtains (Amsler et al., 2008) 9P ^
— 11.5°. The rather small value for 6p is easily understood in terms of our
discussion of the U(1)a problem. In the chiral limit the octet mass-squares
all go to zero, while the m^ ~ m^ induced by the anomaly/instant on effects
is large compared to typical 517(3) mass splittings of < few hundred MeV.
Therefore, one expects |Wp08|,mp88 ^ mPoo an(^ relatively little mixing.
Similar formulas apply to 0 — u mixing. In this case, however, the mixing
angle 6y denned by
(f> = (fig cos 0y _ 0o sin ^v, u = 08 sin 0^ + 0ocos#v, (5.138)
turns out to be large, 6y ~ 38.7°. This is very close to the ideal or magic
mixing angle 61 = 35.3° (tan0/ = l/\/2) for which <f) would be purely ss,
1 - 2 1
This, and similar results for other nonets with sufficient data, is again easy
to understand. The vector mesons are well described by the non-relativistic
quark model. In the chiral limit there is no major distinction between the octet
and singlet, and they should have comparable masses, perhaps dominated by
ZMdyn- This is not exact. For example, the 0o could acquire additional mass
from gluon intermediate states, analogous to the two-gluon diagram in Figure
5.19. However, by charge conjugation and gauge invariance at least three
gluons are required, so the contribution is suppressed by 0(oPs). In any case,
as long as the the unperturbed singlet and octet mass-squared difference is
small compared to the *SI7(3)-breaking effects one expects the heavier mass
eigenstate to be mainly ss since ms ^> mu^ (The precise connection to the
2x2 mass matrix in (5.134) is explored in Problem 5.6.) This ideal mixing
picture is further supported by the expectation
m2p~ml, ml-m2K. ~ n?K+ -m2p, (5.140)
which is obvious from the quark interpretation and agrees reasonably well
with observations. For the vector mesons, unlike the pseudoscalars, the
approximate picture, including the near ideal mixing, still holds if one assumes
the perturbations are linear in the masses.
5.8.4 The Linear a Model
Some of the formal arguments presented above are more transparent when
presented in terms of an effective theory, the linear a model, involving the
composite fields defined in (5.103). We will illustrate this for the SU(2)l x
SU(2)R limit, with fields
T* 1 1 -
** = -iQjl5Q, a=-qq=-(uu + dd), (5.141)

The Strong Interactions and QCD 233
where i = 1, 2, 3 and the normalization of a is chosen for convenience. (Neither
the o nor the 77 field in the next Section corresponds closely to any physical
meson.) We assume that the strong QCD dynamics generates the effective
interactions in
Ca = &&/> + ^(d^a)2 + (<9M7f)2 - g^ [a - m • f75] $
A 2 (5-142>
-y(-2 + -2)-^(-2 + -2) +^
where ip = (ipP ipn)T is the nucleon doublet. Ca is SU{2)L x SU(2)R
invariant except for the linear aa term, which mimics the current quark masses
and explicitly breaks the symmetry. W.l.o.g., we can take a > 0. We see
explicitly from (5.142) that if SU(2)L x SU(2)R were unbroken (a = 0 and no
spontaneous breaking) the nucleons would be massless and the pions would
have a degenerate parity-even partner (the a). The phase in which the axial
generators are spontaneously broken is more realistic.
To see the SU(2)l x SU(2)r invariance of £a explicitly, define a 2 x 2
matrix
„.!+]£*_(■%- «*\ (5,43,
which is reminiscent of our treatment of the adjoint representation of a simple
group in Section 3.2. Then
\d»M)] (3MM)] (5.144)
Ha = $Li Ml + ^Rt Mr + 2^
- V2gn $RMrl>L + JlM^r) - ^TY (MM*) - ^ [IV (MM+)]2 + aa.
ipL,R transform the same way as qL,R, i.e., as (2,1) and (1,2),
SU(2)L: ^^e^'Jfe ^*-^*
.^ (5-145)
SU(2)R: ^l^^l, ^R^e1 * ^R.
From (5.104), M transforms as (2*, 2),
M -> e^'^Me-^'i (5.146)
It is clear by inspection that Ca is invariant except for the aa term. We also
record the infinitestimal form of (5.146),
7T —> 7T — a X 7T + /?CT, <7 —> <7 - /? • 7T, (5.147)
where
0=PR~PL, <J=^+A (5.148)

234 The Stamrd Model and Beyond
are associated respectively with the axial and vector ^aerators. It is apparent
from (5.142) and (5.144) that the meson terms areiMriant under 50(4) for
a = 0, with (<r, if) transforming as a vector. Thii reflects the equivalence
50(4) ~ 517(2) x 517(2). Of course, the vector SV\l symmetry is preserved
even for a ^ 0.
The minimization of the scalar potential
V (<r,it) = Y (°2 + 7f 2) + - (a2 +r|2 - aa (5.149)
is straightforward, i.e.,
(5.150)
da
^ = ^ + A(^+#a)]^=»
at the minimum. In the chiral limit, a = 0, one has
f 0, f>0
<0|tt*|0)=0, (0\v\0)=v={ r^j (5.151)
IV A ' J '
where we assume that any nonzero VEV is in the i direction to ensure a
smooth a —> 0 limit (Dashen, 1971). For a ^ 0, onenas that (0|cr|0) = i/ ^ 0
and (0|7r*|0) = 0. The precise value of v can be obtamed by solving the cubic
equation in (5.150). However, we have no need for an explicit expression for
v. For both a = 0 and a > 0 one can expand V(v+:.if) to obtain
ra2 = /x2 + Ai/2 => ra2*/ = a. (5.152)
Again, this displays the Goldstone alternative, i.e.. sther ra2 —> 0 or i/ —► 0
in the symmetry limit a —> 0. One also has that a (bamical nucleon mass
m^p = g^v (5.153)
is generated for v ^ 0. The a' mass is y/—2/x2 foro = 0. However, the cr' is
not a realistic aspect of the model, so the exact value is not important. From
(3.46) the axial current is
;T*
K = ^75^</> + (V) ^ " (^ (5-154)
From (5.107) this suggests that the pion decay constant is related to v by
(5.111), at least up to renormalization effects. Thafrom the equations of
motion or from (3.50) one recovers the PC AC relation
d'Ai=ani = vml-K1 = ^m2TJ. (5.155)
V2
The phase diagram for the model as a function of /x2 and a is further discussed
in (Lee, 1972).

The Strong Interactions and QCD 235
5.8.5 The Nonlinear a Model
The PCAC relation leads to various soft pion theorems for the behavior of
amplitudes involving low energy pions (Adler, 1965b; Adler and Dashen, 1968;
D'Alfaro et al, 1973). However, these can be made more explicit, and the
relation of the pseudo-Goldstone boson pion fields to the SSB more
obvious, by redefining the fields (Weinberg, 1967b). There are various possible
reparametrizations (see Donoghue et al., 1992, for a very nice discussion).
One that is closely related to the discussion in Section 3.3.5 is to define fields
r] and £ such that
M = U{ft" (^fj WW = (?±f) IU(i[), (5.156)
where
U(€) = e*'*'". (5.157)
Then, £ disappears from the meson self-interaction terms, yielding
Ca = $Li Hl + $ni Hr + \ (dtff + (l/H^} Tr [(cW)+ (d^U)]
- V2gn (^) [$rU1>l + $lUHr) ~ y (^ + V? - \{v + V)\
(5.158)
where we have taken a = 0. However, £ reemerges as derivative interactions
from the |c?M£/|2 term, which can be expanded as a power series in £/v to all
orders. The leading terms are
*+f.T± [(*N7)t iw] = i (^-)2+±-2 [(f. dlt?y - if i2 fay].
(5.159)
Similarly, one can eliminate the (apparently) non-derivative pion-nucleon
interaction by denning new fermion fields
V<l = U^2rPL, r/,'R = U-^rPn, (5.160)
so that the nucleon terms in Ca become
C* = 4>'Li H'L + $& H'r - ^9n i^1) {VrVl + $M
V v2 J (5 161)
+ i&tf1'2 (pu-1/2) i>'L + ifau-1'2 (^r/1/2) yR.
U1/2 can again be expanded in a power series in (?/v. The leading pion-
nucleon interactions are (dropping the primes)
C& = -\^lh\ ■ {djt) * - ^fo»\ ■ (* x <u) </>• (5.162)

236
The Standard Model and Beyond
By comparing the matrix elements between physical nucleon states of the
terms linear in tt and £ in (5.142) and (5.162) one immediately recovers the
Goldberger-Treiman relation (5.115).
The a model, especially as formulated in (5.158), is useful for computing
processes involving low energy pions. The r\ field decouples for large mass
and can be ignored, yielding the nonlinear a model. This and equivalent
representations can be used as a basis for chiral perturbation theory (Pich
1995; Bernard and Meissner, 2007; Bernard, 2008), which summarizes the
implications of spontaneously broken chiral symmetry and the associated soft
pion theorems for low energy hadronic processes.
5.9 Other Topics
QCD and the strong interactions are an enormous subject, and we have only
been able to scratch the surface here. An incomplete list of other topics which
were omitted or only touched on includes the hadron spectrum (see, e.g., Ros-
ner, 2007; Klempt and Richard, 2009, and Chapter 6); gluonium or gluebalh
(bound states of gluons), which are hard to observe because of complications
from mixing with ordinary qq states (Klempt and Zaitsev, 2007; Crede and
Meyer, 2009); the connection to nuclear forces (Epelbaum et al., 2008); the
application of ideas derived from or motivated by string theory, including the
AdS/CFT correspondence (Maldacena, 1998; Klebanov, 2000) and twistor
methods (Witten, 2004) for the calculation of complicated matrix elements;
and computational methods for real hadronic processes. Some of the
packages used for the calculation of matrix elements, event generators, and parton
distribution functions are listed in the website section of the bibliography.
Techniques such as lattice field theory and the AdS/CFT correspondence
have been very useful in extending the study of the strong interactions into
new domains, such as high temperature and/or density (Das, 1997; Kogut and
Stephanov, 2004; DeGrand and DeTar, 2006; Jacobs et al., 2007; Banerjee
et al., 2008; Satz, 2009). Under such conditions there may be a phase
transition to a plasma consisting of quarks and gluons, such as presumably existed
above some critical temperature Tc of order A in the early universe (Kolb
and Turner, 1990). The chiral symmetry, which is spontaneously broken at
T = 0, may also be restored at high T, perhaps at or close to To- These
extreme conditions are also probed in heavy ion collisions in the RHIC
collider at Brookhaven and the ALICE experiment at the LHC, and may be
relevant in extreme astrophysical environments such as the cores of neutron
stars (Armesto, N. , (ed. ) and others, 2008; Alford et al., 2008).

The Strong Interactions and QCD
237
5.10 Problems
5.1 Calculate the spin and color-averaged CM cross sections for qq —> QQ
and qq -» <Mo, where Q is a heavy quark with mass mQ and q0 is a spin-0
color triplet with mass mo. Neglect the mass of the initial quarks.
5.2 Suppose the heavy qo in Problem 5.1 were stable. Then, q0q0 pairs
produced in the early universe could annihilate into quarks and gluons. The
relic density of scalar quarks that do not annihilate depends on the thermal
average of a(q0qo -> qq)Prei and a(qoq0 -> GG)/?re/, where /3rel = \/3qo -
(3q0\ (Kolb and Turner, 1990). Calculate these quantities in the CM (not
the thermal average), assuming that rao is much heavier even than the top
quark. Show that near threshold, s > 4raQ, the rate into quarks is P-wave
suppressed (i.e, is proportional to /?2, where (3 is the (fa velocity), while the
rate into gluons is not.
5.3 The two-loop RGE for the QCD coupling for nq quark flavors is
acts o , 3
where
b2 = —
4-k
dlnQ2
"ll - *±
3
— U2^s f3"s
' b3 = 8^
5
■ 19V
(The one-loop coefficient 62 is -47r6, where b is defined in (5.34), because the
equation is written for as rather than for g^.) Show that the solution can be
written for large Q2 as
ocs{Q2) =
1 63lnln(g)
+ <
1
y%,
5.4 The pion-nucleon a term anN is defined as
m
0~ttN
2MN
(N\uu + dd\N),
where m = (mu + md)/2. It can be thought of as the contribution of the
chiral *SI7(2)-breaking quark masses to the nucleon mass, and it can be
measured independently in nN scattering (e.g., Cheng and Li, 1984). One way to
estimate a^N (Cheng, 1976) is to write the matrix element as
(N\uu + dd- 2ss\N) + 2(N\ss\N),

238
The Standard Model and Beyond
and then invoke the OZI rule (based on simple quark model ideas) to neglect
the second term. The first matrix element can be related to the shift in the
nucleon mass due to 5l7(3)-breaking, as in (3.102). Show that
3 rh
cvn ~ x (MA + ME - 2MN) ~ 25 MeV
2 ms — m
in the OZI approximation. Note that estimates from ttN scattering are
typically higher than this value, suggesting that the OZI approximation is not
very good in this case.
5.5 Derive (5.129).
5.6 (a) Derive the meson GMO formula in (5.133).
(b) Express the elements of the mass matrix in (5.134) in terms of the mixing
angle and eigenvalues.
(c) Show how m|8,moo, and ra^g must be related to obtain ideal mixing.
(d) Show that ideal mixing and the assumption that the octet and singlet are
degenerate in the 517(3) limit lead to (5.140).
5.7 (a) Consider the linear a model in Section 5.8.4. It is claimed that the
most general SU(2)l x SU(2)r invariant renormalizable Lagrangian density
is given by (5.144) with a = 0. Why is there no need for a term
Tr (M^M M*M)?
(b) Suppose we have two sets of four real fields Mi = {a\ + iifi • f)/y/2 and
M2 = (02 + 27?2 • r)/\/2, which transform as
Write the renormalizable SU(2)l x SU(2)r invariant interactions and mass
terms that involve both Mi and M2. Include any quadratic terms that mix
the two.
(c) Find quadratic terms which, when added to the Lagrangian, would break
the symmetry to the diagonal 517(2) group with generators F% = FlL + FlR.
(d) Find quadratic terms which would break the symmetry from SU(2)l x
SU(2)R to SU{2)L.

6
The Weak Interactions
In this chapter we discuss the charged current weak interactions. After a short
overview of the developments that led to the standard model, we describe the
Fermi theory, which in its modern form still gives an excellent description of
a wide variety of weak decay and scattering processes at tree level. (In some
cases, radiative corrections, which require the full structure of the standard
model, are required.) Some representative processes are calculated and
described. Much more detailed treatments may be found in (e.g., Commins and
Bucksbaum, 1983; Renton, 1990; Langacker, 1995; Amsler et al., 2008).
6.1 Origins of the Weak Interactions
We first give a brief history of some of the highlights in the history of weak
interactions and the development of the standard electroweak model (for a
professional history, see Pais, 1986). The story begins with the discovery of
radioactivity by Henri Becquerel in 1896. One of the three types of
radioactive decays that were identified was nuclear (3 decay, in which, apparently,
(N, Z) —> (N — 1, Z + l)e~ (or in modern terms, subsequent to the discovery
of the neutron, n —> pe~). By 1911, experiments by Lise Meitner and Otto
Hahn and others indicated that /? decay violated the conservation of energy.
As we now understand it, momentum and angular momentum would also have
been violated. In 1930 Wolfgang Pauli hypothesized that the missing energy
was carried off by a hypothetical weakly coupled neutral particle, dubbed the
neutrino by Enrico Fermi. The neutrino (or anti-neutrino) would be difficult
to detect because of its very weak interactions, but not impossible. Its
existence was confirmed in 1953 when the electron antineutrino, &e, was directly
observed near a reactor by its rescattering to produce a positron via inverse
8 decay, uep —> e+n (Reines and Cowan, 1953). The second neutrino type,
associated with the muon, was discovered by a group headed by Lederman,
Schwartz, and Steinberger at Brookhaven in 1962, who observed its
rescattering to produce a muon (Danby et al., 1962). Following the observation of the
r* in 1975 (Perl et al., 1975), the existence of the third, tau-type, neutrino,
vr, was unambiguously inferred from the r lifetime and decay properties (see,
e.g., Langacker, 1989b). However, it was not observed directly until 2000 when
239

240
The Standard Model and Beyond
the DONUT collaboration at Fermilab observed its rescattering to produce
r* (Kodama et al., 2001).
In 1933 Fermi proposed a theory of (3 decay, n —> pe~ve, which loosely
resembles QED but involves a zero range (non-renormalizable) four-fermion
interaction and non-diagonal charged currents, as shown in Figure 6.1. The
Fermi Hamiltonian density is
n = GfAj",
(6.1)
where
J\ = Pl^n + i/e7Me (6.2)
is the charge-raising vector current which describes the transitions n —> p and
e~ —> ve. The charge-lowering current is
Jii = ni^P + ^^e, (6.3)
which describes p —> n and ve —> e~ (or the creation of an e~ve pair). The
coefficient
GF ~ 1.17 x 10-5 GeV"2 ~ 1.02 x 10_5m-2 (6.4)
is the Fermi constant, which describes the strength of the interaction. It has
dimensions of mass-2, characteristic of a non-renormalizable theory. In
amplitudes it is typically multiplied by A2, where A = O(MeV) is a characteristic
energy release, so that indeed the interaction is very weak.
FIGURE 6.1
Left: Four-fermion interaction leading to /? decay. The vertices are at the same
spacetime point but are displaced for clarity. Middle: Four-fermion interaction
for vee~ —> vee~. Right: A higher-order correction to vee~ —> vee~.
The original Fermi theory described (3 decay and related processes. It has
been successfully modfied over the years to incorporate new observations*,
*See (Langacker, 1981; Commins and Bucksbaum, 1983; Pais, 1986) for a more complete
list of references.

The Weak Interactions
241
including parity violation (Lee and Yang, 1956; Wu et al., 1957) and the V- A
theory (Feynman and Gell-Mann, 1958; Sudarshan and Marshak, 1958); /i
and r decays; strangeness changing decays (Cabibbo mixing (Cabibbo, 1963));
the quark model; heavy quarks and mixing (the Cabibbo-Kobayashi-Maskawa
(CKM) matrix, which allows the incorporation of CP violation (Kobayashi
and Maskawa, 1973)); and neutrino mass and mixing. In its modified form it
still gives excellent tree level descriptions* of a wide variety of charged current
mediated decays and scattering processes, including
•
Nuclear/neutron /? decay, inverse (n —> pe~ve; ven —> e~p; e~p —> ven)
• [i, r decays (/x~ —> e~9ev^ r~ —> yTv^Vr, uttv~, •••)
• 7T, K decays (tt+ —> /x+iv, ^°e+ve\ K+ —> /x+i/M, 7r°e+i/e, 7r+7r°)
• hyperon decays (A —> pn~\ E~ —> nir~; E+ —> Ae+ue)
• heavy quark decays (c —> se+ve; b —> cyTv^ ctt~)
•
v scattering (y^e —> /x ve\ v^n —> /x p; u^N —> [i X)
"elastic" deep—inelastic
However, the Fermi theory violates unitarity at high energy, reflecting its
non-renormalizability. For example, the cross section for vee~ —> e~ve grows
with energy when computed at tree level. In the modern V — A version the
middle diagram in Figure 6.1 yields
G2 s
a(uee -► e ve) -> —?-, s = E1CM. (6.5)
7T
This is a purely 5-wave process, so 5-wave unitarity (Appendix C) requires
a < ^r. Unitarity therefore appears to fail for
^>*[^f~S00GeV. (6.6)
This is not by itself so serious. In non-relativistic potential scattering, for
example, the Born approximation is not unitary, but unitarity is restored by
higher-order terms. For the Fermi theory, however, higher-order
contributions are divergent, again due to the non-renormalizability. For example, the
diagram on the right in Figure 6.1 involves the integral
MpXp)- (67)
TSome of the processes are now well enough measured that one needs to apply higher-order
corrections, which often require the full renormalizable standard model to make sense.

242
The Standard Model and Beyond
which diverges quadratically for k —> oo (we have neglected the e~ mass
and the external momenta, which is valid for large enough k). Although
vee~ —> e~ve has only been measured at low energies, there is clearly a
theoretical inconsistency; the Fermi theory cannot be the full story.
In the intermediate vector boson theory (IVB) the four-fermion
interaction was eliminated (Yukawa, 1935, the same paper as the meson theory)
and (Schwinger, 1957). Instead, it was assumed that the process was
mediated by a spin-1 particle, analogous to the photon in QED. However, the
intermediate bosons W± were assumed to be very massive (compared to the
energies of the experiments) and electrically charged, as indicated in Figure
6.2. The coupling to fermions is given by
C = gW^J^+gW-J^ (6.8)
where g is the coupling strength. For M^ ^> Q2 = —q2, where q is the
FIGURE 6.2
Top: intermediate vector bosons mediating /? decay and vee~ —> e~ve.
Bottom left: diagram for e+e~ —> VK+VK" in the IVB theory. Bottom right:
additional diagram in the 517(2) theory.
momentum transfer, the denominator q2 — M^ of the W propagator can be
replaced by -M^, and one has effectively a four-fermion interaction. We will

The Weak Interactions
243
see that this reproduces the Fermi theory at tree level with the identification
% ~ mfor Mw >y Q (6-9)
(the factors of 2 and y/2 will become apparent). However, the full propagator
leads to a better behaved amplitude for vee~ —> e~ve at high energies.
Unfortunately, the difficulties reemerge when we consider processes
involving an external W. As was briefly discussed in Section 2.5.1, a vector boson
with an elementary mass term leads to a non-renormalizable theory, and
amplitudes involving the longitudinal degree of freedom are badly behaved at
high energy due to the polarization vectors e^k, 3) ~ k^/Mw In particular,
the amplitude for e+e~ —> VK+VK~ violates unitarity for y/s > 500 GeV due
to the diagram on the lower left in Figure 6.2.
It is possible to resolve these problems by adding more particles in such a
way that the bad high energy behaviors cancel. For example, one may add a
massive, electrically neutral W° boson, so that there is an additional diagram
for e+e~ —> VK+VK", as shown on the bottom right in Figure 6.2. Choosing
the neutral current J° that describes the VK°-fermion coupling and the triple-
vector VK°VK+VK~ coupling so that not only e+e~ —> VK+VK- but also related
amplitudes like vee~ —> VK°VK~ are well behaved, one obtains [J, Jt] oc J° and
3 and 4 point vector vertices that are equivalent to an 517(2) gauge theory!
Thus, one can regard the gauge invariance as a necessary consequence of well
behaved high energy amplitudes.
The 517(2) model just described has no room for electromagnetism and is
not realistic. It was extended by Sheldon Glashow in 1961 (Glashow, 1961)
to an 517(2) x 17(1) model, with the 7 and the prediction of a second neutral
boson (the Z) as well as the W±. The gauge sector of the Glashow model is in
fact the standard model, but at the time there was no satisfactory mechanism
for generating masses for the W± and Z, resulting in difficulties for the
behavior of cross sections such as VK+VK- —> VK+VK- at high energy. This was
remedied by Steven Weinberg in 1967 (Weinberg, 1967a) and independently
by Abdus Salam (Salam, 1968), who invoked the Higgs mechanism to
generate their masses and those of the chiral fermions by spontaneous symmetry
breaking. Weinberg speculated that the SSB would preserve the renormaliz-
ability of the theory. This was proved by 't Hooft and Veltman and others
in 1971 ('t Hooft, 1971b,a; 't Hooft and Veltman, 1972; Lee and Zinn-Justin,
1972).
The original Weinberg model considered leptons only. That was because
only the u, d, and s quarks were known at the time. One could describe
the dominant d <-> u transitions by assuming they transformed as an 517(2)
doublet. The s would have to be an 517(2) singlet because it had no charge-2/3
partner. However, somewhat weaker s <-> u charged current transitions were
observed experimentally, so there would have to be d — s mixing. That in turn
led to strangeness changing neutral currents, i.e., d — s transitions mediated

244
The Standard Model and Beyond
by the Z, which had not been observed experimentally. In particular, both
the tree-level and loop diagrams in Figure 6.3 would lead to K° <-> K° mixing
much larger than what was observed. This difficulty could be remedied if there
existed a fourth, charge-2/3 charm (c) quark. The c and s could transform
as an 517(2) doublet. With the d and s transforming the same way, the off-
diagonal Z vertex disappeared, and the box diagram was strongly suppressed
by the cancellation of the largest parts of the c and u exchange amplitudes
(the GIM mechanism (Glashow et al., 1970)). An early estimate (Gaillard
and Lee, 1974b) of the necessary mass, mc ~ 1.5 GeV, was very close to the
actual value (see (5.125)).
8 K° / °k K°
FIGURE 6.3
Left: K° <-> K° mixing induced by strangeness changing neutral current
vertices in the 3 flavor 517(2) x 17(1) model. Right: Box diagram contribution.
However, physicists at the time were reluctant to entertain the possibility
of additional quarks (after all, the quark model had been invented to simplify
the hadron spectrum). In 1974, however, the J/ip resonance was discovered
simultaneously at Brookhaven and SLAC, and it was soon tentatively
identified as a cc bound state. The existence of charm was subsequently confirmed
by the identification of singly charmed mesons and in neutrino scattering (see,
e.g., Gaillard et al., 1975; Rosner, 1999).
The (flavor diagonal) weak neutral current processes mediated by the Z
were discovered at CERN and Fermilab in 1973, and extensively probed
experimentally in the 1970s and 80s. When combined with the observation of
charm, the basic ingredients of the standard model were in place. QCD was
established during the 1970s as well. The W and Z were discovered with the
expected masses at CERN in 1983, and high precision tests of the Z
interactions at the level of loop corrections were carried out at the LEP and SLC
colliders at CERN and SLAC, respectively, from 1989-2000. The loop
corrections allowed the prediction of the top quark mass, which was confirmed

The Weak Interactions
245
by its direct discovery at Fermilab in 1995. Since ~ 1995 precise tests of the
unitarity of the CKM matrix and verification that it describes the observed
CP violation were carried out in B physics experiments, especially at Cornell
University, SLAC, KEK, and later at the Tevatron at Fermilab. Finally, hints
of nonzero neutrino mass from Solar and atmospheric neutrino experiments
were confirmed in 1998, and the neutrino sector intensively studied since that
time. As of this writing in 2008 the only missing ingredient of the standard
model (extended to include neutrino mass) is the Higgs boson.
6.2 The Fermi Theory of Charged Current Weak
Interactions
Let us now describe the Fermi theory of charged current interactions in more
detail, mainly in the V — A form that was developed prior to the standard
model (the extension to include charm and the third family is straightforward).
More detailed discussions of the experiments in /?, /i, 7r, K, and hyperon decays
that led to its development, including analyses in a more general framework
allowing general «S, P, T, V, and A interactions may be found in (Commins and
Bucksbaum, 1983; Renton, 1990; Langacker, 1995; Severijns et al., 2006). The
Hamiltonian density is
n = ^4J^ (61°)
where the Fermi constant Gf is given in (6.4). The y/2 factor compared to
(6.1) is due to the modification from V to V — A currents. The charge raising
current is given by the sum of leptonic and hadronic currents,
Jt = J? + J?. (6-11)
The leptonic current is
Jf? = Vel» (1 - 75) e" + ^ (1 - 75) M~
= 2(i^L7MeZ +VtiLlpHl),
(6.12)
where of course tpL = Pl^> = ^^-ty. The V^ — A^ (i.e., 7M(1 - 75)) form
corresponds to the maximal amount of parity and charge conjugation violation.
One can add 2i/tl7mT£ to (6.12). The leptonic charge lowering current is
Jl = z~ln (1 - 75) ^ + AOVi (1 - 75) "» = 2 [eLl^eL + Hi Ml] • (6.13)
The simple extension of (6.2) and (6.13) for the hadronic current would be
J£f ~ P7/* (x " 75) ncos0c, j£ ~ n7/, (l - 75) pcos0c, (6.14)

246
The Standard Model and Beyond
where the cos#c Cabibbo angle factor will be discussed below. This would
describe (3 decay. However, there are other observed hadronic transitions
including hyperon decays such as E~ —> n, and meson decays involving 7r+ —►
7r°,iiT+ —> 7T°, or (K+,7r+) —> vacuum. These could be described by adding
additional terms to (6.14). However, a much simpler and more universal form
is obtained by writing it in terms of quark fields. The form prior to the
discovery of the c, 6, and t quarks was
j£t = u7„ (1 - 75) df = 2uLl^L
Jji = <?7„ (1 - 75) u = 2d'Ll^uL, '15)
where a sum over color is implied and 6! is a rotation of the d and s quark
fields
d' = dcos6c + ssin0c. (6.16)
The angle 8C is the Cabibbo angle, with value sin#c ~ 0.23. It describes the
mismatch between the weak eigenstate d', i.e., the field that couples to the
u quark in J^t, and the mass eigenstates d and s. tan#c measures the ratio
between strangeness changing, AS = 1, and strangeness conserving, AS = 0,
transition amplitudes, such as E~ —> ne~ve and n —> pe~ve, respectively.
The fact that the hadronic and leptonic currents have the same strength, up
to the rotation, reflects Cabibbo universality. That is, the squared amplitudes
for /i~ —> v^e'Ve, n —> pe~ve, and E~ —> ne~ue are all different, but they
are universal in the sense that if one could ignore strong interaction and mass
effects one would have
|M(/i" - v^e~ve)\2 = \M(n - pe"Pe)|2 + |M(E" - ne"i/e)|2, (6.17)
since they scale as 1 : cos2 6C : sin 6C. This result emerges naturally in the
quark theory, where it is seen to result from the simple rotation in (6.16), but
is more mysterious otherwise. Universality generalizes easily to the 4 and 6
quark cases (CKM universality), and in the standard model is seen to be the
result of a universal gauge interaction combined with family mixing.
(6.15) incorporates the empirical A«S = AQ rule for strangeness
changing transitions. For example, in the observed E~ —> n transition both the
strangeness and the hadronic electric charge increase by one unit. However,
the AS = —AQ transition E+ —> ne+ve is not observed in nature. Similarly,
despite many experimental searches strangeness changing neutral currents (or
flavor changing neutral currents (FCNC), as they are now called) have never
been observed, with the exception of processes that are consistent with being
of higher order in the electroweak interactions. Thus, there are no5->d
transitions, which could lead, e.g., to E° —> ne+e~. This is illustrated by the
weight diagrams in Figure 6.4.
Using (6.10), (6.12), and (6.15), the four-fermion Hamiltonian is H =
f d3xH, where
H = % (J? + J?) {Je» + Jh")=Hi+ Hsi + Hni. (6.18)

The Weak Interactions
247
S°, A £+
u
• s
FIGURE 6.4
Weight diagrams for the quarks and for the baryon octet. The hadronic charge
raising current Jh^ mediates transitions to the right (AS = 0) or diagonally
up and to the right (AS = AQ), but not diagonally up and left (FCNC) or
to non-adjacent states (AS = —AQ).
In (6.18) the leptonic Hamiltonian density is
GF
Hi
v/2
!>™>(l-75)em) feef,7"(l-75)»'n)
K rn / \ n J
(6.19)
where em = e or /i. It is responsible for such purely leptonic decays and
processes as muon decay (/i~ —> v^e~ue) and inverse muon decay (v^e~ —>
\i~Ve). The semi-leptonic density is
Hsl=7i
£ Vml» (1 - 75) em j (W (1 - 75) u)
+ (u7/i (1 - 75) <f) ( £ em7" (1 - 75) i/m
(6.20)
It is responsible for decays of hadrons into final states which include leptons,
such asn-> pe~De, E~ —> ne~ve, K+ —> /i"1"^, and if+ —> n0e+ve, as well as
for neutrino scattering and inverse (3 decay, ven <-> e~p. In the extension to
three lepton families it also allows semi-hadronic r decays, such as r~ —> vt-k~ .
The non-leptonic term
Km = ^ ( S7m (1 - 75) <*') ( dV (1 - 75) «
(6.21)
drives decays not involving leptons, e.g., K+ —> 7T+7T0, if+ —> 7r+7r+7r°, £+ —>
P7T°, and A0 —> n7r° (Commins and Bucksbaum, 1983). It can also generate

248
The Standard Model and Beyond
a parity-violating perturbation in the NN interaction (Desplanques et al.
1980; Ramsey-Musolf and Page, 2006; Holstein, 2005), leading to effects in
polarized pp scattering, nuclear transitions, and an electromagnetic anapole
(7^75) moment (Haxton and Wieman, 2001). It is difficult to calculate matrix
elements of Hni reliably because two hadronic currents are involved*.
There is an obvious asymmetry between the quarks and leptons in the 3
quark case. That was remedied subsequently with the discovery of the charm
quark, which could partner with the s as an additional term in the hadronic
current
J? = u^ (1 - 75) d' + c7m (1 - 75) s' = 2uL^dfL + 2cl7^'l, (6.22)
where
s' = s cos 0C - d sin 6C. (6.23)
One can also extend to a third family. Then,
Jt = (w^Ml - 75) I M" I + {ucl)^(\ - 75)Vckm Is), (6.24)
where Vckm is the 3x3 unitary CKM quark mixing matrix. Empirically,
the terms in the CKM matrix that mix the third family with the first two are
small and can usually be ignored when considering c, s, or d decays. (They
are of course critical for b decays.) These extensions will be mentioned in
Section 6.2.7 and (along with a further extension to include neutrino mass)
described in the context of the full standard model in Chapter 7.
The charged current Fermi Hamiltonian is of the V — A type, i.e., each term
is of the form
* x
^ (6.25)
H= fd3xH(t,x)= fd
x [(W*,*)) - Vj(M)) (V&{t,£) - A&frZJ) + h.c.
where
VpabfaS) =$a(t,£hpil>b(t,£), A^ab&x) = $a{t, x)<y^<y5ipb(t, x) (6.26)
*For example, the AS = ±1 part can be decomposed into operators with total isospin 1/2
and 3/2. Empirically, the kaon and hyperon decay rates involving the isospin Al = 1/2
piece are much larger than those driven by the Al = 3/2 term, i.e., the Al = | rule,
or the octet rule from the 5(7(3) perspective (see, e.g., Commins and Bucksbaum, 1983;
Donoghue et al., 1992). Especially puzzling are the K —*■ 2n decays, where the ratio of
the relevant amplitudes is around 20. Short-distance QCD corrections can account for a
factor of ~ 3 (Gaillard and Lee, 1974a; Altarelli and Maiani, 1974), but the rest must
be due to long distance non-perturbative effects. There has been some recent progress
understanding this from lattice calculations (Yamazaki, 2009) and from the AdS/CFT
correspondence (Maldacena, 1998; Hambye et al., 2006).

The Weak Interactions
249
are respectively vector and axial currents. We saw in Section 2.10 that under
space reflection
W*>s) "> pV»ab{t,x)P"1 = V£(t, -x)
A»ah{t,x) - PA^t,x)P~l = -A»b(t, -x). (6'27)
Changing the sign of the dummy integration variable and using A^Bn, =
J ^ n (6.28)
[V^i2(t,x) + A^l2(t,x J)(v&{t,2) + A&{t,2))+h.c.
i.e., the interaction changes to V + A and the VA interference terms change
sign. Parity is violated maximally, as is manifested by the fact that left-chiral
fields (corresponding to left-handed particles and right-handed antiparticles
in the massless limit) participate in weak transitions but right-chiral fields do
not. It was not initially suspected that parity was not conserved. However,
in the mid 1950s, the decays K+ —> 7r+7r° and K+ —> 7r+7r+7r° were both
observed. Since the K+ has spin-0 and the pion has negative intrinsic parity,
the first mode would require an even intrinsic parity t]k+ = +1 if parity
were conserved, while the second would imply t]k+ = —1§. This led Lee
and Yang to reexamine the question of whether parity was conserved in the
weak interactions (Lee and Yang, 1956), and soon thereafter C. S. Wu et al.
established parity violation by observing an asymmetry in the direction of the
emitted electron w.r.t. the nuclear spin direction in the (3 decay of polarized
60Co (Wu et al., 1957).
Similarly, under charge conjugation
Vpab -> CV^abC l = -V^ba = ~V^ab
A^ab ~~* (sA^abC = -\~Apba = -\~A^ab.
Thus
H^CHC~1= fd3S^ x [(^i2 + V2)t(v3'i + ^4)t + /i.c.
(6.29)
(6.30)
Charge conjugation therefore changes V—A to V+A, just like space reflection.
This implies that only left-handed particles and right-handed antiparticles are
involved in weak charged current transitions. Even though C and P are each
§ At first, it was thought that the new modes represented the decays of two distinct particles,
0 —► 7r+7r° and r —► 7r+7r+7r°, but experiments indicated that the they must have the same
mass, spin (= 0), and lifetime, but opposite parity. This unexpected state of affairs was
known as the r — 0 puzzle (Commins and Bucksbaum, 1983).

250
The Standard Model and Beyond
violated maximally, the product CP restores H to the V — A form and CP
is conserved,
{CP)H{CP)~1 = H. (6.31)
For example, the charged current weak interactions of the e^ are the same as
those of the e^, while e^ and e\ do not enter the charged current. Actually,
there is observed to be a small amount of CP violation in nature, much
weaker than the normal weak interactions. This cannot be accommodated
in the Fermi theory, or even its extension to 4 quarks. However, as we will
describe in Section 7.6, it can occur in the three family theory.
6.2.1 /x Decay
The muon (p±) is a heavy version of the e±, with mass raM ~ 105.7 MeV
and lifetime rM = 2.20 x 10~6 s. The pT decays nearly 100% of the time into
e~v^ve via the weak charged current, including a small electromagnetic
radiative correction leading to e~v^ue^. (The properties of the /i+ are analogous
by CP invariance.) The diagram for pT —> e~v^ue is shown in Figure 6.5.
The Lagrangian density^
M (Pi)
^e(P2)
M (P2)
*V(pi)
i
FIGURE 6.5
Left: Feynman diagram for pT —> e~v^ue. Right: diagram for tt~ —> pTv^.
-£ = H=^ u^ (1 - 75) /i" e-y (1 - 75) "e
corresponds to a matrix element
M =
—^ u37M (1 - 75) m urf (1 - 75) u2.
(6.32)
(6.33)
^It is sometimes convenient to use the Fierz identity in (2.211) to rewrite this in the charge
retention form —C = —& e~7M (l — 75) fj,~ PM7M (l — 75) ve.

The Weak Interactions 251
The differential decay rate is
Jr _ (2tt)4 <54 (pa + p3 + Pa - Pi) d3p2d3p3d*Pi\M\2 (g ^
2mM (27r)3 2£2 (2tt)3 2£3 (2tt)3 2£4 '
\M|2 can be calculated using the standard trace techniques of Chapter 2.
Neglecting m„,
x Tr
7, (1 " 75) (fa + m,) (1 + ?f^) 7, (1 " 75) fa
r (1 - 75) fal" (1 - 75) (A + me) (^^
(6.35)
where we have kept the pT and e~ spin projections because many experiments
have measured them. For now, however, let us just calculate the total decay
rate, obtained by averaging (summing) over the /i" (e") spins (for mv = 0
only the left (right)-handed v^ip^) is produced),
|M|2 - \M? = \ E lMl2-
(6.36)
Neglecting the electron mass as well (me/m^, ~ 1/200),
|M|2 = ^TY [^ (X - ^) (^ + m^ ^ t1 - 7') *] (6.37)
x Tr [7" (1 - 75) ^27" (1 " 75) fa] ,
where the raM term doesn't contribute because it involves the trace of an odd
number of 7 matrices. Despite the formidable appearance, (6.37) is
straightforward to evaluate using the identities in (2.168). One finds
\M\2 =G2FTr [7„ fa7„ fa (1 + 75)] ^ [7" faY fa (l + 75)]
= 16G^ [pi^i/ + Pli/Jtyi - 9pvPl ' P3 + i€tMTuuPlP3]
X [P^P4 + P2# - fP* ' P4 + tC^^pP^]
= 16G2F [2pi • p2P3 ' P4 + 2pi • P4P2 ' P3 - eM1/Ta;6^^ap[p3P2pP4a] ,
(6.38)
where the contraction of the two tensors has been simplified because the first
three terms in each are symmetric in /i and v, and the last is antisymmetric.
But
c^c^ = "2 (tfgZ ~ &9%) (6-39)
from Table 1.2, so that the last term is just
2 p\ • P2P3 • P4 - 2 pi • p4P2 • P3,
(6.40)

252 The Standard Model and Beyond
and therefore
\M\2 = 64G|pi • p2 Ps • P4- (6.41)
Incidentally, this calculation justifies the identities in (2.171), noting that the
sign of the last term in (6.38) is reversed for the 1 — A75 case.
Assuming that the neutrinos are not detected, we can integrate over their
momenta to obtain the unpolarized differential decay rate
,- 4G2F d3p4tfpZ f d3p2d3p3 ,4 ,
dT = J^f -E^T I -^E^5 iP2 +P3~q) P2pP3°' (6-42)
where q = p\ — p±. To evaluate the integral, we note that
IPa = J ^eT^ {P2 +P3~ti P2>>P*° (6-43)
is a second rank tensor which can only depend on q. It must be of the form
Ip<r = Aq2gpa + Bqpqa, (6.44)
since gpa and qpqa are the only tensors available. We have extracted a q2
in the first term so that A is dimensionless. A and B can be obtained by
evaluating the much simpler integrals (Problem 6.1)
g(>aIpa = AAq2 + Bq2 = nq
qf>q°Ipa = (A + B)q4=*q4.
4 _ T 4 (6'45)
Therefore,
2
A=l' B=\ =* IP- = l[q29pa + 2qpqa], (6.46)
6' 3 pa 6
and
2irG2F d3p4
dT= S&^E^; [{Pl ~Pi) Pl -P4 + 2P1 "(Pi "P4)P4-(P1 -P4)J •
(6.47)
In the muon rest frame, neglecting the other masses,
Pi = (mM, 0) , Pi • p4 = m^Ei, p\ = ml, p\ = 0, (6.48)
yielding
df = JvJs f ™ [(tom _ 2"V^4) m^4 + 2mM (mM - E4) mM£4]
d(27r) i,4mM (649)
= ^$2ndcos0dE4 [3ml - 4m„E4] EJ,

The Weak Interactions
253
where 0 is the polar angle of the e~ and 0 < E4 < ^ is its energy. (£4 = 0
corresponds to two back-to-back neutrinos carrying all of the energy, while
Ea = nfy/2 corresponds to the two neutrinos being in the same direction, so
that p2 • P3 = 0). Integrating over cos0 and denning the dimensionless e~
energy e = 2E4/m^, we obtain the final result for the muon lifetime,
x = Stt2G2f ml f1 2 ro G2Fml
t =r=w^ Ldee [3 -26]=^ (6-50)
which can be used to obtain the Fermi constant Gp = 1.17 x 10-5 GeV-2
from the observed lifetime rM.
The muon lifetime is known extremely well, rM = 2.197019(21) x 10-6 s,
and Gf is a critical parameter for the precision electroweak tests (i.e., for the
prediction of the W and Z masses in the standard model), and also for tests of
CKM universality. To extract a precise value one must include electron mass,
W propagator, and two-loop electromagnetic corrections" (some one-loop
diagrams are shown in Figure 6.6) to the lifetime formula, yielding (Amsler et al.,
2008)
Fermi,with me^0 gM
1+ 25_7r2\o(rnE)+ o^)
4 / 27T 7T2
(6.51)
radiative corrections
where
F(x) = 1 - Sx + Sx6 - x4 - 12x2 lnx
"(-)■'-"■,-shfe)+s~"6 <*>*>
156815 518 o 895 „n. 67 4 53 2l .„.
C2 = ^I8T " «-* - ^6"C^ + 720** + T^ ^'
(C(3) ~ 1.202 is the Riemann Zeta function). One obtains GF = 1.166367(5) x
10-5 GeV-2.
" The QED radiative corrections to Fermi theory are finite to all orders in a and leading order
in Gf (for which there are no InMw effects), provided the usual QED renormalizations
are applied (Kinoshita and Sirlin, 1959; Berman and Sirlin, 1962). On the other hand, the
QED corrections to (3 decay are logarithmically divergent (they become finite in the full
SM). The difference, which has nothing to do with the strong interactions or the neutron
mass, is explored in Problem 6.4.

254
The Standard Model and Beyond
FIGURE 6.6
Representative low order diagrams contributing to /i decay in the Fermi
theory, including one-loop and initial state radiation diagrams. The cross
represents a mass renormalization counterterm.
The expression in (6.50) can be generalized to allow for a polarized muon
and measurement of the electron polarization:
dT
G2Fml
1927T3
[262(3-2e)]
1 +
1-26
3-26
COS0
l-Pe'Se
ded cos 0
(6,53)
where cos 6 = pe • sM is the cosine of the angle between the electron
momentum and the muon spin direction, and pe • se/2 is the electron helicity. These
formulae can of course be extended to include higher-order corrections, the
electron mass, and the W propagator (Commins and Bucksbaum, 1983). The
2e2 (3 — 2e) factor yields ane" energy spectrum characteristic of V — A (see
Section 6.2.7 and Problem 6.2). The pe • sM and pe • se asymmetries reflect
the parity nonconservation. The pe • se term implies that the e~ helicity is
— \ in the limit me = 0, a characteristic of V — A. The cos# term describes a
correlation between the pT polarization direction and the electron momentum
for a given e~ energy (the coefficient of cos# reverses sign for /i+ decay). This
can be used to determine the polarization direction of a sample of muons,
and was used, for example, to determine the muon spin precession in the
Brookhaven experiment measuring the muon anomalous magnetic moment
(Section 2.12.3). Similarly, the polarization of r's produced in e+e~ —> t+t~
in the Z-pole experiments at LEP was determined from the angular
distributions of the decay products in the analogous leptonic r decays, such as
One can also carry out an analysis allowing a more general four-fermion
interaction including 5, P, and T, as well as V and A interaction terms. There
are various parametrizations (see the reviews by Fetscher and Gerber in Lan-

The Weak Interactions
255
gacker, 1995; Amsler et al., 2008), such as
AC
H = —j= {9ll zl 1p VeL »nL Y Ml + 9rr eR 7P veR v^r Y nR
+ 9LRCL'Vpl'eLV»R'VP VR + 9RL ^Rlp "eRV»L Y ML
+ 9LL^L^eR^R^L + 9RR^R^eL V»L V>R ^M>
+ 9LRZLVeRVnLVR + 9RL^RyeL^p,R^L
+ 9LR^LtpayeR^p,LtP<T flR + #£L e~R tpa »eL VpR tP° Ml) + ft.C,
where tpa = apa/y/2. (6.54) is the most general form assuming lepton-number
and lepton-family conservation, but is actually applicable in the more general
case in which arbitrary neutrinos and antineutrinos are emitted, if the
neutrinos are not observed and their masses are negligible (Langacker and London,
1989). The Fermi theory predicts g\L = 1 and others = 0. However, the
other gv 's could be generated in left-right symmetric theories involving a
second Wr which couples to V + A, by W — Wr mixing, and by mixing of the
known leptons with heavy exotic fermions. The gs could be generated from
the exchange of a spin-0 particle, such as a non-standard Higgs boson or by
^-parity violating couplings in supersymmetry. gT could be associated with
the exchange of a spin-2 particle. The coefficients must be relatively real if T
holds. A more detailed discussion of new physics contributions to /i decay is
given in (P. Herczeg, in Langacker, 1995; Kuno and Okada, 2001).
There have been many precise experiments on muon decay and inverse muon
decay, Vp,e~ —> /i_^e, including measurements of the electron spectrum and
helicity, and of correlations involving the muon spin, at PSI, TRIUMF, and
elsewhere. The data are sufficient to establish \g\L\ > 0.960 with (usually
stringent) upper limits on the other couplings (Gagliardi et al., 2005; Amsler
et al., 2008). This excludes the possibility of alternatives to V — A as the
dominant contributor to muon decay and inverse decay. One may also search
for small deviations from V — A, e.g., due to small admixtures of effects
involving the types of new physics mentioned above. It is convenient to generalize
(6.53) to a general four-fermi interaction,
G2Fml D 0 ( , x 4 ,
J\ , dcosO
T-Pjif cose
4(1 -e) + U(8e -6)
(6.55)
where additional terms involving rae, the eT helicity, and radiative corrections
are not displayed. The Michel spectral parameters (Michel, 1950; Kinoshita
and Sirlin, 1957) p, £,J, and 7] (which appears in the rae/raM corrections)
and the overall normalization D are functions of g^ ' , while PM is the /i
polarization from 7rT —> /iT v . The experimental values in Table 6.1 are in

256 The Standard Model and Beyond
impressive agreements with the predictions of the V — A theory,
P = S=l> € = 1, *7 = 0, D = 16, PM = 1. (6.56)
TABLE 6.1
Experimental values (Amsler et al., 2008) of the Michel parameters
and their expectations in the V — A or standard models. Pe+ is the
e+ longitudinal polarization (twice the helicity).
parameter
P
V
S
p*z
P„ &/P
Pe+
experimental value
0.7509 ± 0.0010
0.001 ± 0.024
0.7495 ± 0.0012
1.0007 ±0.0035
> 0.99682 (90%)
1.00 ± 0.04
V -A
3
4
0
3
4
1
1
1
New and even more precise values of p and 6 have recently been reported
from the TWIST collaboration at TRIUMF (MacDonald et al., 2008), leading,
e.g., to a limit \Clr\ on a W — Wr mixing angle. Other tests of the V -I A
theory of charged current leptonic interactions are provided by leptonic r
decays, r —> tvv, and by inverse muon decay.
6.2.2 Ve^T —► vee~
In the Fermi theory the processes vee~ —> i/ce~ and vee~ —> Pee~ proceed via
the leptonic weak charged current interaction in (6.19), by the diagrams in
Figure 6.7. The Lagrangian density is
-C = H = ^ P67m (1 " 75) e~ e~^ (l - 75) ve
= ^ Pc7/i (1 - 75) ve e~r (1 " 75) e~ (6.57)
=> ~| *eln (! - 75) ^e e_7M (^v - ^75) e".
The second (charge retention) form is obtained using the Fierz identity in
(2.211). The last form looks ahead to the full standard model, in which
there is an additional neutral current contribution. The Fermi theory predicts
9v — 9a — 1- The spin-averaged squared amplitude can be calculated in the

The Weak Interactions
257
e (Pa)
^e(p4) e (p3) ^e(p4)
J*4
^e(Pl)
e (p2) *e(Pl)
A
e (p2)
FIGURE 6.7
Diagrams for vee~ —> vee~ and vee~ —> vee~ in the Fermi theory. There are
additional weak neutral current contributions in the full standard model.
same way as for /i decay. One can use (2.171) to obtain
7, ^2 lMl2 = 16G^ \f<9v + 9 a)2 Pi • P2 Pa • P4 + (gv ~ 9a)2V\ • P3 P2 • P4
S1S2S3S4
(Sv ~9\)m\v\ -P4]-
(6.58)
(The incident i/e's all have helicity -1/2, so only the e~ spin is averaged. It
is convenient to include sums over the ve spins, even though the contribution
of the h = +1/2 states vanish.) Let us define the invariant
y
Pi -P4
Pi -P2
(s-m^)(l+cosfli3)
^ 2s '
h ^b
CM
(6.59)
where s = (pi + P2)2 = ml + 2meEu, Ev is the incident v energy in the
electron rest frame (i.e., the lab frame), and Te is the kinetic energy of the
final electron in the lab. y is the fractional energy transfer to the final electron
in the lab, and runs from 0 to (1 + me/2Eiy)~1. Then,
dau
GFvneEv
dy
2tt
me
(gv + gA? + (gv - gA) (i - y) - (gv -gA)-^y
f1
Jo
da
dy
dy
G2FmeEv
2tt
{gv+gA)2 + ^{gv -gA)2
(6.60)
where we have dropped the last (mey/Eu) term in the total cross section for
simplicity. (The limits on y and the (invariant) cross section formula are most
easily derived in the CM and then expressed in lab variables.) Similarly, for
vee~ -> i/ee~
daPe _ G2FmeEv
dy
2tt
-*2
~2^~
me
{gv + <m)2(i - yY + (gv - gA)2 - (gv -gtiirv
E„
GFmeEv
^{9v + 9a)2 + (9v ~ 9a)2
(6.61)

258
The Standard Model and Beyond
ve scattering was not actually observed until after the standard model was
developed, and then most of the experimental studies have involved v^ or v
beams, since they are produced much more easily from ir and K decays at
accelerators. As will be described in Chapter 7, the different y distributions
for the analogous v^e {y^e) neutral current processes in the standard model
were useful for determining the parity structure of the neutral current.
6.2.3 7r and K Decays
The 7T£2 Decays: n —> /xi/, ev
The pions, -k± = ud (du) and 7r° = (uu — dd)/y/2, are much lighter than
the other hadrons, with mn± ~ 139.6 MeV, which is somewhat larger than
ra^o ~ 135.0 MeV due to electromagnetic corrections. As described in Section
5.8, the small value of m^ can be understood because of their role as pseudo-
Goldstone bosons of an approximate global SU(2)L x SU(2)r flavor symmetry
of the strong interactions. The ^ decay via the weak charged current, mainly
into /i±^M(i>M), denoted 7rM2, with a relatively long lifetime of rn± ~ 2.6 x 10~8
s. The 7T° decay is electromagnetic and mainly into 27, with a much shorter
lifetime rno ~ 8.4 x 10~17 s. The role of the chiral anomaly and the number
of colors in the 7r° lifetime was briefly described in Section 5.2. The branching
ratios for the principal decay modes of the pions and charged kaons are listed
in Table 6.2. Neutral K decays will be considered in Section 7.6.
The Feynman diagram for the semi-leptonic decay process tt~ —> /i-^ is
shown in Figure 6.5. The matrix element is
M = "*%^27m (1 "7>i (0|J^t(0)|7r-(g)), (6.62)
where pi = Vv^.Vi — V^ and q = p^- = pi +P2, and the hadronic current is
J? = V^ ~ AhJ =u^(l-f5)[dcos 6C + s sin 0C]. (6.63)
The first (cos#c) term is relevant to n decay. We saw in Section 2.10 that the
pion is a pseudoscalar, P\tt1(p)) = — |7rz(— p)). Reflection invariance** then
implies that only the axial current contributes to {0\Jh^\n~(q)). To see this,
recall that py^p~l = y^ for the vector current, and that the vacuum is
invariant, P\0) = 0. By Lorentz invariance, the matrix element must be of
the form
(01^1^-^))= t^, (6.64)
where a is a constant, since q is the only four-vector available. (The current
is evaluated at (£,x) = 0.) Therefore, by an argument similar to (2.270),
iaq» = (Q\p-lPVrtp-'P\n-{q)) = -<0|K>-(g')> = -<<, (6.65)
**The weak interaction parity violation is explicit in the V — A form of the current. The
matrix elements of each part of the current are (reflection invariant) strong interaction
quantities, up to negligible higher-order weak effects.

The Weak Interactions
259
TABLE 6.2
Branching ratios for the principal decay modes of the pions and charged
kaons. The branching ratios of the conjugate n~ and K~ decays are the
same by CP invariance.
7T+ -> /i+I^
-> e+ve
-> e+ise7r°
-> M+*V7
-7^27
—> e+e 7
—» e+e~ e+e~
"IP -»/x+^
-> e + fe
-» n+ffj,n0
-» e+I/e7T°
->7r+7r°
—» TT+TT+TT~
-» 7r+7r07r°
(^2)
fe)
(*"e3)
(^2)
(*<*)
(^3)
(#e3)
(#2*)
(tfs*)
(#3,r)
99.99%
1.23 x 10-4
1.036 x 10-8
2 x 10-4
98.8%
1.2%
3 x lO-5
63.4%
1.6 x lO-5
3.3%
5.0%
20.9%
5.6%
1.8%
tt beta decay
electromagnetic
/k/U
universality test
universality test
nonleptonic
where q = (Eq, q) and qf = (Eq, —q), i.e., q'^ = </M. Thus, a = 0.
However, PA^P~l = — Ah^, so the axial matrix element can be non-
vanishing. By Lorentz and translation invariance,
(0\Ahrt(x)\n-{q)) = icos^/^^e"^, (6.66)
where the cos 0C is extracted for convenience and fn is the pion decay constant.
We saw in Section 5.8 that fn = 0(A) (denoted / in (5.107)) is related to the
spontaneous breaking of SU(2)l x SU(2)r. Thus,
-fnGFcosec_ ,, , ,Wl 5\ -f*™>i*QFcos6C_ , 5v
M = — u2 (^1+ jh) (1 - 7 ) vi ~ ^ u2 (1 - 7 ) *>i,
(6.67)
where we have neglected mv. Summing over the final spins
f2 2
|M|2 = V |M|2 = ^G2F cos2 0cTY [(1 - 75) A (1 + 75) (A + mM)],
^^ 2 v v •
8pi-p2=4(mJ-m2 j
(6.68)

260
The Standard Model and Beyond
so that
_! _ (2tt)4(54(pi + p2 - q)d3p1d3p2 - 2 _ pi
2m7r(27r)3(2£1)(27r)3(2£2) ' ' 8irm*' '
G2^ cos2 flc/2?^ 2
2
(6.69)
ft mt\l- 2
87r M \ ra*
Using Gf from /i decay, cos 0C ~ 0.974 from (3 decay, and the observed lifetime
this yields fn = (130.4 ± 0.2) MeV ~ 0.93771^-, where the uncertainty is
dominated by higher-order effects (Amsler et al., 2008).
The rate for ^ —> e±ve(ve) is of the same form as (6.69) except raM —> me.
Therefore, one predicts
K* ~ r(7r - M„) m£ U2. - ml J U + U[a))
(6.70)
= 1.28 x 10-4 (1 + O(o)) = 1.24 x 10"4,
where 0(a) is a radiative correction (Commins and Bucksbaum, 1983). This is
in perfect agreement with the experimental value Rn = (1.230+0.004) x 10~4,
and supports the universality of the e and /i charged current interactions.
Neutrino beams at accelerators are obtained mainly by the decays of tt±. The
small value of R^ is the reason that such beams are mainly v^ and J/M, with
little ve or ve (some ve are produced, e.g., from K or secondary ^ decays,
but these are mainly considered as backgrounds).
The suppression of the 7re2 mode ir —> ev is a consequence and test of the V—
A structure of the charged current. V — A is just the left-chiral projection 2P^.
Since chirality and helicity are the same for a massless fermion, weak charged
currents imply helicity h = —1/2 for e~,/i~,i/e, and i/M, up to corrections
of 0(m/E) in amplitude. Similarly, h = +1/2 for e+,/i+,i/e, and v^. Since
the neutrino masses are negligible in this context, the v in tt~ —> £~&e must
have h = +1/2. But the pion has spin-0, so angular momentum in the rest
frame requires h = +1/2 for the charged lepton as well. This is the wrong
helicity, leading to the ml/ml factor in Rn. Thus, the electron mode is
strongly suppressed, despite the partial compensation from the larger phase
space. Things would be different for a scalar-pseudoscalar interaction M ~
u\ (l + a75) ^2, which leads to equal t~ and &e helicities in the massless limit.
A pure 5, P interaction would predict
R„=(ml~ml)2 = 5A9, (6.71)
which is clearly excluded. The value of Rv can also set limits on small S, P
or V + A perturbations on the dominant V — A.

(6.74)
The Weak Interactions 261
The Weak Currents and SU(3)L X SU(3)R Chiral Symmetry
In (5.98) and (5.99) we defined an octet of vector and axial generators of
SU{$)l x SU(3)r and their associated Noether currents, i.e., V1 = q^^q
and A^ = qi^^Q- Tne hadronic weak charged currents are simply related
to some of these, e.g.,
u^d = Al + iAl, u^s = Al + iAl, (6.72)
and similarly for the vector currents. Defining
4 = ^-4» (6-73)
the hadronic weak charged currents are
4f = (4+iJl)cos 0c + (4+iJl)sin e*
4 = (4t)t = (4 - iJl) cosec + (4 - ^l) sin ec.
Similarly, the hadronic electromagnetic current is
J% = \u1»u-l-d1»d-l-s1»s
= -(«7Mu - d^d) + -(u7Mu + d-y^d - 2s^s) (6.75)
^ D
These relations are useful because they allow us to use 517(3) and SI/(3)l x
SU(3)r to constrain and relate their matrix elements. In particular, in the
SI/(2) and SI/(3) limits one can relate the form factors of the weak
interaction vector currents to those measured in electromagnetic transitions, using
(6.75). This all seems obvious in the quark model, but was inferred earlier
and was known as the conserved vector current (CVC) hypothesis. Similarly,
the axial currents and the pseudoscalar mesons in (3.91) both transform as
SI/(3) octets, so in the SI/(3) limit one must have
(0|4(0)|^(g)) = +i^|*V, h3 = 1 • • • 8, (6.76)
where the y/2 is inserted to coincide with the definition in (6.66) of the pion
decay constant^,
<0|«7M75d|7r-> = ^(014 + mi*1 - in2) = +i/,^. (6.77)
Recall that these relations motivated the partially conserved axial current
(PCAC) hypothesis in (5.120). Of course, SI/(3) is broken by ~ 25%, so one
expects that the values fn and /k actually measured in tt and K decays may
differ by up to this amount.
ttSome authors define fn without the \[2 in (6.76), so that it takes the value ~ 92 MeV.

262
The Standard Model and Beyond
The K12 Decays
The K± = us(su) are pseudoscalar mesons with mK± = 493.7 MeV and a
typical weak lifetime of rK± = 1.24 x 10~8 s. The neutral kaons K° = ds
and K° = sd are somewhat heavier (497.6 MeV) because the rrid > mu
quark contribution to the splitting is larger than the electromagnetic one.
The neutral K decays have special properties due to K° — K° mixing and
CP violation, and will be described in Section 7.6. The branching ratios for
the principal K± decays are given in Table 6.2. The dominant mode is K^2,
i.e., K~ —> yTv^. The calculation is identical to tt^- The hadronic matrix
element is
(0| Jh^{x)\K-{q)) = -ifK sin^e-^*, (6.78)
where the kaon decay constant /# may differ from fn due to «SI7(3)-breaking.
Again, only the axial part of Jh^ contributes due to the reflection invariance
of the strong interactions. The rate is
™2
TK-^-^ = G2F sin^ ecf2Km^mK 1 - -f . (6.79)
The Cabibbo angle factor sin#c ~ 0.23 is known independently from Kes,/3,
and hyperon decays, allowing one to determine Jk from the K lifetime and
branching ratio, i.e., rK-_^-^ = B(K~ —> ^~y^)/rK-. The result is fx =
(155.5 ±0.8) MeV~ 1.19/*.
The 7T£s and Kis Decays ;
The nes and Kes decays are the three-body semi-leptonic decays, 7r+ —>
7r°e+i/e (pion beta decay), K+ —> 7r°^+i^, and Kl,s —> 7r±^=F^(^), and their
CP conjugates, where ^ = /ior e. Kl,s refer to the mass eigenstate neutral
kaons ~ (K° ± K°)/y/2 (Section 7.6). '
The matrix element for 7r+ —> 7r°e+i/e is
M = -i^| u„7m(1 - 75)«e <7r°(p')| Jh"(0)|7r+(p)>, (6.80)
where the relevant part of Jh^ is the first term in (6.74). It is
straightforward to show (Problem 6.9) that only the vector current contributes due to
reflection invariance. Furthermore, by Lorentz invariance, the matrix element
must be of the form
(*0(p')\Jh»(x)\n+(p)) = (7r°(p')\Vl»(x) - iV2»{x)\K+{p)) cos6c
= [f+(q2)(p'+pr + f-tfW -PY] cos0ce^'-P>-*,
where f± are form factors analogous to the electromagnetic matrix element
in Section 2.12A. Similarly, the matrix element for K+ —> 7r°^+i^ is
M = -t^| G„7/1(l - j5)ve (n°(p>)\Jh»(0)\K+(p)), (6.82)

The Weak Interactions
263
where
(■K0(p')\Jh^x)\K+(p)) = {^(p')\V^(x)-iV^(x)\n+(p))smec
= [f?(i2)(p'+pr+f«(i2w - py] sin^'-'K (6-83)
Time reversal invariance implies that /+ and /_ are relatively real, as are f±.
It might seem difficult to say anything, given the existence of two form
factors factors for each process. However, in the 517(2) or 517(3) symmetry
limits one can say a great deal. That is because the V* are the Noether
currents associated with the 517(3) generators Fl. In particular, they are
conserved, d^V* = 0, implying f-{q2) = 0. Furthermore, the pseudoscalar
fields transform as an 517(3) octet, i.e.,
[F\ tt»'] = - (Lidj)jk nk = ifijknk, i, j, k = 1 • • • 8, (6.84)
using (3.16). This implies that
FV) = jd3xVi{x)W) = ifijk\«k), (6.85)
since Fl\0) = 0 and since Fl\n^) should not contain states outside of the octet
(this argument will be made more precise in Section 6.2.4). From the general
form analogous to (6.81) or (6.83) this implies
<vrV)|V»kj(p)> = ifijkftfW+p)^-*>■*, (6.86)
with /(0) = 1. For pion (3 decay the q2 dependence can be ignored since
ml <q2 < (771^+ - 77V))2 < m2r- Therefore,
(*0K-iVfr+) = ^^-iV^+in2)
= -JE [ici23 - ^213] (p' +P)M = ~V2(p' +p)M,
(6.87)
i.e., the prediction of CVC is f+(q2) ~ /+(0) ~ —y/2. (See the comment on
the 7r+ phase convention in Problem 3.12.)
Pion Beta Decay
From (6.80) and (6.87), the matrix element for pion beta decay is
M = iGF cos0c (p + p% u^ (1 - 75) ve. (6.88)
A good first approximation to the rate is obtained by neglecting me and
working to leading order in
A = mn+ - m^o ~ 4.6 MeV . (6.89)

264 The Standard Model and Beyond
Since A is much smaller than the 7r+ mass, the 7r° is produced nearly at rest
and the 3-momentum pno = —pe — pu is very small. Therefore,
M ~ 2iGFcos6cm7rUui° (l - 75) ve. (6.90)
The leptonic part can be evaluated by writing out the Dirac spinors explicitly
most conveniently in the chiral basis using (2.191),
uulQ(l-1b)ve = 2ul(-)ve(+)
= -Ay/E^ (j>. (ft)+ X+ (ft) = -4^^ cos °-f, ' l)
where in the massless limit only the +(—) helicity e+(ve) is produced. We
have used the explicit helicity spinors from Table 2.1, and 0ei/ is the angle
between pe and pv. Therefore,
\M\2 = 32G2F cos2 6cmlEeEu (1 + cosOev), (6.92)
which could also have been obtained using standard trace techniques.
The decay rate is
We will evaluate the phase space integral directly, though the more powerful
techniques in Appendix D are more useful in the general case in which A and
me are not small. One can use the S function to do the pno integral, and
approximate
K° = y{Pu + Pe)2 + mlo ~ 77V), (6-94)
as was already assumed in (6.90). The pe and pu integrals are therefore
unconstrained except for an energy-conserving S(A — Eu — Ee) factor. The
cos 0ei/ term integrates to 0 in the overall rate. Therefore,
r =
1 [ d?pu d3pe sfA „ ^,,,,,2
h<Wt££«<*-*-*>""
'- f E2edEe E2vdEv S(A-E„- Ee)
G\ cos2 0,
(6.95)
G%cos2e 5 /'1ilJ2(1 x)2 G2Fcos2flcA5
Including the leading corrections (see (Sirlin, 1978) and Problem 6.7), one
predicts
G2Fcos2fl 5
30tt3
2
" 3 A 5ra'e ,. .
1 - -tt + radiative +
2mw A2
(6.96)
~ 0.399(1) s *, in excellent agreement with the experimental value 0.3980(24)
s_1 by the PIBETA collaboration at PSI (Pocanic et al., 2004).

The Weak Interactions
265
K£S Decays
In addition to pion beta decay, 7r+ —> 7r°e+i/e, the K^ decays K+ —> 7r°^+i^
and #l,s —► T^f^^tipt) are very important as tests of CVC and especially as
a measurement of sin#c. In the 517(3) limit the values of /+(£) at t = q2 = 0
are predicted from (6.86) to be
ff^° = -V2
/+ - ^ u - i. /+ -1-
The /+ ~*7r prediction follows from 517(2), while the others require 517(3).
However, the relative values for K+ and K° follow from isospin, and the
relation between K° and K° follows from the charge conjugation invariance
of the strong interactions. Since isospin is a good symmetry at the percent
level, one expects the 7r+ —> 7r° prediction to be quite reliable. However,
one might expect a large deviation of the 517(3) result for fK~*n, perhaps
as large as the Jk/I-k ~ 1-22 we found for the axial matrix elements from
Ka.1 ^12- In fact, the relation of the vector currents to the generators leads
to the Ademollo-Gatto theorem, which states that the deviations of the /+(0)
due to 517(2) and 517(3) breaking are actually of second order, and should
therefore be small. We will derive the Ademollo-Gatto theorem in the next
section, but first will consider the K^ decays in more detail.
The Ktz decay rate is (Leutwyler and Roos, 1984)
r = i&m^IKs/+(0)|2/^' (6-98)
where
1 ^K-m^f ( m2X / _2x2
(6.99)
A is the kinematic function defined in (2.38) and
^2 ^ 3m2e{m2K-ml)2 ^u,2
\(2t + mj)
m = hit?+°"tzk s:' /°w2, (6-io°)
with
/-<«>-&!■ «'>s«<>+;^fto- (mod
There are also radiative correction factors (Czarnecki et al., 2004) which we do
not display. A special case of (6.98) is derived in Problem 6.8. A complication
in the K^ decays is that t is not necessarily small, so the t dependence of
/+ must be included. Fortunately, this can be directly measured, e.g., by the

266
The Standard Model and Beyond
pion energy distribution (Trippe and Lin in Amsler et al., 2008). Even though
the symmetry breaking effects in /+(0) are second order, they must still be
estimated by chiral perturbation theory or lattice techniques and taken into
account (e.g., Blucher and Marciano in Amsler et al., 2008; Antonelli et al.
2008). Finally, the symmetry breaking may also induce an /_ form factor.
This leads to a contribution proportional to me in the amplitude, which is
non-negligible for K^. Despite these complications the Kez system has been
carefully studied, because it leads to the most precise determination of sin0c
(or its generalization to the CKM element Vus in the three-family case.)
6.2.4 Nonrenormalization of Charge and the Ademollo-Gatto
Theorem
Consider the question of whether electric charge is renormalized by the strong
interactions** for the example of the pion. As discussed in Section 2.12.4, the
matrix element of the electromagnetic current is
(K+(p')\J%(x)\n+(p)) = [<j/ +Prf?(q2) + (p'-pTfSiq2)] eiqx, (6.102)
where q = pf — p. The electric charge, in units of e, is /+ (0), so the
nonrenormalization requires /?(0) = 1 in the presence of the strong interactions.
To see how this comes about, recall that the U(1)q generator is
Q{t)= f d3xJ%{t,x), (6.103)
that current conservation d^ Jq = 0 implies -£*■ = 0, and that the
transformations of the charged pion fields, which can be viewed either as elementary or
as composite as in (5.103), are
[Q,^] ==rt±- (6-104)
All of these statements are obviously true for free fields, but continue to hold
to all orders in the strong interactions since they are U(1)q invariant, i.e.,
[<3, H] = 0. Taking the divergence of (6.102), current conservation
immediately implies q2fS{q2) = 0, and therefore f_{q2) = 0 since strong effects are
not expected to produce a delta function. The nonrenormalization can be
established by taking the matrix element of (6.104) between a single pion state
^The gauge coupling e is renormalized, and may depend on the scale, but that effect
is universal for all charged particles. See (D'Alfaro et al., 1973) for an extension of this
argument to include higher-order electromagnetic effects. Higher-order electromagnetic
effects also induce a small q2 dependence to /+.

The Weak Interactions 267
and the vacuum:
/d3k
(27r)32gfe(7r+(p)|9|7r+(fc))(7r+(fc)|7r"l°)
+ £ (7r+(p)|Q|n)(n|7r-|0) = +{*+(p)\*~\0). ^'^
We have assumed that U(1)q is not spontaneously broken, i.e., Q\0) = 0.
The last J^n term is a sum or integral over all single and multi-particle states
other than the 7r+. But,
<7r+(p)|Q|7r+(/c)) = jd3xe^~k>* [(p + kff%2) + (P-k)°fS(q2
= (2tt)3(53(p - fc)2£?p/<?(0), (6.106)
so the first term in (6.105) is /+(0)(7r+(p)|7r-|0). Similarly,
(7r+(p)|Q|n) = (27r)3(53(p-pn)(7r+|j0(0)|n)ei^-^)£. (6.107)
Since dQ/dt = 0, only (positively charged) states with Ep = En and p = pn,
and therefore m^ = mn, can have nonzero matrix elements. (This can also be
seen from (3.52).) There are no such states, so the last term in (6.105) vanishes
and /+(0) = 1. We note that the strong interactions will, however, introduce
a strong q2 dependence to /+, and also induce a magnetic form factor for
nucleon matrix elements. These effects can be taken from experiment.
Exactly the same reasoning can be used to establish the nonrenormalization
of the 517(2) and 517(3) generators as expressed in (6.85) and (6.86), by
taking the matrix element of (6.84) between <7rfc| and |0). The electric charge
generator is a special case.
Now let us turn on 517(3) breaking and establish that the deviations of the
Kes form factors /+"(0) from the predictions in (6.97) are of second order.
This is most easily seen from the commutator relation
[F4 + 2F5, F4 - iF5] =F3 + V3FS = J3 + |y. (6.108)
This is expected to hold even in the presence of 517(3) breaking mass and
interaction terms, as discussed in Section 3.2.2. Is and y are respectively
the unbroken third component of isospin and strong hypercharge, defined in

268
The Standard Model and Beyond
(3.89). Taking the matrix element,
3.
(n+(p')\I3 + p\*+(p)) = (2n)32EpS3(p'-p)
= <7r+(p')| [Fi+iF\Fi - iF5} \n+(p))
= J (^W^^^4 + iF5\R0(k^R°(k)\F4 ~ iF5\«+(P))
+ J2 (W+\F4 + iF5\n)(n\F4 - iF5\ir+)
- ^(7r+|F4 - iF5\n)(n\F4 + iF5\n+). (6.109)
n
The K° term can be evaluated using the analog of (6.83), but with f£ ~*7r+.
It is a Lorentz invariant, and is conveniently evaluated in the \p | —> oo frame,
to obtain
(/f ^*+ (0))2 (2v)32Ep63(p' - p). (6.110)
The remaining terms, involving the leakage out of the pseudoscalar octet,
are of second order in 517(3) breaking, as is apparent from (3.52). Since
/+" ~*n (0) —> — 1 in the symmetry limit, we obtain the final result
/^'+(0) = -l + O(6|), (6.111)
where e| is the 517(3) breaking parameter defined in (3.106). This is the
Ademollo-Gatto theorem (Ademollo and Gatto, 1964; Behrends and Sirlin,
1960; Fubini and Furlan, 1965; Marshak et al., 1969; D'Alfaro et al., 1973),
which implies that 517(3) is fairly reliable for /+"(0). Similarly, deviations
from the 517(2) prediction in (6.87) are expected to be second order in isospin
breaking, and therefore very small. Similar results hold for the vector (but
not axial vector) form factors in nucleon and hyperon (3 decay.
One can extend these considerations to chiral SU(3)l x SU(3)r. In the
chiral limit the pseudoscalars become massless and deviations from the
symmetry predictions are sometimes dominated by calculable non-analytic terms
generated by infrared singularities (Pagels, 1975). In the case of the /+(0),
the leading corrections to the chiral limit are of order ef/eo (Langacker and
Pagels, 1973), where eo represents the explicit chiral breaking, such as the
common quark mass term in (5.92). This is of first order in chiral 517(3) but
second order in ordinary 517(3), and leads to a 2% reduction in the predictions
for /*(0).
6.2.5 /3 Decay
(3 decay refers both to free neutron decay and to nuclear beta decay. These
and closely related processes are listed in Table 6.3. The amplitude for the

The Weak Interactions
269
TABLE 6.3
(3 decay and related processes. N and Z are the numbers of neutrons
and protons in a nucleus. All of these processes are driven by the cos#c
part of Hsi in (6.20).
neutron decay
nuclear (heavy nuclei, e.g., in reactor)
nuclear (light nuclei, e.g., in Sun)
electron capture (atomic electron)
/i capture (muonic atom)
inverse /? decay
n —» pe
(N,Z)
(N,Z)
e~p —>
|x"p->
ven —>
vep <-> <
^e
-(JV-
-l,Z + l)e"
^{N + l,Z-l)e*
ven
VyU
e~p
s+n
"^
~v<
free neutron decay corresponding to Figure 6.1 is
M = -i^= ue^ (1 - 75) vv (p\u^ (1 - 75) d\n) cos0c. (6.112)
v2
The Vector and Axial Form Factors
We have seen in Chapter 2 that the matrix element of the vector current is
restricted by Lorentz and translation invariance, and by the reflection invariance
of the strong interactions, to be of the form
(p\u^d(x)\n) =up
rF1(q2) + —quF2{q2) + q»F3(q2)
(6.113)
where q = pp — pn and we have used the Gordon decomposition in Problem
2.4 to eliminate possible up(p% +p£)un and uvG^v{pvv -\-pnv)un terms. Time
reversal invariance implies that Fi^,3 are relatively real (Appendix G). We
also have that the hadronic vector current is u^d = Vlfl + iV2fl, where V1'2
are the Noether currents related to the isospin generators. Equivalently, they
are in the same isomultiplet as the isovector part Vs of the electromagnetic
current, Jq = Vs + -75 V8 (CVC). The matrix elements are related by the
517(2) relation (a|V^|6) oc r*6/2, where a,6 = p or n and i = 1,2,3. The
isovector part of Jq can be separated from the isoscalar Vs by considering
the difference between the corresponding proton and neutron form factors,
FX(0) = 2FX» = Ff (0) - F?(0) = 1
F2(0) = 2F2V(0) = Kp-Kn = 1.79 + 1.91 ~ 3.70,
where Ff (0)(F1n(0)) = 1(0) is the proton (neutron) electric charge,
ftp ,n &re
their measured anomalous magnetic moments, and F^2 are the isovector form
factors defined in Section 2.12.4. The ^(0) effect, known as weak magnetism,
has been observed in the e± spectra in 12B —>12Ce~J/e and 12N —>12Ce+i/e,

270
The Standard Model and Beyond
in agreement with the CVC prediction. CVC also predicts that F3(q2) = q
since d^V* = 0. (This also follows from G-parity.) However, its contribution
would be proportional to me and small even if it were present. For most
processes q2 is small enough that its effects can be ignored.
The vector current form factors are therefore under control for neutron
decay. In fact, by the Ademollo-Gatto theorem, corrections are of second order
in isospin breaking and therefore tiny, i.e., (m^ — mu)2, Za2, and Za{rnd -
rau), where we have included a Z for the case of nuclear decay.
The axial matrix element is more difficult, because the axial generators of
SU(2)l x SU(2)r are spontaneously broken. The most general form consistent
with P, Lorentz, and translation invariance is
(p\u^5d{x)\n) = (p\A^{x) + iA2»(x)\n) (6.115)
uneiq*
• ^ti/ 5
7/V<?i(<?2) + -^r^92{q2) + q»l593(q2)
where we have again used the Gordon decomposition in Problem 2.4. Time
reversal requires that <7i,#2, and g$ are relatively real*, and gi is required
to vanish by G-parity (Appendix G). As mentioned following (5.113), the
induced pseudoscalar form factor gs is present because of the chiral symmetry
breaking, which generates a Goldstone boson pole. However, the effect of this
term is usually negligible because it generates a contribution proportional
to me. (It has been measured in /i capture.) #i(0) ~ 1.27 is measured in
the rates and various parity-violating asymmetries in neutron and nuclear /?
decay. Unlike the vector form factor, there is no nonrenormalization theorem
to prevent #i(0) from being changed from its pointlike value of 1, even in
the chiral limit. The proof of nonrenormalization for the vector generators
Fl relied on the fact that Fl\a), where a is a single particle state such as a
pion or nucleon, can only have a nonzero matrix element with another state
with the same invariant mass as a, and there were no such states with the
necessary quantum numbers. For the axial generators, however, the pions
become massless Goldstone bosons in the symmetry limit, so one can have
nonzero matrix elements, e.g., (7rN\Fl5\N).
To understand this better, we outline the Adler-Weisberger relation (Adler,
1965a; Weisberger, 1965). Recall that the commutators of the axial 517(2)
generators are
[Fi\F*]=ieijkF\ ij,k = 1,2,3. (6.116)
Just as we did for the Ademollo-Gatto theorem, take the matrix element of
(6.116) between single particle states. In this case, we will use protons and
*<72 would have to be imaginary for a Hermitian current such as uj^j5u, so it would have
to vanish if time reversal holds. The same is true for the #2 in the matrix element of the
electromagnetic current Jq (where it corresponds to an intrinsic electric dipole moment),
even if parity were violated.

The Weak Interactions 271
ignore isospin breaking. Then,
\ Yl 6S2Sl(p(P2,s2)\2F3\p(p1,s1)) = (2tt)3(53(p2 ~ Pi)2E1
Sl,S2
1 (6-117)
= g E *.a.i<P(P2.*2)| [F15 + iF25,F15 -*F25] |p(p1>Sl)>.
Sl,S2
Now, insert a complete set of intermediate states between the generators. The
single neutron contribution In is readily evaluated using (6.115):
^ = (2^)3^(^-ft)|gl(o)|2-R [7075(A +M„ho75(^i +Mn)}
= {2n)3S3{p2-p1)2E1
^(27r)3«53(p2-p1)2F1|5l(0)|2,
b,(0,P(l-f)
(6.118)
where the last form is valid in the \p\ —> oo frame. The remainder of the sum
is
W« = \ J2S°^ £ {(PlFl5 + iF25\m)(m\F15 - iF25\p)
m^n
-(p\F15 -iF25\m)(m\F15 +iF25\p)\.
(6.119)
The matrix elements in (6.119) can be related to matrix elements of the ^
field using (3.52) and (5.120), which in turn are proportional to the strong
interaction amplitudes for 7r±p —> m, to obtain the Adler-Weisberger formula
|51(0)|2 = 1+^/ -—-j [aw+pH-^-»], (6.120)
where an±p(w) is the total cross section for ir^p scattering (with a massless
pion) at CM energy w. Using the observed cross sections (and including
corrections for the off-shell 7r), (6.120) yields #i(0) ~ 1.24, to be compared with
the current experimental value ~ 1.27. This remarkable agreement was
influential in establishing current algebra, and as we have seen relies not only on
the Fermi theory but also on our understanding of chiral SU(2)l x SU(2)r
as an approximate spontaneously broken global symmetry of the strong
interactions.
Neutron and Nuclear Decay
To a good approximation, the hadronic matrix element for neutron decay is
therefore
<p| J^\n) = uvltl (gv - 9A15) un, (6.121)

272
The Standard Model and Beyond
where
9v = /i(O)cos0c ~ cos0c, gA = #i(O)cos0c. (6.122)
It is then straightforward to work out the neutron lifetime and various decay
distributions (Problem 6.10). The total lifetime is
_i _ r _ G|icos20c
7n — 1 n —
27T3
(\9v\2 + 3|<m|2) / PeEe (A - Eef dEe, (6.123)
«/me
where A = mn — mp ~ 1.29 GeV. The integral*, obtained by a calculation
similar to the one in (6.95) for pion beta decay, is most easily done numerically,
yielding 1.64rag for the observed value of A/me. From the observed lifetime
~ 885.7(8) s one can obtain gy+3g\. The ratio A = gA/gv ~ 1.2695(29) can
be extracted from various asymmetries. For example, for a polarized neutron,
the angular distribution of the e~ w.r.t. the neutron spin direction is
dT _ /A2-A
d cos 0e
ocl - 2/?e (^3^) cos0e, (6.124)
where cos8e = pe • sn and /?e = \pe\/Ee. From gy and gA one can
determine #1 (0) (actually <7i(0)//i(0)) and cos#c. Similarly, the polarization of the
produced electron is just
_Y{he = +\)-T{he = -\) _
which approaches —1 as expected for me —> 0.
For nuclear (3 decay the nuclear matrix element (N—l, Z+l\jft(x)\N, Z) or
(N +1, Z— 1| J^(x)\N, Z) must be determined, either from theory or by
relating to other measured matrix elements. Transitions involving Vfi are known
as Fermi transitions. Especially important are the superallowed transitions
between 0+ states in the same isomultiplet (Hardy and Towner, 2005, 2009).
These are pure Fermi transitions, with the relevant form factors determined
by the isospin generators. Hence, there is little theoretical uncertainty except
for second order isospin violation (the Ademollo-Gatto theorem). The super-
allowed transitions yield the cleanest determinations of cos 0C (i.e., the element
Vud of the quark mixing matrix in the full SM). The axial currents lead to
Gamow- Teller transitions, such as those with A J = ±1. Mixed transitions,
such as neutron decay, involve both. There has been an enormous amount
of experimental and theoretical work on nuclear and neutron /? decay. The
formalism has been developed for arbitrary admixtures of 5, P, T, V, and A
interactions (Jackson et al., 1957). Many different types of asymmetries,
polarizations, and correlations have been measured or searched for. These have
tThis is known as the Fermi integral. For nuclear (3 decay it includes a factor F(Z, Ee) that
corrects for the distortion of the e± wave function by the Coulomb field of the nucleus and
atomic electrons.

The Weak Interactions
273
established the basic V -A nature of the interactions; tested the C, P, T, and
G-parity properties; searched for perturbations on V — A from such types of
new physics as extended gauge groups involving V + A couplings, leptoquark
interactions, and mixing between the ordinary neutrinos or other fermions
with heavy exotic states; tested the radiative corrections, which require the
full standard model; extracted the form factors; and determined Vud f°r
universality tests. Detailed accounts may be found in (Commins and Bucksbaum,
1983; Renton, 1990) and in the articles by Deutsch and Quin and by Herczeg
in (Langacker, 1995). Recent reviews include (Severijns et al., 2006; Abele,
2008).
6.2.6 Hyper on Decays
There have been extensive studies of the semi-leptonic hyperon decays. These
include the AS = 0 decays n —> pe~&e, E^1 —> Ae±ve(Pe), E~ —> E°e~i/e, and
E~ —» H°e~i/e, as well as the AS = AQ = 1 decays A —> p£~Pe, £~ —>
nt'vu S" -> M~vt, E~ -> YPtTvt, and ~° -> Y>+tTvt, where £ = e or fi.
In the 517(3) limit the hyperons transform as an octet, as shown in (3.90)
and Figure 3.1. Since the hadronic weak currents also transform as octets,
one can use 517(3) symmetry to related the matrix elements. For an arbitrary
SU(S) octet of operators O-7, j = 1 • • • 8 one has that
(BklO'lBj) = ifijkOF + dijkOD, (6.126)
where the / and d coefficients are defined in Table 3.2, and Of,d are two
reduced matrix elements that are independent of 2,j, and k. This is the
analog of the Wigner-Eckart theorem for 517(2) and has already been used
in (3.101) in the derivation of the Gell-Mann-Okubo formula. The hadronic
vector currents are associated with the 517(3) generators, so only the fyk
term is non-vanishing, i.e.,
(Bk\VHp)\Bj) = ifijkUk^Ujhiq2), (6.127)
with q = Pk — Pj and /i(0) = 1, exactly analogous to (6.86). The Ademollo
-Gatto theorem applies to the hyperons as well as the pseudoscalars, so one
expects the predictions to be quite accurate, /i(0) = 1 + O(eg), where the
coefficient of the correction can depend on 2, j, k. One can also include weak
magnetism corrections to (6.127), which are also related by 517(3). The
predictions for the vector and axial form factors are summarized in Table 6.4.
The axial current matrix elements can have both / and d type contributions.
It is conventional to define the coefficients F and D by
(5*14(0)1^) = [ifijkF(q2) + dijkD(q2)] «i7/,75«i, (6-128)
where F = F(0) and D = D(0). The corrections are first order in 517(3)
breaking, so the predictions are less reliable than for the vector currents. F

274
The Standard Model and Beyond
TABLE 6.4
517(3) predictions for the values of the vector and axial form
factors at q2 = 0.
A5 = 0
A5 = l
Decay
n —»pe~ue
E± _> Ae±i/e(Pe)
E" -► S°e-Pe
S- - H°e-Pe
A —> p£~De
XT -»n£-P*
S" -► A£"P/
s- _►s°rp«
H° -> E+*-p<
Vector
I
0
v/2
1
-\/t
-1
i/i
i
V2
1
Axial
F+D "
yJlD
V2F
F-D
-y/l(F + §)
-(F-D)
v^-f)
y/2(F + D)
F + D
and D and the g2 dependence must be taken from experiment. One
combination is obtained from the <7i(0) from (3 decay in (6.115),
(p\uw5d\n) = (±(B6 + iB7)\Al + iAl\^(B4 + iB5)), (6.129)
yielding F + D = gi(0). There have been extensive studies of the decay rates,
the £ energy spectrum, correlation between the fte and the initial hyperon
polarization, the £ — v angular correlations, and the final baryon polarization,
both to test the Fermi theory and to obtain an accurate value for sin#c. The
data agree quite well with the theoretical expectations, and yield
F = 0.462(11), D = 0.808(6), Vus (~ sin0c) = 0.226(5) (6.130)
from an overall fit (Gaillard and Sauvage, 1984; Cabibbo et al., 2003; Mateu
and Pich, 2005).
6.2.7 Heavy Quark and Lepton Decays
The r± lepton, with mT ~ 1.777 GeV and lifetime ~ 2.9 x 10-13 s, was the
first member of the third family of quarks and leptons to be discovered. It is
similar to the electron and muon except for its large mass, mT/m^ ~ 17 and
mT/me ~ 3500. It can therefore undergo both leptonic decays, r~ —> vT£~9i,
£ = e or /i, and semi-hadronic (also referred to as hadronic) decays, r —> vTdu
or vTsu, with the quarks emerging as mesons such as n~ or 7r~7r°. The r
decays have been very useful in testing the Fermi theory (and SM) couplings,
extracting the QCD coupling at the r scale, and searching for new physics (see,
e.g., Pich, 1998b; Davier et al., 2006; Pich, 2008; Amsler et al., 2008).

The Weak Interactions
275
The leptonic decays are very similar to /i decay, with branching ratios ~
IS% each for the e and /i channels, consistent with the simple parton model
expectation that the branching ratios for the e, /i, and qq decays should be in
the ratio 1,1,3 (because of the 3 quark colors). They have been extensively
studied (Amsler et al., 2008), to establish the V - A nature of the r current
and to verify weak universality, i.e., that the strength of the interaction is the
same as the one for muon decay, as is expected in the SM extension of the
Fermi theory. For example, the Michel p parameter, analogous to the Michel
parameter for /i decay in (6.55), is measured to be 0.745(8), consistent with
the V — A value of 3/4. Similarly,
^p = 1.0012(53), ^=^ = 0.981(18), ^^ = 1.018(12),
(6.131)
where, e.g., GT_>e is the coefficient of the four-fermi interaction measured
for r~ —> vTe~ve. The values in (6.131) are in impressive agreement with
universality.
The semi-hadronic decays r~ —> h~vT include h = 7r~(~ 11%),
26%), and n~ +n7r°(n < 4), as well as modes involving 3 or 5 charged particles
and modes involving kaons (the latter are suppressed by sin2 6C and by phase
space). The r~ —> 7r~iyT decay rate can be predicted using fn extracted
from 7r —> //z^ (Eq. 6.66), leading to the effective coupling quoted in (6.131).
The semi-hadronic decays can also be used to determine the strong coupling
as(m2), the CKM matrix element VU8 ~ sin#c, and were historically useful
for setting a direct upper bound ~ 18 MeV on the vT mass in modes involving
3 or 5 charged pions (for which little phase space is available).
There have also been extensive searches for rare decays, such as r —> #y,
which are forbidden in the Fermi theory and standard model up to negligibly
small neutrino mass effects, but could be generated by new physics (Section
7.6.6).
In the four-quark model, the c decays via the hadronic weak current
j£ = 57|i(i - 75)c cos0c - d7M(l - 75)c sin0c + u terms (6.132)
in (6.23), i.e., to s and d with relative strengths cos#c and sin#c. In the 3-
family extension, the c decays are little affected because the mixing between
the third family and the first two is small. However, these small effects allow
b quark decays via
J? = t27M(l " 75)bVub + C7M(1 - ^)bVch + • • • , (6.133)
where VUb and Vcb are the relevant elements of the CKM matrix in (6.24).
Empirically \Vub\ < \Vcb\ <^ \VU8\ ~ sin0c. The small values for \VUb\ and
\Vcb\ lead to longer lifetimes for the 6-flavored hadrons than would be naively
expected.
The c quark pole mass, at ~ 1.5 - 1.8 GeV (Eq. 5.125), is marginally heavy
enough to attempt to use use simple parton model (SPM) ideas to describe

276
The Standard Model and Beyond
its decays, especially for the semi-leptonic modes. In the spectator model
one ignores the hadronic aspects of the initial charmed (or 6-flavored) hadron
and simply considers the decay of a free quark, treating the light quarks or
antiquarks as irrelevant spectators. For example, one would expect branching
ratios for c —> (s or d)e+ve ~ 20% by naive counting arguments, compared to
the observed 16% for D+ = cd decays. The D+ lifetime ~ 1.04 x 10-12 s is
also consistent with the naive free quark estimate of (mM/mc)5r///5 ~ 10~12
s, obtained from scaling the /i lifetime, provided one uses mc ~ 1.5 GeV and
neglects the final hadron masses. However, the spectator model is much less
successful for other charmed hadrons. This is easily seen for the D° = cu
meson, with a lifetime ~ 4.1 x 10~13 s. This is less than half of that for
D+, while they would be equal in the spectator model. The mass scale is
clearly too low for the SPM to be reliable, especially for the non-leptonic
modes, and there must be important contributions to the decay amplitudes in
addition to the simple c-quark decay diagram. It is convenient to classify the
contributions in terms of topological diagrams (Chau, 1983; Chau and Cheng,
1986), as shown in Figure 6.8.
Measurements of two-body decays D+ —> /i+^M, D+ = cs —> /i+^M, and
D+ —> t+vt have been made by CLEO, Belle, and other groups, allowing
extraction of the decay constants (analogous to fn and /#)• The results are
in reasonable agreement with the theoretical expectations, mainly based on
lattice calculations (see the Decay Constants review in Amsler et al., 2008).
More detailed discussion of c and 6 decay and also of the charmonium {cc)
and upsilonium (66) systems may be found in (e.g., Kwong et al., 1987; Bhat-
tacharya and Rosner, 2007; Artuso et al., 2008, 2009).
q q d, s q d, s q q q d, s q q q
FIGURE 6.8
Tree-level topological diagrams for D decays, written in terms of an
intermediate vector boson W. Gluons may be attached in all possible ways. The first
(spectator) diagram represents free quark decay, the second (internal
emission) diagram has a reduced color factor (analogous to (5.15)), the third is
the exchange diagram, and the last is the annihilation diagram. There are
additional one-loop (penguin) diagrams (Section 7.6).

The Weak Interactions
277
The SPM estimates for the decay of the b quark are expected to be much
more reliable because of its larger pole mass of 4.7 - 5 GeV. For example,
the 5+ = bu semi-leptonic branching ratio into l+vi + X of 10.99(28)% (this
is the average of e+ve and /i+^M) agrees with the naive SPM expectation of
1/9, based on possible decays into three lepton and two quark families. The
#o = Id -> l+vt + X branching ratio is similar (10.33(28)%), and the B+
and B° lifetimes agree to within 7%. One can use the semi-leptonic
lifetimes to determine Vcb, either using the inclusive decays (summed over all
charmed hadronic final states), with appropriate QCD corrections and quark
masses (Benson et al., 2003), or the complementary exclusive measurements
of specific decays such as5-> D£i>£ or5-> D*£i>e. The latter require
measurements of the shapes of form factors and HQET and lattice calculations of
their normalization (see the review of Vcb and Vub in Amsler et al., 2008). The
results, which are only marginally consistent, are \ycb\ = 0.0416(6) (inclusive)
and \Vcb\ = 0.0386(13) (exclusive), with a combined value
\Vcb\ = 0.0412(11), \Vub\ = 0.00395(35). (6.134)
This small value corresponds to the relatively long lifetime, tb+ ~ 1.6 x 10-12
s, comparable to T£>+ despite the much larger B mass, determined by the
displaced vertex for the decay ~ 0.5 mm from the production vertex. The
much rarer b —> u transitions can also be studied by the inclusive semi-leptonic
decays (exploiting the higher energy leptons that are not kinematically allowed
in b —> ciPe), or in the exclusive B —> -k£vi decays (with similar form factor
complications as in the charm modes), yielding the value for \Vub\ in (6.134).
The K° - K°, D° - £>°, and B° - B° systems exhibit or are predicted to
have interesting (second order weak) mixing and CP violation effects that will
be described along with other aspects of CP violation in Section 7.6.
The t quark, with pole mass ~ 171 GeV, is much heavier than the W±
mass of ~ 80 GeV (Chapter 7). It is therefore predicted in the SM to decay
directly to bW+ (or much more rarely into sW+ or dW+) with the W on
shell. The decay is considered in Problem 7.10 and reviewed in (Demina and
Thomson, 2008). Because it is so heavy, the t decays without hadronizing.
It couples strongly to the Higgs in the standard model and plays an
important role in many alternative models of spontaneous symmetry breaking. Its
production and decays are therefore expected to be an excellent probe of new
physics (Atwood et al., 2001; Bernreuther, 2008).
Lepton Energy Distributions
We conclude this section with a comment on energy distributions. In the
four-fermi V — A theory, neglecting the final fermion masses, strong interaction
effects, etc., the energy distribution of each final fermion i is either 2e?(3 —2e^)
or 12e^ (1 - 6i), where e* = 2Ei/M and M is the mass of the decaying particle.
The first (second) distribution holds for the e~ (De) in /i~ —> e~vilve, as is
seen in (6.50) and Problem 6.2. The difference is due to the sign of the last

278
The Standard Model and Beyond
term in (6.38), which in turn is associated with the relative V — A or V + A
character of the currents (see (2.171)). In particular, the relevant product of
currents for b —> ce~9e is
C7„(l -l5)b c-y*(l -75K- (6-135)
The 6 and e- fields are related the same as for fi decay in (6.32), so the e~
energy distribution will be of the 2e2(3 — 2e) type. For c —> sc+i/e, on the
other hand, the currents are
s7M(1-75)cPe7M(l-75)e"
= -s7M(l-75)ceV(l+75K.
^^n ,^..c (6-136)
where we have used the charge conjugation identity (2.297). The positron
current is therefore V + A. This will give the other sign in (6.38) and a
positron spectrum 12e2 (1 — e).
These considerations are important for probing possible extensions or
modifications of the V — A theory or the standard model. For example, if the
b —> c transition had been V + A, then the e~ spectrum type would have been
reversed.
6.3 Problems
6.1 Calculate the neutrino phase space coefficients A and B denned in (6.44)
for p decay.
6.2 (a) Suppose that the muon weak current were V + A rather than V — A,
so that the interaction for /i decay becomes
rather than (6.32). Calculate the electron energy distribution, and show that
it corresponds to p = 0, where p is the Michel parameter defined in (6.55).
(b) Calculate p for the (parity-conserving) vector interaction
H = \f2GFv^nilpT e~7Mi/e,
where the y/2 is inserted to obtain the same overall rate.
(c) Return to the Fermi V — A theory for unpolarized pT —> e~v^J)e. Give the
energy distributions for the v^ and for the ve. In each case the momenta of
the other two particles are integrated over. Give a simple helicity argument
why one of them vanishes at the endpoint, e = 1, but not the other. Hint:
the results can be written down without additional calculations.

The Weak Interactions
279
6.3 Verify the expressions (6.58), (6.60), and the kinematic limits on y for
elastic ve scattering.
6.4 Discuss the difference between the QED corrections to /i and (3 decay in
the Fermi theory. Hint: Use an effective
~£0 = ^§ P7„ (1 - 75) n e-7" (1 - 75) »e
interaction for (3 decay, and write the four-fermi interactions in each case in
charge retention form.
6.5 The formula (6.53) for polarized /i decay implies that the angular
distribution is 1 — cos# for a massless e~ of the maximum allowed energy /iM/2,
where 6 is the angle between the e~ direction and the muon spin. The
analogous formula for /i+ decay is 1 + cos 6. Interpret these results in terms of
angular momentum conservation.
6.6 Use the spin projection in (2.174) to verify the cos# = pe -sM dependence
of the expression (6.53) for polarized /i decay. Hint: show how the calculation
leading to (6.49) is modified by the spin projection.
6.7 Use the formalism developed in Appendix D to verify the 3A/2m7r
correction in (6.96) to the pion beta decay rate. Neglect rae.
6.8 Verify the K^ decay rate in (6.98) in the special case m^ ~ me ~ 0,
/+(t)~/+(0),and/_(t)~0.
6.9 Use reflection invariance to show that (^(P2)|^j7rfc(pi))> the matrix
element of the axial current between single pion states, is zero.
6.10 Calculate (a) the neutron lifetime, (b) the e~ polarization, and (c) the
electron asymmetry with respect to the neutron spin direction in terms of
gv and qa defined in (6.121). Work in the approximation that the proton
momentum is negligible compared to its mass. Hint: write out the amplitude
explicitly using the Pauli-Dirac representation for the spinors and 7 matrices.
Use the helicity basis for the leptons. (d) Repeat the lifetime calculation,
this time using trace techniques, summing and averaging over all spins, and
calculate the ev angular correlation coefficient aeiy. The latter is defined by
the angular distribution 1 + aei//3epe • pv, where pv can be measured using
Vv = -(pp+Pe)'
6.11 Calculate the predicted rate and the angular distribution of the n~ with
respect to the r spin direction for r~ —> n~i/T in the Fermi theory. Show,
using the measured lifetime tt = 2.9x10-13s, that it leads to a branching
ratio in agreement with the observed value of 11%. Neglect m^.

280
The Standard Model and Beyond
6.12 The J/?/;(3100) is a narrow charmonium (cc) resonance with orbital
angular momentum L = 0 and spin-1, which was predicted (Appelquist and
Politzer, 1975) before the observation of charm. Assuming that the J/ip can
be considered a non-relativistic bound state, show that the lifetime for J/tp -+
e+e~ by one photon is
where we have taken raj/^ ~ 2rac and neglected the electron mass. Nc = 3 is
the number of colors, qc = 2/3 is the electric charge of the c quark, and ^(0)
is the wave function at the origin.

7
The Standard Electroweak Theory
In this chapter we describe the standard 517(3) x 517(2) x 17(1) model in detail,
including the structure of the Lagrangian density, the spontaneous symmetry
breaking mechanism, and the rewriting of the Lagrangian after SSB. The
electroweak gauge interactions, the properties of the W and Z, the Higgs,
CP violation, and neutrino mass are described. More detailed treatments
may be found in the books and reviews listed in the bibliography.
7.1 The Standard Model Lagrangian
The standard model is based on the gauge group G = 517(3) x 517(2) x
1/(1). The 517(3) (QCD) factor has gauge coupling gs and 8 gauge bosons
(gluons) Gl,i = 1---8. It is non-chiral, and acts on the color indices of
the L and i?-chiral quarks qra, where a = 1,2,3 refers to color and r to
flavor. QCD was described in detail in Chapter 5. However, the bare (current)
masses introduced in (5.1) are not allowed in the context of the full standard
model, but must be generated by the Higgs mechanism. QCD itself is not
spontaneously broken, and the gluons remain massless.
In contrast to QCD, the electroweak 517(2) x 17(1) factor is chiral. The
51/(2) group has gauge coupling g, gauge bosons Wl,i = 1,2,3, and acts
only on flavor indices of the L-chiral fermions. It leads to the charged current
interactions of the Fermi theory, and also includes a neutral boson W°
associated with a fermion phase symmetry. The abelian 17(1) factor has gauge
coupling g' and gauge boson B. It is also chiral, acting on both L and R
fermions but with different charges. After spontaneous symmetry breaking
(SSB), 517(2) x 17(1) is broken to a single unbroken £/(1)q, incorporating
QED with the photon a linear combination of W° and B. The orthogonal
combination (Z), as well as the W±, acquire masses. G is sometimes
written as SU(S)C x SU(2)L x U(l)y- The subscripts have no group-theoretic
significance, but refer to the physical application, i.e., c refers to color, L to
the left-chiral nature of the 517(2) coupling, and Y to the weak hypercharge
quantum number.
281

282
The Standard Model and Beyond
The standard model Langrangian density is
+ £/+£^ + £yufc, (7.1)
which refer respectively to the gauge, fermion, Higgs, and Yukawa sectors of
the theory. There are in addition ghost and gauge-fixing terms which enter
into the quantization. The gauge terms are
Cgauge = ~G^G^ ~ -W^W*"* ~ -B^B*™, (7.2)
where the field strength tensors for 517(3), 517(2), and U{\) are respectively
G\v = d^Gl - dvG^ - gsfijkGffi, i, j, k = 1 • • • 8
W;„ = W ~dvWi- gtiikWiW$, i, j, k = 1 • • • 3 (7.3)
B^v = d^Bv — dvB^.
These include the gauge boson kinetic energy terms as well as the three and
four-point self-interactions for the Gl and Wl, as shown for an arbitrary non-
abelian theory in Figure 4.1. The abelian 17(1) gauge boson has no self-
interactions. We ignore the possibility of FF terms such as the 0qcd one in
(5.1).
The fermion part of the standard model involves F = 3 families of quarks
and leptons. Each family consists of
I — doublets :
R — singlets :
a0 -W
7/° d° p-°
amR'> ainRi tmR->
°mL
vmR->
m I
e-°)
m / i
(7.4)
in which the L-chiral fields are 517(2) doublets and the R fields are singlets,
leading to parity breaking in 517(2). The superscripts ° refer to the fact that
these fields are weak eigenstates, i.e., they have definite gauge transformation
properties, with the elements of each doublet transforming into each other
under 517(2), and m = 1,2,3 labels the family. After spontaneous symmetry
breaking, these will become mixtures of mass eigenstate fields. The u° and
d° are quarks which will (after SSB) be identified as having electric charges
2/3 and -1/3, respectively. There are 2F = 6 quark flavors (u° and d° for
each family) all together. Each carries a color index u^nL Ra or a^rnL Ra not
displayed in (7.4), so there are really 3 quark doublets per family. The 517(2)
and 517(3) commute, so the QCD interactions do not change the flavor, and
vice versa. v° and e~° are the leptons. They are color singlets and will have
electric charges 0 and —1. We have tentatively included 5l7(2)-singlet right-
handed neutrinos u^nR in (7.4), because they are required in many models for
neutrino mass. However, they are not necessary for the consistency of the
theory or for some models of neutrino mass, and it is not certain whether

The Standard Electroweak Theory
283
they exist or are part of the low-energy theory. All of these fields except the
i/° R carry weak hypercharge Y, which is denned by
Y = Q-Tl (7.5)
where T3 is the third generator of SU(2)L and Q is the electric charge*.
Weak hypercharge should not be confused with the strong hypercharge y
denned in (3.89). U(1)Y commutes with SU(3)C and SU(2)L, so it has the
same value for all members of 517(3) x 517(2) multiplets. The Y eigenvalue
is y = q - t\ = i, §, and -| for g^L, w^, and dj^, respectively. For the
leptons, y = -±,0, and -1 for ^L,i/^, and e^°R.
The representations can be summarized by the symbol {7^3,712,3/}^ for
fermion ip, where 713 and rii are the 517(3) and 517(2) representations and
y is its hypercharge. Thus, the 16 fields (15 without v^r) of each family
transform as
+ 1/2+1/6 = 2/3
-1/2 +1/6 = -1/3
+ 1/2- 1/2 = 0
-1/2- 1/2 = -1
2/3 (7.6)
-1/3
-1
0,
where q = t\ + y is the electric charge eigenvalue.
The fermion representation is highly reducible, since each weak eigenstate
family transforms only into itself. The existence of three families is empirical.
£ gauge is actually invariant under a global U (3)6 family symmetry, in which
the 3 families of q^L, ^L, u^, d^R, e~^, and i/^R transform into each other.
However, there is no evidence that these symmetries are gauged. In any case
most of the generators are broken by the Yukawa interactions. (The unbroken
vector generators are baryon and lepton number.)
The SM is anomaly free (Section 4.5) for the assumed fermion content.
There are no 5I7(3)3 anomalies because the quark assignment is non-chiral,
and no SU(2)3 anomalies because the representations are real (Section 3.1.3).
The 5l7(2)2y and Y3 anomalies cancel between the quarks and leptons in
<3>2'6>^
t1'2'-^
q<L
qd°mL
^L
^L
f3'1'^*
q-°mR
<1'1'-1>«™0H
{M,0}„o
qd°m*
q*-Jn
q-°mR
*Some authors define Q - Tf = V/2.

284
The Standard Model and Beyond
each family, by what appears to be an accident. The SU(3)2Y and Y
anomalies cancel between the L and R fields, ultimately because the hypercharge
assignments are made in such a way that U(1)q will be non-chiral.
The SU(2)l and U(l)y representations are chiral, so no fermion mass terms
are allowed*. Cf therefore consists entirely of gauge-covariant kinetic energy
terms,
Cf=J2^0mLm0mL + e0mLiPC
mL
m=l
(7.7)
+ i'Rfi + d0mRiPd°mR + ^mRi^e°mR+P^RiPu^R),
where we have allowed for an arbitrary number F of fermion families. The
first term in (7.7) is
QmLL
q,/3=1
[(V + fr • #M + %IB^j Sa0 + ^-Xa0 • (?„/]
UmLl3
(7.8)
where the / is the 2x2 517(2) identity matrix. It is clear from (7.8) that the
5I7(3)C and 517(2)l x U(1)y groups commute. We will simplify the notation
by suppressing the color indices on the quark fields. Then, the fermion gauge
covariant derivatives become
D.ql
mL '
D„e°mL
i9' Pt\n°
*$'
BH ) LrnLi
D,u°mR = (d, + ^BM) u°mR
xmR
D„e°mR = (dli-ig'B^e°mR
D^°mR = dymR, (7.9)
where it is understood that there are also gluon couplings for the c^L^mR^
and dmR. The fermion gauge interactions can be read off from (7.7).
The Higgs part of C is
C4, = {D»<l>)iDtA<l>-V{<l>), (7.10)
where <j> = I ,0 ) ls a complex Higgs scalar, transforming as {1,2, ^}^. Its
adjoint {1,2*,—|}^t is $ = I ,0f )• The SauSe covariant derivative is
W = (^ + i{r-Wli + ^-B^ J>. (7.11)
+The possibility of Majorana mass terms for the i/^R will be considered in Section 7.7.

The Standard Electroweak Theory
285
The square of the covariant derivative leads to three and four-point
interactions between the gauge and Higgs fields. V(</>) is the Higgs potential. The
combination of 517(2) x 17(1) invariance and renormalizability restricts V to
the form
F(0) = +/iV0 + A(0+0)2. (7.12)
For /i2 < 0 there will be spontaneous symmetry breaking, and the VEV of
(O|0°|O) will generate the W and Z masses. The A term describes a quartic
self-interaction A(</>~</>+ +</>°t</>0)2 between the Higgs fields. Vacuum stability
requires A > 0.
The last term in (7.1) represents the Yukawa couplings between the Higgs
doublet and the fermions, which are needed to generate fermion masses by the
spontaneous breaking of the chiral gauge symmetries. For F fermion families
it takes the form
F
CYuk = - ^ [r^nQ0rnL^UnR + rtnQ°rnL4>d^R
m,n=l (7.13)
+ TemnPmncl>e0nR +r»mne0mJvZR\+h.c,
where
4>=(fo), 4> = iTa^ = (^_) (7.14)
-r)
are respectively the Higgs doublet and a conjugate form which will be
discussed below. Tu,rd,re, and Yv are completely arbitrary F x F matrices,
which ultimately determine the fermion masses and mixings. They do not
have to be Hermitian, symmetric, diagonal, or real* (the hermiticity of Cyuk
is ensured by the addition of the h.c. to the displayed terms). They are the
most arbitrary aspect of the SM, introduce most of the free parameters, and
break most of the 17(F)6 family symmetries of the rest of C. The Tu term
would of course be absent if there are no i^rt- ^or example, the ran term of
Td yields
^dmnQ°mLH°nR = ^rnn Kl4>+d°nR + d°mL4>°(%R] (7.15)
and its Hermitian conjugate
rild°nR^q°mL = Ttn [d°nR4>-u0mL + d0nR^d°mL} , (7.16)
which lead to the interaction vertices in Figure 7.1. Electric charge is
conserved at the vertices (guaranteed since Q is embedded in G), while chirality
is flipped, which is characteristic of a Yukawa vertex. The family is changed
*One can even generalize to non-square matrices of dimension Fl x Fr, where Fl # Fr
are the numbers of L and R chiral fermions of a given type ip. This would guarantee the
existence of one or more massless 2-component fermions tpL or ipR unless there are also
Majorana mass terms. Neither option is experimentally viable except for the neutrinos.

286
The Standard Model and Beyond
dmL
</>+- —< -iTdmn </>0-—< -<
dnH dnR
dnR dnR
+-—+.--/ _«*, ^..._./ _ir^
tt,
j0
'ml amL
FIGURE 7.1
Yukawa couplings associated with r^ing^lL</)d^ + /i.e.
for ra ^ n. After SSB, the VEV (0|</>°|0) will generate effective mass terms
for the d° and other fermions.
It is straightforward to write the Yukawa couplings involving Td,e in (7.13).
However, the Tu and Tv terms require the existence of a complex scalar Higgs
( $°\
doublet 3> = I _ J transforming as {1,2,— ^}<j> so that one can write
invariant terms qmL®unR + h.c. and ^l^v^r + ^-c- Neither <j> nor </>* have the
needed quantum numbers. However, in the SM one can avoid the need to
introduce a second Higgs doublet 3> by utilizing the "tilde trick," namely that
4> defined in (7.14) indeed transforms as {1,2, —\}i- That is because the 2*
representation of 517(2) is equivalent to the 2, as shown in (3.17). Thus, <j>
can play the role of 3>, with the identification </>°t = 3>° and —(j>~ = 3>~,
rumnq°mJu°nR = rumn [u°mL^u°nR - d°mLru°nR} . (7.17)
The equivalence of the fundamental and its conjugate does not generalize
to higher unitary groups. Furthermore, in supersymmetric extensions of the
standard model the supersymmetry forbids the use of a single Higgs doublet in
both ways in £, and one must add a second Higgs doublet. Similar statements
apply to many theories with an additional U(l)' gauge factor, i.e., a heavy
Z' boson, or to the 50(10) grand unified theory. One can get by with a

The Standard Electroweak Theory
287
single Higgs multiplet in the extensions of the SM to the non-supersymmetric
versions of SU{2)L x SU{2)R x 17(1) and 517(5).
7.2 Spontaneous Symmetry Breaking
The SM as written is not realistic because bare mass terms are not allowed for
the electroweak gauge bosons or for the fermions. However, as described in
Sections 3.3 and 4.3 effective masses may be generated by spontaneous
symmetry breaking, and in fact the Higgs doublet <j> was introduced for that
purpose. If its neutral component </>° acquires a nonzero VEV the SU(2)l xU(1)y
electroweak gauge symmetry will be broken to electromagnetism, I7(1)q,
generating masses for the chiral fermions. By the Higgs mechanism the Goldstone
bosons will be absorbed to become the longitudinal components of the massive
W± and Z.
7.2.1 The Higgs Mechanism
Similar to the case of a single complex scalar in Section 3.3.3 it is convenient
to rewrite 0 in a Hermitian basis as
*=W = l£(*s+*4)J' (7-18)
where fa = <j>\ represents four Hermitian fields. In this new basis the Higgs
potential becomes
V{<t>) is clearly 0(4) ~ 517(2) x 517(2) invariant. This is an example of an
accidental symmetry, the most general potential consistent with the 517(2) x 17(1)
gauge invariance and renormalizability exhibits a higher symmetry. The extra
(global) generators are explicitly broken by the Yukawa and gauge
interactions.
Without loss of generality we can choose the axis in this four-dimensional
space so that (0|<fc|0) =0, i = 1,2,4 and (0|</>3|0) = v > 0. (Other directions
can be transformed into this form by an 517(2) x 17(1) rotation.) Thus,
(7.20)

288
The Standard Model and Beyond
which must be minimized with respect to v, analogous to the case of a single
complex scalar in (3.145) or a Hermitian scalar in (3.132). Two important
cases are illustrated by the dashed and solid curves in Figure 3.4. For /i2 > 0
the minimum occurs at v = 0 and 517(2) x 17(1) is unbroken at the minimum.
On the other hand, for /i2 < 0 the v = 0 symmetric point is unstable, and the
minimum occurs for v ^ 0, breaking the 517(2) x 17(1) symmetry. The point
is found by requiring
V'(is) = v(ii2 + \is2) = 0, (7.21)
which has the solution
(7.22)
at the minimum. The dividing point /i2 = 0 cannot be treated classically. It
is necessary to consider the one-loop corrections to the effective potential, in
which case it is found that the symmetry is again spontaneously broken
(Coleman and Weinberg, 1973).
We are interested in the case /i2 < 0, for which the Higgs doublet is replaced,
in first approximation, by its classical value v in (7.20). The generators
corresponding to L1, L2, and L3 — Y are spontaneously broken,
^ = ^7l(°)^0'* = 1'2'3' Yv=I2T2^)^ (7-23)
However, the vacuum carries no electric charge (Qv = (L3 + Y)v = 0), so the
U(1)q of electromagnetism is not broken^, and SU(2)l x U(1)y —> U(1)q. !
We therefore expect that the photon A, associated with the unbroken
generator Q = L3 + V, as well as the eight gluons, to remain massless, while
W± = (W1 T iW2)/V2 and Z, associated with T3 - Y, to become
massive. To see this, one must quantize around the classical vacuum, i.e., write
4> = v + <//, where <\J are quantum fields with zero vacuum expectation value.
To display the physical particle content it is useful to rewrite the four
Hermitian components of <// in terms of a new set of variables using the Kibble
transformation, as discussed for a general gauge theory in Section 4.3.
*=^'<E!-{'i")(,+0«)' f"4'
where the Ln are the three broken generators L1, L2, and L3 — Y, and H is a
Hermitian scalar field, the physical Higgs boson.
If we had been dealing with a spontaneously broken global symmetry the
three Hermitian fields £l would be the massless pseudoscalar Goldstone bosons.
§It is automatic that at/(l) remains unbroken in the minimal theory with a single Higgs
doublet. However, in extended versions of the SM, e.g., with two or more Higgs doublets,
all four generators might be broken, depending on the scalar potential. See Problems (7.3)
and (7.4).
-

The Standard Electroweak Theory
289
These would have no potential and would only appear as derivatives. In a
gauge theory they disappear from the physical spectrum, as we saw in Section
4.3. It is useful to quantize in the unitary gauge,
*-<->™"*-ji(.i*y *-w+o*y c-
25)
along with the corresponding transformations on the other fields. The unitary
gauge is simplest for displaying the particle content of the theory, because the
Goldstone bosons disappear and only the physical degrees of freedom remain.
As described in Section 4.4 it is better to use other gauges for calculating
higher order contributions to physical processes because the unitary gauge is
highly singular, and delicate cancellations between diagrams are required.
7.2.2 The Lagrangian in Unitary Gauge after SSB
Let us rewrite the Lagrangian in (7.1) after spontaneous symmetry breaking
in the unitary gauge.
The Gauge and Higgs Sectors
The Higgs covariant kinetic energy term takes the simple form
|2
(DM0)tD^=i(Oi/)
2 " + 2 M
° ) + H terms. (7.26)
We will return to the kinetic energy and gauge interaction terms of the physical
H field later, but for now we concentrate on the part depending only on v.
(7.26) can be rewritten using
where
to obtain
TiWi = T3^3 + y/2T+W+
y/2 '
+ V2t~W-,
T1 ± iT2
2 '
g-^w^w- + \(92 + 9'2)^-
'-JBp+gW*'
(7.27)
(7.28)
2
(7.29)
M2
where W± are the complex charged gauge bosons which will mediate the
charged current interactions, and
-a'B + aW3
Z= y v^JL = -smewB + cos6wW3 (7.30)

290
The Standard Model and Beyond
is a massive Hermitian vector boson which will mediate the new neutral
current interaction predicted by 517(2) x 17(1). In the second form, 0w is the
weak angle, denned by
a' a' Q
tan0w = — => sm6w = —, cos0w = —, (7.31)
9 9z 9Z
where^
9z = \/92+9'2- (7.32)
The combination of B and W3 orthogonal to Z is the photon (7), with field
A = cos 6WB + sin6wW3, (7.33)
which remains massless. The (tree-level) masses are predicted to be
Mw = 9i, Mz = if = ^-, MA = 0, (7.34)
2 2 cosffw
implying the relation
M,
sin^ = l-^. (7.35)
One can think of the generation of masses as due to the fact that the W and Z
interact constantly with the condensate of scalar fields and therefore acquire
masses, as in Figure 7.2, in analogy with a photon propagating through a
plasma. Each Goldstone boson has disappeared from the theory but has re-
emerged as the longitudinal mode of a massive vector. The number of fieki
degrees of freedom is unchanged. Before SSB, there are 4 massless electroweak
gauge bosons, each with 2 helicities, and 4 Hermitian scalars, fa, for a total
of 12. After SSB, there are 3 massive bosons, each with 3 helicities, the
photon with 2, and one Hermitian scalar H, again totalling 12. One sees from
(7.34) that in the limit g' —> 0 one would have Mw = Mz> That is because the
global 0(4) symmetry of (7.19) is broken to 0(3) ~ 517(2) by SSB. This global
custodial symmetry is respected by the 517(2) gauge interactions in (7.26) for
gf = 0, so that Mw± = Mw3 = Mz- On a deeper level, the custodial 50(3)
ensures that the coefficient v2 is the same for the W± and Z mass terms
in (7.29), even for gf ^ 0, implying Mw = MzcosOw Since this relation
is well satisfied experimentally, alternative models of spontaneous symmetry
breaking often involve a custodial 517(2) global symmetry to maintain it.
It will be seen below that GF/y/2 ~ g2/SM^, where GF = 1.166367(5) x
10~5 GeV~ is the Fermi constant determined by the muon lifetime. The
weak scale v is therefore
v = 2Mw/g ~ (V2GF)~1/2 ~ 246 GeV. (7.36)
* There is a confusing variety of notations for g, g', \l\g', and gz in the literature (even
in the author's own papers). Similarly, there is no uniformity as to whether symbols such
as v represent (<j>°) or y/2{(f)0).

The Standard Electroweak Theory
291
W
,2^^
w
v
en
FIGURE 7.2
Effective masses for the W and electron generated by the VEV v of the neutral
Higgs field.
Similarly, we will see that g = e/sin#w, where e is the electric charge of the
positron. Hence, to lowest order
Mw = Mz cos 6w
(ttq/v^Gf)1/2
sin 6w
(7.37)
where a ~ 1/137.036 is the fine structure constant. We will see that sin2 6w
can be measured from the predicted neutral current scattering to have a value
~ 0.23, so one expects Mw ~ 78 GeV, and Mz ~ 89 GeV. (These predictions
are increased by ~ 2 GeV by loop corrections, including the running of a.)
The W and Z were discovered at CERN by the UAl (Arnison et al., 1986)
and UA2 (Ansari et al., 1987) collaborations in 1983, in the reactions pp —>
W+X —> £v+X andpp —> Z+X —> £+£~+X, where X represents unobserved
hadrons. Subsequent measurements of their masses and other properties have
been in excellent agreement with the standard model expectations, including
the higher-order corrections (Amsler et al., 2008). The current values are
Mw = 80.398 ± 0.025 GeV, Mz = 91.1876 ± 0.0021 GeV.
The full Higgs part of C is
C4f = (D^DIA4>-V{<l>)
(7.38)
-M\Z»Za [ 1 + —
(7.39)
+ \ (d,Hf - V{4>),
where in unitary gauge the Higgs potential (7.12) becomes
V{fl>) -> -^ - M2#2 + Ai/ff3 + \H\
(7.40)

292
The Standard Model and Beyond
The second line in (7.39) also describes the ZZH2, W+W~H2 and the
induced ZZH and W+W~H interactions, as shown in Table 7.1 and Figure 7.3.
The last line includes the canonical kinetic energy term and potential for the
H. The first term in the Higgs potential V is a constant, (0| V(i/)|0) = -//4/4A.
It reflects the fact that V was defined so that V(0) = 0, and therefore V < 0 at
the minimum. Such a constant term is irrelevant to physics in the absence of
gravity, but will be seen in Section 8.1 to be one of the most serious problems
of the SM when gravity is incorporated because it acts like a cosmological
constant much larger (and of opposite sign) than is allowed by observations.
TABLE 7.1
Feynman rules (~ iC) for the gauge and Higgs interactions after SSB,
taking combinatoric factors into account. The momenta and quantum
numbers flow into the vertex. Note the dependence on M/v or M2/v.
W+W-H: \ig^92u = 2ig^ W+W~H2: \ig^g2 = 2ig^
lA„ „2,,_o„„ Mi 7 7 TJ2. 1;„ „2 _ n- Ml
Z^Z^H: \igtil,gzv = 'ligiiv-f- Z^ZVE2: \ig^g2z = ^9^-^f
H3: _6»Ai/ = -3i^ H4: -6iA =-3i^J
. TUf
Hff: -ihf = -i-
W+(p)-yv(q)W-{r): ieC^(p,q,r)
W+(p)Zu(q)W-(r): ig cos 9wC^(p,q,r)
W+W+W-W-: ig2Qiivpa
W^Zv^aW~: -ieg cos OwQfipva
W+ZuZaW-: -ig2 cos2 dwQwv<J
W+WvW- : -ie2Q,ilpVa
<W(p, q, r) = g^iq - p)„ + g^(p - r)„ + gva{r - g)M
^lp,vpo = ^QiLvQpcr ~ Qp,pQvo ~ Qp,aQvp
The second term in V represents a (tree-level) mass
MH = v/-2//2 = V2\v (7.41)
for the Higgs boson. The weak scale is given in (7.36), but the quart ic Higgs
coupling A is unknown. A priori, A could be anywhere in the range 0 < A <
oo, so there is no prediction for M#. There is an experimental lower limit
MH > 114.4 GeV from LEP (Barate et al., 2003). Otherwise, the decay
Z* —> ZH would have been observed. Theoretical upper (from unitarity
and perturbativity) and lower (from vacuum stability) limits on Mh will be
described in Section 7.5. The last two terms in V are, respectively, the induced
cubic and the quart ic self-interactions of the Higgs.

The Standard Electroweak Theory 293
H
H
H \
Aftr fc r-r
/
/
H '
w^ i
w+ <
ZV> 1
Zu c
H ,
H '
V «^
^ \»
\ /"
X """^
/ \»
/ w
x H
H
FIGURE 7.3
Higgs interaction vertices in the standard model.
The 517(2) gauge kinetic energy terms in (7.2) lead to 3 and 4-point gauge
self-interactions for the W's,
cW3 = - ig{dpwi)w+w- [<r»<r - <r<r»\
- i9(dpw+)w^w- [<r<r - <r<r\ (7.42)
- ig(dpW;)W*W+ [gTgT - 9^9"°],
and
CW4 = 9- [W+W+W-W-Q^° - 2WJ;W?W*W-QW} , (7.43)
where Q^upa is denned in Table 7.1. These carry over to the W, Z, and 7
self-interactions provided we replace W3 by cosOwZ + sinOwA using (7.30)
and (7.33) (the B has no self-interactions). The resulting vertices follow from
the matrix element of iC after including identical particle factors and using
g = e/sinOw They are listed in Table 7.1 and shown in Figure 7.4. (They
also follow from Problem 4.7.)
The Yukawa Sector
The fermions acquire masses by the SSB, as illustrated in Figure 7.2. To be
explicit the unitary gauge expressions for <j> and </> in (7.25) can be inserted in

294
The Standard Model and Beyond
1
wp,
W2
w?t
w?t
Wn
Wn
wn
~r<r
w„
ieC^l/(7(p,q,r)
e
tan^vy
K„»o C»v<j{p,q,r)
1 • 2 /) Qv>vpo
sin Vw
1, n ^z>iipv<y
tan Vw
tan2 0w
Qu
2 /) ^VPva
W?t
~rv
W~n
~r<r
16 ^fipva
FIGURE 7.4
The three and four point-self-interactions of gauge bosons in the standard
electroweak model. The momenta and charges flow into the vertices.

The Standard Electroweak Theory
295
(7.13). For F families this yields
p>
-Cyuk = V u^L^mn ( —■=- ) uQnR + (d, e, i/) terms + /i.e.
m^! V V2 / (7.44)
= u°L {Mu + huH) u°R + (d, e, v) terms + /i.e.,
where in the second form u°L = (u^Lu2L • • • u°FL) is an F-component column
vector, with a similar definition for u°R. Mu is an F x F fermion mass matrix,
M^n = Tumn^=, (7.45)
induced by spontaneous symmetry breaking, and
Mu aMu
hu = — = |£=- (7.46)
i/ 2Mvy v y
is the Yukawa coupling matrix. We have already emphasized that Tu, and
therefore Mu and hu, need not be diagonal, Hermitian, or symmetric. To
identify the physical particle content it is necessary to diagonalize M by
separate unitary transformations Al and Ar on the left- and right-handed fermion
fields. (In the special case that M is Hermitian one can take Al = Ar, while
Al = A*R for a symmetric M.) Then (for F = 3),
(mu 0 C
AU^MUAUR = Ml = \ 0 mc 0 (7.47)
is a diagonal matrix with real non-negative eigenvalues equal to the physical
masses of the charge | quarks. Similarly, we denote the eigenvalues of the
Yukawa matrix hu by hu, hc, and ht. One also diagonalizes the down quark,
charged lepton, and neutrino mass matrices by
Ad^MdAdR = Mi, Aej*MeAeR = Mf>, AV^MVAVR = MVD. (7.48)
In terms of these unitary matrices we can define mass eigenstate fields Ul =
^lu°l = (ul cl ^l)T, with analogous definitions for uR = ARu°R, dL,R =
A<L,Rdi,R, eL,R = Aej*Re°LR, and vl,r = ^Rv\R, so that, e.g.,
u°LMuu°R = uLMluR. (7.49)
For the quarks, the mass eigenvalues in M^' are the masses that appeared
as bare or current masses in the QCD Lagrangian density (5.1), even though
in the context of the full SM they are really spontaneously generated. They
are not to be confused with the dynamical masses of (9(300 MeV) generated
by the spontaneous breaking of the global SU(3)L x SU(3)r chiral symmetry
of QCD, as described in Section 5.8.

296
The Standard Model and Beyond
So far we have only allowed for ordinary Dirac mass terms of the form
^rnL^nR f°r tne neutrinos, which can be generated by the ordinary Higgs
mechanism. Another possibility are lepton number violating Majorana masses
which (for v°L) require an extended Higgs sector or higher-dimensional
operators. It is not clear yet whether Nature utilizes Dirac masses, Majorana
masses, or both. These issues will be discussed in Section 7.7. What is known
is that the neutrino mass eigenvalues are tiny compared to the other masses
< (9(0.1) eV, and most experiments are insensitive to them. In describing
such processes, one can ignore Yu, and the vR effectively decouple. Since
Mu ~ 0 the three mass eigenstates are effectively degenerate with eigenvalues
0, and the eigenstates are arbitrary. That is, there is nothing to distinguish
them except their weak interactions, so we can simply define i/e, i/M, vT as the
weak interaction partners of the e, /i, and r", which is equivalent to choosing
AVL = AeL so that vL = A^V£. Of course, this is not appropriate for physical
processes, such as oscillation experiments, that are sensitive to the masses or
mass differences.
The unitary matrices Al,r can be constructed by noting that MM^ and
M^M are Hermitian. From (7.47) and its conjugate one has
AU^MUMU^A\ = Au^Mu]MuAuR = Mf =
(m2Ul 0 0 0 \
0 m2U2 0 0
0 0 m2U3 0
V 0 0 0 '••/
(7.50)
^l r and the eigenvalues can be constructed by elementary techniques. The
Hermiticity of MM^ and M^M guarantees that the m\v are real and that the
eigenvectors are orthogonal, while their special form** implies that m2u > 0.
However, the A^ R are not unique, as is indicated by the hats. We have
implicitly assumed in writing (7.50) that the eigenstates are ordered in the
same way for MM^ and M^M. Even then, A^ R are only determined up to
phases. That is, if A^ R are solutions to (7.50), then so are
AU — AU TfU AU — AU TfU
(7.51)
where
TSU
KL,R
(el<p^,R o 0 0 \
0 ei(t>2L,R o 0
0 0 e^L.H o
\ 0
0
o '■•/
(7.52)
"As discussed in Section 7.6.6, this implies that the lepton flavors Le, LM and LT are
separately conserved for mu = 0.
**That is, m1r = {Muxr\Muxr) > 0, where xr is the normalized eigenvector of MU^MU.

The Standard Electroweak Theory
297
are arbitrary diagonal phase matrices which correspond to the unobservable
phases of the mass eigenstate urL,R fields. (If there are degenerate
eigenvalues, then there is additional freedom in K% R associated with arbitrary
unitary transformations in the space of degenerate eigenvectors.) For a
particular choice of transformations, Aa^MuA\ will be diagonal (provided that
the eigenvectors are ordered the same way), but there is no guarantee that the
elements will be real or positive. The usual prescription is to choose K^ for
convenience, e.g., to remove unobservable phases from the CKM matrix. Then
one can choose the phases in KR so that the rajf are real and non-negative.
The A\ are not observable in the SM, i.e., they don't enter the Lagrangian,
though in principle they could be observable in some extensions of the SM
involving new interactions of the R fields.
Finally, the fermion terms in C must be rewritten in terms of the mass
eigenstate fields. For the u quarks, the kinetic energy and Yukawa terms are
Cu = u°Li@u0L + u0Ri@u°R
„-.o
Muu°R(l + — \ +h.c.
AU\
Ui All
= uLiAu^Al 0uL + uRiA^Al @uR -
iuh
uLAuLJMuAuRuR ( l + - J + h.c.
H
= uLifluL + uRifluR -
F
uLM%uR ( 1 + — j + h.c.
y^r Up- rnUr I H J
(7.53)
where uT = utl + urR. The kinetic energy terms remain in their canonical
form since A^ R are unitary, and the mass terms are of canonical form since
Mp is diagonal. The coupling of the physical Higgs boson H to ur is —ihUr =
—imur/v, i-e., for a given v the Higgs coupling is proportional to mass, just
as for the gauge and self-couplings in Table 7.1.
All of these considerations apply to the d quarks and leptons as well, so
£</> = ^2 ^r
10 — mr
1 +
!)]
Ipr,
(7.54)
where the sum runs over all of the fermions ip = u,d,e, and v (with the
usual v caveats). The coupling of H to ipr is —imr/v = —igmr/2Mw, which
is very small except for the top quark. The decay H —> bb will dominate
for Mh <^C 2Mw> The branching ratios for H —> cc and t+t~ are smaller
and the others negligible in this case, making the Higgs difficult to produce
and difficult to observe. The Higgs Yukawa couplings are scalar and are
flavor-diagonal in the minimal model: there is just one Yukawa matrix for
each type of fermion, so the mass and Yukawa matrices are diagonalized by
the same transformations. In generalizations in which more than one Higgs
doublet couples to each type of fermion there will in general be flavor-changing

298
The Standard Model and Beyond
Yukawa interactions involving the physical neutral Higgs fields (Glashow and
Weinberg, 1977). There are stringent limits on such couplings; for example
the Kl — Ks mass difference implies hds/Mn < 10~6 GeV-1, where hds is the
ds Yukawa coupling (Gaillard and Lee, 1974b; Langacker, 1991; Nir, 2007).
One must also be careful that U(1)q is not broken in such extensions^.
The Weak Charged Current (WCC)
We now rewrite the fermion gauge interactions in terms of the mass eigenstate
fields. These can be read off from (7.7), using the expressions for the gauge
covariant derivatives in (7.9); the expressions in (7.27), (7.30), and (7.33) for
the gauge boson mass eigenstates; and the unitary transformations defined
following (7.48), to obtain the fermion part of £,
cf + cYuk = A/> - ^ (-W + j#w+)
99'
-jzau - J±±3Lj»za (7-5*)
where C^ is given in (7.54) and the other terms are the fermion gauge
interactions. The W± terms are the charged current interaction, with charge raising
and lowering currents
J# = E KV(1 - 75)e°m + u°mr(l ~ 7 VJ
m=Fl (7-56)
J<w = E [&7"(1 - 75)^ + ^7"(1 - 75)<]
m=l
in the weak eigenstate basis. We saw in Chapter 6 that these violate P and
C maximally, but are CP conserving. The fermion gauge vertices are shown
in Figure 7.5.
The amplitude for a ^-channel four-fermion interaction in the SM is second
order, as shown in Figure 7.6. It is
For small momentum transfer, \q^\ <C Mw, this leads to the effective Hamil-
tonian density
Heff = iM = ^Jl^, (7.58)
ttThe minimal supersymmetric extension of the SM does not involve flavor-changing neutral
Higgs couplings or electric charge violation due to the Higgs fields even though there are two
Higgs doublets. There are, however, potential problems associated with the scalar quark
and lepton fields in some regions of parameter space.

The Standard Electroweak Theory
299
— ieq* ~fh
-0.
*2cos<
' vV(af _ af ~5\ —i-2—~V>(i ,5^ .
'W
,y~
w
'qtj
FIGURE 7.5
The fermion gauge interaction vertices in the standard electroweak model.
glr = t)L - 2sm20wqf and gfA = t3fL, where t*uL = t\L = +±, while t3dL =
= — ^. The djU{W~ vertex is the same as for UidjW+ except Vqij -^
/3
(Vj) .. = Vqij' The lepton-VK111 vertices are obtained from the quark ones by
Ui
vu dj
e, , and Vq -> Ve.
which reproduces the effective Fermi theory provided we identify the Fermi
constant as
Gi
v/2 8M^ W
(7.59)
(Of course, the correspondence carries over to decay processes and 5 and u
channel exchanges.) (7.59) allows us to determine the weak scale v ~ 246 GeV,
as in (7.36). The phenomenology of the weak charged current interactions was
discussed in Chapter 6.
2V2 uw
FIGURE 7.6
Charged and neutral current four-fermion ^-channel exchange processes in the
standard model. The W~ or Z carries four-momentum q.

300
The Standard Model and Beyond
Rewriting the currents in terms of mass eigenstates
J$ = 2vLl» A*A\ eL + 2uLl» Au^AdL dL
Ve Vq (7.60)
J& = 2eLl»V}vL + 2dL7'%tuL,
where v>L,dL,eL, and vl are the F-component column vectors defined
following (7.48). The F x F unitary quark mixing matrix
Vq = A^Af (7.61)
describes the mismatch between the unitary transformations relating the weak
and mass eigenstates for the up and down-type quarks. It is ultimately due to
the mismatch between the gauge and Yukawa interactions. Ve is the analogous
leptonic mixing matrix. It is critical for describing neutrino oscillations and
other processes sensitive to neutrino masses, and will be described in Section
7.7. As commented above (7.50), however, for processes for which the neutrino
masses are negligible we can effectively set Ve = I (more precisely, Ve will only
enter such processes in the combination V^Ve = I, so it can be ignored).
An arbitrary complex FxF matrix involves 2F2 real parameters. However,
Vq is unitary, which imposes F2 constraints, (V(jVq)Tnn = Smn, so it can be
described by F2 real parameters. Not all of these are observable, however.
Recall that the F fields Ul each have an arbitrary unobservable phase, as
do the dL fields, described by the matrices K^' in (7.52). One can use the
freedom in choosing K^ to remove 2F — 1 phase differences from Vq, so that
altogether there are
F>-(2F-l) = (F-lf=F(F-1h{F-1)!;F-l) (7.62)
angles phases
observable parameters. These consist of F(F — l)/2 rotation angles (the
number of angles in an O(F) rotation), with the remainder being observable
CP-violating phases. To see that such phases are CP-violating, one has that
under CP
~ M " M t* (7'63)
/U,mL/yn*qmnU"nL y ~Q"nLl ^qmn^mL — ~Q"nLl ^qnm^"rnLi
where it is understood that (t,x) —> (£, — x) as well. (The transformation
of W±fl in (7.63) was defined so that the Lagrangian would be invariant in
the absence of phases.) The hadronic part of the charged current interaction
therefore transforms into
- --^ [uL^V*dLW+ + dL-fV^uLW-} ,

The Standard Electroweak Theory
301
which differs from C^(t,-x) if there are observable phases** in Vq. One
must have F > 3 families to obtain CP breaking in the weak charged current
interactions, and all three must be involved in a given process to lead to an
observable effect. These issues will be described in detail in Section 7.6.
For F = 2 families, there is one angle and no phase, so Vq is just the
Cabibbo rotation
_ / cos0c sin0c\
VcaMbbo- {_s[nocCOSoc)' (7.65)
where sin#c ~ 0.23, as in (6.22). Thus, in the four quark theory, the
amplitudes for £L —> vzL, (dL —> uL or sL —> cL), and (dL —> cL or sL ^ uL)
are proportional to l,cos#c, and sin#c, respectively. The Cabibbo rotation
in (7.65) is an excellent approximation to transitions amongst the first two
families, even in the full 3 family case, because the elements of Vq connecting
the first two families to the third are very small.
For F = 3 one has Vq = Vckm, which involves 3 mixing angles and one
observable CP-violating phase,
(VudVusVuh\ / 1 A A3\
Vckm = Vcd Vcs Vcb ~ A 1 A2 , (7.66)
\Vtd Vts VtbJ \X3X2 1 /
where A = sin#c. The second form is an easy to remember approximation to
the observed magnitude of each element, which displays a suggestive but not
well understood hierarchical structure. These are order of magnitude only;
each element may be multiplied by a phase and a coefficient of 0(1). There
have been extensive studies of K, D, and B decays as well as other decays and
scattering processes to determine the elements of Vckm, to test its unitarity
(which could appear to be violated in the presence of new physics), and to test
whether the observed CP violation can be described by the phase in Vckm-
These tests and parametrizations of Vckm will De described in Section 7.6.
Here we just note that Vckm can most likely account for the CP violation
observed in particle processes, but an additional source of CP breaking is
required to account for baryogenesis, i.e., the origin of the baryon (matter-
antimatter) asymmetry of the universe.
QED
The second gauge interaction term in (7.55) is the QED interaction
CQ = // JgAM, (7.67)
**There are no such phases in the weak basis, so the charged current vertices are CP-
conserving while the quark mass matrix (and therefore the propagators) are CP-violating.
The situation is reversed in the mass eigenstate basis, but the two descriptions are equivalent
for physical amplitudes.

302
The Standard Model and Beyond
where A^ in (7.33) is the massless photon field and the (Hermitian)
electromagnetic current
^ = E<^V</v°=£
m=l
1
d° <y»d°
\dP^d° - eV e°
(7.68)
is the sum over all fermion vector currents weighted by their charges qi. J^ is
purely vector since we chose the weak hypercharge assignments y = q —1\ for
both the left and right chiral fields to yield the same q even though their t\
differ. Comparing with (2.214), we identify the positron electric charge with
e =
99'
g sin 6w
1
1
1
Jq takes the same form in the mass eigenstate basis,
F
e2 g2 ga'
*
m=\
, ^m7 ^m
1
1
^m7 U"n
"mi ^m
-u^u - ^d^d - e^e,
(7.69)
(7.70)
since, e.g.,
uV«° = «° 7M«2, + u°R^uR
= uL^A^A\uL + uRrAujARuR = «7"«.
(7.71)
This is ultimately due to the fact that only fields of the same charge and
chirality, such as the u^nL, are able to mix with each other. Jq is therefore
flavor diagonal and family universal, as well as P, C, and CP preserving.
The standard model incorporates QED and all of its successes, as described
in Section 2.12.3.
The Weak Neutral Current (WNC)
The Fermi theory of charged current weak interactions and QED were well
established before the development of the standard model and were
incorporated into it. However, the weak neutral current (WNC) interaction (along
with the W and Z bosons) was a new ingredient predicted by the 517(2) x 17(1)
unification.
The weak neutral current interaction is described by the last term in (7.55),
VF+F
^Z^M
'2CO8 0W, z »~ 2Jz^
(7.72)

The Standard Electroweak Theory
303
where ZM is the massive neutral boson denned in (7.30). The (Hermitian)
weak neutral current is
4 = E W-f [*?l(1 - 75) - 2qr sin2 6W ] tf
= £&>? VU - 75)VV° - 2sin2 6W Jg, (7'73)
r
summed over all fermions. Specializing to the SM quantum numbers,
3% = u°Lru°L - d°L^d°L + vl^vl - e» tM - 2 sin2 0W J%
= uL^uL - dL^dL + urfvL - eL^eL - 2 sin2 0W Jq,
(7.74)
where the t\ = ±^ have been absorbed into the Pl, e.g., ^ti7M(l — 75)n =
ul1^ul- The neutral current has two contributions. The first only involves
the left-chiral fields and is purely V — A. The second is proportional to
the electromagnetic current with coefficient sin2 6w and is purely vector. P
and C are therefore violated in the neutral current interaction, though not
maximally. There are no phases, so CP is conserved. J^ is sometimes written
in terms of its chiral or its V, A components as
J» = 2^ 1/V7" [eL(r)PL + eR(r)PR] r/,r = J^V [grv - <£75] t/v, (7.75)
where gyA = €L(r)±€R(r). This form is especially convenient for generalizing
to alternative or extended gauge theories, and also for incorporating radiative
corrections. In the SM
clM = t*rL - sin2 0W qr, eR(r) = - sin2 0W qr
grv=t3rL-2sm2ewqr, grA=t3rL.
Like the electromagnetic current J% is flavor-diagonal and has the same
form in weak and mass bases in the standard model; all fermions which have
the same electric charge and chirality and therefore can mix with each other
have the same SU(2) x 17(1) assignments, so the form is not affected by the
unitary transformations that relate the bases. It was for this reason that the
GIM mechanism (Glashow et al., 1970) was introduced into the model, along
with its prediction of the charm quark. Without it the d and s quarks would
not have had the same 517(2) x 17(1) assignments, and flavor-changing neutral
currents would have resulted. To see this, suppose there were only one left-
chiral quark doublet, (u°L d°L)T, with s°L and the right-handed quarks being
SU(2)l singlets. Then, in terms of the mass eigenstates ul = u°L and
dL
sl
\ _ /cos 0C - sin 6C\( d°L \ ,_ __.
)-\^n9c cos6c)\sl)' {7J7>

304
The Standard Model and Beyond
the hadronic part of J% would be
Jh/ =u°L>fu0L - d°L^d°L ~ 2 sin2 0W J$*
=uL^uL - dL^dL cos2 0C - srfsL sin2 0C (7.78)
- (drfsL + sL^dL) cos 6C sin 0c-2 sin2 0W J^f',
leading to unacceptable strangeness changing neutral current transitions (as
well as large box diagram effects), as shown in Figure 6.3. The absence of
such effects is also a restriction on extensions of the standard model involving
exotic fermions (Langacker and London, 1988b).
A typical four-fermion process mediated by the Z in the t channel is shown
in Figure 7.6. In the limit that the momentum transfer is small compared
to Mz one can neglect the g-dependent terms in the propagator, and the
interaction reduces to an effective four-fermi interaction*
nefCf = -kJpZv (7-79)
The coefficient is the same as in the charged current case because
GF g2 _g2+g'2 (
That is, the difference in Z couplings compensates the difference in masses
in the propagator. However, unlike the charged current, J% is Hermitian, so
there is an extra combinatoric factor of 2 when one takes a matrix element !of
njNC
neff
The weak neutral current was discovered at CERN in 1973 by the Gargamelle
bubble chamber collaboration (Hasert et al., 1973) and by HPW at Fermi-
lab (Benvenuti and al., 1974) shortly thereafter, and since that time Z
exchange and 7 — Z interference processes have been extensively studied in many
interactions, including ve —> ve, vN —> i/N, vN —> vX\ polarized e~-hadron
and /i-hadron scattering; atomic parity violation; and in e+e~ and Z-pole
reactions. Along with the properties of the W and Z they have been the
primary quantitative test of the unification part of the standard electroweak
model.
7.2.3 Effective Theories
Let us digress briefly on effective theories, of which the four-fermion WCC
and WNC interactions derived from the SM are an example. An effective
*Care must be taken with factors of 2 from ^l,-Pl,h5 the coefficient in (7.72), etc, in
deriving (7.79). Also, an additional factor of 2 will arise in the off-diagonal terms in the
square of Jz or in taking matrix elements of the diagonal terms. Finally, some authors
absorb the | in (7.72) into the definition of J£, but fortunately the 2 is explicitly removed
from the r.h.s. of (7.75) so that the e£/)/?(r) and 9y A are the same.

The Standard Electroweak Theory
305
geld theory is a description of physics at a given energy scale in terms of the
degrees of freedom that can actually appear as physical states at that energy.
In particular, it is often convenient to integrate out the fields corresponding
to particles too heavy to produce, so that they do not appear explicitly in
the theory but whose effects are described by non-renormalizable operators*.
This can be done easily in the path integral formalism, but the results can
also be obtained by examining Feynman diagrams (as we have done for WCC
and WNC interactions in (7.58) and (7.79)), or by solving the Euler-Lagrange
equation for the heavy field, neglecting the kinetic terms. To illustrate the
latter method, let us rederive the fermion WNC interaction in (7.79). If we
ignore the Higgs and self-interaction terms, the terms in C involving the Z
are 2
CZ = -\z^ + ^Z^ - ^Z^z. (7.81)
The Euler-Lagrange equation for the Z is therefore
e- (ssfc) ~ it - ° * <D+M«z- ~ 6-d-z" - i^sr'*- <™>
At low energies compared to Mz the derivatives will ultimately be replaced
by factors of E, where E is a typical external energy for the process being
considered. Therefore, (7.82) can be solved, yielding
Z»=2MF^+°(]S)- <"»
This can be reinserted into Cz, to yield the effective four-fermion operator
£?/} = ~KCf = -%Mlco£Qw3*Jz>i = " vf"7^*7*"- (7"84)
More generally, in an effective theory (e.g., Weinberg, 1995; Manohar, 1996;
Pich, 1998a; Rothstein, 2003; Burgess, 2007) one writes the most general La-
grangian density Ceff for the low energy fields that is consistent with the
symmetries, including higher-dimensional (non-renormalizable) terms such as
those in (7.84). Powers of derivatives 3M/M, where M is the heavy particle
scale, can also be included, so £e// is a systematic expansion in powers of
energy. The coefficients of the terms in Ceff can be obtained from
experiment, or can be computed in terms of the underlying theory (as in (7.84)).
One can treat the low energy effective theory like any other if one uses a
tAccording to the decoupling theorem (Appelquist and Carazzone, 1975) heavy particles
do not enter the effective low energy theory except through renormalized parameters and
non-renormalizable operators. However, care must be taken in the application of this result
when the heavy particles violate a symmetry and therefore require strong coupling, as we
will see in Section 7.3.6 in connection with the oblique parameters.

306
The Standard Model and Beyond
mass-independent renormalization scheme (such as MS), with a finite
number of counterterms for any given power of energy. Major applications of
effective theories are to describe the low energy limit of a known theory,
focusing on the important aspects and symmetries (as in the example above),
or to parametrize the observable effects of still unknown new physics
associated with a higher scale M (such as the use of the four-fermi interaction to
describe the WCC before the SM was developed). Another major example is
chiral perturbation theory, touched on in Section 5.8, in which the low energy
strong interactions are expressed in terms of an effective theory of mesons
and baryons incorporating the spontaneously broken 517(3) x 517(3) flavor
symmetry, the soft pion theorems, etc. The heavy quark and soft collinear
effective theories were mentioned in Section 5.6.
7.2.4 The i% Gauges
It is straightforward to write the interaction vertices and propagators in an
arbitrary R^ gauge, as was discussed for a general non-abelian theory in
Section 4.4. Following SSB, instead of the Kibble representation in (7.24) the
Higgs doublet can be rewritten in terms of shifted fields
(t>= I v+H+iz J , (7.85)
just as in the 517(2) example in Section 4.4. H is the physical Higgs scalar,
while 2, w+, and w~ = lu+t are the Goldstone boson fields that disappear in
the unitary gauge. We also define the ghost fields
7]z = cos 6w Vz ~ sm 0\v Vy, Vj — sm Qw Vz + cos Qw Vy
V = -p—, Vc = cos 26w r\z + sin20w 7?7,
where rji and r\y are associated with SU(2)l and J7(l)y, respectively. The
anti-ghost fields are rfz, 77^, and
„±t = ^A (7.87)
The propagators for the mass eigenstate fields are listed in Table 7.2.
The Higgs potential in (7.12) takes the form in (4.81), which is repeated
here for convenience
V{4>) = -n2H2 + XuH [H2 + z2 + 2w+w~] + ^ [H2 + z2 + 2w+w~]2 .
(7.88)

The Standard Electroweak Theory
307
TABLE 7.2
Propagators for the standard model mass eigenstate fields in the R^
gauges. C = 0> 1> and oo correspond respectively to the unitary,
't Hooft-Feynman, and Landau (renormalizable) gauges. A +ie is implicit
in each denominator. Neutrino masses are neglected. (The propagators for
massive Dirac and Majorana neutrinos are considered in Section 7.7.2.)
iD^{k) :
iDg{k) :
iAH(k) :
iAw±(k) :
iVv±{k):
iSf^u{k) :
—i
k2
av fc"fc»(i-i/Q
y k2
k2-M^
— i
k2-M2
k2-Mfj
nV>v
fc"fc"(l-l/Q]
k2-M2wli \
„„ _ fc"fc"(i-i/Ql
y k2-M\n \
k2-M^/Z
ty—mf k2—m2t
iAz(k) :
W^{k) :
Wnz(k):
iSv(k) :
k2-M%/s
— i
fc2-M|/£
iPL= i&Pl
The gauge interactions of the Higgs fields are
;9z
(w-^w+^j CM + ^ (z*d»H) Z„
- i| ([H - iz]~d»w+) W- - i9- (w-*$»[H + iz\) W+
+ uH (£w+»W^ + °fZl\ + v9-^- (w+W-» + w~ W+») BM
+ ^- («;+(# - faJW-** +w~(H + iz)W+^B„, (7.89)
where
C„ = /* * * = cos 26w Z„ + sin 26w j4„.
Vr + 9'2
(7.90)
Eq. 7.89 is a generalization of the 517(2) example in (4.84) and of the unitary
gauge expression in (7.39). (The gauge boson mass term is the same as (7.26).)
One can similarly extend the discussion of the Higgs Yukawa couplings in
Section 7.2.2 to include the Goldstone bosons. Analogous to (7.53) the quark

308
The Standard Model and Beyond
mass and Yukawa terms are
V2
, H ■ 5Z
1 + tj5-
V V
u + dMf)
i + —+n -
V V
+ ^d [-V]M%PR + MdDV}PL] uw~
+ —u [-M%VqPL + VqMdDPR] dw+,
(7.91)
fU.d
where u and d are column vectors of mass eigenstate fields, M^' are the
diagonal matrices of mass eigenvalues, and Vq = Au^ AdL is the quark mixing
matrix. A similar result applies to the leptons, with u —> v, d —> e, and
Vq -^ Ve, provided the neutrino masses can be ignored or they are Dirac.
The ghost interactions with the scalars and gauge fields are
Cghost = -ig [(^*?-f) rf - (o>Vf) V+] Wl
- ig[(&>r,+1) W+ - (^r/-t) W~]m
-ig(d^vt)[v+w--v-w+]
+
2£
gMw
(7.92)
2£
(v+^+ + V-^-) + 9-^vUz]h
. gMw / ++ + -t -\ •
7.3 The Z, the W, and the Weak Neutral Current
After the discovery of the weak neutral current in 1973 there were generations
of weak neutral current experiments, eventually at the precision of a few %
and in a few cases ~ 0.5%. The motivation was in part to determine whether
its properties agreed with the predictions of the 517(2) x 17(1) model, and
whether the latter could be distinguished experimentally from a number of
alternative theories. This goal was largely achieved by the model-independent
analyses of the data, which allowed arbitrary V and A interactions (but
generally assumed family universality and V — A couplings for neutrinos). These
were consistent with the SM but eliminated competing gauge theories
predicting four-fermi interactions very different from the SM predictions. Other
possibilities involving purely S, P, and T interactions were excluded by the
observation of weak-electromagnetic interference in processes involving charged

The Standard Electroweak Theory
309
particles. More complex gauge theory competitors with similar 4-fermi
interactions but different gauge bosons were largely ruled out by the later discovery
of the Z and W. As later generations of more precise results (including the
Z pole, Tevatron, and other high energy experiments) became available, the
emphasis turned more towards precision tests of the SM at the loop level,
leading to predictions for the top quark and eventually the Higgs masses,
precise measurements of couplings for comparison with unification predictions,
and testing the underlying structure of renormalizable gauge theories. They
also allowed searches for and constraints on small deviations from the SM
predictions that could be attributed to new physics beyond the standard model.
In this Section we describe some of these tests and their implications. More
detailed discussions may be found in, e.g., (Kim et al., 1981; Amaldi et al.,
1987; Costa et al., 1988; Langacker et al., 1992; Langacker, 1995; Amsler et al.,
2008; Erler and Langacker, 2008). For a historical perspective, see (Langacker,
1993).
7.3.1 Purely Weak Processes
ve~ —► ve~ Elastic Scattering
The neutral current reactions v^e- —> v^e~ and v^e~ —> v^e~ do not occur
in the Fermi theory, but are predicted to proceed by ^-channel Z exchange, as
shown in Figure 7.7. They were observed and studied in a number of
experiments at CERN, Fermilab, and Brookhaven, most precisely by the CHARM II
collaboration at CERN (for a review, see J. Panman in Langacker, 1995). The
momentum transfers are tiny compared to the Z mass, so the four-fermion
effective interaction in (7.79) applies. From (7.74) the effective Lagrangian
density is
-C" = ^| PM y (1 - 7>„ e l^ye - 9%l5)e. (7.93)
In the SM one expects,
«#? ~ - ± + 2 sin2 6W, tff ~ -1- (7.94)
gye is predicted to be small, since sin2 8w ~ 0.23. (7.93) is valid in any gauge
theory, provided that any right-chiral neutrino vr does not have significant
gauge interactions and that any v mass is negligible. We use the symbols gyeA
to represent the coefficients in the 4-fermi interaction. These could differ from
the coefficients gyA relevant to the Zee vertex in more complicated theories
involving multiple gauge bosons. Differences can also be generated when one
includes higher-order radiative corrections in the SM. In practice, however,
the radiative corrections to elastic ve scattering (Sarantakos et al., 1983) are
small compared to the experimental precision except for top quark effects.
Radiative corrections will be discussed in Section 7.3.4.

310
The Standard Model and Beyond
We saw in Section 6.2.2 that there is a charged current contribution to
vee~ —> vee~ scattering, which can be put in the same form as (7.93) by a
Fierz transformation. The results in (6.60) and (6.61) can therefore be applied
to elastic v^e~ and P^e~ scattering provided we substitute gv,A —> 9\fA.
(The mey/Eu terms are not important at accelerator energies.) Elastic vee~
and vee~ scattering have both charged and neutral current contributions, as
shown in Figure 7.7. Formulae (6.60) and (6.61) still apply to those reactions
provided we take gy,A = 9yeA + 1-
One can determine gye and gv£ separately by measuring either the energy
(y) distributions or the total cross sections for both v^e and P^e, up to a
fourfold ambiguity associated with the overall sign and with the interchange of g^€
and guA. Of course, the neutrino beams are not monochromatic, so the v^ and
v^ spectra must be modeled. As shown in Figure 7.8, one of the four solutions
is consistent with the SM for sin2 6w ~ 0.23. If one assumes the validity of
the SM then sin2 0W can be best determined from the ratio of the v^e and
v^e cross sections because many of the systematic uncertainties cancel. The
precise value of the extracted sin2 6w depends on the renormalization scheme,
as will be discussed in Section 7.3.4. The current value of sin2 6w from v M e
elastic scattering, in the on-shell scheme denned by sin2 6w = 1 — M^/M^,
is 0.2232(77) (Amsler et al., 2008). The extracted values of g^A for the SM-
like solution are compared with the SM expectations, obtained using the SM
parameters from the global best fit, in Table 7.3. The agreement is excellent.
Limits on small deviations from the SM value can be used to limit certain
types of possible new physics, such as additional heavy Z' gauge bosons or
mixing with exotic heavy fermions (see, e.g., Amaldi et al., 1987).
There were also measurements of vee at the Savannah River reactor and of
vee at Los Alamos. These are not as precise as the v^e measurements, but
because of the charged current contribution they can help resolve the four-fold
ambiguity. As is seen in Figure 7.8 two of the four are excluded. The fourth
solution, obtained for gye <-> gv£, is excluded by e+e~ —> ^fi~ data under
the plausible assumption that the weak neutral current is dominated by the
exchange of a single Z boson. The LSND collaboration at Los Alamos also
demonstrated interference between the charged current and neutral current
contributions to vee scattering (Auerbach et al., 2001), obtaining the value
-1.01(18) for / = 2aT/awcc, where a1 and awcc are respectively the
interference and WCC contributions to the cross section, compared with the SM
expectation of —1.09. (Other observations of WCC-WNC interference are
described in their paper.) This is significant because it provides a confirmation
of the conclusion from WNC-electromagnetic interference (Section 7.3.2) that
the WNC is really vector and axial.
Deep Inelastic Neutrino Scattering
There have been many experiments measuring neutrino hadron scattering,
including elastic v^p —> v^p scattering, inelastic exclusive processes such as

The Standard Electroweak Theory
311
VG IA,
XX
FIGURE 7.7
Left: Diagram for is^e~
right: Diagrams for vee~
v^e' in the SU{2) x C/(l) model Middle and
vee
o
T-T
&
:>r. -
CVJ
9
o
$
001
or
q'
T1.0
CO
Q..
0-
o
o
reactor
vee(LANL)
] all
o
o
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.(
9a
FIGURE 7.8
(-)
Allowed regions in gye and gv£ from v M e~, ^ee~, and reactor j/ee~
elastic scattering, compared to the predictions of the SM as a function of
sin2#vy. The v^e regions are at 90% c.L, and the bands are at la. Updated
from (Amaldi et al., 1987).

312
The Standard Model and Beyond
TABLE 7.3
Values of the model-independent neutral current parameters
compared to the SM expectations for the global best fit values of the
parameters. From the Electroweak Review in (Amsler et al., 2008),
where one can also find the full SM expressions including radiative
corrections and the correlations on the experimental uncertainties.
Experiment SM Expectation
0.040 ±0.015
0.507 ±0.014
3012 ± 0.0013
0310 ± 0.0010
50 ± 0.033
co+0.41
oo-0.28
0.1526 ±0.0013
0.514 ±0.015
0.23 ±0.57
0.077 ± 0.044
-0.0397(3)
-0.5064(1)
0.3039(2)
0.0300
2.4630(1)
5.1765
0.1528(1)
-0.5298(3)
-0.0095
-0.0623(5)
vv!P -* v^-kN (dominated by the A resonance), and deep inelastic v^N —>
v^X scattering, as well as their v^ and charged current analogs (for reviews,
see F. Perrier in Langacker, 1995; Conrad et al., 1998; Amsler et al., 2008).
Deep inelastic e^p —> e^X (involving both 7 and Z exchange) and e^p —>
ve(9e)X (via W^ have also been studied at the HERA ep collider at DESY.
The WCC experiments have been especially important as tests of QCD and
probes of parton distribution functions, providing information complementary
to charged lepton scattering. They also provide information on the CKM
matrix elements \Vcd\ and \VCS\. The neutral current experiments have mainly
functioned as tests of the WNC couplings and determinations of parameters
such as sin2 8\y •
Here we will focus on the deep inelastic neutrino scattering, for which many
experiments have been performed, mainly at CERN and Fermilab. The
neutrinos are mainly produced by 71^2 and K^, where the tt and K emerge from
proton collisions on a target. These are almost entirely v^ and J/M, though
enough ve{p^) are produced, mainly in Kes decays, that these must be taken
into account as background. More 7r+ and K+ are produced than 7r~ and
K~, so v^ beams are more intense than v^. The typical neutrino energies at
the CERN and Fermilab experiments are in the 10's-100's GeV range with
a broad spectrum. The spectrum becomes somewhat narrower in energy in
the narrow band beams (NBB), in which the energy of the parent hadron is
selected. These actually lead to two peaks in energy, associated with the pion
Quantity
9ve
97
9l
9\
0l
Or
C\u + C\d
C\u — Cid
Ciu + Cid
Ciu — Cid

The Standard Electroweak Theory
313
(lower energy) and kaon decays. Even in the NBB there is considerable
uncertainty in the incident neutrino energy. Ev can be determined in WCC events
such as VpN -> //"X by measuring the outgoing /i" and hadron momenta,
but this is not possible in WNC current events such as v^N —> v^X because
of the unobserved final neutrino. One therefore typically concentrates on the
total WNC cross sections. Care is still required to determine or model the
incident energy spectrum and average over it appropriately.
We will consider the charged and neutral current processes
WCC: i/MJV->/x-X, v^N^n+X
WNC: VpN - i/MX, P^N -+ v^X, K ' j
where TV can be a proton, neutron, or nucleus. The nuclear targets, especially
heavy ones, yield more events. They are also easier to interpret theoretically,
especially if they are isoscalar (i.e., equal numbers of p and n), such as 12C, or
close to isoscalar, such as 56Fe. The p and n-target experiments, performed,
e.g., by observing the relevant tracks in a deuterium or neon bubble chamber
such as BEBC at CERN, have lower statistics but provide information on the
isospin structure of the WNC or of the parton distributions.
Deep Inelastic Charged Current Scattering
The kinematics for deep inelastic neutrino scattering are the same as those for
e~ scattering, as described in Section 5.5. The calculation of the cross section
is also similar. In the relevant regime Q2 <C M^ the WCC cross section is
where
LjJ" = 1* [7M(1 -75) *7"(1 -T5)^]
t
2
a2 1 (7 97)
is the leptonic tensor, analogous to the one for e~ scattering in (5.46) with me
neglected. (Note that the symbol v enters several ways: as a Lorentz index, to
indicate a neutrino, and as the invariant p-q/M. The meaning should be clear
from the context.) The antisymmetric e^pva is due to the parity-violating
vector-axial interference. The tensor V^ for v^N —> /i+X is obtained by
interchanging k and A;', so that the e^pu<T term changes sign. Similarly, the
hadronic tensor WY£ is defined as in (5.55), except Jq is replaced by the
hadronic charge-raising current J^J. W^ can be written in terms of the

314 The Standard Model and Beyond
structure functions W^z m a form similar to (5.58):
Wu'p
"flit
-a +Ml
+
i
M2
+ ie
*V
pq
Vv
p-q
~^rqv
(7.98)
lipvcr
2M2
W£D{Q\v)
where the final term is again due to vector-axial interference*. The
superscripts on the structure functions W\^ indicate that those for e~, v, and v are
all different, while Wg can differ from W%. They also depend on the target
(p, n, or TV). However, to the extent that one can ignore the contributions of
the second and third families and of Cabibbo or CKM mixing, one has from
isospin that
Wu
rr ip
Win, W?n = I K + W?n] ~ W?N, (7.99)
where TV is an isoscalar target. In analogy with e DIS, we introduce new
structure functions
Fr (x, Q2) = MWr (Q2, v), F^ (x, Q2) = vWZ$ (Q2, v), (7.100)
anticipating that the quark model or QCD will imply an approximate scaling
behavior F""v (x, Q2) ~ F"'v (x), up to higher-order corrections. In that case,
the differential cross section is
d2a^cD _ G\MEV
dxdy 7r
xy2Fr(x) + (l-y)F^p(x)±xy(l
y
)Fr(x)],
(7.101)
where the ± is for v{v), and the total spin-averaged cross section
,1 ,i d2duj
ag = f dx f dy
Jo Jo
dxdy
(7.102)
is predicted to scale with the neutrino energy in the lab frame, Ev.
Let us now consider the simple quark parton model approximation, as
indicated in Figure 7.9, and initially ignore the heavy families and quark mixing.
Then, similarly to (5.70) for e~ scattering, one has
F%{x) = 2xF»(x) = 2x [d(x) + u(x)], F£(x) = 2x [d{x) - u(x)]
F£{x) = 2xF[(x) = 2x [u(x) + d(x)] , Fg{x) = 2x [u{x) - d(x)] ,
(7.103)
■'■Strictly speaking, there are additional possible terms in the hadronic tensor since J^ is
not exactly conserved. However, they lead to effects proportional to the lepton masses and
are therefore negligible in the deep inelastic regime.

The Standard Electroweak Theory
315
where F2 = 2xFi is just the Callan-Gross relation, i.e., that the partons have
Spin-^, and the - sign for the antiquarks in F3 is due to V - A. Comparing
with the analogous relations for e" scattering in (5.70), we see that v DIS
allows one to separate q from q by measuring F2 and F3 separately. (7.103) is
easily extended to include three families and CKM mixing. However, for the
energies and precision of the existing experiments it suffices to consider just
the first two families, since the b and t content of the nucleon is small. Using
the Cabibbo approximation to the mixing,
F£3{x) =2x[d(x) (cos20C + sin26C£C)
+ s(x) (sin2 0C + cos2 0C£C) ± u(x) ± c(x)],
(7.104)
where the terms correspond respectively to d —> u, d —> c, s —> u, s —> c, u -^
(d+s), and c -^ (d+s). The ± signs are for ^2(^3). £C(Q2) is a kinematic
suppression factor associated with the non-zero c quark mass (we ignore mu^s)-
One expects £c —> 1 for Q2 ^> m2, while the small deviations from unity for
finite Q2 can be extracted from the c production data or estimated
theoretically. Similarly,
Fl3(x) =2x [u(x) + c(x) ± d{x) (cos2 0C + sin2 6C£C)
± s(x) (sin2 6C + cos2 0C£C)].
To include the full CKM matrix, one can replace
(7.105)
(cos2 6C + sin2 0C£C)
(sin2 8C + cos2 0C£C)
{\Vud\2 + \Vcd\2£c)
{\Vus\2 + \Vcs\2tc)-
(7.106)
(In principle, the coefficient of u(x) in F£3 is changed to \Vud\2 + |KiS|2, etc.,
but in practice the difference is negligible.)
FIGURE 7.9
Typical diagrams for neutrino deep inelastic scattering in the quark parton
model. The initial and final lepton momenta are k and h', and q = k — k'.
An interesting test of the quark model and charge assignments is the 5/18£/l
rule, which concerns the ratio of F2 in e~ DIS from an isoscalar target TV

316
The Standard Model and Beyond
(Eq. 5.79), to F%N. In the simplified model in which one ignores the s and c
quarks in the nucleon and sets £c ~ 1 one predicts
FhN_ = \ (F2p + F2n) = (l + l)ix(uP + dP + uP + d") =5_
*2N \ {F%p + Ffn) x(dP + UP + UP + dP) 18' ^ 7>
where qp is the quark distribution function in the proton and we have used
the isospin relation (5.83). (7.107) agrees well with the data, but not with
the expectation ~ 0.5 of an early competing model with integer charged
quarks (Nambu and Han, 1974).
The parton model expressions for the WCC cross sections are
d2ovcc _2G2FMEU
dxdy
\xd [\Vud\2 + \Vcd\%] +xs [\VUS\)2 + \VCS\%]
Ad
+ x{u + c)(l-y)2} (7-108)
~ 2a2?ME» \x(d + s) + x(u + c) (1 - yf]
7T L J
and
d2aDcc _2G2FMED
dxdy 7r
2G2FMED
x(u + c)(\—y) + xdXd + xs\s
x (u + c) (1 — y) + xd + xs\ ,
(7.109)
where the target labels are not displayed. The second form is valid at
sufficiently high energy that the c-quark threshold effects are negligible, and
ignores the small CKM mixing with the third family. One sees explicitly that
the total cross sections are predicted to grow linearly with energy to the extent
that one can ignore the Q2 dependence of the quark distribution functions, W
propagator effects, £c, etc. One sees in Figure 7.10 that this is the case. The
v^ and v^ data can be combined with e~ DIS to separate the various quark
distributions. Of course, QCD predicts logarithmic violation of scaling, which
is tested in detail (for a review, see Conrad et al., 1998). The observed
structure functions are also an important input in the global QCD fits to parton
distribution functions described in Section 5.5.
Neutrino induced opposite sign dimuon production is associated with the
production of charm (c) quarks from d or 5, via v(p)N —> ^X + c(c), with
the subsequent decays c —> (5,d)/i+i/M or c —> {s,d)^L~u^. By measuring
these dimuons for both v^ and v^ and in different Eu ranges, one can extract
l^cdl^SIV^I2, and £c separately from the experimental data, where S is the
momentum carried by s quarks, S = J xs(x)dx.
Deep Inelastic Neutral Current Scattering
The neutral current processes v^v^N —> v^{p^)X have been especially
useful in testing the WNC predictions of the standard 517(2) x 17(1) model. They

The Standard Electroweak Theory
317
0 10 20 30 50 100 150 200 250 300 350
I ■ I ■' i ■ ■■ i | ■ i ■ i j i i i i | i i i i | i i i i | i i i i n 1-0
T
0.8
>
O
I
vN
if ^
i?f il fy^frf***'-*-* '"*
J
F
m
[31
[41
•
D
X
o
NuTeV
CCFR(96)
CCFR(90)
CCFRR
[51
[61
[71
[81
o CDHSW
n ggm-sps
0 BEBCWBB
O GGM-PSv
[91
[101
[111
[121
O
A
V
A
GGM-PSv
IHEP-JINR
IHEP-ITEP
SKAT
[131
[14]
[15]
[161
T
♦
®
H
CRS 1
ANL
BNL-7ft
CHARM
01) U-
T i i i i i i i i i
0.8
-10.2
10 20 30 50
100 15
£v[GeV]
250
FIGURE 7.10
Total charged current neutrino cross sections a*£' jEv vs Eu. The simple
parton model or QCD predict that these should be ~ constant at high energy.
Plot courtesy of the Particle Data Group (Amsler et al., 2008).
were extremely important in establishing the correctness of those predictions
to leading order and in limiting small deviations. Prior to the Z-pole
experiments at LEP and SLC they provided the most precise measurements of
sin2#w
From (7.75) and (7.79) the effective four-fermi interaction for WNC v-
hadron scattering in the SM is
yvh
^/(l-T5)"
x £ K(r) qT 7„(1 - l5)Qr + e£h(r) qr 7„(1 + 75)<Zr] ,
r
where from (7.76)
(7.110)
^) = +2-3
sin2^, eg(u)
3
sin 6w
(7.111)
e"Rh(d) = +-sm2ew.
Just as for ve scattering, (7.110) will continue to hold in an arbitrary gauge
theory with neutrinos coupling to V — A, or can be extended to include
radiative corrections to the SM tree-level predictions (Marciano and Sirlin, 1980;

318
The Standard Model and Beyond
Sirlin and Marciano, 1981), for appropriate values of the e^1^. We therefore
allow for arbitrary e^^u.d), but will assume family universality, i.e., that
*LhR{d) = e£R(s) and'c^(tt) = e^(c).
It is then straightforward to show that in the simple parton model,
dxdy
7T L J
\e?(d)\2 + \e»Rh(d)\2(l-y)2](xd + xs)
\e%(u)\2 + \eih(u)\2(l-y)2](xu + xctc)
+
+
+
(7.112)
Kh(d)\2 + \elh(d)\2(l-y)2\{xd + xs),
while ef (r) <-> efcV) fori/.
Now, consider WCC and WNC scattering from an isoscalar target, and
ignore the 5 and c content of the nucleon and take £c ~ 1. (One must correct
for these in the actual analysis.) Then, from (7.108), (7.109), and (7.112), the
total cross sections are
<*rr =
-xq + xq
_„ 2g2fmev r1 r i :
a~=-^^h [xq+rq_
2G2fmep r1 n
7T Jo [3
2GFMEV f
K Jo
2GFMED f1 (
* Jo V
dx
dx
A
A
1
xq + -xq
-xq + xq
+ 92r
+ 9r
1
-xq + xq
xq + -xq
■])
(7.113)
dx
J dx,
where q = (up + dP)/2 and q = {vP + dp)/2. The effective L and i? couplings
#i r are defined as
<?£ = kf (u)|2 + |ef (d)|2 = 5 " sin2 ^ + I sin4^
(7.114)
where the expressions on the right are those for the tree-level standard model.
It is useful to consider the ratios of WNC and WCC cross sections, because
many of the uncertainties associated with the strong interactions, neutrino
§ Similar to gye^, we use the symbols cj^^r) to indicate that they are the coefficients in the
four-fermi interaction, and can differ from the Zqq vertex factors in (7.75) by higher-order
corrections or new physics.

The Standard Electroweak Theory
319
°nc 2 , /I „
— ~ 9l + 9r^
R~7
/0 xq(x)dx
So xq(x)dx
fluxes, and systematics cancel in the ratios. Under the same approximations
as (7.113) one finds
Ry^^L-gl+gfa Rv^~g\ + m, (7.115)
where
7tv 4- -I- f IS T.n(nrAdnr.
(7.116)
can be measured directly, e is the ratio of the part of the nucleon's momentum
carried by antiquarks to the part carried by quarks. One would have r = ^
for q/q = 0> birt the observed value r ~ 0.44 corresponds to 6 ~ 0.125. It is
remarkable that within the stated approximations the details of the quark
distributions cancel out, except for r, which can be measured directly. Although
this was shown here within the simple part on model, the result essentially
follows from isospin symmetry alone, except for a term involving Y2 which is
weighted by a small coefficient of sin4#w (Llewellyn Smith, 1983).
The cross section ratios have been measured to 1% or better by the CDHS
and CHARM collaborations at CERN and by CCFR at Fermilab during the
1980s and 1990s. Although (7.115) gives a good first approximation, in
practice one must correct for nonisoscalar target effects (Nn ^ Np in 56Fe); s(x)
and c(x); the c threshold, £c; third family mixing; the W and Z propagators;
radiative corrections; experimental cuts; and the QCD evolution of the quark
distributions. Since these are relatively small corrections for the 12C and 56Fe
experiments, it suffices to use the QCD improved parton model described in
Section 5.5 to estimate them. The same estimates can be used for the n and
p target data, for which the isospin argument does not apply: even though
the theoretical uncertainties are larger for such targets, the model is adequate
since the experiments are less precise.
The isoscalar target experiments yielded the most precise determinations
of sin2 6w prior to the Z-pole era, sin2 0\y = 0.233(3) (5), where the first
error is experimental. The second error is theoretical and is largely due to
the uncertainties in the charm quark threshold effect £c. Even though c
production represents less than 10% of the WCC cross section and £c can be
determined to ~ 15% from dimuon data, the remaining uncertainty
dominated the error. The theoretical uncertainties can be minimized by using the
Paschos-Wolfenstein ratio (Paschos and Wolfenstein, 1973),
IT = !~~!ir ~ & - 92r ~ \ - sin2 ew, (7.117)
acc acc Z
for an isoscalar target, which follows from isospin. (There is still a small
residual uncertainty from the charm threshold, but it is suppressed by sin2 6C.)
Utilizing R~ requires a high intensity and high energy v^ beam, which became
possible in 1996 using the sign-selected beam at Fermilab.

320
The Standard Model and Beyond
The NuTeV collaboration (Zeller et al., 2002) has utilized the sign-selected
beam and a version of (7.117) to obtain a more precise value sin2#w =
0.2277(16), which is insensitive to the top quark and Higgs masses and has
little uncertainty from the c threshold. However, this value is ~ 3a above
the current value of sin2 0W = 0.2231(3) from the global fit to all data (Am-
sler et al., 2008). The origin of the discrepancy is not understood. It could
be an indication of some sort of new physics, such as contributions from a
Z' (Davidson et al., 2002). However, it could also be associated with a subtle
QCD effect, such as an asymmetry between the s(x) and s(x) distributions^,
larger than expected isospin breaking effects, nuclear shadowing, NLO QCD
effects, or electroweak radiative corrections. It is hard to draw any conclusions
until there is a full global analysis of all of these issues.
The neutrino-hadron data can be used in a global analysis to determine
eL%(uid)- The results are shown in Figure 7.11 and Table 7.3 (where 8LiR =
t&n~1[et£hR(u)/e'£hR(d)]). It is seen that the deep inelastic isoscalar data
determine g\ and g2R quite well. The isospin structure, which depends on DIS from
p and n targets and other reactions, is well-determined for the L couplings,
but only poorly for the evR . Nevertheless, the i/-hadron data is consistent
with the SM for sin2 6W ~ 0.23.
0.4
0.2
j 0.0
-0.2
-0.4
1-0
L i^f
I \ i i i
-0.4 -0.2 0.0 0.2 0.4
eL(u)
-0.2 -0.1 0.0 0.1 0.2
FIGURE 7.11
(-)
90% c.l. allowed regions in e*£R(u) vs €*£R(d) from V^ N elastic, exclusive
inelastic, and deep inelastic scattering, compared to the predictions of the SM
as a function of sin2 0w, updated from (Amaldi et al., 1987).
^The discrepancy is reduced to ~ 2.1cr if one applies the asymmetry in fQ dxx[s(x) — s(x)\
recently measured by NuTeV (Mason et al., 2007).

The Standard Electroweak Theory
321
7.3.2 Weak-Electromagnetic Interference
Following the discovery of the weak neutral current in the early 1970s, it took
some years to establish that it was uniquely consistent with the predictions
of the 517(2) x 17(1) model. As described in Section 7.3.1, ve and v- hadron
scattering eventually zeroed in on the SM predictions when analyzed in a
general V, A framework. However, it was conceivable that the WNC could
be due to 5, P, and T couplings, e.g., due to the exchange of spin-0 or 2
particles. These all involve a helicity flip of an initial left-handed neutrino (or
right-handed v) produced in a charged current decay, into a neutrino of the
opposite helicity (Eq. 2.209 and 2.210). However, it is not feasible to measure
the final helicity. In fact, there is a confusion theorem that the final electron or
hadron distributions in v^e~ elastic scattering or deep inelastic i/^N scattering
predicted by any V, A theory can be duplicated by some combination of 5, P,
and T (Kayser et al., 1974).
The easiest way to resolve this ambiguity was to observe the interference
between WNC amplitudes and those involving other interactions known to be
V and/or A in character, which could not occur for 5, P, or T at high
energies where fermion masses can be ignored. In fact, some interference between
WNC and WCC amplitudes was eventually observed, such as in vee~ and
vee~ elastic scattering. However, the more precise tests (and also the earliest
chronologically) involved interference between Z exchange and electromag-
netism in eq, e+e~, and (recently) in e~e~ interactions. At low energies, such
as in atoms, the WNC is only a tiny perturbation on the Coulomb or other
electromagnetic effects. However, QED is purely vector and parity conserving,
so observation of VA interference effects is a clear sign of the WNC. (These
are usually parity violating, but in some cases can be parity conserving if they
involve the product of two axial currents from the WNC.) At higher energies,
the 7 and Z exchange amplitudes may be comparable and Z propagator
effects may be observable as well. There have by now been many observations
of WNC-electromagnetic interference, all of which are in excellent agreement
with the predictions of the standard model. They include:
• Polarization asymmetries in deep inelastic eD —> eX (SLAC) and fiC —>
fiX (CERN); in low energy elastic or quasi-elastic polarized electron
scattering at Bates, Mainz, and Jefferson Lab; and in polarized e~e~
M0ller scattering (SLAC).
• Atomic parity violation in cesium (Boulder, Paris), thallium, and other
atoms.
• Cross sections and forward-backward asymmetries in e+e~ —> ££,qq,cc
and bb at PEP, PETRA, TRISTAN, and LEP 2.
• Forward-backward asymmetries in pp —> e+e~ by the CDF and DO
collaborations at Fermilab.

322
The Standard Model and Beyond
Parity Violating £±-Hadron Interactions
The parity-violating part of the effective WNC eq Lagrangian density is
-Ceq = --— J2 [di g7^75e qrfqi + C2ie 7Me fc-yVft] , (7.118)
i=u,d
where the tree-level expressions in the SM are
Ciu = -- + - sin2 6W, C2u = ~x + 2 sin2 Ow
12 1
Cld = 2 ~ 3 Sin2 0W* C2d= 2~2 Sin2 6w'
Early attempts to measure the effects of WNC-electromagnetic interference
in atoms were unsuccessful, leading to considerable confusion and doubts
concerning the correctness of the 517(2) x 17(1) model. However, those results
turned out to be erroneous. The first significant interference observation was
in the SLAC measurement (Prescott et al., 1979) of the parity-violating
polarization asymmetry Apy = {o-r—o~l)/{o-r+o-l) for deep inelastic e~ scattering
on deuterium, where gr,l refer respectively to the cross section for a right-
or left-handed e~. One expects
Apy l-{l-y)2
^2-=a^^1 + {l_yr (7'119)
where Q2 and y are the usual kinematic variables for DIS, and
al = 7—^ I Clu - -jCid J 77^ 7-7= (-7+0 sin °W
3Gf / 1 \ 3GF ( 3 5
5^2™ V * ) ™ 5^2™ V 4 + 3
a2 = V~tk— C2u - -zC2d -> —-7=— sin 0W - - )
h\[2-KOt \ 2 ) b^/2'Ka V 4/
(7.120)
They observed an asymmetry, consistent with the SM expectations for sin2 6w
= 0.224(20), agreeing with the neutrino experiments and resolving the
confusion. Subsequent measurements of polarization asymmetries in DIS ^
scattering confirmed the result, and more recent precise low energy electron
scattering polarization asymmetries have constrained Ciu,d to a narrow strip
consistent with the SM, as can be seen in Table 7.3 and Figure 7.12. (The C2u,d
are also consistent, but not so well determined.) These results are reviewed
by P. Souder in (Langacker, 1995) and in (Young et al., 2007). Implications
for new physics are described in (Erler et al., 2003).
Atomic Parity Violation
The parity-violating WNC can induce mixing between S and P wave states
in atoms, leading to such effects as the rotation of the polarization plane of

The Standard Electroweak Theory
323
linearly polarized light as it passes through an atomic vapor, or differences
in transition rates induced by left- and right-circularly polarized photons.
Observations have been made in bismuth, lead, and thallium. For reviews,
see (Masterson and Wieman in Langacker, 1995; Bouchiat and Bouchiat, 1997;
Ginges and Flambaum, 2004; Amsler et al., 2008), but the most precise have
been measurements of 65 —> 75 transitions in cesium in Paris and Boulder,
with the most recent Boulder results at the 0.4% level. Furthermore, cesium
has a single valence electron outside a tightly bound core, allowing accurate
calculation of the matrix elements needed to interpret the results.
The effective interaction in (7.118) leads to a non-relativistic potential for
the e~ in an atom Z, TV:
V(re) ~ ~^f-Qw {63(fe),ae.pe} , (7.121)
where pe is the electron momentum operator and Qw is the weak charge
Qw = -2 [Clu (2Z + TV) + Cld(Z + 2TV)]
o (7.122)
~Z(1- Asm2 0W)-N
(see Problem 7.8). In (7.122) we have kept only the C\u,d terms. This
involves an axial e~ current and vector hadronic current, with the latter adding
coherently over the nucleons in a heavy atom because it is spin independent.
In fact, the effects of this term scale as Z3. One factor is obvious from the
coherence effect in Qw, while the others involve the electron wave function
and momentum near the nucleus. There are no such enhancements for the
C2u,d (nucleon spin-dependent) terms, which require unpaired nucleons", so
the Ciu,d strongly dominate. Similarly, experiments in hydrogen or deuterium
would be extremely clean theoretically, but the effects are much smaller and
to date no measurements have been completed.
As mentioned, Qw has been measured in cesium at the 0.4% level.
Neither the theoretical expression in (7.121) nor the treatment of the atom as
hydrogen-like are adequate for such precision. In practice, one must correct
for the finite nuclear size, treat the electron relativistically using the Dirac
equation, carry out a many-body Hartree-Fock calculation of the wave
functions, and include both electroweak and QED radiative corrections (Blundell,
Johnson, and Sapirstein in Langacker, 1995; Ginges and Flambaum, 2004;
Amsler et al., 2008). The latter are difficult due to the large nuclear charge,
but careful treatments have now yielded a precision of ~ 0.5%, allowing a
determination Qw{Cs) = —72.62(46), in impressive agreement with the SM
expectation —73.16(3) (Amsler et al., 2008). The corresponding weak angle
issin2<V = 0.2212(19).
"Nevertheless, the small spin-dependent effects can be separated by measuring different
hyperfine transitions. They have been observed in cesium, but are dominated not by the
WNC but by the larger electromagnetic anapole moments mentioned in Section 2.12.4.

324
The Standard Model and Beyond
The results from polarized lepton asymmetries and atomic parity violation
can be combined in a global analysis. The combinations C\u ± C\d and C2u ^
Cid are now very well determined and in agreement with the SM, as can be
seen in Table 7.3 and Figure 7.12.
0.18
SLAC: D DIS
0.16
U0.14
+
c
PVES
0.12
0.1
-0.8 -0.7 -0.6 -0.5 -0.4
C\ U-C\ d
FIGURE 7.12 :
Allowed regions in C\u — C\d vs C\u + C\d from atomic parity violation (la),
recent parity violating electron scattering (PVES) experiments (la), older
e~- hadron asymmetries (la), and the global 95% cl ellipse. The dotted
lines are the allowed region without the new PVES. The SM prediction (Erler
et al., 2003) for the global best fit value of sin2#w is shown as a star. Plot
reproduced with permission from (Young et al., 2007).
e~e+ —► // Below the Z
There have been many measurements of e~e+ —> //, where / = e,/^,r, 6,c,
or q (i.e., an unidentified quark flavor), below the Z pole at SLAC (SPEAR,
PEP), DESY (DORIS, PETRA) and KEK (TRISTAN), as well as
measurements at or near the Z-pole (LEP, SLC), and above it (LEP 2). The PEP
and PETRA measurements (for reviews, see (Wu, 1984; Kiesling, 1988) and D.
Haidt in (Langacker, 1995)) were able to observe effects of the s-channel Z
exchange as a perturbation on the dominant one photon contribution, but were
at low enough energy that they could be treated as a four-fermi interaction.
TRISTAN was at a higher energy, where the virtual Z propagator effects were
APVTi
APVC

The Standard Electroweak Theory
325
non-negligible (e.g., Mori et al., 1989). The principal observables relevant to
the Z were the total cross sections and the forward-backward asymmetries
a = aF + aB, AFB = ■ , (7.123)
0~F +&B
where
f1 da , n f° da , „
/ -j -dcosO, aB = -j -dcosO, (7.124)
J0 dcosv J_x dcosv v '
and 6 is the angle between the e~ and / in the center of mass. It is difficult to
present the results in a model independent form because of the Z propagator.
Instead, we will write the tree-level formulae (for arbitrary s) assuming that
only the s-channel photon and Z exchanges are relevant, but allow arbitrary Z
couplings, (^-channel contributions must be included for Bhabha scattering,
e-e+-e-e+).
Polarization and forward-backward asymmetry effects for massless fermions
are easily calculated using the helicity amplitudes calculated in Section 2.9.
The helicity amplitudes in (2.245) are easily generalized. Suppose the
amplitude for e~(pi) e+(p2) -> /(ft)/(ft) is
M = - ieLL uz^^Plv^ v2^Plux - ieRR us^Prv^ v2^PRui
(1.12b)
- ieLR U3<y^PLV4 v2^PRm - ieRL u3'y^PRv4 v2^PLuu
where the first (second) subscript on the e is the chirality of the f(e~). In the
massless limit the four terms in (7.125) do not interfere with each other, and
coincide respectively with the helicity amplitudes M (—h, —h), M (H—, H—),
M(—h, H—), and M (H—,—h) defined below (2.235). Reflection invariance
only holds in the special case ell = eRR and €lR = eRL> Nevertheless, we
can use (2.245) to obtain
M (-+, -+) = ieLL s (1 + cos<9), M (+-, +-) = ieRR s (1 + cos<9)
M(-+,+-) =ieLRs (l-cos0), M (+-,-+) =ieRLs (l-cos0),
(7.126)
from which any cross section or polarization effect can be calculated. For
example, the spin-average cross section and forward-backward asymmetry are
° = l\ lM^sdC°Se = 4^ (|€LL'2 + kflfl|2 + HR? + kfiL|2) (7-127)
and
Afb=3 f|^i;+|^-|^i;-|^in. (7.128)
4 \\€ll\2 + \€rr\2 + \eLR\2 + \eRL\2J
Assuming that only the s-channel 7 and Z exchange are relevant, the
coefficients in (7.125) are given by
tAB
Ql*L - 4V2GFM2zD(s)eA(f)eB(e), (7.129)

326
The Standard Model and Beyond
where A, B = L or R. The €L,R{r) are the chiral couplings of the Z to
fermion r. They are denned in (7.75) and their SM values given in (7.76)
D(s) = (s — M\ + iMz^z) is the Breit-Wigner form for the propagator of
an unstable Z of width T^, as discussed in Appendix F. It is introduced here
so that the expression is valid below, near, and above the Z pole. (7.129) can
be rewritten by factoring out the pure QED piece
AirQfa
cab =
4vq
1 + -pr- (cos SR + i sin SR) eA (f)eB (e)
Qf
(7.130)
where
Gf sM\ , . MZTZ
cos Sr —> 1 at low energies for Tz/Mz <C 1, while cosSr —> -1 for 5 > M\.
At 5 = M\ we have that cosJ^ —> 0 and the interference term vanishes.
Substituting into (7.127) and (7.128), we find
a = a0Fu AFB = 3F2/4Fi, (7.132)
where gq = AttQ'JqP/Ss is the QED cross section in (2.228), and
Fi = 1 + ^ gfv 9<v cosSR + || (gf? + g?) (tf + gf)
f 4 2/ (7-i33)
F2 = +^j g{ gA cosSR + -^gfAgA gfv gev.
We have used that 6l(^) ± €R(r) = gTv A and therefore e2L(r) — e2R(r) = gygTA.
For 5 < M\ we have that xo = 0(s/M%) < 1 and SR ~ TZ/MZ < 1, so only
the first term in F2 is relevant. This involves only the axial couplings, which
in the SM are independent of sin2#vy. The leading contribution to Afb is
therefore an absolute prediction. For f = fi~ the SM couplings are
9a=9a = -\, g^=gev = -\ + 2sm2ew. (7.134)
The observed asymmetry, shown in Figure 7.13, agrees with the SM prediction.
Asymmetries for / = r~ and f = b also agreed well, and were important in
establishing the canonical doublet assignments of the t£ and b^ and singlet
assignments of tr and 6r (i.e., that the third family is sequential), prior
to the direct discovery of the vT and t. This was quite important, because
the third family could well have been different from the first two. As an
example, the JADE collaboration at PETRA obtained the first significant
measurement of the bb asymmetry, at y/s = 35 GeV (Bartel et al., 1984), from

The Standard Electroweak Theory
327
<s>
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
i
r
SPEAR PEP PETRA TRISTAN LEP ,
1ARKI ** IY •IJ
*MAT OJADfc a'IOPA/___^^^
MUtKIl MlRKIvVIMUS ^i—-_^a
©piimo / * * •"
TAW) /
A- | J
s. I /
\*J e+e~—>fn+[T
^_^ . . . . i .
25 50 75 100 125 150 175 200
Vs [GeV]
FIGURE 7.13
Experimental results on the forward-back asymmetry in e+e~ —> /i"1"//- as
a function of CM energy, compared with the SM prediction. Updated plot
reprinted with permission from (Mnich, 1996).
li+lf
which one can extract** gbA = -0.54 ± 0.15. In the context of SU(2) x C/(l),
but allowing arbitrary assignment of the 0l and bR to SU(2) multiplets, one
expects gbA = t\L — t\R, where t*L R are their T3 eigenvalues. The JADE
value is consistent with the SM assignment t%L = -1/2, tlR = 0, but not with
several alternative models for the third generation, such as the anomaly-free
models listed in Table 7.4. These include a mirror family (Maalampi and Roos,
1990), involving L-singlets and jR-doublets, and non-chiral (vector) models.
The four models in Table 7.4 predict gbA = -1/2, +1/2,0, and 0, respectively,
so only the sequential model was viable. In particular, the "topless" singlet
vector model was excluded, strongly suggesting that the the top quark had
to exist (barring extremely exotic alternatives). A similar result was also
obtained by the absence of flavor changing neutral current decays B —> t+t~ X
by the CLEO collaboration at CESR, which would have been generated by
the violation of the GIM mechanism if the bLiR were singlets that decayed
by bL - dL or hL - sL mixing (Kane and Peskin, 1982). Similarly, the r
couplings were obtained by observations of the r lifetime, decay distributions,
FB asymmetry, and absence of flavor changing neutral current decays r —> £££,
requiring the existence of the vT. In fact, the canonical assignments for all of
the known fermions could be extracted from the data (Langacker, 1989b).
** Subsequent LEP experiments allowed a much more precise determination of both
9yA (Schaile and Zerwas, 1992).

328
The Standard Model and Beyond
TABLE 7.4
Anomaly-free SU(2) assignments for the third family. A right-chiral vr
singlet could be added. The leptons could be interchanged in the two vector
models, e.g., vector doublet quarks and vector singlet r. More complicated
assignments are also possible.
Sequential family
Mirror family
Singlet vector
Doublet vector
PL W.
r£ tL bL
rL bL
(r~\ (0.
TR
(*).
TR
(&
tR
bR
(0.
bR
_0k
(topless)
A forward-backward asymmetry does not by itself require parity violation,
and in fact a non-trivial asymmetry is generated by higher-order QED
corrections that must be taken into account in the analysis. As an example, the
effective interaction in (7.125) would be reflection invariant for €ll = €rr and
€lr = €rl — 0, though that would be difficult to achieve in a simple gauge
theory.
The 7 — Z interference effects in e~e+ —> // have also been observed above
the Z-pole at LEP 2 (Alcaraz et al., 2006). The results are in agreement' with
the SM. Asymmetries in the inverse process, pp —> e+e~, have been measured
by CDF and DO at the Tevatron.
The polarization asymmetry Apy, in which one reverses the helicity of
the incident e~ in e~e~ —> e~e~ M0ller scattering, has been measured at
SLAC at low Q2 ~ 0.026 GeV2 using a nearly 90% polarized beam (Anthony
et al., 2005). The process is dominated by t and u-channel 7 exchange, with
Apy generated by interference with Z exchange. They obtained the value
sin20|^ = 0.2397(13) for the (running) weak angle at the relevant scale in
a particular renormalization scheme (to be discussed in Section 7.3.4), to
be compared with the SM expectation (Czarnecki and Marciano, 2000) of
0.2381(6).
7.3.3 Implications of the WNC Experiments
Even before the era of the high precision Z-pole experiments at LEP and SLC
began in 1989, the weak neutral current experiments and observation of the
W and Z had done much to establish the standard electroweak model. Global
analyses of the data were essential to the program, especially in
constraining alternatives or extensions of the SM, because no one type of experiment
was sensitive to all of the possible parameters. They also allowed a unified

The Standard Electroweak Theory
329
theoretical treatment of similar experiments. The caveat is that one must be
careful with the treatment of systematic and theoretical uncertainties and in
correlations between uncertainties.
Major results included:
• Model-independent fits showed that the four-fermion interactions
relevant to vq, ve, and eq were for the most part uniquely determined. They
were consistent with the SU(2) x [7(1) SM to first approximation,
eliminating many competing gauge theories and alternatives involving spin-0
or 2 exchange, and limiting small deviations. Measurements of Mw and
Mz agreed with the expectations of the SU(2) x U(l) gauge group and
canonical Higgs mechanism, eliminating more complicated alternative
models with the same four-fermi interactions as the standard model.
• The combination of WCC and WNC results allowed the SU(2) x U{\)
representations for all of the known fermions to be uniquely determined,
i.e., that all of the /l transformed as SU(2) doublets and the Jr as
singlets. That required that the t and vr had to exist, as partners of
the b and r~.
• The precision of the deep inelastic neutrino data was high enough to
require the application of QCD-evolved structure functions.
• Both WNC and WCC processes were measured precisely enough to
require QED and electroweak radiative corrections in their
interpretation, and attention had to be paid to the definition of the renormalized
sin2#ws as will be discussed in Section 7.3.4.
• sm26w = 0.230 ± 0.007 (using the on-shell definition), and mt < 200
GeV from the electroweak radiative corrections.
• The SU(S) x SU(2) x U(\) couplings were determined well enough to
exclude the gauge coupling constant unification predictions of the simplest
form of non-supersymmetric grand unified theories (GUTs). However,
they were consistent with the predictions of the simplest supersymmetric
GUTs, as will be discussed in Section 8.2.6.
• Significant limits could be placed on many types of new physics which
could perturb the SM predictions, including a heavy Z', new fermions
with exotic quantum numbers, exotic Higgs representations, leptoquarks,
and many types of new four-fermi operators generated by other types of
underlying new physics. In many cases both WNC and WCC constraints
were critical.

330
The Standard Model and Beyond
7.3.4 Precision Tests of the Standard Model
The precision electroweak program entered a new phase in the late 1980s with
the advent of the Z-pole experiments at LEP (CERN) and SLC (SLAC), which
eventually allowed tests of the SM at the 0.1% level. With such precision, care
was needed in the application of QED, electroweak, QCD, and mixed radiative
corrections, and in the definition of the renormalized weak angle sin2 6W. The
radiative corrections were sensitive to the top quark and Higgs masses and to
the QCD coupling as, and therefore allowed these quantities to be constrained
or predicted by the data. Here, we briefly survey some of the relevant issues
(which were also necessary, though less critical, for the WNC experiments
described in Sections 7.3.1 and 7.3.2).
Input Parameters
The basic input parameters relevant to WNC, WCC, and Z/W properties in
the SM are: (a) the SU(2) x [/(l) gauge couplings g and g'\ (b) the weak
scale, v = \/2(0|v?0|0) ~ 246 GeV, that characterizes the SSB; (c) the Higgs
mass Mh defined in (7.41) (or, equivalently, the quartic coupling A), which
enters the radiative corrections; (c) the heavy fermion masses mt,mb, etc.,
which are needed in radiative corrections and phase space factors; and (d) the
strong fine structure constant as, which enters radiative corrections.
Especially important are g, gf, and v, which are needed for weak amplitudes and
mass formulae at the tree level. However, it is convenient to trade them for
other quantities that are precisely known and more directly measured, and to
express the theoretical expressions for other observables in terms of them. A
conventional choice is:
• The fine structure constant, a = 1/137.035999679(94), determined from
the anomalous magnetic moment of the electron and other low energy
QED tests, as described in Section 2.12.3. It is related by e = gsinOw
(7.69). However, the running a must be extrapolated to Q2 = M|,
as described in Section 2.12.2. With the conventional (on-shell) QED
renormalization scheme,
a-\M%) = 1-AQ(M^ = 128.91(2), (7.135)
a
where the uncertainty is mainly from the hadronic contribution Aa£ad
to Aa. The value of Aa^ is correlated with a3, but for as(M§)=0.12
one finds % 0.02786(12) (recent estimates are reviewed in the
Electroweak article in Amsler et al., 2008). In the MS scheme one expects
d_1(M|) ~ 127.93(2). Even these tiny uncertainties dominate the
theoretical uncertainty in the precision program.
• The Fermi constant, GF = \/y/2v2 = 1.166367(5) x 10~5 GeV, defined
in terms of the muon lifetime using (6.51).

The Standard Electroweak Theory
331
• The third parameter was traditionally sin20w, defined at tree level by
g,2/{g2 + 9'2) and determined by WNC processes or properties of the Z.
However, following the ultra-precise determination of Mz = 91.1876(21)
GeV at LEP, it has become more common to view Mz as the third input
parameter. At tree level Mz = ev/2s\\\6w cosOw by (7.34).
Definitions of the Renormalized sin2 Ow
As discussed in Section 2.12.1, higher-order corrections to amplitudes and
other observables are usually divergent. In renormalizable theories the
divergences can all be absorbed into renormalized (physical) charges, masses, and
wave function renormalizations. To actually carry out the renormalization
one must first regularize the divergent integrals in a way which preserves the
symmetries, such as Pauli-Villars or dimensional regularization. It is also
necessary to define the renormalized quantities. Two popular definitions are the
on-shell and the modified minimal subtraction (MS) schemes. The on-shell
scheme was illustrated in Section 2.12.1, where the renormalized electron mass
m was defined as the position of the propagator pole, which coincides with
the kinematic meaning of mass. The renormalized electric charge is defined
in terms of the electron-photon vertex when all of the external particles are
on shell, v\ — V\ — m2 and q2 = (P2 — Pi)2 = 0. An alternative is the
MS scheme (see, e.g., Peskin and Schroeder, 1995). One can define the
renormalized charge e(ji) at scale ji as eo — 5e, where eo is the bare charge
and the counterterm Se is chosen to absorb the (n — 4)_1 poles that arise
in the dimensional regularization of higher-order corrections, as well as some
associated constants, ln47r — 7^, where 7^ ~ 0.5772 is the Euler-Mascheroni
constant. Renormalized masses are defined similarly. The MS schemes are
simple computationally and minimize some higher-order effects, and they lead
automatically to running (//-dependent) couplings and masses. However, they
are not as directly related to what is actually measured as the on-shell
quantities. Of course, one can write theoretical expressions for all observables
in terms of either set of renormalized quantities, and there are a-dependent
translations between them.
Now, let us consider the renormalization of sin2 0\y in the SU(2) x U(l)
theory (for reviews, see W. Hollik in Langacker, 1995; Amsler et al., 2008). In
Section 7.2 we encountered a number of expressions for sin2 0w, all of which
are equivalent at tree-level. These included
M2
sin2 6w =1 —tj|- (on-shell)
z
sin2 6w cos2 6w = —7= 7 (Z — mass)
\I1GfMz

332
sin 6W =
g2 + g'2
gev = --+2sm20w
where gy is the vector Zll coupling defined in (7.75). Each of the four
expressions in (7.136) can be the basis of the definition of the renormalized sin2 Qw
which will be denoted s^, s2Mz, s2z, and s|, respectively, and in each case all
observables can be expressed in terms of the renormalized sin2 9w as well as
mt, Mh, »s, etc. The four schemes are related by calculable corrections of
O(a), which can, however, depend on mt and Mh . Each has its advantages
and disadvantages.
The on-shell scheme (Sirlin, 1980) defines s^ = 1 - Mj^/Mf to all
orders in perturbation theory, where M\y,z are the physical (pole masses). It
is therefore based more on the spontaneous symmetry breaking (SSB)
mechanism than on the WNC vertices. It is the most commonly used scheme, is
simple conceptually, and is used in the program ZFITTER (Arbuzov et al.,
2006). However, observables1"1" involving the Zff vertex receive somewhat
artificial mt and Mh dependence, and the dependence of Mz on mt is also
somewhat enhanced, so the experimental value s^ = 0.22308(30) has a larger
uncertainty than the other schemes. Other drawbacks are that the mixed
QCD-EW radiative corrections are large in the scheme, and it becomes
awkward in the presence of any "beyond the standard model" physics that affects
Mz or Mw.
The Z-mass scheme is an alternative on-shell scheme (Novikov et al., 1&93),
which essentially uses the tree-level relation between Mz and s2Mz, except the
value of the running a(M|) is used. It is simple conceptually, and the value
s2Mz = 0.23105(5) extracted from the measured Z mass has the smallest
uncertainty because there is no mt or Mh dependence in the relation. (The
uncertainty is mainly from a(M|).) However, the mt and Mh dependence and
their resultant uncertainties reemerge when other observables are expressed
in terms of s2Mz. The other difficulties of the on-shell scheme are shared.
The other schemes considered here are based on the coupling constants and
therefore the Z vertices more than on the SSB mechanism. The MS scheme
defines a running
*^2) = ~4~F' (7'137)
where g and g' are defined by modified minimal subtraction (Marciano and
Sirlin, 1981). The scheme, which is used in the radiative correction program
GAPP (Erler, 1999), is simple theoretically, is convenient for comparison of
running couplings with grand unification theories, and is closely connected
The Standard Model and Beyond
(MS)
(effective), (7.136)
ft It should be emhasized that even in the on-shell scheme the most precise experimental
value of Syy is obtained from a global fit of s^ and the other parameters to all of the
precision observables, and not just the value obtained from the defining relation.

fhe Standard Electroweak Theory
333
with the scheme used in QCD calculations, thereby minimizing the
uncertainties in the mixed QCD-EW radiative corrections. Other advantages are
that the value of sz is usually insensitive to new physics, the Z asymmetries
are almost independent of mt and MH for fixed sz (except for Z —> bb, which
involves special vertex corrections involving the t), and the sensitivity of the
value obtained from Mz is reduced compared to the on-shell value, leading to
a smaller uncertainty in the experimental value, s\ = s2z(Mz) = 0.23119(14).
The running of s2z(Q2) can be tested by determinations based on low
energy WNC experiments, as shown in Figure 7.14. The agreement is generally
good, though the deviation of the NuTeV deep inelastic measurement (Section
7.3.1) is evident. The drawback of s2z is that it is a "theorist's" definition.
All observables can be expressed in terms of it, but there is no simple
defining relation. Just as in the on-shell scheme, the experimental value of s2z is
generally obtained by a global fit to all of the data.
The effective scheme, which has been used very successfully in expressing
the results of the Z-pole experiments, defines an effective s2 for each type of
fermion / in such a way that the formulae for the Z —> // partial widths and
asymmetries take their tree level form when written in terms of effective axial
and vector couplings
9a = V^7*/l, 9fv = V?~f (*/l " Zsfaf) , (7-138)
where pf is common to the vector and axial couplings. This form is very
simple conceptually and has the advantage that the Z —> // asymmetries are
almost independent of mt and Mh (except for f = b). The most precise is the
charged lepton effective coupling, s2 = 0.23149(13), which is closely related
to s2z. Drawbacks are that it is a phenomenological definition, which in detail
depends on how all of the radiative corrections to the processes observed near
the Z-pole have been applied. Also, s? is different for each /, and the scheme
is not very useful for describing non-Z-pole observables.
Radiative Corrections
There are several classes of radiative corrections that must be applied to
electroweak quantities. The first are the reduced QED corrections consisting of
diagrams in which real and virtual photons are attached to charged particles
in all possible ways, but not including vacuum polarization (self-energy)
diagrams, as illustrated in Figure 7.15. For Z exchange processes this set is
often finite and gauge invariant. However, these diagrams depend on the
energies, acceptances, and cuts, and need to be calculated and applied for each
individual experiment.
Gauge self-energy {oblique) diagrams for the 77, 7Z, ZZ, and WW
propagators involving fermion, gauge, and Higgs loops, as illustrated in Figure 7.16,
are the most important. They lead to large mt and Mh dependence of M^,
Mz, partial widths, and four-fermi amplitudes, and to the major differences
between the renormalization schemes. The dominant effects are quadratic in

334
The Standard Model and Beyond
FIGURE 7.14
Running s2z(Q2) measured at various scales, compared with the predictions
of the SM (Czarnecki and Marciano, 2000; Erler and Ramsey-Musolf, 2005).
The low energy points are from atomic parity violation (APV), the polarized
M0ller asymmetry (PV) and deep inelastic neutrino scattering (corrected; for
ans-s asymmetry). Qweak shows the expected sensitivity of a future
polarized e~ measurement at Jefferson Lab. Plot courtesy of the Particle Data
Group (Amsler et al., 2008).
FIGURE 7.15
Typical reduced QED corrections to Z exchange. Vacuum polarization
diagrams are not included.

The Standard Electroweak Theory 335
FIGURE 7.16
Typical one-loop gauge self-energy diagrams, also known as vacuum
polarization or oblique diagrams.
mt and logarithmic in MH, i.e., of O(am^) and 0(a\nMH). One-loop self-
energy diagrams that are sensitive to the top and Higgs masses are shown in
Figure 7.17. These were significant in constraining the values of mt and Mh
from the precision data.
t(b)
Hi
FIGURE 7.17
Oblique (self-energy) diagrams involving the top and the Higgs.
Electroweak vertex and box diagrams include those in which a W or Z is
exchanged across a vertex, or box diagrams involving WW,WZ, or ZZ, as
illustrated in Figure 7.18. These are usually small. However, they are needed
for gauge invariance and are not always negligible, e.g., in effective four-fermi
operators. They are generally absorbed into the parameters in the effective
interactions. Especially important are corrections to the Zbb vertex involving
the t quark, shown in Figure 7.19. Like the oblique corrections, these scale
as amf for large mt and therefore make a significant contribution to Z —> bb
decay, especially the width. The Zbb vertex corrections were especially useful
for distinguishing the effects of mt from the Higgs mass and from possible
new physics contributions to the oblique corrections.
QCD and mixed QCD-electroweak diagrams involve gluon exchange be-

336
The Standard Model and Beyond
FIGURE 7.18
Electroweak vertex and box diagrams, involving W and Z bosons.
FIGURE 7.19
One-loop contributions to the Zbb vertex involving the t.
tween quarks in electroweak processes, as illustrated in Figure 7.20. They
affect the hadronic W and Z decay widths at one-loop level, and can
therefore be used to determine as(M|) from the Z decays. They also affect the
oblique corrections at two-loop level, where the dominant effects are of order
aasm2.
Mw, Mz, and WNC Amplitudes at Higher Order.
The tree-level expressions for Mw,z in (7.34) can be rewritten in terms of the
independent variables a, Gp, and sin2 0w as
Mw = -
A0
sin0
Mz =
w
cos 6w'
(7.139)
where A0 = (7ra/y/2GF)1/2- All of the quantities in (7.139), i.e., Mw,z,
a, Gf, and sin 6w, are unrenormalized, though we do not indicate that
explicitly for notational simplicity. Including radiative corrections, (7.139) is
modified in the MS renormalization scheme to
Mw
Ao
sz(l-Arw)1/2'
Mz = -
Mw
l/2£
cz
(7.140)
where s\ is the MS weak angle, c| = 1 - s|, A0 = (ttq/v^Gf)1^2 =
37.28057(8) GeV using the physical values for a and Gf, and Arw collects

The Standard Electroweak Theory
337
FIGURE 7.20
QCD corrections to the Wqq or Zqq vertices and mixed corrections to an
electroweak self-energy diagram.
the radiative corrections to Mw (For comparison with Z-pole physics it
is convenient to define Arw so that the running s\ is evaluated at M#.)
The largest contribution to Arw is due to the running of a-1 to the value
(M§) ~ 128 at the electroweak scale, and yields
d-1
Ar
w
Ar0 = 1
a
d(Mf)
0.06649(12).
(7.141)
Including smaller effects, one expects Arw ~ 0.06962(12) in the SM**. The
large mt — r^b mass difference breaks SU(2) symmetry and leads, via the
oblique corrections from the diagrams in Figure 7.17, to an increase that
depends quadratically on mt in the ratio Mw/Mz compared to the bare
ratio. The quadratic dependence does not enter Arw because the shift in
Mw has been absorbed into the observed strength of the WCC, i.e., into the
measured Gp- Rather, it is shifted into the parameter p in (7.140). There are
other contributions to p, but the largest is from mt,
P~l + Pt, (7.142)
where the ubiquitous quantity pt is (Veltman, 1977; Chanowitz et al., 1978)
Pt =
3GFm?
SV2tt2
0.00915 (
mt
Y
170.9 GeVJ
(7.143)
In the on-shell scheme (7.139) becomes
A0
Mw
swil-Ar)1/2'
Mz
Mw
cw '
(7.144)
where s^ = 1 — M{y/M| is the definition of the renormalized angle, c^ =
1 ~ s\yi and Aq is the same as in the on-shell scheme. Ar collects all of
"All of the explicit numbers for the radiative corrections (and others not given here) are
from (Amsler et al., 2008).

338
The Standard Model and Beyond
the radiative corrections relating a, a(Mz), Gp, Mw, and M^. The largest
contributions to Ar are
Ar~ Aro-^-, (7.145)
where £w = sw/cw- Aro is due to the running of a-1 and is the same as in the
MS scheme. The pt dependence of Ar is enhanced by the t^ in the on-shell
scheme, due to the fact that the physical mt effect of increasing Mw/Mz
forces a downward shift in s^ which must be compensated by Ar. There
are additional contributions to Ar from bosonic loops, including those which
depend logarithmically on Mh- Altogether, one expects Ar ~ 0.0369(14) in
the SM.
Mw has been measured separately at the Tevatron and at LEP 2 at the
0.04% level using event shapes and also the behavior of the e"e+ —> W~VK+
cross section near threshold. The results agree and can be combined to obtain
Mw = 80.398(25) GeV, in agreement with the SM expectation of 80.375(15)
GeV using the global best fit parameters. This is about 3.7% higher than
the value of ~ 78 GeV that one would expect (for the same sz) at tree-level,
therefore confirming the radiative corrections, especially the running a. The
value of Mz will be discussed in Section 7.3.5.
The tree-level amplitude A® for a WNC process i is of the form
G°
A? = -^Wn20wO, (7.146)
where Fi is a function of z, such as defined in (7.79), and G°F and sin2 8w
are the unrenormalized quantities. The effect of higher-order corrections is to
replace A® by the amplitude
A = PlR ^pi(44) + AlR, (7.147)
where Gp is the renormalized Fermi constant defined by (6.51), Fi takes the
same functional form as in (7.146), and s2R is the renormalized quantity in
scheme R. The three parameters plR, k1r, and A^ depend on the process and
on the renormalization scheme, and one expects plR — 1,k1r — 1, and A^ to
be of 0(a). They are defined operationally, i.e., by the way that they modify
the tree-amplitude (for an example, see Marciano and Sirlin, 1980). Oblique
corrections directly affect plR, but may also enter k1r through their effects on
s^y in the on-shell scheme. Vertex corrections typically affect plR and k*r,
while box diagrams may yield contributions to A^. The A^ are usually small
enough to be neglected.
In the MS scheme one may write
P* = PPlad, (7-148)
where p is a universal part due to the shift of Mf in (7.140). p contains
the quadratic mt dependence in (7.142) and the dominant Mh dependence.

The Standard Electroweak Theory 339
a* and k% are close to unity and (like Arw) depend only weakly on mt
and Mh (witn tne exception of processes involving the Zbb vertex). In the
0n-shell scheme both pl and k% have universal terms containing the quadratic
mt and the dominant MH dependence, and much weaker process-dependent
contributions. The leading mrdependent term is
pi ~ 1 + pt, k{ ~ 1 + pt/tfy, (7.149)
where (similar to Ar) the k% dependence is present to cancel the mt
dependence of the vertices introduced by the definitions of s^.
The W Decay Width
Consider the coupling
£ = "/i7M {gv ~ <M75) hV* + h.c, (7.150)
of a massive vector V to fermions /i^- (If V is Hermitian and f\ = f2
then gv,A are real and there is no need for the +h.c.) The partial width for
V —> /1/2 is easily shown to be (e.g., Problem 2.18)
r(v - a/2) = ^ (|<M2 + I<mI2) (7.i5i)
at tree-level and neglecting the fermion masses. Applying this result to lep-
tonic W decays using (7.55) and (7.60), this implies
g2Mw _ aMw
48tt " 12sin2<V '
r(W+ -> e^e) = y-^ = ^™7 , (7.152)
and similarly for p^v^ and t+i/t. It is convenient to rewrite (7.152) as
T(W+ -► e+i/c) = F w « 226.20 ± 0.10 MeV (7.153)
6v27T
using g2Mw/^y/2 = GfM^. This form absorbs the dominant radiative
corrections, including the running of a. The numerical values for the partial
widths include the small residual electroweak corrections and fermion mass
effect. Similarly, the hadronic partial widths are
T(W+ - Uidj) = F- w \Vij\2 « (705.97 ±0.31) |V^|2 MeV, (7.154)
6V27T
where Vlj is the CKM matrix element, and
C = 3 (1 + ^Mwl + L4Mo| _ 12 7?o|\ (7155)
includes the color counting factor and the three-loop QCD vertex correction
for massless quarks. Again, there are small QED and electroweak corrections

340
The Standard Model and Beyond
and one can include fermion mass effects. Adding the contributions, one
predicts IV ~ 2.0902(11) GeV, in agreement with the experimental world
average IV = 2.141(41) GeV. Early measurements of IV from the Tevatron
and the SppS (CERN) pp colliders were indirect, relying on the measured
ratio
o~(pp —> W —> ivi)
a(pp^ Z-> el)
-'^r^mc^B^raj^ ^
theory LEP
where the ratio of production cross sections and the leptonic width were taken
from theory and the Z —> ^branching ratio from LEP. Recent determinations
from CDF (Aaltonen et al., 2008) (with the most precise value), DO, and the
LEP experiments are more direct and based on kinematic decay distributions.
7.3.5 The Z-Pole and Above
As can be seen in Figure 7.21 the cross section for e+e_ annihilation is greatly
enhanced near the Z-pole. This allowed high statistics studies of the
properties of the Z at LEP (CERN) and SLC (SLAC) in e~e+ -► Z -► £'£+, qq,
and vv (for reviews, see the articles by D. Schaile and by A. Blondel in Lan-
gacker, 1995; Grunewald, 1999; Schael et al., 2006a). The four experiments
ALEPH, DELPHI, L3, and OPAL at LEP collected some 1.7 x 107 events
at or near the Z-pole during the period 1989-1995. The SLD collaboration
at the SLC observed some 6 x 105 events during 1992-1998, with the lower
statistics compensated by a highly polarized e~ beam with Pe- > 75%.
The basic Z-pole observables relevant to the precision program are:
• The lineshape variables Mz, Tz, and o-peak (Appendix F).
• The branching ratios for Z to decay into e_e+, /x_/x+, or t~t+; into qq,
cc, or bb\ or into invisible channels such as vv (allowing a determination
of the number Nv = 2.985 ± 0.009 of neutrinos lighter than Mz/2).
• Various asymmetries, including forward-backward (FB), hadronic FB
charge, polarization (LR), mixed FB-LR, and the polarization of
produced r's.
The branching ratios and FB asymmetries could be measured separately for
e, //, and r, allowing tests of lepton family universality.
LEP and SLC simultaneously carried out other programs, most notably
studies and tests of QCD, and heavy quark physics.
The Z Lineshape
One of the most important observables is the Z lineshape, i.e., the cross
section for e"e+ —> //, where / = e,/x, r, 6, c, or hadrons, as a function

The Standard Electroweak Theory
341
~io:
mr
10:
10J
10
*i ' ' 1
(\
1 ~~SR
E* ORIS
[ !
L KFKB
| PFP-II
r ■ . »
\
PIP
PFfRA
Z i '
1
TRISTAN SLC
LEPI
■ 11.. ■ i. . i i ■
eV
. . I ■
~->hadrons 1
s. W+W J
LEPII ^
1 - . H
0 20
60 80 100 120 140 160 180 200 220
Centre-of-mass energy (GeV)
FIGURE 7.21
Cross section for e+e- —> hadrons as a function of CM energy. At low energies
the cross section is dominated by 1 photon exchange in the s channel, while
at high energies the Z dominates. The approximate energy ranges of various
e+e- colliders are also shown. Reprinted with permission from (Schael et al.,
2006a).
s = EcM near the Z-pole. The expected Breit-Wigner form* (slightly more
sophisticated than the one in (F.2)), is
af(s)~ar
sT\
(s
MlY
+
£fr|
(7.157)
where Of is the peak cross section and Tz is the total Z width. By measuring
the cross section at a number of energies near the peak one can determine
Mz, Tz, and <7/, as illustrated in Figure F.l. For example, the cross section
o~had for e~e+ —> hadrons is shown in Figure 7.21. The results for the Z are
generally expressed in a model-independent way in terms of the partial widths
T(//) for Z —> // and in terms of effective couplings to be defined below.
From Problem 2.20 the expression for the peak cross section is
af =
12tt r(e-e+)r(//)
Ml
r2
1 z
(7.158)
* There are non-negligible corrections from reduced QED diagrams, including initial state
photon radiation and s-channel photon exchange, as well as 7 — Z interference. These are
taken into account, usually assuming the SM expressions, in determining the Z parameters.

342
The Standard Model and Beyond
Using (7.72) and (7.151), the width for Z —> // at tree-level is
(' /2 , f2\ _ CfaMz ( /2 /2\
r(//) " 48^ l^ +^ ) ~ 12sin20wcos20w l^ +^ J '
(7.159)
where C/ = 1 (leptons) and 3 (quarks) is the color factor, Qv A are defined
in (7.75), and we have neglected the fermion masses. Including radiative
corrections, T(ff) becomes
where for quarks Cf now includes QCD corrections similar to (7.155)
evaluated at M§, a is evaluated at M\, and gv A are effective vertex factors
9a = >/p/*/l, 9v = y/h (*/l ~ 2*fQfS2z) , (7.161)
which are the same as the tree-level expressions in (7.76) except for the elec-
troweak radiative correction factors pj and kj. Note the similarity of this
parametrization to the one in (7.147) for WNC amplitudes. The second form
in (7.160) is obtained using (7.140), and contains the quadratic mt
dependence of (7.142) as well as the largest Mh dependence in the universal factor
p. Many types of corrections enter into pf and £/, but the overall corrections
are small and only weakly sensitive to mt and Mh- (The pf are analogous ,to
the piad of (7.148).) For example, one expects pt ~ 0.9981 and hi ~ 1.0013.
The one exception is / = b because of the t contributions to the vertex in
Figure 7.19. For the known mt one has the rather large corrections pb ~ 0.9874,
k\, ~ 1.0065. In practice, (7.160) must be extended to include fermion mass
effects (including those within the radiative corrections), two-loop QED
corrections, two-loop mixed QED-QCD corrections, etc. The partial width can
also be written in the on-shell scheme as
with
9a = VPft3fL, 9Sv = sfp] {t)L - 2Kfqfs2w). (7.163)
We have used the same notation for gv A in both schemes to avoid a
proliferation of symbols, but they differ in detail. In particular, there is no analog
of p in (7.162), with the effects now absorbed into the pf. The dominant mt
dependence is pf ~ 1 + pu Kf ~ 1 + pt/tw (°f (7.149)).
It is convenient to define an effective angle si
s2f = KfS2v = kfs2z. (7.164)

The Standard Electroweak Theory
343
h^b is insensitive to mt and MH, so one has sj ~ s\ + 0.00029. The effective
couplings can then be written as in (7.138), i.e., they take the form of tree-
level expressions up to an overall factor of y/pj, where pf depends on the
renormalization scheme. The Z-pole analyses often extract the gL A from the
data in a model-independent way.
Using (7.160) or (7.162) and additional corrections, the SM predictions are
{300.10 ± 0.09 MeV (uu), 167.18 ± 0.02 MeV (yv)
382.89 ±0.08 MeV (dd), 83.97 ±0.03 MeV (e+e~) (7.165)
376.01 T 0.05 MeV (66)
for rnt = 170.9(1.9) GeV, MH = 117 GeV, and as = 0.1200(17). The small
partial width into e+e~ is due to the small value of (jy, which would vanish for
s\ = 1/4. The total width is predicted to be Tz ~ 2.4952(9) GeV, compared
with the experimental value 2.4952(23).
The results of the four LEP experiments have been carefully combined by
the LEP Electroweak Working Group (LEPEWWG) (Schael et al., 2006a,
LEPEWWG website), which took into account common systematics,
corrected the data for non-Z effects, applied radiative corrections, and carried
out SM and model-independent analyses of the Z-pole and other data, in
collaboration with SLC, Tevatron and other LEP working groups. For the Z
lineshape, results were typically presented in terms of a conventional set of
observables: M^, Tz, Chad, Re, Rb, and Rc, where
_ 12tt r(e+e-)T(Z^ hadrons)
ohad = -z f| (7.166)
is the peak cross section into hadrons, and
^-fUS-"-** R'-s¥m- ,,=(e'"'r) <7167)
are ratios of partial widths to the total width into hadrons, in which many of
the radiative corrections and uncertainties cancel. This set had the benefit of
being weakly correlated, though the precision still required careful treatment
of the full error correlation matrices.
The principal Z-pole observations from LEP and SLC are listed in Table 7.5,
along with the standard model expectations using the global best fit values for
the input parameters. The precision of the Mz determination, to ~0.0023%,
is unprecedented in this high energy regime. The energy calibration was done
using a resonant depolarization technique, and corrections had to be made for
the tidal effects of the Sun and Moon, the water table and the water level in
Lake Geneva, and leakage currents produced by nearby trains! Many of the
other quantities are measured to the 0.1 — 1% level, and the agreement with
the predicted values from the fit is generally quite good, though there are one
or two exceptions.

344
The Standard Model and Beyond
TABLE 7.5
Principal Z-pole observables, their experimental values, theoretical
predictions using the SM parameters from the global best fit with Mh free
(yielding Mh = 77+22 GeV), Pu^ (difference from the prediction divided
by the uncertainty), and Dev. (difference for fit with Mh fixed at 117
GeV, just above the direct search limit of 114.4 GeV), as of 11/07,
from (Electroweak review in Amsler et al., 2008; Erler and Langacker,
2008). T(had), T(inv), and r(£+£~) are not independent.
Quantity
Mz [GeV]
Tz [GeV]
T(had) [GeV]
T(inv) [MeV]
r(£+£-) [MeV]
crhad [nb]
Re
Rp
Rt
Rb
Re
A0,e
^FB
^FB
j0,r
^FB
A0,b
^FB
j0,c
^FB
A0,s
^FB
s*(A%) (LEP)
sj(A°/B) (CDF)
Ae (hadronic)
(leptonic)
(Pr)
A,
AT (SLD)
(Pr)
Ab
Ac
As
Value
91.1876 ±0.0021
2.4952 ± 0.0023
1.7444 ±0.0020
499.0 ±1.5
83.984 ± 0.086
41.541 ±0.037
20.804 ± 0.050
20.785 ±0.033
20.764 ±0.045
0.21629 ± 0.00066
0.1721 ±0.0030
0.0145 ±0.0025
0.0169 ±0.0013
0.0188 ±0.0017
0.0992 ± 0.0016
0.0707 ± 0.0035
0.0976 ±0.0114
0.2324 ±0.0012
0.2238 ± 0.0050
0.15138 ±0.00216
0.1544 ±0.0060
0.1498 ±0.0049
0.142 ±0.015
0.136 ±0.015
0.1439 ±0.0043
0.923 ± 0.020
0.670 ± 0.027
0.895 ± 0.091
Standard Model
91.1874 ±0.0021
2.4968 ± 0.0010
1.7434 ±0.0010
501.59 ±0.08
83.988 ±0.016
41.466 ±0.009
20.758 ±0.011
20.758 ±0.011
20.803 ±0.011
0.21584 ±0.00006
0.17228 ±0.00004
0.01627 ± 0.00023
0.1033 ±0.0007
0.0738 ± 0.0006
0.1034 ±0.0007
0.23149 ±0.00013
0.1473 ±0.0011
0.9348 ± 0.0001
0.6679 ± 0.0005
0.9357 ±0.0001
Pull
0.1
-0.7
—
—
—
2.0
0.9
0.8
-0.9
0.7
-0.1
-0.7
0.5
1.5
-2.5
-0.9
-0.5
0.8
-1.5
1.9
1.2
0.5
-0.4
-0.8
-0.8
-0.6
0.1
-0.4
Dev.
-0.1
-0.5
—
—
—
2.0
1.0
0.9
-0.8
0.7
-0.1
-0.6
0.7!
1.6
-2.0
-0.7
-0.4
0.6
-1.6
2.4
1.4
0.7
-0.3
-0.7
-0.5
-0.6
0.1
-0.4

The Standard Electroweak Theory
345
The individual lepton channels could be measured separately, allowing the
lepton universality prediction Re = R^ ~ RT (up to a small r mass
correction) to be tested. The data are in agreement with universality within the
uncertainties, as can be seen in Figure 7.22. The three can also be combined
to form an average lepton ratio Re. The LEP values of Re are shown in Figure
7.23 and compared with the predictions of the SM as a function of the Higgs
mass Mh , obtained using the known values for Mz, mt, and as.
0.022
0.018 H
*<?
0.014 H
0.01
20.6
20.7
20.8
20.9
RV=rhac/r..
FIGURE 7.22
Measurements of Re. vs A^fe for ti = e,/i, and r, as well as the
combined results and the standard model predictions. Reprinted with permission
from (Schael et al., 2006a).
Other quantities can be derived from the conventional set, such as the
hadronic and leptonic widths, T(had) and T(££). Especially important is the
invisible Z width determined in principle by subtracting the hadronic and
charged lepton widths from the total (lineshape) one,
T(inv) = TZ- T(had) - $^r(£A).
(7.168)
In the SM T(inv) is due to the (unobserved) neutrinos. It can be predicted
from the other observables provided one knows the number Nv of neutrino
flavors, implying that the lineshape observables are not all independent for a

346
The Standard Model and Beyond
given Nv. The hadronic cross section is shown as a function of y/s in Figure
7.24 compared with the SM prediction for 2, 3, and 4 flavors of light neutrinos.
It is seen that the data are only consistent with Nu = 3. One can quantify
this by allowing Nv = 3 + AA^ to be a free parameter, with
T(inv) = (3 + AW„)r(i/i/), (7.169)
where T(vv) is the theoretical width in (7.165) for a single neutrino flavor.
One obtains Nv = 2.985 ±0.009 or AN„ = -0.015(9)+. Thus, a fourth family
with a light neutrino is clearly excluded. (The exclusion is valid for vnv
almost as large as Mz/2. However, it does not apply to right-handed (sterile)
neutrinos, which do not couple directly to the Z.) The result for T(inv)
is also important for constraining other types of possible extensions to the
SM, because it would receive contributions from any decays into unobserved
particles. This is conveniently parametrized by (7.169), even when AN„ has
nothing to do with extra neutrinos. For example, AN„ would be \, for a single
flavor of light scalar neutrino (Z —> viv\) in supersymmetry, while ANV = 2
would be expected in triplet Majoron neutrino mass models with spontaneous
L violation, from the decay of the Z into a pseudo-Goldstone boson (Majoron)
and a light scalar (Section 7.7).
Z-Pole Asymmetries
Most of the Z-pole asymmetries* can be expressed in terms of the quantities
9v +9a
where gv A are the effective couplings defined in (7.138). The asymmetries
are easily calculated (ignoring the fermion masses) using (7.125) and (7.126)
and the effective coupling
^ = -^E/V(^-^75)/^- (7-171)
For example, the forward-backward asymmetry A/B for e_e+ —> // at the
Z-pole, defined by (7.123) just as in the low energy case, is
A% = \AeA}. (7.172)
tThis value was obtained from the global fit including all correlations. A nearly
identical result is obtained using the definition used by the LEP collaborations, Nu —
Y(ll\ ( t(u9) ) ' w^ere T(^) is measured and the last factor is the theoretical ratio
in the SM.
■I-The Born asymmetries A0 considered here are the idealized asymmetries obtained after
subtracting reduced QED and off-pole effects from the data, and dividing by the beam
polarization Pe- for the SLC results.

The Standard Electroweak Theory
347
Ratio of Hadronic to Leptonic Width
Experiment
ALEPH
DELPHI
L3
OPAL
LEP
common error
B10
X
2
10 4
20.65
20.729 * 0.039
20.730 t. 0.060
- 20.809*0.060
- 20.822*0.044
X2/dof = 3.5/3
20.767 ± 0.025
0.007
20.
= 0.118*0.003
linearly added to
■
"M, = 172.7*2.9 GeV
20.85
Forward-Backward Pole Asymmetry
Experiment
ALEPH
DELPHI
L3
OPAL -
LEP
common error
10 3-
^10'
10
■f—
0.017
0.0173 ±0.0016
0 0187 ±0.0019
■0.019210.0024
0.0145 ±0.0017
X2/dof = 3.9/3
0.017110.0010
0.0003
0.0275810.00035
linear! added to
;■
|M<=178.0±4.3GeV
0.021
FIGURE 7.23
R£ and A^B from the four LEP experiments, compared with the SM
predictions as a function of MH using the observed M^. Reprinted with permission
from (Schael et al., 2006a, with LP05/EPS05 updates).
30
*20
10
ALEPH
DELPHI
L3
V OPAL
L ♦ average measurements, II
I error bars increased III
l by factor 10 ff/
i 1 i i i 1 1 1 1 L-
2v
/3v\ 1
A
_i—i—i—i—i—. i i
86
Ecm[GeV]
92
94
FIGURE 7.24
Lineshape for Z —► hadrons, compared with the expectation for 2, 3, and 4
ordinary neutrinos lighter than ~ Mz/2. The asymmetry is due to the explicit
factor of s in the numerator of the Breit-Wigner formula (cf. Eq. F.7) and to
radiative corrections. Reprinted with permission from (Schael et al., 2006a).

348
The Standard Model and Beyond
The LEP experiments measured AjB for / = e, //, r, 6, c, and (less precisely)
s. These and other asymmetries are listed in Table 7.5. The individual
measurements of A°pB, A£B, and A°jTB, shown in Figure 7.22, are in reasonable
agreement with lepton universality, allowing them to be combined into an
average A°pB. The LEP experiments also measured a forward backward
asymmetry between positive and negative charge in hadronic Z events. This is
more complicated to interpret because it depends on the fragmentation of the
final quarks, but nevertheless leads to a measurements of the effective leptonic
angle that is denoted s\{ApqB) in Table 7.5. The CDF collaboration at the
Tevatron also measured the forward-backward e~ asymmetry in the inverse
process pp —> e_e+, obtaining the angle s\{ApeB) (Acosta et al., 2005).
As discussed following (6.53), the polarization of a r in e_e+ —> t~t+ can
be measured from the angular distribution of the r decay products in the r
rest frame. The r~ polarization, averaged over the CM angle 6 of the r~, is
0 _ a(hT- = +i) - a(hT- = -\)
a(hT +2 J +<J{tiT -2)
which allows the extraction of AT. Similarly, the polarization of r~ produced
at a definite angle 0 is
where z = cosO, allowing Ae to be determined from the angular distribution.
The SLC had a longitudinally polarized e~ beam, with a reversible
polarization of > 75%. This allowed them to measure the polarization asymmetry
A° -*<*«- =-s)-*(fte-=+£) _ ,4 /7mx
<*\K- = ~V +^"(^e- = + 5)
which projects out the electron couplings. The measurements were done
separately for hadronic and leptonic final states. They also measured mixed
forward-backward polarization asymmetries for various final states,
A0FB = 4F-4B-°fRF+°fRB = 3^ (? ^
<F + °LB + aRF + °RB 4
in an obvious notation, allowing them to determine the final fermion couplings.
Af involves the ratios of effective couplings, so pf cancels. Aj and the
asymmetries are therefore insensitive to mt and Mh for a fixed s\ or s%, allowing
the weak angle to be extracted with little ambiguity. (Similar statements
apply to Mw and to Rc and Re in (7.167), although some mt dependence enters
the ratios through the Zbb vertices.) However, Mz and the other lineshape
variables do depend on them through p, so the m* and Mh constraints come
mainly from comparing Mz with the asymmetries and other observables, or,

The Standard Electroweak Theory
349
H 0 23099 ± 0.00053
F 0.23159 2:0.00041
0 2o098 000026
-*- 0.23221 ± 0.00029
H * 0.23220 ± 0.00081
~J * 0 2324*00012
•4 0.23153 ±0.00016
J x*Mo.l. 11.8/5
iry*1727±29GeV
_ , , , ,___, , ____,
023 0232 0234
s.n2eeff
FIGURE 7.25
s| from various leptonic and hadronic asymmetries, compared with the SM
predictions as a function of Mh using the observed Mz. Reprinted with
permission from (Schael et al., 2006a).
equivalently, by expressing the other quantities in terms of Mz rather than
sf. As an example, the observed ApB as well as Re are compared to the SM
expectation as a function of Mh in Figure 7.23, where the SM predictions
utilize the observed Mz. These observables favor relatively low values for
Mh , but with large uncertainty due to the logarithmic dependence.
There is some tension between the leptonic and hadronic asymmetries, as
can be seen in Figure 7.25. The most precise determinations of sf are from
the polarization asymmetry Alr and from the b FB asymmetry APpB. Alr
is proportional to gy ~ — | + 2sf, and is therefore extremely sensitive to sf.
On the other hand, the leptonic FB asymmetry APpB is proportional to (jy£
and is therefore less sensitive since gy is small. A°pB = ^AeAt does not have
this difficulty, and is much more sensitive to the e vertex than the b vertex
assuming the SM. As can be seen from Table 7.5 and Figure 7.25, there is a
possible discrepancy of A^B from the value predicted from the best fit, and
the corresponding s| is about 2<j higher than the average. Similarly, the sf
from Alh is nearly 2a lower. This could well be due to fluctuations, but the
other (less precise) hadronic and leptonic asymmetries tend to follow the same
pattern. Also, the hadronic asymmetries favor a larger Higgs mass, while the
leptonic ones (and Mw) favor a small Mh, so the average value discussed
below relies on a delicate balance. It is possible that there is some kind of
new physics which affects these measurements. For example, there could be
a tree-level effect that couples preferentially to the third family and accounts
for the A°pB value (it would be hard for a new physics effect that only enters
Average
>
CD

350
The Standard Model and Beyond
at loop level to lead to the needed 4% shift). Possibilities would include a
flavor off-diagonal coupling of a heavy Zf gauge boson or the mixing of the 6
with a heavy exotic fermion. However, there would have to be a compensation
between the L and R couplings to avoid a large change in R^.
Precision Tests at High Energy
The second phase of LEP, LEP 2, ran at CERN from 1996-2000, with energies
gradually increasing from ~ 140 to ~ 209 GeV (Alcaraz et al., 2006). The
principal electroweak results were precise measurements of the W mass, as
well as its width and branching ratios; a measurement of e+e_ —> W+VK",
ZZ, and single W, as a function of center of mass (CM) energy, which tests
the cancellations between diagrams that is characteristic of a renormalizable
gauge field theory, or, equivalently, probes the triple gauge vertices; limits
on anomalous quartic gauge vertices; measurements of various cross sections
and asymmetries for e+e_ —> // for / = e_,/x_,r_,g, b and c in reasonable
agreement with SM predictions; and a stringent lower limit of 114.4 GeV on
the Higgs mass, and even hints of an observation at ~ 116 GeV. LEP 2 also
studied heavy quark properties, tested QCD, and searched for supersymmetric
and other exotic particles.
The Tevatron pp collider at Fermilab has run from ~1987, with a CM
energy of nearly 2 TeV. The CDF and DO collaborations there discovered
the top quark in 1995, with a mass consistent with the predictions from the
precision electroweak and B/K physics observations; have measured the t
mass, the W mass and decay properties, and leptonic asymmetries; carried
out Higgs searches; observed Bs — Bs mixing and other aspects of B physics;
carried out extensive QCD tests; and searched for anomalous triple gauge
couplings, heavy W and Z' gauge bosons, exotic fermions, supersymmetry,
and other types of new physics (Acosta et al., 2005; Amsler et al., 2008).
The HERA e±p collider at DESY observed W propagator and Z exchange
effects, probed electroweak couplings, searched for leptoquark and other exotic
interactions, and carried out a major program of QCD tests and structure
functions studies (e.g., Aktas et al., 2006; Zhang, 2009).
7.3.6 Implications of the Precision Program
The principal Z-pole observables are listed in Table 7.5, and a number of
important non-Z-pole observables in Table 7.6. The direct measurement of
mt is important for controlling the radiative corrections in the analysis more
precisely than could be done from the indirect constraints alone, and in
allowing sensitivity to the Higgs mass. Mw provides information on radiative
corrections independent of the Z lineshape and asymmetries. The low energy
WNC measurements play a critical role in constraining many types of new
physics even though they are less precise than the Z-pole observations,
because the latter are essentially blind to any new physics that doesn't directly

The Standard Electroweak Theory
351
affect the Z or its couplings to fermions (such as heavy particle exchanges,
new box-diagrams, or four-fermi operators).
The Standard Model Fit
As of November, 2007, the result of the global fit to the precision data yielded^
Mh = 77-ll GeV> mt = 171.1 ± 1.9 GeV (7.177)
as = 0.1217(17), a(Mf)-1 = 127.909(19), Aa^ = 0.02799(14)
s2z = 0.23119(14), sj = 0.23149(13), s^ = 0.22308(30),
with a good overall x2/df of 49.4/42. The three values of the weak angle s2
refer respectively to the MS, effective Z-lepton vertex, and on-shell values,
defined after Eq. 7.136. The latter has a larger uncertainty because of a
stronger dependence on the top mass.
Aaihad is defined after (7.135).
TABLE 7.6
Principal non-Z-pole observables, as of 11/07. mt is from the direct CDF
and DO measurements at the Tevatron; Mw is determined mainly by
CDF, DO, and the LEP 2 collaborations; g\, corrected for the s — s
asymmetry, and g\ are from NuTeV; gye are dominated by the CHARM II
experiment at CERN; Apy is from the SLAC polarized M0ller asymmetry;
and the Qw are from atomic parity violation. Other non-Z-pole
observables are included in the fits but not listed. See Table 7.5 for details
concerning the fits.
Quantity
mt [GeV]
Mw (pp)
Mw (LEP)
A
&
9ve
9Ae
Apy X 107
Qw(Cs)
Qw(Tl)
Value
170.9 ±1.8 ±0.6
80.428 ± 0.039
80.376 ± 0.033
0.3010 ±0.0015
0.0308 ±0.0011
-0.040 ± 0.015
-0.507 ±0.014
-1.31 ±0.17
-72.62 ± 0.46
-116.4 ±3.6
Standard Model
171.1 ±1.9
80.375 ± 0.015
0.30386 ± 0.00018
0.03001 ± 0.00003
-0.0397 ± 0.0003
-0.5064 ± 0.0001
-1.54 ±0.02
-73.16 ±0.03
-116.76 ±0.04
Pull
-0.1
1.4
0.0
-1.9
0.7
0.0
0.0
1.3
1.2
0.1
Dev.
-0.8
1.7
0.5
-1.8
0.7
0.0
0.0
1.2
1.2
0.1
§The fit results are from the Particle Data Group (Amsler et al., 2008), which uses the fully
MS program GAPP for the radiative corrections (Erler, 1999). The results are generally
in good agreement with those of the LEPEWWG (Schael et al., 2006a, LEPEWWG
website), which uses the on-shell renormalization program ZFITTER (Arbuzov et al., 2006).
However, the PDG fits use a more complete set of low energy data.

352
The Standard Model and Beyond
The precision data alone yield^ mt = 174.7+7°g° GeV from loop corrections
in impressive agreement with the direct Tevatron value 170.9 ±1.9 GeV. The
fit actually uses the MS mass mt{rrit), which is ~ 10 GeV lower, and is
converted to the pole mass at the end.
The Z-pole data yield as = 0.1198(20) for the strong coupling. This
determination has negligible theoretical uncertainty if one assumes the exact
validity of the standard model, and is also insensitive to oblique (propagator)
new physics. However, it is very sensitive to non-universal new physics, such
as those which affect the Zbb vertex. The Z-pole value is in good agreement
with the world average of 0.1176(20), which includes other determinations at
lower scales, most of which have their own theoretical uncertainties (Amsler
et al., 2008; Bethke, 2007)".
The prediction for the Higgs mass from indirect data, Mh= 77^2 GeV,
should be compared with the direct limit MH > 114.4(95%) GeV obtained
at LEP 2 (Barate et al., 2003). There is no direct conflict given the large
uncertainty in the prediction, but the central value is in the excluded region,
as can be seen in Figure 7.26. Including the direct LEP 2 exclusion results
one finds Mh < 167 GeV at 95%. MH enters the expressions for the radiative
corrections logarithmically. It is fairly robust to many types of new physics,
with some exceptions. In particular, a much larger Mh would be allowed
for negative values for S or positive values for T, where S and T are the
oblique parameters (Peskin and Takeuchi, 1990, 1992) to be discussed below.
The predicted value would decrease if new physics accounted for the value of
Af$ (Chanowitz, 2002).
As will be discussed in Section 7.5 there is a theoretical range 115 GeV <
Mh ^180 GeV in the SM provided it is valid up to the Planck scale, with
a much wider allowed range, 85 GeV < Mh < 700 GeV, otherwise. The
experimental constraints on Mh are encouraging for supersymmetric
extensions of the SM, which favor low Higgs masses because A in (7.41) is not
an independent parameter (it is related to the electroweak gauge couplings)
^ Prior to LEP and the SLC the WNC current experiments and early W and Z masses could
set an upper limit on mt of O(200) GeV (see, e.g., Amaldi et al., 1987). The more precise
Z-pole experiments led to a prediction for mt that was correlated with Mf/, which was
consistent (within the rather large uncertainties) with the value later observed directly by
CDF and DO. For example, the first measurement of Mz by MARK II at the SLC already
implied mt = 1401^2 GeV f°r ^H = 100 GeV, with the central value changing to 128
(165) GeV for MH = 10 (1000) GeV (Langacker, 1989a). By the late 1980s observations
of B — B oscillations and improved analyses of CP violation in K decays made possible by
measurements of the relevant CKM matrix elements allowed independent lower limits on
mt to be set, e.g., mt > 130 GeV (Buras, 1993).
"The higher value in (7.177) is due to the inclusion of data from hadronic r decays, which
is less sensitive to new physics but is dominated by theoretical uncertainties. A recent
revaluation of the theoretical formula (Maltman and Yavin, 2008; Baikov et al., 2008;
Beneke and Jamin, 2008) lowers the r decay value to 0.1187(16), consistent with the other
determinations, but this has not yet been incorporated in the global fit.

The Standard Electroweak Theory 353
and because the direct LEP 2 lower limit is weakened in some regions of the
supersymmetry parameter space.
1000
500
200
50
20
1° ...
140 150 160 170 180 190
mt [GeV]
FIGURE 7.26
1<t allowed regions in M// vs mt and the 90% cl global fit region from precision
data, compared with the direct exclusion limits from LEP 2. Plot courtesy of
the Particle Data Group (Amsler et ah, 2008).
Beyond the Standard Model
New physics modifies the SM predictions for the precision electroweak observ-
ables in several ways. For example, new heavy W or Z' gauge bosons (Section
8.3); exotic Higgs particles (Accomando et al., 2006); leptoquark bosons
(Cheung, 2001; Chemtob, 2005; Barbier et al., 2005; Alcaraz et al., 2006), such as
occur in supersymmetric models with imparity violation; or supersymmetric
particles in box diagrams (Section 8.2) can lead to new four-fermi operators,
which are especially important below or above the Z-pole (Cho et al., 1998;
Cheung, 2001; Han and Skiba, 2005; Alcaraz et al., 2006). The most
important effects for the Z-pole experiments are: (a) new physics which affects the
Zff vertices, and (b) new physics which modifies the masses and
propagators of the W and Z. The first class includes such tree-level effects as the
mixing of the ordinary with heavy exotic fermions (Langacker and London,
1988b) and Z-Z' mixing, as well as actual vertex corrections from new gauge
bosons, leptoquarks, superpartners, etc. Their implications are model
dependent and hard to parametrize in general, since they affect each type of chiral
—i—i—i ■ ■ ■
„-r
r„a-r.r
Z had' J q
asymmetries
low-energy
exclu

354
The Standard Model and Beyond
fermion in a different way. For this reason the precision constraints are usually
studied for specific models or types of models. The second class, the oblique
corrections, are described below.
The Oblique Parameters
The oblique corrrections are universal in that they are independent of the
fermions in the process, and can be described by a small number of
parameters. They are especially important for describing new particles that only
or mainly affect the precision observables through vacuum polarization loops,
such as new fermion or scalar multiplets with little or no mixing with or direct
coupling to the ordinary particles. They can also be used to parametrize the
effects of the t quark and Higgs on the radiative corrections (although for the
top one must also include the Zbb vertex diagrams in Figure 7.19). However,
we will follow (Amsler et al., 2008) and focus on the new physics contributions
to the oblique parameters, treating the mt and Mh effects separately.
The po parameter
»-$% <7178>
extends the MS parameter p in (7.140). The SM expression for Mw in (7.140)
is unmodified, po is associated with new physics which breaks the SU(2)
vector generators and therefore violates the custodial symmetry relation between
Mw and Mz (and therefore between the WNC and WCC amplitudes) in
(7.139). We assume that the new physics which leads to po 7^ 1 is a small
perturbation which does not significantly affect other radiative corrections**.
Then po can be viewed as an extra parameter that also multiplies Gf in the
expressions for the effective WNC interaction in (7.79), the quantity xo m
(7.131), and the partial Z-widths in (7.162).
po 7^ 1 can be generated by non-degenerate multiplets of heavy particles,
analogous to the effect of the non-degenerate t and b quarks in (7.142). Non-
degenerate SU(2) doublets of scalars and/or fermions with masses rani, ran2,
such as fourth family (sequential), mirror (involving right-chiral doublets and
left-chiral singlets), or vector pairs of quarks and leptons, heavy Higgs doublets
not involved in symmetry breaking, or unmixed squark or slepton doublets in
supersymmetry, yield (Veltman, 1977; Chanowitz et al., 1978)
n
where
Ami = m2nl + m2n2 - Af^m\ in !M > (TOnl - mn2)2, (7.180)
**One must still define a fourth renormalized parameter. A convenient set is a, Gf, Mw,
and Mz-

The Standard Electroweak Theory
355
with Cn = 1(3) for color singlets (triplets). There is enough data to determine
p0 and the other parameters, with the global fit yielding
po = i.ooo4i8;g8g, (7.i8i)
with little change in the other parameters except that the 95% cl upper limit
on Mh 1S relaxed to 215 GeV. The po value corresponds to
£^Am2n<(98GeV)2. (7.182)
n
Such non-degenerate multiplets lead to po > 1- However, additional Higgs
doublets which participate in SSB or heavy lepton multiplets involving Majo-
rana neutrinos can yield po < 1 (for references, see the Electroweak article in
Amsler et al., 2008). Tree-level effects, such as triplet or higher-dimensional
Higgs representations with non-zero VEVs can contribute to po with either
sign (Problem 7.1).
Heavy chiral fermions break the axial SU(2) generators as well as the
vector ones, so their effects are not adequately parametrized by po alone. These
can be taken into account by a generalization to the 5, T, and U
parameters (Peskin and Takeuchi, 1990, 1992). Related parametrizations are given
in (Kennedy and Langacker, 1990; Marciano and Rosner, 1990; Golden and
Randall, 1991; Altarelli and Barbieri, 1991). Define the vacuum polarization
functions
iU^(q) = iltriLvtf) - q^Aijiq2)] (7.183)
as the one-particle irreducible propagator corrections shown in Figure 7.27.
The second term is irrelevant for the existing precision experiments because
the g^ or qv always yield light lepton masses when contracted into the external
fermion lines. We are concerned with the new physics contributions Ti^w(q2)
to the vacuum polarizations. As long as the new physics is much heavier than
Mz it is sufficient to assume that the H??w(q2) are approximately linear from
q2 = 0 to M|. The effects on the precision observables can then be expressed
FIGURE 7.27
Vacuum polarization bubbles for the gauge boson propagators. Examples are
shown in Figure 7.17.

356
The Standard Model and Beyond
in terms of the three parameters
a(Mp ns^(MD-ngf(o)
lip|A = m| (7-184)
-&r{S+u)= mi •
Eq. 7.184 assumes an MS renormalization scheme as defined in (Marciano
and Rosner, 1990). The more general expressions are given in (Amsler et al.
2008), along with additional details and references. These quantities have a
factor of a removed, so new physics contributions are expected to be of 0(1).
The T parameter is equivalent to po5
Po = r^r~1+aT' (7-185)
while S and U are associated with the axial currents and can be generated
even by degenerate heavy chiral multiplets. The contribution of a degenerate
multiplet to S is
r
where t\L (tfR) is the T3 eigenvalue of Vvl (Vv#)- A heavy degenerate fourth
family or mirror family would contribute 2/37T to S. Similarly, in technicolor
models (Weinberg, 1979; Susskind, 1979) with scaled up QCD-like dynamics,
one expects S ~ 0.45 for an SU(2) doublet of technifermions with Ntc = 4
technicolors, and S ~ 1.62 for a full technigeneration (Peskin and Takeuchi,
1992; Golden and Randall, 1991). Most types of new physics yield U = 0.
The effects of T on the precision observables are the same as po (using
(7.185)). S and U mainly modify the SM expressions for the Z and W masses
Ml = M2Z0 ! ~ aT ^
l-GFM*Z0S/2V27r (7j8?)
M*=M* !
LW - 1V1WQ
1-GfM^0(S + U)/2V2tt'
where Mzo and Mwo are the SM expressions in the MS scheme (see Amsler
et al., 2008, for the modifications to the widths, for which wave function
renormalizations must be included). The global fit yields
S = -0.04 ± 0.09 (-0.07), T = 0.02 ± 0.09 (+0.09) (7.188)
for [7 = 0, where the central value and error are for Mh = 117 GeV, and
the number in parentheses is the shift in the central value for Mh = 300

The Standard Electroweak Theory
357
GeV. The allowed contours in the S - T plane are shown in Figure 7.28. We
emphasize that these values are for the new physics contributions only; the
mt and M# dependence is included separately. (Most of the original papers
fixed mt and MH at some reference value and included the effects of their
variation in S and T.) A heavy degenerate fourth or mirror family, which
corresponds to T = 0, is strongly excluded. This complements the invisible
Z width constraint below (7.169) on a fourth neutrino with mass < Mz/2.
A heavy fourth family with large mass splittings may be marginally allowed
due to a compensation between the effects of S > 0 and T > 0, as can be
seen in Figure 7.28 (e.g., Kribs et al., 2007; Amsler et al., 2008), though at
the expense of large Yukawa and Higgs couplings that may lead to Landau
poles at low scales. Similarly, the QCD-like technicolor models are excluded,
but versions which do not resemble QCD in their dynamics may be allowed.
Supersymmetric extensions of the SM usually give very small contributions
to S and T (or to the precision observables).
S and T are strongly correlated with the Higgs mass M#. It is clear from
Figure 7.28 that large values of M# could be compensated by new physics with
S < 0 and T > 0 (see Peskin and Wells, 2001, for some specific examples).
From (7.186) degenerate chiral multiplets yield S > 0, but negative values are
possible in other cases.
The approximation that the vacuum polarizations are linear in q2 breaks
down for new physics that is not much heavier than M^, such as a light Higgs.
The oblique parameter formalism can be extended to accomodate this by the
inclusion of more parameters (Maksymyk et al., 1994). The formalism can
also be extended to include effects relevant to LEP 2 (Barbieri et al., 2004).
The oblique parameters can also be redefined in terms of higher-dimensional
operators (Barbieri et al., 2004; Han, 2008).
Summary
The precision Z-pole, LEP 2, WNC, and Tevatron experiments have
successfully tested the SM at the 0.1% level, including electroweak loops, thus
confirming the gauge principle, SU(2) xU(l) group, representations, and the
basic structure of renormalizable field theory. The standard model parameters
sin20vv, mt, and as were precisely determined. In fact, mt was successfully
predicted from its indirect loop effects prior to the direct discovery at the
Tevatron, while the indirect value of as, mainly from the Z-lineshape, agreed
with more direct QCD determinations. The combined direct and indirect data
imply Mfj < 167 GeV at 95% c.l. This range is consistent with, but does not
prove, the expectations of the supersymmetric extension of the SM (MSSM),
which predicts a light SM-like Higgs for much of its parameter space. The
agreement of the data with the SM imposes a severe constraint on possible
new physics at the TeV scale, and points towards decoupling theories (such
as most versions of supersymmetry and unification), which typically lead to
0.1% effects, rather than new TeV-scale dynamics or compositeness (e.g., Lit-

358
The Standard Model and Beyond
1.00
0.75
0.50
0.25
H 0.00
-0.25
-0.501-
-0.75
rz><WRrRq
asymmetries
— - Mw
v scattering
— • Qw
E158
I I I I 1 I I I I I
| i i i i | \y i i L'i i i i [ i i r
/ //
s7
/
fc
%
_JL_L
/ '/
''\ \'/'\ I I I I I I I I I I I I I \/\ I I I
all:MH= 117 GeV
al!:MH= 340 GeV -q
all: M = 1000 GeV
■ i , , , , i . ■ ■ ■ i . ■ ■ ■
-1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25
S
FIGURE 7.28
90% c.l. contours in the S-T plane (with U = 0) for MH = 117, 340, and
1000 GeV. Also shown are the la allowed regions for individual types of
measurements for M// = 117 GeV. The S and T shown here parametrize the
effects of new physics only. Plot courtesy of the Particle Data Group (Amsler
et al., 2008).
tie Higgs, dynamical symmetry breaking or composite fermions), which
usually (but not always) imply deviations of several %, and often large flavor
changing neutral currents. Finally, the precisely measured gauge couplings
were consistent with the unification expected in the simplest form of grand
unification if the SM is extended to the MSSM (Section 8.4). Most of these
results could be made even more precise by a high intensity Z-pole option
(GigaZ) for a possible future e+e~ linear collider (Erler et al., 2000).
7.4 Gauge Self-Interactions
The predicted gauge self-interactions of the SM, listed in Table 7.1, are
essential probes of the structure and consistency of a spontaneously-broken

The Standard Electroweak Theory
359
non-abelian gauge theory. Even tiny deviations in their form or value would
destroy the delicate cancellations needed for renormalizability, and would
signal the need either for compensating new physics (e.g., from mixing with other
gauge bosons or new particles in loops), or of a more fundamental breakdown
of the gauge principle, e.g., from some forms of compositeness.
Experimental tests of the self-interactions are therefore important, although in practice
many types of new physics that could modify them would show up sooner in
the Z-pole and WNC experiments (e.g., De Rujula et al., 1992; Burgess et al.,
1994).
In the SM, the triple gauge couplings (TGC) for WWZ and WWj are
W^{p)lv{q)W-{r) : ieC^a(p,q,r)
W+(p)Zv(q)W;{r): iecot OwC^p, q,r),
where the charges and momenta flow into the vertex and
C»vcr(p, g, r) = g^v(q - p)a + g^{p - r)v + gva{r - g)M. (7.190)
These have been constrained by measuring the total cross section and
various decay distributions for e~e+ —> W~W+ at LEP 2, and by observing
pp —> W+W~, WZ, and W^ at the Tevatron. Possible anomalies in the
predicted quartic vertices in Table 7.1, and the neutral cubic vertices for ZZZ,
ZZ^, and Z77, which are absent in the SM, have also been constrained by
LEP 2 (Alcaraz et al., 2006).
The three tree-level diagrams for e_e+ —> W~W+ are shown in Figure 7.29.
The cross section from any one or two of these rises rapidly with center of mass
energy, but gauge invariance relates these three-point vertices to the couplings
of the fermions in such a way that at high energies there is a cancellation. It
is another manifestation of the cancellation in a gauge theory which brings
higher-order loop integrals under control, leading to a renormalizable theory.
It is seen in Figure 7.30 that the expected cancellations do occur.
H X X
e e+ e e+ e e+
FIGURE 7.29
Diagrams for e"e+ -> W~W+ in the SU(2) x 17(1) model.

360
The Standard Model and Beyond
~ 30
ot»-
20-
10-
0-
LEP ' I
PRELIMINARY J
-\
♦ . * • •
4
YFSWW/RacoonWW
. no ZWW vertex (Gentle) i
* only ve exchange (Gentle)
I j , , 1
160
180
200
Vs (GeV)
FIGURE 7.30
Cross section for e~e+ —> W~W+ compared with the SM expectation. Also
shown is the expectation from t channel ve exchange only, and for the ve and
7 diagrams only. Reprinted with permission from (Alcaraz et al., 2006).
More detailed information can be obtained from the angular distributions.
It is useful to parametrize the TGCs in a form more general than the SM
predictions. Neglecting terms that are irrelevant for light fermions, there
are only 7 possible Lorentz invariant form factors for the effective VW~W+
vertex, where V=^ or Z (Gaemers and Gounaris, 1979; Hagiwara et al., 1987;
Gounaris et al., 1996; Mele, 2004):
<Ff w =gvww [gYv* {w-vw^ - w+,w-")
+ kv w+w-v»v + -%- v^w+m:
m
w
ph
+ igYW+W-iW + STY")
(7.191)
2 rV v p°
2 ouvv v*- v-tl
2ra
w
PV
where F^v = dilFv — dvF^ for F = W±,Z, or 7 is the non-interacting field
strength. The form factors must be real (up to rescattering effects, which are
small for a weakly coupled theory), and it is reasonable to neglect any
momentum dependence and treat them as constants. The coefficients gvww are

The Standard Electroweak Theory
361
arbitrary normalizations, and can be set to the conventional values g1ww = e
and gzww = ecotOw- The SM expectations are then
9i = 9i =k7 = kz = 1, \1 = \z = 0, (7.192)
and the others zero as well.
Going beyond the SM, the g\, Ky, and Xy are C and P invariant, g§
violates C and P but preserves CP, and the others violate CP. The C, P,
or CP violating couplings are relatively weakly constrained at present and
are usually neglected. It is reasonable to impose U(1)q invariance, since it is
believed to be unbroken, in which case g\ = 1 and gj = g$ = 0. In fact, gj,
k7, and A7 can be interpreted in terms of the electromagnetic multipoles of
qw = e9i> V>w = e(l + K1 + X1)/2Mw, Qw = -e(«7-A7)/M^, (7.193)
which are respectively the W+ charge, magnetic moment, and quadrupole
moment, predicted in the SM to assume the values e, e/Mw, and —e/M^.
If one makes the stronger assumption of a global (custodial) SU(2) x U(l)
symmetry in the limit g' = 0, one obtains two more constraints
kz = 9i ~ tan^vv(^7 - 1), Az = A7. (7.194)
Making all of these assumptions, one is left with three real parameters, gf,
k7, and A7, which are used in most analyses. (In fact, the data is usually
analyzed allowing only one to vary from the SM value at a time.) The current
values are
g?-l = -0.016(21), k7 - 1 = -0.027(45), A7 = -0.028(21), (7.195)
in agreement with the SM expectations of zero.
W Helicity Measurements
The polarizations of produced W±, which can be measured by the angular
distributions of their decay products in the W rest frame, probe the couplings
of the W in the production process. In particular, the fraction of
longitudinally polarized W's may differ from the SM prediction in alternative models
to the Higgs mechanism for spontaneous symmetry breaking (e.g., Chen et al.,
2005). Polarization studies have been carried out at LEP 2 using e_e+ —>
W~W+ (Mele, 2004), and at the Tevatron using t -> Wb. In t -> bW+ one
expects the fraction Po of longitudinally polarized W+ to approach unity for
mt ^> Mw, due to the large numerical value of the components of the
longitudinal polarization vector e(pw,3) ~ pw/M\y for Ew ^> Mw in Eq. 2.123,
while for the actual masses one expects P0 = m^/(m^ + 2M{^) ~ 0.70.
(See (Kane et al., 1992) and Problem 7.10.) Similarly, the fraction P+ of
helicity +1 W+ should vanish for small mj, due to the V — A coupling but
could be non-vanishing for an admixture of V + A. A recent CDF
observation (Chwalek, 2007) obtained P0 = 0.59(14) and P+ < 0.10 at 95% c.l., in
agreement with the SM expectations.

362
The Standard Model and Beyond
7.5 The Higgs
The spontaneous symmetry breaking sector is the most uncertain part of the
standard model. The simplest possibility is that the SSB is accomplished by
the VEV of an elementary Higgs doublet, as described in Section 7.2.1, or by
a more complicated Higgs sector involving two or more Higgs doublets (as in
the supersymmetric extension of the SM) or other representations. It is also
possible that the SSB is associated with a dynamical mechanism not involving
elementary scalar fields, as will be touched on in Chapter 8. In this section
we will describe some of the theoretical and experimental constraints on the
SM Higgs and prospects for its detections. For more detailed discussions,
see (Gunion et al., 1990; Carena and Haber, 2003; Reina, 2005; Aglietti et al.,
2006; Gomez-Bock et al., 2007); the Higgs Boson review in (Amsler et al.,
2008); and the sites devoted to Higgs production, decay, and experimental
constraints listed in the Web Sites section.
In the standard model there remains one physical Higgs particle H after
spontaneous symmetry breaking, with a potential given by (7.40) and gauge,
self-interaction, and Yukawa couplings to fermions given in Table 7.1. The
mass is Mh = \[7Z\v, where A is the quartic self-coupling and v ~ 246 GeV
is v^Ol^lO). The couplings to a particle of mass M are always proportional
to Mjv, M2/v, or (M/v)2. Thus, the couplings to fermions are small except
for the top quark, making the H hard to produce or detect in that way. More
optimistic are gauge couplings, such as VVH where V = W^ or Z, which are
proportional to g2v ~ My/v.
7.5.1 Theoretical Constraints
The Higgs mass and quartic coupling are related by
Mh = 2X^\ A = ^ = ^L, (7.196)
H SM^ ^/2
where v ~ 246 GeV and Gf ~ 1.2 x 10~5GeV2 are known. The only
constraint we have imposed so far on A is (tree-level) vacuum stability, A > 0,
which would allow any Mh from 0 to oo. However, there are a number of
more stringent theoretical constraints which lead to nontrivial upper and lower
bounds. In this section we consider issues related to the Higgs and gauge
boson self-it eract ions. Other theoretical issues, involving the Higgs contribution
to the vacuum energy and the higher-order corrections to the Higgs mass, are
discussed in Section 8.1.

The Standard Electroweak Theory
363
Renormalization Group Constraints
The renormalization group equations for the running gauge couplings are given
for an arbitrary gauge theory in (5.33) in Section 5.4.1. For the SM the /?
function one-loop coefficients in (5.34) are
bgs " 16tt2
bg 16tt2
^ = +l6^
11-^
3
1
F=3,nH = l 167T2
22
y
4F
~3~
6
(-7)
1
F=3,nH = l 167T2
19
6
(7.197)
20F nH
-\ h —
9 6
1
F=3,nH = l 167T2
41
+ -
for SU(3), SU(2), and E/(l), respectively^. These are valid at momenta
much larger than mt and v, and assume the existence of F fermion families
and Tin Higgs doublets. For F = 3 and uh = 1, both 517(3) and SU(2) are
asymptotically free.
A and the Yukawa couplings also run. At one-loop (see, e.g., Cheng et al.,
1974; Gunion et al., 1990)
d\(Q2)
1
d\nQ2 32tt2
24A2 + 24A/i2 - 24/i^ - 3A (3g2 + g'2)
+5(2,4 + (,2+^2)2)]
dht{Q2) = 1 r^,^ , /. 9 9 9 17
dlnQ2 ~32tt2
(7.198)
^-^(SflJ + ^ + g^)],
where it is understood that the couplings on the r.h.s. are the running cou-
plngs at Q2. The quantity in (7.196) is the low energy value A(i/2), while
htiy2) ~ ™>t/v is the t-quark Yukawa coupling defined in (7.46) or (7.54); we
have neglected all of the other Yukawas. The running of ht is due to vertex, t
quark, and Higgs self energy diagrams (for a scalar coupling there is no analog
of the Ward-Takahashi identity described in Section 2.12.1). Typical one-loop
diagrams contributing to the running of A are shown in Figure 7.31.
The first bound that we consider is the triviality upper limit on Mh (Cabibbo
et al., 1979). From (7.196), \{v2) is larger than unity for MH > 350 GeV,
while ht(v2) ~ 0.7 for mt ~ 170 GeV and the electroweak gauge coupling
terms at low energy are small (Table 1.1). For large Mh one can therefore
approximate the A equation by the first term on the r.h.s., which is
immediately solved to obtain
W)
.2\ _
A(^2)
i-^mr
(7.199)
ttin considering grand unified theories it is conventional to introduce the GUT-normalized
U(l) gauge coupling g\ = J\g', which has the coefficient bgi = |bg> —► j^ (+75)-

364
H
y'
A
H{ )H
x
H
H
The Standard Model and Beyond
H H
O
i-^-i
/ * \
H H
H
FIGURE 7.31
Typical diagrams contributing to the running of A.
This diverges at the Landau pole
QLP = ve2*2^v2\ (7.200)
Presumably, it does not make sense for A to diverge within the domain of
validity of the theory**. It suffices that Qlp is larger than the scale A at
which new physics sets in (and above which the RGE in (7.198) no longer
apply). Requiring Qlp > A leads to the triviality limit
■I /Q |
/ 2v/2tt2 \ f 0(140) GeV, A ~ MP '
H < \3GF\n(A/is)J " \ 0(650) GeV, A ~ 1500 GeV ' (7*2U1)
where MP = G^1/2 ~ 1.2 x 1019 GeV is the Planck scale. This limit is
somewhat fuzzy, because the one-loop approximation is not valid when A is
large, and in fact perturbation theory breaks down. However, it provides
a reasonable estimate (as a function of the new physics scale) of how large
a value of Mh is consistent with having a weakly coupled field theory. In
particular, the observation of a SM-type Higgs at a mass scale much larger
than 200 GeV would strongly suggest that new physics sets in at a rather low
scale. A more detailed evaluation including two-loop effects and the neglected
terms in (7.198) is shown in Figure 7.32. One finds that MH < 180 GeV for
A rsj MP, while MH < 700 GeV for A < 2MH- (It would not make much
sense to consider an elementary Higgs field for a lower A.) The latter upper
bound can be justified by non-perturbative lattice calculations (Hasenfratz
et al., 1987; Kuti et al., 1988; Luscher and Weisz, 1989), which suggest an
absolute upper limit of 650 — 700 GeV.
**In the pure XH4/4 theory the only way for the one-loop RGE solution to remain finite
for Q2 —y oo would be to take A(i/2) = 0, justifying the term triviality.

The Standard Electroweak Theory 365
n l~ i i I i i I i jj_| I I I I I I
103 106 109 1012 1015 1018
A [GeV]
FIGURE 7.32
Theoretical limits on the SM Higgs mass as a function of the scale A at
which new physics enters. The upper limit is from the absence of a Landau
pole below A, while the lower limit is from vacuum stability. Reprinted with
permission from (Hambye and Riesselmann, 1997).
There is also a lower limit on the Higgs mass from vacuum stability (e.g.,
Cabibbo et al., 1979; Altarelli and Isidori, 1994; Casas et al, 1996; Isidori
et al, 2001). To see this, consider the small A limit of the first equation in
(7.198). If we keep only the (dominant) h\ term*, and also treat ht as a
constant, then one finds
A(<?2)~A(„2)-?|ln^ (7-202)
which goes negative at Q2 for which
*»-£-£ "I-StA <»«>
A negative A within the domain of validity of the theory would suggest an
unstable vacuum, so we have a lower limit on A(^2) and Mjj coinciding with
the terms on the r.h.s. in (7.203) for Q = A. In particular, MH > 85 GeV for
A = 1500 GeV. For large A it is not valid to neglect the running of ht. One
needs to integrate the coupled equations, include two-loop effects, and include
loop contributions to the effective potential in addition to those in X(Q ).
* Lower limits that would have been relevant for a much lighter t quark are reviewed in (Gu-
nion et al., 1990).

366
The Standard Model and Beyond
Typical results are shown as a function of A in Figure 7.32. Combining the
triviality and vacuum stability limits, Mh is constrained to the rather limited
range 130 — 180 GeV for A = Mp, with a somewhat larger range for smaller
A. The lower limit is weakened somewhat (to around 115 GeV for A = MP)
if one allows a sufficiently long-lived metastable vacuum (Casas et al., 1996-
Isidori et al., 2001).
These limits do not apply in the minimal supersymmetric extension of the
SM (MSSM), because A is not an independent parameter and there are
additional contributions to the RGE. It is interesting that there is a complementary
upper limit of ~ 130 GeV on the lightest Higgs scalar in the MSSM, which is
close to the SM vacuum stability bound for most values of A, so the
observation of a Higgs would help distinguish between models. However, the upper
bound increases to ~ 150 GeV in extensions of the MSSM.
Tree Unitarity and the Equivalence Theorem
Another theoretical upper limit is based on the (tree-level) unitarity for W+VK~
—> W+W~ scattering and related processes such as ZZ, ZH, and HH
scattering (Lee et al., 1977) (cf. the discussions of unitarity breakdown in high
energy vee —> vee and e+e~ —> W+W~ scattering in Section 6.1). The
potential difficulty involves the longitudinal polarization states for the W
or Z, which dominate in high energy processes since the polarization
vector eM(£, 3) ~ k^/Mw,z grows with k at high energy. The amplitude for
W^W^ —> W^Wl (the subscript indicates longitudinal) is
M = -i^GFMl(^ + T^) (7.204)
for s,Mfj ^> M{y, with the relevant diagrams shown in Figure 7.33. If one
takes Mh —> oo this grows linearly with s, leading to violation of 5-wave
unitarity (Appendix C) at y/s ~ (16tt\/2/Gf)1/2 > 2.4 TeV, analogous to
the problem in (6.6) for vee —> vee scattering in the Fermi theory. This
illustrates why the existence of the Higgs (or some alternative form of spontaneous
symmetry breaking) is essential to the consistency of the theory, since taking
Mh —> oo is equivalent to removing it from the theory. However, there are
problems even for finite Mh- It is straightforward to show (Lee et al., 1977;
Gunion et al., 1990) that the S-wave amplitude obtained from (7.204) grows
with Mjj for s > M^, i.e.,
GFM2H
where ao is the 5-wave projection of the amplitude T = —iM, as defined in
(C.5) and (C.7). The unitarity condition |ao| < 1 then leads to an upper limit
on Mh- This can be strengthened somewhat by considering the coupled
channel analysis and a more precise unitarity constraint, leading to the unitarity
(7.205)

The Standard Electroweak Theory
367
bound
1/2
MH < f -^^ ) ~ 700 GeV,
/W2\
\3GF J
(7.206)
which is comparable to the lattice version of the triviality bound.
W"
w~ w^
w~ w^
z,~r,H
w+
w~ w*
w~ w+
w~
FIGURE 7.33
Tree-level diagrams for W+W~ —> W+W~. The zigzag lines can represent a
Z, 7 or H.
Of course, (7.206) is based on the tree-level amplitude, and unitarity only
applies rigorously to the full amplitude. One should therefore interpret (7.206)
as the condition for a weakly coupled (perturbative) Higgs and gauge self-
interaction sector, for which higher-order corrections to the amplitude are not
expected to be important. Conversely, a violation of (7.206), or the nonobser-
vation of a Higgs below this scale, would suggest that spontaneous symmetry
breaking is associated with a strongly coupled Higgs sector or some strong
coupling alternative to the elementary Higgs mechanism. This would
presumably be manifested by enhanced WW cross sections at high energy (Chanowitz
and Gaillard, 1985) and effects such as WW (bound state) resonances.
In Section 7.2.1 we applied the Kibble transformation to the Higgs doublet
following SSB to go to the unitary gauge, in which it is manifest that the
Goldstone bosons are eaten to become the longitudinal degrees of freedom
of the W± and Z. We also saw above that amplitudes involving the W
and Z at high energy are dominated by their longitudinal components. It is
therefore not surprising that such high energy amplitudes can be calculated
more easily in terms of the original Goldstone degrees of freedom using the
equivalence theorem (Lee et al., 1977). (More rigorous discussions in more
general gauges and including higher-order effects are given in (Chanowitz and
Gaillard, 1985; Chanowitz et al., 1987).) We will illustrate this with a simple
example, using the expressions in (7.85) and (7.88) for the Higgs doublet </>
and Higgs potential V((f>) in an R$ gauge, in which H is the physical Higgs
field, and w± and z are the Goldstone degrees of freedom that disappear in
the unitary gauge. (The gauge and Yukawa interactions for </> are also given

368
The Standard Model and Beyond
in Section 7.2.4.) The equivalence theorem states that amplitudes involving-
high energy longitudinal Z's and W's can be obtained by the much simpler
calculation of the corresponding amplitudes involving z and w±. As a simple
example, the w+w~ —> w+w~ amplitude is given at tree-level by the four-
point w+w~w+w~ vertex derived from (7.88), and by H exchange in the s
and t channels. One finds
M = -4zA + (-2zAi/)2
= -2i\
s-M% + t-M%
s t
s-M2H + t-M2H\
(7.207)
which reproduces (7.204) after using (7.196).
7.5.2 Experimental Constraints and Prospects
As described in Section 7.3.6, the precision electroweak data, which depend
logarithmically on M#, favor a relatively low value, Mh < 167 GeV at 95%
cl, with a central value Mh= 77^2 GeV.
The Higgs was searched for directly at LEP, through the Higgstrahlung
process e~e+ —> Z —> ZH, which probes the ZZH vertex in Table 7.1. At
LEP 1, the 5-channel Z was on-shell and the final one virtual, while at LEP
2, the s-channel Z was virtual. Various combinations of final states were
searched for, including H —> bb or t~t^ and Z —> qq,£~£+, or vv. LEP was
able to exclude a SM Higgs with mass below 114.4 GeV at 95% cl (Barate
et al., 2003; Amsler et al., 2008). There were hints of a signal at ~ 115
GeV, but this was inconclusive. As seen in Figure 7.26 there is some tension
between the precision fits and the direct lower limit, but no real conflict. LEP
2 also obtained limits on possible Higgs-like states with masses below 114.4
GeV but with reduced rates compared to the SM expectation, as would be
expected in some extended models as discussed below.
There are a number of production mechanisms for the Higgs at hadron
colliders, some in association with other particles, as indicated in Figure 7.34.
The gluon fusion mechanism GG —> H, which proceeds via a virtual £-quark
loop, has the largest cross section at both the Tevatron and the LHC. The
associated production of W±H or ZH is also important. Vector boson fusion
(W+W~ —> H or ZZ —> H), and associated ttH production should also be
important at the LHC.
The Higgs decay branching ratios are shown as a function of Mh in Figure
7.35. For large MH, the decays H -> W+W~ and H -> ZZ dominate. Of
course, the cleanest signature would be for the "golden" mode H —> ZZ —>
4£, but this is suppressed by the low leptonic branching ratios. The chains
H —> WW —> qq£v and H —> ZZ —> qq££ are therefore also critical. Below the
WW threshold, down to ~ 135 GeV, the decays WW* (or ZZ*), where V*
is off-shell, dominate. For still lower masses H —> bb is most important. This

The Standard Electroweak Theory
369
w, z
H t
H
FIGURE 7.34
Representative Higgs production diagrams at a hadron collider, including
gluon fusion (left), W or Z fusion (second left), and typical diagrams for
associated production of WH, ZH, or tiH.
mode would be important for a relatively light Higgs (produced in association
with a W or Z) at the Tevatron, but is difficult at the LHC because of the
huge b background. The most promising light Higgs channel at the LHC
is iif —> 77, which proceeds at one-loop. Other possibilities involve various
associated productions, and vector boson fusion with tagging of the forward
quark jets associated with the radiated W or Z.
The partial Higgs decay widths into fermions are
T(H - //) = Cf
Gprnj
AV2tt
P3fMH,
(7.208)
where /?/ = (1 — 4m^/M^)1/2 is the fermion velocity and Cf = 1 (leptons) or
3 (quarks) is the color factor. The dominant QCD corrections are included by
evaluating the running masses at M^, e.g., rab((100 GeV)2) ~ 3 GeV (Gomez-
Bock et al., 2007). These fermionic widths are very small (except H —> ti for
a very heavy H). For example, T(bb)/MH ~ 2 x 10-5, so that T(bb) ~ 2 MeV
for MH = 100 GeV. The partial widths for on-shell W+W~ or ZZ are
T(H - VV*) = Sv^j^il ~ xyf'2 (\ - xv + \x2v^ M* , (7.
209)
where 6w = 2, Sz = 1, and xy = AMy/Mjj. The partial width grows as Mfj.
This is because for Mh ^> My the decay is dominated by the longitudinal
vector states, and their polarization vectors are of order Mn/My (see Problem
7.16). Asymptotically,
r(H^W+W- + ZZ)/MH~±(j^) , (7.210)
so a heavy Higgs is expected to be very broad. The partial widths for virtual
decays, T(H —> VV*), and for the loop-induced decays H —> 77, Z7, and GG,
are given, e.g., in (Gunion et al., 1990; Gomez-Bock et al., 2007).

370 The Standard Model and Beyond
100 200 300 500 700
mha| (GeV)
FIGURE 7.35
Branching ratios for the principal SM Higgs decays as a function of M#. Plot
reproduced with permission from (Carena and Haber, 2003).
As of the Summer 2008, the combined CDF and DO Higgs searches were
becoming sensitive to the Higgs boson at the SM expected rate for Mh in the
160 GeV region*. They have an excellent chance of discovering (or excluding)
the Higgs during the final years of Tevatron running. The LHC is expected
to be sensitive to a SM or MSSM Higgs over the entire relevant region. If the
Higgs is discovered, the LHC should be able to obtain a precise measurement
of its mass. Determinations of the Higgs width and self-couplings would most
likely require a luminosity upgrade of the LHC or future colliders, such as a
possible International Linear Collider (ILC) (Weiglein et al., 2006). Non-
observation of a Higgs would imply the existence of an alternative symmetry
breaking mechanism, with the implications mentioned in the previous section.
Extended Higgs Sectors
Many theories beyond the SM involve extended Higgs sectors*, involving
additional Higgs doublets (such are required in the supersymmetric extension
of the SM) and/or Higgs fields transforming under different SU{2) repre-
tln Spring 2009 the CDF and DO combined working group obtained a preliminary exclusion
of the SM Higgs in the range 160 - 170 GeV at 95% c.l. (TEVNPH, 2009).
*We are using Higgs in the sense of any spin-0 color-singlet fields which are not forbidden
by additional symmetries from mixing with the SM Higgs doublet.

The Standard Electroweak Theory
371
sentations, such as singlets or triplets (for reviews, see Gunion et al, 1990;
Accomando et al, 2006; Barger et al, 2007; Djouadi and Godbole, 2009).
Such additional multiplets allow new physical scalar particles, including
extra electrically-neutral states, singly-charged states for additional doublets
or triplets, and even doubly-charged states for SU(2) triplets with y = ±1.
Electrically neutral states may mix with the H (even for SU(2) singlets),
modifying the couplings of the mass eigenstates hi. In some cases, there are
two or more mass eigenstates with scalar couplings to fermions, and one or
more with pseudoscalar (7s) couplings, and in other cases (involving CP
violation) the scalars and pseudoscalars mix. The LEP 2 lower limit of 114.4 GeV
is often weakened, e.g., because the strength of the ZZhi coupling may be
reduced due to the mixing between two scalars or because the lightest scalar
has new decay modes which are difficult to observe, such as into two light
pseudoscalars (Chang et al., 2008). In the MSSM, for example, there are two
Higgs doublets. In the decoupling limit, in which the second Higgs doublet
is much heavier, there is little mixing and the direct LEP 2 lower limit is
similar to the standard model. However, the direct limit is considerably lower
in the non-decoupling region in which the new supersymmetric particles and
second Higgs are relatively light (Heinemeyer et al., 2006, 2008; Barate et al.,
2003). (This is partially compensated by a new search channel, Z* —> scalar
+ pseudoscalar.) This will be further discussed in Chapter 8.
In addition to the implications for the Higgs spectrum and couplings,
extended Higgs sectors can lead to flavor changing Higgs-fermion couplings (e.g.,
in non-supersymmetric two-doublet models in which both doublets couple
to the same fermions), or to modifications to the po parameter (i.e., to the
Mw jMz relation) due to the VEVs of Higgs triplets or higher-dimensional
representations (Problem 7.1) unless a custodial symmetry is somehow
imposed. There are also new possibilities for explicit or spontaneous CP
violation in the Higgs sector (with possible implications for baryogenesis (Barger
et al., 2007)), and even the danger of the spontaneous violation of electric
charge (Problems 7.3 and 7.4).
It is also possible that the elementary Higgs field is replaced by some sort
of composite field, as in the spontaneous breaking of the chiral symmetries in
QCD, or by other mechanisms, e.g., involving higher dimensions of space-time.
7.6 The CKM Matrix and CP Violation
We saw in Section 2.10 that CP violation can occur in a field theory
whenever the Lagrangian density involves more complex parameters than can be
removed by field redefinitions. Nevertheless, CP violation is observed to be

372
The Standard Model and Beyond
very tiny, and it came as a surprise to most physicists when such a violation
was first observed in 1964 (Christenson et al., 1964). It is straightforward to
write down phenomenological models of new interactions which accomodate
CP violation (for a review of early attempts, see Commins and Bucksbaum
1983). However, when the SM was developed it was difficult to directly
incorporate CP violation, because for one or two families the SM is sufficiently
simple that CP emerges as an accidental symmetry. One possibility was to
extend the Higgs sector, allowing CP-violating effects associated with the
scalar propagators (e.g., Weinberg, 1976). In this case the weakness of CP
breaking would be attributed to the small Higgs Yukawa couplings. Another
was to assume the existence of an entirely new very weak interaction to
mediate the CP violation, such as in the superweak model (Wolfenstein, 1964). A
third possibility, apparently realized in nature, is to introduce a third fermion
family (Kobayashi and Maskawa, 1973). Observable CP violation would only
occur when all three families are relevant to a process, accounting for the
weakness.
On the other hand, most extensions of the SM have potential new sources
of CP violation that are not naturally suppressed, so CP violation studies
are important. Moreover, CP violation is necessary to explain baryogenesis.
It is likely that the origin of the baryon asymmetry is related to CP phases
beyond those in the CKM matrix, possibly those related to neutrino mixing
or perhaps associated with "beyond the standard model" effects observable
at the LHC.
This section will give a brief introduction to CP violation and mixing effecjts,
mainly in the K and B meson systems, and the closely related status of the
CKM matrix. Much more detailed treatments may be found in (Commins and
Bucksbaum, 1983; Jarlskog, 1989; Branco et al, 1999; Bigi and Sanda, 2000;
Kleinknecht, 2003; Ibrahim and Nath, 2008; Sozzi, 2008; Amsler et al, 2008).
Electric dipole moment constraints will be considered in Section 7.6.5. Strong
CP violation, associated with the parameter Oqcd introduced in (5.1), and
baryogenesis will be touched on in Section 8.1.
It should be emphasized that most of the processes considered involve non-
trivial complications from the strong interactions. Enormous effort involving
perturbative techniques such as HQET and SCET, as well as non-perturbative
lattice calculations and chiral perturbation theory have been required to
overcome those obstacles.
7.6.1 The CKM Matrix
In Section 7.2.2 the quark and lepton mixing matrices Vq and Ve were seen to
arise from the mismatch between the fermion gauge and Yukawa interactions,
i.e., between the weak and mass eigenstates. They also required a mismatch
between the u and d (or e and v) sectors, as is apparent in (7.60). The
empirical forms of Vq were briefly discussed for two and three quark families
in (7.65) and (7.66), and in Chapter 6. Here, we consider Vq for 3 families,

The Standard Electroweak Theory
373
ie., the CKM matrix, in more detail.
As already described, after imposing unitarity and removing unobservable
phases by redefinitions of the chiral quark fields, Vq involves 3 mixing angles
and 1 CP-violating phase. There are a number of possible parametrizations,
but we will follow that of the Particle Data Group (see the article on the CKM
matrix in Amsler et al., 2008),
(Vud Vus Vuh\ /l 0 0
Vq = Vcd Vcs Vcb \ = 0 C23 523
\Vtd Vta Vtb) \0-823C23
^12^13 512^13
-S12C23 - Ci2S23Sl3e*5 C12C23 - Sl2S23«13e*5 ^23^13 ), (7.211)
S12S23 - ci2c23Si3el6 -C12S23 - si2c23Si3el6
where Cij = cos Oij and Sij = sinfl^. The mixing angles are #12, #13, and #23,
and 5 is the CP-violating phase. Assuming unitarity, the magnitudes of the
elements of Vq are (Amsler et al., 2008)
/0.9742 0.226 0.0036\
|V^|~ 0.226 0.973 0.042 , (7.212)
\0.0087 0.041 0.9991/
which implies 812 ~ sin^c ~ 0.23, while 813 <C 523 <C S12. As we will see,
the observed smallness of CP violation is not because 6 is small, but rather
because observable violations require that all three families contribute to the
relevant transition amplitude, and therefore are suppressed by small
mixing angles. It is often convenient to employ the approximate Wolfenstein
parametrization (Wolfenstein, 1983)
/ l-A2/2 A A\3(p-i<n)\
Vq=\ -A l-A2/2 A\2 +0(A4), (7.213)
\A\3(l-p-i7]) -A\2 1 /
where A ~ sin0c ~ 0.23. The powers of A incorporate the suggestive
hierarchical pattern in (7.66), while A ~ 0.81 (from Vcb), p, and 77 are real and
of O(l). The displayed terms are unitary through 0(\3). The higher-order
corrections ensure unitarity to higher order, but are not important in practice.
CP violation is associated with 77, i.e., tan5 = r\jp. There are various possible
conventions for the CP phase in Vq. However, the quantity
p + ifi=-Ys&* (7.214)
vcb Vcd
is independent of the convention. From (7.213)
p = p(l - A2/2), V = 17(1 " A2/2), (7.215)

374
The Standard Model and Beyond
up to corrections of 0(X4). Vq takes the same form as (7.213) when written
in terms of p and f\ to that order. It is also convenient to define the Jarlskoq
invariant
J = %m(vusvcbv:bv;s)
2 • X A2X6- (7.216)
which is a convention-independent measure of CP violation.
Measurements of the elements of Vq are important to test the consistency of
the SM and determine its parameters, which are relevant for other quantities
and processes such electric dipole moments or the CKM contribution in models
of baryogenesis. It is especially important to verify that Vq really is unitary,
i.e., VqV^ = VqVq = i". A violation or apparent violation would signal the
presence of new physics. This could take the form of a fourth family or heavy
quarks with exotic SU(2) assignments, so that the 3x3 CKM submatrix
would not by itself be unitary. Alternatively, it could involve new interactions
like super symmetry, leptoquarks, compositeness, or a heavy W' (e.g., coupling
to right-handed currents) or Z1 gauge boson that were not properly included
in the analysis and therefore led to an incorrect determination of some of the
elements. Unitarity studies include tests of weak universality (the diagonal
elements), especially
(VqV])n = \Vud\2 + \VUS\2 + \Vub\2 = 1, (7.217)
as well as unitarity triangle tests (off-diagonal elements), such as !
(v2 v«) 3i = vubVud + v;bvcd + v;bvtd = o. (7.218)
Magnitudes of the CKM Matrix Elements and Weak Universality
We first briefly survey the magnitudes of the CKM matrix elements. For more
detail, see the recent reviews in (Battaglia et al., 2003; Di Lodovico, 2008)
and the articles on the CKM matrix, on Vud,3, and on VUiCb m (Amsler et al.,
2008).
\VUd\i As discussed in Section 6.2.5 the most precise determination is from
superallowed 0+ —► 0+ /? decay, which only involves the vector current.
The Ademollo-Gatto theorem ensures that the theoretical uncertainties
are of second order in isospin breaking, e.g., Za2, though these must be
taken into account, allowing a clean extraction of Gp\Vud|- Taking Gf
from muon decay yields \Vud\ = 0.97418(27). The neutron lifetime yields
an independent determination of G|r|V5xd|2(l + 3A2), but the precision
of the extracted \VUd\ is limited by the measurement of A = gA/gv ~
1.2695(29) from the decay asymmetries. Pion beta decay, 7T+ —► Tpe+Ve,
also yields a theoretically clean measurement (Section 6.2.3), but with
a larger uncertainty.

The Standard Electroweak Theory
375
\VUs\: The Ke3 decays also involve only the vector current, implying by the
Ademollo-Gatto theorem that corrections to the form factor /+(0) are
second order in ST/(3)-breaking. Nevertheless, there are still
theoretical uncertainties in the determination of \VUS\, as described in the cited
reviews and in Section 6.2.3. Recent high statistics experiments at Fer-
milab, CERN, Frascati, and IHEP§ yield f+(0)\Vus\ = 0.21668(45), with
/+(0) normalized to unity in the SU(3) limit. Using the value /+(0) =
0.961(8) (Leutwyler and Roos, 1984), one finds \VUS\ = 0.2255(19).
Other determinations involving hyperon, Ke2, and hadronic r decays
each involve their own theoretical uncertainties, as detailed in Chapter
6, but yield reasonably consistent results.
IVcdU^csh The value \Vcd\ = 0.230(11) is obtained from the deep-inelastic
processes v{v)N —> ji^X + c(c) as described below (7.108). Consistent
values are obtained from semi-leptonic D decays. \VCS\ is more
difficult. The best determination comes from measurements of leptonic and
semi-leptonic D decays. The average of the BaBar, Belle, and CLEO-c
results is \VCS\ = 1.04(6). There is also a much less precise DELPHI
determination from tagged W+ —> cs decays. An accurate but indirect
determination from the four LEP 2 experiments utilized the leptonic
branching ratios W —> £i?£ measured in e~e+ —> W~W+ —> ////.
From (7.153)-(7.155),
f1 + ^Mk) £ \Vmnf~-l--l, (7.219)
n=d,s,b
where B^£ is the average of the £ = e, //, and r branching ratios.
Assuming lepton universality, this implies (Amsler et al., 2008, LEPEWWG
website),
^ \Vmn\2 = 2.002(26), (7.220)
consistent with the expectation of 2 from CKM universality. Using.the
measured values for the other elements, one obtains \VCS\ = 0.977(14).
Deep-inelastic neutrino scattering only determines S'lVcsl2, where S is
the s quark momentum fraction.
iKblJVd,!: The values \Vub\ = 0.00395(35) and \Vcb\ = 0.0412(11) are
obtained from inclusive and exclusive semi-leptonic B decays, as described
in Section 6.2.7.
I^dl? |Vis|, |V^&|: These are mainly constrained by the mass differences Ara^s
obtained from B% — B% and B® — B® mixing, which are driven by box
^Older experiments gave considerably lower values, but it is not clear how radiative
corrections were treated. For discussions, see (Czarnecki et al., 2004; Amsler et al., 2008;
Antonelli et al., 2008).

376
The Standard Model and Beyond
diagrams involving two W's, similar to Figure 6.3 (Section 7.6.4). The
results are |Vid| = 0.0081(6) and \Vts\ = 0.0387(23), with the
uncertainties dominated by the unquenched lattice calculations used to
obtain the matrix elements. These uncertainties are reduced in the
ratio Amd/Ams, implying the more precise |Vid/Vis| = 0.209(6). Other
constraints from radiative B decays are described in the reviews. The
K+ —> is^vv rate would allow a theoretically precise determination of
|VidV^* | (Buras et al., 2005), but so far only three events have been
observed. There are so far no accurate measurements of \Vtb\- From the
cross section for single top production measured by CDF and DO, which
involves a virtual W in the s or t channel, one obtains \Vtb\ > 0.78 at 95%
c.l. (Abazov et al., 2009; Aaltonen et al., 2009). Somewhat more
stringent constraints are found from B(t —> Wb)/B(t —> q) if one assumes
unitarity (^n=d,s b \^tn\2 = !)• Unitarity combined with the measured
\Vtd,ts\ or the measured \Vub,cb\ yields \Vtb\ ~ 0.999.
Combining these results one obtains
\Vud\2 + \Vus\2 + \Vub\2 =0.9999(10), (7.221)
(3 Ke3 negligible
in impressive agreement with weak universality (Czarnecki et al., 2004; Amsler
et al., 2008)^. This is especially remarkable because in confirms the theory
at the loop level: without radiative corrections applied to //, /?, and K decays
the sum would have been ~ 1.04, and those corrections are only finite and
meaningful in the full SU(2) xU(l) gauge theory. The agreement also strongly
constrains such new physics as a heavy WR coupling to right-handed currents,
W — Wr mixing, and mixing between ordinary and heavy fermions.
Combining (7.220) with (7.221) one finds £n \Vcn\2 = 1.002(26), also in
agreement with universality. For the third row one has ^n \Vtn\2 > 0.55.
Weaker constraints apply to the columns, ^m |Knn|2-
7.6.2 CP Violation and the Unitarity Triangle
Universality tests whether the diagonal elements of V^Vq and VqVj are equal
to unity. The unitarity triangle tests involve whether the off-diagonal
elements, such as (V^Vq)31 in (7.218), vanish. The name derives from the fact
that each term in the expression for (V^V^) is a complex number, which
can be thought of as a vector in the complex plane. Unitarity implies that
they should sum to zero, i.e., form a triangle.
^For many years there was an apparent 2-3<r deviation, which was often (wrongly) assumed
to be due to some problem with the radiative corrections to superallowed (3 decays. The
problem was resolved only recently by the new generation of high precision K&
measurements which resulted in a higher \VUS\, as mentioned above.

The Standard Electroweak Theory
377
There are six unitarity triangle tests, but we will concentrate on (VWq)
which is illustrated in Figure 7.36. The goal is to overconstrain the triangle by
redundant tests, to check whether unitarity really is satisfied and to determine
the parameters. In particular, the lengths of the sides may be determined from
QP conserving rates, such as the b —> u decay rate or B° — B° mixing, as
described above, while the angles can be determined independently by CP-
violating effects. Any deviation would signal the presence of new physics,
such as new fermions, new sources of CP-violation (e.g., loops involving su-
persymmetric particles), or new flavor changing neutral current interactions
(e.g., from extended Higgs sectors or a family-non-universal heavy Z').
v;Lhvud
v;bvcd
VtlVto
(plv)
p + ir)
(0,0)
(1.0)
FIGURE 7.36
Left: the unitarity triangle for (VqVq)31 = 0. Right: the triangle scaled by
V*bVcd, so that the vertices are at (0,0), (1,0), and (p,fj).
In the Wolfenstein parametrization,
(vjvq)31 = v:bvud + v;bvcd + v;bvtd
~ (l - y) A*3 (P + "?) - A*3 + AX3(l-p- it]) (7.222)
~ ^A3 (p + if)) - AX3 + AX3(l-p- if)) = 0.
l^c{,Vcd| ~ A A3 is known accurately, so it can be divided out,
AX3
P + IT]
-1 +1 — p — ir\.
v;bvud/A\* vc*bvcd/A\s v*bvtd/A\*
(7.223)
The rescaled unitarity triangle is therefore the sum of the vectors p + ifj, — 1,
and 1 — p — if}. The sides are of length ^/p2 +ry2, 1, and y/(l — p)2 + ^2,
respectively, while the angles a, /?, and 7 defined in Figure 7.36" satisfy •
Vu = \Vtd\e~i0, Vub = IKfele^.
(7.224)
"The alternative notation </>i = 0, <p2 = ot, and <j>3 = 7 is used frequently, especially by the
Belle collaboration.

378
The Standard Model and Beyond
7 is the same as the CKM phase 6 in the parametrization in (7.211). The
Jarlskog invariant J in (7.216) is twice the area of the original (unrescaled)
version of the triangle. In fact, all six unitarity triangles can be shown to have
the same area.
The consistency of this picture can be tested by determining the points
(p, fj) by redundant means to see whether they agree. One constraint follows
from
VubVud
v:hVcd
= ^p2 + fj2~ 0.41(4),
(7.225)
which yields the circular annulus centered at the origin marked \Vub\ in Figure
7.37. To proceed we need further inputs from the K and B systems.
0.7
\~ a>
0.6 h-JJ
F «
I
M 1
0.3
0.2
0.1
0.0
-0.4
Amd & Arr^
AlTL
.sin
%
fitter
ICHEP08
-0.2
0.0
0.2
0.4
FIGURE 7.37
The unitarity triangle, showing the consistency of various CP-conserving and
violating observables from the K and B systems. Plot courtesy of the CKM-
fitter group (Charles et al., 2005, http://ckmfitter.in2p3.fr), with kind
permission of The European Physical Journal (EPJ).
7.6.3 The Neutral Kaon System
The neutral K° — K° system has been extremely important in particle physics
(for reviews, see, e.g., Chau, 1983; Commins and Bucksbaum, 1983; Winstein
and Wolfenstein, 1993; Kleinknecht, 2003;_Amsler et al., 2008). The
magnitude of the mixing between the K° and K°, induced by second-order weak
effects, led to the prediction of the charm quark and of its mass, and has been

The Standard Electroweak Theory
379
a stringent constraint on new sources of flavor-violating physics at the tree
and loop level. CP violation was first observed and studied in the neutral
kaon system, and provides even more stringent limits on some kinds of new
physics.
The states K° and K° defined in Section 3.2.3 are the isospin partners of
the K+ and K~, respectively, and carry strangeness +1 and —1. In the quark
model,
\K°) = \ds), \K°) = \sd). (7.226)
Under CP
CP \K°) = Vk\K°), CP \K°) = Vk\K°), (7.227)
where t]k is a phase. As discussed in Section 2.10 one can always perform a
field redefinition on the kaons (or on the quarks) to change t]k] we use this
freedom** to choose j]k = — 1. The strong interactions conserve strangeness
and are CP invariant, so K° and K° are the relevant states for describing
strong interaction transitions. For example, n~p —> K°A is allowed, where
|A) = \sdu) is the S = — 1 isoscalar hyperon, while n~p —> K°A is forbidden.
The two states are degenerate by CPT with mass mKo = 397.6 MeV. This
is 3.9 MeV larger than mK±. Similar to the n — p mass difference, this is of
opposite sign from the expected electromagnetic contribution, and is due to
the quark mass difference rrid — mu > 0.
However, the weak charged current interactions can violate strangeness and
lead to K°—K° mixing at second order (in Gp)- This implies that the K°—K°
system is described by a 2 x 2 mass matrix,
where MKo = M^o (~ mKo) by CPT invariance^. MKx is the weak mixing
term, generated by the diagrams in Figure 7.38. It is tiny, but its effects are
important due to the degeneracy of the diagonal terms.
If we first ignore the third family then MKx is real (i.e., CP is conserved),
and the eigenstates of M are the maximal admixtures
K°ia= J5 ~ks,l- (7.229)
The expression for K®2 will continue to hold as a definition even after CP-
violation is turned on, while Ks,l {S and L stand for "short"and "long,"
** Other frequently used conventions are t)k = +1, and the Wu-Yang convention (Wu and
Yang, 1964), which makes the amplitude Aq defined in (7.279) real. The latter is somewhat
awkward in the standard model. The tjk = — 1 convention follows from (7.226) if we define
the d and s quarks to have the same intrinsic C and P phases (Problem 2.5).
t^See (Commins and Bucksbaum, 1983; Sozzi, 2008) for a general discussion allowing for
the possibility of CPT violation.

380
The Standard Model and Beyond
d ~~^^n~5Cw^~-*~-"~ * rf-—^ u,c,t
j>0 U,C,tk U, C, t¥ j^Q j£0
FIGURE 7.38
Second-order diagrams leading to K° — K° mixing in the standard model.
respectively) will represent the mass eigenstates in the more general case.
K® and K% are CP eigenstates, with eigenvalues +1 and —1, respectively.
The states |7r+7r~) and |7r°7r°) with definite angular momenta are also CP
eigenstates,
CP|7T+7T-) = +I7T+7T-), CP|7r°7r°) = (-l)L|7r°7r°). (7.230)
But L = 0 for K decays, so CP conservation allows K® = Ks —► 2n but
forbids K% = Kl —> 27r. Similarly,
CPItt+tt-tt0) = (-l)/3-|7T+7r-7r0), CPIttVtt0) = -IttW), (7.231)
where ^3^ = 0,1,2,3 is the total isospin of the 7r+7r_7r0 system and we have
assumed that the total angular momentum is zero. 7*3^ = 0 or 2 can only
occur when there are internal orbital angular momenta for the 3tt system,
leading to a strong centrifugal suppression (Amsler et al., 2008). Therefore,
Kl —► 37r is allowed, while Ks —> Stt is strongly suppressed. Since the phase
space for the 2tt decay modes is much larger than for 37r, one expects a much
shorter lifetime for Ks than for Kl, motivating the terminology. This picture
is indeed approximately valid: The lifetimes tks and tkl are respectively
0.8953(5) x 10"10 s - (2.7 cm )~l and 5.12(2) x 10"8 s - (15.3 m )_1, with
their non-leptonic decays almost exclusively 2tt and 3tt. The K^ modes are
competitive for Kl, but are only of (9(10-3) for Ks, as can be seen in Table
7.7. However, the assumption that CP is absolutely conserved was invalidated
by the observation of Kl —► 2tt decays at the 10~3 level (Christenson et al.,
1964). We will return to the subject after discussing the calculation of K°—K°
mixing in the standard model.
Calculation of Am^
The mass difference between the eigenstates in (7.229) is
AmK = mK2 - mKl = 2MkR. (7.232)
We will see below that it is predicted to be
AmK ~ ^m2c\Vcd\2\Vcs\2f2KmKBK, (7.233)

The Standard Electroweak Theory
381
TABLE 7.7
Principal branching ratios for the neutral K decays. The
charged K branching ratios are given in Table 6.2.
Ks - 7r°7r°
—► 7T+7T~
—> 7r^e±jye(ue)
KL —> /KTe±ve(ve)
->7TT^±'/m(^)
—> /K+/K~/KQ
^37T°
—»ir+n~
^7r°7r°
(K2ir)
(K2„)
(*<*)
(K*)
(^3)
(K3n)
(*3»)
30.7%
69.2%
7 x 1CT4
40.6%
27.0%
12.5%
19.5%
2.0 x 10~3
8.7 x 10-4
CP-violating
CP-violating
up to third family and QCD corrections, where Jk ~ 1.22/n- ~ 160 MeV is the
kaon decay constant defined in (6.78), and Bk = ^(1) is associated with the
matrix element of the effective operator corresponding to Figure 7.38. Ara^
is much too small to be measured kinematically, but it can be determined
indirectly by the regeneration technique described below to have the extremely
small value 3.483(6) x 10-6 eV. Using (7.233) with BK = 1 one obtains the
prediction mc ~ 1.5 GeV (Gaillard and Lee, 1974b), in agreement with the
value in (5.125). In fact, the QCD corrections are non-negligible and there is
still a non-trivial uncertainty from Bk, but it is clear that the SM prediction
is qualitatively correct. Without the GIM mechanism (Glashow et al., 1970)
the box diagram contribution would have been orders of magnitude larger,
and there would also have been a tree contribution from the flavor changing
sdZ vertices in (7.78). Assuming that no new physics contributions to Attik
can be much larger than the experimental value also significantly constrains
new tree-level physics leading to flavor changing couplings and certain types
of new box diagram effects, as will be further considered in Section 7.6.6.
Now, let us derive (7.233). Box diagrams similar to Figure 7.38 will lead to
an effective |A5| = 2 operator
£|A5|=2 = C|A5|=2 Jyz(1 _ y,^ J^(1 _ ^ + ^ (? ^
We will work in the 't Hooft-Feynman gauge, which means that we must also
include the box diagrams in which one or both of the W's are replaced by
the Goldstone bosons w±, as described in Section 4.4. We start with the
schematic expression
i£J5;i=2 = ir[(iC)4], (7.235)
where C includes both the WCC couplings in (7.55) and the Yukawa couplings

382
The Standard Model and Beyond
in (7.91), i.e.,
£ = - ^= {(dV(l - 75)V> W~ + «y (1 - lbWqd W+)
Mlk
+
*5£<1+->*>
uw + u
Mu
diu"1
(7.236)
where we have neglected m^s (the b terms are irrelevant), and we have used
y/2/u = g/y/2Mw- Expanding £4 and contracting the appropriate pairs of
W+W~, w+w~, and unun fields, using the W and w propagators in (4.68)
and (4.63), and neglecting external momenta, we find
,|AS|=2
2%
£
m,n=u,c,t
2V2
f d4k / 1
\ * \ *
AmAn
M2w
1
k2-
m±
1
k2 — m2
(7.237)
- 2xmxnM^d^(l - 75)s d7M(l - 75)^
+xmxnJ #(1 - 75)s J #(1 - 75)s] + /i.e.,
where
and
xn —
M2W
An — VndVns->
Y, A"=°-
n=u,c,t
(7,238)
(7.239)
The three terms are associated with the WW, Ww, and ww diagrams
respectively. In the first and last terms the fi parts of the numerators of the fermion
propagators survive, while in the Ww term the fermion mass term survives.
The coefficient reflects the fact that there are 4!/2 equivalent terms for the
WW and ww diagrams, 4! for the Ww, and that (1 — 75)2 = 2(1 — 75). One
can easily show that
d^ #7^(1 - 75)s d7„ #7M(1 - 75)s = Ad #(1 - 75)s d #(1 - 75)s (7.240)
by using the Fierz identities in (2.211) or by using the identity in Problem
2.5. Furthermore, one can replace kpka by gpak2/A, since there is no other
four-vector in the d4k integral. Finally, one can employ the Wick rotation
and angular factor as in (E.15) to obtain
16tt2 ^
\ * \ * rpmn
AmAnr >
(7.241)
m,n=u,c,t

The Standard Electroweak Theory
383
where
Fmn = (l + ^) B^n + 2xmxnB™
zkdz (7-242)
r>mn __
Jo
(2 + l)2(2 + Xm)(2 +
The integrals are elementary, and yield
?mn _ ^fc(xm) ^fc(^n)
Br = ^ .v^a (7 243)
with
. / x 1 x2lnx „ , v 1 xlnx , ^ x
To verify (7.233), let us neglect the mixing with the third family, so that
Xu = —Ac = cos 0C sin 0C, (7.245)
and r2 m2
CIAS\=2 = _^F^WX2 [Fuu + Fcc _ ^uc] (?>246)
xc is very small (~ 3.5 x 10~4 for mc ~ 1.5 GeV), and xw is neglible for either
the current or dynamical mass. To leading order in xc,
F™~1, Fcc-l + 3xc + 2xclnxc, Fwc - 1+xc+ xclnxc, (7.247)
so the quantity in square brackets in (7.246) is xc**.
Now consider the matrix element
(K°\CleAfsfl=2\K°)
MKR = -* ^ '. (7.248)
The \/(2nriK) follows from our covariant state normalization, noting that
(p'\H\p) = Ep{p'\p) = (2n)363(p' -pW\H(0)\p), (7.249)
and using (2.3) and translation invariance (1.14). To estimate the matrix
element, we will use the vacuum saturation approximation, in which one inserts
the vacuum state between the ds operators in all possible ways, i.e.,
<tf<W(l -l5)sd^(l -l5)s\K°)vac
=2{K°\dV(1 - 75)sa|0) (0|JV(1 - 75)^l^°) (7-250)
+ 2(K°\dar(l - 75)^|0) (0|JV(1 - l5)sa\K°),
** Without the c quark, the corresponding factor would be Fuu ~ 1, about 3000 times
larger, as stated below (7.233).

384 The Standard Model and Beyond
where we have reintroduced the color indices a and /? to keep track of them-
the second term, which corresponds to the second diagram in Figure 7.38 is
obtained by performing a Fierz transformation before inserting the vacuum.
The 2 comes from interchanging the two ds factors. The matrix elements of the
vector currents vanish by reflection invariance, and the axial matrix elements
are given in the SU(3) limit by (6.76), where <?7M75s = A^ + iA7^ = (s7M75d)t
Our phase choice t\k = — 1 in (7.227) corresponds to
\K°) = ^|6 + i7>, \K°) = ^|6-i7>, (7.251)
implying
<0|dV(l - l5)s0\K°(q)) = (0| A"(1 - lb)d0\K\q)) = iff. (7.252)
In fact, SU(3) breaking is important, but from isospin one has that / should
be identified with the Jk from K~ —> /x~PM in (6.78). Similarly,
<0|dV(l - l5)s0\K°(q)) = -f ifKq", (7.253)
so that*
(K°\d>f(l - j^sdj^l - j5)s\K°)vac = \f2Km2K. (7.254)
Vacuum saturation is not expected to be a good approximation, but is useful
as an order of magnitude estimate. It is therefore conventional to define a
factor Bk to describe the departure from vacuum saturation, i.e,
(K°\d<y»(l -75)^"7m(1 -75)s|^°> = \f2Krn2KBK. (7.255)
Combining (7.232), (7.246), (7.248), and (7.254), one reproduces (7.233).
QCD corrections (e.g., Buchalla et al., 1996; Buras, 2005) to C'A5'=2, which
also involve additional penguin diagrams, are significant. Including the t
quark,
C|A5|=2 = _^^[A'V5o(xc) +A;2Th5o(xt) + 2A;A;%5o(xc>xt)]^)>
(7.256)
where 771,2,3 = 1.32(32)[1.3 GeV/mc(mc)]1-1,0.57(1),0.47(5) are short
distance QCD corrections, defined to be independent of the renormahzation scale
//, and J-(fi) contains scale, scheme-dependent, and higher-order factors. Bk->
defined in (7.255), also becomes //-dependent, but
BK = BK(^T(fi) (7.257)
* Older references often quote a value 3/2 larger. This was based on a pre-QCD calculation
involving only one color of quark.

The Standard Electroweak Theory
385
is an observable that is scale independent. BK may involve both short distance
and long distance contributions, where the latter are associated with virtual
hadrons rather than quarks. Estimates using lattice, large-iVc, and chiral
perturbation theory techniques are consistent with BK = 0.85(15), with a
recent review (Juttner, 2007) of lattice calculations giving the lower and more
precise value 0.720(39). The So functions in (7.256) are obtained from (7.241)
after eliminating \u = — (Ac + Xt). After a somewhat tedious calculation, one
finds
^0\XC) = Xc
4xt - lis? + x\ 3x3t\nxt
SoiXt) = 4(1 -zt)2 " W^V3 ~ (7-258)
D0{XCi Xt) — Xc
In
xt 3xt 3xf In xt
xc 4(1 -xt) 4(1-xt)2
3.0 x 10"3
to leading order in xc (Inami and Lim, 1981). The numerical values are for
xc ~ 3.5 x 10~4 and Xt ~ 4.52 (for mt = 170.9). Complex phases in the An's
lead to CP violation. This requires not only the existence of three families,
but that all three are relevant to the process. Otherwise, the factors could be
made real by a redefinition of the fields.
Ks,l Decays, Oscillations, and Regeneration
We saw following (7.231) that, assuming CP conservation, Ks decays rapidly
to 27T, while Kl —► Stt and to semi-leptonic modes with a lifetime nearly
three orders of magnitude longer. However, in 1964 the decays Kl —+ 2tt
were observed, suggesting a tiny violation of CP. To see how this was done,
let us consider an idealized experiment in which one starts with a pure K°
beam, which could be produced for example by the strong interaction process
7T~p —► K°A in a target. In particular, consider a single particle state |^(T))
which is initially a K° with definite velocity /?,
|V(0)> = \K°) - -^ (\KS) + \KL)). (7.259)
If one could ignore the decays, then at a later proper time r = d//?7, where d
is the distance travelled, the state would have evolved to
Mr)> = -i= \\Ks)e-im'T + \KL)e-im^}
/A X /A X (7-26°)
= [cos (^) |*°> -sin (^1) lK^}e-^.
That is, there is a probability sin2 Ar^KT that the K° will have oscillated into a
K°. This is of course an example of the two-state problem familiar in quantum
mechanics, which is itself reminiscent of the classical coupled oscillator system.

386
The Standard Model and Beyond
We will encounter it again for B° — B° and neutrino oscillations. However, in
the neutral kaon system the oscillation time 27r/ArriK ~ 1.2 x 10~9 s is long
compared to tks, so we must take the decays into account. This can be done
by replacing (7.260) by
Mr)) = -^ [\Ks)e-im'Te=^ + | tfL)e-iro^e^]. (7.261)
The factors exp(—Ts,lt/2) represent the depletion of the beam into the 2n
and 37r channels, respectively, with rates Ts,l = 1/tkS)L • (These factors could
be incorporated in a more elegant manner by considering the multi-channel
system including the 2tt and 3tt states or by introducing a density matrix,
but the simple exponential factors are adequate for our purposes.) Clearly,
one observes 2tt decays near the source from the Ks component. Far away,
for tks <^ t <C tkl one has an essentially pure Kl beam and one expects to
observe only 3tt decays.
Before turning to CP violation, let us consider the kaon regeneration
technique that can be used to measure Atuk- The basic idea is familiar from
measurement theory in quantum mechanics, or from the use of polarizers in
optics. Suppose one places a piece of matter, known as a regenerator, a
distance d\ downstream from the source. If the corresponding n is much larger
than tks the beam entering the regenerator is a pure Kl, i.e.,
Mt1)) = \Kl)=IK°)^K°), (7.262)
where we have renormalized the coefficient to unity. The two components can
scatter or be absorbed in the regenerator with different amplitudes, so the
state emerging on the other side at r2 = t\ + e is
|V(r2)) = a\K°) + b\K°) = ^\Ks) + ^\Kl), (7.263)
where we can again renormalize so that \a\2 + \b\2 = 1. For a ^ b the Ks
component has been regenerated. One typically expects |6| <C \a\ since the
K° component can be strongly absorbed by K°p —> 7T+A in the regenerator,
while there is no analogous reaction for K°. In the extreme case of b = 0, a
pure emerges.
Now, suppose the regenerator is placed close to the original source, so that
|^(ti)) has a non-negligible Ks component. The Ks,l components can then
interfere with each other so that the (unnormalized) state emerging from the
regenerator is
IVKt2)> = [(a + b)X + (a - b)] \KS) + [(a - b)\ + (a + b)} \KL), (7.264)
where A = exp^Ara/^ — rsr2/2), we have removed a common phase, have
approximated exp(—TlT2/2) ~ 1, and have neglected the thickness of the

The Standard Electroweak Theory
387
regenerator. The intensity of Ks in the regenerated beam can be measured
by observing the 2tt decays, with a rate proportional to \{Ks\^(t2))\2• For
fr ~ 0 this is
|<tf5h/>(T2)>|2oc [l + e-r5T2+2e-r5r2/2cos(AmKr2)] . (7.265)
Arax can then be determined by varying d\ ~ d2. In practice,
regeneration experiments usually involve two regenerators, allowing Atuk and CP-
violating parameters to be determined by varying the distance between them
or utilizing the energy spread of the beam.
CP Violation in K decays
In 1964, Fitch, Cronin, and collaborators observed 2tt decays in a neutral kaon
beam produced at the Brookhaven AGS some 300 Ks decay lengths from the
source, implying the CP-violating Kl —► 27r decays at the 2 x 10~3 level.
Subsequent observations utilized the regeneration technique, which allowed
the measurement of both the magnitudes and phases of the CP-violating
parameters (as well as Am^). Since regeneration measures interferences, it
also eliminated possible alternative explanations of the events, such as the
emission of an unobserved third exotic particle in the decay.
There are two possible sources of the CP violation:
• CP violation in the K° — K° mixing (indirect violation). This would be
induced by an imaginary part in MKf: defined in (7.228), so the mass
eigenstates Ks,l no longer coincide with the CP eigenstates K\2 in
(7.229).
• CP-violation in the decay amplitude (direct violation).
We now understand that both effects are present, though the indirect CP-
violation is much larger. They are both understandable in the SM as being
due to the CP-violating phase in the CKM matrix for 3 families. This phase
can be large, but the observed effects are strongly suppressed by small mixing
angles.
Define the CP violating parameters
(7.266)
where A(Ks,l —> 7r+7r~,7r°7r°) are the decay amplitudes. The difference
between them is due to the fact that the 2tt state can have isospin 0 or 2. As
commented briefly in Section 6.2, the 1 = 2 amplitude is much smaller than
the / = 0 one (the AI = \ rule), e can be generated by mixing (indirect)

388
The Standard Model and Beyond
or by direct breaking in the 1 = 0 amplitude, though the former is more
important for our phase convention, e' is due to a phase difference between the
1 = 0 and 2 amplitudes; it indicates direct violation but is suppressed by the
A/ = \ rule.
|?7+_| and |t7oo| can be measured from the rates for the various 2tt decays
while the phases </?+_ and ipoo can be measured by interference effects in
regeneration experiments. A fit to the results yields (Amsler et al., 2008)
|77oo| = 2.222(10) x 10"3, y>0o = 43.7(8)°
|77+_| = 2.233(10) x 10"3, </>+_ = 43.4(7)°, ^ '^
and the corresponding values
|e| = 2.229(12) x 10"3, ftefc'/c) - ^ (1
?7oo
V+-
= 1.65(26) x 10-3.
(7.268)
The 5fte(e;/e) measurement was especially difficult since it involves differences
between two small effects, and was not carried out precisely until fairly
recently, by the KTEV (Fermilab) and NA48 (CERN) collaborations (Sozzi and
Mannelli, 2003). CP breaking is also observed in the difference between the
Kes decay rates,
SL = r (kl -> n-e+v) + r(KL^ n+e-p) - 3-32(6) x 10 • (7-269)
To interpret these results we need to find the eigenstates \Ks,l) (in (he
presence of CP violation) of the operator
ff = M-«|, H\V)=i^\1>), (7.270)
which governs the time evolution of the system. M = M* is the 2x2 mass
matrix defined in (7.228), while r = r* is an analogous decay matrix. It is
defined as
ra6 = Y. pfM(a -> /)*M(6 -> A (7-271)
/
where pf are the coefficients and phase space factors for a decay into
channel / defined in (D.7). In the absence of CP violation the eigenvalues of T
are Ts,l and (7.261) is the solution to (7.270). CPT conservation ensures
that the diagonal entries are equal, i.e, MKo = M^o and T^o = T^o. It is
straightforward to find the eigenstates and eigenvalues (e.g., Commins and
Bucksbaum, 1983),
\Ks)=P\K<>)-q\K°)=]K°l+"l«p
V^0 (7.272)
\KL)=p\K°)+q\K°) = i*K?}+}**),
V^
ie|2

The Standard Electroweak Theory
389
where
4 _ 1-e
MKK-^
Mkr-^i
(7.273)
p 1 + e
so that nonzero e requires imaginary parts for MkR and/or TkR. The
eigenvalues are
Ts,l
ms,L - t-ir- = [ mk°
so that
AmK = mL-ms ~ 2&eMKR, TS-TL~ -2^ieTKR, (7.275)
neglecting corrections of 0(e). \KL) and \Ks) are not orthogonal, but satisfy
(KL\KS) = ff^ ~ 2$fte c. (7.276)
1 + \e\z
This is due to the non-Hermitian nature of H in (7.270). From (7.273) and
(7.274) one finds to 0(e) that
€. — —zn ^ zr~. I — e sin u ^
Ts - TL - 2iAmK V AmK 2AmK
(7.277)
where
^ = tan-i 2Am* = 43.51(5)° (7.278)
is - rL
is the superweak phased
Finally, let us define the amplitudes for K° and K° to decay into 2tt states
of definite isospin / = 0 or 2 and J3 = 0 by
(II3 = 0\'Hnl\K0)=AIeiS>
(II3=0\Hnl\K°) = -A*Iei61,
where Hni is the Hamiltonian density for non-leptonic transitions_defined in
(6.21). A/ is the weak part of the amplitude. It changes sign for K° because
of (7.227) (with t]k = -1) and (7.230), and the complex conjugation reflects
the CP transformation, as described in Section 2.10. exp(i5i) represents
the S-wave phase shift induced by the strong interaction final state
interactions of the two pions, and is the same for both K° and K°. Empirically,
Se^/SeAo = a; ~ 0.045 < 1 (the AI = \ rule), and
0'e = 52- (50 + |=42.3(1.5)°. (7.280)
fIn the superweak model (Wolfenstein, 1964) all CP violation is due to a new superweak
interaction that only contributes to the mixing. It was finally excluded by the observation
of e'.

390
The Standard Model and Beyond
The 5-wave 7T+7T and 7r°7r° states are related to the isospin states by
-L (!„+„-) + Itt-tt^)) = -yjl\l = 0 73 = 0> - -^|/ = 2 73 = 0>
ft ft (7'281)
k°O = -^-|/ = 0/3 = 0> + ^-|/ = 2/3 = 0).
Combining (7.266), (7.272), (7.279), and (7.281), one obtains
c'
2e, /V (7.282)
^oo = e -
26',
1 - V^e^2"*0)
where
"9mi2 9mi0
_ .3mA0 , 1 i5/
KeA0 V2
KeA2 KeA0
(7.283)
Under the reasonable approximation that the 2tt state with 1 = 0 dominates
the decays the i^smTK^ term in (7.277) becomes
so that !
Both e and e' involve phase differences, which cannot be rotated away by field
redefinitions. However, the relative importance of the two contributions to
e can be changed by choosing a different convention for t]k in (7.227). For
the common conventions the i^sm Aq/^Rc Aq contribution is < 5 — 10% of the
total (Andriyash et al., 2004) and is usually neglected. In any case, the phase
of e is predicted to be the superweak phase, 0€, defined in (7.278). If one
neglects the second term in e and approximates 0e by 7r/4 one obtains the
commonly used approximate form
e~^f »*."*«» V (7.286)
y/2 \ AmK
where
MKlK2 = (K°\M\K$) = i%mMKR. (7.287)
e' is predicted to be small because of the u factor. For that reason and because
of the accidental near equality of 0e and 0f€ one expects
<A)o ~ <p+- =0e= 43.51(5)°. (7.288)

The Standard Electroweak Theory
391
prom (7.267) we see that this is indeed the case. This is often viewed as a
test of CPT invariance, which was critical in the derivation. When one uses
the SM expressions for the WCC, the Ke3 asymmetry parameter in (7.269) is
predicted to be
SL ~ 25te e = 25tee = 3.23 x 1(T3, (7.289)
in agreement with the experimental value.
The SM prediction for e is obtained by combining (7.256) and the related
results with (7.286) (Buras, 2005),
e
G2FM^f2KmK ^Kein/4
6y/27r2AmK
1 3.84;io4 (7-29°)
X
SsmXthfleXc [f]iS0(xc) - %So(xc,xt)] - dieAt^SoO&t)),
where we have used ^n An = 0 and the conventions for Vq in (7.211). The
last term dominates. Using the experimental value for e and expressing the
result in terms of the Wolfenstein parametrization,
fj[(l - p)A27]2S0{xt) + 0.29(7)] A2BK = 0.187, (7.291)
which corresponds to the hyperbola labelled €k in Figure 7.37. The
uncertainty in the band is dominated by that in Bk-
e' is due to direct CP violation in the decay amplitudes. It is believed to be
dominated by the penguin diagrams (so-named for their appearance) shown
in Figure 7.39. The gluon penguin contributes only to Aq in the SU(2) limit,
and may be partially responsible for the A J =1/2 rule. However, because
of the small value of lj, such corrections as isospin breaking and the Z and 7
penguins are also important. There is a cancellation between the diagrams,
increasing the theoretical uncertainty. Furthermore, the matrix elements are
sensitive to the s quark mass (which is rather uncertain—see (5.124)), which
enters in some approaches using the PC AC equations in (5.117) and (5.120).
Estimates have utilized large Nc, chiral perturbation theory, lattice, and other
techniques for the long distance effects (for recent reviews, see D'Ambrosio
and Isidori, 1998; Bertolini et al, 2000; Buras and Jamin, 2004; Pich, 2004).
The SM predictions for die (e'/e) oc Sra A* oc 77 are roughly consistent with the
experimental value in (7.268), but the uncertainties are too large to usefully
constrain the unitarity triangle.
7.6.4 Mixing and CP Violation in the B System
Mixing and CP violation in the neutral B system have been extensively
studied, especially at LEP and the SLC, and more recently by CLEO at
CESR (Cornell), at the asymmetric B factories BaBar at PEP-II (SLAC) and
Belle at KEKB (KEK), and by CDF and DO at the Fermilab Tevatron (for

392
The Standard Model and Beyond
FIGURE 7.39
The penguin diagrams for | AS\ = 1 processes. Left: the gluon penguin, which
only contributes to Aq. Right: electroweak penguin, which contributes to A2.
reviews, see Barberio et al., 2008; Amsler et al., 2008)*. The
asymmetric B factories employ high intensity e± beams of unequal energy so that
e~e+ —> T(45) —> B%B% yields B^B^ pairs that are boosted along the beam
direction. This allows larger and more easily measured distances between the
production and decay vertices, and allows the time between the decays to be
measured by their spatial separation along the boost direction. The rates at
the Tevatron (and at the LHC) are much higher, but are in the more difficult
hadronic environment. The higher-energy machines allow study of the B®
system as well as the B%.
The formalism for B® — B® mixing and CP violation, with i = d,s, is similar
to that for the neutral kaons (see, e.g., Carter and Sanda, 1981; Buras, 2005;
Nir, 2007; Amsler et al., 2008). One important distinction is that in the kaon
system there is a very large difference between the Kl and Ks lifetimes,
because in the absence of CP breaking the 2tt state is only accessible to the
Ks. There is no analog for the neutral B decays because of the many decay
channels, and one expects similar lifetimes for the two mass eigenstates, which
are instead labelled jB/^ and Bl{ , where H and L denote "heavy" and "light."
Similar to (7.226) and (7.227), we define the strong interaction eigenstates
\B°d) = \db), |BS> = \bd) = -CP\B°d)
\B°S) = \sb), \B°8) = \bs) = -CP\B°S), (7.292)
where we have chosen an j]bx = — 1 phase convention. The B® and B® are
mixed§ by box diagrams similar to Figure 7.38, but in this case only the t
* Mixing in the neutral D system has recently been observed by BaBar, Belle, and CDF (e.g.,
Meadows and collaboration, 2008; Amsler et al., 2008). The results are consistent with the
SM expectations, though the latter have significant long distance uncertainties. New physics
implications are considered in (Golowich et al., 2007).
§i?2 — B® and B% — B® mixing are negligible because the diagonal terms are not degenerate.

The Standard Electroweak Theory
393
quark exchange is significant and the long-distance corrections are expected
to be small. The eigenstates are
\BHi)=Pi\B°) + qi\B?),
where
\BLi)=Pi\B°)-qi\B?)
Pi
1 + ei
1VIBiBi
MstBi
r* -
_?- BiBi
1 2
4 2
-I 1/2
i = d,s, (7.293)
(7.294)
Ignoring rB.£., which corresponds to ignoring the difference in lifetimes, one
finds that qi/pi is a pure phase and that
Ami = rnHi - mLi = 2|MB.5.|,
Ami
(7.295)
Ami can be measured by observing B® — B® oscillations. Suppose one starts
with an initial state tagged as a B®. The tagging is typically done by observing
the decay of the other meson^, e.g., from a B° — B° or B° — B~ pair. In
some cases, such as T(45) —> B^B^ (or <j> —> K°K°), the two are produced in
a coherent state, but we will ignore that complication. Similar to (7.259),
IV'(O))
At a later proper time r,
\B?)
2pi
(\BHi) + \BLi)).
(7.296)
|^(r))
±.(\BHl)e-im»iT + \BLl)e-im^)t
2
COS
f Arriii
V 2
wi-*(t)^:
2
(7.297)
where rrii = (raj/. +771^ )/2 and we have ignored the lifetime difference. Ami
can then be obtained by observing the proper time dependence for Bf and/or
Bf decays, e.g., by b -> c^"^ (from B°) or 6 -> cT^ (from B°). The
results are Amd = 0.507(5) ps"1 and Ams = 17.77(12) ps"1. The latter
was especially difficult to measure (by CDF and DO at the Tevatron) because
of the much shorter oscillation time 27r/Ams. (Early studies utilized time-
integrated effects, such as the relative number of same-sign and opposite-sign
lepton pairs resulting from associated B° — B° production.)
The standard model calculation from the box diagram yields
C2 M2
Am, = ^^VBmBo SQ{xt)\V^Vtb\2 BBJ%t, (7.298)
"Same-sign tags, utilizing other 6-jet fragments, are also possible.

394
The Standard Model and Beyond
where mBo ~ 5.28 GeV, mBo ~ 5.37 GeV, tjb ~ 0.55(1) is the short distance
QCD correction (Buras, 2005), and So(xt) is defined in (7.258). BB% and
fst are analogous to Bk and Jk> From unquenched lattice QCD
calculations (Amsler et al., 2008), fBsB1^ ~ 275(17) MeV and fBdB1^ ~ 223(18)
MeV. Some of the theoretical uncertainties cancel in the ratio £ = 1.23(4).
The values for \Vtd\, |Vis|, and the ratio quoted in Section 7.6.1 utilized (7.298)
with Vtb ~ 1. Alternatively, one can use the Ami as a constraint on the uni-
tarity triangle. In the Wolfenstein parametrization
\v;dvtb\2 = A2\%\-p)2 + ffl \v;svtb\2 = a2\\ (7.299)
so that Arrid yields the circular annulus centered at (p, 77) = (1,0) shown in
Figure 7.37. The uncertainty is dominated by BBdf%d, so one can obtain a
tighter constraint from the ratio Amd/Ams, also shown in Figure 7.37. One
sees that the \Vub\ and Amd/Ams constraints intersect at (p, 77) ~ (0.2,0.35),
with large uncertainties, and that the €k hyperbola is consistent.
CP Asymmetries
Many CP-violating asymmetries have been measured or searched for in the
neutral (and charged) B system, which have established indirect CP violation
and strongly supported the validity of the unitarity triangle predictions (for
reviews, see Buras, 2005; Nir, 2005; Barberio et al., 2008; Amsler et al., 2008).
We will illustrate with one especially clean and important case, i.e., the time-
dependent asymmetry !
a'(T) = r(B°(T)^f) + r(B°(T)^fy (7-300)
where we have suppressed the label i = d or s. |jB°(t)) and \B°(r)) are defined
as the states that were initially tagged as B° or B°, and / is a CP-eigenstate,
such as J/ipKs, 7r+7r~, or pKs- One observes asymmetries associated with
the interference between decays with and without mixing, and in some cases
asymmetries associated with the decays themselves. Similar to (7.297),
where
|B°(t)> = |B0>/+(t) + |B0>^/-(t)
P
\B°(r)) = |jB°)/_(t)? + |B°)/+(t)1
f_(T) = -ism(^1)e-i™e-Sr,
(7.301)
(7.302)

The Standard Electroweak Theory 395
and we have again ignored the lifetime difference. Also, define
Af = (f\Hnl\B°), Af = (f\Hnl\B°) (7.303)
in analogy with (7.279). Af and Af may be written
Af = Y,\A°\eiSaei,fia, Af = -VfJ2\Aa\eiS°e-i^, (7.304)
a a
where the sum is over the diagrams a which contribute to the decay, 5a is the
strong final state interaction phase associated with Aa, (pa is the weak CP-
phase generated (in the SM) by the phases in the CKM matrix, and 77/ = ±1
is the CP parity of /, i.e., CP\f) = 7]f\f). The decay amplitudes at proper
time r are then
(f\Hnl\B°(T)) = Af[f+(T) - \ff-(r))
</|W„,|B°(t)> = A,* [/_(t) - Xff+(T)], (?-305)
where
\f = -^. (7.306)
1 pAf
„-rv
The decay rates are
r(jB°(r) ->/) = \Af\2 [c+ + c_ cos(Amr) -9mA/ sin(Amr)]e"
T(B°(r) ->/) = |A/|2[c+-c_cos(Amr) + 9mA/ sin(Amr)] e"rr,
(7.307)
where c± = (1 ± |A/|2)/2 and we have used \p/q\ ~ 1. (7.307) leads to the
asymmetry
af(r) = Cf cos(Amr) - Sf sin(Amr), (7.308)
where
c/ = rrp^p' 5/ = itp^f- (7-309)
(There are a number of conflicting notations for Cf and Sf and their signs in
the literature.) Since \p/q\ ~ 1, a nonzero Cf implies direct CP-breaking in
the decay, i.e., \Af/Af\ ^ 1, while Sf is the mixing-induced term.
Especially simple to interpret are decays for which one diagram dominates
(or two diagrams have the same weak phase). In that case the strong
interaction effects cancel, and \f is just
\f = +71f<±e-2i* = 7]fe2i^e-2iv, (7.310)
where <p is the weak phase and 2ipM 1S the phase of q/p, so that
Cf = 0, Sf= 7]f sin 2(<pM - <p). (7.311)

396
The Standard Model and Beyond
In particular, for the golden modes / = J/ipKs,L (with 77/ = =f1,
respectively), one has Sf = ± sin 2/3, which corresponds to
Q
P
M
BB
M
BB
1/2
— Ilk.
vttvtd
VtbV.
= e-2i?
td
(7.312)
as can be seen from Figure 7.36. From Figure 7.40 the contribution of the
weak phase is
(7.313)
D-2iy> _
vcsv
1.
cb
Neglecting other small effects associated with T BB and the CP breaking in the
kaon system, one obtains Sj/^ks,l = ±sin(2/3). The J/i/jKs,l asymmetries
have been measured by BaBar and Belle, leading to sin 2(3 = 0.681(25). As
can be seen in Figure 7.37 this agrees well with the region in the p — 77 plane
determined by the other methods (other observations largely eliminate other
branches for 2(3).
J/rp
K°
b
d-
■ u
d
+ 7T-
FIGURE 7.40
Top: tree-level diagram for B® —> J/ipKs,L (left) and B® —> tt^tt
Bottom: Gluon penguin diagrams for B® —> (j)K° (left) and B®
(right). Electroweak penguins involving Z and 7 are also possible.
(right).
The direct measurement of the angles a and 7 is more difficult. For example,
the asymmetry for B° —> tttt would yield sin 2a if it were due entirely to
the tree-level diagram for b —> duu shown in Figure 7.40. However, there
is also a significant contribution from the b —> d penguin diagram, which

The Standard Electroweak Theory
397
has a different phase. The tree and penguin effects can be sorted out with
some effort, using different isospin channels (Gronau and London, 1990) for
go __> 7r+7r",7r07r° and B+ -> 7T+7T0 decays, as well as isospin, polarization,
and Dalitz plot information on related B —> pit and pp decays, yielding a ~
88(6)°. Measurements of 7, with larger uncertainties, are reviewed in (Amsler
et al., 2008).
Another class of charmless decays, such as B° —> (j)Ks or tt~Ks are driven
by the b —> sqq penguin diagrams shown in Figure 7.40. These are also
sensitive to sin 2(3 in the SM. Since the SM amplitude is loop-suppressed, these
modes are a good place to search for the effects of new physics (Buras et al.,
2004; Buchalla et al., 2005), which could enter in penguin diagrams, such
as in supersymmetry (e.g., Artuso et al., 2008), or at the tree-level, such as
in models with an extended Higgs sector or a Z' with flavor changing
couplings (e.g., Barger et al., 2004). Each measurement is individually consistent
with the J/ipK° value, but there is a hint of a systematic shift to lower values
of the effective sin 20. The effect is too small to make a definitive statement
at present.
As of this writing, asymmetries in Bs decays are being measured by CDF
and DO. There are some indications of non-standard model effects, but again
the situation is in flux.
The Wolfenstein Parameters
The constraints from the magnitudes of the CKM elements; K, D, and B
mixing; e, e'; and CP violation in the B decays are generally consistent with
the SM predictions and with the unitarity triangle, although there is still room
for new physics and even hints of discrepancies. A global fit (see the CKM
article in Amsler et al., 2008) to the CKM parameters assuming unitarity
yields
A = 0.226(1), A = 0.81(2), p = 0.14(3), 77 = 0.35(2) (7.314)
as well as the results in (7.212). The J parameter defined in (7.216) is found
to be J = 3.1(2) x 10~5. For more information on the unitarity triangle, see,
e.g., (Ali, 2003; Charles et al., 2005; Rosner, 2004; Buras, 2005; Nir, 2005,
2007; Barberio et al., 2008; Amsler et al., 2008), as well as the websites listed
in the bibliography. Future prospects for the unitarity triangle and related
physics studies include the LHCb detector at CERN and a possible future
Super B factory.
7.6.5 Time Reversal Violation and Electric Dipole Moments
Time reversal and CP violation (Section 2.10) are closely related, in the sense
that any CPT invariant interaction that violates one must violate the other.
We are assuming CPT here, so we will continue to use CP and T
interchangeably for theoretical discussions. Nevertheless, CP and T are different symme-

398
The Standard Model and Beyond
tries with different physical consequences, so possible T-violating observables
open a new window on standard model tests and new physics searches. In
particular, the effects of true CP violation are essentially limited to flavor
changing processes such as K and B decays, while T-odd observables such as
electric dipole moments are also relevant for flavor-diagonal channels.
The only direct observation of T violation to date is a difference in the K°
and K° oscillation probabilities by the CPLEAR collaboration at CERN (An-
gelopoulos et al., 2003). Another possibility is to directly observe T violation
in scattering or decay processes. However, this is very difficult because T
not only reverses spins and three-momenta, but also interchanges initial and
final states. For example, consider the matrix element (f\Hw\i) of the weak
Hamiltonian (or relevant effective operator) between an initial state i and final
state /. Using (2.312),
(f\Hw\i) = (Tf\THwT-l\Ti)* = (Ti\THwT~l\Tf), (7.315)
where Ti and Tf have their momenta and spins reversed compared to i and /
and also have the roles of "in" and "out" states interchanged, and we have used
that Hw and THwT~l are Hermitian. A direct test of whether THwT~l =
Hw requires the comparison of two different reactions, which is very difficult
(or impossible for decays).
It is, however, possible to search for pseudo T violation by measuring such T
odd quantities as the triple correlations p\ • (p2 x p3 ) or S\ • (p2 x p3 ), where Sj
or pj are spins or momenta in the process". However, care must be taken with
strong or electromagnetic final (or initial) state interactions, which can mimic
the effects of T violation or at least complicate its determination. For diefi-
niteness, we will consider the triple product r(p) = p\ • (p2 xps ). Correlations
involving spin and more detailed derivations are described in (Gasiorowicz,
1966; Sozzi, 2008). The expected value of r is
_ fDfT(p)\(Ww\i)\2 (? ....
<r(p)>- /£/K/l^>l2 ' (7-316)
where Df contains the appropriate phase space and flux factors. Any CP or
T violating effect requires interference between contributions to the amplitude
with different phases, analogous to (7.304). Since r{p) is also odd under space
reflection, one needs the interference of amplitudes with difference parities.
Let us therefore assume that
(f\Hw\i) = A{p)eiSAei(fiA + B{p)ei^+5B)eiipB, (7.317)
where we have suppressed the spin indices. 5a,b are the strong phases
associated, e.g., with final state interactions; <Pa,b are the weak phases associated
"Such correlations must be associated with Lorentz invariant quantities, such as
£nvp<rPoPiP2P3' so tne Process must be sufficiently complicated to allow enough indepen-
dent four-vectors.

The Standard Electroweak Theory
399
with the T and CP-violating parts of the interaction; and the real amplitudes
A and B have opposite parities, i.e., A(-p) = A(p) and B(-p) = -B(p).
The phase n/2 could have been absorbed in 6B, but is pulled out for
convenience. Such a relative phase always occurs in the interference between even
and odd parity amplitudes when summing over spins, such as can be seen in
(2.168). It is clear that
(r(p)> oc cos(6A - 6B) sin((pA - <Pb) + sin(SA - SB) cos((pA - <pB), (7.318)
so a non-zero value can be induced by final state interactions (6a~6b / 0), by
T and CP violation (y>A — Vb ¥" 0)> or both. For this reason, it is difficult to
isolate T violating correlations in processes involving more than one strongly
interacting final or initial state particle. There is, however, some possibility of
observing such effects in cascade decays in supersymmetric theories in which
there are mainly leptons and neutralinos in the final state (e.g., Bartl et al.,
2004; Langacker et al., 2007; Ellis et al., 2008).
More promising are electric dipole moments (EDMs) df for particle / (for
recents reviews, see Pospelov and Ritz, 2005; Sozzi, 2008; Ibrahim and Nath,
2008), defined by its interaction with an electric field E,
Hedm = -df • E. (7.319)
For a spin-^ field this corresponds to an effective Langrangian density
Cedm = Y^f^^fFnv (7.320)
If we define the electromagnetic form factors F{ and g{ of 'ipfy^ipf analogous
to those of the proton in (2.388) and (2.389), then df is related to g{ by
j = _{ll^Ma = ^Sl^iMs, (7.321)
2m/ rrif
where qfe and rrif are the electric charge and mass of / (Problem 2.22). For
quarks one can also define analogous chromoelectric dipole moments (CEDMs)
d? by
Ccedm = ^Qra^^qr^. (7.322)
It is obvious from (7.321) that an electric dipole moment (or chromoelectric
dipole moment) violates time reversal and space reflection invariance, since
by classical reasoning df must be even (odd) under T (P), while S is just the
opposite. This extends to any non-degenerate system using the Wigner-Eckart
theorem, and will be shown formally for spin-| in Appendix G.
There are stringent experimental limits on possible EDMs for the electron
and neutron, and also on the (paramagnetic) 205J7 and (diamagnetic) 199Hg

400
The Standard Model and Beyond
atoms:
de = 0.07(7) x 10-26, \dn\ < 2.9 x 10~26 (90%)
\dHg\ = 2 x 10"28 (95%), |dn| < 9 x 10~25 (90%) (7.323)
all in units of e-cm. Future experiments may improve the sensitivities
significantly, by as much as four orders of magnitude for de. There are also
prospects for new or greatly improved sensitivities to other EDMs, such as
the muon or deuteron. The atomic** EDMs are complementary to those of
the e~ and n because they receive contributions not only from the EDMs of
their constituents, but also from CP-violating eN or ttN interactions.
The predicted EDM effects due to the CKM mixing in the standard model
are extremely small. The quark EDMs are only generated at the three-loop
level, and are expected to be of O(l0~34e — cm), while de only receives
contributions from four-loop diagrams (at least for massless neutrinos) and should
be smaller than ~ 10_38e-cm. The contributions of the quark EDMs to dn are
expected to be negligible, with the largest CKM contribution d%KM ~ 10~32e-
cm from a two-loop diagram involving a gluon penguin. One major
complication, however, is from the strong CP violation (Oqcd) term in the QCD
Lagrangian density in (5.1), which contributes den ~ 4.5 x 1O_150 (Kim and
Carosi, 2008). One therefore requires |0| < 10-11, while CKM effects are
expected to shift 6 by O(l0~3). It is not known whether 6 is small due to an
accidental or other cancellation between the bare and shift values, or to some
dynamical mechanism, and therefore whether or not d6n = 0. This strong
CP problem will be elaborated in Section 8.1. The atomic EDMs are also
expected to be small in the SM.
Because the SM contributions are expected to be so small (except possibly
for d6n) EDMs are an excellent place to search for the effects of new physics.
These typically have new CP-violating phases and allow EDMs at one-loop
level, leading to values already excluded or within reach of future EDM
experiments. For example, in the MSSM the e" and quark EDMs and CEDMs
(which feed into EDMs) can be generated by one-loop vertex diagrams
involving neutralinos, charginos, gluinos and the fermion scalar partners (e.g.,
Ibrahim and Nath, 2008; Ellis et al., 2008), with new CP phases possible from
the // and jB/x terms, gaugino masses, and A terms. There may also be
significant effects from induced three-gluon operators (Weinberg, 1989a), and from
Higgs exchange and two-loop diagrams involving Higgs fields (Barr and Zee,
1990). For large CP phases the existing EDM limits typically require SUSY
masses ^> 0(1 TeV) unless there are fine-tuned cancellations. Conversely,
superpartner masses < 1 TeV require small CP phases, creating some
tension with models of electroweak baryogenesis (see Section 8.1). At any rate,
**The Schiff theorem (Schiff, 1963) states that atomic EDMs vanish in the non-relativistic
limit for a point like nucleus because of electron screening effects. Fortunately, it is violated
by relativistic and finite nuclear size effects.

The Standard Electroweak Theory
401
future EDM experiments have an excellent chance of observing EDMs if TeV
scale supersymmetry, or many other SM extensions, exists, and would be a
powerful diagnostic.
7.6.6 Flavor Changing Neutral Currents (FCNC)
Flavor changing neutral currents are similar to electric dipole moments and
CP violation in that they are strongly suppressed in the SM, but can be much
larger in most extensions.
In the standard model the couplings of the Z to fermions are flavor
diagonal at tree level because of the GIM mechanism (Glashow et al., 1970),
i.e., because all fermions that have the same charge, color, and chirality and
are therefore able to mix with each other are assigned to the same kind of
SU(2) x U(l) representation. Off-diagonal Z-fermion couplings are induced
at loop level, mainly by penguin diagrams such as in Figure 7.39, but these
are small. The GIM mechanism also significantly suppresses the contributions
of box diagrams such as in Figure 7.38. Similarly, the couplings of the Higgs
to fermions are flavor diagonal in the SM because the same transformations
that diagonalize the fermion mass matrices automatically diagonalize the H
Yukawa couplings. This continues to hold in models with multiple Higgs
doublets provided that only one doublet couples to (pLu°R, one to (f[d>0R, and
similarly for the leptons (Glashow and Weinberg, 1968).
Many types of new physics lead to FCNC, often including new sources of
CP violation (see, e.g., Langacker, 1991). In many cases the new effects
enter at tree level, where they are especially significant because they compete
with SM loop effects. Other types enter at loop level, but may be enhanced
compared with SM loop effects by stronger couplings or by not having the
same cancellations. In some cases the new physics leads to processes that are
forbidden or nearly so in the SM. For example, the SM exhibits an
approximate lepton flavor symmetry: to the extent that the v masses can be ignored
there are separately conserved Le, LM and LT numbers, with Le = +1 for
(e~,2/e), Le = 0 for (//~,z/M) and (r_,2/r), and analogously for I/M)T.
Processes such as // —> ey are therefore essentially forbidden in the SM up to
negligible (mv/MwY effects (Cheng and Li, 1977), but are allowed in many
extensions.
One type of new physics which leads to FCNC effects involves heavy Z'
gauge bosons with GIM-violating family non-universal couplings (Langacker
and Plumacher, 2000), as frequently occur in string constructions (e.g., Blu-
menhagen et al., 2005). Closely related are Kaluza-Klein excitations of neutral
gauge bosons in models in which the fermion families are located at different
positions in large ((9(TeV-1)) or warped extra dimensions (Delgado et al.,
2000), which therefore have family non-universal couplings. The mixing of
ordinary fermions with heavy ones with exotic SM quantum numbers (e.g.,
left-chiral singlets or right-chiral doublets) can also lead to off-diagonal Z
couplings (Langacker and London, 1988b). FCNC may also be mediated by

402
The Standard Model and Beyond
heavy gauge boson exchange in alternatives to the Higgs mechanism, such
as extended technicolor (Eichten and Lane, 1980; Hill and Simmons, 2003-
Appelquist et al., 2004). Another possibility is multiple Higgs doublets which
couple to the same types of fermion (Hall and Weinberg, 1993; Atwood et al.
1997). Leptoquarks are possible particles with couplings to quarks and lep-
tons, such as (e^ — ^lR)QLdc, while diquarks have couplings like dfiucLdc. In
this example, the antilepton fields are expressed in a conjugate form
analogous to the conjugate Higgs doublet in (7.14), and dc is a heavy spin-0 color
anti-triplet with electric charge +^. The tilde is suggestive that dc could be
the scalar partner of the sc (or the dc for the leptoquark case) in il-parity
violating versions of supersymmetry (Hewett and Rizzo, 1989). The two types
of coupling violate lepton number and baryon number, respectively, and rapid
proton decay would result if both were present simultaneously. Even if one
is absent (stabilizing the proton) the other could mediate FCNC (Chemtob,
2005; Barbier et al., 2005). We finally mention that models involving
composite quarks or leptons lead to FCNC by constituent interchange (Bigi et al.,
1986).
Loop effects may also generate significant effective FCNC interactions by
box diagrams analogous to Figure 7.38 or by penguin-like vertex corrections
that lead to off-diagonal Z vertices. These can be very important in
supersymmetry (Gabbiani et al., 1996; Misiak et al., 1998; Ellis et al., 2007;
Hamzaoui et al., 1999; Gorbahn et al., 2009), for example, especially in
diagrams involving gluinos (which couple strongly) or in the limit of large tan/?
(for which the b Yukawa is large), unless the scalar partners of the same type
are nearly degenerate or the mixings are aligned with the CKM mixings.
Similarly, there can be important loop-induced effects in SU(2)l x SU(2)r x U(\)
models involving a heavy Wr coupling to V + A (Beall et al., 1982; Langacker
and Uma Sankar, 1989; Zhang et al., 2007), or in extensions involving heavy
neutrinos (Cheng and Li, 1977).
There are stringent constraints on new sources of FCNC and CP violation
from K° - K°, D° - D°, and B° - B° mixing (Sections 7.6.3 and 7.6.4).
There have also been extensive searches for and observations of rare K
decays (Ritchie and Wojcicki, 1993; Buras and Fleischer, 1998; Barker and Ket-
tell, 2000; Buras et al., 2008; Artuso et al., 2008; Amsler et al., 2008). Many
focus on lepton flavor-violating decays such as Kl —> /x±e=F or K+ —> 7r","/x±e:F,
which are completely negligible in the SM. Others are useful for determining
CKM parameters (e.g., K+ —> tt^i/u in Section 7.6.1), searching for direct
CP violation (e.g., KL —> tt°£+£~ or 7r°2/i/), testing CPT, or probing
longdistance strong interaction physics. Charmless B decays are also sensitive to
new tree and loop contributions (Section 7.6.4), as are other rare or forbidden
processes such as b —> sj, B —> /x+/x~, or B —> /xe. Similarly, there have
been extensive searches for other types of lepton flavor-violation, such as the
leptonic processes // —> 3e, // —> e7, \i —> e conversion in interactions with a
nucleus, and analogous r decays (Pich, 2008; Marciano et al., 2008), as well
as in B, D, and Z decays.

The Standard Electroweak Theory
403
As will be described in Chapter 8 there are good reasons to suspect that
new physics emerges at the TeV scale. As emphasized above, most types are
likely to lead to observable FCNC, CP violation, and EDM effects. In fact,
it is surprising that no such effects have yet been observed. The constraints
on new physics relevant to FCNC can be parametrized by effective higher-
dimensional operators, especially four-fermi operators (e.g., Buchmuller and
Wyler, 1986) such as
4/5'=2= E ^(darsa)(db^sb) + h.c. (7.324)
a,b=L,R ab
For example, Atuk and e (defined in (7.266)) imply (Nir, 2007)
-^ > 106 GeV [Am*], ~i== > 2 x 107 GeV [c], (7.325)
assuming only that the new physics part is not larger than the experimental
values. There are even stronger limits on the LR operators. The analogous
limits from B and D mixing are in the 105 — 106 GeV range. The implication
is that new physics at the A ~ TeV scale should have been seen by now
unless the coefficients like cab are very small. This could be due to very weak
coupling, but the motivations for new TeV physics such as the Higgs/hierarchy
problem or gauge unification suggest that it should have at least electroweak
scale coupling. One therefore presumably requires that the coefficients of the
flavor changing operators are strongly suppressed, similar to those from the
SM (e.g., the factor oim2c\Vcd\2\Vcs\2/M^ in (7.233)). Since most new physics
models do not have such strong suppressions, there is a strong tension between
attempts to solve the Higgs/hierarchy problem and constraints from FCNC,
CP, and EDMs. This tension has led to much recent discussion of minimal
flavor violation (MFV), which is the hypothesis that all flavor violation, even
that which is associated with new physics, is proportional to the standard
model Yukawa matrices (D'Ambrosio et al., 2002; Nir, 2007), leading to a
significant suppression of flavor changing effects similar to the SM. Examples
include supersymmetry with anomaly or (simple forms of) gauge mediation
of super symmetry breaking (e.g., Chung et al., 2005).
We have already emphasized that for massless neutrinos the SM possesses
an accidental lepton flavor symmetry. An extension of this idea, which
motivates minimal flavor violation, is that there are no flavor changing or CP
violation effects at all in the SM when one turns off the Yukawa
interactions (Oqcd becomes unobservable as well). In fact, in that limit there is a
global U(3)5 flavor symmetry associated with the three families each of #l,
^l, Ur, dR, and eR (U(3)6 if we include the vR), with the diagonal generators
corresponding to, e.g., conserved Ur, Cr, and tR numbers. Of course, this
symmetry is strongly broken by the third family Yukawas, but those of the
first two families are small.
It is still uncertain whether MFV is employed by nature, or whether FCNC
are suppressed by some other mechanism. It is clear, however, that FCNC,

404
The Standard Model and Beyond
ED Ms, CP violation, and rare decays have an enormous reach in searching for
new physics. Effects involving the t quark and other third family members
are especially interesting (Section 6.2.7), since it is so much heavier than
the other two and may play a role in new physics, especially new physics
associated with spontaneous symmetry breaking. The t quark decays will be
extensively studied at the LHC, and hopefully there will be new generations
of lower-energy experiments on rare //, r, K, and B decays and processes
and on EDMs. Lepton number and lepton flavor violation associated with
neutrinos will be discussed in Section 7.7, and baryon number violation, e.g.
baryogenesis, proton decay, and neutron oscillations, in Chapter 8.
7.7 Neutrino Mass and Mixing
Neutrinos are a unique probe of many aspects of physics, geophysics, and
astrophysics on scales ranging from 10-33 to 10+28 cm. Neutrino scattering and
decays involving neutrinos have been essential in establishing the Fermi theory
and parity violation, determining the elements of the CKM matrix, and
testing the weak neutral current predictions of the standard model, and therefore
played a significant role in the precision electroweak program as described
in Section 7.3.6. Deep inelastic scattering involving neutrinos and charged
leptons has also been critical in establishing the existence and properties! of
quarks, the structure of the nucleon, and the predictions of the short distance
behavior of QCD. Similarly, neutrinos are important for the physics and/or
probes of the Sun, Earth, stars, core-collapse supernovae, the origins of cosmic
rays, the large scale structure of the universe, big bang nucleosynthesis, and
possibly baryogenesis.
Neutrinos are also interesting because their masses are so tiny and because,
unlike the quarks, some of the leptonic mixing angles are large. Small neutrino
masses are sensitive to new physics at scales ranging from a TeV up to the
Planck scale, but because of their unusual nature there is a good chance
that they are somehow connected with the latter, possibly shedding light
on an underlying grand unification or superstring theory. The neutrinos are
also unique in that they do not carry either color or electric charge. It is
therefore possible (and many physicists think probable) that their masses
are Majorana (lepton number violating) rather than Dirac (lepton number
conserving, analogous to the quark and charged lepton masses). Establishing
the nature of the neutrino masses, as well as understanding the origin of the
small masses and large mixings, is of fundamental interest.
The original version of the SU(2) x U(l) model did not have any
mechanism to generate nonzero masses at the renormalizable level. However, it is
straightforward to extend the original model by the addition of SU(2)-sing\et

The Standard Electroweak Theory
405
right-chiral neutrinos, allowing Dirac mass terms. These could either yield
light Dirac neutrinos if the Yukawa couplings were extremely small, or light
Majorana neutrinos through the seesaw mechanism. One could also generate
small Majorana masses without right-handed neutrinos via extended Higgs
sectors or higher-dimension operators. Most extensions of the standard model
(with the notable exception of the minimal SU(5) grand unified theory)
involve one or both of these mechanisms, though they do not necessarily explain
the smallness of the masses.
In this Section we review the basic issues related to the neutrino masses
and mixings, the major classes of models, and some of the experiments. More
detailed discussions may be found in a number of books (Boehm and Vogel,
1987; Kayser et al., 1989; Bahcall, 1989; Langacker, 2000; Mohapatra and Pal,
2004; Giunti and Wook, 2007) and review articles (Vergados, 1986; Cvetic
and Langacker, 1991; RafTelt, 1999; Gonzalez-Garcia and Nir, 2003; Dolgov,
2002; Barger et al., 2003; Altarelli and Feruglio, 2004; Langacker et al., 2005;
Strumia and Vissani, 2006; Mohapatra and Smirnov, 2006; Mohapatra et al.,
2007; Gonzalez-Garcia and Maltoni, 2008; Camilleri et al., 2008; Amsler et al.,
2008).
7.7.1 Basic Concepts for Neutrino Mass
Active and Sterile Neutrinos
We saw in Chapter 2 that the minimal fermionic degree of freedom is a Weyl
two-component field, as defined in (2.196) and in Section 2.11. A Weyl field
can be represented either in four-component notation, e.g., by -0L — Pl^P —
., or in two-component notation as *&l- We will use both in this section.
It is useful to first distinguish between active and sterile neutrinos. Active
(a.k.a. ordinary or doublet) neutrinos are left-chiral Weyl neutrinos which
transform as SU(2) doublets with a charged lepton partner, and which
therefore have normal weak interactions. The electron-type L doublet and its
right-handed partner is
*-(*)■*'-(£)• (7-326)
in four-component notation, where tycR = C^)\ is the field related by CP to -0L,
as in (2.301) or (2.328), and we have also carried out a "tilde" transformation
on the SU(2) doublet indices analogous to the one for the Higgs in (7.14) so
that £ transforms as a 2 rather than a 2*. We reemphasize that we define
7°, i.e., the Dirac adjoint acts on V>l and not on V>-
Sterile (a.k.a. singlet or "right-handed") neutrinos, which are present in
most extensions of the SM, are SU(2) singlets. They do not interact except
by mixing, Yukawa interactions, or beyond the SM (BSM) interactions. In
four-component notation, a sterile right-chiral Weyl field will be written as

406
The Standard Model and Beyond
vr and its conjugate as z/£,
v*. -£? "I (7-327)
In two-component notation, the L and R chiral fields will be written as J\fL
and Nr, respectively, with their CP conjugates Nr and jV£:
Ml-^Mr (active), JsrR—^Jsri (sterile). (7.328)
Dirac Masses
A fermion mass term converts a Weyl field of one chirality into one of the
opposite chirality,
-C = m $aLrl>bR + ^bR^aL) = m (v^bR + * J* ^l) , (7.329)
as in (2.342). A physical interpretation is that a massless fermion has the
same helicity (chirality) in all frames of reference, while that of a massive
particle depends on the reference frame and therefore can be flipped.
A Dirac mass connects two distinct Weyl fields, i.e., ^bR ¥" ^aR- For
a single type of neutrino, a Dirac mass connects an active neutrino with a
sterile one^,
—Cd = mD {yLvR + vrvl) = mDvDvD
= mD(j^R+J^RATL), (?-330)
where i/£> = i/^ + vr is a Dirac field. It has four distinct components, z/^, V^,
vr and i/£, and there is a conserved fermion number (or lepton number L in
this case), corresponding to the global phases symmetry vl,r —► ^vl^r. This
L conservation ensures that there is no mixing between vi, and i/£, or between
vr and 2/^. However, when embedded in the SM context the neutrinos are
chiral, so mo violates the third component T\ of weak isospin by fcd\ = ^.
It can be generated by the Higgs mechanism, as described in Section 7.2.1
and illustrated in Figure 7.41, and is in principle analogous to the quark and
charged lepton masses. Dirac masses can be easily generalized to three or
more families. However, the tiny values of the neutrino masses requires that
the Higgs Yukawa couplings hv = mv/v defined in (7.46) would have to be
extremely small if they are due to a simple Dirac-type Higgs coupling: mv ~ 1
eV would correspond to hv ~ 10-11, for example, to be compared with the
t quark coupling ht = 0(1) or the electron coupling he ~ 10-5. Of course,
we do not understand the ratio he/ht either, so some caution should be taken
with such statements. In any case, most particle physicists believe that either
an alternative mechanism or some explanation for a small hv is needed, as
will be discussed below.
ft There are variant forms of Dirac neutrino masses involving two distinct active or two
distinct sterile neutrinos, as will be discussed below.

The Standard Electroweak Theory
407
vl «* • * Vr ul m ® + Vr
1 j *
<0°) <**> <</>°>
FIGURE 7.41
Mechanisms for generating a Dirac neutrino mass. Left: an elementary
Yukawa coupling to the neutral Higgs doublet field <jp. Right: a higher-
dimensional operator leading to a suppressed Yukawa coupling.
Majorana Masses
Majorana mass terms are more economical in that they only require a
single Weyl field, i.e., ^bR = ^caR m (7.329). They are not as familiar as Dirac
mass terms because they violate fermion number by two units. For the quarks
and charged leptons such mass terms are forbidden because they would
violate color and/or electric charge. However, the neutrinos do not carry any
unbroken gauge quantum numbers, so Majorana masses are a possibility.
For an active neutrino, a Majorana mass term describes a transition between
a left-handed neutrino and its conjugate right-handed antineutrino. In four-
component language, it can be written
-CT = ^j- {pLvCR + VrVl) = ^j- {phCvl + vTLCvL) = ^vmvm. (7.331)
As is clear from the second form, Ct can be viewed as the annihilation or
creation of two neutrinos, and therefore violates lepton number by two units,
AL = 2. In the last form in (7.331), vm = "l + ^r is a self-conjugate**
two-component (Majorana) field satisfying vm = "m = C^m- A Majorana
v is therefore its own antiparticle and can mediate neutrinoless double beta
decay (/?/?oi/), in which two neutrons turn into two protons and two electrons,
violating lepton number by two units, as shown in Figure 7.42. A Majorana
mass for an active neutrino also violates weak isospin by one unit, At\ = 1
(hence the subscript T for triplet), and can be generated either by the VEV
of a Higgs triplet or by a higher-dimensional operator involving two Higgs
doublets (such as the minimal seesaw model), as in Figure 7.43. The \ in Ct
is needed to yield the correct expression for the Hamiltonian. It is somewhat
**Unlike a Hermitian scalar, a Majorana state still has two helicities, corresponding to vl
and vcR. They only mix by the Majorana mass term, so there is still an approximately
conserved lepton number to the extent that tut is small. For example, there could be
a cosmological asymmetry between vl and i/^, even for Majorana masses, if the rate for
transitions between them is sufficiently slow compared to the age of the universe (Barger
et al., 2003).

408
The Standard Model and Beyond
analogous to the extra \ in the free-field Lagrangian density for a Hermitian
scalar. This is more obvious from the kinetic energy term
Cke = -jipMi $vM) = ^ (Phi 0vl + P%i fivcR) = vLi $vL, (7.332)
where the two terms are equal because of (2.297). The Majorana mass term
can be rewritten in two-component language using (2.326) as
-CT = ^ (N[N<R + N$NL) = ^ (Ml^Ml - NltfNL) . (7.333)
FIGURE 7.42
Diagram for neutrinoless double beta decay (/?/?o^)«
A sterile neutrino can also have a Majorana mass term of the form,
ms
!I^ fr.c_rr.cT , ltcTrilc\ _ ^S.
-£s = -f (PIvr + WD = -^ iplCvf + vfCvl) = ^vms'Ms,
(7.334)
where vms = v\+vr = v%fs. In this case, weak isospin is conserved, At^ = 0,
so ms can be generated by the VEV of a Higgs singlet*. In two-component
form
-CS = ^ (m?Mr + J^f_) = ^ (NUia^l* -Mfia2Af_) ■ (7-335)
The four-component fields for Dirac and for active and sterile Majorana
neutrinos can therefore be written
(7.336)
VD =
VM =
(ML\
UiJ"
(Nb\
Wr)~
( Mb \
~W*?)
( Nb \
~Wk)'
vms =
(Ml\
UJ~
( Ml
- Wni
*ms could also be generated in principle by a bare mass, but this is usually forbidden by
additional symmetries in extensions of the SM.

The Standard Electroweak Theory
409
Vh <*-
Ht
vl
"l
-*r
vl
(<t>s)
(<f>°)
•
<</>°>
(<p°)
VL
—*—A—*~•-
vr vr
(<f>°)
VL V^l
(4>°)
e
hr
4
i
i
e
(<p°)
"eL
FIGURE 7.43
Mechanisms for a Majorana mass term. Top left: coupling to a neutral Higgs
triplet field 0^. Top right: a higher-dimensional operator coupling to two
Higgs doublets. Botton left: the minimal seesaw mechanism (a specific
implementation of the higher-dimensional operator), in which a light active
neutrino mixes with a very heavy sterile Majorana neutrino. Bottom right: a
loop diagram involving a charged scalar field h~.
where two of the components are not independent in the Majorana cases.
The free-field equations of motion with a Majorana mass term obtained
from the Euler-Lagrange equation (2.18) are
i $vL - mTCvTL = 0, i^d^NL ~ mTio-2Afl = 0. (7.337)
This leads to the free-field expression
»m(x) = I ^ £ [u(p, s) a(p, s)e'^ + v{p, s)a\p, s)e^"] ,
(7.338)
which is of the same form as the free Dirac field in (2.155) except that there
is no distinction between a and b operators. Any spin basis can be used, but
the helicity basis is usually most convenient.
A more compact notation for dealing with Majorana masses and fields will
be developed in Section 8.2.2.
A Comment on Phases
We implicitly assumed that the masses mo and tut in (7.330) and (7.331) are
real and positive. More generally, however, they can be negative or complex,

410
The Standard Model and Beyond
but can be made real and positive by field redefinitions. In the Dirac case for
an arbitrary ip, one has generally
-CD = mD$LipR + m^R^L = mDty[vR + m*DVjR VL. (7.339)
There is freedom to redefine both ipL and -0R by separate phase
transformations to remove any phase in mo and make it positive (see Section 3.3.5 and
Problem 3.28). In the SM, only the WCC interactions depend on the phases
of the left-chiral fermion fields, and no SM interaction involves the right chiral
phases. It is therefore convenient to choose the phases of the L-chiral mass
eigenstates (-0L in the simple example in (7.339)) by any convenient
convention, and then adjust the Vr fields to make the mass eigenvalues real and
positive. This was done in Section 7.2.2 to remove unobservable phases from
the CKM matrix, and a similar procedure can be applied to the lepton mixing
if there are only Dirac masses.
However, for a general Majorana mass term,
-CM = g {rriM^LCtpl + m*Mil>lCil>L) = - (m^*l**2*l " ™m^1>2^l) ,
(7.340)
there is only one independent field -0L (or \I>l). One usually chooses to use
the phase freedom in -0L to make tum real and positive, in which case there is
no remaining freedom. This will imply the existence of additional Majorana
phases in the leptonic mixing matrix Ve for the WCC for the case of Majorana
neutrino masses. However, such phases are only observable in processes such
as /?/?ot/ which involve the phases of the neutrino masses explicitly, as can be
seen by working in an alternative convention of a simple Ve but leaving trm
complex.
Even after a phase redefinition we will define the conjugate fields by the
same convention as for the original ones. That is, for
u'L = e*vL, N'L = e^AfL, (7.341)
we define
Mixed Models
When active and sterile neutrinos are both present, there can be Dirac and
Majorana mass terms simultaneously. For one family, the Lagrangian density
has the form
-'-iwiz'zK'Xh1^ <7M3)
(7.342)

The Standard Electroweak Theory
411
where 0 refers to weak eigenstates, and the masses are
rriT : |AL| = 2, At3L = 1 (Majorana)
mD : |AL| = 0, At3L = - (Dirac) (7.344)
ms : |AL| = 2, At3L = 0 (Majorana).
The two terms involving mo are equal since
^aL^bR = ^bL^aR, ^PaR^bL = ^bR^aL
^aL^bR = ^bLlpaR, ¥aL^bR = ^bL^aR
(7.345)
for arbitrary ^a,6 by (2.291). Diagonalizing the matrix in (7.343) yields two
Majorana mass eigenvalues rrii and two Majorana mass eigenstates^, v^m =
ViL + viR = viM^ w^n ^ = 1, 2. The weak and mass bases are related by the
unitary transformations
,0c
similar to (7.47). Al and Ar are generally different for Dirac mass matrices,
which need not be Hermitian. However, the general 2x2 neutrino mass
matrix in (7.343) is symmetric because of (7.345). This implies that AVL =
A*£K, where K is a diagonal matrix of phases analogous to (7.52). (There
is additional freedom in K in the presence of degeneracies.) K is in general
arbitrary, but our phase convention, in which v£R = CvJL, implies K = I.
There are several important special cases of the mixed model in (7.343)
(a) Majorana: mo = 0 is the pure Majorana case: the mass matrix is
diagonal, with rai = mr, rri2 = ms, and
.. ..0 ,,c _.0c
V\L - ^L> V\R - VR
"2L = V°L, ^R = "%
(7.347)
(b) Dirac: The Dirac limit is mr = rns = 0. There are formally two
Majorana mass eigenstates, with eigenvalues vrt\ = mo and rri2 = —mo and
eigenstates
'For Majorana masses, and especially in mixed models, there is no conserved lepton number,
and it is just a matter of definition to refer to the L states as particles, Vil, and the R states
as antiparticles, v£R. Unfortunately, this becomes awkward in the Dirac limit where the
Weyl states vr and u^ are labelled as particle and antiparticle because they carry lepton
number +1 and —1, respectively. There is no notation known to the author that is not
awkward in some circumstances.

412
The Standard Model and Beyond
VlL = ±={vl + v°L% v\R = -L(*# + uR)
i i (7.348)
u2L = ^=("° " "£), "C2r = ^=("£ ~ "&)•
Note that 1/1,2 are degenerate in the sense that \m\\ = |ra2|, but the
actual eigenvalues have opposite sign. To recover the Dirac limit, we
will depart from our usual procedure of redefining the phase of v2 to
make m2 positive. Rather, let us expand the mass term
-C = ^-{p1LulR - v*lvc2R) + h.c. = mD(v°Lis°R + i/&i/£), (7.349)
which clearly conserves lepton number (i.e., there is no v\ — z/£c or
u*r — v% mixing). Thus, a Dirac neutrino can be thought of as two
Majorana neutrinos, with maximal (45°) mixing and with equal and
opposite masses. This interpretation is useful in considering the Dirac
limit of general models.
(c) Seesaw: The limit ms ^> rriD,T (e.g., mr = 0, mo = 0(mu,me,md),
and ms = O(Mx), where Mx ~ 1014 GeV) is known as the seesaw (or
minimal seesaw) (Minkowski, 1977; Gell-Mann et al., 1979; Yanagida,
1979; Schechter and Valle, 1980). The eigenstates and eigenvalues in the
seesaw limit are
0 mD oc 0 mD
ms ms
V2L ~ —1/£ + v°l ~ v°l, m2 ~ ms, (7.350)
ms
with \m\ I <C mo for mr = 0. At energies low compared to ms the v2
decouples and one obtains an effective theory involving a single active
Majorana v\m ~ ^l + ^rC- The minimal seesaw mechanism is illustrated
in Figure 7.43.
(d) PseudoDirac: this is a perturbation on the Dirac case, with mr, ms <
mp. There is a small lepton number violation, and a small splitting
between the magnitudes of the mass eigenvalues. As an example, mr =
e, ms = 0 leads to Irai^l = rnD ± e/2. The pseudo-Dirac case is also
sometimes encountered for the variant Dirac forms involving two active
or two sterile neutrinos.
(e) Mixing: The general case in which mo and ms (and/or mr) are both
small and comparable leads to non-degenerate Majorana mass
eigenvalues and significant ordinary-sterile (z/£ — z/£c) mixing, such as were
suggested by the LSND results to be described below. Only this and
the pseudo-Dirac cases allow such mixings.

The Standard Electroweak Theory
413
Extension to Two or More Families
These considerations can be generalized to two or more families, or even to the
case of different numbers of active and sterile neutrinos. For F = 3 families,
define the three-component weak eigenstate vectors
•4l\ (Al
"9 = I Al , A° = Al
<4l) \Al,
(7.351)
and similarly for vRc, i/R and the two-component fields. In the Dirac case,
(7.330) generalizes to
-£D = (ylMoi/lt + P°RM^°L) = (N^MdN% + A/^M+X) , (7.352)
where Mo is a completely arbitrary 3x3 matrix. Mb can be diagonalized
by separate left and right unitary matrices AUL R, just as in (7.47),
(mi 0 0 \
0 m2 0 , (7.353)
0 0 m3J
with mass eigenstate fields
vL = AvyL, uR = AR^R. (7.354)
The leptonic part of the weak charge raising current is then
J^ = ^L^VeeL, (7.355)
where
Ve = Av^Ai, (7.356)
is the leptonic mixing matrix, analogous to the quark mixing matrix Vq in
(7.61). V = V\ is also know as the PMNS matrix, after Maki, Nakagawa, and
Sakata (Maki et al., 1962) and Pontecorvo (Pontecorvo, 1968). The counting
of angles and phases in Ve is the same as for the CKM matrix (see Eq. 7.62).
One can choose the arbitrary phases in AUL (i.e., the KVL matrix analogous
to (7.51)) to remove unobservable phases from Ve (leaving 3 angles and one
phase for F = 3), and then choose those in AVR (i.e., K^) to make the mass
eigenvalues real and nonnegative.
For Fa = 3 active and Fs > 3 sterile neutrinos (and only Dirac masses),
three linear combinations of the sterile fields will join with the active ones
to form three massive Dirac fields, leaving Fs — 3 massless Weyl fields. The
reverse situation occurs for Fs < Fa-
In the Majorana case the triplet mass term (7.331) generalizes to
-Ct = \ (i>lMTvRc + i#MM) = \ ("iMrCvf + ^TCM^°) ,
(7.357)

414
The Standard Model and Beyond
where Mr is symmetric, Mr = M%, by (7.345). This symmetric property
holds for all Majorana mass matrices. The corresponding two-component
expression is
-Ct = \ (N?MTio2M°L* ~ M2Tiv2M^MZ) ■ (7.358)
Since Mt is symmetric, it can be diagonalized by a transformation analogous
to (7.353), but with AVL = A*£K, where as usual K is an undetermined
diagonal phase matrix in the absence of degeneracies. We will choose K = J
in order to maintain v\R — CvJL for the eigenstates (cf (7.342)). In that case
the phases in AVL are uniquely determined by the requirement that the mass
eigenvalues mi are real and positive*. Consequently, there is less freedom
to remove phases from Ve than in the Dirac case (or in the CKM matrix).
The counting in (7.62) is modified in that one can only remove the F phases
associated with the charged lepton fields from the F2 parameters in a general
unitary F x F matrix, so that there are F(F — l)/2 mixing angles and F(F~
l)/2 observable CP-violating phases. The upshot is that Ve can be written
V£ = KvMVt, (7.359)
where Ve involves ^(F—1)(F — 2) phases analogous to those in the CKM
matrix, such as the single phase for F = 3 displayed in (7.211). KVM is a diagonal
matrix of Majorana phases. Only the F — 1 phase differences are observable,
so one often takes it to be of the form KVM — diag(e~*ai, • •' , e~2aF-1,l).
Since the Majorana phases only multiply the mass eigenstate fields, they do
not enter any amplitude involving only external neutrinos or those involving
an ordinary (lepton-number conserving) internal neutrino line. They do affect
amplitudes involving lepton number violation, such as the /?/?ot/ amplitude in
Figure 7.42.
To summarize, for F = 3 families, the adjoint of the leptonic mixing matrix
can be parametrized by
/l 0 0 \ / ci3 0 s13e-i6\ ( cia s12 0\ /V0* 0 0\
Vt = 0 c23 523 0 1 0 -Sl2 cia 0 0 e™* 0 ,
\0 -523 C23/ \-Sl3eW 0 Ci3 / V ° ° V V ° ° V
(7.360)
where dj = cos 0^ and sij = sin 0^ and 5 are leptonic mixing angles and a
CP-violating phase similar to (but numerically different from) the angles in
the CKM matrix in (7.211), and ai52 are Majorana phases. The same form
holds for Dirac masses, except the last factor becomes the identity. Note that
we are following the conventions in (Amsler et al., 2008), and that (7.211)
refers to Vq while (7.360) refers to V^ . To conform to standard notations, we
will also define the mixing matrix
V = V}. (7.361)
■l-The phase would be undetermined but unobservable for a zero eigenvalue.

The Standard Electroweak Theory
415
In the general multi-family case, with both Dirac and Majorana masses,
-H<^)(«|m:)(|)+<- <««>
where Mr = MT and Ms = Ms. There are six Majorana mass eigenvalues
and eigenvectors. The transformation to the mass eigenstate basis is
"i = 4?(^c), (7-363)
where AVL is a 6 x 6 unitary matrix and vL is a six-component vector. The
analogous transformation for the R fields involves AVR = A"L*)C* because the
6x6 Majorana mass matrix is symmetric. Our phase convention vcR = Cv^
again implies the choice JC = I.
Analogous to (7.350), the seesaw limit of (7.362) occurs when the three
eigenvalues of Ms are all large compared to the elements of MD and Mr (the
latter is usually assumed to be zero). Then, one has
A« = Ag = (^ A;at) Bl\ (7.364)
where AVL and A^f are 3x3 unitary matrices, J and 0 are respectively the
3x3 identity and zero matrices, and
One finds
"H (%*) B" ~ (^ " "f*'"1 MS) ■ <? 366»
That is, there are three light (approximately) active neutrinos with an effective
Majorana mass matrix Mr — MdM^Mq, diagonalized by AVL = AVR . There
are also three heavy (approximately) sterile neutrinos with mass matrix Ms,
which is diagonalized by Au^ = AUR*. The latter decouple at energies small
compared with their masses.
In the more general case in which one or more of the eigenvalues of Ms
are not large, some or all of the light states will include non-negligible
sterile components. In particular, the active and sterile neutrinos of the same
chirality will mix significantly (as was suggested by the LSND experiment) if
both Majorana and Dirac masses are of the same order of magnitude or in
the pseudo-Dirac case. Constructing models with these features presents a
special challenge compared to the Dirac, Majorana, and seesaw cases because
one must find an explanation as to why two different types of mass terms are

416
The Standard Model and Beyond
small. If all three sterile states remain light, the leptonic mixing matrix in
the charge raising current in (7.355) becomes 6x3 dimensional, i.e.,
Vt = A*P\A\, (7.367)
where A^ is the 6x6 neutrino mixing matrix and Pa = {I 0) is the 3x6
dimensional matrix that projects onto the active neutrino subspace. One
can easily generalize to the case of Fa = 3 active neutrinos and Fs sterile
neutrinos, in which case there are 3 + Fs Majorana mass eigenvalues and PA
becomes a 3 x (3 + Fs) dimensional projection.
Our original definition of a Dirac mass term as one which couples two
distinct Weyl fields and a Majorana mass as one which couples a Weyl field
to itself is not very useful in the multi-family case. In the above discussion
we implicitly referred to couplings between active and sterile neutrinos as
Dirac, and active-active or sterile-sterile couplings as Majorana. However, an
alternate definition, which we will now adopt, is to define Majorana or Dirac
masses on the basis of the form of the mass eigenvalues and eigenvectors. One
can then view Majorana mass terms as the generic case, and reserve the term
Dirac for special or limiting cases in which there is a conserved lepton number.
We already saw an example of the Dirac limit of the one family mixed model,
which trivially generalizes to the F family case when Mr = Ms = 0, MD =^ 0.
A less obvious example involves the two family Majorana mass matrix
M f 0 mZKM\ ( }
\mzKM 0 J v }
This has the same form as the Dirac limit of (7.343), except that in this case
the L and R components are both active. Let us be even more explicit, and
assume that (7.368) holds in the basis in which the charged leptons (e and /x)
are already diagonal. Then we can identify v\L = veL, v\cR = v^R, v\L = v^i,
and v\R = vc R, so that
r ™>ZKM /_ c , - c , -c , -c \
-£>T = g K^Mft + V^VeR + "»R"eL + ^eR^L)
= mzKM {peL + ^R) iyeL + I/£R) .
This Zeldovich-Konopinski-Mahmoud model (Zeldovich, 1952; Konopinski and
Mahmoud, 1953) involves a Dirac neutrino, in that there is a conserved
quantum number (Le — LM, rather than L = Le + LM) and because two distinct
Weyl neutrinos are involved. However, in many ways it is more closely related
to the Majorana case, i.e., it violates weak isospin by one unit and is a
limiting case of the general 2x2 Majorana matrix. Any perturbation involving
nonzero diagonal elements of Mr would break the "degeneracy" mi = — mi
of the two eigenvalues. A modern F = 3 version of the ZKM model involves
the matrix
/0 11\
MT = mZKM 10 0, (7.370)
Vioo/

The Standard Electroweak Theory
All
which leads to one massive Dirac neutrino VeL + ^{i%R + i4R), and one mass-
less Weyl neutrino ^ [u^L -vtL), with Le-L^,- LT conserved. This actually
yields a spectrum somewhat similar to the observed one, but would require
nontrivial perturbations both in MT and the charged lepton mixing to be fully
realistic. An example of a perturbation on MT leading to a generalization of
the pseudo-Dirac case is considered in Problem 7.21.
An analogous situation sometimes occurs (especially in complicated models
in which Fs > Fa) when two sterile neutrinos pair to form a Dirac neutrino,
e.g., with Ms of a form analogous to (7.368).
Let us conclude this section by reemphasizing that there is no distinction
between Dirac and Majorana neutrinos except by their masses (or by new
BSM interactions). As the masses go to zero, the active components reduce
to standard active Weyl neutrinos in both cases. There are additional sterile
Weyl neutrinos in the massless limit of the Dirac case, but these decouple
from the other particles. We also repeat the comment from Section 7.2.2 that
one can ignore Ve in processes for which the neutrino masses are too small to
be relevant. They are then effectively degenerate (with vanishing mass) and
one can work in the weak basis.
7.7.2 The Propagators for Majorana Fermions
For free Dirac fields there is a conserved fermion number and therefore only
a single type of propagator, given in (2.178). For Majorana neutrinos, on
the other hand, there is no conserved fermion number and there are three
non-zero propagators,
**-(*-*')5F(fc)
<M*-*')ct5F(fc) (7.371)
ik.(x-x') SF(k)(-C),
where Sr(k) is the usual fc2_^X_i€- The first Majorana propagator is
analogous to the Dirac case, while the second and third can be thought of as
annihilating or creating two neutrinos, respectively. They can easily be
derived from
Y2 u&s) v(Pi S)T = (& + m) (-Q
(7.372)
y^ u(p, s)T v(p, s) = C* (i> - m)
s
and two similar identities with u <-> v and m —> —m, which in turn follow
immediately from (2.173), (2.285), and (2.286). Similar expressions apply to
the Majorana fields which occur in super symmetry.
(0\T[vM(x),i>M(x')]\0)=iJ
(0\T[vM(x),VM(x')]\0)=iJ
(0\T[vM(x),vM(x')]\0)
-/
d4k
d*k
(2^'
d4k

418
The Standard Model and Beyond
Expressions for amplitudes involving the second and third terms in (7.371)
take an unusual form, but they can usually be rendered more familiar by use
of (2.285) and (2.286), or by the use of (2.291) along with vM = vcM. As
a simple example, consider the process W~W~ —> e~e~, assuming that the
ve is Majorana and ignoring all family mixing effects. This proceeds via the
diagrams in Figure 7.44, which form a critical part of those for (3(3qv in Figure
7.42. The relevant WCC interaction is
C = -^JwW;> (7-373)
where
J£ = e7"(l - 75H = n»(l - 7>m, (7.374)
since P^z/^ = 0. The amplitude from the ^-channel diagram is therefore
2 r.# + mT
k2
(-Q
(7.375)
where k = ps — p\, tut is the Majorana neutrino mass, the propagator follows
from the last expression in (7.371), and we have assumed \k2\ > m?n. This
expression can be simplified using (2.284), (2.286), and (2.287),
[U47"(l - 75)]/J(-C)W = - [(1 - l5huv4]s , (7.376)
so that
M= (^|)2^2,(^:)S37-(l-75)7%, (7.377)
which can be evaluated in the usual way. Note that the u^ spinor has been
replaced by v^. The ft part of the propagator has dropped out because of the
pure V — A interaction, but would be allowed if there were an admixture of
V + A. The same result can be obtained more directly by rewriting
Jvw = e7"(l - 7>M = -VM-f(l + 75)ec (7-378)
obtained from (2.291) for one of the vertices. Then, the first propagator in
(7.371) is the relevant one, and we must use a v spinor for ej since ec is the
positron field.
7.7.3 Experiments and Observations
In this section we describe the principal laboratory and astrophysical
constraints on the number of light active and sterile neutrinos, and their masses
and mixings. The constraints on possible heavy Dirac or Majorana neutrinos
and on neutrino decays^ are reviewed in (Kolb and Turner, 1990; Cvetic and
Langacker, 1991; Raffelt, 1999; Mohapatra and Pal, 2004; Amsler et al., 2008).
§ Radiative decays, such as V2 —> e+e i/i, if kinematically allowed, or V2 —► 1/17, are possible
in the SM. The latter is loop suppressed and extremely slow. BSM physics could allow faster

The Standard Electroweak Theory
419
ttit T T ttit
+ X »—( \—« X »
W~ W~ W~ Wr
2
FIGURE 7.44
Diagrams for W~W~ —> e~e~ assuming a Majorana neutrino. Left: £-
channel. Right: u-channel. The cross represents a Majorana mass insertion.
Neutrino Counting
As discussed below (7.169), the width for the Z to decay invisibly implies
that there are only 3 active neutrinos with masses < Mz/2. More precisely,
Ninv = 2.985 ± 0.009 from the global fit to precision data1, where Nlvnv does
not include sterile neutrinos or very heavy active neutrinos, but does include
the effects of other possible invisible decay channels asssociated with new
physics. Precision constraints also exclude additional heavy active neutrinos
if they belong to a complete degenerate chiral family, but may be evaded for
non-chiral doublets, nondegenerate families, etc.
The other major constraint comes from big bang nucleosynthesis (BBN) (e.g.,
Kolb and Turner, 1990; Dolgov, 2002; Steigman, 2007; Amsler et al., 2008),
which has also been a critical test of hot big bang cosmology and of many
possible types of nonstandard particle physics. The basic point is that the
reactions
n + ve <-> p + e~, n + e+ <-> p + ve (7.379)
kept the ratio of neutrons to protons in thermal equilibrium - = exp( nT p)
~ exp(—mn~mp) in the early universe as long as the reaction rate T ~ G2FTb
was larger than the expansion rate (Hubble parameter) H ~ 1.66yJ~glT2 jMp,
where Mp is the Planck mass. H2 is proportional to the energy density
P — ^*tt2T4/30, where g* = gs + \qf and gs,F are the number of rela-
tivistic bosonic and fermionic degrees of freedom in equilibrium at
temperature T. The equilibrium was maintained until the freezeout temperature
v2 —> 1/17 or invisible decays, such as V2 —► ^l^i^i, V2 —► v\F, or V2 —► ^M, where F
is a familon (a Goldstone boson associated with a hypothetical broken family symmetry),
and M is a Majoron (a Goldstone boson associated with a spontaneously broken lepton
number).
'There is also a more direct determination of the invisible width from e~eJt —► 7 4- invisible,
yielding Nlnv = 2.92 ± 0.05.

420
The Standard Model and Beyond
Tf ~ (y^/G^Mp)1/3 = 0(few MeV) when r ~ H, at which time the n/p
ratio was frozen at the value exp(— mn~mp) except for neutron decay, and
most of the neutrons were eventually incorporated into AHe. By apparent
coincidence Tf is close to mn — mp, so the expected abundance depends
sensitively on g*. In the SM, one expects g* = 43/4 for me <T < raM (gs = 2 from
two photon helicities and gp = 10 from 3(i/l + ^h) + two helicities each of e*).
This leads to the prediction that the ratio of primordial 4He to H by mass
should be ~24%, in agreement with observations". However, any additional
contribution to the energy density for T > few MeV would increase H and
Tf, and therefore the predicted helium abundance. This can be parametrized
by writing
gF = 4 + 2NBBN (7.380)
where any deviation of the effective NBBN from 3 could indicate new light
degrees of freedom in equilibrium, or such effects as neutrino masses of (9(MeV)
or neutrino decay. There has long been some uncertainty and controversy
in the observational primordial abundance, but most recent estimates yield
ANbbn = nbbn _ 3 < i, e.g, ANBBN = 0.2 ± 1.2 (Amsler et al., 2008)
or ANBBN = 0.3±q[76 (Simha and Steigman, 2008). This constrains not only
additional active neutrinos with masses < 1 MeV, but also light sterile
neutrinos of the type suggested by the LSND experiment, which could be produced
by mixing with active neutrinos for a wide range of mixing angles (Dolgov,
2002; Cirelli et al., 2005). However, ANBBN does not include the sterile vR
components of light Dirac neutrinos, which could not (for the currently
.relevant mass ranges) have been produced in equilibrium numbers unless they
have new BSM interactions (Barger et al., 2003).
Most new physics effects increase the predicted 4He abundance. One
important counterexample is a possible large asymmetry between v and v,
which would preferentially drive the reactions in (7.379) to the right,
decreasing the n/p ratio and possibly compensating the effects of ANBBN.
(Only the ve — ve asymmetry directly affects the reactions, but the observed
neutrino mixing would probably have equilibrated the asymmetries between
the families.) However, such an asymmetry would have to be enormous,
{nv — n^/n^ ~ ^(1), compared to the baryon or charged lepton
asymmetries to have much effect. Even allowing it, the present constraints from BBN
along with the CMB and large scale structure data imply that such
asymmetries would not significantly perturb the constraints on ANBBN (Simha
and Steigman, 2008). A large asymmetry could lead to important nonlinear
effects in the case of active-sterile neutrino mixing, however.
The WMAP collaboration (Dunkley et al., 2009) has recently obtained a
nontrivial constraint on the number of relativistic degrees of freedom that were
II There is also a weak dependence on the baryon density relative to photons, which is
determined independently by the D abundance and by the cosmic microwave background
(CMB) anisotropics (Dunkley et al., 2009).

The Standard Electroweak Theory
421
present at a redshift of z ~ (1000-3000) from the CMB. When combined with
other cosmological observations this implies (Komatsu et al., 2009) N^MB =
4.4 ± 1.5 for the number of active neutrinos (and sterile neutrinos weighted
by their abundance) that were relativistic at a temperature T ~ (0.3 - 1) eV.
The BBN and CMB constraints on the cosmological neutrinos are
indirect. Following their decoupling at T ~ few MeV the cosmological
neutrinos presumably were redshifted so that their momentum distribution should
at present have a thermal form, characterized by an effective temperature
fu = (4/H)1/3^ - 1.9 K, where T7 - 2.73 K is the CMB temperature and
the (4/11)1/3 factor is because the 7's but not the neutrinos were reheated
by e"e+ annihilation at T < me (Steigman, 1979; Kolb and Turner, 1990;
Weinberg, 2008). This corresponds to a number density of ~ 50/cm3 for each
neutrino degree of freedom, i.e., ~ 300/cm3 for 3 flavors with two helicity
states. Local clustering and modifications of the momentum distribution are
not expected to be large unless the masses are > 0.1 eV (e.g., Ringwald and
Wong, 2004). Direct detection of these relic neutrinos appears extraordinarily
difficult, whether by macroscopic torques or forces (Stodolsky, 1975; Cabibbo
and Maiani, 1982; Langacker et al., 1983); Z bursts, which could occur if there
were a source of ultra high energy cosmic ray neutrinos that could annihilate
on relic neutrinos to produce Z's (Weiler, 1982; Eberle et al., 2004), allowing
observation of the decay products or observed as absorption dips in the
spectrum; or by 2/-induced e± emission by nuclei (e.g., Lazauskas et al., 2008).
For reviews, see (Gelmini, 2005; Ringwald, 2005; Strumia and Vissani, 2006).
Neutrino Mass Constraints
A stringent kinematic limit on the effective ve mass-squared
3
< = £lV«|2m' (7-381)
1=1
can be obtained from the shape of the e~ spectrum near the endpoint in
tritium /? decay, 3H —> 3He e~ve. In (7.381) ra* is the ith mass eigenvalue,
independent of whether it is Dirac or Majorana, and V is the leptonic mixing
matrix as defined in (7.361). Anomalies in earlier experiments have been
largely resolved, with the current value from experiments in Mainz and Troitsk
yielding ra2e = (1.1 ± 2.4) eV2. The Karlsruhe KATRIN experiment should
improve the sensitivity on mVe down to around 0.2 eV. We also mention the
historically important observation of a burst of (9(20) neutrinos (presumably
mainly Pe's) from the core-collapse Supernova 1987A by the Kamiokande and
1MB collaborations, which implied (amongst many other things) that mVe <
20 eV, because otherwise the arrival times of the detected neutrinos would
have spread out more than was observed. It was hard to make the limit
precise, however, because it depended on theoretical details of the neutrino
emission (e.g., Bahcall, 1989; Raffelt, 1999).

422
The Standard Model and Beyond
The kinematic limits on the v^ and vT masses, defined analogously to
(7.381), are much weaker: mV[i < 0.19 MeV from 7r+ —> /x+^M and mVr < ig 2
MeV from r~ —> 37r±i/r, 57r±(7r°)2/r. These bounds are now superseded by
much more stringent ones from neutrino oscillations and cosmology. However
it is interesting that the combination of this bound on mVT from ALEPH with
the BBN constraint (which becomes relevant because an ~ (1 — 20) MeV
neutrino contributes more than 1 to NBBN) excluded the possibility of a stable
or long-lived vT above 1 MeV (e.g., Fields et al., 1997).
Light massive neutrinos would contribute to the cosmological energy
density, and they would close the Universe, 0^ = 1, for E ~ 35 eV, where
X = Xi\mi\ (7.382)
is the sum of the masses of the light active neutrinos, as well as sterile
neutrinos weighted by their relative abundance. Observationally, some 20% of the
energy density is dark, but it is most likely cold dark matter (CDM) that was
non-relativistic at decoupling, such as weakly interacting massive particles
(WIMPs) or axions. Light neutrinos would be hot dark matter
(relativists at decoupling), which would free-stream away from density perturbations,
preventing the formation of the observed smaller scale structures during the
lifetime of the universe**. The combination of cosmological data on the CMB,
large scale structure, supernova expansion data, and the Lyman alpha forest
(Lya) can be used to set an upper limit on E. A recent study (Fogli et al,
2008) obtained 95% c.l. upper limits on E ranging between 0.19 and 1.2 eV,
depending on the data set and other assumptions. Using future Planck
satellite data, it may be possible to extend the sensitivity down to 0.05 — 0.1 eV,
close to the minimum value 0.05 eV ~ \/|Ara^m| allowed by the neutrino
oscillation data.
Majorana masses can lead to (3(3ov, i.e., nn —> ppe~e~, which violates
lepton number by two units, by the diagrams in Figure 7.42 (for reviews, see
Vergados, 1986; Elliott and Vogel, 2002; Avignone et al., 2008). Since there is
no missing energy tt, events should show up as a peak of known energy in the
e~e~ spectrum from a sample of (3(3ov-unstable nuclei. However, the process
would have an extremely long half-life, so problems of backgrounds are severe.
The amplitude for (3(3qv is M ~ Anucmpp, where Anuc contains the nuclear
matrix element. Anuc cannot be directly measured and therefore introduces
**Warm dark matter, e.g., from keV mass sterile neutrinos, is an intermediate
possibility (Asaka et al., 2006). Other astrophysical and cosmological implications of intermediate
mass sterile neutrinos are described in (Fuller et al., 2009).
ttTwo-neutrino double beta decay, /?/?2i/, is the process nn —► ppe~e~PP, which can occur
by ordinary second-order weak processes in some /^-stable nuclei. It leads to a continuous
e~e~ spectrum and has been studied in a number of nuclei (Barabash, 2006). It is helpful
for testing calculations of the nuclear matrix elements entering /3/3ou- A third possibiity is
Majoron decay, nn —> ppe~e~ M or /3/3qum, where M is a Majoron (Goldstone boson). It
would lead to a spectrum intermediate between /3/3qu and fifow

The Standard Electroweak Theory
423
considerable uncertainty into the interpretation of any upper limit or future
observation (Simkovic et al., 2008; Vogel, 2008). m00 is the effective Majorana
mass in the presence of mixing between light Majorana neutrinos,
mP0 = ^(Veifrrii- (7.383)
i
It is just the (1,1) element of tut or of the effective Majorana mass matrix in
a seesaw model. It involves the square of Vei rather than the absolute square,
allowing for the possibility of cancellations between terms. Such cancellations
could occur even if the original mass matrix were real because some of the
eigenvalues could be negative. (In our phase convention the mi are taken to be
positive, but the signs would appear because some of the Majorana phases in
(7.360) would then be ±i.) This also shows why the /?/?ot/ amplitude vanishes
for a Dirac neutrino, which can be viewed as two Majorana neutrinos which
give equal and opposite contributions. The cancellations could in principle
allow the determination of (CP-violating) Majorana phases different from 0 or
±2, though this is difficult in practice because the other parameters including
the matrix elements would have to be known rather well (Barger et al., 2002;
P&scoliet al., 2002).
An observation of (3(3qv in 76Ge with a half-life ~ 2 x 1025 yr has been
claimed (Klapdor-Kleingrothaus and Krivosheina, 2006) by members of the
Heidelberg-Moscow experiment, but this has not been confirmed by other
experiments. It would correspond to mpp ~ (0.16 — 0.52) eV (Fogli et al.,
2008). The best current upper limit is from the 130Te Cuoricino
experiment (Arnaboldi et al., 2008) in the Gran Sasso Laboratory, which obtains
wpp < (0.19—0.68) eV at 2<j, with the range due to the nuclear matrix element
uncertainties. Future experiments should be sensitive down to 0(0.02 eV) or
better.
A heavy Majorana neutrino could also contribute to ra^, but its
contribution would be suppressed by finite range effects (i.e., the m2 in the
denominator of the propagator would be important),
(e~mir /r)
m00^22{Vei)2miF{mi,A), F(rm,A)= x ' ', (7.384)
i \ I I
where A is the nucleon number. F(rrii,A) is ~ 1 for m* <C 10 MeV, but
falls rapidly for larger values (e.g., Vergados, 1986). Lepton-number violating
effects other than Majorana neutrino masses can also lead to (3(3$v (although
the existence of a (3(3ov amplitude implies the existence of a Majorana mass at
some level (Schechter and Valle, 1982)). For example, models involving both
V — A and V + A interactions as well as lepton number violation can induce
fifov amplitudes not directly proportional to neutrino masses (Problem 7.22).
There could also be effects from new interactions such as leptoquarks or
imparity violation in supersymmetry. If (3(3qv is observed, it would be useful to
study it in several different nuclei, both to help control nuclear matrix element
uncertainties and to shed some light on the underlying mechanism.

424
The Standard Model and Beyond
Electromagnetic Form Factors
Neutrinos have no electric charge, but they can acquire magnetic and electric
dipole moments by diagrams analogous to the weak corrections to the muon
magnetic moment in Figure 2.20 or from new physics. These lead to effective
electromagnetic interactions
1 i
KvQ = -^HjfijG^niF^ + -dijUja^^riiF^, (7.385)
where rii can represent either a Dirac (isid) or Majorana (z^m) mass eigenstate
field. The first (second) terms are magnetic (electric) dipole interactions, as
can be seen from (2.353), (7.320), and Problem 2.22. The flavor-diagonal
terms i = j are known as direct or intrinsic moments, while those for i ^ j
are transition moments.
The magnetic and electric dipoles flip chirality. In the Majorana case,
ViM = viL + i/fR, so that
ViMO^ViM = Vjl^vZr + ^V^i/iL. (7-386)
However, using i/?R = CvJL,
W^r = vfRCa^CvJL = -ViLa^ucjR, (7.387)
so that fiij = —fij{. (This can also be seen from (2.298).) Majorana
neutrinos therefore cannot have direct magnetic moments, but can have transition
moments jiij ^ 0 for i ^ ji, which can mediate decays such as ViM —► ^jMl-
Similar statements apply to electric dipole moments.
Both direct and transition moments are possible for Dirac neutrinos v^ =
ViL + vm. For a single flavor
vdg^vd = vlo^vr + vro^vl* (7.388)
One can write vl and vr in terms of two degenerate Majorana neutrinos using
(7.348),
»l = -ftiviL + "2l), vr = -k(v1r - v2r)> (7-389)
where the unnecessary superscript 0 has been dropped. Then
vlo^vr = -^=vlLa^vc2R + l=92La^v{R. (7.390)
That is, a direct Dirac magnetic (or electric) moment is an antisymmetric
combination of transition moments between degenerate Majorana states.
In the simplest extension of the SM with a small Dirac mass m*, the direct
neutrino magnetic moment is (Marciano and Sanda, 1977; Lee and Shrock,
1977)
"~W~MxUr',(r^)'-- <739I)

The Standard Electroweak Theory
425
where hb is the Bohr magneton. This is negligibly small compared with
laboratory limits < 10_10/xb (Vogel and Engel, 1989; Amsler et al., 2008), and
various astrophysical limits, e.g., from stellar cooling, pv < few x 10"12 (Raf-
felt, 1999; Giunti and Studenikin, 2008). (The latter also applies to electric
dipole moments and to both Dirac and Majorana transition moments.)
However, it is possible to construct models with much larger magnetic moments,
as was motivated by an alternative solution to the Solar neutrino problem
involving resonant spin-flavor precession in an assumed strong Solar magnetic
field (Akhmedov, 1988; Lim and Marciano, 1988; Gonzalez-Garcia and Mal-
toni, 2008).
7.7.4 Neutrino Oscillations
Neutrino oscillations are analogous to the neutral K and B meson oscillations
described in Sections 7.6.3 and 7.6.4, and occur due to the mismatch between
weak and mass eigenstates. They do not mix the neutrino helicities, and are
therefore independent of whether the masses are Majorana or Dirac. First
consider two neutrino flavors, ve and z/M, related to the mass eigenstates by
We) = |^i)cos0 + |j/2)sin0, ||/M) = -|i/!) sinff+ |i/2) cos ff, (7.392)
where 0, which corresponds to 0i2 in (7.360), is the neutrino mixing angle.
Suppose that one starts at time t = 0 with a pure state |^(0)) = |^M) of
definite momentum** \p\ from the decay 7r+ —> /x+^M. The two mass eigenstate
components each develop with their own time dependence, so that
\v(t)) = -ji/i) sm6e-iElt + \v2) cosfle"^2*
-|i/i)sin0e l 2E +|i/2)cos0e * *e
-iEt
(7.393)
at a later time t, where we have assumed that the neutrinos are extremely
relativistic, so that Ei = y/\p\2 + mf ~ E + mf/2E where E ~ \p\. After
traveling a distance L there is a probability
P^-n^L) = \(Ve\"(t))\2 =sin20COS20
. 2^ . 2 /Am2!/'
= sin2 20 sin2
= sin2 20 sin2
-■mil* •m2t *
AE
1.27Am2(eV2)L(km)
£(GeV)
(7.394)
^This simple approximation yields the correct result. However, a complete treatment
requires consideration of the coherence of the initial and final wave packets, the relation
between t and the location L of the detector, etc. (see, e.g., Giunti, 2002; Lipkin, 2006;
Strumia and Vissani, 2006).

426
The Standard Model and Beyond
for the neutrino to have oscillated into a z/e, where Am2 = m\—m\ and L ~ t.
The ve could be detected, e.g., by the reaction ven —> e~p. The oscillation
length is defined as
4tt£
L- = A^ (7'395)
Such vacuum oscillations depend only on |Ara2| and not on the absolute mass
scale or on the hierarchy (which mass is larger). In appearance experiments
one searches for the production of a different neutrino flavor than one started
from, such as by the production of an e~ or r~ in an initial v^ beam. In an
ideal experiment with definite L and E one observes not only the appearance
of the new flavor, but the characteristic L/E dependence. However, for large
Am2L/E the oscillations are averaged by the spread in neutrino energy E
and by finite detector/source sizes, so that
Pu„-+u.{L) ->!sin220, (7.396)
which is the same result one would have obtained from an incoherent
superposition of v\ and V2- In a disappearance experiment, one searches for the
reduction in the flux of the initial v^ (or other flavor) as a function of L and
E, making use of the probability PVyi^Vyi (L) = 1 — Pv _+Ve (L) for the state to
remain a v^. For both types of experiments, careful attention has to be paid
to the initial flux and spectrum (obtained from other measurements, theory,
or by an initial calibration detector) and to backgrounds.
Even with more than two types of neutrino, it is sometimes a good
approximation to use the two-neutrino formalism in the analysis of a given
experiment, e.g., if some of the mixings are small or if some of the A^ = m2 — m2
are small compared to E/L, and most results are presented in terms of
allowed or excluded regions in the sin2 20 — Am2 plane or the tan2 0 — Am2
plane*, whether or not that is really valid. However, a more precise or general
analysis should take all three neutrinos into account. It is straightforward to
show that the oscillation probability for va —> v^ after a distance L is
PVa^(L) =<Jafe-4^$fte(v:iVfeiVajV^)sin2 (^)
^ /A L\ (?-397)
+ 2 £ SmtoW^V£) sin ( -^- ,
*One usually labels the mass eigenstates so that Am2 > 0. The cases 0 < 6 < 7r/4 and
7r/4 < 6 < ir/2 are physically different. However, they cannot be distinguished by vacuum
oscillation experiments, so it was traditional to use the variable sin2 26 to describe the
results. Matter effects, however, can distinguish the two cases, so it is better to use tan2 6
instead. The region tan2 0 > 1 is sometimes known as the "dark side"(e.g., Gonzalez-Garcia
and Nir, 2003).

The Standard Electroweak Theory
427
where va and vh are weak (flavor) eigenstates and V is the leptonic mixing
matrix in (7.360). It is apparent from (7.397) that Pvb-+Va{L) is the same
as Pva-vb(L) except that V -> V*, and that any difference between them is
due to CT-violating phases in V. The combination V^V^VajV^ is a Jarlskog
invariant, i.e., independent of phase conventions. The Majorana phases do not
enter, so CP-violation in neutrino oscillations requires mixing between at least
3 families, just as in the CKM matrix. For F = 3, it is given by the phase S in
(7.360), and all CP-violating effects would vanish for S13 = 0. In practice, it
is extremely difficult to compare PUa^Uh{L) and PUh^Ua{L) directly, because,
e.g., ve are mainly produced at reactors, and v^ipy) at accelerators. However,
CPT implies that PUh^Va(L) = PDa_Db{L). Thus,
P„a-„b(L) ^-^ fttt^(£). (7.398)
The oscillation rates for va vs va can be compared, e.g., by using v^ (p^)
beams from tt+ —> /x+^M ( tt~ —> /x~^M), and any difference must be due to
leptonic CP violation (assuming CPT holds).
Oscillations between active neutrinos of different flavors are known as first
class (flavor) oscillations. The results in (7.397) and (7.398) generalize to
second class oscillations (Barger et al., 1980) between light active and sterile
neutrinos of the same helicity, which can occur when there are both Majorana
and Dirac mass terms. For example, mixing between 3 active and 3 sterile
neutrinos would still be described by (7.397), except V is now a 6 x 6 unitary
matrix. Other phenomena, which can sometimes mimic neutrino oscillations,
include neutrino-antineutrino transitions (involving new interactions) (Lan-
gacker and Wang, 1998), and massless neutrinos that are non-orthogonal due
to mixing with heavy states (Langacker and London, 1988a).
The Mikheyev-Smirnov-Wolfenstein (MSW) Effect
Eq. 7.393 or its generalization to 3 or more flavors describes the time evolution
of an (initial) weak eigenstate in vacuum. However, for propagation through
matter, such as the Sun or Earth, one must take into account the phase
changes associated with the coherent forward scattering of the neutrinos with
the matter, very much like index of refraction effects in optics (Wolfenstein,
1978). Under appropriate conditions, the matter effects can combine with the
neutrino masses to yield an effective degeneracy and therefore an enhanced
transition probability, the MSW resonance (Mikheyev and Smirnov, 1985).
To see how this works, consider the time evolution of a neutrino state
K')) = £ca(*)k), (7.399)
a
in vacuum. The weak eigenstates \i/a) are related to the mass eigenstates \i/i)
by
l"«> = £ h><*> = £Kik>- (7-400)
i i

428
The Standard Model and Beyond
(In (7.400) and the following, we ignore the momentum integrals and delta
functions, as well as the (2tt)32E normalization factors, all of which cancel
between the matrix elements and the sums over intermediate states.) Evolving
the mass eigenstates analogously to (7.393), one finds that the coefficients
satisfy the Schrodinger-like equation
where the vacuum Hamiltonian (due to the masses) is
l2E
(isa\HvWb) = £ VaiEiK ~ Y, V«<SV« + ESab. (7.402)
The last term only affects the irrelevant overall phase of the state. This and
similar multiples of the identity can be dropped in what follows. For two
flavors, Hy becomes
_ Am2 /-cos20 sin20\
Hy-~AE~{ sin 20 cos 20J (7-403)
using the conventions in (7.392), with Am2 = m^ — m\.
In the presence of matter, (7.401) must be modified by the addition of the
effective (matter) Hamiltonian
d3x J2 -^ *aL-fi>aL £<Vv7„(0k - 9aJ5)A), (7.404)
which describes the scattering of va from fermions r = e,p, n, where g™A are
the effective vector and axial couplings which receive contributions from Z
exchange and (in the case of r = e) from W exchange, as in (7.93). The
brackets on the last term indicate an expectation value in the static medium.
Assuming the medium is unpolarized,
(^7m(Sv - 9Alb)i>r) = 9V"r%, (7-405)
where nr is the number density of particle r (cf., Eq. A.19). Using also
that J d3xiylLiyaL is the va number operator, NVaL — Nvo , we obtain the
propagation equation^
ijtca(t) = (va\Hv\isb)cb(t) + y/2GFl Y, 9vrnr)ca(t). (7.406)
\r=e,p,n /
tThe numerical factor and sign for the matter term were derived in (Barger et al., 1980)
and (Langacker et al., 1983), respectively.

The Standard Electroweak Theory
429
The vector coefficients are
gf? = +-+2sm2ew,
tf
<P
+ --2sin^H,,
1 9
■ +--2sm20w,
9v --J
/" _ _ *
(7.407)
sr
9v
0,
for veL, v»,tL, and sterile vsL, respectively. The signs are reversed for vcR, as
can be seen from the number operator or from (2.297). g^ contains an extra
+1 from the WCC, which makes the effect important for ve <-> v^^.
Specializing to two families,
jl(
Ca(t)
Cb(t)
where
/-A-iCos20 + ^fn ^ sin 20 \
wsin2^ wcos20_^n/
n = na-nb, na = 2J 9\fnT,
r=e,p,n
Ca(t)
cb(t)
(7.408)
(7.409)
and we have symmetrized the diagonal elements by subtracting the common
term \/2Gf n/2. For an electrically neutral medium, i.e., ne = np, this yields
{ne for veL <-> ^l^tL
ne - \nn for veL <-> vsL , (7.410)
-\nn for v»l,VtL^vsl
with the signs reversed for vcR.
(7.411)
Under the right conditions, the matter effect can greatly enhance the
transitions. In particular, if the Mikheyev-Smirnov-Wolfenstein (MSW) resonance
condition ^g- cos 20 = \[2Gfu is satisfied, the diagonal elements vanish and
even small vacuum mixing angles lead to a maximal effective mixing angle.
Because of the sign switch, an enhancement for v^ corresponds to a
suppression for is ft and vice-versa. The matter effect breaks the sign degeneracy for
vacuum oscillations, and allows a determination of the sign of Am2. Because
of the E dependence it can lead to a distortion in the final neutrino spectra.
Finally, if the matter density varies significantly along the neutrino path, as
is the case for Solar neutrinos produced near the Solar core, one may
encounter a level-crossing at the position for which the resonance condition is
satisfied (for a given E), as illustrated in Figure 7.45. If the density varies
sufficiently gradually, the transition is adiabatic, i.e., the neutrino remains
on one level, with a maximal flavor transition probability. A more abrupt
(non-adiabatic) transition will have a non-neglible probability to jump from
one level to another, reducing the flavor transition probability (see e.g., Kuo
and Pantaleone, 1989; Gonzalez-Garcia and Nir, 2003; Barger et al., 2003;
Strumia and Vissani, 2006; Gonzalez-Garcia and Maltoni, 2008).

430
The Standard Model and Beyond
*Nw I"2> ~ We)
rrti(r) ^sSs^^
|i/i> - |i/c>
1
7*0 r _
FIGURE 7.45
A level-crossing (resonance) at r^ in the presence of matter density that
decreases with distance r from the neutrino source. The rrii(r) are the
eigenvalues of the matrix in (7.408).
Oscillation Experiments
There have been many experimental searches for and observations of
neutrino oscillations and transitions (for recent reviews, see Gonzalez-Garcia and
Maltoni, 2008; Amsler et al., 2008), including experiments at accelerators
and reactors, and those involving Solar neutrinos and atmospheric neutrinos
(from the decay products of particles produced by cosmic ray interaction^ in
the atmosphere). Major observation and exclusion regions are plotted in'the
two-neutrino formalism in Figure 7.46.
Solar Neutrinos
Neutrinos (i/e) are produced by fusion reactions in main sequence stars by
the pp and CNO chains, which ultimately lead to Ap —> a + 2e+ + 2ve. The
standard solar model (SSM) (Bahcall, 1989; Bahcall et al., 2005), which is well
tested and constrained by helioseismology and other Solar observations, and
by the properties of other stars, is dominated by the pp chain and leads to the
predicted Solar ve spectrum in Figure 7.47. The most important reactions are
p + p^> D + e+ + ve, D+p^> 3#e + 7, 2 3#e-► a + 2p. (7.412)
The first step leads to the most abundant (pp) neutrinos, which however, have
low energy and are hard to detect. About 15% of the time, however,
3#e + a-> 7£e + 7, 7Be + e~ -> 7Li + i/c, (7.413)
for one of the 3He, leading to the two intermediate energy discrete 7Be lines
in the ve spectrum. Approximately 0.02% of the chains involve the sequence
7Be + p^ 8£ + 7, SB^ 8Be*+e++ve, (7.414)

The Standard Electroweak Theory
431
10"
m
V ~lJ&f
C^""CD^SW^>'
CHORUS ^>f^C
all solar 95%
CI 95%
X ^^
MINOS
amLANDl
95%
uper-K
95%
All limits are at 90%CL
unless otherwise noted
io-12
10
HI 10-2 100
tan26
http://hitoshi.berkeley.edu/neutrino
W
FIGURE 7.46
Neutrino oscillation results, showing the Solar/KamLAND and atmospheric
neutrino oscillation regions, the LSND region, and various exclusion regions.
Plot courtesy of H. Murayama.

432
The Standard Model and Beyond
which leads to the 8B neutrinos. These are insignificant numerically, but
because of their much higher energy are easiest to detect. The flux of pp
neutrinos is well constrained by the observed Solar luminosity, but the predicted
7Be and (especially) 8B fluxes are much more uncertain because of their strong
dependence on the temperature of the Solar core, low energy cross sections,
and the Solar composition.
Bahcall-Serenelli 2005
Neutrino Spectrum (±la)
Neutrino Energy in MeV
FIGURE 7.47
Spectrum of Solar neutrinos predicted by the standard solar model. The
gallium experiments are sensitive to the pp and higher energy neutrinos, the
Homestake chlorine experiment to the higher energy 7Be line and above, and
the water Cherenkov experiments to the 8B neutrinos. The other (minor)
reactions are described in (Bahcall, 1989). Plot reproduced by permission of
the AAS from (Bahcall et al., 2005).
The first Solar neutrino experiment was the radiochemical 37Cl experiment,
which used a 105 gallon tank of cleaning fluid placed deep underground in the
Homestake gold mine in South Dakota to shield from cosmic rays. Ray Davis
and collaborators observed the decays of the 37Ar atoms produced in the
reaction ve +37Cl —> e~ + 37Ar and separated chemically from the tank. The
original goal was to probe the Solar interior, but by the early 1970s it was
apparent that they were only observing ~ 1/3 of the expected flux, beginning
a 30-year enterprise that ultimately explored both Solar and neutrino physics.

The Standard Electroweak Theory
433
The Homestake results were confirmed in the late 1980s by the Kamiokande
II water Cherenkov experiment in Japan, which also searched for proton decay
and observed neutrinos from Supernova 1987A and atmospheric neutrinos.
The reaction ve~ —> ve~ was mainly sensitive to i/e's, but because of WNC
scattering had about 1/7 sensitivity to v^T, which could be produced if the
i/e's oscillated. They observed about 1/2 of the expected (without oscillations)
Solar flux, and also confirmed that the z/s actually came from the Sun because
the e~ direction was correlated with that of the neutrino. These results were
later confirmed by the successor SuperKamiokande experiment.
The water Cherenkov experiments were only sensitive to the upper part of
the SB spectrum and the much rarer hep neutrinos. The Homestake
experiment had a lower threshold and was sensitive to more of the SB spectrum
and to some extent the 7Be and pep neutrinos. The reduced fluxes that they
observed could have been due to uncertainties in the standard solar model
(e.g., by a 5% reduction in the temperature of the core) or other astrophysical
effects, or to neutrino oscillations/transitions into z/M, vT, or sterile neutrinos.
To distinguish these possibilities, radiochemical experiments on gallium, using
the reaction ve +71 Ga —> e~ + 71Ge, were carried out in the 1990s. This
has a much lower threshold (233 keV) than the chlorine reaction (814 keV),
allowing detection of the much more numerous pp neutrinos. An observation
comparable to the SSM prediction would have suggested that the Homestake
and (Super)Kamiokande deficits were due to astrophysical effects, while a
reduction comparable to that for SB would have indicated neutrino oscillations.
Eventually three gallium experiments, GALLEX and GNO in the Gran Sasso
and SAGE in the Baksan lab in Russia, were carried out, indicating > 50%
of the predicted SSM flux.
No one type of experiment by itself could definitively exclude an
astrophysical explanation, especially allowing for large modifications of the SSM, but
the three types together constituted a rough measurement of the distortion of
the neutrino energy spectrum. Including the Solar luminosity constraint and
assuming that plausible astrophysical effects would not significantly modify
the shape of the 8B spectrum, it was found that the 7Be neutrinos would have
had to be suppressed much more than the 8B ones (Hata et al., 1994). Because
the 8B is made from 7jBe, this effectively excluded astrophysical explanations,
but allowed neutrino oscillations or transitions (which could modify the shape
of the SB spectrum).
The case was clinched a decade later by the SNO (Sudbury Neutrino
Observatory) heavy water experiment in Canada. SNO observed Cherenkov
radiation from electrons and photons from neutron capture, allowing then to
measure both WCC and WNC scattering from deuterium by
ve + D->e~ +p + p, vx + D^n/x+p + n, (7.415)
where vx is any active neutrino. They also measured the electron scattering
reaction ve~ —> ve~, consistent with but less precisely than SuperKamiokande.

434
The Standard Model and Beyond
The combination of these measurements allowed the SNO collaboration to
separately determine the fluxes of ve and of v^ + vT arriving at the Earth, as
shown in Figure 7.48. The result was that the sum of the three was
consistent with the SSM flux prediction, and that about 2/3 had oscillated or been
converted into z/M)T (assuming that there are no sterile neutrinos involved),
confirming both the SSM and neutrino oscillations.
At various stages the Solar neutrino experiments allowed a number of
solutions for neutrino oscillation parameters. These included the Am2 ~ 10~10
eV2 vacuum oscillation solutions, for which the Earth-Sun distance was of
0(L0SC). The matter effects were unimportant for these solutions and
oscillations during the propagation between the Sun and Earth dominated. There
were also several solutions with higher Am2, for which matter effects were
significant. Eventually, the combination of the observed rates from the
different reactions, as well as limits on the distortion of the 8B spectrum and
on possible day-night asymmetries (due to reconversion to ve in the Earth),
established the large mixing angle (LMA) solution characterized by
Am| - (3 - 11) x 1(T5 eV2, sin2 0© - (0.25 - 0.35), (7.416)
where Am© ~ Ara^ and 00 ~ 0i2 in the notation of (7.360). (These ranges
correspond qualitatively to the extent of the 90% c.l. region in the oscillation
plot.) Note that the Solar angle 00 is large but not maximal, i.e., 00 ^ 7r/4, so
the ve is predominantly v\, while V2 consists mainly of a linear combination of
v^ and vT. The characteristics of the LMA solution are that a higher energy ve
encounters an adiabatic MSW resonance, emerging from the Sun in an
essentially pure V2 mass eigenstate. This does not oscillate, and has a probability
sin2 0Q ~ 0.31 of interacting as a ve. The density is too low for a resonance
for the lower energy neutrinos, so they emerge from the Sun as a ve, arriving
at the Earth with an average survival probability ~ 1 — \ sin2 200 ~ 0.57.
The sign of Am© > 0 is determined from the matter effect. Subsequently,
the LMA solution was dramatically confirmed by the Kamland reactor
experiment in Japan (see below), and recently by observations of the upper 7Be
line by the Borexino experiment in the Gran Sasso. Future experiments could
possibly map out the spectrum in more detail, verifying the transition from
the high energy MSW transition to the low energy neutrinos, and further
constraining the standard solar model.
A number of other particle physics interpretations for the Solar neutrino
deficits and other observations have been advanced, including oscillations into
sterile neutrinos (Cirelli et al., 2005); resonant spin-flavor transitions involving
a large magnetic moment (Pulido, 1992; Shi et al., 1993); neutrino decay (Bea-
com and Bell, 2002); new flavor changing or conserving interactions (Friedland
et al., 2004; Miranda et al., 2006); mass varying neutrinos due to
interactions with the environment (Kaplan et al., 2004); Lorentz, CPT, or
equivalence principle violations (Glashow et al., 1997; Gonzalez-Garcia and Maltoni,
2008); or CPT-induced decoherence between the quantum components of the

The Standard Electroweak Theory
435
8 I
2 s
3-
2\-
\\
<j£c°68%CL.
<jg,°68%CL.
| <jg 68% CL
0.5
1
- (J)„T 68%, 95%, 99% CI
J
2.5 3 3.5
<|>e (x 10 cm2 S1)
FIGURE 7.48
Fluxes from the WCC and WNC reactions in the SNO heavy water
experiment, and on electron scattering (ES) from SuperKamiokande and SNO,
compared with the standard solar model expectation. Plot reproduced with
permission from (Aharmim et al., 2005).
wave function (Barenboim and Mavromatos, 2005). All of these are now
excluded as the dominant effect for Solar neutrinos, although they could still
exist as perturbations on the basic picture.
Atmospheric Neutrinos
Although the first indications of neutrino oscillations involved the Solar
neutrinos, the first unambiguous evidence came from the oscillations of atmospheric
neutrinos. Atmospheric neutrinos result from pion and muon decays, which
are produced in the upper layers of the atmosphere due to the interaction of
primary cosmic rays. The data from the Kamiokande and SuperKamiokande
water Cherenkov detectors indicated the disappearance of v^ and v^ (which
will not be further distinguished in this paragraph). This was first seen in
the ratio of the v^/ve fluxes, and later confirmed dramatically by the zenith
angle distribution of v^ events, with SuperKamiokande officially announcing
their results in 1998 (Fukuda et al., 1998). Results from other experiments
such as 1MB, MACRO, and Soudan confirmed the results. The details of the
SuperKamiokande ve and v^ events as well as constraints from other
experiments show that the dominant effect is the oscillations of v^ into i/T, and not
ve or vs, with the oscillation parameters
lA™Lnl ~ (1-2 - 3.2) x 10"3 e\\ sin2 6atrn ~ (0.35 - 0.65), (7.417)
Ara§2 and 6atm ~ #23- The data is consistent with maximal
7r/4), and in fact the best fit is for that value. There are no
where Am2atm ~
mixing {0atrn ■

436
The Standard Model and Beyond
Earth matter effects for v^ —> vT, so the sign of Ara2im is not determined. The
more recent long-baseline experiments described below confirm these results.
Comments similar to those for Solar neutrinos concerning alternatives to
neutrino oscillations apply to the atmospheric results. It is interesting that
the atmospheric neutrino oscillations are a quantum mechanical coherence
effect on the size scale of the Earth.
Reactor Neutrinos
There have been a number of reactor disappearance experiments, which
compared the flux of ve at short distances, L < 1 km, with the theoretical
expectations based on the known reactor energy output. In particular, the Palo
Verde and Chooz experiments excluded significant ve mixing for | Ara2| > 10~3
eV2. Since the atmospheric neutrino results already established | A7Ti§2| > few
xlO-3 eV2, this implies that sin2 2#i3 < 0.19 at 90% c.l., which is small
compared to the other leptonic mixing angles but comparable to quark mixings.
The value of #13 is critical, not only for models of neutrino mass but also
because the possibility of observing leptonic CP violation (and significant matter
effects in terrestrial experiments) depends on having a sufficiently large 013.
Future experiments should be able to extend the sensitivity significantly, in
part by having a near detector to give a more precise direct determination of
the initial ve flux.
Because of the characteristic Am2L/E dependence of neutrino oscillations,
long-baseline (L ~ hundreds of km) experiments can probe to much lower
Am2 than the traditional short-baseline ones. The Kamland experiment is a
liquid scintillator detector at the location of the original Kamiokande
detector. It observed a ve flux from a number of Japanese reactors, at a typical
distance L ~ 200 km, which allows one to probe down into the region of the
LMA Solar neutrino solution. The Kamland results dramatically confirmed
the LMA interpretation of the Solar neutrino deficit and gave a much more
precise value for Am^, as can be seen in Figure 7.49. They were also able to
directly observe the L/E dependence expected from neutrino oscillations. An
important background is from neutrinos from radioactive decays in the Earth.
This opens the possibility of future experiments to determine how much such
decays contribute to the interior heating of the Earth.
Accelerator Neutrinos
Accelerator experiments have included v^ (or v^) disappearance experiments
and searches for ve or vT appearance at CERN, Brookhaven, and Fermilab,
typically with sensitivities to Am2 > 10_1 eV2. Recently, there have been
long-baseline experiments involving beams from KEK to SuperKamiokande
(K2K), Fermilab to the Soudan mine (NuMI/MINOS), and CERN to the Gran
Sasso Laboratory (CNGS and the OPERA detector). These are sensitive to
much lower Am2 than the traditional short-baseline accelerator experiments,
down into the atmospheric neutrino region ~ 10~3 eV2. The K2K and MINOS

The Standard Electroweak Theory
437
:
_
-
Solar
.-95%C.L.
-- 99% CL
99.73% CL.
* solar best fit
1 i
ri
* :
±i
i mill
KamLAND
^H|95%C.L.
99% CL
nM73%CL.
• KamLAND ft
1 l 1 1 1-LJ.Ll j l
• Data-BG-Geo vc
Expectation based on oscillation parameters
* determined by KamLAND
20 30 40 50 60 70 80 90 100
L(/E_ (km/MeV)
FIGURE 7.49
Left: Oscillation parameters determined by Kamland compared to the LMA
Solar neutrino solution. Right: L/E dependence of the ve survival probability
as determined by Kamland. Plots reproduced with permission from (Araki
et al., 2005; Abe et al., 2008).
results have confirmed the SuperKamiokande atmospheric oscillation results
and significantly reduced the uncertainty in |Ara2tm|, as can be seen in Figure
7.46.
Future long-baseline experiments may (depending on Ves) be able to observe
CP violation effects and determine the type of neutrino hierarchy via matter
effects.
The LSND experiment at Los Alamos has claimed evidence for oscillations,
especially v^ —► ve obtained from //+ decay at rest, with lAra^^^l > 1 eV2
and small mixing. This was not confirmed by the KARMEN2 experiment at
Rutherford, but there was a small parameter region allowed by both. The
LSND result, along with the Solar and atmospheric oscillations, would imply
three or more distinct Ara2,s, and therefore at least four light neutrinos that
mix with each other. The extra neutrinos would have to be sterile because
the invisible Z width result N™v = 2.985 ±0.009 does not allow a fourth light
active neutrino.
Recently, the Fermilab MiniBooNE experiment, searching for ve appearance
in a v^ beam, did not observe any excess events (Aguilar-Arevalo et al., 2007).
This excludes the LSND parameters in a simple two-neutrino interpretation
(i.e., four neutrinos altogether), but leaves open the possibility of CP-violating
effects in a scheme involving two or more sterile neutrinos. MiniBooNE is
currently running using a p-induced v^ beam, which should allow a direct test
of this possibility.

438
The Standard Model and Beyond
7.7.5 The Spectrum
Three Active Neutrinos
All data other than the LSND results can be accommodated by three active
neutrinos. One recent global three-neutrino analysis (Schwetz et al., 2008)
obtained
Ato^ - 7.65(22) x 10-5 eV2, |Am§2| - 2.40(12) x 10-3 eV2 (7.418)
sin2 02i ~ 0.30(2), sin2 023 ~ 0.50(7), sin2 013 < 0.035 at 90% c.L.
Other analyses (Gonzalez-Garcia and Maltoni, 2008; Fogli et al., 2008) obtain
similar results. The atmospheric mixing 023 is compatible with maximal (7r/4),
while it is now clear that the Solar angle 0i2 is large but not maximal. The
third mixing, 0i3, is small and consistent with zero*.
There are several possible patterns for the spectrum. Assuming that the
absolute masses are comparable to the mass splittings, the sign ambiguity
in Am§2 allows either the normal hierarchy or the inverted hierarchy, as
illustrated in Figure 7.50. The normal hierarchy is most similar to the quark
spectrum, but the analogy is poor since the CKM mixing angles are all small.
In both cases, the data is compatible with a zero value for the lightest mass, m\
or m3, respectively. For mx = 0 in the normal hierarchy, one finds m2 ~ 0.009
eV and ms ~ 0.049 eV, with 7712/7713 ~ 0.18, which is quite large compared to
the analogous quark and charged lepton mass ratios. The effective (3(3$v mass
I TO/3/31 is predicted to be very small, ~ few xlO-3 eV, for the normal
hierarchy because of the small s23, and it could even vanish due to cancellations
for nonzero m\. This range is below the sensitivity of the next generation of
experiments. For the inverted hierarchy, TO3 = 0 implies TOi ~ 0.048 eV and
TO2 ~ 0.049 eV. For small TO3 one has
|TO/3/3| ~ |c22TOi + 522TO2|, (7.419)
which can vary from ~ (c?2 - s?2)|Ato§2|1/2 to l^m2^1/2, i.e., (0.01 - 0.05)
eV, depending on the relative phases of mi)2. This should be within the reach
of existing and planned (3(3$v experiments, at least in their later phases.
It is also possible that the absolute masses are larger than the mass
differences, perhaps as large as nearly 1 eV each if one stretches the cosmological
limits on E in (7.382). This degenerate spectrum allows values for \mpp\ from
(ci2 — 5i2)l7r^l t° |m|, where |m| ~ E/3 is the average mass. Either sign
for ATO32 is still allowed, and there is a smooth interpolation between the
degenerate case and the normal and degenerate hierarchies as |to| becomes
small.
tOne analysis claims weak evidence for a non-zero value, sin2 #13 = 0.016 ± 0.010 (Fogli
et al., 2008).

The Standard Electroweak Theory
439
2 iiiniiiiiiiiiii
1 Mil
2 '"mi i
A™Ln
Am0
1 iiiiii
Am|
A™>ltm
FIGURE 7.50
Left: the ordinary hierarchy for 3 neutrinos. Right: the inverted hierarchy.
The vertical-dashed, open, horizontal-ruled regions indicate the central values
of |Veil2, |VMi|2, and |Vri|2, respectively, except for |Ve312, which is the 90%
c.l. limit. The degenerate case corresponds to adding a large common mass
to each state. Note that in this context 2/3 is usually defined as the isolated
state rather than heaviest.
Implications of LSND
The LSND results, if interpreted in the context of neutrino oscillations,
require the existence of one or more light sterile neutrinos that mix significantly
with active neutrinos of the same helicity. One additional sterile neutrino
could be accomodated in either the 2 + 2 or the 3+1 patterns, illustrated
in Figure 7.51. The 2 + 2 case would require that the Solar and/or
atmospheric results involve a significant admixture of oscillations into the sterile
neutrino. However, it is well established that neither the Solar nor the
atmospheric oscillations are predominantly into sterile states, and this scheme
is excluded. In the 3+1 schemes a predominantly sterile state is separated
from the three predominantly active states. This is consistent with the Solar
and atmospheric data, but excluded when reactor and accelerator
disappearance limits are incorporated. The recent MiniBooNE results increase the
tension. Generalizations of the 3+1 pattern to allow mixing with 2 or 3
sterile neutrinos allow compatibility between LSND and either MiniBooNE or the
disappearance experiments, but apparently not both simultaneously (Maltoni
and Schwetz, 2007). (Other possibilities, e.g., involving CPT violation, de-
coherence, new interactions, extra dimensions, mass-varying neutrinos, and
hybrids, are surveyed in Gonzalez-Garcia and Maltoni, 2008). These schemes
also run into serious cosmological difficulties, though there are some highly
speculative loopholes (Cirelli et al., 2005; Langacker, 2005).

440
The Standard Model and Beyond
AmLSND
> 1
Amltm
A™|
ArnLSND
J ^matm
«Am^
FIGURE 7.51
Left: a 2 + 2 pattern. Right: a 3 + 1 pattern. The dashed (solid) lines indicate
qualitatively the fractions of sterile (active) neutrinos. Other patterns
correspond to the inverted hierarchy for the atmospheric neutrinos or to placing
the largely sterile state in the 3 + 1 case on the bottom.
7.7.6 Models of Neutrino Mass
There is an enormous number of models of neutrino mass and mixing, as
reviewed in (Gelmini and Roulet, 1995; Gonzalez-Garcia and Nir, 2003; Barger
et al., 2003; King, 2004; Langacker, 1999, 2005; Altarelli and Feruglio, 2004;
Mohapatra and Smirnov, 2006; Mohapatra et al., 2007; Albright and Chen,
2006; Chen and Mahanthappa, 2009). Here we mention the major possibilities.
Dirac Masses
There are several possibilities for suppressing hv in the Yukawa interaction
-CYuk = y/2h„ (Il&r + vr^Il) (7.420)
(cf., Eq. 7.13) to obtain a small Dirac mass. Usually one assumes that hv is
forced to vanish at tree-level by some additional symmetry, such as an extra
U(iy gauge symmetry and/or due to an underlying superstring construction.
It is possible, however, that higher-dimensional operators of the form
->HDO
= jj^jf (fsh4»>R + ^vr&Il) (7.421)
are allowed by the symmetries (Cleaver et al., 1998), where (fis is some SM-
singlet spin-0 field and M is a new physics scale, such as the Planck mass.
This is illustrated for k = 1 in Figure 7.41. If (fis acquires a VEV, an effective
Yukawa coupling V2hVeff = (((ps)/M)k will be generated, which can be very
small for (</>s) <C M. For example, M = Mp, k = 1, and (<j>s) at an
intermediate scale rsj 108 GeV yields mv ~ 1 eV. Another possibility is that hVeff is
only generated at loop level, e.g., in a theory with an extended Higgs sector,
or in a supersymmetric theory with heavy exotic quarks (Masiero et al., 1986)

The Standard Electroweak Theory
441
or with non-holomorphic soft scalar interactions (Demir et al., 2008). Small
Dirac masses may also emerge in higher-dimensional theories, e.g., in which
Vh is confined to our 4-d brane, but vR is free to propagate in the extra
dimensions, in which case hv is determined by the (possibly very small) overlap
of their wave functions (Dienes et al., 1999b; Arkani-Hamed et al., 2002).
Related string-inspired possibilities involve large intersection areas in intersecting
brane constructions (Blumenhagen et al., 2005) or string instantons (Cvetic
and Langacker, 2008), both of which yield exponential suppressions.
Majorana Masses
Models which generate Majorana masses are most popular, because no
unbroken standard model gauge symmetry forbids them. (However, models
descended from string constructions may be more restrictive because of the
underlying symmetries and selection rules (Langacker, 2005; Giedt et al.,
2005; Mohapatra et al., 2007).) Majorana masses for active neutrinos violate
weak isospin by one unit. A simple way to generate them at the renormal-
izable level is to extend the SM by the addition of a complex Higgs triplet
(j)T = (0^, 0~, </>^~)T, with tr = 1, and yr = —1. The Yukawa couplings of
(f)T to the leptons are
-£-4>t = 5 hr ^ '<!>t£r + h.c.
-^«U-^)(A)+- <7'422)
with (py = 0^, (fij, = -7x(4>t ~ i<f>T)i and 0t~ = 772(^T ~*~ ^t)> an<^ wnere ^R
is defined in 7.326. This yields a Majorana mass mr = —hr^fo if (fix acquires
a VEV Vfo/y/2, as illustrated in Figure 7.43. The constraints on hr are less
stringent than those for a Dirac mass since one can have \v^T \ <C v ~ 246 GeV
(the parameter p0 = M^/(Mj; cos2 0w) requires |^T| < O(10~2)is
(Problem 7.1)). The original version of the model (Gelmini and Roncadelli, 1981;
Georgi et al., 1981) involved a global lepton number symmetry (the coupling
in (7.422) is invariant since (\>t can be defined to have L = 2). Therefore,
the spontaneous breaking implies a massless Goldstone boson, known as the
triplet Majoron. Astrophysical constraints (see, e.g., RafTelt, 1999) on stellar
cooling from Majoron emission required |&tyT| < O(10) keV, consistent with
small neutrino mass. However, there was no special reason for v^T to
actually be that small. The triplet Majoron model was eventually excluded by
the invisible Z width, as already discussed below (7.169), because the decay
into a Majoron and light scalar would have a partial width equal to that of
two additional active neutrinos. The triplet model could survive, however, by
adding a coupling
-CUt = V^T =KftT-<£T(j) + h.c. (7.423)

442
The Standard Model and Beyond
between cj>t and the Higgs doublet 0, where k has dimensions of mass. The
clash between (7.423) and (7.422) implies that the lepton number is explicitly
violated, yielding a mass for the Majoron which can be taken large enough
to evade observational bounds. A currently popular version (the type II
seesaw (Ma and Sarkar, 1998; Hambye et al., 2001)) includes a mass term
\li$r<\)j> - 4>t for the triplet, where ^ is very large and positive and associated
with a new physics scale. Substituting (jp —> v/y/2 in (7.423)§, V^T becomes
linear in 0^, so that it is forced to acquire a VEV
kv1 . kv2 .
v4>t = 2"> mT = hT^2~. (7.424)
This is very small in the seesaw limit, ^Jk ^> v.
Instead of a Higgs triplet, a Majorana mass for active neutrinos can be
generated by the Weinberg operator (Weinberg, 1980), which is a higher-
dimensional operator in which two Higgs doublets are combined as an isospin
triplet, as illustrated in Figure 7.43. The form of the interactions can be
obtained from (7.422) if we identify <j>xT with (ftr1^, where the Higgs doublet
<j> and its tilde form </> are <j> = ((/>+ (jP)T and </> = (0°* — (j)~)T as defined in
(7.14). Then,
-£** = ^ (hrIR) • (fr$) + h.c. = jj (lL $) (f Ir) + h.c.
C -0 ( 0°t0- 0°t0Ot \ (7-425)
where M is the relevant new physics scale and C is a coefficient which can
be absorbed into M if desired. The second form is obtained using the SU(2)
Fierz identity in Problem 1.1. (The second term in the Fierz identity vanishes
for a single Higgs doublet.) Thus, the Majorana mass is tut = —Cv2/M. For
example, M ~ 1019 GeV (the Planck scale) and C ~ 1 implies tut ~ 10-5
eV. £</></> describes an effective theory that can be generated by many different
underlying models, including the type II seesaw mentioned above. It is easily
generalized to three families, and is considered by many to be the favored
description of small neutrino masses. The scale of M/C (around 1014 GeV
for rriT ~ 1 eV) is suggestive of, though a few orders of magnitude below, a
grand unification or superstring scale.
The most familiar implementation of (7.425) is in the minimal or type I
seesaw (Minkowski, 1977; Gell-Mann et al., 1979; Yanagida, 1979; Schechter
and Valle, 1980)^, in which the active neutrinos mix with heavy sterile Majo-
§One should in principle minimize V w.r.t v and v^T simultaneously. In practice the
modification to v from the coupling in (7.423) is usually negligible in the seesaw limit, and
one can always adjust the other Higgs parameters to maintain the observed value of v.
^Some authors reserve the term "seesaw" for the type I version, but we will use the term
to refer to any model which leads to a small Majorana or Dirac neutrino mass suppressed
by one or more powers of a large new physics scale.

The Standard Electroweak Theory
443
rana neutrinos, as in (7.350) or (7.366) and Figure 7.43, leading to Majorana
masses of 0{m2D/ms). In some versions there is a spontaneously broken
global lepton number, leading to a very weakly coupled Goldstone boson,
the singlet Majoron (Chikashige et al., 1981). Type I seesaw models have
been constructed with many different scales, depending on mD. For example,
choosing mo comparable to the electron mass and mv ~ 0.1 eV implies Ms ~
few TeV. However, most popular are those based on grand unified theories,
where the Dirac masses mD are typically of 0(mu^t). The sterile masses
in Ms may be generated by large Higgs multiplets (e.g., the 126 of 50(10))
or by higher-dimensional operators, and must typically be several orders of
magnitude below the grand unification scale Mx, which is > 1016 GeV in the
supersymmetric case (Section 8.4). Such constructions are usually combined
with family symmetries which restrict and relate the elements of M&, Ms,
and the quark and charged lepton mass matrices, and sometimes combine the
type I and II seesaws (using the Higgs triplets in the 126). Generating the
large observed leptonic mixings in the same GUT context which yields small
quark mixings is a challenge, which requires carefully chosen family
symmetries, and/or non-symmetric (lopsided) mass matrices (Albright et al., 1998),
which lead to large mixing in the unobservable A^ and in the observable AeL
mixing matrices. One especially attractive aspect of the minimal seesaw is
that it opens the possibility of the leptogenesis (Fukugita and Yanagida, 1986;
Davidson et al., 2008) scenario for generating the observed baryon asymmetry
of the Universe (Section 8.1).
Most of these GUT type models are hard to embed in known classes of su-
perstring constructions, where any underlying grand unification is frequently
broken in the higher-dimensional theory, and in any case it is difficult or
impossible to obtain large representations like the 126. More promising is that
the Majorana mass terms in Ms are generated by higher-dimensional
operators in the superpotential" of the form (<f)s/Mp)k vclvr, in which k SM singlet
fields (fis (which need not be the same) acquire VEVs somewhat below the
Planck scale Mp. This mechanism can lead to a sufficiently suppressed ras,
but is likely to lose the GUT and family symmetry structure of the 50(10)
models. String constraints sometimes forbid the simultaneous existence of
such operators and those needed for Mq (Giedt et al., 2005), but there are a
few successful examples (Lebedev et al., 2008). Other possibilities for
generating ms involve higher-dimensional operators in the Kahler potential (Arkani-
Hamed et al., 2001) or string instanton effects (Blumenhagen et al., 2007;
Ibanez and Uranga, 2007).
Seesaw models with lower scales have also been considered. For example,
in some gauge extensions of the SM (e.g., with an extra U(\)' (Kang et al.,
"The Weinberg operator could also emerge directly in a superstring construction. However,
C/M would typically be ~ 1/Mp, which is too small. This difficulty could be avoided if
some of the extra dimensions are large compared to the inverse of the string scale (Conlon
and Cremades, 2007).

444
The Standard Model and Beyond
2005)) the "sterile" neutrinos are charged under the extended group, so any
Majorana masses for them cannot be much larger than the symmetry breaking
scale. For a TeV scale, the extended seesaw model (Mohapatra and Valle
1986), in which ms ~ ^P+1/Mp,p > 1, where // is some scale smaller than
the Majorana mass M, remains a possibility. Small Dirac masses may also
occur in such cases.
We briefly mention two other classes of models which lead to Majorana
masses. One involves masses that are only induced at loop level, e.g.,
associated with an extended scalar sector. In one well-known example (Zee,
1980) an 5[/(2)-singlet charged scalar field h~ is introduced which couples to
both leptons and Higgs doublets, leading to a loop-induced Majorana mass as
shown in Figure 7.43. This example actually leads to off-diagonal masses such
as ^eL^H' *-e' tne ZKM model of (7.369), and also requires a second Higgs
doublet, because the h~ coupling is antisymmetric in lepton and Higgs family
indices. Another possibility occurs in supersymmetric theories with il-parity
violation (Section 8.2), which allows mixing and a type of seesaw between
active neutrinos and neutralinos. The mixing can generate one small
neutrino mass at tree level, with the other two masses entering at loop level (e.g.,
Grossman and Rakshit, 2004; Rakshit, 2004).
Mixed Mass Models
The LSND experiment suggests the possibility of mixing between active and
sterile neutrinos of the same chirality. Many models involve sterile
neutrinos, but generating significant mixing is more difficult, because one requires
Majorana and Dirac masses which are both small and probably comparable.
Possibilities include higher-dimensional operators associated with an
intermediate scale (Langacker, 1998) or sterile neutrinos from a mirror world (Foot
and Volkas, 1995; Berezhiani and Mohapatra, 1995).
Textures
There are also many texture models** (e.g., Barbieri et al., 1999; Altarelli and
Feruglio, 2004), involving specific guesses about the form of the 3x3 neutrino
mass matrix, or of the Dirac and Majorana mass matrices entering seesaw
models. These are often studied in connection with models also involving
quark and charged lepton mass matrices, such as grand unification, family
symmetries, or left-right symmetry (e.g., Raby, 2009). A major complication
is that the form of a mass matrix depends on the basis chosen, e.g., whether the
charged lepton mass matrix is diagonal, and any underlying family symmetries
may take a different form depending on the basis.
**On the other extreme are the anarchy models, involving random entries in the neutrino
mass matrices (Hall et al., 2000).

The Standard Electroweak Theory
445
The observed values of 92s ~ tt/
mixing matrix is close to
/
^ = v-f
4 and 013 ~ 0 i
Cl2 5i2
$12 C12
$12 C12
v/2 72
0
V2-
0 suggest that the leptonic
(7.426)
up to field redefinitions and possible Majorana phases that are irrelevant for
oscillations. The special case of ci2 = y/2/3 and s\2 = \/l73, known as
tri-bimaximal mixing (Harrison et al., 2002; Ma, 2004), is consistent with the
mixing angles in (7.418) within ~ 1<j, while bimaximal mixing, c\2 = 512 =
■v/i/2 was at one time a serious possibility. This and similar patterns may well
be associated with underlying permutation symmetries, such as 54, or groups
of even permutations, such as AA (e.g., Altarelli, 2007). Precise measurements
of the mixing angles to search for deviations from tri-bimaximal are clearly
important.
In practice, small deviations from any specific form such as tri-bimaximal
mixing are possible. For example, they may be associated with broken discrete
symmetries, and/or they could apply only to the neutrino mixing, with
additional small mixing comparable to the CKM angles induced by the charged
leptons (Cabibbo haze (Datta et al., 2005)). It has been observed that #12
is close to the quark-lepton complementarity (Raidal, 2004; Minakata and
Smirnov, 2004) value 6\2 + 6c = 7r/4, where 6C is the Cabibbo angle, reopening
the possibility of bimaximal mixing in the neutrino sector. Renormalization
group evolution of couplings is another complication, since one might expect
underlying symmetries to apply at a large GUT or string scale rather than at
low energies, and mass degeneracies (in sign as well as magnitude) at a high
scale may be destabilized by the running (e.g., Antusch et al., 2005).
In most quark texture models the small mixing angles are associated with
the small ratios of mass eigenvalues, as in Problem 7.26. Two of the neutrino
mixings are large, however, suggesting the possibility that the mixings and
mass eigenvalues are decoupled. This can easily occur. In particular, any
Dirac, triplet Majorana, or seesaw-induced 3x3 neutrino mass matrix of the
form
/ A B-B\
Mv = I B C -D\ (7.427)
\-B -D C)
will lead (e.g., Altarelli, 2007) to the leptonic mixing matrix in (7.426),
corresponding to s 13 = 0 and 523 = C23 = 1/v^, with
sin22*12=[.4-(C + D)]2 + 8fl2- (?-428)
The mass eigenvalues are related by
A = cl2mi+sl2m2, C + D = sl2m1+cl2m2, C-D = m3, (7.429)

446
The Standard Model and Beyond
which because of the phase conventions chosen in (7.426) may be negative or
complex. Clearly, the parameters can be chosen to yield any of the types of
mass hierarchy. Tri-bimaximal mixing occurs whenever
A + B = C + D, (7.430)
implying
2 2
512 = 3' Cl2=3 (7.431)
m1= A- B, m2 = A + 2B, m3 = C -D.
Bimaximal mixing corresponds to
A = C + D, (7.432)
so that
s\2 = c212 = -, rai,2 = A t \/2#, m3 = C-D. (7.433)
The form of (7.429) exhibits an obvious v^ <-> vT interchange symmetry (up
to signs depending on our phase convention).
7.7.7 Implications of Neutrino Mass
Most extensions of the SM predict non-zero neutrino masses at some level, so
it is difficult to determine their origin. Many of the promising mechanisms,
such as the minimal seesaw, involve very short distance scales, e.g., associated
with grand unification or string theories, and are therefore difficult to verify
directly. Some models lead to other predictions. For example, lepton flavor
violating processes^ like ji —> ey, jiN —> eN, or ji —> 3e, by the exchange
of sneutrinos v in supersymmetry or other mechanisms (e.g., Masiero et al.,
2004), may be observable, but their connection to the neutrino mass
generation mechanism is very model dependent. TeV-scale models for neutrino
mass, on the other hand, are more likely to lead to consequences observable
at the LHC (e.g., Atre et al., 2009).
There are many unanswered questions. These include:
• Are the neutrinos Dirac or Majorana? Majorana masses, especially if
associated with a type I or II seesaw, would allow the possibility of lep-
togenesis. The observation of (3(3qv would establish Majorana masses (or
at least L violation), but foreseeable experiments will only be sensitive
to the inverted or degenerate hierarchies. If the neutrinos are Dirac,
this would suggest that additional TeV scale symmetries or string
symmetries/selection rules are forbidding Majorana mass terms.
t^The nonzero neutrino masses and mixings themselves violate lepton flavor, but their
effects are negligible except for neutrino oscillations.

The Standard Electroweak Theory
447
• What is the absolute mass scale (with implications for cosmology)? This
is very difficult, but ordinary and double beta decay experiments, as well
as future cosmological observations, may be able to establish the scale.
• Is there leptonic CP violation? What is 0i3? Is the hierarchy ordinary
or inverted? Leptonic CP violation is a necessary ingredient in lepto-
genesis. The CP phases in V are different from the ones relevant to
leptogenesis, though they are often related in specific models. For three
families, the phase 5 in (7.360) may be observable, e.g., in differences
between the v^ —> ve and v^ —> ve oscillation rates in long-baseline
experiments such as T2K or NOz/A (for a recent review, see Nunokawa
et al., 2008). However, such effects are proportional to #13, and also
depend on the sign of Am^ because of matter effects in the Earth. #13
may be determined in long-baseline reactor experiments, such as Double
Chooz and Daya Bay. The nature of the hierarchy may be determined
simultaneously with CP breaking in long-baseline experiments, or in the
observation of a future supernova, the observation of (3(3qv if the
neutrinos are Majorana, and possibly in future Solar neutrino experiments.
On a longer time scale, these issues may also be addressed at a dedicated
neutrino factory (from a muon storage ring), or in beta beams involving
e~ ore4" emission from accelerated heavy ions. All of these possibilities
are reviewed in (Camilleri et al., 2008). Ultra-high energy neutrinos
from violent astrophysical events can be observed in large detectors in
ice or water, such as IceCube and ANTARES, possibly shedding light
on neutrino oscillations or decay, nonstandard high energy effects, and
on the astrophysical sources (e.g., Hooper, 2006).
• If the LSND result is confirmed, it will suggest mixing between ordinary
and sterile neutrinos, presenting a serious challenge both to particle
physics and cosmology, or imply something even more bizarre, such as
CPT violation.
• Are there any new v interactions or anomalous properties such as large
magnetic moments? Most such ideas are excluded as the dominant
effect for the Solar and atmospheric neutrinos, but could still appear as
subleading effects.
7.8 Problems
7.1 Consider a generalization of the SU(2) x [/(l) model involving k multi-
plets fa, i = 1 • • • A;, of complex scalars. The dimension of the ith multiplet is
2^ + 1, where ti can be 0,1/2,1,3/2 • • •, and the elements have T3 eigenvalues
t\ = -ti, -ti + 1 • • • U (cf., the rotation group). Also, the ith multiplet has

448
The Standard Model and Beyond
weak hypercharge yi. Assume that each multiplet has one electrically neutral
component 0^, i.e., with qi = tf + yi = 0, and that that component acquires
a vacuum expectation value ($) = Vi/y/2.
(a) Show that the mass eigenstates W±, Z, and A are the same as in the
standard model.
(b) Calculate the W and Z masses in terms of g, g', ti, £f, and vi.
(c) The po parameter, p0 = ^w/(^z cos2 ®w) is predicted to be unity at the
tree level in the standard model and in extensions involving additional Higgs
doublets. Show that in the more general case
m2w =Zi=Mti+i)-(mwi\2
Po M*co**0w 2£-=1(i?)>i|2
(d) Specialize to the case of one doublet and two triplets
where i/^ = y/2(</>°) > v$> = >/2(*°) and i/^ > w = y/2(T,°). Calculate pQ to
leading nontrivial order in v^jv^ and isz/vq.
(e) Now consider the case of multiple Higgs doublets but no higher-dimensional
representations. Argue that the couplings of neutral physical Higgs bosons
to fermions will no longer be flavor-diagonal. (Do not attempt to write the
Higgs potential or find the exact Higgs mass eigenstates.)
7.2 Verify that there are no Y or y3 anomalies in the SM.
7.3 In the SM with a single Higgs doublet <j> one can always perform an
SU(2) x U(\) transformation so that ((/>) = A= I J, with v real. Therefore,
SU(2) x U(l) —> U(\)q and there is a conserved electric charge, Q. However,
for two or more doublets with the same hypercharge yn = — \ one has
/ , v _^n_ /exp(zpn)sinan
y/2 \ exP (ton) cos an
where a\, <Ji, and pi can be chosen to be zero by an SU(2) x U(l)
transformation, but the other angles are determined by the potential. If any of
the an are non-zero then SU(2) x U(l) is completely broken and there is no
conserved electric charge. Nontrivial values of o~n and pn may be associated
with CP violation, although in some cases they can be rotated away by field
redefinitions. Analyze the two doublet case and show under what conditions
it leads to the spontaneous breaking of U(1)q. To simplify the analysis,
consider only renormalizable terms, impose CP invariance (so the coefficients in
the potential can be taken to be real), impose the Z^ symmetry fa —> ( — l)l(f)i,
and assume that the parameters are such that v\ ^0, V2 ^ 0. Hint: it is not
necessary to actually determine the values of the Vi.

The Standard Electroweak Theory
449
7.4 Suppose the scalar sector of the standard model is extended, so that
it includes not only the ordinary Higgs doublet 0, but also a new complex
field a. Assume a transforms as a singlet under the SU(2) gauge group (and
also under SU(3) of color), but that it carries weak hypercharge ya = 1 and
therefore electric charge qa = 1.
(a) What is the most general renormalizable gauge invariant potential V (0, a)?
(b) Show that there are some choices of parameters for which not only cj> but
also o will have non-zero vacuum expectation values.
(c) Suppose that the parameters are such that in unitary gauge,
where va = y/2{a) and v — y/2((f)0) are the (real) vacuum expectation values,
and E and H are physical real scalars of definite mass. Assume that va <C
v. Show that the photon acquires a small mass, and calculate it to leading
nontrivial order in vajo.
(d) Show that Z can decay into 7E if it is kinematically allowed, and calculate
the rate to leading nonzero order in vajo.
7.5 Let V7? and -02 be two fermion fields, and define
Let
C = ^Pi pip0 - (zc^°75r^° + h.c) ,
where c is real and positive and T = ( 9 n ) • Calculate the physical fermion
masses, and express the mass eigenstate fields i^l,r in terms of -0^ r-
7.6 Suppose one adds to the standard model an exotic non-chiral pair of
charged leptons, E~[°R, which are both SU(2) singlets with ys = — 1. Then,
ignoring the second and third families, the leptons are
( Ul\ -0 jji-0 jji-0
I -0 ) eR ^L hR •
The mass matrix for the charged leptons is (dropping the superscript —)
-'-w«)C;£)($)+^.
where x and y are generated by the VEVs of the Higgs doublet, and A and B
are bare masses (or can be generated by the VEVs of an SU(2) singlet Higgs).
e°R and E°R have the same quantum numbers, so w.l.o.g. we can take linear

450
The Standard Model and Beyond
combinations such that A = 0. We also assume that B ^> x,y.
(a) Find the mass eigenvalues and eigenstates to order x/B and y/B.
(b) Find the weak neutral current J% to the lowest nontrivial order in x/B
and y/B for each term, and show that it includes flavor changing components.
7.7 A hypothetical future accelerator allows the collisions of e~ with Higgs
particles.
(a) Draw the tree-level diagrams for e~ H —> e~Z, and show that only one of
them is nonzero for me = 0.
(b) Calculate the spin-averaged center of mass differential cross section for
e~H —> e~Z as a function of the CM scattering angle 0, s = EgM, Mz, MH, g^
and the weak angle 0w Take me = 0.
7.8 Use the C\u^ part of the parity-violating e-hadron interaction in (7.118)
to calculate the corresponding non-relativistic potential in (7.121). Use the
formalism developed in Problem 2.24.
7.9 Verify the expression in (7.172) for the forward-backward asymmetry
in e_e+ —> // at the Z-pole directly using trace techniques. Neglect the
fermion masses. Hint: it is not required to calculate F and B separately, only
the combination Afb-
7.10 From (7.55), the interaction Lagrangian for the coupling of the top
quark t to the bottom quark b and the W+ is
£ = -gW-trf*1^ ~ 75)* + h.c,
where g = gV*h/2y/2.
(a) Calculate the differential decay rate dT/d cos 6 for t —> bW+ in the t rest
frame, where 0 is the angle between the t spin direction and the b momentum.
Sum over the b and W+ spins. Neglect nib but keep Mw-
(b) Calculate the fractions Fo, F+, and F_ of all t decays that are into a W
with helicity 0 (longitudinal), +1 (R) , or — 1 (L), respectively.
(c) One of the Fq5± should vanish. Interpret that fact in terms of angular
momentum.
7.11 Show that the effective interaction in (7.191) reproduces the SM result
in (7.189) for the couplings in (7.192).
7.12 Integrate the one-loop RGE in (5.33) and (7.198) numerically (using
Mathematica, Maple, Fortran, C++, etc) and verify qualitatively the upper
and lower limits on Mh in Figure 7.32.
7.13 The renormalization group equations sometimes have infrared stable or
ultraviolet stable fixed points, which are constant values for couplings or their

The Standard Electroweak Theory
451
ratios that are approached asymptotically for Q2 —> 0 or Q2 —> oo. As a simple
example, show that the ratio ht(Q2)/gs(Q2) has an infrared stable fixed point
in the standard model if one neglects g and gf, and find its value. The relevant
RGE equations are given in (5.33) and (7.198). (The exact solution can be
found in (Pendleton and Ross, 1981).) In practice the fixed point may not be
reached until Q2 is too small for the one-loop equations to be valid.
7.14 Calculate the amplitude and cross section for ZLZL —> W£W£ at high
energy using the equivalence theorem.
7.15 Use the equivalence theorem to rederive the leading term in rrtt/Mw
for the polarized differential decay rate for t —> bW+ considered in Problem
7.10a. Neglect ra^.
7.16 Derive (7.209), and use the equivalence theorem to verify the leading
term for large MH/MV.
7.17 Choose the phase conventions for the K° and K° fields according to the
SU(3) convention in (3.91), and furthermore express the octet of pseudoscalar
fields in terms of the quark fields by fa = —iq^-^q, where q = (uds)T (as
in (5.103)). Calculate the CP phase j]k in (7.227) in terms of the CP phases
for the d and s quarks defined in Section 2.10. Show that t]k = — 1 under the
additional convention that the d and s have the same CP phases.
7.18 Derive (7.230), (7.231), and the statement that the I3n =0,2 states
must have nonzero internal angular momenta.
7.19 Derive (7.240) using the Fierz identities in (2.211).
7.20 Show that (7.338) is a solution to the free Majorana field equation
(7.337).
7.21 Show that adding a small 23 element to the ZKM matrix in (7.370)
(a generalization of the pseudo-Dirac case), will lead to three Majorana mass
eigenvalues even though none of the diagonal elements is nonzero.
7.22 Suppose the WCC in the second form in (7.374) contained a small
admixture of V + A,
j£ = e7"[(l-75)+€(l+75)W,
with e< 1. Calculate the leading correction to the ^-channel amplitude in
(7.377) for W~W~ —> e~e~ and show that it survives even for tut = 0. The
interpretation is that the amplitude for the diagrams in Figures 7.42 or 7.44
requires a chirality flip on the internal neutrino line, which can be due either
to a Majorana mass or to a new interaction. (This mechanism does not occur
in normal SU(2)l x SU(2)r x U(l) models, because the V + A interactions
involve i/r rather than z/^.)

452
The Standard Model and Beyond
7.23 Show that the neutral current coupling vlI^^l in (7.74) for an active
massive neutrino can be written as |^d7m(1 —75)^d (i.e., V — A) if it is Dirac
and as -^m7m75^m (pure axial) if it is Majorana, where vd = vL + vR and
vm = "l + vcr.
7.24 Suppose there existed a fourth stable active neutrino with mass mv in
the GeV range but small compared with Mz/2 (this is now excluded by the
measurements of the invisible Z width, but was once considered an interesting
possibility). Similar to the discussion in Problem 5.2, the relic density of such
neutrinos depends on the thermal average of g(vdVd —> X)/3rei if the neutrino
is Dirac, or g{vm^m —> X)/3rei if it is Majorana, where (3rei is the relative
velocity, X represents a sum over accessible final states, the energy is at or
just above threshold, s > 4ra£, and one averages over the (thermalized) initial
spins. Calculate the leading contribution to a/3rei in the CM via the s-channel
Z annihilation diagram at threshold in each case, where X = ff is a light
fermion with neutral current couplings given by (7.75). Show that it is 5-wave
(constant) for the Dirac case and P-wave (ex /?2 = Prei/ty m tne Majorana
case. Neglect rrif. (This calculation was used in the derivation of the Lee-
Weinberg (and others) bound (e.g., Kolb and Turner, 1990; Dolgov, 2002),
mu > 2 GeV (Dirac) or mu > 8 GeV (Majorana), to avoid overdosing the
Universe, assuming that the neutrino is stable and heavier than (9(100) eV
and that fl^ti2 < 1.)
7.25 Derive (7.397).
7.26 As a toy model of a quark texture, consider the empirical Oakes relation
sin0c ~ rn^/mK (Oakes, 1969), where 6c is the Cabibbo angle. Show how
this might emerge from the three-quark Fermi theory described in Section 6.2
if one assumes that for some reason mu ~ Mfx ~ 0. Hint: recall the relation
between the quark and pseudoscalar masses described in Section 5.8.

8
Beyond the Standard Model
8.1 Problems with the Standard Model
The structure of the electroweak part of the standard model was explored in
detail in Chapter 7, but for convenience we summarize the Lagrangian density
after spontaneous symmetry breaking*:
£ = Cgauge +£</> + ^2 *l>r [i 0 ~ ™>r " ) VV
(8.1)
where the self-interactions for the W±, Z, and 7 are given in (7.42) and (7.43),
£0 is given in (7.39), and the fermion currents in (7.60), (7.70), and (7.74).
The standard electroweak model is a mathematically-consistent renormal-
izable field theory which predicts or is consistent with all experimental facts.
It successfully predicted the existence and form of the weak neutral current,
the existence and masses of the W and Z bosons, and the charm quark,
as necessitated by the GIM mechanism. The charged current weak
interactions, as described by the generalized Fermi theory, were successfully
incorporated, as was quantum electrodynamics. The consistency between theory
and experiment indirectly tested the radiative corrections and ideas of renor-
malization and allowed the successful prediction of the top quark mass.
Although the original formulation did not provide for massive neutrinos, they
are easily incorporated by the addition of right-handed states vr (Dirac) or
as higher-dimensional operators, perhaps generated by an underlying seesaw
(Majorana). When combined with quantum chromodynamics for the strong
interactions, the standard model is almost certainly the approximately correct
description of the elementary particles and their interactions down to at least
10~16cm, with the possible exception of the Higgs sector or new very weakly
coupled particles. When combined with general relativity for classical gravity
*C„L is given by J2r vrhi $vrL - \mvr l^rL^rR + h.c.j f 1 + ^ J for Majorana vL masses
generated by a higher-dimensional operator involving two factors of the Higgs doublet, as
in the seesaw model.
453

454
The Standard Model and Beyond
the SM accounts for most of the observed features of Nature (though not for
the dark matter and energy).
However, the theory has far too much arbitrariness to be the final story. For
example, the minimal version of the model has 20 free parameters for mass-
less neutrinos and another 7 (9) for massive Dirac (Majorana) neutrinos*, not
counting electric charge (i.e., hypercharge) assignments. Most physicists
believe that this is just too much for the fundamental theory. The complications
of the standard model can also be described in terms of a number of problems.
The Gauge Symmetry Problem
The standard model is a complicated direct product of three subgroups,
SU(3) x SU(2) x C/(l), with separate gauge couplings. There is no explanation
for why only the electroweak part is chiral (parity-violating). Similarly, the
standard model incorporates but does not explain another fundamental fact
of Nature: charge quantization, i.e., why all particles have charges which are
multiples of e/3. This is important because it allows the electrical neutrality
of atoms (|gp| = |ge|). The complicated gauge structure suggests the existence
of some underlying unification of the interactions, such as one would expect
in a superstring (e.g., Green et al., 1987; Polchinski, 1998; Becker et al., 2007)
or grand unified theory (Section 8.3). Charge quantization can also be
explained in such theories, though the "wrong" values of charge emerge in some
constructions due to different hypercharge embeddings or non-canonical
values of Y (e.g., some string constructions lead to exotic particles with charges
of ±e/2). Charge quantization may also be explained, at least in part, by
the existence of magnetic monopoles (e.g., Preskill, 1984) or the absence of
anomalies*, but either of these is likely to find its origin in some kind of
underlying unification.
The Fermion Problem
All matter under ordinary terrestrial conditions can be constructed out of the
fermions (z/e, e~, u, d) of the first family. Yet we know from laboratory studies
that there are > 3 families: (z/M,//_, c, s) and (vt,t~,t,b) are heavier copies
of the first family with no obvious role in Nature. The standard model gives
no explanation for the existence of these heavier families and no prediction
for their numbers. Furthermore, there is no explanation or prediction of the
fermion masses, which are observed to occur in a hierarchical pattern which
varies over 5 orders of magnitude between the t quark and the e~, or of the
quark and lepton mixings. Even more mysterious are the neutrinos, which
1"12 fermion masses (including the neutrinos), 6 mixing angles, 2 CP violation phases (+
2 possible Majorana phases), 3 gauge couplings, Mh, v, Oqcd, Mp, A
cosmj minus one
overall mass scale since only mass ratios are physical.
*The absence of anomalies is not sufficient to determine all of the Y assignments without
additional assumptions, such as family universality.

Beyond the Standard Model
455
are many orders of magnitude lighter still. It is not even certain whether the
neutrino masses are Majorana or Dirac. A related difficulty is that while the
CP violation observed in the laboratory is well accounted for by the phase in
the CKM matrix, there is no SM source of CP breaking adequate to explain
the baryon asymmetry of the universe.
There are many possible suggestions of new physics that might shed light
on these questions. The existence of multiple families could be due to large
representations of some string theory or grand unification, or they could be
associated with different possibilities for localizing particles in some higher-
dimensional space. The latter could also be associated with string compact-
ifications, or by some effective brane world scenario (Hewett and Spiropulu,
2002; Csaki, 2004; Sundrum, 2005; Amsler et al., 2008). The hierarchies of
masses and mixings could emerge from wave function overlap effects in such
higher-dimensional spaces, or as exponential suppressions associated with
intersection areas in the internal dimensions in intersecting brane
constructions (Blumenhagen et al., 2005). Another interpretation, also possible in
string theories, is that the hierarchies are because some of the mass terms
are generated by higher-dimensional operators and therefore suppressed by
powers of (0|5|0)/Mx, where S is some standard model singlet field and Mx
is some large scale such as Mp. The allowed operators could perhaps be
enforced by some family symmetry (Froggatt and Nielsen, 1979). Radiative
hierarchies (e.g., Babu and Mohapatra, 1991), in which some of the masses
are generated at the loop level, or some form of compositeness are other
possibilities. Despite all of these ideas there is no compelling model and none of
these yields detailed predictions. Grand unification by itself doesn't help very
much, except for the prediction of mj, in terms of mT in the simplest versions.
As discussed in Section 7.7 the small values for the neutrino masses suggest
that they are associated with Planck or grand unification physics, as in the
seesaw model, but there are other possibilities.
Almost any type of new physics is likely to lead to new sources of CP
violation.
The Higgs/Hierarchy Problem
In the standard model one introduces an elementary Higgs field to generate
masses for the W, Z, and fermions. For the model to be consistent the Higgs
mass should not be too different from the W mass. If Mh were to be larger
than M\y by many orders of magnitude the Higgs self-interactions would be
excessively strong. Theoretical arguments suggest that Mh < 700 GeV (see
Section 7.5.1).
However, there is a complication. The tree-level (bare) Higgs mass receives
quadratically-divergent corrections from loop diagrams such as those in
Figure 8.1. One finds (Section 8.2.1)
M2H = {M2H)hare + 0(\g\h2)k\
(8.2)

456 The Standard Model and Beyond
h( ) wf )
H -•*—'- H H -^-^ H
* 92
W f
h-sQ.-h „.....Q....„
W f
FIGURE 8.1
Radiative corrections to the Higgs mass, including self-interact ions,
interactions with gauge bosons, and interactions with fermions.
where A is the next higher scale in the theory. If there were no higher scale, one
could simply interpret A as an ultraviolet cutoff and take the view that M#
is a measured parameter, with (Mn)bare not observable. However, the theory
is presumably embedded in some larger theory that cuts off the momentum
integral at the finite scale of the new physics^. For example, if the next
scale is gravity, A is the Planck scale Mp = G^ ~ 1019 GeV. In a grand
unified theory, one would expect A to be of order the unification scale Mx ~
1014 GeV. Hence, the natural scale for Mh is (9(A), which is much larger
than the expected value. There must be a fine-tuned and apparently highly
contrived cancellation between the bare value and the correction, to more
than 30 decimal places in the case of gravity. If the cutoff is provided by a
grand unified theory there is a separate hierarchy problem at the tree-level.
The tree-level couplings between the Higgs field and the superheavy fields lead
to the expectation that Mh is close to the unification scale unless unnatural
fine-tunings are done, i.e., one does not understand why (Mw/Mx)2 is so
small in the first place.
One solution to this Higgs/hierarchy problem is TeV scale supersymme-
try, in which the quadratically-divergent contributions of fermion and boson
loops cancel, leaving only much smaller effects of the order of supersymmetry-
breaking. (However, supersymmetric grand unified theories still suffer from
the tree-level hierarchy problem.) There are also (non-supersymmetric)
extended models in which the cancellations are between bosons or between
fermions. This class includes Little Higgs models (Arkani-Hamed et al., 2002;
§ As discussed in Section 2.12.2, there is no analogous fine-tuning associated with logarithmic
divergences, such as those encountered in QED, because aln(A/rae) < O(l) even for A =
MP.

Beyond the Standard Model
457
Perelstein, 2007), in which the Higgs is forced to be lighter than new TeV
scale dynamics because it is a pseudo-Goldstone boson of an approximate
underlying global symmetry, and Twin-Higgs models (Chacko et al., 2006).
Another possibility is to eliminate the elementary Higgs fields, replacing
them with some dynamical symmetry breaking mechanism based on a new
strong dynamics (Hill and Simmons, 2003). In technicolor, for example, the
SSB is associated with the expectation value of a fermion bilinear, analogous
to the breaking of chiral symmetry in QCD. Extended technicolor, top-color,
and composite Higgs models all fall into this class.
Large^ and/or warped extra dimensions (Arkani-Hamed et al., 1998; Dienes
et al., 1999a; Randall and Sundrum, 1999) can also resolve the difficulties, by
altering the relation between Mp and a much lower fundamental scale, by
providing a cutoff at the inverse of the extra dimension scale, or by using
the boundary conditions in the extra dimensions to break the electroweak
symmetry (Higgsless models (Csaki et al., 2004)). Deconstruction models,
in which no extra dimensions are explicity introduced (Arkani-Hamed et al.,
2001; Hill et al., 2001), are closely related.
Most of the models mentioned above have the potential to generate flavor
changing neutral current and CP violation effects much larger than
observational limits. Pushing the mass scales high enough to avoid these problems
may conflict with a natural solution to the hierarchy problem, i.e., one may
reintroduce a little hierarchy problem. Many are also strongly constrained
by precision electroweak physics. In some cases the new physics does not
satisfy the decoupling theorem (Appelquist and Carazzone, 1975), leading to
large oblique corrections (Section 7.3.6). In others new tree-level effects may
again force the scale to be too high. The most successful from the precision
electroweak point of view are those which have a discrete symmetry which
prevents vertices involving just one heavy particle, such as il-parity in super-
symmetry, T-parity in some little Higgs models (Cheng and Low, 2003), and
KK-p&rity in universal extra dimension models (Appelquist et al., 2001).
A very different possibility is to accept the fine-tuning, i.e., to abandon
the notion of naturalness for the weak scale, perhaps motivated by anthropic
considerations (Agrawal et al., 1998). (The anthropic idea will be
considered below in the discussion of the gravity problem.) This could emerge, for
example, in split supersymmetry (Arkani-Hamed and Dimopoulos, 2005).
The Strong CP Problem
Another fine-tuning problem is the strong CP problem (e.g., Peccei, 1989;
Dine, 2000; Ramond, 2004; Kim and Carosi, 2008). One can add an
additional term ££% g^GG to the QCD Lagrangian density which breaks P, T
^See (Horava and Witten, 1996; Witten, 1996; Lykken, 1996b) for early explorations of
large dimensions associated with low superstring scales.

458
The Standard Model and Beyond
and CP symmetry, as in (5.1)". Gl^v = e^vpaGlpa/2 is the dual field strength
tensor introduced in (5.6). This term, if present, would induce an electric
dipole moment dn for the neutron. The rather stringent limits on the dipole
moment lead to the upper bound |0qcd| < 10~n. The question is, therefore,
why is Oqcd so small? It is not sufficient to just say that it is zero (i.e.,
to impose CP invariance on QCD) because of the observed violation of CP
by the weak interactions. As discussed in Chapter 7, this is believed to be
associated with phases in the quark mass matrices. The quark phase
redefinitions which remove them lead to a shift in Oqcd by O(l0~3) because of the
anomaly in the vertex coupling the associated global current to two gluons, as
discussed following (5.131). Therefore, an apparently contrived fine-tuning is
needed to cancel this correction against the bare value. Solutions include the
possibility that CP violation is not induced directly by phases in the Yukawa
couplings, as is usually assumed in the standard model, but is somehow
violated spontaneously. Oqcd then would be a calculable parameter induced
at loop level, and it is possible to make Oqcd sufficiently small. However,
such models lead to difficult phenomenological and cosmological problems**.
Alternately, Oqcd becomes unobservable (i.e., can be rotated away by the
quark phase redefinition) if there is a massless u quark (Kaplan and Manohar,
1986). However, most phenomenological estimates (Nelson et al., 2003) are
not consistent with mu = 0. Another possibility is the Peccei-Quinn
mechanism (Peccei and Quinn, 1977), in which an extra global U(l) symmetry is
imposed on the theory in such a way that Oqcd becomes a dynamical variable
which is zero at the minimum of the potential. The spontaneous breaking of
the symmetry, along with explicit breaking associated with the anomaly and
instanton effects, leads to a very light pseudo-Goldstone boson known as an
axion (Weinberg, 1978; Wilczek, 1978). Laboratory, astrophysical, and
cosmological constraints suggest the range 109 — 1012 GeV for the scale at which
the U(l) symmetry is broken (e.g., Kim and Carosi, 2008).
The Gravity Problem
Gravity is not fundamentally unified with the other interactions in the
standard model, although it is possible to graft on classical general relativity by
hand. However, general relativity is not a quantum theory, and there is no
obvious way to generate one within the standard model context. Possible
solutions include Kaluza-Klein (Chodos et al., 1987) and supergravity (e.g., Nilles,
1984; Wess and Bagger, 1992; Terning, 2006) theories. These connect gravity
with the other interactions in a more natural way, but do not yield renormal-
izable theories of quantum gravity. More promising are superstring theories
II One could add an analogous term for the weak SU(2) group, but it does not lead to
observable consequences, at least within the SM (Anselm and Johansen, 1994; Dine, 2000).
** Models in which the CP breaking occurs near the Planck scale may be viable (Nelson,
1984; Barr, 1984).

Beyond the Standard Model
459
(which may incorporate the above), which unify gravity and may yield finite
theories of quantum gravity and all the other interactions. String theories are
perhaps the most likely possibility for the underlying theory of particle physics
and gravity, but at present there appear to be a nearly unlimited number of
possible string vacua (the landscape), with no obvious selection principle. As
of this writing the particle physics community is still trying to come to grips
with the landscape and its implications. Superstring theories naturally imply
some form of supersymmetry, but the supersymmetry relevant to our sector
of Nature could be manifest only at a high scale and have nothing to do with
the Higgs/hierarachy problem (split supersymmetry is a compromise, keeping
some aspects at the TeV scale).
In addition to the fact that gravity is not unified and not quantized there
is another difficulty, namely the cosmological constant. The cosmological
constant can be thought of as the energy of the vacuum. However, we saw
in Section 7.2.2 that the spontaneous breaking of SU(2) x U(l) generates a
value (0|V(i/)|0) = —/x4/4A for the expectation value of the Higgs potential
at the minimum. This is a c-number which has no significance for the
microscopic interactions. However, it assumes great importance when the theory is
coupled to gravity, because it contributes to the cosmological constant. The
cosmological constant becomes
Acosm = Afeare + ASSB, (8.3)
where A&are = SttGnV(0) is the primordial cosmological constant, which can
be thought of as the value of the energy of the vacuum in the absence of
spontaneous symmetry breaking. (The definition of V((/>) in (7.12) implicitly
assumed A&are = 0.) Assb is the part generated by the Higgs mechanism:
I Assb I = 87rG„|(0|y|0)| ~ 1056Ao6s. (8.4)
It is some 1056 times larger in magnitude than the observed value A0bs ~
(0.0024 eV)4/87rGjv (assuming that the dark energy is due to a cosmological
constant), and it is of the wrong sign.
This is clearly unacceptable. Technically, one can solve the problem by
adding a constant +//4/4A to V, so that V is equal to zero at the minimum
(i.e., Abare = 27tGjvA*4/^)- However, with our current understanding there
is no reason for A&are and Assb to be related. The need to invoke such an
incredibly fine-tuned cancellation to 50 decimal places is probably the most
unsatisfactory feature of the standard model.
The problem becomes even worse in superstring theories, where one expects
a vacuum energy of 0(Mf>) for a generic point in the landscape, leading to
IAssbI > 10123Aobs. The situation is almost as bad in grand unified theories.
Finally, any solution must deal with other contributions to the vacuum
energy, such as zero-point energies (which cancel in supersymmetry), the smaller
but still very large energy associated with the the chiral symmetry breaking
in QCD, or other QCD or QED vacuum condensates. Contributions from

460
The Standard Model and Beyond
strings, electroweak breaking, QCD, and QED are associated with very
different physics scales, but somehow they must all add up to a negligible value.
So far no compelling solution to the cosmological constant problem has
emerged. One intriguing possibility invokes the anthropic (environmental)
principle (Barrow and Tipler, 1986; Rees, 2004; Hogan, 2000), i.e., that a much
larger or smaller value of |Acosm| would not have allowed the possibility for
life to have evolved because the Universe would have expanded or recollapsed
too rapidly (Weinberg, 1989b). This would be a rather meaningless argument
unless (a) Nature somehow allows a large variety of possibilities for |Acosm|
(and possibly other parameters or principles) such as in different vacua, and
(b) there is some mechanism to try all or many of them. In recent years it
has been suggested that both of these needs may be met. There appear to be
an enormous landscape of possible superstring vacua (Bousso and Polchinski,
2000; Kachru et al., 2003; Susskind, 2003; Denef and Douglas, 2004), with no
obvious physical principle to choose one over the other. Something like eternal
inflation (Linde, 1986) could provide the means to sample them, so that only
the environmentally suitable vacua lead to long-lived Universes suitable for
life. These ideas are highly controversial and are currently being heatedly
debated.
The New Ingredients
It is now clear that the standard model requires a number of new ingredients.
These include
• A mechanism for small neutrino masses. The most popular
possibility is the minimal seesaw model, implying Majorana masses, but
there are other plausible mechanisms for either small Dirac or
Majorana masses, as described in Section 7.7.
• A mechanism for the baryon asymmetry. The standard model
has neither the nonequilibrium condition nor sufficient CP violation to
explain the observed asymmetry between baryons and antibaryons in
the Universe (Sakharov, 1967; Bernreuther, 2002; Dine and Kusenko,
2004)tt. One possibility involves the out of equilibrium decays of
superheavy Majorana right-handed neutrinos (leptogenesis (Fukugita and
Yanagida, 1986; Davidson et al., 2008)), as expected in the minimal
seesaw model. Another involves a strongly first order electroweak phase
transition (electroweak baryogenesis (Trodden, 1999)). This is not
expected in the standard model, but could possibly be associated with
ft The third necessary (Sakharov) ingredient, baryon number nonconservation, is present
in the SM because of non-perturbative vacuum tunneling (instanton) effects ('t Hooft,
1976b), which violate B and L but preserve B — L. These are negligible at zero
temperature where they are exponentially suppressed, but important at high temperatures due
to thermal fluctuations (sphaleron configurations), before or during the electroweak phase
transition (Klinkhamer and Manton, 1984; Kuzmin et al., 1985).

Beyond the Standard Model
461
loop effects in the minimal supersymmetric extension (MSSM) if one of
the scalar top quarks is sufficiently light (Carena et al., 2008).
However, it is most likely in extensions of the MSSM involving SM singlet
Higgs fields that can generate a dynamical // term, which can easily lead
to strong first order transitions at tree-level (e.g., Barger et al., 2007).
Such extensions would likely yield signatures observable at the LHC.
Both the seesaw models and the singlet extensions of the MSSM could
also provide the needed new sources of CP violation. Other
possibilities for the baryon asymmetry include the decay of a coherent scalar
field, such as a scalar quark or lepton in supersymmetry (the Affleck-
Dine mechanism (Affleck and Dine, 1985)), or CPT violation (Cohen
and Kaplan, 1987; Davoudiasl et al., 2004). Finally, one cannot totally
dismiss the possibility that the asymmetry is simply due to an initial
condition on the big bang. However, this possibility disappears if the
universe underwent a period of rapid inflation (Guth, 1981; Kolb and
Turner, 1990; Lyth and Riotto, 1999).
• What is the dark energy? In recent years a remarkable concordance
of cosmological observations involving the cosmic microwave background
radiation (CMB), acceleration of the Universe as determined by Type la
supernova observations, large scale distribution of galaxies and clusters,
and big bang nucleosynthesis has allowed precise determinations of the
cosmological parameters (Kolb and Turner, 1990; Peebles, 1993; Dunk-
ley et al., 2009; Amsler et al., 2008): the Universe is close to flat, with
some form of dark energy making up about 74% of the energy density.
Dark matter constitutes > 21%, while ordinary matter (mainly baryons)
represents only about 4-5%. The mysterious dark energy (Peebles and
Ratra, 2003; Frieman et al., 2008; Martin, 2008), which is the most
important contribution to the energy density and leads to the acceleration
of the expansion of the Universe, is not accounted for in the SM. It could
be due to a cosmological constant that is incredibly tiny on the particle
physics scale, or to a slowly time varying field [quintessence). Is the
acceleration somehow related to an earlier and much more dramatic
period of inflation? If it is associated with a time-varying field, could it be
connected with a possible time variation of coupling "constants" (e.g.,
Uzan, 2003)?
• What is the dark matter? Similarly, the standard model has no
explanation for the observed dark matter, which contributes much more
to the matter in the Universe than the stuff we are made of. It is likely,
though not certain, that the dark matter is associated with elementary
particles. An attractive possibility is weakly interacting massive
particles (WIMPs), which are typically particles in the 102 - 103 GeV range
with weak interaction strength couplings, and which lead naturally to
the observed matter density. These could be associated with the
lightest supersymmetric partner (usually a neutralino) in supersymmetric

462
The Standard Model and Beyond
models with il-parity conservation, or analogous stable particles in
Little Higgs or universal extra dimension models. There is a wide variety
of variations on these themes, e.g., involving very light gravitinos or
other supersymmetric particles. There are many searches for WIMPs
going on, including direct searches for the recoil produced by scattering
of Solar System WIMPs, indirect searches for WIMP annihilation
products, and searches for WIMPs produced at accelerators (Jungman et al.,
1996; Gaitskell, 2004; Bertone et al., 2005; Hooper and Baltz, 2008)!
Axions, perhaps associated with the strong CP problem or with string
vacua (Svrcek and Witten, 2006), are another possibility. Searches for
axions produced in the Sun, in the laboratory, or from the early universe
are currently underway (Hannestad and Raffelt, 2004; Asztalos et al.,
2006; Kim and Carosi, 2008).
• The suppression of flavor changing neutral currents, proton
decay, and electric dipole moments. As discussed in Sections 7.6.5
and 7.6.6, the standard model has a number of accidental symmetries
and features which forbid proton decay at the perturbative level,
preserve lepton number and lepton flavor (at least for vanishing neutrino
masses), suppress transitions such as K+ —> k+vv at tree-level, and
lead to highly suppressed electric dipole moments for the e~, n, atoms,
etc. However, most extensions of the SM have new interactions which
violate such symmetries, leading to potentially serious problems with
FCNC and EDMs. There seems to be a real conflict between attempts
to deal with the Higgs/hierarchy problem and the prevention of such
effects.
Possible Types of New Physics
There is an enormous range of possibilities for new physics beyond the
standard model, many of which were mentioned above or in Chapter 7. Many
types are "bottom-up," i.e., motivated by attempts to resolve some of the
problems of the standard model or to understand the dark matter. These
include supersymmetry, dynamical symmetry breaking, extended Higgs
sectors, little Higgs models, large or warped extra dimensions, family symmetries,
and extended TeV-scale gauge groups. Other "top-down" types, such as grand
unification and superstring theories, attempt to achieve a more fundamental
unified understanding of the microscopic interactions, to incorporate gravity,
to describe the dark energy, or to make contact with ideas concerning the early
universe. Finally, additional or exotic heavy fermions, additional Higgs
particles, and new types of interactions may arise as remnants of an underlying
theory that is characterized by a much larger scale, such as a superstring
theory in which most but not all degrees of freedom achieve Planck scale masses.
Such remnants are quite plausible from a top-down perspective, though they
may appear superfluous and non-minimal from a bottom-up view. In the

Beyond the Standard Model
463
following we describe three important examples of possible new physics: su-
persymmetry, extended (TeV-scale) gauge symmetries, and grand unification.
8.2 Supersymmetry
Supersymmetry is a hypothetical symmetry very different from those we have
so far encountered, viz., between bosons and fermions. It is especially
motivated by attempts to unify gravity with the other interactions, especially in
supergravity and superstring theories, though that does not by itself imply
that supersymmetry is relevant at the TeV scale. However,
supersymmetry broken at the electroweak-TeV scale is one of the leading contenders for
solving the Higgs/hierarchy problem discussed in Section 8.1. Other
motivations include gauge coupling unification, which is much more successful in
the supersymmetric extension than in the SM, and the existence of
natural candidates for cold dark matter in some versions. The possible anomaly
in the anomalous magnetic moment of the /x± could also be accounted for
by relatively light supersymmetric partners, as mentioned in Section 2.12.3.
The technology and phenomenology of supersymmetry are extremely
complicated, and here we can only give a brief introduction. These topics are
thoroughly described in a number of articles (Fayet and Ferrara, 1977; Nilles,
1984; Haber and Kane, 1985; Haber, 1993; Bagger, 1996; Dine, 1996; Lykken,
1996a; Martin, 1997; Raby, 2004; Peskin, 2008) and books (e.g., Kane, 1998;
Weinberg, 2000; Polonsky, 2001; Drees et al., 2004; Baer and Tata, 2006; Bi-
netruy, 2006; Terning, 2006; Dine, 2007). The original idea of a symmetry
between bosons and fermions involved the two-dimensional worldsheet of a
string theory (Ramond, 1971; Neveu and Schwarz, 1971; Gervais and Sakita,
1971). Supersymmetry for a four-dimensional field theory was introduced
in (Wess and Zumino, 1974a,b; Salam and Strathdee, 1974), and applied to
the SU(2) x U(l) model in (Fayet, 1975). The history of the development of
supersymmetry is described in detail in (Weinberg, 2000; Wess, 2009).
8.2.1 Implications of Supersymmetry
Corrections to the Higgs Mass
As described in Section 8.1 the SM leads to troubling quadratically-divergent
contributions to Mjj. For example, the top quark contribution (the first

464
The Standard Model and Beyond
diagram in Figure 8.2), implies a radiative correction**
-. (A*i), - -NA-ik,f / |t n (_J_ _J_) , (M)
where —1 is due to the closed fermion loop, Nc = 3 counts the £-quark colors
and the vertices follow from (7.46) with ht = mt/v. The quadratically
divergent part can easily be calculated by applying a Wick rotation, as in (E.14)
to yield
(AM2\ - AiK(h* [Aik?BdkBd£l4 -k% + m2t Nc(ht)2 2
since f dQ,^ = 2tt2.
Now, suppose there are two complex color-triplet scalar fields 0r,r = 1,2,
with mass and couplings to H given by
C^h = -XrH24>l4>r - ktH<\>\<\>t - m2rc\>\<\>T. (8.7)
Each contributes to Mjj by the middle and right diagrams in Figure 8.2. The
middle diagram is quadratically divergent, yielding
-i(AM!V = -2W(0_i_ - <iU4)fc~^A».
(8.8)
The quadratic divergences cancel if we choose Ai = A2 = (ht)2. This would be
a remarkable accident if not enforced by some symmetry. Fortunately, in the
supersymmetric extension of the standard model it occurs naturally: in that
case (/>i?2 are the scalar partners of the t^ and tcL (i.e., £#), respectively, and
the necessary coupling constant relation is enforced by the supersymmetry. In
fact, in the supersymmetric limit, one also has that kt = 2v(ht)2 and m2 =
m2, which implies that the logarithmically divergent contributions to Mjj also
cancel, as shown explicitly in (e.g., Terning, 2006). This is an example of the
non-renormalization theorem in the supersymmetric limit. In the presence of
soft supersymmetry breaking the quadratic divergences continue to cancel,
but there are finite contributions to Mfj of the order of the supersymmetry
breaking scale. (See Eq. 8.160.)
Supersymmetry is therefore successful at solving the part of the hierarchy
problem associated with the loop corrections. However, as we will see below
it introduces a new tree-level hierarchy problem, the \i problem.
t^The diagram yields the t contribution to (H\iA£\H), where we are being careless
about external momenta since we are considering the divergent contribution. But C =
— ^MJjH2 H , so the matrix element is — i (AM|) .

465
<t>r
^r \ i Kr
<t>r
FIGURE 8.2
t quark and <j>k contributions to M]f.
The Supersymmetric Spectrum
Supersymmetry is elegant in its principles but not economical in its particle
content: it requires a more than doubling of the SM spectrum. Each standard
model particle must have a superpartner (or sparticle) differing in spin by 1/2
unit, which we will denote with a tilde. The sparticles have the same SU(3) x
SU{2) x U(l) assignments as their partners, and for phenomenological reasons
none of the SM bosons or fermions can be each other's partners. In particular,
each left (right)-chiral quark q^ (qn) is predicted to have a spin-0 scalar quark
(squark) partner qL ((Jr). Of course, the L and R labels for a spin-0 particle are
not directly related to spin—they simply mean that the scalar is the partner of
the corresponding quark. As discussed in Section 2.11, it is convenient to work
in terms of the left-chiral particles and antipartides, using the correspondence
qcL = Cq^ (Eq. 2.296) and qcL = q*R. In the supersymmetric limit, the masses
of the quarks and squarks would be the same and their interactions related
in a definite way, as we saw in the example of the cancellation of the Higgs
quadratic mass divergence. No light squarks or other superpartners have
been observed, so the supersymmetry must be broken, with the sparticles all
relatively heavy, e.g., several hundred GeV*. Similarly, the leptons £ must
have spin-0 slepton partners £, and the gauge bosons must have spin-1/2
gaugino partners, such as the gluinos (G), the winos (W), and the bino (B).
Extensions to supergravity also predict a spin-3/2 partner of the graviton (g),
the gravitino (#3/2)-
Supersymmetry also requires an extended Higgs sector. We saw in Section
7.1 that a single Higgs doublet <j> could have the needed Yukawa couplings to
generate masses for both the u and d quarks by making use of the conjugate 0
defined in (7.14) by </> = ir2^ (this tilde does not represent a superpartner).
However, supersymmetry does not allow the needed 0 Yukawa couplings, so
we must instead introduce two separate Higgs doublets
*It is fairly natural for the superpartners to be much heavier than the SM particles, because
the former can all acquire large masses without the SSB of SU(2) x U(l), while the SM
particle masses other than the Higgs all require SSB.
peyond the Standard Model
"-kOc-" " r: "
tR

466
The Standard Model and Beyond
where <j)d is similar to the SM </> and has the couplings needed to generate
masses for the d quarks and charged leptons, while (fiu plays the role of the
SM <f>, and can lead to masses for the u quarks and Dirac neutrino masses. It
is actually more convenient to introduce the conjugate doublets
ku = ( hP) = ~^u' kd = ( h~ ) ~ ^' ^8"10)
which will allow us to write the Higgs Yukawa interactions in a manifestly
supersymmetric form. In any case, the minimal supersymmetric standard
model (MSSM) involves 3 neutral and one conjugate pair of charged Higgs
particles (not including the Goldstone bosons), as opposed to the single Higgs
scalar of the SM.
Supersymmetry also does not allow the quartic Higgs self-interaction A(0t</>)2
of the SM. Rather, its role is played by (known) gauge couplings. This
removes one of the most arbitrary aspects of the SM, and leads to a theoretical
upper bound (at tree level) on the mass of the lightest neutral mass eigenstate
Mlo <cos22/3M|, (8.11)
where tan/? = K0|ft°|0)|/|(0|ft2|0)| is the ratio of neutral Higgs VEVs. (8.11)
violates the LEP lower limit of 114.4 GeV on the mass of the SM Higgs,
which applies to most but not all parameter regions for the MSSM as well.
Fortunately, there are large radiative corrections to the effective potential
associated with top and scalar top loops, which increase the upper bound to
~ 130 GeV (Haber and Hempfling, 1991; Okada et al., 1991; Ellis et al., 1991),
consistent with the Higgs mass range suggested by the combination of direct
searches and upper bounds from precision electroweak physics.
The two Higgs doublets must also have spin-^ partners [Higgsinos). (If
there were only one Higgs doublet the Higgsinos would introduce triangle
anomalies, providing another rationale for the second doublet.) The
Higgsinos can mix with the winos and bino to produce two mass eigenstate
Dirac charginos (x^, r = 1,2), and 4 mass eigenstate Majorana neutralinos
(Xr> t = 1 • • • 4). (These are sometimes denoted C-jr and N® instead.)
The MSSM particles are listed in Table 8.1
Other Implications and Difficulties
There are a number of other implications of supersymmetry, both good and
bad.
• We have not yet observed any superpartners, so supersymmetry must
be broken, probably with sparticle masses in the several hundred GeV
range or higher. To avoid the reintroduction of the hierarchy problem
the breaking should be soft, i.e., appearing only in scalar and gaugino
mass terms and in cubic scalar couplings. However, general soft break-

Beyond the Standard Model
467
TABLE 8.1
Standard model particles and their supersymmetric partners. Family
indices and mixing are ignored for the fermions and their scalar partners.
The right-chiral fermions and their partners are related by tpR = CjpcT and
i>R — ^i- The gauge and Higgs particles are listed in the weak basis. The
graviton g and its partner the gravitino, #3/2, are also listed.
yL CL
spi,o (£) nn (£) «|
spin-i (fy u%dl (^) WLei GW'B
spin-1
spin-§ £3/2
spin-2 g
ing terms introduce an enormous number of free parameters^. Moreover,
there are mass sum rule constraints that would be violated for
spontaneous or dynamical breaking that occurs directly in the ordinary sector
associated with the MSSM particles. Supersymmetry breaking
therefore most likely occurs in some hidden sector which is only very weakly
coupled to the ordinary sector. How this breaking occurs (e.g., Intrili-
gator and Seiberg, 2007) and how the information is transmitted to our
sector (the mediation mechanism) (e.g., Chung et al., 2005) introduce
considerable uncertainties.
• Many supersymmetric models involve or impose a discrete il-parity
symmetry (Farrar and Fayet, 1978), Rp, which requires that every allowed
interaction vertex involves an even number of superpartners. This
implies that the lightest superpartner (the LSP) is absolutely stable, and
therefore a candidate to be dark matter. Neutralinos are the most
promising possibility, although scalar neutrinos or the gravitino cannot
be excluded.
• In the decoupling limit, in which the superpartners and extra Higgs fields
are all much heavier than the electroweak scale, the contributions of the
new particles to electroweak precision observables is small, consistent
^The MSSM with ^-parity conserved has 124 free parameters (Dimopoulos and Sutter,
1995), not including neutrino masses and mixings or right-handed scalar neutrinos.

468
The Standard Model and Beyond
with the excellent agreement with the SM predictions. Also, the
lightest neutral Higgs (h°) acts very much like the SM Higgs in this limit.
The mass range suggested by the precision data is consistent with the
range expected for Mho, but certainly does not prove the existence of
supersymmetry.
• On the other hand, the sparticles lead to possible new sources of FCNC
and of CP violation, e.g., in the K and B systems and in EDMs. The
non-observation of such effects suggests that sparticles should be very
heavy and/or that sparticles of a given type are nearly degenerate and/or
that there is some kind of alignment between the mixing effects in the
quark and scalar quark sectors.
• The combination of gauge invariance and supersymmetry does not allow
masses for any of the MSSM particles (prior to the SU{2) x [/(l) and
supersymmetry breaking), with the exception of the Higgs scalars and
their Higgsino partners, which are allowed to have a common arbitrary
mass ji. If the supersymmetry derives from an underlying string theory
one might expect // to be comparable to the Planck or string scales,
or possibly 0 if it is forbidden by some extra symmetry. However,
neither 0 nor a very large value for // is allowed phenomenologically. The
fi problem (Kim and Nilles, 1984), which is a tree-level form of the
Higgs/hierarchy problem, is to understand why // should be nonzero
and comparable to the soft supersymmetry breaking scale. Possible
explanations are discussed in Section 8.2.6.
• The Higgs soft mass parameters are typical soft supersymmetry breaking
parameters. Assuming that the ji problem is somehow solved, the scale
of electroweak symmetry breaking is therefore tied to the
supersymmetry breaking scale, up to an order of magnitude or so. (A larger splitting
reintroduces a more moderate version of the hierarchy problem.)
• Many, but not all, supersymmetry breaking and mediation mechanisms
imply that the Higgs and other scalar mass-squares are positive at some
large scale, such as at Mp. However, electroweak symmetry breaking is
most easily accomplished for a negative Hu mass-square Mjj , though
this is not absolutely essential. These constraints can be reconciled by
the fact that the soft mass-squares are running quantities, just like
coupling constants. A positive Mjj at a high scale can be driven negative
at a low scale by a large top-Yukawa coupling ht. This radiative breaking
mechanism (e.g., Ibanez and Ross, 2007) therefore requires a heavy top
quark mass.

Beyond the Standard Model 469
8.2.2 Formalism
In this section we summarize some of the formalism used in the construction
of supersymmetric field theories.
The Lorentz and Poincare Groups
Let us briefly survey the Lorentz and Poincare groups, which describe the
classical spacetime symmetries. A Lorentz transformation can be defined by
its action on four-vectors,
xm _> jv = A^ Xm _> ^ = Xv (A-i)^ ^ (8 12)
with (A-1) = A^. The group of such transformations is known as 50(1,3),
i.e., it leaves invariant x^y^ = g^x^yv', where g^v = diag(l, —1,-1,-1). A
can be expressed in terms of its generators* (i.e., elements of the Lie algebra)
MPa by
A = e-*wP*M'-, (8.13)
where the 6 group parameters uopa are real and antisymmetric in the indices,
as are the 6 generators Mpa. The group action and commutation rules can be
obtained using the classical representation
Mpa = XpVa - XaVp, (8.14)
where pa = +ida is the position space representation of the momentum
operator Pa. For small uop<T one finds
rf* ~ {S*v - u^x". (8.15)
It is useful to define the group parameters
<S = \*ijk<»3ki C* =<A iJ,k = 1,2,3, (8.16)
and generators
ji = L^Mjk = (xx p)\ K{ = M0i = xV - x'p°, (8.17)
so that
luf°M/Hr=C3'J + C'K. (8.18)
J1 are just the rotation generators (angular momenta), and K% generate
Lorentz boosts in the i direction. The rapidity £ is related to the velocity
*These elements actually represent the proper orthochronous Lorentz group, with det A =
+1 and A°0 > +1. It must be supplemented with space reflection and time reversal to
obtain the full Lorentz group.

470 The Standard Model and Beyond
(3 of the Lorentz boost by C = /3tanh_1 /?, which is approximated by 0 f0r
small /?. Then
x'° ~ x° - /?V, xfi nx'-iux xy - ?x0, (8.19)
which indeed represent § an infinitesimal rotation by Q and boost by /?.
One can extend to the Poincare group by including the translations
xf = Tx = e+iaPppx = x- a, (8.20)
with a field $(x) transforming as in (1.13), i.e.,
$(x) -► &(x) = e+iaPp^(x)e-iaPp^ = $(x + a) = ^(T^x). (8.21)
The Lie algebra of the Poincare group can be derived from [xp,pa] = —igpa,
[PM, Pv\ = 0, [MMI/, Pp] = -z(<7MPP„ - <7„PPM)
[AfMl/, Mp(T] = i{gvpM^ - gvaM^p - g^pMva + g^M^),
and correspondingly
[JV] = ieijkj\ [K\J1] = ieijkK\ [K\Kj] = -ieijkJk. (8.23)
This implies that Jl± = \[Jl ± iK{] form a commuting 517(2)L x SU(2)R
algebra,
[4,4] = ieijkJ^ [4, Ji] = 0, (8.24)
which means that the compact representations of the Lorentz group can be
labelled by (j+,j_), where j± = 0, |, 1, • • • are defined by the Casimirs j£ =
j+(j+ +1) and J_? = j-(j- +1). A field &(x) transforming under the (j+, j_)
representation goes into
$(x) -► $'(x) = e-4wP'M^$(x)eiwP'M^ = A0+ii_)$(A-1x), (8.25)
under a Lorentz transformation A, where A(j j_) is the representation matrix
of giu^Afp* (cf (328) and (3.30)). For example, the four-dimensional (\,\)
representation is just the defining four-vector representation, as will be shown
in an example below. Similarly, the (|,0) and (0, |) representations^ are
two-dimensional, with J = a/2 and K = =R<?/2, respectively,
§ These are passive transformations, in which x' represents the coordinates of an event in a
transformed coordinate system. For an active transformation, in which an event is rotated
and boosted in a fixed reference system, the signs of Q and £ must be reversed.
^These representations define the classical group SL(2, C) of complex 2x2 matrices with
unit determinant. It is obvious from this form that the Lorentz group is non-compact, i.e.,
C can take any real value, and Tr (KlK^) = -\5ij < 0.

Beyond the Standard Model
471
They describe the transformations of L and R chiral Weyl spinors \J>L and
$r, respectively, as is shown in Problem 8.1. The conjugate representations
A(1..o, = «-*■*-** = AJJ,, A(0,n = e-«^ * = %0) = A£0),
(8.27)
are equivalent but useful for constructing Lorentz invariants.
Spinor Notation for Two-Component Fields
The notations for L and R chiral fermions that we introduced in Chapter 2 can
be confusing when discussing C and CP transformations, Hermitian
conjugation, Majorana masses, or independent fields, especially in four-component
notation. In supersymmetry one works mainly in terms of L-chiral particle
and antiparticle fields. It becomes tedious to express Dirac or Majorana mass
terms or Yukawa couplings using the notation developed so far (e.g., the Dirac
mass terms in (2.342) or the Majorana ones in (7.333)), so it is worthwhile
here to introduce a more streamlined version of the two-component
notation. The key points are that Hermitian conjugation reverses the chirality of
a field, and that the id1 that enters charge conjugation can be viewed as a
raising/lowering operator.
We have used the notation that tpL and i\)cL (or \I>l and ^CL) are left-chiral
_ T
fields, while Hermitian or Dirac conjugated fields, such as CipJ^ = C(iPl)
(or icr2l&*L = £0"2(^l)*T), and the corresponding conjugates of i\)cL, are right-
chiral. A more compact notation is to write all left-chiral spinors by a symbol
such as £ (without a bar), and right chiral spinors by a symbol with a bar",
such as fj. (The bar has nothing to do with Dirac adjoint or with antiparticle).
One can, if desired, refer to a left-chiral antiparticle spinor with a superscript**
c. Thus, for example,
*L^£, n^e, *r^v, *R^nc- (8.28)
It is often useful to display the components. It is conventional to use a lower
undotted index for an L-spinor and an upper dotted one for an il-spinor, e.g.,
*La^^a, ^fla^f, (8.29)
where a = 1,2. The dotted index simply indicates that it is an il-spinor,
and also serves as a warning since dotted and undotted indices are never
contracted. The dot and bar are superfluous since they both indicate R, but
both are useful depending on the context. The il-spinor is introduced with
an upper index for convenience in the construction of Lorentz invariants and
"Some authors use a f or *.
**As we will see, however, £c and ff are not independent of rjt and £t.

472 The Standard Model and Beyond
covariants. Under Lorentz transformations, £ and fj transform as (|,0) and
(0, |) respectively, i.e.,
& - Ma%, ff - {M-^r^ff, (8.30)
where
with A(i)0) defined in (8.26).
We next introduce raised and lowered indices, associated with the conjugate
representations of the Lorentz group. Let
€ap = _€pa = _€^ a, /? = 1,2, (8.32)
be an antisymmetric tensor with e12 = — 612 = 1, with a similar definition for
eaP. Note that ea@ is just the a(3 component of ia2. Then
e°T€7/, = -e^e^ = 6%. (8.33)
We define the L and R spinors £a and Tfo with raised or lowered indices by
r = <?%, Va = tatiff- (8-34)
These transform as (| ,0) and (0, | ) respectively, i.e.,
ia - £"(Ar V- % - ^V^ (8.35)
Lorentz invariants can then be formed by contracting two L or two R spinors,
66 = ff &a = -&«ff = -eotxTizUn = Skin = 66, (8-36)
which commute since the fields themselves are anticommuting fermions. The
shorthand £1^2 always implies that the first index is upper and the second is
lower. Similarly,
mm = via^ = m. (8.37)
where the convention for the barred spinors is that the first index is lower
and the second is upper. Another way of saying this is that £a and fja are
interpreted as column vectors, while £a and fja are row vectors. From (8.30)
and (8.35) both £x£2 and fj\f\2 are Lorentz scalars.
Next, we recall that Hermitian conjugation converts L spinors into R spinors.
Therefore, (&*)* is a barred spinor with a lower index, and similarly for (77° )*,
Id = tea)1, rf = {t)\ (8.38)
so that
in = faOf. (8-39)

Beyond the Standard Model 473
We can use (2.196) to express a four-component Dirac field as
*■(::)■(?)■ (fc)-OO- <->
The mass terms for tp can therefore be written as
$LlpR = V[Vr =^V = SaV*
$R^L = ^R*L = Vt = V°ta (8.41)
^ = $LlpR + $Rtl>L = *[*R + ^h^L = & + l£,
which is most closely connected with the last form of (2.342). For two Dirac
fields Vi,2,
1plLlp2R = &T/2, 1plRlp2L = ^1^2- (8.42)
We also introduce the vector bilinear forms
fjd*i = fja d>*% = (£a»ri)\ t**fj = e^ f = (Ve»&, (8.43)
for arbitrary spinors £ and 77, where we interpret the first (second) index of ov
as a dotted (undotted) upper index, and the first (second) of <jm as undotted
(dotted) lower. It is straightforward to show
-a^ = e«V^, ^ = ^H^\ (8.44)
which imply
f)^i = -^f). (8.45)
The bilinear forms in (8.43) transform as Lorentz four-vectors. For example,
fj^^fjM^a^M^ (8.46)
One can show (Problem 8.2) that
M]a^M = t^vav, (8.47)
establishing the result.
Vector currents for a Dirac field can be written
^ir7M/02R = ^\R^^2R = Vi^V2 = -V2^Vi-
The QED Lagrangian density in (2.214) in spinor notation is
C = &» [id^ + eA^] £ + na" [idp + eA„] 7) - m(£fj + V0 ~ \fiauF»1/, (8.49)
(8.48)

474 The Standard Model and Beyond
where the electron field is i\) = (£ f))T. The second term in (8.49) can be
rewritten as f)a^ [id^ — eA^] 77, corresponding to the second form of the
electromagnetic current in (2.341), i.e., in terms of ^l and ^CL rather than vJ/L
and Vr. Similarly, the tensor current is
W>2R = *lL S^*bR = 26 5^772, W>2L = 2lfc 5^6, (8.50)
where s^v and s^ are defined in (2.333). Other identities can be found in (e.g.
Chung et al., 2005; Dreiner et al., 2008).
The expressions for the C, P, and T transformations can easily be translated
into the spinor language
*-($)-*•-(?)• *r(?)- *•?-(! (85i)
up to possible intrinsic phases^. It is understood that the appropriate
transformations are made on the space-time variable x. We see from (8.51) that
charge conjugation interchanges £ and 77, i.e., £c = 77 and 77c = 6 As simple
examples, the space reflection transformations expressed in 4 and 2 component
language include
(8.52)
PlplLlp2FlP * = ^\R^2L ^^ PZ1V2P 1 = *7l6
P^IL1^^2lP~1 = $\Rl^2R ^=> Pll^&P'1 = 17l<Vfc,
while charge conjugation leads to
CjpiL^RC'1 = ^iL^2R = ^2L^\R ^^ C'fiTfcC"1 = fji£2 = 6?7l
C^li^lC-1 = riLl^C2L = -$2r1^ir (8.53)
«=* Cha^C-1 = f}i^V2 = -772^77!.
The Fierz identities in (2.335) and (2.336) become
(6<^6) (6^4) = (6<7M6) (6^6) = 2(66) (66)
(m^m) (6^4) = -2(7716) (6*72).
(8.54)
Finally, a Majorana fermion, such as the triplet Majorana neutrino
introduced in (7.331) and (7.333), can be written as
—GO-(f)- (£)-(£)■ (8-55»
ttThe transformations P, CP, and T raise or lower indices, implying an extra minus sign
when acting on a spinor with the index in the "wrong" location. E.g., P^P'1 = fja, while
p^ap-i _ _fjd> There is also an extra sign in PxpcP~1 because of the opposite intrinsic
parity for an antifermion, as discussed below (2.327).

Beyond the Standard Model 475
so the free field Lagrangian density in (7.332) and (7.333) becomes
£ = £^0M£-^ [« + «], (8.56)
where vM = (f W -
The Supersymmetry Algebra and Representations
Supersymmetry transformations connect fermions and bosons, and the
associated generators Q must therefore be fermionic**, i.e., anticommuting. In
analogy with the Weyl fields £ and 77, we will introduce fermionic charges
Qa,a = 1,2 and their conjugates Q& = (Qa)^ The N = 1 supersymmetry
algebra* is
{Qa,0/j} = 2a^PM, [Qa,P»] = [Oa,PM] =0
{Qa,Q/3} = {0d,0/j}=0 (8.57)
[Qa, MM„] = (a^)jQ^ [Qd, MM„] = (*^)%Qe.
The non-vanishing commutators of the supersymmetry charges with M^v are
an indication that supersymmetry connects states of different spin and
statistics. Because of the antisymmetry, the product of three of more Q's must
vanish
QaQpQ-y = 0, (8.58)
and similarly for the Q. The indices can be raised and lowered using ea&, just
as for the Weyl spinors, e.g., Qa = €a(3Qp. The first relation in (8.57) implies
<0|ff|0> = J(0|Qi(Qi)t + (QxjtQ! + Q2(Q2)t + (Q2)tQ2|0> > 0, (8.59)
that is, the ground state energy of a supersymmetric field theory must have
nonnegative energy. Furthermore, if the supersymmetry is not spontaneously
broken, i.e, for Qa|0) = (QQ)t|0) = 0, the vacuum energy must vanish,
(0|i/|0) = 0. In the case of a scalar potential V((j>), for example, the
condition for the spontaneous breaking of the supersymmetry is that V(i/) > 0
at the minimum. This is to be contrasted with internal symmetries, for which
the relevant issue is whether v ^ 0 at the minimum, as illustrated in Figure
8.3. The requirement (0|i/|0) > 0 can be violated in supergravity (gauged
supersymmetry) or in the presence of explicit supersymmety breaking terms.
**The Coleman-Mandula theorem (Coleman and Mandula, 1967) states that under
reasonable assumptions the space-time symmetry of a field theory cannot be extended beyond the
Poincare algebra except for internal symmetry generators if the generators obey
commutation rules. However, it can be extended by the inclusion of fermionic generators (Haag
et al, 1975).
The algebra can be extended to N fermionic charges Qza, i = 1 • • • N. The cases N = 2,4,
and 8 are of considerable interest for theoretical discussions, but do not appear to be directly
applicable as extensions of the SM.

476
The Standard Model and Beyond
4> \ <t>
FIGURE 8.3
Left: potentials which preserve internals symmetries. Right: potentials which
break internal symmetries. In both cases, supersymmetry is preserved for
V\min = 0 (solid) and spontaneously broken for V\min > 0 (dashed).
The Qa and Qd acting on a single particle state either annihilate it or create
a state with spin and helicity differing by \ unit. The most important massless
irreducible multiplets for our purposes are: (a) the chiral supermultiplets,
which consist of a Weyl fermion and a complex scalar*. Examples are the
chiral quark or lepton fields and their partners, or the Higgs scalar and its
partner. Chiral supermultiplets can be paired to form Dirac fermions. (b)
The vector supermultiplets, consisting of a massless vector, e.g., with helicity
+1, and a Weyl fermion partner. These are generally elements of the adjoint
representation of a gauge group. A vector supermultiplet can be combined
with its CP conjugate to form a massless vector with helicities ±1 and a
(massless) Majorana fermion*. (c) The gravity supermultiplet, consisting of
the spin-2 graviton with helicity 2 and its spin-3/2 gravitino partner. This can
again be combined with its CP conjugate involving a graviton with helicity
= -2.
The massless chiral and vector supermultiplets are all that are needed
for the MSSM. There are also massive representations, that are occasionally
needed, e.g., for extensions of the SM involving an extended gauge
symmetry broken at a scale large compared to supersymmetry breaking. We will
illustrate the most important ones in Section 8.2.3. Before constructing the
transformations of the supermultiplets and the rules for invariant Lagrangian
densities, it is convenient to introduce Grassmann variables and superspace§.
tThe supermultiplets also involve unphysical (non-propagating) auxiliary fields.
*It is convenient in this context to combine a Weyl spinor and its conjugate as in (8.55),
even if the Majorana mass is zero.
§Our notations and conventions follow most closely those in (Wess and Bagger, 1992; Drees
et al., 2004).

Beyond the Standard Model
All
Grassmann Variables and Superspace
Supersymmetry transformations mix particles with different spins, and
therefore can be considered an extension of the rotation and Poincare groups. It
is therefore convenient to extend the notation of four-dimensional spacetime
with coordinates xM into a larger superspace, involving two additional anti-
commuting {Grassmann) coordinates 9a, with a = 1,2, and their conjugates
0a = (9ay• (Such Grassmann variables are similar to the ones introduced
in field theory for dealing with fermions using functional methods.) The
indices can be raised, lowered, or contracted using aa@, just as for chiral fields
£ and x, or the supersymmetry generators Q and Q. Because of the anticom-
muting nature, the product of three or more 0's vanishes, 9ot9p91 = 0, and
ga0a = 0 (no sum). This will enormously simplify the task of constructing
general functions of superspace variables by expanding around 9 = 9 = 0. Of
course,
92 = 9a9a = -20102, 02 = 0d0d = 20*02. (8.60)
One can introduce the concepts of differentiation and integration in super-
space, as are described in the more detailed books and articles. We will only
utilize integration, defined by
f d9a = 0, J d9a90 =6a(3, J d9jp = 0, (8.61)
and similarly for d9. Thus, for example
/ d9a92 = e^ f d9a919(3 = e0a90 - eotl91 = -29a. (8.62)
It is useful to define
d29 = -jd9ad9a, d29 = -id0dd0d, d49 = d29d29, (8.63)
from which one finds
/ d2002 = / d2002, / d400202 = 1, (8.64)
with all other integrals vanishing. These forms will be useful as projection
operators onto the supersymmetric parts of operators.
The Grassmann variables can also be contracted with and anticommute
with spinor fields and supersymmetry generators, and can be formed into
vectors. Two useful identities are
6i6x = 6aia60X0 = -\oHx, eo»oeove = +^veW, (8.65)
which can be derived by writing out the components and by using the Fierz
identity in (8.54).

478
The Standard Model and Beyond
A Simple Example
Before we consider the general supersymmetry transformation rules, let us
give a simple example. Consider the free field theory for a massless complex
scalar cj>, a massless left-chiral Weyl spinor £, and an auxiliary complex field
F, with Lagrangian density
C = {d^d^cj) + i^d^ + F+F. (8.66)
F does not represent a physical particle (the Euler-Lagrange equation in this
case is just F = 0), but it is needed to ensure a supersymmetric action. The
supersymmetry transformations of the fields into each other turn out to be
<t> -+ </> + 6</>, 6<J) = y/2e£
£-►£ + <*£, &Z = V^cF - iV^a^ed^ (8.67)
F -+ F + 6F, 5F = -iV2ea»d^,
where e and e are Grassmann parameters that define the transformation,
analogous to aM and u)pa for Poincare transformations, or (3l for the internal
global symmetries discussed in Chapter 3. It is straightforward to prove that
C is invariant under this transformation (Problem 8.5). It can also be shown
that these transformations indeed form a representation of the supersymmetry
algebra in (8.69). We note that the shift in F is a total derivative (since e is a
constant). Therefore f d4xF(x) is invariant under the transformation. This
will be very useful in more realistic examples.
Superfields
A superfield is an operator $(x,6,0) that is a function of the superspace
coordinates x, 0, and 0. One can expand $ in a power series in 0 and 0, but
no terms involving more than two factors of 0 or 0 survive because of the
antisymmetry. Consequently, the most general Lorentz invariant superfield is
$(x, 0,0) = <t>{x) + y/20£(x) + V2§x{x) + 00F{x) + 00G{x)
1 __ (8.68)
+ 6a^6A^x) + eee\(x) + 066k(x) + -0000£>(x),
where 0, F, G, and D are Lorentz scalar functions of x; £, x, ^, and A are
Weyl spinors; and A^ is a vector.
We are now ready to interpret the supersymmetry transformations as
translations in superspace. The algebra in (8.57) can be rewritten in terms of
commutators involving 9Q = 0aQa and 6Q = 0dQa,
[0Q, 0Q] = 20^0 PM, [0Q, PM] = [0Q, PM] = 0
[0Q,9Q] = [0Q,0Q] = 0 (8.69)
[0Q, M^] = 9a^Q, [SQ, M^] = 6aQ.

Beyond the Standard Model
479
Just as we defined the translation operator exp(+za • P) in ordinary space in
(8.20), we can define a supersymmetry translation operator as
S(a,e,3 = ei<*+*«+fl-p>, (8.70)
where e and e are Grassmann variables like 0 and 0. The multiplication of two
group elements can be calculated exactly using the Baker-Campbell-Hausdorff
construction in (3.6) because all of the commutators after the first vanish, with
the result
S{a», e, e)S{V, 6,6) = S(a» + b» + iea»6 - i6a»e, e + 5, c + 6). (8.71)
The transformation of a superfield $(#,0,0) can then be found in analogy
to (8.25),
$ _► & = SQS'1 ~ $ + % [eQ + eQ + a • P, $] , (8.72)
where the last form assumes infinitesimal translation parameters. The action
of the generators can be found by casting them in the form of covariant
derivatives with respect to the Grassmann variables. The derivation is carried out
in the standard references, but here we will just give the result,
$(af,6,0) -+ fc'Cx", 0,6) = $(x» + a» + iea^S - i9a»e, 9 + e, 0 + e). (8.73)
A general superfield such as that in (8.68) is highly reducible, but special
cases are irreducible and close under the transformations. We are most
interested in left-chiral superfields 3>l(£,0, 0), which only involve L-chiral spinors;
right-chiral superfields (J)r (such as ($l) )*, and vector superfields, which
satisfy V = V^ (such as $£,$l or $l + $£). A left-chiral superfield and its
associated left-chiral supermultiplet involve two complex scalars, <j>(x) and an
auxiliary field F(x), and an L-spinor £(x), with specific relations between the
components. It can be written
$L(x, 0,0) = cj)(x) + y/20£(x) + 99F(x) - i9a^9df,(p(x)
1 — i - (8.74)
- -eeeea^d^ix) + -p00 (d^(x)) a»e.
The supersymmetry transformation of 3>l can be obtained from (8.73) (with
a = 0), noting that, e.g.,
0(x" + iea»9 - iOa^e) ~ </>(x) + i(ea»9 - 9a^e)d^(x), (8.75)

480 The Standard Model and Beyond
to obtain
£0 = V2ei
5i = V2eF - -^cr^ed^ - -^cr^ed^cj) = y/2eF - iy/lo^ld^
^^*s v2 V2
sof{x) > v ' * v ' (8.76)
4>(x) OavOd^x)
SF = - A=^d^ + -^(c^)^ = -iV2e<J»d^
v v ' N v '
K{*) ee{d^{x))o»o
where the origin of each term is indicated and we have used (8.65). This
result reproduces the transformations in (8.67). $l can be written in a more
compact form by introducing the variable
y^=x^ - i0o»0, (8.77)
to obtain
*L(x, 0,0) = $L(i/, 0) = <t>{y) + y/20£(y) + 00F(y), (8.78)
which depends explicitly only on 0. (8.78) is easily verified by expanding the
fields and using (8.65).
From now on we will drop the subscript L on a left-chiral superfield. Some
important aspects to (re)emphasize are: (a) a chiral superfield transforms' as
an IRREP of the supersymmetry algebra, (b) The (auxiliary) F-component
of a chiral superfield transforms as a total derivative, so that f d4xF(x) is an
invariant, (c) Superfields commute, $i$2 = $2$i> because each term has an
even number of Grassmann variables and spinors. (d) The product $i$2 or
sum $i + $2 of two chiral superfields is also a chiral superfield. This can be
seen for the product by multiplying out the expressions for $1,2,
$1$2 = 0102 + V20 (602 + 601) + 00 (01*2 + 02 *\ " 66) , (8.79)
where all of the fields are functions of y. This clearly has the form of a left-
chiral superfield. The same is obviously true for the sum. We also record the
product of three left chiral superfields for later use:
$1$2$3 = 010203 + V20 (60203 + 60301 + 60102) (8.80)
+ 00 (0102*3 + 0203^1 + 0301*2 ~ 6603 ~ 6601 ~ 6602) •
The superpotentiafl W($a) is a holomorphic function of left-chiral super-
fields <$a, i.e., it depends only on the 3>a and not on their adjoints or on other
^An important consequence of supersymmetry is that the ultraviolet divergences are milder

Beyond the Standard Model
481
right-chiral or vector superfields. We will mainly be concerned with superpo-
tentials which are third order polynomials of the superfields, but in principle
W could be any holomorphic function expressible as a power series. It follows
from the above that W is itself a left-chiral superfield, and that its F
component is a supersymmetric invariant of the action. The F component, denoted
[W]f, is given by
J „ u^a 19=9=0 z „ u
*=0=O a,b
»fla*6
\ 0=0=0
(8.81)
where we have used (8.64). The second form is most easily seen from the
examples in (8.79) and (8.80).
A vector superfield V = V^ is another important special case of (8.68), with
the restrictions
</> = </>+, £ = X, F = G\ AM=4, A = /c, D = Dl (8.82)
One can use (8.73) to show that the change in the (auxiliary) D component of
a vector superfield under a supersymmetry transformation is a total derivative,
and therefore D(x) in an invariant when integrated over x. From (8.64) and
(8.68),
[V]D= f dA6V=l-D{x). (8.83)
The special case V(x, 0,6) = £a [*o(2/, #)]+ *o(2/, 0) leads (Problem 8.7) to
/V0^[<M^)]+*a(^,0)=^
** a a
(8.84)
which we recognize as the kinetic energy terms for <f)a and £a and the auxiliary
field term from (8.66). Vector superfields are also used to describe gauge
bosons and their partners, as will be discussed below.
The supersymmetry transformations associated with 5(a, e, e) in (8.70) are
global, i.e., e, e, and a are constants. They can be promoted to local (gauge)
supersymmetry by allowing the parameters to be functions of spacetime, just
as in an ordinary gauge symmetry. Local supersymmetry necessarily implies
the existence of gravity, and is therefore known as supergravity. Supergravity
is nonrenormalizable, and would presumably be an effective theory below some
cutoff scale, e.g., the Planck scale, where it could emerge from an underlying
superstring theory.
than in non-supersymmetric theories due to cancellations. One aspect of this is the non-
renormalization theorems (Grisaru et al., 1979; Seiberg, 1993), which include the statement
that the superpotential is not renormalized to any order in perturbation theory. This
restricts the form of renormalization group equations for the coefficients, since couplings
are only renormalized by wave function effects. It also means that if a term is absent from
the superpotential or very small, it will not be generated by renormalization.

482 The Standard Model and Beyond
8.2.3 Supersymmetric Interactions
Yukawa and Scalar Interactions
The Lagrangian density for a supersymmetric theory of scalars and fermions
(but no gauge interactions) can be written
C = [K]D + ([W]F + h.c.)= f d49 K{$,&) + ( f d2 9W($) + h.cY (8.85)
where W(&) is holomorphic and the Kahler potential K($,&) is Hermitian.
We will focus on the simplest case"
K(*,&)=Y,[*a(vMJi*a(v,0), (8.86)
a
which yields the canonical kinetic energy terms in (8.84). The expression for
[W]f is given in (8.81). The auxiliary fields Fa can be eliminated by using
the Euler-Lagrange equations of motion,
I =-\wa(^.
10=0=0
(8.87)
The terms involving Fa therefore lead to the scalar potential V((/>), with
V(4>) = - E (FaF* + F«W« + F1WD = E l^l2 = E |W«MI2- (8-88)
a a a
As expected, V{<j>) is non-negative, and the condition for supersymmetry ,to
be unbroken is for Wa{<f>) = 0 at the potential minimum**. We also define'
wab{4>)
Fi dW
d$a
dW
= -Wa{4>), Fa = - —
6=0=0 u^a
(8.89)
e=e=o
so that
where
d*ad*b I
£ = £KE + Cf-V(<i>), (8.90)
CKE = E[(^) + 5/V* + iZa^d^a]
a
(8.91)
a,b
are respectively the kinetic energy and fermion mass/interaction terms.
II More general forms occur in supergravity and in models of supersymmetry breaking. Also,
some authors reserve the term Kahler potential for related functions relevant to
supergravity. _
**This is generalized to V = ei<t>l<t>a/M2p) [\wa\2 - 3\W\2/Ml] in supergravity with a
minimal Kahler potential, where Wa = Wa + 4>aW/M'p and Mp is the Planck mass. The
condition for supersymmetry to remain unbroken turns out to be Wa = 0 at the minimum,
which allows V < 0.

Beyond the Standard Model 483
A Majorana Fermion
As a first example, consider the Wess-Zumino model (Wess and Zumino,
1974a), which contains a single left-chiral superfield 3>. We choose
W{Q) =a$ + ^$2 + -$3, (8.92)
with a, m, and h real. This implies
£ = {d^d^ + i^d^ - j(« + «) - M«0 + $&f) - V(4>). (8.93)
The potential is
V((/)) = |a + m(/) + /i(/)2|2, (8.94)
which has supersymmetry preserving minima at the zeros of a + ra(/> + hcj)2.
Let us take a = 0 and consider the minimum with (0) = 0, so that
C =Cke - ^ (« + «) - fc(«^ +«^) /Q ncN
z (8.95)
- m2\(j)\2 - hm\(j)\2((j) + (/>+) - /i2|0|4.
We recognize the ££ + ££ term as a Majorana mass term, as in (8.56), so
the spectrum consists of a Majorana fermion V>m — I ? ) = V>l + ip% and a
complex scalar 0, both with mass m. The Majorana fermion mass and kinetic
energy terms in four-component notation are
— Tfl — — 1 — Tfl —
^hi fi^L - y {ipLlpR + iI>r1I>l) = l^Ml fl^M ~ ^M^M, (8-96)
as in (7.332) and (7.334). There are Yukawa interactions
Cyuk = -h {^L<t> + McR<t>]) = -h $MPLrl>M<l> + JmPriPm^) , (8.97)
as well as cubic and quartic interaction terms for cj>, with coefficients related
by supersymmetry to the Yukawa coupling. It is sometimes useful to write <f>
in terms of Hermitian components, <j> = (S-\-iP)/y/2, where S and P indicate
scalar and pseudoscalar. Then,
V(4>) = ^{S2 + P2) + ^S (S2 + P2) + ^ (S2 + P2)2
2h V2 4 (8.98)
It is interesting to consider the special case m = 0. From (8.95) we see
that £ has a global 17(1) phase symmetry under £ —► ei<7^£, <j> —► e~2iq^(p.
However, the superfield $ does not appear to have a simple transformation

484 The Standard Model and Beyond
property. This is an example of an il-symmetry (e.g., Dine, 2007), which
does not commute with supersymmetry because different components have
different charges. One can formally express the il-symmetry by assigning a
transformation to the Grassmann variable 0, i.e.,
6 - ei0e, d20 - e~2i0d26, W - e2i0W,
which implies
S-^e^*, 0-*e*"V, e-»e-*'^, F^e'^F.
(8.99)
(8.100)
A Dirac Fermion
As a second example, consider three chiral superfields [/, £/c, and H. The
symbols are chosen to be suggestive that the fermionic components of U and
Uc will combine to form a Dirac field (i.e., the u quark), while H will play the
role of the Higgs, but at this stage they are three independent superfields and
there are no gauge interactions or chiral symmetries. We denote the scalar
and spinor components as (u,£u), (uc,T]u), and (h, h), respectively, where the
use of T]u rather than ££ is motivated by (8.51) and the subsequent discussion,
and the notations h for the H spinor and u, uc, and h for the scalars are
suggestive of the MSSM. We choose the superpotential
W = mUUc + huUUcH + W{H), (8.101)
where m and hu are real and W depends only on H. We first take hu = 0
and W = 0, so that
£ = CKe ~ rn (iuVu + ZuVu) ~ rn2 (\u\2 + |iic|2)
= £>ke ~ m (uRuL + uLuR) - m2 (\u\2 + |iic|2) ,
2/,.,.2^,.,c,2N (8-102)
which corresponds to a Dirac fermion u = ( _ I = ul + ur, as well as two
\VuJ
complex scalars, u and uc, all with mass m. Including h and W,
C = CKE + Cu + Ch- V(u, uc, h), (8.103)
with
Cu=-m {iuriu + iufiu) - hu [iuTiuh + iuVuh])
- hu [€uhuc + iJiu^ ) - hu (r]uhu + fjuhu^)
, +x (8.104)
= - m(uRuL + uLuR) - hu (uRuLh + uLuRlv)
- hu (hcRuLuc + uLhcRuc]\ - hu (uRhLu + hLuRvl) ,
where Jil is the L-chiral Higgsino field corresponding to h, and hcR is its CP
conjugate. Similarly,
CH = ~\ (hhWHH(h) + hhWHH(h)i) , (8.105)

Beyond the Standard Model
485
and
huuuc + WH(h)\ . (8.106)
Again, u = I _ ) is a Dirac field with a bare mass m and/or a spontaneously
\VuJ
generated mass hu(h) if h acquires a VEV (which we assume is real), h is either
a massless Weyl spinor or acquires a Majorana mass, depending on W (we
assume that (u) = (uc) = 0). From (8.104) we see that the Yukawa couplings
urUiJi and ULURh) are accompanied by analogous couplings in which one
fermion and one scalar are replaced by their superpartners, as illustrated in
Figure 8.4.
--- u
Ul ' Ul
FIGURE 8.4
Yukawa vertices from (8.104). There are three more diagrams in which the
incoming and outgoing particles are reversed.
Abelian Gauge Interactions
Let us first consider a U(\) gauge symmetry. The gauge boson A^(x) is a
component of a vector superfield V, which can be written
V{x, 9,6) = 4>{x) + y/20x(x) + y/20x{x) + 66G(x)i + 66G(x)
1 __ (8.107)
+ ea^oA^x) + oee\{x) + eee\{x) + -eeeeD{x),
where 0, A^, and D are Hermitian. The form of V can be simplified by noting
that V + iA — zA* is also a vector superfield for any chiral superfield zA. In
particular, if (f)\ is the scalar component of A, then from (8.74) the supergauge
transformation
V-+V = V + iA-iA^ (8.108)
generates a new vector superfield with vector component
A'lt = AM+dp(<l>A + 4h), (8.109)

486
The Standard Model and Beyond
which is just an ordinary U(l) gauge transformation of the form in (4.9).
The D component [V]d = \D(x) is not only supersymmetry invariant (when
integrated over x), but is invariant under supergauge transformations.
The other components of A can be used to put V in the form
v(x, e, §) = Oo^o a^x) + eeex(x) + m\{x) + )-eeeeD{x). (8.110)
This form, known as the Wess-Zumino gauge (WZ) is not manifestly su-
persymmetric, i.e., a supersymmetry transformation takes one back to the
general form. However, it is extremely useful because it only involves the
relevant physical degrees of freedom, i.e., the gauge boson A^, its superpart-
ner the gaugino A, and the auxiliary real D field, and is therefore analogous
to the unitary gauge. One still has the freedom to perform ordinary gauge
transformations while remaining in the WZ gauge. It is useful to note that in
the WZ gauge
v2(x, e, 6) = ]-eeeeA^{x)A^x), v3 = v4 = • • • = o. (8.111)
Under a supergauge transformation, a chiral superfield 3>a and its conjugate
transform as
$a ~+ e~2i9q"A $fl, $+ - $t e+2WaAt ^ (8 U2)
where g is the gauge coupling and qa is the charge of (j)a. It is easy to see
that the special case of an ordinary gauge transformation reproduces (4.9),
with (3{x) = -g[<t>\{x) +<t>\{x)*\. Combining (8.108) and (8.112), we see that
&ae29qaV$a is supergauge invariant, and that
£g= f d49 Y, *U29qaV*a = \52 *U29qaV*a] D, (8-113)
a a
which generalizes the chiral kinetic energy term in (8.84), is supersymmetric
and supergauge invariant. Cg can be written in terms of the component fields
by expanding the exponential and using (8.111). One obtains
(8.114)
where Dafl is the gauge covariant derivative, i.e.,
Da»<t)a = (#M + igqaA^cpa, Da^a = (^ + 1^^)^. (8.115)
We recognize the first two terms as the gauge covariant kinetic energies for <j>a
and £a, while the third is a related fermion-scalar-gaugino interaction required
by the supersymmetry.
The construction of the kinetic energy terms for A^ and A is rather involved.
It is possible to construct a supersymmetric derivative of V which contains

Beyond the Standard Model 487
the field strength FM„ and a derivative of A and which transforms as a chiral
supermultiplet but with a spinor index. The F term of the contraction of this
field with itself^ yields the gauge kinetic terms
CA = ~\f^F<" + iXa^X + ±D2. (8.116)
One can also write a superpotential including any terms that are 1/(1)
invariant, such as $a$b$c provided qa + qb + qc = 0. Finally, for the special case of
a E/(l) gauge symmetry, one can use the fact that D(x) is gauge and super-
symmetry invariant to add a Fayet-Iliopoulos (FI) term (Fayet and Iliopoulos,
1974)
CFI = kD(x) = 2k[V]d, (8.117)
where k is a constant. There is no analog of the FI term for a non-abelian
gauge symmetry or for a U(\) embedded in a non-abelian group.
The auxiliary field D enters Cg, Ca, and Cfi- The Euler-Lagrange equation
implies
D(x) = -(/c + ^OTfl0t0fl). (8.118)
a
Putting everything together,
C = - ^F^+iXa^X - \^T(UbWab(<j)) +UbWab((j))^ (8.119)
a,b
+ J2[(Da^a)i DMa + i^a^Da^a - y/2gqa{lMa + fata] ~ V(</>),
a
where
V(rh) = ^T\FJ2 4- ,
V(</>) = J2 \Fa\2 + \\D\2 = VF + VD. (8.120)
Fa and D are given by (8.87) and (8.118), respectively.
As an example, consider a single chiral superfield $ with q = 1. There is no
gauge invariant holomorphic term in $, so the superpotential vanishes, and
C = - \f^vF^ + iXa^X + |(0M + igAJt
+ i£7"(0M + igAJt - V2g{i~X4> + <fiX£) - V(<j>), (8.121)
where
V{<t>) = \\D\2=\\K + 9tf<t>\2- (8-122)
t^More general supersymmetric but non-renormalizable gauge kinetic terms, encountered
in supergravity and string constructions, involve an additional gauge kinetic function that
is holomorphic in the chiral superfields.

488 The Standard Model and Beyond
For K/g > 0 the minimum of the potential is at 0 = 0, so the gauge symmetry
is unbroken. However, V = \D\2/2 = k?/2 ^ 0 at the minimum, so the
supersymmetry is broken. This is manifested by the fact that the complex
scalar <j> acquires a mass
m:
l=9DLin=9"> (8-123)
while the Weyl spinor £ and the vector supermultiplet remain massless. The
massless £, known as the Goldstino, is characteristic of a spontaneously broken
supersymmetry. It is the analog of the Nambu-Goldstone boson of an internal
symmetry.
For KJg < 0 the minimum of the potential will be at (<j>) ^ 0, so the
gauge symmetry will be spontaneously broken. Writing k = —gv2/2, where
v is real and positive, one can choose ((f)) = v/y/2 and <f> = (y + h)/y/2 (in
unitary gauge), where h is Hermitian. Since D = 0 at the minimum, the
supersymmetry is preserved. The spectrum consists of one massive scalar
(ft), a massive vector A^, and a massive Dirac fermion formed by combining
the fermion from the chiral supermultiplet with the gaugino, ip = I t J. All
of these have mass gv, i.e., they form a massive vector supermultiplet.
Non-abelian Gauge Interactions
(8.119) generalizes in a fairly obvious way to the non-abelian case (which is
given for a nonsupersymmetric theory in (4.31)). One finds
£ = - \FlvF^v + iWD^X -±J2 (t«ZbWab(4>) + UbWab(4>)i) (8.124)
a,b
+ [(£>«M0)f D%4> + «£a"D«M£ - VlgiZXLU + ^A'L&J - V(</>),
where
V{4>) = VF + DD=Y,\Fa\2 + \Y,\Di\2' D^-gtfLU- (8-125)
a i
14 is the group representation matrix for the matter fields (which is the same
for <j> and £ but which may be reducible). The D term can be written more
explicitly as
Vd = \ £ l^l2 = \ Eh^i {L%)ab 4>b\2. (8.126)
i i
In (8.124) D$> and D\ are respectively the covariant derivatives for $ and A.
0*m = ^1 + ig^L%, DXfl = dj + igA\Uadj, (8.127)

Beyond the Standard Model 489
where (Lladj)jk = -icijk is the adjoint representation matrix. Thus,
(Dx^i = d^Xi - gcijkA^Xk, (8.128)
which should be compared with (4.27). The gaugino-gaugino-gauge
interaction can be rewritten in four-component notation as
-igcijkXWAiX* = -igcijkyL^Ai\kL = -igcijkX^A^Xh£
(8.129)
= -TiQCijk^M^A^X
Mi
with X1 —► XlM = I r-i J = XlL + Ag. The vertices for the gauge interactions
are shown in Figure 8.5.
^aL
-OI^Hjk
^aL
<f>a x
\
/
f
4>bL ' <t>b '
-i9^pL (4)o6 -*9wa+Pbr (4)o6
<f>a x
*M
AM
-iV29PR (L%)ah -iV29PL (4)^
FIGURE 8.5
The three-point gauge and gaugino vertices corresponding to (8.129), (8.124),
and (8.130). (There are two ways to contract the gauginos in the first
diagram.) The notation pa,b is defined in Figure 4.1. The double lines represent
the Majorana gaugino XlM = XlL + Ag. There are additional scalar seagull and
gauge self interactions, which are shown in Figure 4.1.
Similarly, the fermion-scalar-gaugino interactions are
- V2g [iay (L%)ab 4>b + ^y (4)a6 &
(8.130)

490
The Standard Model and Beyond
where the second form is in four-component notation, with £a —► ipaL-
Frequently, we consider a reducible representation involving pairs of left-chiral
superfields $ and 3>c whose fermionic components will eventually pair to form
Dirac fields. Similar to the example described above (8.102), we write their
components as (</>,£) and (</>c, 77), respectively. Allowing for the possibility of
a chiral gauge symmetry, we take L<j> = Ll and L$c = —L^ for their
representation matrices, where Ll and Lr are the representations for the L and R
chiral fermions tpL <-► £ and i/jr <-> 77. The fermion-scalar-gaugino interactions
in this case become
- >/2</[&A« (LiL)ab4>b + 4>lXi (4)a66 - fiX (LR)bafja - V^ {Ur)^
= -y/^g^aL^M {VL)ab4>bL + 4>tLXM {Li)abTPbL
-4r^m (LR)ba i>aR - ^«A'M (LR) ba 4>aR], (8.131)
where in the second expression we have introduced the suggestive notation
<t>a -+ <t>aL and <f)ca -H- <f)caL = 4>\R.
8.2.4 Supersymmetry Breaking and Mediation
Since we do not observe degenerate supermultiplets, supersymmetry (if it is
present at all) must be broken (for recent reviews, see Chung et al., 2005; Luty,
2005; Intriligator and Seiberg, 2007). Just as with an ordinary symmetry, the
breaking can be explicit or spontaneous. There are a number of reasons that
the breaking should be spontaneous, including the possibility of a naturally
small supersymmetry breaking scale (compared to the Planck scale), limiting
the number of parameters, suppressing flavor changing effects, and allowing a
consistent extension to supergravity or superstring theory. Nevertheless, the
effective theory relevant at low energies may well involve explicit symmetry
breaking terms. These should be soft (i.e., mass terms and cubic scalar
interactions, with dimension < 4), rather than hard to avoid the reintroduction of
the Higgs hierarchy problem.
Spontaneous breaking occurs when the F and D terms cannot all vanish
simultaneously. We already saw one example of D-term breaking in the abelian
gauge model in (8.121) involving a single charged superfield. For n/g > 0
the D term could not vanish, so the complex scalar acquired mass while
the Weyl fermion (the Goldstino) remained massless. It is also possible to
construct F-term breaking (O'Raifeartaigh) models, in which one or more of
the |Fa| is nonzero (O'Raifeartaigh, 1975). Consider three chiral superfields
with superpotential
W = m$2$3 + /i$i ($3 - fi2) , (8.132)
where for simplicity we will take m, /i, and ji2 to be real and positive. Then
F* = -h (0| _ ^ ^ p* = _m(f)^ F* = _m(f)2 _ 2Hx^ (8.133)

Beyond the Standard Model
491
which obviously cannot all vanish simultaneously, so Vp > 0 at the minimum.
For example, for ra2 > 2h2fi2 the minimum occurs for (02) = (03) = 0, with
(cj)i) undetermined (i.e., there is a flat direction at tree level), for which
VF = \Fl\2 = h2^>0. (8.134)
Quantizing around the minimum (taking (<j)\) = 0) yields scalar mass-squares
m2 = 0, 771?, = ra2, mjR = m2-h2n2, mli = ra2+/i2//2, (8.135)
where foRj are the Hermitian components of 03, i.e., 03 = (4>3R + i(t>3i)/V2-
The fermion mass terms are from
Cf = — m^2^3 + h.c. + Yukawa terms, (8.136)
so that £2 and £3 combine to form a Dirac fermion with mass ra, while £1
remains massless. Similar to the D-term breaking example, £1 is the mass-
less Goldstino associated with the spontaneous supersymmetry breaking. As
should be intuitive, it is associated with the supermultiplet with the nonzero
F term.
The existence of the Goldstino is a generic feature of spontaneous breaking.
In supergravity theories it is "eaten" to become the helicity ±| components of
the massive gravitino (the super-Higgs mechanism). These issues are discussed
in detail in the more extensive treatments listed in the bibliography.
Tree-level spontaneous supersymmetry breaking implies sum rules relating
the mass-squares of the fermionic and bosonic degrees of freedom. For
example, the sum of the scalar mass-squares for the O'Raifeartaigh model in
(8.132),
m2B = 2ra2 + 277*2 + m\R + mjj = 4ra2, (8.137)
is the same as that for the fermions, where in each case one must weight by the
number of degrees of freedom (2 for a complex scalar, 4 for a Dirac fermion).
There is a similar sum rule for the D term example in in (8.121), which
however, involves a contribution from \D\. For this reason such tree-level
breaking is not a phenomenologically viable option for the MSSM, because
the sum rules would require that some of the superpartners are light, contrary
to observational limits.
These constraints can be evaded if the breaking is radiative, i.e., associated
with loop effects, or if it occurs in a hidden sector that is only weakly coupled
to the SM particles. The hidden sector models are especially promising, e.g.,
because they may allow for suppressed flavor changing effects (although FCNC
are still problematic in some cases). Breaking could occur in the hidden sector
by tree level F or D mechanisms, or the breaking could be dynamical, i.e.,
**It is likely that such a supersymmetry breaking vacuum would be metastable, with a
nearby supersymmetric minimum (Intriligator et al., 2006).

492
The Standard Model and Beyond
associated with some strong dynamics in the hidden sector (e.g., Witten
1981, 1982; Affleck et al., 1985). The latter possibility could explain why the
supersymmetry breaking scale is small compared to the Planck scale. For
example, an asymptotically free gauge group that is weakly coupled at the
Planck scale might become strong at some intermediate scale As <C Mp,
just as QCD becomes strong at Aqcd, leading to an effective F term and
a gravitino mass (in supergravity) m3/2 ~ F/Mp. The breaking could be
associated, for example, with a gaugino condensate,
(A* A'') = 5ijA3s, (8.138)
analogous to the dynamical breaking of the chiral SU(3)l x SU(3)r flavor
symmetry of QCD or with some dynamical alternatives to the Higgs
mechanism for electroweak breaking (Hill and Simmons, 2003). In this case, one
expects F ~ A3S/MP.
The supersymmetry breaking in the hidden sector must somehow be
communicated to the MSSM particles, leading to effective soft masses for the
gauginos, Higgsinos, squarks, and sleptons, as well as soft cubic scalar
couplings, with a typical scale msoft. This should be in the 100 GeV-few TeV
range if supersymmetry is relevant to the Higgs/hierarchy problem. There are
a number of possibilities for the mediation mechanism. One is supergravity
mediation, in which the sectors are connected by higher-dimensional
operators with coefficients suppressed by inverse powers of Mp. One usually finds
mSoft ~ F/Mp or msoft ~ A|/Mj,, suggesting y/F ~ 1011 GeV. Simple
versions of supergravity mediation may lead, e.g, to squark masses that are
family universal at the Planck or GUT scale. However, RGE effects lead to
splitting at lower energies, with possible difficulties for FCNC. The gravitino
mass in supergravity theories is comparable to msoft.
In the original gauge mediation (e.g., Giudice and Rattazzi, 1999) models*,
the information about supersymmetry breaking is transmitted by messenger
fields that interact with the hidden sector and also are charged under the SM
gauge group. The effective soft breaking (due to loop effects) leads to
msoft ~ ±£, (8.139)
where M is the mass of the messenger particles. For y/F ~ M one finds the
much lower scale y/F ~ 104 — 105 GeV, implying a very light gravitino. Since
F is low and the SM gauge interactions are family universal, gauge mediation
is much less problematic for flavor changing effects.
There are many other possibilities, including anomaly mediation, gaugino
mediation, radion mediation, and D-term mediation, as reviewed in (Chung
et al., 2005; Heinemeyer et al., 2006). Z' mediation is mentioned in Section
8.3.
*A recent general treatment is given in (Meade et al., 2009).

Beyond the Standard Model
493
8.2.5 The Minimal Supersymmetric Standard Model (MSSM)
It is straightforward in principle though complicated in practice to write the
Lagrangian density for the MSSM. Here, we will show the main features, but
for the details of the squark, slepton, neutralino, and chargino mixings, CP
phases, etc., the reader is referred to the reviews and books that have already
been cited. We will follow a notation similar to the example in (8.101), using
upper case letters for left-chiral superfields and upper case letters with a
superscript c for the left-chiral conjugates of right-chiral fields. Thus, the
quark and lepton superfields will be written
Q=^J Uc Dc; L=(^) N° E° (8-14°)
respectively. (Some authors use a hat notation, such as Q, to indicate a chiral
superfield.) Their scalar components will be written as, e.g.,
u = uL, uc = ucL = u\i (8.141)
for U and Uc, and the fermion spinor components as
£u <"> UL, T]u <-> UCL [f}u <-> Ur] . (8.142)
We will usually not display family indices, or superscripts ° for weak eigen-
states, as in (7.4), but these can easily be added. The Higgs chiral supermul-
tiplets are
with scalar and fermion components such as ft+ and ft J <-> ft+L for H+. The
indices for SU(2) doublets are contracted using ea6, just as for the spinor
indices of the Lorentz group. Unlike the spinor case, however, the order is
significant since chiral superfields commute. For example,
QHU = eabQaHub = UH°U - DH+ = -HUQ. (8.144)
SU(3) color indices are unambiguous, e.g., QUC = QaUca or eal3lUcotDc(5Scl'.
The superpotential for the MSSM (assuming a conserved il-parity) is
W = iiHuHd - TdQHdDc + TUQHUUC - TeLHdEc + TVLHUNC
= » (H^H- - H°uH°d) - Td (UH- - DH°d) Dc + Tu (UH°U - DH^) Uc
- Te (NH~ - EH°d) Ec + r" (JVffJJ - EH+) Nc, (8.145)
where the first term yields supersymmetric masses // for the Higgs and Hig-
gsino fields, and the others yield the Higgs Yukawa vertices and their super-
symmetric partners. The signs are chosen so that the fermion mass terms

494
The Standard Model and Beyond
have the "correct" signs for positive T{h^d). Family indices such as in (7.13)
are suppressed. The kinetic energy and gauge interactions are derived in an
obvious way from (8.124) and will not be displayed explicitly here.
The effective soft breaking terms for the MSSM include mass-squared terms
for the scalars, Majorana mass terms for the gauginos, a scalar analog of the
H term (the Bji term), and scalar cubic interactions (A terms) similar to the
superpotential Yukawa couplings. Thus,
3
Csoft = - Y: ml4>t4>r -^(BB + BB)-^Y1 {W + l*^-)
r i=\
8
" ^ Y. (?'& + &&) + [-B^ihuhd + AdTdqhddc
-AuTuqhuuc + AeTelhdec - AvYvlhuvc + /i.e.], (8.146)
where ^r sums over all of the scalars (Higgs, squarks, sleptons), q = I -J,
1=1 f ], and the same conventions for the SU(2) indices hold as in the
superpotential. A and B have dimensions of mass, and we have extracted factors
of ji and the Yukawa couplings T from the coefficients, because this occurs
naturally in some (but not all) models of supersymmetry breaking/mediation^.
(8.146) is almost* the most general soft breaking allowed for the ^-conserving
MSSM. !
In general (8.146) contains many free parameters after including family
indices, but specific models, motivated by theoretical considerations, phe-
nomenological constraints (e.g., from FCNC and CP violation), or simplicity
typically have fewer. For example, minimal supergravity models assume
universal values for the soft parameters at the Planck or GUT scale,
mB = mw = mG = m\/2, mr = m0, Au = Ad = Ae = Av= A,
(8.147)
where the universal A parameters multiply the full Yukawa matrices for each
sector. It is usually further assumed that the soft parameters and // are
all real, so that altogether there are only 5 dimensionful parameters, rai/2,
mo, A, jB, and //. Some specific models further relate the universal soft
breaking parameters ra^, mo, and A, as well as jB, to each other and to
tThere is no uniformity in the literature about the signs of the A terms or whether to
extract \i and I\
J An explicit Dirac mass term for the Higgsinos can always be absorbed in \i as long as
B(jl is free. One could in principle add non-holomorphic cubic terms involving the "wrong"
Higgs field, such as q(ia2hu)dc, but in most symmetry breaking/mediation schemes these
are suppressed by an additional factor of msoft/M and are therefore negligible. (A possible
application for small Dirac neutrino masses is considered in (Demir et al., 2008)).

Beyond the Standard Model
495
the gravitino mass m3/2 (see, e.g., Martin, 1997). These parameters must
then be run down to the electroweak scale (e.g., Chung et al., 2005), which
induces mass splittings and flavor off-diagonal couplings to the mass eigenstate
fermions. Two of the parameters, // and £?, are then typically traded for Mz
and tan/? = {h°u)/{h°d). These are actually derived quantities, but Mz is
known and tan/? is closely related to observables. The basic parameters are
then mi/2, m0, A, Mz, and tan/?, as well as the sign of //, which is not
determined, as well as the gauge couplings and fermion spectrum. Much of
the phenomenological study of the MSSM has involved minimal supergravity
or versions with even more restrictions. It is extremely simple and is a useful
benchmark, but it is likely that the real world is more complicated (assuming
low energy supersymmetry exists). Recently, there has been considerable
interest in some of the alternatives mentioned in Section 8.2.4, as well as in
somewhat more general versions of supergravity, e.g., allowing Higgs masses
at the Planck scale that differ from those of the squarks and sleptons.
The Higgs Sector
The scalar potential V(ftu, hd, q, £, uc, dc, vc, ec) includes F-term, D-term, and
soft contributions^,
V = VF + VD + Vsoft. (8.148)
We will assume that the minimum of V does not violate color, electric charge,
B, or L, i.e., that the VEVs of the squark and slepton fields are zero*. Keeping
only the Higgs doublets in V, the contributions are
VF=\n\2{\hu\2 + \hd\2)
V°=J
9 ^hlThu + h\fhd\ + 9T\\hu\2 -\hd\2
(8.149)
=9-^ \\K\2 - \hd\2\2 + 9\ \h?hd _ ftotft-|2
Vsoft =m2hu\hu\2 + m2hd\hd\2 - [B/x (h°uh°d - h+ha) + h.c] ,
where
\hu\2 = K\2 + \h°u\2, \hd\2 = \hd\2 + \h^\2, (8.150)
we have used the Fierz identity in Problem 1.1 to rearrange Vd, and have
assumed the U(1)y FI term is zero. One can make a field redefinition so that
Bfi is real and positive, and an SU(2) xU(l) transformation so that (ftJ) = 0
and vu = v^2(ftu) 1S rea^ an<^ positive. Vp is then minimized for (ft^") = 0,
§For more detailed discussions of the Higgs sector of the MSSM and extended supersym-
metric models, see (Gunion and Haber, 1986; Gunion et al., 1990; Carena and Haber, 2003;
Accomando et al., 2006; Amsler et al., 2008).
^This is not automatic and leads to restrictions on the soft parameters, especially the A
parameter associated with the top quark Yukawa (e.g., Casas et al., 1996).

496
The Standard Model and Beyond
while the Bfi term is minimized for V& = y/2(h^) real and positive. That
is, electric charge and CP conservation in the Higgs sector are automatic
at tree-level in the MSSM, provided the squarks and sleptons do not have
VEVs". This is to be contrasted with general extensions of the SM involving
additional Higgs multiplets which do allow electric charge violation (Problem
7.3 and 7.4).
The potential in terms of vu^ is
■\r< \ ^ 2 2 , ^ 2 2 i-> ,9 ~^~ 9 / 2 2\2
V(vu, vA) = jVu + 2md"d ~ BWd + 32 K ~ vd)
(8.151)
where
ml ee |^|2 + ml, m\ = |^|2 + m\d, M2 3 (_< "*{j) , (8.152)
and it is understood that vu^ are real and positive (or at least nonnegative).
We see that at tree level V depends only on three parameters, ra2, m2d, and
Bfi (as well as known gauge couplings). One combination of these will be
fixed by the requirement that v2 = v\ + v\ ~ (246 GeV)2 at the minimum.
All aspects of the tree-level Higgs sector are therefore expressible in terms of
the two remaining parameters, which we will take to be
ma = ml + md and tan/3 = —. (8.153)
"d
The quartic term in V vanishes for vu = z/^, so vacuum stability requires
m2u + m2d>2Bfi. (8.154)
Electroweak symmetry breaking requires that the origin is not a minimum, i.e.,
that one of the eigenvalues of M2 is negative (the vacuum stability condition
(8.154) does not allow both to be negative), so
det M2 < 0 => m2um2d < (£//)2. (8.155)
If (8.154) and (8.155) are satisfied, one can easily minimize V to obtain
-N2 = ^ +
Mz , TOL tan I3 ~ ml
d
2 ' tan2/?-J (8.156)
sin 2/3= ——^—5- = —-f-,
ml + m2d M2'
II CP violating VEVs may be induced by loop effects in the effective potential associated
with the Yukawa and soft couplings.

Beyond the Standard Model
497
where Mf = (g2 + g'2)v2/±. (See Problem 3.31 for a very similar calculation.)
The first equation relates v2 to the other parameters. We have separated
|^i|2 from the soft masses to emphasize that electroweak breaking requires a
relation between fi and the soft parameters, which often requires a nontrivial
fine-tuning. Note that 0 < sin 2/3 < 1 (0 < (3 < n/2) by our phase conventions
and vacuum stability. Most radiative breaking schemes lead to m| < 0 (or
at least m2^ < ra2^), because of the large top-Yukawa which drives m|
to lower values at low energy, and tan/? > 1. The latter is also favored on
phenomenological grounds. A non-zero Bjj, is required for both doublets to
acquire VEVs, as is needed to generate all of the fermion masses.
It is straightforward though somewhat tedious to expand the potential
around the minimum to find the Higgs mass eigenstates and eigenvalues (see
Problem 3.31) and to write their Feynman rules, so we will only give the
main results. We first write the physical neutral fields in terms of Hermitian
components by
h = — ( ^hu \ h =J_("d + hdR + ihdI\
yftW + KR + iKi)' d V2\ V2hd j
hd-%Thd~ J2 \-yd-hdR + ihdI)
In the last form, the tilde refers to the SU(2) conjugation and not a super-
partner, and /ij = (hj) . The mass eigenstates consist of two neutral (CP
even) scalars h and H (with the convention Mh < M#), one neutral (CP-
odd) pseudoscalar A, and one charged pair H±, as well as three Goldstone
bosons z and w±. The latter are analogous to those in the SM (Eq. 7.85) and
disappear in the unitary gauge. A does not mix with h and H at tree level,
but CP-violating loop effects can induce mixing between all three. The mass
eigenvalues are
M2A=m2u+m2d, M2H± =M2A + M2^
M2 X
lh,H
2
M2A + M2Z^ yJ(M2A + M2f - AM2 M\ cos2 2(3
(8.158)
so that Mfl + Mjj = M\ + M\. One also finds the very important (tree-level)
constraint
Ml < M\ cos2 2/? < M|, (8.159)
with the inequality saturated for MA ^> M^. This is to be contrasted with
the SM, where in principle there is no upper limit on the Higgs mass. For
large stop masses there are important radiative corrections to this tree-level
result, dominated by the top/stop loops in Figure 8.2. At one loop, the upper
limit in (8.159) is replaced (under reasonable assumptions) by (Haber and
Hempfling, 1991; Okada et al., 1991; Ellis et al., 1991)
f2 ^ ^2 , W™$
Mi <Mi +
z ^ S^M2,
/Mf\ X2 ( X2 X
{li\m2) +M2\l 12M2)
(8.160)

498 The Standard Model and Beyond
where Mj = ^(m~ + m~ ) is the average of the masses of the two stop
mass eigenstates (mixtures of II and Ir), and Xt is a stop mixing parameter
that will be defined in the discussion of sparticles below. The largest value
occurs for maximal mixing, Xt = V6MS, which yields Mh < 130 GeV** after
including other corrections (Heinemeyer et al., 2006, 2008). The no mixing
scenario, Xt = 0, implies Mh < 115 GeV. In both cases, the upper limit is for
large tan/? > 5-10.
The mass eigenstate fields are
h \ _ /cosa — sina\ /Iiur\
H J ~ \sina cos a J \hdRj '
(8.161)
where the mixing angle a is given by
2/„ , Ml(M27-Ml)
cos2(/? - a)= *;,,f rM, 8.162
M\(M2H-Ml) }
with —7r/2 < a < 0. The massive pseudoscalar and charged Higgs fields are
A = cos (3hui + sin /?/id/
+ (8.163)
H+ = cos/?/i+ +sin/?/i+, if" = (#+)T,
while the Goldstone bosons are the orthogonal combinations. A simple way
to derive (8.163) is to introduce a new basis
hi \ - /^cos/? -sin/A fhu\ (
hn)-{sin(3 cos/?JUJ' ( }
By construction, (h°j) = 0 and (h^j) = \[2v, so the massive A and H+ are
associated with /ij, while the Goldstone bosons are in /i/j. In unitary gauge
Combining (8.161) and (8.164), the CP-even components of hjjj are
( hIR\ _ fcos(P - a) - sin(/3 -a)\fh\ ,„ ,
\ him) ~ ^sin(/3 - a) cos(/3 -a))\H)- {^lbbj
The hiji basis is very convenient for deriving the Feynman rules for the
gauge-Higgs interactions. huR has the same gauge interactions as the SM
**The bound on the lightest Higgs is weakened to Mh < 150 GeV in extensions of the MSSM
which remain perturbative up to the Planck or GUT scale (Kane et al., 1993; Espinosa and
Quiros, 1993; Accomando et al., 2006).

Beyond the Standard Model
499
H in (7.39) and Table 7.1. For example, the induced SM HVV(V = Z,W)
vertices in (7.39) become
Chvv - U^-W^W- + *&z*z\ [sin(/? - a)h + cos(/? - a)H
(8.167)
There are no induced AW or E±W^Z vertices, since A and #=*= are
associated with the other doublet. One can similarly read off the contributions of
hu to the four-point vertices, hi is a second scalar doublet unrelated to SSB.
It leads to new hAZ, HAZ, h^W*, and H^W^ interactions, as well
as additional contributions to the four-point vertices. These can be read off
from the SM R^ gauge interactions in (7.89), with the substitutions z —> A,
w± —> H±, H —> hm, and v —> 0. (All of the rules are given in detail in
Gunion et al., 1990). We give only one example here,
Chiraz = y [A*$»(cas{l3 - a)h - sin(/? - a)ff)]ZM, (8.168)
with gz = (g2 +^/2)1/2, which leads to the decays Z —> hA and Z —> HA, for
which there are no SM analogs.
The Higgs Yukawa couplings differ from the SM by the presence of two
doublets. The quark terms derived from (8.145) are (the leptons are similar)
-CYuk = -Td (dRuLh- - dRdLh°d)+Tu (uRuLh°u - uRdLhl)+h.c, (8.169)
which is to be contrasted with (7.44). Therefore, the Yukawa matrices defined
in (7.46) are modified to
Yu Mu Mu j Yd Md Md
hu = ±- = — = -^—, hd = ±= = — = -5i-. (8.170)
V2 vu i/sin/3 V2 vd i/co&0
The Yukawa couplings of t and b to huR and hdR become
ht = ^-, hb = ^-, (8.171)
v sin p v cos p
respectively, rather than the SM values of mt^/v. Since mt is large, ht would
diverge (generate a Landau pole) below the Planck or GUT scale unless sin /?
is sufficiently large, requiring tan/? > 1.7 (Dedes et al., 2001). hb is small in
the SM, but can be considerably enhanced in the MSSM for large tan/?, with
implications for FCNC (Hamzaoui et al., 1999; Gorbahn et al., 2009). In fact,
it equals ht for tan/? ~ 45. If we keep only the t and b Yukawa couplings and
use (8.161) and (8.163), the Yukawa couplings to the mass eigenstate Higgs
fields are
—£>Yuk = ~it (cos ah + sin aH) -\ bb (— sina/i + cos aH)
usmp v cos p
-' mt rftA-i—tzn(3h5bA (8.172)
v tan /? v
r y/2mt - + V2mb
L v tan /?
tRbL H+ + tan /? bRtL H+ + /i.e.

500
The Standard Model and Beyond
As expected, h and H couple as scalars and A as a pseudoscalar. (8.172) is
easily extended to include the r, the lighter fermions, and fermion mixing.
A very important special case is the decoupling limit, Ma > Mz, in which
H, A, and H± form a degenerate heavy doublet and h acts like the SM Higgs.
Prom (8.162), one has cos(/? - a) -> 0, i.e, a -> (3 - §. Prom (8.167), (8.168),
and (8.172) h has the couplings of the SM Higgs, and the first tree-level
inequality in (8.159) is saturated.
In the decoupling limit, the LEP lower limit of 114.4 GeV on the mass of
the SM Higgs is applicable to Mh as well. The tension between this direct
lower limit and the best fit from the precision electroweak data described in
Section 7.3.6 carries over, provided the other sparticles are also much heavier
than Mz (e.g., in the several hundred GeV range), so that their effects on the
precision observables are small (e.g., Erler and Pierce, 1998; Ellis et al., 2007;
Heinemeyer et al., 2008). On the other hand, for smaller Ma the direct LEP 2
search limits are modified. The limit from Higgstrahlung (e~e+ —> Z* —> Zh)
is weaker than in the SM because of the sin(/? — a) factor in the coupling.
However, the complementary process (Problem 8.8) e~e+ —> Z* —> Ah, with
an amplitude proportional to cos(/? — a) becomes important for light enough
h and A. By combining the channels, one obtains Mh,A ^ 90 GeV assuming
CP conservation (e.g., Schael et al., 2006b; Amsler et al., 2008), relaxing
the tension with the precision electroweak data (Heinemeyer et al., 2008).
Allowing loop-induced CP violation, the h,H, and A can mix to form three
neutral mass eigenstates Hi. The lightest usually still decays primarily into
66, but the ZZHi and ZHiHj vertices are modified by the mixing, allowing
values as small as 45 GeV for the lightest scalar (e.g., Williams and Weiglein,
2008).
Squarks and Sleptons
The squarks and sleptons acquire masses from a number of sources. Consider
the Ul,r for a single family. The relevant mass terms in (8.148) are
v(uL,uR) = \ruh°u\2 (\uL\2 + \uR\2) + \-i*h°d + r*fiLfiJj|2
+ mlL I^lI2 + rnlR\uR\2 + (Auruh°uuLu*R + /i.e.),
where the three lines represent Vf, Vd, and Vsoft, respectively. The
coefficients SuL R in Vd are
K\2-K\\2
^ 9z
1Il,r ~ sin2 °w Qul.r] -» cos2/3M|
*&L.R ~ Sln2 0W 1*L.*
(8.174)

Beyond the Standard Model
501
where t\L (t\R) = \ (0), and q^L R = § are the third component of weak
isospin and charge. Replacing \ruh%\ -> mu, the 2 x 2 mass term is therefore
-U = U fiU (< +< + ^ 2 fumu \tuL\ (8 175)
where
Xu = Au-/i*cot/?. (8.176)
Similar expressions hold for d, z>, and e, except the appropriate values of t3
and q must be inserted, and cot/? —> tan/? for d and e. One can easily extend
the discussion to allow mixing between the families, in which case the mass-
squared matrix in (8.175) would be 6 x 6. Most consideration has been given
to models in which such inter family mixings are minimized (by choosing or
generating family universal boundary conditions at the GUT or messenger
scale) in order to suppress FCNC.
In realistic scenarios the soft mass-squares m2? dominate. The supersym-
metric (but SU{2) x[/(l) breaking) mj terms are negligible except for the
stop, and to a lesser extent the sbottoms and staus; the 5? contributions
are also small but need to be included. The Xfrrif terms lead to mixing
between /l and Jr. These are also small except for the third family, especially
the stops. The latter may be split significantly for large enough Xt, and could
even lead to an unphysical unstable vacuum with a negative m ~ for large At.
The significance of Xt for the lightest Higgs mass was discussed below (8.160).
In the simple minimal supergravity and gauge mediation scenarios, the
squarks are usually heavier than the sleptons because of RGE running and/or
because of their stronger coupling to messengers. One exception is the lighter
stop eigenvalue, which can be the lightest sparticle for large enough Xt.
Gluinos
The eight gluinos G% can acquire Majorana masses tuq from Csoft in (8.146).
Gluinos can be produced by ordinary QCD processes, as well as by processes
such as ^-channel squark exchange in qq scattering (the relevant cross sections
are collected in the Cross Section article in Amsler et al., 2008). The gluino
can decay into a quark and squark, G —> qq^ or qq, where the q may be real or
virtual. This initiates a cascade of decays, ultimately leading to a number of
q, q,£, and I as well as the LSP (e.g., a neutralino or the gravitino/Goldstino)
in i?-parity conserving models. Since the G is Majorana, conjugate pairs of
decays such as G —> qqe+ve+ LSP and G —> qqe~Pe+ LSP occur with equal
rates (up to CP violating effects), so pair-produced gluinos should lead to the
same number of same-sign charged leptons as opposite sign. Prom (8.131) the

502
The Standard Model and Beyond
gluino-quark-squark interactions (for a single family) are
CGad = - ^9s
A* A*
QL^PRG^qL - qR — PLGlMqR
+qI^&MPlQl ~ <?r^&mPr<1R
(8.177)
where GlM = GlL + G^, and we have inserted the Pl,r to make clear the
chirality.
The experimental limits and future prospects for sparticle masses depend
on the ordering of the masses, e.g., whether the gluino is lighter than all of
the squarks, and other model dependent assumptions, so they will not be
described in detail here (for recent reviews, see, e.g., Amsler et al., 2008; Feng
et al., 2009). However, the current Tevatron lower limits on the squark and
gluino masses are in the 300-400 GeV range, while the LHC reach should be
into the TeV range (Aad et al., 2009; Bayatian et al., 2007). The limits on
the non-strongly interacting sparticles are weaker. LEP 2 excludes any new
charged particle below ~ 100 GeV, while some of the neutral particles could
be even lighter.
(8.177) is easily extended to three families, provided one interprets the
quark and squark fields as weak eigenstates. When the interaction is rewritten
in terms of mass eigenstate fields, the vertices will in general be flavor-changing
due to a (probable) mismatch between the quark and squark unitary
transformations, much like the generation of the CKM matrix due to a mismatch
between the u and d type quarks. This can lead to potentially dangerous
FCNC effects (Gabbiani et al., 1996; Misiak et al., 1998; Ellis et al., 2007)
due to diagrams such as those in Figure 8.6, unless the squark masses are
nearly degenerate, the quark and squark mixing matrices are approximately
aligned, or the squarks are very heavy (multi-TeV). The gluino vertices are
especially problematic because of the strong QCD coupling, but there are
analogous problems involving neutralino and chargino vertices and the slep-
tons.
Neutralinos, and Charginos
Prom (8.145) and (8.146) we see that the neutral Higgsino pair h^ d have a
Dirac mass —/i, while the neutral gauginos B and W3 have Majorana masses
rrig and m^, respectively. In addition, the gaugino-Higgsino-Higgs
interactions from (8.130),
-C = V2g
4>tw'>
\K + $\wiT-hd
+ V2g'
+ /i.c,
(8.178)

Beyond the Standard Model
503
K°
t,Z,G
- . -i : - . -
d, s, 6* Yd, J, &
! !
G
d, s, b / \ d, s, b
/ G \
/ \
FIGURE 8.6
Typical gluino-quark-squark diagrams leading to FCNC effects. Left: a new
contribution to the Kl — Ks mass difference. Right: a contribution to b —>
5 + (7, Z, G). There are analogous diagrams involving neutralinos, charginos,
and sleptons.
lead to gaugino-Higgsino mixing when the neutral Higgs fields acquire VEVs.
The relevant terms are
1
-C
-gvuW*hl + gudW3h°d + tfvuBh°u - g'vdBh°d + /i.e., (8.179)
leading to the 4x4 Majorana neutralino mass term
/
-£*, = \(BW3 h°d hi)
ms
0 -9-^A
u 2
"lW 2 2
9 v<i gvd
2 2
V
0
f^\ ( B \
W3
0
+ h.c.
2 2
(8.180)
(8.180) implies four Majorana mass eigenstate neutralinos, x^r — 1---4,
with masses moo.
Similarly, the charged Higgsinos /i+ and hd have a Dirac mass +/i, while
W1 and I^2 can be combined to form a Dirac pair W± = (Wl =F Wl)/y/2 (cf
Eq. 7.28). The charged Higgsinos and gauginos are mixed by (8.178), leading
to the 2x2 Dirac mass matrix
£^d>
-*-<** K){-1* %U-
The mass eigenstate Dirac charginos Xi,2 nave mass eigenvalues
1
m^)2=2(|m^|2 + l/i|2 + 2M^
5 ((\™W\2 + N2 + 2M^)2 " 4|*W - M2, sin2/?|2)
(8.181)
(8.182)
1/2

504
The Standard Model and Beyond
Some models predict gaugino unification, i.e., that the gaugino masses are
in the ratio
ttiq : m^ : m^ ~ a3 : a2 : c*i ~ 7 : 2 : 1, (8.183)
where
a3 = as, a2=>L-, a, =-^. (8.184)
ai (the GUT-normalized U(l) coupling) is rescaled because J\Y has the
same Dynkin index for a fermion family as the SU(2) and SU(S)
generators. This occurs, for example, in minimal supergravity schemes in which
the gaugino masses are equal at the GUT scale, as are the GUT-normalized
gauge couplings. The RGE equations preserve the ratios. Simple forms of
gauge mediation lead to the same relation, while anomaly mediation predicts
an entirely different pattern with the m^ the smallest. Unfortunately, the
actual measurement of the gaugino masses is complicated by mixing with the
Higgsinos. We finally comment that gaugino mass terms break continuous R
symmetries, but allow a discrete imparity.
The neutralinos and charginos are expected to be produced efficiently at
hadron colliders through the decays of squarks and sleptons. Initial pairs of
squarks and/or gluinos produced by QCD processes could cascade decay into
multiple jets and leptons, e.g.,
q^qxl xl^^F, ^=-***?. (8-185)
If the x? is the LSP it will escape from the detector, implying large numbers
of cascades with multiple jets and leptons and missing transverse energy.
Detailed studies of the distributions of the number of jets, numbers of same and
opposite sign leptons, multiple leptons, lepton flavors, etc., should yield
information on the spectrum and test the Major ana character of the gluinos and
neutralinos. If the intermediate particles in a cascade are on-shell, it should
be possible to determine the masses by kinematic edge techniques (Problem
8.10). Some spin information may also be obtainable from the angular
correlations of the decay products (e.g., Wang and Yavin, 2008). This would
be important to distinguish supersymmetry from other scenarios with similar
cascades, such as some versions of Little Higgs and extra dimensional models,
as mentioned in the Higgs/hierarchy discussion in Section 8.1.
Neutralinos and charginos (and sleptons) may also be produced directly,
but at a lower rate, by Drell-Yan processes, ^-channel squark exchange, etc.
For example, trileptons plus missing energy could result (e.g., Barger and
Kao, 1999) from
W+* - xtxl xt - t+"X°i, X°2 - ^"X?- (8.186)
In supergravity and anomaly mediated scenarios with imparity conservation
the x? is usually the LSP, typically with a dominant B composition in minimal

Beyond the Standard Model
505
supergravity or W° for anomaly mediation**. There are at least some
parameter regions which yield dark matter consistent with observations (for recent
discussions, see, e.g., Arkani-Hamed et al., 2006; Ellis et al., 2007; Hooper,
2009). In models with a lower supersymmetry breaking scale, such as gauge
mediation, the LSP is expected to be the gravitino/Goldstino, <73/2. The next
lightest sparticle (NLSP) may be the x? or it could be a charged particle, such
as the tr. The NLSP may decay promptly, with a displaced vertex, or outside
the detector, e.g., by x? —► 7#3/2 or I —> ^3/2. Possibilities for dark matter
and cosmological constraints from NLSP decays are discussed in (Giudice and
Rattazzi, 1999).
8.2.6 Further Aspects of Supersymmetry
K-Parity Violation
The superpotential in (8.145) assumed a conserved imparity, a discrete R
symmetry under which each particle of spin 5 has Rp = (_i)3(s-L)+25.
All of the ordinary particles (quarks, leptons, Higgs, and gauge bosons) have
Rp = +1, while their superpartners have Rp = — 1. imparity therefore ensures
that the LSP is stable and suppresses precision electroweak vertex corrections.
However, it is possible add terms to W that violate imparity,
Wftp ~ HUL , LLE^ , LQDC , jTITjy, (8.187)
# mixing # dilepton ^ leptoquark # diquark
where we have suppressed the coefficients (the LLEC and UCDCDC terms
must be antisymmetric in the LL or DCDC family indices). The first three
terms violate lepton number, but are otherwise allowed because L and Hd
have the same SM gauge quantum numbers. The fourth term violates baryon
number. The LQDC and UCDCDC operators imply leptoquark and diquark
couplings, respectively, for the squarks (Section 7.6.6), while the cr in LLEC
is a dilepton (Cuypers and Davidson, 1998). imparity can also be violated
spontaneously if (i>l,r) ¥" 0-
The simultaneous presence of the leptoquark and diquark couplings would
lead to rapid proton decay via dc exchange, which would generate a B — L
conserving operator £qud. One of the two must therefore be absent or
incredibly small, but the other would be allowed. The diquark operators could
lead to neutron oscillations, i.e., n —> n transitions or nn annihilations in
a nucleus (e.g., Kuzmin, 1970; Mohapatra and Marshak, 1980; Dover et al.,
1992). The various operators are also strongly constrained by precision
electroweak measurements, FCNC, EDMs, and direct collider searches, especially
for the first two families (Chemtob, 2005; Barbier et al., 2005). The HUL
t^The dominantly W^ chargino is expected to be somewhat heavier due to loop
corrections (Pierce et al., 1997), allowing W*1 —> W0n±.

506
The Standard Model and Beyond
operator or sneutrino VEVs would lead to neutrino-neutralino (and charged
lepton-chargino) mixing, which could give mass to one neutrino. The other
neutrinos could obtain mass at loop level from the trilinear operators (e.g.
Grossman and Rakshit, 2004; Rakshit, 2004).
In the presence of i?-parity violation the LSP would no longer be stable
so other candidates (such as an axion) would be needed for the dark matter.
For example, the couplings in (8.187) could lead to such decays as jft —>
l+l~v, P^qq, vqq, or qqq. Such decays would resemble the imparity conserving
case if they occurred outside the detector. There would also be the possibility
of other types of LSP, such as a f.
The fi Problem
The \x problem was already introduced in Section 8.2.1. A nonzero \x is not
actually required for the Higgs sector of the theory, since the needed terms in
the tree-level potential can be generated by the soft terms m\u and Bfj, in
(8.149). However, a significant fi is required to generate large enough masses
for the charginos, as can be seen from (8.182).
One possibility is to generate \i at the same time as the soft terms by non-
renormalizable operators coupling the observable and hidden sectors (Giudice
and Masiero, 1988; Casas and Munoz, 1993). To illustrate how this works, it
is convenient to introduce a spurion superfield X = 62Fx, with a non-zero F
component. A spurion is usually an effective or composite field (or superfield
in this case) which is introduced with the right transformation properties to
parametrize the effects of symmetry breaking. ;
One typically expects string or supergravity effects to generate allowed
higher-dimensional operators involving X and the other superfields in the
Kahler potential or superpotential, presumably associated with inverse
powers of the Planck scale Mp. For example, a term -crX^X$l$r/Mp in the
Kahler potential would lead to the soft mass-squared term
d4ecr r T = -crL^4>ict>r, (8.188)
with m2, = cr\Fx\2/Mp. Gaugino masses and A terms can be similarly
generated, e.g., a superpotential term — c\23X^1 $2$3/Mp leads to a corresponding
A term with A123 = ci23^x^1 <f>24>3/Mp. In the Giudice-Masiero mechanism
one also introduces a term c^X^HuHd/Mp + h.c. into the Kahler potential,
which yields a \x term
W„ = / d2ec„^f± = c^HuHd, (8.189)
with \x = c^Fx/Mp the same order of magnitude as the soft terms.
In this discussion, we have implicity assumed a supergravity mediation, so
that msoft ~ Fx/Mp. In schemes with a lower Fx, such as gauge mediation,

Beyond the Standard Model
507
such operators are still present but are negligibly small compared to msoft,
which is now < Fx/M, where M < MP is the messenger scale. It is also
sometimes the case, even in supergravity mediation, that there are new
discrete or continuous symmetries that allow the operators such as (8.188) but
not the one in (8.189).
An alternative solution is to utilize renormalizable operators. In the singlet-
extended versions of the MSSM** one replaces //by a dynamical variable, by
introducing a SM singlet superfield S with a superpotential coupling
WSHuHd = XsSHuHd (8.190)
to the doublets. If S acquires a VEV an effective \x parameter /ie// = As(S)
is generated. The singlet extensions usually impose additional symmetries on
the theory to forbid an elementary jj, term. There are a number of
realizations of this mechanism (see Accomando et al., 2006; Barger et al., 2007, for
reviews). The best known is the next to minimal model (NMSSM), in which a
discrete Z3 symmetry forbids \x but allows the cubic terms \sSHuHd and k,S3
in W (Ellis et al., 1989). The original form of the NMSSM suffers from cosmo-
logical domain wall problems because of the discrete symmetry. This can be
remedied in more sophisticated forms involving a broken R symmetry (Pana-
giotakopoulos and Tamvakis, 1999b). A variation on that approach yields
the new minimal model (nMSSM), in which the cubic S3 term and its soft
analog are replaced by tadpole terms linear in S with sufficiently small
coefficients (Panagiotakopoulos and Tamvakis, 1999a). A U(l)' symmetry, which
is perhaps more likely to emerge from a string construction, is another
possibility (Suematsu and Yamagishi, 1995; Cvetic and Langacker, 1996; Cvetic
et al., 1997). This avoids the domain wall problem by embedding the discrete
symmetry of the NMSSM into a continuous one. The singlet extended models
involve an additional Higgs scalar and (except for the U{\)' case) an
additional pseudoscalar which may be light. The experimental lower limits on the
lightest Higgs may be weakened by decays into two pseusocalars (Chang et al.,
2008) or by reduced couplings due to an admixture of singlet component (e.g.,
Barger et al., 2006). The theoretical upper limit in (8.160) is also weakened
(to around 150 GeV) by new F term and (for U(l)') Z)-term contributions
to the potential. As mentioned in Section 8.1, the singlet extended models
also make it relatively easy to achieve the strong first order electroweak phase
transition needed for electroweak baryogenesis.
Gauge Unification
As will be discussed in Section 8.4 simple grand unified theories predict that
the properly normalized gauge couplings in (8.184) should all be equal at and
above the GUT (unification) scale, above which the symmetry breaking can be
**For other possibilities, see (Komargodski and Seiberg, 2009).

508
The Standard Model and Beyond
ignored. Such unification may also occur in some superstring theories which
compactify directly to the SM or the MSSM. Unification can be tested using
the observed gauge couplings at Mz and the known RGE equations for the
gauge couplings. Prom (5.36), at one loop, one has
a<(Q2) ai{Ml) 47rfUnMf
where
(bl\
b2
16tt2
' 10 >
6
v-7/
33 \
5 x
or Tfi^ I _l
SM MSSM
(8.191)
(8.192)
in the SM or the MSSM, respectively. There were early indications that
unification was more successful in the MSSM than in the SM (Dimopoulos et al,
1981; Amaldi et al., 1987), but precise tests became possible (Ellis et al,
1990; Amaldi et al., 1991; Langacker and Luo, 1991; Giunti et al., 1991) after
the gauge couplings at Mz were determined accurately at LEP. (In practice,
one needs to include two-loop effects, and corrections or uncertainties
associated with the t, Higgs, sparticle, and GUT particle thresholds (Langacker
and Polonsky, 1993, 1995; Carena et al., 1993; Bagger et al., 1995).) As can
be seen in Figure 8.7, the couplings do not precisely unify when extrapolated
assuming the SM, but do approximately meet at around 3 x 1016 GeV when
the effects of the new sparticles in the MSSM are included in the (3 functions.
8.3 Extended Gauge Groups
There have been many proposed extensions of the gauge symmetry of the
standard model or of the MSSM. One class of extensions, which is the focus of
this Section, involves extending the electroweak SU(2)xU(l) gauge symmetry.
Such extensions are sometimes motivated by bottom-up considerations, e.g.,
solving the \x problem of the MSSM in U{\)' extensions, or restoring P and
C invariance in SU{2)l x SU(2)r x U(l) models. Another, and perhaps
stronger, motivation is top-down: such extra gauge symmetries often appear
as accidental remnants of the breaking pattern of an underlying theory, such as
a superstring theory or a theory in which new TeV scale physics is responsible
for SU{2) x £7(1) breaking.
Another class of extensions* involves the introduction of horizontal (or fam-
*For a general discussion, see (Ramond, 1979). Recent examples include (Altarelli et al.,
2008; Luhn and Ramond, 2008).

Beyond the Standard Model
509
§
Standard Model
2 4 6 8 10 12 14 16 18 20
log10n(GeV)
Supersymmetric Standard Model
2 4 6 8 10 12 14 16 18 20
log10n(GeV)
FIGURE 8.7
Extrapolation of the observed gauge couplings in the standard model and in
the MSSM, from (Langacker and Polonsky, 1993).

510
The Standard Model and Beyond
ily or flavor) symmetries, which relate or distinguish between different fermion
families and hopefully shed light on the spectrum. The horizontal symmetry
could be discrete or continuous. If it is continuous and spontaneously
broken it is presumably a gauge symmetry to avoid unwanted Goldstone bosons.
FCNC, which can be mediated, e.g., by gauge bosons, are a serious constraint
on models with family symmetries broken at the electroweak or TeV scale.
These are not necessarily present in the original weak eigenstate basis.
Family transitions at one vertex may be compensated at the other, or there may
be family diagonal but nonuniversal transitions. However, the interfamily
mixing associated with the transformation to the mass eigenstate basis
almost inevitably leads to FCNC. Unfortunately, there are no really compelling
models of horizontal symmetries.
Grand unified theories, which unify the strong and electroweak interactions,
will be discussed in Section 8.4.
8.3.1 SU(2) X U(l) X U(l)f Models
Many extensions of the SM and MSSM involve additional U(l)' factors and
associated Z1 gauge bosons (for reviews, see Hewett and Rizzo, 1989; del
Aguila, 1994; Cvetic and Godfrey, 1995; Cvetic and Langacker, 1997; Leike,
1999; Rizzo, 2006; Langacker, 2009). These include superstring theories, grand
unified theories, and many models involving new TeV-scale physics, such as
dynamical symmetry breaking and little Higgs models. As a simple example
of why £/(l)'s so often survive, consider SU(m) broken by a real adjoint Higgs,
fa. Define the m x m matrix <f> = faL^, analogous to (3.31). The VEV ((f))
can be diagonalized by an SU(m) transformation, which makes it clear that
the unbroken subgroup is £/(l)m (or a larger group if some of the diagonal
entries are equal).
The extra gauge bosons could be extremely heavy, massless or very light,
or anywhere in between. Here we will mainly consider electroweak/TeV scale
Z's. These are especially motivated in supersymmetric models, where the
SU(2) x U(l) and U(\)' breaking scales are usually both driven by msoft>
In addition, some models involving extra dimensions of space allow the Z
and other SM gauge bosons to propagate in the bulk, implying Kaluza-Klein
excitations with masses ~ i?_1 ~ 2 TeV x (10_17cm/i?), where R is the scale
of the extra dimension. These excitations would be very similar to the bosons
from new gauge symmetries, except that their couplings would be essentially
the same as the SM ones.
U(iy Gauge Interactions
Consider the case of a single additional U{\)' factor, with family universal
couplings and (initially) no kinetic mixing (Section 2.13). The SM coupling

Beyond the Standard Model
511
of the three neutral gauge bosons to fermions in (7.55) generalizes to*
-CNC = gJ£Wl + g'J^ + g2 J%Zl = eJ^ + 9l J{*Z?M + g2 J£Z°„
(8.193)
where Z^ is the new gauge boson, J% is the U{\)' current, and g2 the gauge
coupling. If we were to work in terms of W3, B, and Z§, then following SSB
one would have to deal with a 3 x 3 gauge boson mass matrix. However, this
can be avoided (so long as electric charge is not broken) by first transforming
to A and Z\ = Z, which are related to W3 and B by (7.30) and (7.33), i.e.,
they are the neutral SM gauge bosons. Similarly, J^ = J^/2 where J^ is the
SM current defined in (7.73), and gY = (g2 +#/2)1/2 = gz*. This second form
is especially convenient when the mixing with the Z§ is a perturbation. The
currents J£,a = 1,2 are
r r
(8.194)
where the elL R are the SM couplings in (7.76), and the e2L R depend on the
U(l)f model. The currents take the same form in the fermion weak and mass
bases because of our assumption of family universality. When working in
terms of left chiral fermion fields, it will be convenient to define the charges
Qaf = €?(/), Qafc = -eaR(f). (8.195)
for fermion /l and its charge conjugate /£. We similarly define U(l) charges
Qai for a complex scalar fa, with Qu = tsi — sin2 9w Qi- The neutral current
part of its gauge covariant derivative is therefore
D^i =[d^+ ieq^ + i ^ gaQaiZ°afl J &. (8.196)
All members of an SU(2) multiplet must have the same Q2 since the two
groups are assumed to commute.
Gauge Boson Masses and Mass Mixing
When some of the scalars acquire VEVs, they will generate masses for the
neutral (and charged) gauge bosons. Assuming that electric charge is not
broken, i.e., qi = 0 for all scalars with (fa) ^ 0, one finds from (8.196) that
tin some cases, the two U(l) currents are linear combinations of Jy and a second current.
However, if the model contains the SM as a subgroup, it can be written in this form by a
rotation of the two abelian gauge bosons.
*The g\ used here differs from the y/5/3g' used in connection with gauge unification.

512 The Standard Model and Beyond
the photon A^ remains massless, while Z\ 2 develop a mass-squared matrix
M2 _, 292iZit23i\(4>i)\2 2g1g2j:it3iQim)\2\ = fMlo A2
z~z' \29i92ZihiQi\(<j>i)\2 2522EiQfl(<Ai)l2 rU2Mi,
(8.197)
where Qi = Q<n. Mzo would be the Z mass in the absence of mixing. If the
Higgs fields are all 51/(2) doublets and singlets, then
ti—2
where Vi = V2((f>i) and v ~ 246 GeV, as in the SM. The mass eigenstates
corresponding to (8.197) are
rzO \ / a „\„ a \
(8.199)
(Z1
with eigenvalues
M.% - \
)-"{%)■ "-
M2Zo+M2z,Ty/(M2zo-
( cos 6 sin#
y — sin 0 cos 6
-M2,)2+4A4
(8.200)
The mixing angle is given by
6 = -1 arctan f *A° ) , tan2 0 = ^f ° "ff ■ (8.201)
2 \M2, - M|0 / Af| - M|0 v • '
An important limit is Mz> ^> (M^o, |A|), which typically occurs because an
SU(2) singlet field 5 has a large VEV ^> v and contributes only to Mz>> One
then has
A4
M\ ~ M|0 - VT2- < M2, M2 - M§, (8.202)
and
* ~ ~T77T ~ C~772 wltn ^ = ^ ^o i/jL \i9 • (8.203)
C is model dependent, but typically \C\ < O(l). Prom (8.201)-(8.203) one
sees that both |0| and the downward shift (Mzo — M\)/Mzo are of order
Mil Ml.
As an example, consider a complex SU(2) singlet field S and two SU{2)
doublets 4>Uyd or their conjugates hu^. The doublets are defined as in (8.9)
and (8.10) for the MSSM, but we are not necessarily requiring supersymmetry.
Then,
m2o = \g\i\vu\2 + Wd\2), A2 = IgmiQuKl2 - Qd\M2) (8 204)
Ml=g22(QlWu\2 + Q2dWd\2 + Q2s\s\2),

Beyond the Standard Model
513
with Qu,d = Qhu,hd and s = V2(S). In the supersymmetric version of this
example (e.g., Cvetic et al, 1997) one usually assumes that the 11(1)' symmetry
does not allow an elementary fi term ^HuHd in W, i.e., Qu + Qd^ 0, but does
allow the coupling \sSHuHd in (8.190), i.e., Qs + Qu + Qd = 0. The potential
for 5 (which denotes the scalar component as well as the superfield), /i°, and
h°d is then V = VF + VD+ Vsoft, where the MSSM expressions in (8.149) are
replaced by
VF = X2s(\h0u\2\hd\2 + \S\2\h0u\2 + \S\2\h°d\2)
Vd = 9J (KI2 - \h°d\2)2 + f (QuK\2 + QM\2 + Qs\S\2)2 (8-205)
Vsoft = m2hu\h°u\2 + m2d\h°d\2 + m2s\S\2 - (\sAsSh°uhd + h.c.).
fi and Bfj, from the MSSM are replaced by /ie// = As(S), and (B/j,)eff =
XsAs(S). There is no MSSM analog of the first (second) term in VF (Vb)-
One can make field redefinitions and gauge transformations so that (B(j,)eff
and the VEVs are real and positive, and it is automatic (at least in the tree-
level Higgs sector) that the minimum preserves U(1)q invariance. (The
contributions of /i+ and h~l to V are similar to the MSSM and are not displayed.)
We see from (8.205) that all of the dimensional parameters in V are given by
soft supersymmetry breaking terms, resolving the fi problem and suggesting
that both the electroweak and U(\)' breaking scales should be close to msoft,
up to an order of magnitude or so§.
Kinetic Mixing
Kinetic mixing was discussed generally in Section 2.13. For a U(\)\ x £7(1)2
gauge theory, the most general gauge kinetic term^ is (Holdom, 1986)
Ckin - -^-F^AV - f^^V - C-fF°rPlv, (8.206)
where F®^ are the field strength tensors. Without loss of generality, we
can rescale the Z\2 so that c\ = C2 = 1 and (since the eigenvalues must
be positive) c\i = sinx- Kinetic mixing term is not expected initially if the
U(l)2 is embedded in a simple group, but it can be generated at the loop
level due to multiplet mass splitting, e.g., it can arise from RGE effects if
Trm<M((5i(52) 7^ 0, where the notation indicates that the trace is restricted
to the particles lighter than the scale \x (e.g., Babu et al., 1998). It can
also come about by superstring loop effects (Dienes et al., 1997) or by other
constructions.
§More complicated models involving additional S fields with opposite sign charges may
allow intermediate scale U(l)' breaking along flat or nearly flat directions (Cleaver et al.,
1998; Erler et al., 2002).
^Analogous kinetic mixing terms are not allowed for non-abelian theories.

514
The Standard Model and Beyond
Kinetic mixing could provide a means to induce a weak coupling between the
ordinary sector and a supersymmetric hidden sector. For example, a massless
Z1 mixing with the photon would result in a small fractional electric charge
for the hidden sector particles (Holdom, 1986). Here we restrict ourselves to
the case in which kinetic mixing is a small (e.g., RGE induced) perturbation
on the effects of a TeV scale U{\)'. Suppose one has
C = -Jf^AV - ^2^2% - ^r^V + \z^M\_z,Z% (8.207)
where M\_z, is induced by Higgs VEVs, and has the same form as (8.197).
One can put Ckin into canonical form using the non-unitary transformation
(I;M;r/rJ(4)-(t)' <™
in which case the mass-squared matrix becomes
M\_z, - M2Z_Z, = VTM2Z_Z,V, (8.209)
which must be diagonalized as in (8.199). It is straightforward to show that
in the limit (M^o, |A|) <C Mz> and |x| <^C 1 the effects of \ on M\ and 6 are
second order and negligible (i.e., A -> A-M|0x in (8.202) and (8.203)). The
only important change is then that the neutral current couplings of the Z\ in
(8.193) are shifted to include a first-order component proportional to Ji,
-CNC - eJ^Ap + 9xJ?Z^ + (g2J2 - glXJ?) Z%. (8.210)
The light boson couplings are not affected at this order, and one must still
include the further effects of mass mixing (the 0 rotation). This shift in the Z\
couplings is especially important because in many models the fermion charges
are orthogonal (i.e., Tr((5iQ2) = 0) prior to kinetic mixing; it can be used,
e.g., to suppress the couplings of the Z2 to leptons (leptophobic models).
U(iy Models
There are an enormous number of possible U{\)' models, distinguished by
the Z' mass, the chiral couplings to the quarks and leptons and whether they
are family universal, the extended Higgs sector, the possible exotic fields that
may be necessary to avoid anomalies from the extended gauge sector, possible
couplings to a hidden sector, possible kinetic mixing, etc. Even assuming elec-
troweak/TeV scale masses, couplings comparable to electroweak, and family
universal charges there are many models in the literature. Here we list a few
of the major classes.
• The sequential Zsm boson is defined to have the same couplings to
fermions as the SM Z boson. This is difficult to achieve in an ordinary

Beyond the Standard Model
515
gauge theory", but is a useful benchmark.
• The electric charge Q and the weak hypercharge Y = Q — T? in (7.5)
can be written (at least for the SM particles) as
Q = Tl+Y = Tl+Tl+ TBL, (8.211)
where T| takes the values t3R = +^ for vr and ur\ t\ = — ± for e#
and dR; and 0 for tpL. Tbl is defined as (B — L)/2, where B (L)
are baryon (lepton) number. The left-right (LR) models, which can
descend from the SU(2)l x SU(2)r x £/(l) models considered in the
next section or from 50(10) grand unified theories, involve the neutral
current interaction
-CNC = gJ&Wlp + 9rJSrW^ + 9blJ£lWbl,>, (8.212)
in an obvious notation. Anticipating that U(1)sr x U(1)bl will be
broken to U(l)y at a scale Mz> ^> M^o, it is convenient to put (8.212)
into the form (8.193) by rotating W\ and Wbl to a new basis B and
Z§. This leaves invariant the kinetic energy terms (which we assume
are canonical). One can take B = cos jW%r + siwy Wbl and choose 7
so that B couples to g'Y. One then finds (Problem 8.12) that (7.69)
becomes
? = ? + ^ = ? + i + 5ST' (8'2I3)
Z\ = sin j W3r — cos 7 Wbl is associated with the charge
«"-V§
<xTsr TBl
a
(8.214)
where a = tan 7 = gR/gsL = V^2 c°t2 Ow — 1 > with «: = ^h/^. The
coupling has been normalized to #2 = \/§0tan0w ~ 0.46. For
example, the left-right symmetric version of the SU(2)L x SU(2)R x £/(l)
model (Mohapatra, 2003), breaking at the TeV scale, implies qr = g so
that a ~ 1.53. Realistic breaking patterns for 5O(10) typically lead to
a ~ 0.7 - 0.9 (Robinett and Rosner, 1982b). The \ model in Table 8.2
coincides with the LR model with a = \/2/3 ~ 0.82. Generalizations of
the LR models, e.g., to J2 = Jbl, are described in the reviews.
The LR models and their generalizations are attractive in that they are
the unique nontrivial U(l)' extension of the SM that does not require
the introduction of additional chiral fermions (other than vr) to cancel
"it could occur in a complicated constructions involving new fermions, or as a Kaluza-Klein
excitation in theories with extra TeV-1 scale dimensions.

516
The Standard Model and Beyond
anomalies. However, they are perhaps less interesting in a
supersymmetric context, because the Higgs superfields Hu,d form a vector pair
with T^ = ±\ and TBl = 0. Therefore, an elementary fi term in (8.145)
is not forbidden by the extra U(l)f. One must also introduce SM singlet
supermultiplets to break the U{\)'. They would most likely be
introduced as non-chiral vector pairs to avoid anomalies, leading to their own
"//" problem. (One could instead give large VEVs to the scalar partners
of the i/c, but this would break imparity and would be challenging for
neutrino phenomenology.)
• Many Z' studies focus on the two extra U(\)'s which occur in the
decomposition of the Eq GUT (Robinett and Rosner, 1982a; Langacker
et al., 1984; Hewett and Rizzo, 1989), i.e., E6 -> 50(10) x (7(1)^ and
50(10) —> SU(5) x U(l)x. We consider them only as simple examples of
anomaly-free U{\)' charges and exotic fields, and do not assume a full
underlying grand unified theory**. In E$, each family of left-handed
fermions is promoted to a fundamental 27-plet, which decomposes
under E6 -> 50(10) -> SU(5) as
27 -> 16 + 10 + 1 -> (10 + 5* + 1) + (5 + 5*) + 1, (8.215)
as shown in Table 8.2. In addition to the 15 fermions charged under the
SM gauge group and the i/c, each 27-plet contains a second SM singlet,
the 5. Both the vc and the 5 may be charged under the U{\)'. There is
also an exotic color-triplet quark V with charge —1/3 and its conjugate
Pc, both of which are SU(2) singlets, and a pair of color-singlet SU(2)-
doublets, hu = I , q ) and hd = [ h- ) with yhUid = ±1/2. The exotic
fields, which are needed to cancel anomalies, are all singlets or non-chiral
under the standard model, and therefore do not yield large corrections to
the precision electroweak observables. However, they are usually chiral
under the U(l)'', i.e., they are quasi-chiral.
The Eq models can be considered in both non-supersymmetric and
supersymmetric versions. In the supersymmetric case, the scalar partners
of the 5 (or the vc) can develop VEVs to break the U{\)' symmetry.
Similarly, the scalar partners of one hu^ pair can be interpreted as the
two Higgs doublets of the MSSM. The two additional hu^ families may
be interpreted either as additional Higgs pairs or as exotic-leptons (hd
has the same SM quantum numbers as an ordinary lepton doublet, while
hu would be conjugate to a right-handed exotic doublet).
U(l)^xU(l)x may survive to low energies, though most studies assume
that only the £/(l)^,, the U(\)x, or a U{\) associated with a linear
**In a full supersymmetric grand unified theory with a TeV-scale U{1)' the scalar partners
of the V and Vc would have Yukawa couplings related to those of the Higgs, and could
therefore mediate rapid proton decay.

Beyond the Standard Model
517
TABLE 8.2
Couplings of the Z°, Z°, Z°, and Z°N to an £6 27-plet of left-chiral
fermions or superfields.
50(10)
10
16 5*
1
5
10 5*
1 1
SU(5)
(u,d,uc,e+)
(dc,v,e~)
(P,hu)
{Vc,hd)
s
2y/WQx
-1
3
-5
2
-2
0
\/24^
1
1
1
-2
-2
4
tVlSQn
-2
1
-5
4
1
-5
2\/10<2at
1
2
0
-2
-3
5
combination of their charges does so. Examples of linear combinations
include Qv = J\QX ~ y\Qi>i which occurs in some heterotic string
compactifications, or QN = \QX + ^-Q^, which allows a neutrino
seesaw with a large vc mass (Ma, 1996) or can avoid cosmological and
astrophysical constraints on Dirac neutrinos.
Except for the \ model, all of these £/(l)'s forbid an elementary \x term.
They allow all of the other (Yukawa) interactions in the MSSM super-
potential in (8.145), as well as the terms SHuHd and SWC, where
Hu,d are the Higgs/exotic lepton superfields, and V and Vc represent
the superfields as well as the fermions. Therefore, the U(l)' breaking
induced by (5) can also generate masses for the Higgs and exotic fields.
Leptoquark or diquark operators, such as LQVC or UCDCVC, which can
allow V decay and also FCNC, are also allowed by the U(\)'. (They
cannot both be present because they would then lead to rapid proton
decay.) One problem with the EQ models is that in its minimal form the
particle content is not consistent with the minimal MSSM-type gauge
coupling unification. This can be restored by adding a vector pair of
Higgs-type doublets (e.g., an additional Hu + H£) from an incomplete
27 + 27*, but only at the cost of introducing a /i-type problem for this
new pair.
• The supersymmetric LR and Eq models all either involve a vector pair
of chiral superfields, therefore leading to a version of the \x problem, or
else are not consistent with the simple form of gauge unification found
in the MSSM. The minimal unification models (Erler, 2000) remedy this
by starting with the MSSM particle content and then choosing exotics
in sets which preserve the MSSM unification at tree-level, with U(\)'
charges chosen to cancel anomalies. At least two SM singlets Si with
different U(\)' charges are required to given mass to all of the exotics,
however. It is possible but nontrivial to ensure that enough of them

518
The Standard Model and Beyond
acquire VEVs, and to avoid unwanted accidental global symmetries and
the associated Goldstone bosons (Langacker et al., 2009).
Implications of a U(l)'
A Z' at the electroweak-TeV scale would have important implications for
WNC, Z-pole, and LEP 2 experiments, as well as for direct searches at
colliders, e.g., in pp or pp —> Zf —> £+£~ by the Drell-Yan (s-channel) process.
The low energy and LEP 2 experiments are influenced not only by Z — Zf
mixing, which modifies the Z couplings, but by Z2 exchange and by the shift
in the Z\ mass from its SM value. The effective four-fermi interaction relevant
to low energy processes in (7.79) is replaced (recalling that J\ = Jz/2) by
Tiffi = -C?f} = ^(PeffJ? + 2W,J2 + VJI), (8-216)
where
Peff = Pi cos2 6 + pi sin2 0 ~ pi, w = — cos0sin0(pi — p2) ~ —0
9\ 9\
y=(^) (Pisin2e + p2cos26)~(il\ P2. (8.217)
In (8.217) pa = Myy/iM2 cos2 0\y), and the second expressions are for small
p2 and 6. In the same limit, the Z\ mass shift in (8.202) results in
Pl~l + 02/P2 = l+(^) C2p2, (8.218)
where C is defined in (8.203). The modification of (8.217) relevant to LEP 2
is straightforward. The Z-pole experiments are insensitive to Z2 exchange,
but are very sensitive to Z — Z' mixing. This shifts the Z\ mass downward
according to (8.218), and therefore shifts the value of sin2 6w obtained from
Mz relative to the values from the asymmetries and other observables. The
mixing also modifies the tree-level vector and axial couplings of the Z\ to
fermion ipr. The couplings in (7.76) become
gry _> cos^(r) + ^sin^(r) - glv{r) + g-6g2v(r)
91 9l (8 219)
9a - cos^i(r) + ^ sin^2 (r) - g\(r) + ^ff^(r).
9\ 9\
These depend on the U(l)' charges, so the oblique parameter formalism
described in Section 7.3.6 is usually not a good parametrization of Z' effects.
To the extent that the U{\)' is a small tree-level perturbation it is usually
sufficient to use the SM radiative corrections, though some care is needed
in the definition of sin2^, e.g., by using the MS rather than the on-shell
definition.

Beyond the Standard Model
519
In addition to the precision tests, there have been direct searches for a Z'
by the CDF and DO collaborations at the Tevatron, seeking resonances or
forward-backward asymmetries in pp —> e+e~ or /i+/i~.
Currently, the limits on M2 are dominated by the Tevatron direct searches,
with lower limits on the models described here typically in the 800 GeV-1 TeV
range. The Z-pole experiments limit the mixing to |0| < few xlO-3.^
The LHC will have sensitivity for electroweak coupled Z's up to around 4-5
TeV by the Drell-Yan process, but can be lower by ~ 1 TeV if the sparti-
cle/exotic channels are open. If a Z' were discovered with mass up to around
2.5 TeV it would be possible to carry out significant diagnostic studies of
its couplings by forward-backward asymmetries, rapidity distributions, heavy
quark decays, associated productions, and rare decays. A possible future e+e~
ILC could extend the LHC reach somewhat and provide complementary
diagnostics.
The observation of a TeV scale Z', especially in the context of supersym-
metry, could have a significance far beyond the Z' itself. The possible physics
implications include: (a) an extended Higgs/neutralino sector, complicating
and modifying the collider physics signatures and expanding the range of
possibilities for neutralino cold dark matter; (b) the possible existence of heavy
(Z1 scale) exotic particles, which may decay rapidly or be quasi-stable; (c)
possible FCNC effects if the Z' couplings are family non-universal (which
often occurs in string constructions); (d) an enhanced possibility for electroweak
baryogenesis; (e) a number of possibilities for exponential or power law
suppressed Dirac or Majorana neutrino masses; (f) a possible efficient source for
sparticle production; and (g) a possible connection to an otherwise hidden
sector, either by kinetic mixing or by direct U{\)' couplings (which often occurs
in string constructions). The latter possibility could lead to Z' mediation of
supersymmetry breaking by a Z' — Z' mass difference. All of these possibilities
are reviewed in (Langacker, 2009).
8.3.2 SU(2)L x SU(2)R x E/(l) Models
The simplest SM extension involving an additional charged gauge boson is the
SU(2)L x SU(2)R x U(l) model (Pati and Salam, 1974; Mohapatra and Pati,
1975; Senjanovic and Mohapatra, 1975), in which the first (second) SU(2)
couples to L (R) chiral fermions. A left-right (LR) interchange symmetry
between the factors is usually imposed, so that space reflection and charge
conjugation invariance hold at the Lagrangian level. The original version of
the model, in which the extended gauge and LR symmetries are broken at the
TeV scale, is disfavored by gauge unification and neutrino mass constraints.
Most modern treatments assume that one or both symmetries are broken at
ttThese results assume an electroweak scale p2> with the GUT-motivated value p2 ~
y/5/3g' = y/5/Sg tan 9\y often assumed.

520
The Standard Model and Beyond
a large scale, and they are usually considered in the context of a subgroup
of the supersymmetric 50(10) grand unified theory. The model has also
been revived (Csaki et al., 2004) in the context of warped extra-dimensional
models to provide a custodial symmetry to protect the po parameter. Here,
we will give a short overview of the original non-supersymmetric TeV scale
model as useful background material. Modern developments are reviewed,
e.g., in (Mohapatra, 2003).
The SU(2)L x SU(2)R x U(1)Bl Fields and Gauge Interactions
We assume that the fermion fields
QmL - [ d0
QmR
Kn
dt
lmL —
transform under SU(2)L x SU(2)R x U(1)Bl as
^mR —
(8.220)
(*l,**,**l) = (2,1,£), (1,2, |), (2,1,--),
(M,--),
(8.221)
generalizing (7.4). The SU(2)L and SU(2)R gauge bosons W%R,i = 1,2,3,
therefore couple to V — A and V + A currents, respectively. The U{\)bl
gauge boson WgL couples to the charge Tbl = (B — L)/2 that was defined
in (8.211). The gauge couplings #l = g, gn, and gsL are related to the g' of
the SM and to e by (8.213). The fermion covariant kinetic energies are
Cf = fLiV>q°L + &&& + PLij>PL + PRij>PR,
(8.222)
where the sum over families is implied. Cf includes the fermion gauge
interactions
-//
-#7m
- Pm
9l
9l
f
t-W£
2
D
BL
«£
9r7„
9bl
W£
BL
ll ~ Al.
9r
9h
T -
t-W£
1 2
+ i^W£
BL
&
9bl
Tw»BL
1°
lR'
(8.223)
A Higgs multiplet transforming as (2, 2*,0) is needed to generate fermion
masses and eventually break the SU(2) x[/(l) symmetry. We introduce $
and its tilde form $,
$ =
4>Ut
<t>2 <t>2
$ = T2$*T2 =
0°* -0+
-4>i 0?*
(8.224)
in analogy to (7.14). Like most extensions of the SM, there are two separate
SU(2) doublets. Unlike the MSSM, both neutral Higgs fields will have Yukawa

Beyond the Standard Model
521
couplings to both u and d when one includes the tilde couplings, implying
Higgs mediated FCNC at some level. $ transforms as
$ -> $' = UL0L)QUR0R)i = e^iQe-^i, (8.225)
and similarly for $, and the corresponding gauge covariant derivative is
/ ?. w^ r • Wit \
^ = ^HiUL^-^^ . (8.226)
The $ kinetic energy is
C* = Tr
(DM$)f (£>"*)] , (8.227)
and the potential can be constructed in an obvious way from the invariants
Tr ($\$j) and Tr($\&j$li&e), where ®i,j,k,£ can be $ or $. We assume that
the neutral components of $ acquire VEVs,
At least one additional Higgs field with tsL 7^ 0 is needed to break the elec-
troweak symmetry to U(1)q, and it should be an SU{2) singlet with a neutral
component to ensure MwR ^> M^L. There are a number of possibilities, but
two are usually considered.
In the doublet model, one introduces the pair of doublets
-(I)
*n Sr=(%]' (8'229)
which transform as (2,1, |) and (1,2, |) respectively. The Sl is not really
essential for SU(2)L x SU(2)R x £/(l), but is introduced to allow the possibility
of a left-right interchange symmetry. We assume that the potential for <£, 5l,r,
which we do not display, is such that \vsr\ = \(S%)\ ^> v/V2, where the
electroweak scale v ~ 246 GeV is given by v2 = 2(\vsl\2 + N2 + |«;|2)j
with \v6L\ = \(S°L)l In that case SU(2)L x SU(2)R x U(l) is broken to
SU(2)l xU(l)y at scale \vsr\. The 5l,r do not contribute to fermion masses.
The other common model employs Higgs triplets transforming as (3,1,1)
and (1,3,1),
A+ , AR = A+ (8.230)
A£ / V A° J
One assumes that \vAR\ = \(A°R)\ > v/y/2 = (\k\2 + |k/|2)1/2 and that
|^al| = |(Ai)l <^ v/y/2. The second condition is required by the po
parameter in (7.181) and Problem 7.1. The triplet model is popular because AR can

522 The Standard Model and Beyond
yield a |i>Ai?|-scale Majorana mass term, ~ ^Ar^m^l^/^ + h.c, for the
right-handed neutrino (Mohapatra and Senjanovic, 1980), similar to (7.422).
For Dirac masses comparable to the charged leptons and h^Rv^R = 0(1 TeV)
this would yield ve^,r seesaw masses of order eV, keV, and MeV, respectively,
which was quite acceptable when the model was proposed. These values are
now excluded, but similar considerations apply when the model is
reinterpreted as a GUT scale model, probably embedded in 50(10). Of course,
a VEV for Al would yield a direct type II seesaw Majorana mass term
~ hALVALVLVji/V2 + /i.e. The models also have the interesting feature of
doubly charged Higgs fields Aj"^ (Accomando et al., 2006).
The Gauge Bosons
The charged and neutral gauge boson mass matrices are obtained by a
calculation similar to Problem 7.1. For the charged bosons,
2
Cw = (M^)abW+Wb- =J2MfWt+Wtr, (8.231)
i=\
where a,b = L,R and
2 _/ Ml MlRe^\
w~\MlRe-^ MR J' (8-232)
The elements are expressed in terms of the Higgs VEVs as
Ml,R = \92l,r (M2 + W? + K.hI2 + 2\vAl,r\2) ee \g
MlR = -9l9rWk*
(8.233)
|k'k*|'
where the mixing term is in general complex. The mass eigenstate fields are
[W+\_f cosC e-ia,sin<\ (W+\
\w+)-\-smt e-™ cost jyw+j ' (8"234)
where the mixing angle C and phase uj are
M2
tan2C = qF A/f2 ™ , eiw = ±einr. (8.235)
The ± signs represent alternative phase conventions for W2±. The mass
eigenvalues M22 are given by a formula analogous to (8.200). In the limit vr > vl,
2 _ * 2 2 w2 _ A/r2 _ * 2 2
M{ ~M2L~ -g2Lv2L, Ml ~ M2R ~ -g2Ru2R
<
c^±^l4|™'|
9r \"r\2
9r,.
9l
x 2 A/f2 (8.236)
9r\ M{
9lJ M2
2'

Beyond the Standard Model 523
The neutral gauge boson sector is the same as for the LR model in Section
8.3.1. The mass-squared matrix is given by (8.197), with
Mh = -gzzui, M\, = - {g\ + g%L) P%
4 (8.237)
A2 = -^-a(N2 + |Kf),
where v\ R are the same as v\ R defined in (8.233) except 2|i;ai,iR|2 —>
4|^al,hI2- The Z' charges and a are defined in (8.214). For vR > i/L,
"i j1+i)P4±P4. (8.238)
Ml,, \ «VI«.I2+2|«A.I2'
The Yukawa Couplings
The $ and $ Yukawa couplings to the quarks and leptons are
-CYuk = ^2fmL (rmnQ + Smn^) QnR + J2 ^nL (^mn^ + <W<l) £°nR + h.C.
mn m,n
(8.239)
There may be additional couplings of the Al,h to the leptons, generating
Majorana neutrino masses. The $ and <£ Yukawa matrices are independent,
allowing nontrivial quark and lepton mixings in the charged current
interactions**. This leads to the quark mass matrices
Mu = rn + sk'*, Md = rd + sk\ (8.240)
and similarly for the leptons. These can be diagonalized just as in the SM in
Section 7.2.2, yielding a hadronic charged current interaction that generalizes
(7.55),
-4V = A=u^ [gLVqLW£+PL + gRVqRW£+PR] d + /i.e. (8.241)
in terms of the mass eigenstate quark fields. Vq = A^Af is just the CKM
matrix, while VqR = ARAdR is the analogous mixing matrix for the SU(2)R
currents, which (unlike in the SM) is observable. A similar expression holds
for the leptonic currents, except that the right-handed current is absent in
the low energy effective theory if the vR acquires a large Majorana mass from
AR. In terms of the mass eigenstate gauge fields,
~£hw = ^ {**. [9LVqLPL + 9R tan C e* V*PR] d W+" (g ^
+ ulti [-9l tan C VqLPL + gReiu VqRPR] d W+" } + h.c.
** Supersymmetric and 50(10) extensions require a second independent <$ field.

524 The Standard Model and Beyond
The effective four-fermi interaction at low energies in (7.58) is replaced by
K
eff
4GF
V2
^L^L + a^L^R + Jr^Jl + a^R^R
where the apparent (measured) Fermi constant is
Gf = 92L cos2 C a
The V ^p A charge raising currents are
4,/Im = ^VqL'RPLtRd + v^VeL'RPL,Re,
(8.243)
(8.244)
(8.245)
with the usual caveat about vR. The SM current j]^^ in (7.60) is 24M. The
coefficients in Heff are
a = l+/?tan2C,
2
6* = c = eiw^taiiC(l-/J)
<*=(g) (tan2C + /?), with/?=^.
(8.246)
Left-Right Symmetry
Our discussion of SU(2)l x SU(2)r x £/(l) has so far been quite general.
However, most studies invoke a left-right interchange symmetry, under '
wi
wR,
Wbl <-> Wbl,
$l <-► SR, AL
<&
<&,
4
4
The Lagrangian density is invariant under LR* for
(8.247)
9r = 9l, r
rt
«t
u = uT, hAL = hAR, (8.248)
as well as appropriate constraints on the scalar potential. In this case, there
is a formal P and C invariance at the Lagrangian level. However, the invari-
ance is of a different character than for SU(S)C or U(1)q because the gauge
symmetries are still chiral under SU(2)l x SU(2)r.
The gauge symmetry breaking also breaks the LR symmetry. In particular,
even though r and s are assumed to be Hermitian, Mu = tk + sk,'* is not
Hermitian if k and k' are complex, implying V£ ^ V^. However, many
* A discrete LR symmetry broken at the TeV level would lead to problems with cosmological
domain walls, similar to Section 3.3.2, would not allow the development of a baryon
asymmetry while C is conserved, and is not consistent with gauge coupling unification, giving
more reasons why current studies generally assume that the gauge symmetry and/or the
LR symmetry is broken at a high scale.

Beyond the Standard Model
525
studies assume either: (a) manifest LR symmetry, i.e., r and s are complex
(so that CP is explicitly broken) but k and k! are real (or at least their phases
are small). If they are real, VqL = VqR. (b) Pseudo-manifest LR symmetry,
i.e., r and s and the other Yukawa matrices are real and symmetric, but
ft and k' are complex (spontaneous CP breaking). The mass matrices are
therefore complex-symmetric, so that A^ = A^KU, where Ku is a diagonal
phase matrix, and similarly for the other fermions. Unlike the somewhat
similar Majorana neutrino case (Eq. 7.358), it is more convenient to choose
the phases in Au^d to put Vq in the canonical CKM form. Then Ku'd must
be chosen to make the mass eigenvalues real and positive. The result is that
the elements of VqL and VqR may differ in phase, but they have the same
magnitude, |V^n| = |V^n|.
Experimental Constraints
There are many modifications to the SM physics due to the Wr. The mass of
the W ~ W\ boson is lowered by the mixing effects (corrections to (8.236)).
That and Z — Z' mixing can modify the predicted M\y/Mz ratio and the
relation of the masses to other precision electroweak observables. Low energy
observables are affected both by Wl — Wr mixing and by Wr exchange,
which lead to an admixture of V + A in the hadronic and perhaps in leptonic
interactions. Such effects have been searched for extensively in \x and (3 decay,
in weak universality tests, in new box diagram contributions to the Kl — Ks
mass difference, as mentioned in Chapters 6 and 7, and in direct searches
for a W' at the Tevatron. No definitive evidence has been observed for such
effects, but the extracted limits on Mi and £ are dependent on the assumptions
concerning gR/gL, VqR, <*>, and the nature of vr. Assuming manifest or pseudo-
manifest left-right symmetry there is a stringent limit M2 > 2.5 TeV from the
contribution of the box diagram with one Wl and one Wr to Atuk defined in
(7.232) (Beall et al., 1982; Zhang et al., 2007). However, M2 in the sub-TeV
range would be possible more generally (even with qr = #l) (Langacker and
Uma Sankar, 1989; Amsler et al., 2008). The limits on \C\ are stringent even
for general VqR. E.g., universality implies \(\ < (9(0.001) for gR = gL and
u; = 0 (for heavy vr), but the constraint is weaker for u ^ 0 (Problem 8.13).
8.4 Grand Unified Theories (GUTs)
The gauge problem described in Section 8.1 suggests the possibility of grand
unification (Pati and Salam, 1974; Georgi and Glashow, 1974), in which the
SM gauge group SU(S) x SU(2) x U(l) is embedded in a simple group G,
with the quarks and leptons combined in the same multiplets. The Pati-Salam
model achieved a partial unification linking quarks and leptons, based on the

526
The Standard Model and Beyond
group SU(4)C x SU(2)l x SU(2)r. The electroweak part incorporates the
left-right symmetric SU(2)L x SU(2)R x U(1)bl model, with the TBl
generator embedded in SU(4)C. The color SU(A)C group extends QCD to include
a fourth "color," identified with lepton number, so that the fundamental
representations for the first family are (uR,UG,UB,Ve) and (dR,dG,dB,e). The
SU(4:)C symmetry would have to be spontaneously broken to SU(S)C x UBL
at a sufficiently high scale. Alternative versions of the model involved integer
charged quarks and extended electroweak groups.
The first full unification of SU(S) x SU(2) x U(l) into a simple group
was the Georgi-Glashow SU(5) model (extensively developed in Georgi et al.,
1974; Buras et al., 1978), in which the strong, weak, and electromagnetic
interactions are unified above the scale Mx ^> Mz of SU(b) breaking. Thus,
the properly normalized SM gauge couplings extrapolated from low energy
should meet at Mx. At the time the model was written the gauge unification
worked reasonably well, suggesting Mx ~ 1014-15 GeV. As mentioned in
Section 8.2.6 the unification (using the more precise couplings determined
subsequently) works much better in the supersymmetric extension of the SM,
yielding Mx ~ 1016 GeV.
The left chiral q, qc, £ and £c are unified in the same multiplets in SU(5)
(though each family is still reducible). This implies electric charge
quantization since there are no U(l) factors. It also predicts proton (and bound
neutron) decay mediated by the new gauge bosons that connect quarks with
leptons and quarks with antiquarks, by the diagrams in Figure 8.8. One
expects decays into modes such as p —> e+7r°, with a lifetime
- -■ M* 1029yr- (8.249)
a2m^ Mx~1014 GeV
More detailed estimates allowed a somewhat longer lifetime, but nevertheless
the original model was eventually excluded by searches at SuperKamiokande
and elsewhere*. The gauge mediated lifetime is much longer in the
supersymmetric version, ~ 1038 yr, but such models lead to faster decays into modes
such as vK+ by processes involving sparticles. The simplest forms of SU(5)
also make predictions relating the quark and lepton Yukawa couplings, that
are only partially successful.
In this section we will describe the structure and some of the implications
of the non-supersymmetric SU(5) model, and briefly comment on some
extensions. Much more detailed discussions can be found in a number of
reviews (Langacker, 1981; Ross, 1985; Hewett and Rizzo, 1989; Mohapatra, 2003;
Amsler et al., 2008).
^The current lower limit on the partial lifetime into e+7r° is 1.6 x 1033 yr (Shiozawa et al.,
1998). For a recent review, see (Nath and Fileviez Perez, 2007).

Beyond the Standard Model
527
FIGURE 8.8
Diagrams leading to proton decay in the SU(5) model. X and Y are color
antitriplet gauge bosons with electric charges 4/3 and 1/3 respectively.
8.4.1 The SU(5) Model
The Fields
SU(5) is a rank 4 group (like the SM), with fundamental 5x5 representation
matrices U = Az/2, i = 1 • • • 24, which generalize the SU(3) matrices in Table
3.1 in an obvious way. The upper left 3x3 block corresponds to the SU(S)
subgroup (with indices a, b = 1,2,3 denoted by a and /?), while the lower
right 2x2 block (with a, b = 4, 5 denoted by r and s) corresponds to SU(2).
The normalized hypercharge generator Y\ = yJ\Y is diagonal,
Ly,
Vm
diag(-2 -2-2 3 3),
TrL2Yl
1
2'
(8.250)
and commutes with 51/(3) and 51/(2). 51/(5) contains 12 additional
generators not contained in 51/(3) x 5(7(2) x U(l), with nonzero values for Llar or
Llra. We will use the tensor notation introduced in (3.114) for the fields, with
a fundamental 5 written with a lower index, such as 3>a, and an antifunda-
mental 5* as *a.
The 24 adjoint gauge fields Ai decompose under 5(7(3) x SU{2) x U(\) as
24 -_(8,1,0) + (1,3,0) + (1,1,0) + (3,2*,-^) + (3*, 2,+|), (8.251)

528
The Standard Model and Beyond
where the third number is hypercharge, y = q — t3. This can be written as a
5x5 matrix
24
£^-
z=l
V2'
(8.252)
It is sometimes convenient to define the fields Aha = Aab, These are non-
Hermitian for a ^ b, with Aba = {Affi. The diagonal elements are not
independent and are constrained by Aaa = 0. Displaying the entries explicitly,
(8.253)
where W± = (Wl qp iW2)/y/2, W° = W3, and B are the SU{2) x 1/(1)
bosons, and Gf = ^Li QiK^I^ are the Sluon fields with Ga = °- There
are 12 additional gauge bosons A? and Ara, which carry both SU(2) and color
indices. These can be written
(G\~% G\ G\
r<\ r*2 2s ^3
/^>1 r-2 ^r3 2B
t*3 G3 ^3-730
X1 X2 X3
\ Y1 Y2 Y3
Xi
x2
x3
Wu , 3B
s/2 V30
w~
?1 )
%
%
w+
W° , _3B_
s/2 V30 /
A^ = Xa[Z*,2,qx = i],
At=Ya [r,2,qY = i\,
Al = Ya [3,2*,9y = -i],
(8.254)
where the 5(7(3) and 5(7(2) representations and the electric charge are shown.
Each fermion family of left-chiral fields is assigned to a reducible 5* +10
representation, where the 5* is anti-fundamental and the 10 is the antisymmetric
product of two 5's:
5*
xa
10
V>a6 = —</>6a
(3*,l,i) + (l,2*,-i)
(3*,l,-?) + (3,2,i) + (l,l,l),
(8.255)
Tpccp
Tpccr
tp45
where a, b = 1 • • 5; a,(3 = 1, 2,3; and r = 4,5. We emphasize that the \ and
ip are L-chiral, but we omit the subscript L to simplify the notation. The
field content can be written explicitly as
(8.256)
Xa = (d
cl dc2 dc3 e~
/ 0 uc3
^6 = 71
-u03 0
uc2 -ucl
U\ U2
\ di d2
~Ve
-uc2
ucl
0
u3
d3
)l
—u\ —d\ \
-u2 —d2
-u3 -d3
0 -e+
e+ 0 /

Beyond the Standard Model 529
where the family indices and weak-eigenstate superscript ° are suppressed.
The lepton doublet ( _^ J =ir2 ("lj in x transforms as a 2*, and the
l/\/2 in ip is introduced to avoid double counting. The antisymmetric part of
3 x 3 in SU(S) is a 3*, so the upper 3x3 block of ip is ipap = ea(3luc^'/y/2.
Similarly, the antisymmetric part of 2 x 2 in SU{2) is a singlet, ^54 = —-045 =
e^/y/2. The fermion family multiplet can be extended to 5* + 10 + 1, where
the singlet is the vcL.
We also introduce a real adjoint Higgs multiplet $, and a fundamental
5-plet H that contains the SM Higgs doublet (0+ = if4 and (f>° = if5),
24 u (Ha\
*=x>> ««=(;:] <8-257»
The $ fields have the same quantum numbers as the gauge fields in (8.253),
and are introduced to break SU{5) down to SU(3) x SU(2) x U(l). The
Higgs doublet f ,0 ) transforms as (1,2, |), while its SU(5) partner Ha =
(3,1, — |) is a color triplet with qn = — |.
The Fermion Gauge Interactions
The gauge covariant derivatives for the fermion fields are
{D^)ab = d^ab + i^A^cb + ii^AX^ad,
(8.258)
where g§ is the SU(5) gauge coupling and we have used L\. = —L1^. The
fermion gauge interactions are therefore
cf = xj wx)a+rbi m)ab = 4M+cfY> (8-259)
where the fields are weak eigenstates and a sum over families is implied. £?M
includes the kinetic energy terms and the SM fermion gauge interactions in
(5.1) and (7.7), with the restriction that the gauge couplings are unified,
93 = 92= 9\= 95, where g3 = g8, g2 = g, 9\ = J-^. (8.260)
C*Y contains the leptoquark and diquark interactions of the X and y,
CfY = -^= [g+ £«dRa + e\ fiadLct - vcR YadRa - e\ y«uLa]
> v /
leptoquark vertices
a - _ (8.261)
- i| [e«^ucLl £auLp+^ucLl %dw\ +h.c.
N v '
diquark vertices

530
The Standard Model and Beyond
These expressions can be transformed using the identities in (2.296) and
(2.297). For example,
e+R fiadRa = -dcLct £«e-L, e*^ucLl £auL(3 = -e^uTRlC~l £auL(3.
(8.262)
The transitions can be shown schematically (suppressing color) as
X,Y X,Y
5* 10
Since the X and Y act both as leptoquarks and diquarks, both baryon and
lepton number are violated (B — L is conserved), leading to proton and bound
neutron decay by the diagrams in Figure 8.8. Prominent modes predicted by
SU(5) included p —> e+7r°,e+p°,e+u;, e+77,e+7r+7r~, i>7r+, Pp+, P7r+7r°. Bound
neutrons can also decay into P7r°, Pp°, Puj, P77, e+7r~, e+p~, etc. One expects
Ml
-, where a$ = |^ (cf. the muon lifetime rM ~ M^/g4m5^). More
detailed early estimates predicted rp ~ 1031±2 yr, with the lower end of the
range favored by current input parameters. The SU(5) prediction of proton
decay motivated a number of sensitive searches involving large underground
detectors. These eventually excluded most of the predicted range of lifetimes,
i.e., they require M\y > 1015 GeV, somewhat larger than expected from the
gauge couplings. Proton decay continues to be a stringent constraint on more
modern versions of grand unification, which usually predict longer lifetimes.
Spontaneous Symmetry Breaking
The adjoint Higgs $ introduced in (8.257) has a potential (Buras et al., 1978)
V{*) = ^TV ($2) + I [TV ($2)]2 + ^TY ($4) + |lY (*3), (8.263)
with Tr($2) = $J$J, etc. Invariance under $ —> —$ is often imposed,
implying c = 0. For /i2 < 0 one has (0|$|0) ^ 0, and one can always take
(0|$|0) to be diagonal by performing an SU(5) transformation. One can
show (Li, 1974) that for b > 0, c = 0 the minimum is at*
3 3 —2/i2
($) = diag(i/<j> i/* i/* - -!/<*> - -I/*) with v\ = ——*——. (8.264)
z z 15a + lb
*The minimum is of the form diag(j/^ v'^ v'^ v'^ — 4v^) for b < 0, c = 0, implying that
SU(b) -> SU(4) x U(l).

Beyond the Standard Model
531
(Vacuum stability requires a > -76/15, so v% is positive.) Therefore, SU(5)
is broken to SU(3) x SU(2) x £/(l) at the GUT scale
25
M2X = M$ = -g-^4- (8.265)
Twelve of the scalars in $ are eaten by the X and Y. The others attain
masses of C?(|//|), but are of little phenomenological importance because they
do not couple to fermions.
To further break SU(S) x SU{2) x U{1) down to SU(S) x U(1)Q one must
utilize the 5-plet Ha in (8.257). The potential in (8.263) for $ is extended to
V(*,if) = ^TY(*2) + \ [IV(*2)]2 + ^(*4) + |l>(*3) + f #+#
+ ^ (iftff)2 + aH\HTY ($2) + (3H^2H + £#+$#, (8.266)
where the SU(5) indices are contracted in an obvious way, and $ —> — $
invariance implies c = 5 = 0. Two severe fine-tuning problems immediately
arise: (a) One needs to have (0°) = -j= ~ 10~13i/<j>, where i/ ~ 246 GeV is
the weak scale (the GUT hierarchy problem), (b) The color triplet mass M^
must be > 1014 GeV > lO^M^ to avoid too rapid proton decay mediated by
Ha (the doublet-triplet splitting problem). (Note that the /? and S terms are
the only ones in V($, H) that can split the doublet mass from the triplet.)
Both of these enormous hierarchies must emerge from the parameters in
V($,H). If the dimensional parameters are all of O(Mx) and the dimen-
sionless ones of (9(1) then it is natural for M^ to be comparable to i/<j>. One
must then fine-tune the parameters to one part in v%/v2 ~ 1024 to obtain
sufficiently small v and M<p. The fine-tuning problem is actually even worse,
because it must apply to the entire effective potential. That is, one must
"retune" the parameters at each order of perturbation theory to avoid
destabilizing the hierarchy^.
Yukawa Couplings and Fermion Masses
The SU(5) symmetry does not allow couplings of $ to the fermions, but allows
CYuk=lrnnXaJC^abH^
+ Tmn €abcd€ ^ah C </W He + Kmnv£LCXanHa + /l.C,
where m and n are family indices, T is symmetric, e is antisymmetric with
e 12345 _ i an(j fae weak eigenstate superscripts ° have been suppressed. The
§A small parameter is said to be technically natural if a new global symmetry emerges
when it is set to zero, such as the chiral symmetry which is introduced when a fermion
mass vanishes. That ensures that only one fine-tuning is needed, because renormalization
effects are proportional to the original small parameter ('t Hooft, 1980). Unfortunately,
there is no such new symmetry in the present case.

532
The Standard Model and Beyond
fermion masses are generated from (Ha) = SaTTo- The ^rst term yields
Cde = -dLMddR - e+MeTe+ + /i.e. = -dLMddR - eLMeeR + /i.e., (8.268)
where
Md = MeT = it
2
(8.269)
That is, the SU(5) symmetry predicts that the d quark and charged lepton
mass matrices are the same up to a transpose^, because they derive from the
same coupling. (8.269) implies that the eigenvalues are equal at Mx, i.e.,
rrid = rae,
ms
mu
mb
mr.
(8.270)
This appears to be a disaster. However, the situation is partially remedied
because the Yukawa couplings are running quantities, mainly from the gauge
loops. The largest effects are from the gluons, which make the quark masses
larger than the leptons' at low energies. To lowest order (Buras et al., 1978)
In
md(Q2)
mc(Q2)
= ln
md{Ml)
+-
me(M%)\ 11-4F
In
*s(Q2)
OB(Af2)
+ 4F [ab{MD
(8.271)
where F is the number of families. This implies m& ~ 5 GeV for mT ~ 1.7
GeV and F = 3, consistent with the observed value. However, (8.271) fails
for the light families. To avoid uncertainties in the overall scale of the light
quark masses, one usually considers the predicted ratio,
me
md
(8.272)
(8.272) fails by an order of magnitude, with m^/me ~ 200 and ms/md ~ 20
from (5.123). This suggests a more complicated Higgs structure, such as the
introduction of an additional Higgs 45-plet with family symmetries to restrict
the form of the Yukawa matrices (Georgi and Jarlskog, 1979). The second and
third terms in (8.267) lead to the u quark and Dirac neutrino mass matrices
Mu
M«T = i£l\
V2
Hi - -#/.
(8.273)
^The transpose is exploited in the lopsided neutrino mass models mentioned in Section
7.7.6, where a nonsymmetric 7 induces large mixings in AeL and the unobservable A^. The
mechanism is more difficult to implement in 50(10), because the simplest analog involving
Higgs 10's is symmetric.

Beyond the Standard Model
533
Gauge Unification
We have seen in (8.260) that the properly normalized 577(3) x SU(2) x U(l)
gauge couplings should all be equal at Mx, implying that the value of the
running weak angle is predicted at Mx,
s2z(Mx
) =
3/5
g2+g'2 1 + 3/5 8'
(8.274)
where we use the MS angle defined in (7.137). To compare with experiment
the couplings must be run to low energy (Georgi et al., 1974), using (8.191).
The SM coefficients are given in (8.192), but it instructive to follow the
historical approximation, using the expressions in (7.197) for an arbitrary number
F of families but ignoring the (small) contribution of the Higgs,
12tt2'
16tt2
22
y
4F
»3 =
167T2
n - ^
3
(8.275)
These can be solved to yield
a(M2)
as(M2) 8
1-
llq(Afj) M2X
2tt M\
s2z(M2z) = - +
5 a(M2)
9as(M2)'
(8.276)
independent of F. (Corrections do depend on F.) Plugging in as(Mz) ~ 0.12
and a(M|) - 1/128 yields s2z{M2) ~ 0.20 and Mx ~ 1015 GeV, which
is a reasonable zeroth order prediction. However, as already discussed in
Section 8.2.6, more precise two-loop calculations with the current inputs do
not quite work, as can be seen in Figure 8.7. Starting with a and as, one finds
Mx ~ 1014 GeV and s\{M\) ~ 0.21, which is too low. Taking the modern
view of using a and s2z(Mz) as inputs yields a very low as(Mz) ~ 0.07 and
Mx ~ 1013 GeV, both of which are unacceptable.
Cosmology
The SU(5) model apparently offered an elegant explanation for the observed
baryon asymmetry of the Universe (Section 8.1), i.e., the out of equilibrium
decays of the superheavy partners Ha of the Higgs bosons (Eq. 8.257) (Yoshimura,
1978). Although the lifetimes of the Ha and their conjugates H^ must be
equal by CPT, the rates into specific corresponding modes could differ by C
and CP violating effects (a partial rate asymmetry (Okubo, 1958)), e.g.,
T(H - qcqc) ^ Y{HC - qq), Y{H - ql) ^ Y{HC - qcF). (8.277)
Combined with the intrinsic B violation and the nonequilibrium (i.e., the
decays occurred at T <. My) all of the Sakharov ingredients to generate a
baryon asymmetry were fulfilled and an adequate asymmetry (with B—L = 0)
could be generated (though only in non-minimal models). Unfortunately, it

534
The Standard Model and Beyond
was later recognized that the symmetry would be subsequently erased by non-
perturbative sphaleron effects. The idea could be resurrected in more
complicated models in which a nonzero B — L was produced in the decays, the most
plausible being leptogenesis". Other possibilities for the baryon asymmetry,
such as electroweak baryogenesis, do not involve out of equilibrium decays.
A serious difficulty for grand unification involved magnetic monopoles,
which are topologically stable gauge/Higgs configurations present when a
simple or semi-simple group is broken to a group including a [/(l)
(Coleman, 1985). One expects a superheavy monopole mass Mm ~ Mx/q, and
most likely an efficient production during the phase transition resulting in the
symmetry breaking. Although MM pairs would annihilate efficiently, enough
would be left over to greatly overdose the universe (Preskill, 1979; Kolb and
Turner, 1990). Suggestions for solving this monopole problem included the
possibility that electric charge was not conserved at intermediate
temperatures, leading to rapid annihilations (Langacker and Pi, 1980). However,
much more compelling was the suggestion that a subsequent period of
inflation would dilute the density to negligible values, as well as resolving other
cosmological problems (Guth, 1981; Kolb and Turner, 1990; Lyth and Riotto,
1999).
8.4.2 Beyond the Minimal SU(5) Model
Supersymmetric Grand Unification
It is straightforward to extend the SU(5) model to supersymmetry (Dimopou-
los and Georgi, 1981). Just as in the MSSM, one must add an additional Higgs
multiplet transforming as a 5*. The Higgs part of the superpotential becomes
W*,H = V*Tr ($2) + A*TY ($3) + iiHHuHd + \*HHu$Hd, (8.278)
where
(na\ (Hc*\
Hua=[hi]i Had = [ h°d , (8.279)
$ is defined as in (8.257), and we are using the same symbols for the su-
perfields and their scalar components. The Yukawa couplings to the matter
supermultiplets are analogous to (8.267), with H —> Hu and H^ —> Hd.
As has already been emphasized in Section 8.2.6 and Figure 8.7, the gauge
unification is much more successful when the couplings are extrapolated using
the MSSM (3 functions (Dimopoulos et al., 1981), with the couplings
approximately meeting at Mx ~3x 1016 GeV. The agreement is not perfect: using
a and s\[M\) as inputs yields the prediction as(Mz) ~ 0.13, which is
somewhat above the observed value ~ 0.12. However, the discrepancy is small
"Another difficulty with GUT baryogenesis was that any asymmetry would be erased by a
subsequent period of inflation. This continues to be a constraint on leptogenesis models.

Beyond the Standard Model
535
enough that it could be due to threshold corrections due to multiplet splitting
or higher-dimensional operators at the GUT scale (Langacker and Polonsky,
1993, 1995; Carena et al., 1993; Bagger et al., 1995). Low-scale threshold
corrections associated with the sparticle masses can also play a role, though
they tend to be of the wrong sign for most breaking/mediation schemes (in
which the colored sparticles are usually heavier). The rrib(Mx) = mT(Mx)
prediction in the supersymmetric case works well for tan/? = vu/vd ~ 1 or
40-50 (e.g., Barger et al., 1993; Langacker and Polonsky, 1994; Carena et al.,
1994), with the larger value consistent with the LEP 2 limits on the Higgs
mass.
Because of the large value of Mx expected in the supersymmetric case,
the contribution of dimension-6 operators such as uude/M\ (from X and Y
exchange) are greatly suppressed, leading to proton lifetimes of O(103S) yr.
However, there are dangerous new dimension-5 operators (Weinberg, 1982;
Sakai and Yanagida, 1982; Dimopoulos et al., 1982; Ellis et al., 1982) such as
usdv/Mx or cdue/Mx • These are generated by the exchange of the fermionic
partners of the H and Hc from the Higgs multiplets Hu,d, which have the
same Yukawa couplings as those that generate the fermion masses. The
operators must involve a second or third family because of the commuting nature
of the super fields. They may be dressed by the exchange of a gluino, neu-
tralino, or chargino, to produce a proton decay operator such as usdv/Mx-
Since there is only one inverse power of the large mass scale, the proton
lifetime rp ~ rrip/Mx, which is dominantly into modes such as p —> vK+ or
n —> PAT0, is relatively short even after taking into account the small Yukawa
couplings and additional couplings and loop factors from the dressing. The
SuperKamiokande limit, r(p —> vK+) > 2.3x 1033 yr (Kobayashi et al, 2005),
excludes the minimal supersymmetric SU(5) and 50(10) models, while
versions with non-minimal Higgs sectors predict lifetimes within a factor of 5
or so of the current limits (see, e.g., the Grand Unification review in Amsler
et al., 2008). Of course, one must also exclude proton decay from squark
exchange with Rp violating couplings, just as in the MSSM. These are known
as dimension-4 operators because they are not suppressed by any powers of
the GUT scale.
The SU(5) e 5O(10) G E6 Chain
Each left-chiral fermion family in SU(5) is assigned to a reducible 5* + 10
representation, or 5*+ 10+1 including a singlet i/£. These are combined in
an irreducible 16-plet when SU(5) is embedded in the larger rank 5 5O(10)
group (Pritzsch and Minkowski, 1975; Georgi, 1975), which requires & i/£. In
addition to the breaking pattern 5O(10) —> SU(5) x U(l)x described in
Section 8.3.1 and considered here, 5O(10) has alternative breaking patterns into
(a) flipped SU(5), in which the hypercharge generator Y includes a component
of Qx, leading to different identifications of the particles in the 16-plet (e.g.,
Jiang et al, 2007); (b) the Pati-Salam group.

536
The Standard Model and Beyond
In SU(5) the mass matrices Md = MeT, Mu, and M£ are all independent.
In the simplest extension to 50(10) the fermion masses are generated by
a single Higgs field in the vector (10) representation, which decomposes into
5 + 5* under SU(5). This implies two Higgs SU(5) multiplets Hu and Hd with
two Higgs doublets, hu^, just as in (8.279), even in the non-supersymmetric
version. (In SU(5) a single Higgs can play both roles because the Yukawa
couplings in (8.267) involve both H and H^, analogous to the SM tilde
couplings in (7.13).) There is only a single ip\6ip\6(t>io Yukawa matrix, which is
symmetric in family indices, with the immediate consequences
Md = MdT = Me, Mu = tan/?Md, Mu = MVD. (8.280)
The first relation is similar to the SU(5) case (except for being symmetric),
while the second is a consequence of Yukawa unification, i.e., the 6, r, and
t Yukawas are all equal at Mx- In the supersymmetric case the Yukawa
unification for the third family works well for large tan/? ~ 40 — 50. However,
both relations fail badly for the first two families. Not only do they give
incorrect predictions such as m^/me = ms/md = mc/TnU) but the second
relation implies a trivial CKM matrix, Vq = I. The last relation equates the
(Dirac) neutrino masses to the u quark masses, which is obviously a disaster.
A realistic spectrum therefore requires an extended Higgs sector.
Introducing additional 10-plets resolves the problem with the CKM matrix, but
not the other difficulties. Similar to the addition of a 45 in SU(5) one can
obtain a realistic spectrum by introducing very large 120 or 126 dimensional
multiplets. The 126 also allows couplings that can generate Majorana masses
for the i/£, allowing a type I seesaw, and/or Majorana masses for vl (the type
II seesaw), as described in Section 7.7.6. For that reason, 5O(10) models are
frequently used, in connection with family symmetries, to generate models of
neutrino mass, or more generally, models for fermion mass textures. Instead
of large representations (which are very unlikely if there is an underlying su-
perstring construction) one can supplement the ip\6ipi6(f>io Yukawa couplings
with higher-dimensional operators (Babu et al., 2000; Pati, 2006).
The even larger Eq (e.g., Slansky, 1981; Hewett and Rizzo, 1989), which
has been suggested in some heterotic string compactifications, is rank 6 and
includes the 5O(10) x U(l)^ subgroup mentioned in Section 8.3.1. It has
an alternative maximal SU(S)C x SU(3)l x SU(3)r subgroup, referred to as
trinification when a permutation symmetry on the factors is imposed (e.g.,
Choi and Kim, 2003). The 5O(10) x U(l)^ embedding extends each family to
a 27-plet, which breaks to 5O(10) and 5*7(5) as in (8.215). The 16 contains
a SM family, while the 10 are new exotic fermions,
10=(|-) +(e-) +Vl+VR> (8-281)
and. the 1 is a singlet Sl, similar to the vcL. The 10 and 1 are quasi-chiral, i.e.,
vector or singlet under the SM but charged under the extra U{\)' s (Table

Beyond the Standard Model
537
8.2). The VLiR are exotic charge -± quarks, while the vector pair of doublets
are usually considered to be exotic leptons in non-supersymmetric studies.
However, in the supersymmetric case they are often considered to be Higgs
doublet superfields instead, as mentioned in Section 8.3.1. (The interpretation
depends on what global symmetries are imposed on the superpotential.) The
Ee model has an especially rich phenomenology because of the U(l)' s and
the exotic multiplets, and is often studied for that reason (e.g., King et al.,
2006; Kang et al, 2008).
Extra Dimensions and Strings
In orbifold GUTs (Kawamura, 2001; Raby, 2009) the grand unification is
present in a higher-dimensional space, but the gauge symmetry of the
effective four-dimensional theory is (usually) that of the MSSM. Such
constructions allow new symmetry breaking mechanisms associated with boundary
conditions in the extra dimensions, and they may retain many of the
desirable features of grand unification (such as gauge coupling unification, third
family Yukawa relations, etc.) while avoiding some of the difficulties (e.g., the
doublet-triplet problem, too rapid proton decay, and the need for large Higgs
representations).
Grand unified theories do not incorporate gravity, and are therefore not as
ambitious as superstring theories. Heterotic string theories include underlying
grand unification symmetries. They may compactify into an effective four-
dimensional GUT, although it is difficult to generate the adjoint and other
large Higgs multiplets introduced in most bottom-up constructions. They
may also lead to versions of orbifold GUTs (e.g. Raby, 2009), or compactify
directly to the SM or MSSM, or to an extended version, with limited memory
of the underlying GUT. Constructions may retain simple MSSM-type gauge
unification, or the unification may be modified (and complicated) by the
effects of new matter multiplets that survive to low energy and/or by the string
scale gauge coupling boundary conditions, especially for the U(l)y (e.g., Di-
enes, 1997). The fermion families or the elements of the families may have
different origins in the construction, breaking or modifying GUT Yukawa
relations and possibly leading to family nonuniversal couplings to new U{\)' s.
Intersecting D-brane constructions usually do not involve a full underlying
GUT, but they often descend to four dimensions using a Pati-Salam group
(e.g., Blumenhagen et al., 2005).
8.5 Problems
8.1 The left and right chiral projections Ul,r of the u spinors in the chiral
representation in (2.188) both reduce to the form Ul,r{0, s) = y/m(f)s in the

538
The Standard Model and Beyond
particle rest frame, p —> 0. Show that the forms in (2.188) for arbitrary p =
(3jm can be obtained by acting on 11^(0, s) with the (active) Lorentz boost
A(£ = — /3tanh-1/?), using the representation matrices A^i 0) and A/0 i.
respectively.
8.2 Prove (8.47) using the explicit form for A/10) in (8.26).
8.3 Let ipiM and tp2M be two Majorana fields written in 4-component
notation, as in (8.55). Use the two-component spinor formalism developed in
Section 8.2.2 to prove
1plM^1p2M = Vr1p2MripiM,
with ar = +1 for T = 1,75, and 7^75, and ar = -1 for T = 7^\cr^v'. Note
that this rederives the relations (7.345) (symmetric mass matrix), (7.387)
(antisymmetric magnetic transition moments), and Problem 7.23 (no vector
neutral current coupling) for Majorana fields, derived previously using four-
component techniques.
8.4 Prove the Grassmann variable identities in (8.65).
8.5 Prove that the Lagrangian density in (8.66) is invariant under the su-
persymmetry transformations in (8.67). Hint: the easily derived identities
a»ov + ovo» = 2g*"'I1 o»ov + ovo» = 2gtA"I
are useful.
8.6 Verify (8.78).
8.7 Derive (8.84) and (8.114).
8.8 (a) Calculate the cross sections for e~e+ —> Zh and e~e+ —> Ah via a
virtual Z in the s-channel in the MSSM. Assume that one is well above the
Z pole, and neglect the e± masses.
(b) You should find that
"W..,,* -4h/\ . 3M|
= W(/?-a)IfMl +
a(Ah) ^ 'k\h\ *1
where kzh and kAh are the magnitudes of the final CM momenta for each
reaction. Interpret this result in terms of the equivalence theorem.
8.9 Derive (8.174).

Beyond the Standard Model
539
8.10 Consider the cascade decay
in the MSSM, where x?,2 an(* ^l are a11 on-shell, with (unknown) masses
mi,2 and me-, respectively. Assume that the x? is not observed and that the
only kinematic information obtained for the event are the four-momenta p± of
the e+ and e~. Show (neglecting the e± masses) that the maximum possible
invariant mass of the e+e~ pair is
Mmax = Vimj-mDimj-mj)
me
Note that the actual Mee distribution is expected to show a very sharp cutoff
(the kinematic edge) at M™ax, allowing a good measurement of that
combination of superpartner masses.
8.11 Consider the U{\)' extension of the MSSM, with the neutral Higgs
potential given in (8.205). There are now six neutralinos: the four MSSM ones,
the singlino 5, and the U{\)' gaugino, Z'. (a) Write the 6x6 neutralino
mass matrix corresponding to (8.180), and analyze the spectrum in the U(l)'
decoupling limit, s = y/2(S) ^> (msoft, v,iieff), with g2Qs,u,d fixed and
comparable to g and g'.
(b) Show that the gauge/Higgs spectrum in the decoupling limit includes the
the MSSM particles and a degenerate massive vector supermultiplet
associated with the Z', Z', and 5.
8.12 Derive (8.213) and (8.214).
8.13 Show how the universality test in (7.221) is modified in the SU(2)l x
SU(2)r x U(l) model. Assume that ( and /? are small, gR = gL, V^ = Vq',
and that the vr are very heavy. What are the approximate limits on £ for
uj = 0 and for to = 7r/2?
8.14 (a) Assuming (<£) takes the form in (8.264), verify the expression for
(b) Verify (8.265).

Canonical Commutation Rules
In this Appendix we briefly summarize the commutation rules and other
aspects of standard field theories.
Spin-0 Fields
Consider a spin-0 field <f> and Lagrangian density £, as defined in Section 2.2.
The conjugate momentum operator to <f> is defined as
where </> = -^ and the second definition is for the case in which (f> is complex.
For a spin-0 field, the canonical equal time commutation rules are
[(j>(t, X), <f>{t, X')] = 0, [7T(t, £), 7T(t, 2')] =0
[7r{t,x),<t>(t,x')} = -i53(x-x').
If 4> is complex, similar rules hold for <ft and 7r+, while [ir(t,x), (ft{t,x')] = 0.
If there are distinct fields 0f, then \-Ki(t,x),(j)j(t,x')] = —iS3(x — x')6ij.
The Hamiltonian density is
W(7T, (f>) = 7T</> + TT^ - £(0, 0M0). (A.3)
The Hamiltonian H(t) and momentum P(t) operators are
H(t) = f d3xH{Tr,<t>)
P(t) = f d3x [ttV0 + 7rfV0f], (A.4)
so the four-momentum operator is
P»(t) = (H, P)= I d3x [7rd^(f> + rfff1^ - g^C}. (A.5)
(The upper case P is used to distinguish the momentum operators from the
ordinary c-number momentum p^.) If (j) is real then the second terms in (A.3)-
(A.5) should be omitted. H(t) and P(t) are actually independent of t unless
there is explicit t dependence in C.
541

542
The Standard Model and Beyond
The Hermitian scalar field is described in Section (2.3), with the Lagrangian
density given in (2.19). The conjugate momentum is tt(x) = SC/S<p(x) = </>(#),
and the Hamiltonian density is
W(tt, <j)) = \ [j? + (V0)2 + mV] + VM). (A.6)
Similarly, the Lagrangian density for the complex scalar field of Section 2.4 is
given in (2.84), implying n(x) = </>*(#), ir^(x) = </>(#), and
«(*,*)= [(^£) (f )+(V0t).(V0)+mV0
+ V>(0,0+)- (A.7)
For the free Hermitian field one can use the explicit form for 4>o(x) in (2.24)
to obtain
pv
=\J
= jcPpjf
d3p
(27r)32£j
p»[a(p)ai{p)+ai(p)a{p)]
M(p) c53(0)
(2n)32Ep
+
(A.8)
where
Af(p) = ai(p)a(p)
(A.9)
is the number operator, which counts the number of particles carrying
momentum p. The S3(0) term in (A.8) integrates to 0 for \x ^ 0. For \x = 0 it
is the simple harmonic oscillator zero-point energy. It is infinite due to the
integration over p, e.g., it leads to an infinite vacuum energy
(0\H\0)
63(0)
J <?p^
p2 + m2
oo.
(A.10)
(The S3(0) is an artifact of our use of continuum state normalization. For
box normalization in a volume V it would be replaced by the density of states
V/(2tt)3.) This constant of the energy is unobservable when considering the
microscopic interactions*, so it is customary to ignore it. Such terms can
be systematically removed by a technique known as normal-ordering, which
means that creation operators are always written to the left of annihilation
operators in a product (with an appropriate minus sign for fermions). The
normal ordering of an operator O is sometimes indicated by the symbol \0\, so
that :a\a2:=:a2a\:= a\a2, while :a\a2'.= a\a2- We will always assume normal-
ordering for momentum and other related operators, such as the currents and
*It would become observable when gravity is included, so more care is needed in that case.
The zero-point energies cancel between bosons and fermions in supersymmetric theories.

Canonical Commutation Rules 543
charges associated with internal symmetries, but will not display the ::. Thus,
(A.8) is replaced by
Similarly, for the free complex scalar field in (2.89),
d3p
where
Af+(p) = aHp)a(p), M-(p) = b*(p)b(p)- (A.13)
The Noether current in (2.96) corresponds, for the free field, to the charge in
(2.100), i.e.,
d3p
where Nn± are the overall number operators for tt±.
p" s^= / esfe ' •'ww - / jMep **M{f)- (A11)
ie free complex scalar field in (2.89),
p" = IwfW/" [*V(iO +*"-(?)]. (A-12)
Q = I {2*f2Ep W+&)-*-&)] = **+-**-, (A.14)
Spin-1 Fields
The Lagrangian density for a Dirac fermion in (2.148) implies that the
conjugate field is na = SC/Sipa = itp^. Fermion fields satisfy canonical equal time
anticommutation rules
{rPa(t,x),rP0(t,x')} = O, {rpUt,S),ipl(t,x")} = 0
{tpa{t, x), ^(t, x')} = S3{x - x')6a0.
The expression for 7ra implies
n = 7rtp-C = i^tp, Pfl= J dzxipUd'1iP, (A.16)
which hold even in the presence of gauge and Yukawa interactions. For the
free field in (2.155), this implies
2 ' d3p
P» = £ / (a^s ^ [fl+ & S)°® S) + 6+ ® SWP> S)]
(A.17)
The Noether current for fermion ip is
J» = jyyUfa (A.18)
while the corresponding charge for the free Dirac field in (2.155) is

544 The Standard Model and Beyond
Spin-1 Fields
The Lagrangian density for the free electromagnetic field is given in (2.107).
Discussion of conjugate variables is complicated due to gauge issues, so we
will only quote the Hamiltonian,
H=\jd?x(E2 + B2). (A.20)
For the free field in (2.109),
(A.21)

B
Derivation of a Simple Feynman Diagram
Let us sketch the derivation of the first term in (2.29), i.e., the tree-level
amplitude Mfi = (p3P4\M\pip2) corresponding to the first diagram in Figure
2.3 for the Hermitian scalar field with k = 0.
Our starting point is to consider a transition from an arbitrary initial state
i to an arbitrary (but different) final state / with the same four-momentum.
As shown in texts in quantum mechanics and field theory, the unitary
transition matrix element Ufi and transition amplitude Mfi can be calculated in
perturbation theory using the interaction picture, to yield*
Ufi = (fM+oo, -oo)|i) = (2n)4S4 (j^Pf ~ 51 Pi) Mfi, (B.l)
up to subtleties concerning disconnected diagrams. In (B.l),
U(t2,ti) = (f\T [e-i/*?"'(<)<"] |^ (B.2)
is the time evolution operator, T is the time-ordering operator, and
Hj(t) = -Lj(t) = -Jd3x £/(<M*)) (B.3)
is the interaction part of the Hamiltonian in (A.4). The fields occurring in Hi
are free fields. For the non-gauge interactions of spin-0 particles, £/ = — V/.
For our example, we can expand the exponential and keep just the linear term
in Hi (the identity term doesn't contribute to the scattering amplitude),
Ufi ^ / d4x (p3P4|^/(0o(^))|piP2>, (B.4)
where
id(M*)) = -iViiM*)) = -i^|^. (B.5)
Substituting the expression (2.24) for the free field 4>o(x), one finds
d3Pm
Ufi ^ -i
-/ft —
4! J ii (2k)
f2Em
r d
/ d4x(0\a3a4 Y[ [ane-ip"x + ale+ip»-x] a\4\0),
m=a
d
(B.6)
n=a
*Some authors extract a factor of i from their definition of M.
545

546 The Standard Model and Beyond
where a\ = a(pi), etc. One can now move the a's to the right and the a^'s to
the left. The nonvanishing terms in the matrix element in (B.6) are*
4! c<(p.+Pb-Pc-Pd)-x<0|a3ai a±a\ aca\ ad<4|0), (B.7)
where the 4! is due to the fact that there are 4! non-zero terms that differ only
by the relabeling of the dummy indices a • • - d. Finally, one has
ada\ = (27r)32EdS3(pd - p2) + a\ad -> (27r)32EdS3{pd - p2), (B.8)
where the a\ad term vanishes since a|0) =0, and similarly for the other three
terms. The momentum integrals can then be done using the delta functions,
while the Jd4x integral yields (27r)4S4(p4 + P3 — Pi — P2), so that
Ufi = -i\(27r)46\p4 +P3-Pl -p2), Mfi = -i\. (B.9)
The Feynman rules for more complicated diagrams involving internal lines
can be derived using the Wick ordering theorem.
^The terms involving the commutators [ap,aj], q = a • • • d, vanish for non-forward
scattering. More generally, it can be shown that they should be ignored in such calculations.

c
Unitarity, the Partial Wave Expansion and
the Optical Theorem
The S matrix is a unitary operator describing the evolution of an initial state.
It is given by S = £/(+oo, — oo), where the time evolution operator U(t2,t\)
is defined in (B.2), i.e., Sfi = Ufi. The transition matrix T is related by
S — I + iT. It is convenient to extract a momentum-conserving 6 function
from a matrix element,
</|S|i) = 6fi + i(f\T\i) = Sfi + (2ir)464 (pf - Pi) iTfi, (C.l)
where the amplitude M defined in (B.l) is related by Mfi = iTfi. The
unitarity of the S matrix,
(S^S)fi = J2s}nSni = Sfi, (C.2)
n
implies the unitarity formula (for pf = Pi)
-i (Tfi - Tjt) = 2%mTfi = ^(2tt)454 (pn - Pi) TjnTni. (C.3)
n
Applying this to the special case f = i and using (2.50) yields the optical
theorem,
2%mTii = 4k^atot(k), (C.4)
where crtot(h) is the total (elastic plus inelastic) cross section at CM energy
V^ and k is the center of mass three-momentum defined in (2.36). Thus, the
optical theorem relates the imaginary part of the forward elastic scattering
amplitude to the total cross section.
For elastic two-body scattering T is related to the traditional scattering
amplitude /(&, 9) (especially familiar in quantum mechanics) by
T = -iM = 87TV/i/(A:, 6) (C.5)
(see, e.g., the Kinematics article in Amsler et al., 2008). Thus, the CM cross
section is
547

548 The Standard Model and Beyond
The partial wave expansion for spinless or spin-averaged particles (see
Weinberg, 1995, for the general case) is
1 °°
■K*'*) = T^(2* + l)a,(AOP<(cos0), (C.7)
£=0
where Pi is the £th Legendre polynomial. The partial wave amplitude is
ae(k) = -j dcosef(k,e)Pe(cose)= L/A ; ^ ±, (C.8)
where Se is the £th phase shift, and 0 < r)e < 1 is the inelasticity parameter.
Prom unitarity
\ae(k)\2 <%mae{k) < 1, (C.9)
with the first inequality saturated for purely elastic scattering (rje = 1). The
partial wave amplitudes, and their graphical representation in terms of the
Argand plot, are especially useful for parametrizing the behavior of low
energy amplitudes (where few partial waves are important), searching for and
studying the properties of resonances, etc., as described in introductory and
quantum texts. It is also useful for unitarity bounds on the elastic cross
section a(k) (integrated over angle), which are useful in connection with the
breakdown of the Fermi theory and with theoretical constraints on the Higgs
mass:
oo . oo
a(k) = ^>(fc) = -£ £(2£ + l)|a,(*)|2, (CIO)
so that
.«(*) < ^12. (an)
In terms of /(&, 9) the optical theorem is
Air Att *—>
°tot{k) = —$mf{k,0) = — ^(2Hl)9maK^) >cr{k). (C.12)
e=o
The optical theorem, combined with some general considerations involving
analyticity and crossing, can be used to derive the Froissart bound (Froissart,
1961) on the high energy behavior of the total cross section,
atot<C\n2s, (C.13)
for s —> oo, where C is a constant.

D
Two, Three, and n-Body Phase Space
Techniques for evaluating 2-body phase space were considered in Section 2.3.5.
Here we formalize that discussion and consider some techniques for the 3-body
case. More general discussions are given in (Barger and Phillips, 1997) and
in the Kinematics article in (Amsler et al., 2008).
Let us define the Lorentz invariant 2-body phase space factor
d2(m],m2a,rnl) = S4 (pi - pa -pb) -^ -^, (D.l)
where pi is the total four-momentum. For a decay process, m2 = p2 is the
mass-squared of the decaying particle, while for a scattering process m2 =
p2 = s. We saw in (2.54) that in the rest frame or CM (where pi = (m/,0)),
d2 (mj,m2m2b) = dtt^- -> 27rdcos(9^ -> 4tt^, (D.2)
v 7 m/ mi mi
where dft = dip cos 6 is the solid angle element of pa, and
Pf = \Pa\ = \Pb\ = ^ , (D.3)
with A(x, y, z) = x2 + y2 + z2 — 2xy — 2xz — 2yz. The differential cross section
for 2-body scattering is
where the initial momentum pi is given in (2.37). Similarly, a 2-body
differential decay rate is
dr=^lMl2 = l6^lMl2"OT9- (D'5)
These results can be immediately generalized to n-body phase space,
relevant to I —> fi • • • fn. Defining
dn=S4(Pl-±Pf)f[d^, (D.6)
549

550
The Standard Model and Beyond
the differential cross section and decay rate are respectively
da = ^LCn\M\2 dn, dT = ^-Cn\M\2 dn, (D.7)
with cn = [2{27rf)}-n.
The case n = 3, with
A ( 2 2 2 2\ r4 / x ^Pa d3pb d3pc
^3(m7,ma,mfe,mc) =5 (pj - pa - pb - pc)-g--^--^-, (D.8)
is very important, especially for decay processes. We will sketch two
techniques. The first method readily generalizes to n bodies. It involves the
factorization of ds into a product of 2-body factors, one of which describes
b + c and the other describes a + be, where be represents the collective b + c
system. The technique is especially useful when b and c are not observed (a
simplified version was used for \x decay in Section 6.2.1), when there is a
resonance in the be channel, or when a is the first particle emitted in a cascade.
To begin, we introduce the factor
r(mj—ma)
Mmi-rna) , v
1= / dm2bcS (m2bc-(pb+pcy)
J(mb+rncy V '
/ d4pbc S4 (pbc -pb- pc) 9 (Ebc)
(mb+mc)2 N ' (D.9)
in the expression for cfa. We can then rearrange the expression to yield
A ! /, 2r4/ x d3Pa d3pbc .4 / x d3Pb d3pc
dz = ~ / dmbc S (Pl -pa- Pbc) -= =— 8 (Pbc ~ Pb - Pc) -= F~
* J &a &bc &b &c
= 2 / dmbc d2 {m2j,ml, m2bc) d2 (mgc, mg, m2c).
(D.10)
Thus, ds is just the product of b + c phase space, restricted to invariant mass
mbc, and phase space for a + 6c, integrated over the allowed range shown in
(D.9) for mbc. Of course, the matrix element \M\2 occurs under the integrals.
The separation is especially useful if \M\2 also factorizes. It should be
remembered that the cfo's are Lorentz invariants, so it is often convenient to evaluate
the factors in different Lorentz frames, such as the / and be rest frames. As
a simple example of factorization, suppose there is a narrow resonance in the
be channel, with, e.g.,
|M|2=, 2 /(^;"lcLr2 -/«"fojrr-*«-m«)> <d-u>
«c~mr) +MrTr MrTr
as shown in Figure D.l. (D.ll) is the Breit-Wigner resonance formula for an

Two, Three, and n-Body Phase Space
551
FIGURE D.l
Three-body final state with a resonance in the be channel. / is the initial
decay or scattering state, gfa is the amplitude to produce aR, and g^c is the
resonance decay amplitude. /(mf^m2^) in (D.ll) is given by |<7/^|2|<7^|2.
unstable particle R of mass Mr and width YR = 1/tr propagating in the be
channel, as described in Appendix F. /(mj,m^c) describes the splitting of
the initial state into a + R and the decay of R into be. The final form is the
narrow resonance approximation, valid for Yr <C Mr. In this case
, ,,,,2 l/,J2J/2 2 2W/2 2\ ^ (mfec> mb > mc)
d*W? = -2 j dml d2 (ml ml, ml) /« ml) x ^ M\/+J^
^d2(m],mlMl)f(mlMl) x _]jZ_d2 (Mg.mg.m?). (D.12)
Now let us further suppose that / factorizes into terms involving the
production and decay, which would be the case for a spin-0 resonance, i.e.,
/(rrijiMji) = |^|2|^|2. Then, incorporating the relevant factors of 2 and
2ir from (D.7), one finds
a(I -> abc) = a{I -> aR)B{R -> 6c), T{I -> abc) = T{I -> aR)B{R -> 6c),
(D.13)
where B(R —> be) = T(R —> be)/Yr is the branching ratio for R —> be. This
reproduces the intuitive result that a cross section or decay rate involving an
intermediate narrow (i.e., long-lived) state factorizes into the cross section or
rate to produce the narrow state times the branching ratio into the final state.
An interesting feature of this result is illustrated by assuming that R can only
decay into 6c, so that B = 1. In this case the overall rates in (D.13) are
independent of the coupling strength gffc. This is completely different from a
virtual intermediate state, and is due to the fact that once R is produced it
always decays. The same holds if there are other decay channels so long as
B(R —> be) is held constant. Of course, the rates do depend on the production
coupling \gfa\.
An alternative method is useful when \M\2 only depends on pa • Pb, Pa • Pa
and Pb'Pa such as a decay process summed and averaged over spins. In the /

552 The Standard Model and Beyond
rest frame pa,fe,c lie in a plane. By assumption, \M\2 does not depend on the
orientation of the plane or the direction of pa in the plane, so we can integrate
over them. The spatial S function can be used to eliminate pc. The angle
between pa and pb can be expressed in terms of the energies of a and b using
the remaining energy S function, so that ds becomes an unconstrained integral
over Ea and Eb within an allowed region. The result, derived in (Barger and
Phillips, 1997), is
27T2
ds (m|, m^,ml,ml) = 2-K2m2 dXadXb = —^ ^mbcdm2ac, (D.14)
7717
where we define
Xi = —-, m = —|, i = a,b,c, (D.15)
mi rrij
and
m% = (Pi + Pjf = {pi - Pkf =mj+m2k- 2miEk, (D.16)
where i,j, and k are all different. The invariants can be expressed as
Pi • Pj = -m2- (1 + fik - m - Hj - Xk), (D.17)
which follows from (D.16). The energy or invariant mass variables satisfy
Xa + Xh + Xc = 2, m2ab + m2ac + m2bc = mj + m2a + m2b + m2c. (D.18)
However, the actual boundaries for Xa^b are rather complicated (see Barger
and Phillips, 1997), so we only give them for some special cases.
™>a,b,c = 0 :
0 < Xa < 1, 1 - Xa < Xb < 1
2/ia/2 < Xa < 1 + /ia, a - (3 < Xb < a + (3
(D.19)
ma,b = 0,mc 7^ 0 :
0 < Xa < 1 - /ic, 1 - Xa - ixc < Xb < X ~ Xa ~ Mc
l-Xa
An approximation to this technique was used for pion beta decay in Section
6.2.3. The method is applied to calculate the leading correction to that result
in Problem 6.7.

Two, Three, and n-Body Phase Space
553
This method is closely connected to the Dalitz plot, in which individual
events observed in a 3-body process are plotted in the Xa—X^ or the m\c—m2ac
plane. Prom (D.14) the phase space is uniform within the allowed region, so an
excess of events would correspond to an enhancement of the matrix element.
A resonance in a two-body channel would show up as an excess along a line
parallel to the x or y axis, or along a line x + y = constant.

E
Calculation of the Anomalous Magnetic
Moment of the Electron
In this Appendix we sketch the derivation of the one-loop (Schwinger)
contribution of a/2n to ae = (ge — 2)/2, the anomalous magnetic moment of the
electron, in part to illustrate some of the techniques for calculating Feynman
integrals. The relevant contribution is from the diagram in Figure E.l, which
Pi
FIGURE E.l
One-loop vertex correction contributing to the anomalous magnetic moment
of the electron. The photon momentum is q = p2 — P\ •
will be of the form
ie0u2[a^ -\-b(px + p2Y\ui, (E.l)
> v '
1 OL
where r£L = Zf1^ - 7^ in the notation of Section 2.12.1. From the
discussion following (2.354) and using the first Gordon identity in Problem 2.4,
we can identify ae = F2(0) ~ —2m6, which will turn out to be non-divergent.
555

556 The Standard Model and Beyond
Prom the Feynman rules,
r" -de)2 f d*k ~* * i
OL y > J (2tt)4 k2+ie (p2-k)2-m2+ie (pi - fc)2 - m2 + i€
x [iv Qfc- jK + m) y (A- jK + m) y] (E.2)
)4 7 D
where
(27T)
D=(k2 + ie) (k2 -2p2-k + ie) (k2 - 2Pl • k + ie) . ( '^
We have used p^2 — m2 and have replaced e^ by e2 and m0 by m since
the differences are of higher order. (We set Zi = 1 in the identification of
F2(0) for the same reason.) It is understood that the external electrons are
on-shell and that T^L is sandwiched between u spinors. The d4k integral
appears formidable, but it can be handled by a series of tricks. First, one uses
Feynman parametrization to combine the denominators. There are a number
of ways to do this, but we will use the identity
(E.4)
(E.5)
1 _,_ 1V f1 dz1---dznS(l-j:izi)
a1a2'"an J0 VL,iaA\
for any a\- • • an. For example,
i _ r1 dz
d\d2 Jo [aiz + a2{l - z)}2'
Applying (E.4),
1=2 f1 dz1dz2dz3S(l-Zzi) =2 f1 Vz
D J0 [k2 - 2p2 • kz2 - 2pi • kzx + ie]3 Jo [k'2 - c + ie]3 '
where
Vz = dz\dz2dzz8 (1 — Y^ Z{)
k' = k - p2z2 - pizi (E-7)
c= (p2z2 -\-piZi)2 =m2 (1 - z3)2 -a2 zxz2,
with c > 0 for q2 < 0. Therefore,
FM = Z^! f1 f d4kB»(Pl,p2,k)Vz
OL (2tt)4 Jo J [k'2-c + ie]3
= -He2 f1 f d4k,B»(Pl,p2,k' +p2Z2+plZl)Vz
(2tt)4 Jo J [k'2-c + ie]3
(E.8)

Calculation of the Anomalous Magnetic Moment of the Electron 557
where we have shifted the integration variable d4k —> d4k' in the second line.
The terms linear in k! integrate to zero
/
d4k' k'
^=0- (E.9)
[fc'2 - c + ie
The quadratic terms involving k'^kl must be of the form,
since the l.h.s. of (E.10) is a Lorentz tensor, and g^v is the only tensor
available. Contracting each side with g^v implies that C = \. Finally, using
the 7 matrix identities in (2.170) as well as U2 i>2 = fnu2 and i>\U\ = mux,
BtM(p1,p2,k' +p2z2 +Pizx) = 2m(p1 +P2Y
-2ziz2
+z2{l -z2)+zi (1 -zi)
+ 7^ terms + vanishing terms.
(E.ll)
Therefore, dropping the prime on A;,
F9(0) = ®^ /•^ib[-2** + *(l-*) + *(l-*i)]P*> (E.12)
V ' (2tt)4 Jo J [k2-c + ie]3 K '
where c in (E.7) is evaluated at q2 = 0, i.e., c = m2 (1 — Z3) .
Now, consider the integral
T = f <*4fc _ [+°° Au f <*3fc
ln~J [V-c + ier-J^ dk°J [kl-V-c + ie]
(E.13)
where c > 0. In is convergent for n > 2. The ko integral can be viewed as a
contour integral along the real axis in the complex ko plane. The integrand has
poles at &o = ±( v k2 + c — ie), which lie in the second and fourth quadrants.
One can therefore perform a Wick rotation, in which the contour is rotated
to lie along the imaginary axis,
,oc dkoJ \e-&-cr=iL dk0EJr>
d3k
[k2-k2-c]n J-oo J [-k2E-k2-c]n
(E.14)
where koE = —iko is the Euclidean energy, and we have taken e —> 0 since the
rotated contour is far from the poles. This can then be written
T I ""• f d4kE ( I^V /"WO f°° kEdkB
( ' (n-l) (n-2) c"-2'
(E.15)

558
The Standard Model and Beyond
where ks is a Euclidean four-vector with k\ = k^ + k2, k^dks = k\dk\l2
and J dft.4 = 2n2 is the area of a unit 3-sphere in four dimensions (Problem
1.3). The integral has poles for n = 1 and n = 2, indicating the divergence of
the integral, but is well behaved for n > 2. These techniques can be readily
generalized to ones such as /d4kk2p/[k2 — c + ie]n.
Putting everything together, the anomalous magnetic moment is
F2(0)
a f1 [-2zi
7T Jo
Z2 + Z2(l~ Z2) + Z1(l- Zi)
U-23)2
dzidz2dzsS (l — y^^Zj)
zi + z2
dzidz2Q (1 — z\ — z2) =
a
2n'
(E.16)

Breit- Wigner Resonances
An unstable particle or resonance of mass Mr and width T = 1/r can be
treated in Feynman diagrams by replacing the denominator of its propagator
by the Breit- Wigner resonance form
so that
W)|2=tf-^+^P- (R2)
If T is large, then additional energy dependent corrections may be important
away from the peak (see, e.g., Gounaris and Sakurai, 1968). Away from the
resonance peak a rate involving a resonance is suppressed compared to the
on-shell rate by M^T2/(q2 — M^)2 <C 1, and far enough off shell the finite
width effect is irrelevant. (There may be additional energy dependent factors
associated with phase space, etc.)
For a narrow resonance, T/Mr <^C 1, it is sometimes useful to employ the
narrow width approximation
\D(q2)\2 ^ j^6(q2 - M%), (F.3)
which effectively treats the production of the long-lived state as if it were
stable. This was used in a 3-body phase space example in Appendix D,
where it was shown that the production rate of a given final state through a
narrow intermediate spin-0 resonance factorizes into the rate to produce the
resonance, times its branching ratio into the final state. This result generalizes
to more complicated production and decay processes and to differential cross
sections and decay rates. One can still use (F.3) for narrow resonances with
nonzero spin. However, the factorization into production rate times branching
ratio only holds if one averages or sums over the external spins and integrates
over all of phase space. Otherwise, there are spin and angular correlations
that do not factorize.
Let us consider an example of an unstable Hermitian spin-0 field (f> which
can couple to fermions a or 6, with
£ = {ga^a^a + 9bMb) 4>, 9a,b = real. (F.4)
559

560
The Standard Model and Beyond
The amplitude for aa —> bb via the s-channel (f> resonance is
M = i3ga9bUbViVaUa D(s), (F.5)
where we take MR = m<f> and r = Y^ in D{s). Neglecting ma and m^,
|M|2 = i5^2 |U(S)|2^(A ^TY^ fi&) = g2ag2bs2 \D(s)\2, (F.6)
so that the cross section is
'M'lib^t^-A^W'- (F7)
Note that it is usually not useful to use the narrow resonance
approximation for an s channel process unless one is integrating over s. (F.7) can be
conveniently rewritten using
Taa = r (<f> - ad) = ^-L-|Maa|2, (F.8)
with
\Maa\2 = \i9a\2 TV(ja ffe) = 2M2Rg2a, (F.9)
so that
Taa = ^^, (F.10)
with a similar formula for Tbb. If there are no other decay channels, then
r = ra&+Tb-b. From (F.7) and (F.10),
a{8) ~ (s-M*R)> + MRT> H^l M%B<*Bbb> (F-U)
where Baa = raa/r is the branching ratio into ad. This convenient form
absorbs the couplings ga,b into the partial width or branching ratio factors.
It also illustrates that the peak cross section at s = m2R is independent of the
coupling strengths for fixed branching ratios because they cancel between the
partial and total widths. Similarly, for a massive vector resonance V^ with
interactions
C = [gj>a-fil>a + 9b*Pbl^b] V^ (F.12)
one finds the same result as (F.ll) except that 47r —> 12tt (Problem 2.20).
By measuring a(s) as a function of s one can determine M\ from the
position of the peak, T from the width of the peak, and a combination of branching
ratios from the peak cross section o~peak = &{Mji), as illustrated in Figure F.l.
This was extremely powerful, for example, in the Z lineshape measurements at
LEP. (In practice, one must apply additional energy dependent corrections.)
In some cases, however, the detector energy resolution is not good enough to

Breit- Wigner Resonances
561
resolve the peak and determine T. One then effectively integrates over s to
obtain (for the spin-0 case in (F.ll))
/
4?r2
°(s)ds ~ — TaaBb-b.
(F.13)
&peafc
FIGURE F.l
The cross section vs s for the Breit-Wigner resonance form in (F.12),
illustrating the peak position, width, and peak cross section.
The expression in (F.ll) is typical for an s-channel resonance (see, e.g.,
Weinberg, 1995), provided that spins are averaged (summed) and the phase
space is integrated over. The overall coefficient depends on the spins of the
resonance and of the initial particles. Sufficiently near the peak, the cross
section for a^a2 —> b\b2 is
a(s) =
A(2SR + l) 4Tr(s/M%)faia2Tblb2
(2501 + l)(25a2 + 1) (s - Ml)2 + MlP '
(F.14)
where Sr, Sai, and Sa2 are respectively the spins of R, a\, and a2 (with 25 + 1
replaced by 2 for a photon). The prefactor is because the 2Sr-\-1 intermediate
spin states contribute equally to (and do not interfere in) the total spin-
averaged rate. There may be additional s dependence away from the pole,
due, e.g., to vertex factors or threshold effects.

G
Implications of P, C, T, and G-parity for
Nucleon Matrix Elements
In this appendix we summarize the implications of various discrete symmetries
for the matrix elements of vector and axial currents between on-shell nucleon
or hyperon states. Recall that even though the weak interactions violate
these symmetries, e.g., because of the V — A structure of the charged current,
the hadronic matrix elements are dominated by strong interaction effects,
which are believed to be invariant under P, C, and T (Section 2.10), and
approximately so under G-parity (Section 3.2.5). However, higher-order weak
corrections could lead to small deviations from the symmetry predictions.
Crossing Symmetry
Consider an amplitude or matrix element involving an initial particle b(p,s),
(a\0\/3b(p,s)) = 0(p)u(p,s), (G.l)
where a and (3 are other external particles. The spinor-valued function 0(p)
depends on p = p and on the momenta and spins of the other particles.
Crossing symmetry implies that the corresponding matrix element for a final
bc is described by the same function, i.e.,
(abc(p,s)\O\0) = ±0(p)v(p,s), (G.2)
where in this case p = —p, and the ± sign depends on how many fermion
interchanges are involved in the crossing. In both (G.l) and (G.2), p and s
refer to the physical momentum and spin, while p refers to the momentum
flowing in the direction of the arrow in a Feynman diagram. Although the
same function 0(p) describes both processes, the physical values of p are in
different regions, i.e., the two processes are related by analytic continuation.
A similar relation applies to the interchange of a final particle and initial
antiparticle. The crossing symmetry is obvious from Feynman diagrams but
holds more generally, as can be proved using the LSZ reduction formalism.
We will be concerned here with matrix elements of an operator O between
single particle states,
(a(pa)\0\b(pb)) = ua(pa)0(pa,pb)ub(pb), (G.3)
563

564 The Standard Model and Beyond
where the spin labels are not displayed. Applying crossing to both a and 6,
(bc{pb)\0\ac(pa)) = -va(pa)0(-pa,-pb)vb(pb)
= +ub(pb)Oc(-pa,-pb)ua(pa),
where Oc = COTC~1 (Table 2.2) and the second form follows from (2.293).
Vector and Axial Form Factors
We will be considering matrix elements of the vector and axial vector currents
VZb=jal»i},b = v£, <6 = 6,7*7% = <. (G.5)
Analogous to (2.389), assuming only Poincare invariance and the Gordon
identities, the most general matrix element of V^ between the corresponding
single particle states is of the form
(a\V:b(x)\b)=ua
1»FU<12) + ^Atf) + Wlttf)
+ ri59?ab(Q2) + ^^(S2) + q^59UQ2)
ubetqx
= uaT^(q)ub, (G.6)
where q = pa — pb and the mass m can be taken to be (ma + mb)/2. The
labels a and b on the states and spinors refer collectively to momentum, spin,
and flavor. When we specify the special case a = b we mean that the flavore
are the same, but the initial and final momenta and spins may differ. We
will henceforth take x = 0. A similar expression holds for (a|A^6|6), with
superscripts A on the form factors and matrix, F^fe, gfah, Ta£(q).
Space Reflection
As was already discussed in Section 2.10, space reflection invariance requires
r^te) = iKabWh0, r%b) = -rXaoWh0 (G.7)
where q^ = p'afJl - p'h^ = q». This implies gYab = F£b = 0, up to weak
interaction corrections (such as the anapole moment mentioned below (2.391)). All
of the results are summarized in Table G.l.
Charge Conjugation
Prom (2.291) the currents transform under charge conjugation as
CVSfi-l = -V^, CA^C-^+A^. (G.8)
Therefore, using (G.5) and charge conjugation.
(aKb(x)\b) = (b\V£{x)\ar = - <&TO(x)|ay. (G.9)

Implications of P, C, T, and G-parity for Nucleon Matrix Elements 565
(G.6) and the crossing relation in (G.4) then yield
uar2(q)ub = - [ubr^(-q)Ua\* = -uaT^b(-q)ub. (G.10)
Applying similarly reasoning to A'1, one obtains
r^"(«) = -r^(-g), r%(q) = +r^(-«), (G.ii)
which are satisfied for real FY and gf, and for imaginary F* and gY. As we
will see, time reversal invariance implies real form factors, in which case both
P and C independently imply F/4 = gY = 0.
In the special case of a = 6, one has CV^C~l = -V" and CA^C'1 = +A»,
from which it follows that
r£"(9) = -r££(«), r%(q) = +r&(«). (G.12)
(This also follow from (G.ll) and the Hermiticity of V^ and AM, which imply
that r%A»(q) = TVa^(-q).) Thus, the only G-allowed terms are i^2aa, <#aa,
dtzaai and ^sia- (dLa and Fsia would have to be imaginary.)
Time Reversal Invariance
It follows from (2.312), (2.313), (2.320), and (2.321) that all of the form factors
must be real (or at least relatively real for non-standard phase conventions)
if time reversal holds. We already saw that for diagonal (a = b) transitions,
Hermiticity requires g\aa and F^aa to be imaginary. Thus, observation of an
intrinsic electric dipole moment ($Xaa 7^ 0) would imply the violation of both
space reflection and time reversal invariance.
G-Parity
The G-parity transformation G = Cel7rT in (3.130) is useful for the proton-
neutron currents associated with the weak charged current. One has
G^yV-nG"1 = +t/v7"V<n, G^W'nG-1 = -^p-yVVv. (q ,v
G\p) = \ne), G|n) = -|pc). ( ' '
Combining with the crossing relation in (G.4), G-parity implies (Weinberg,
1958)
r#(«) = -r^(«), r#(g) = +r#,(«) (G.w)
for the weak charged current form factors. (G.14) is satisfied by F^2pn and
gf^pn, which are known as first class form factors, and violated by the second
class ones F^pn and g^pn- (The situation is reversed for the parity-violating
form factors.) There was once considerable activity in (3 decay searching for
second class form factors (including some short-lived positive indications).

566
The Standard Model and Beyond
Such effects, if significantly larger than those expected from isospin breaking,
would presumably be due to second class currents with the opposite G-parities
from those in (G.13). It is almost impossible to construct viable models
involving second class currents within the general quark model framework,
however (e.g., Langacker, 1977).
TABLE G.l
Restrictions imposed by discrete symmetries on the form factors associated
with vector and axial vector currents.
Space Reflection:
Charge Conjugation:
Charge Conjugation (a
Hermiticity (a = b):
Time Reversal:
G-parity:
= 6):
9u* = 0
pv _ pv*
iab iab
pA _pA*
iab iab
pv _ nv _ n
r3aa — 9\,3aa ~ u
pV,A _ _pV,A*
3aa 3aa
others real
pVyA _ pV,A*
iab iab
^3pn — 92pn = 0
FtAab = 0
yiab yiab
yiab yiab
^l,2aa — 9laa = 0
V,A V,A*
9laa ~ 9laa
QV,A _ QV>A*
yiab yiab
^l,2pn — 9\,3pn = 0

H
Collider Kinematics
Here we collect a few definitions and results relevant to kinematics at colliders.
For more detail, see (Barger and Phillips, 1997; Han, 2005; Amsler et al.,
2008).
Single Particle Variables
In Chapter 2 and in considering decay processes we have usually used spherical
coordinates to describe the final particle phase space. However, for studying
short distance processes in QCD one is usually interested in large transverse
momenta for the final particles*. Moreover, in a hadron-hadron collision the
CM of the incident hadrons does not in general coincide with the CM of the
hard scattering subprocess of interest. For these reasons, it is often convenient
to work in cylindrical coordinates, with the z axis along the beam direction, pr
and (p represent the magnitude of the transverse momentum and the azimuthal
angle of a produced particle, i.e.,
P*1 = (E,Px,Py,Pz) = (E,pTcos(p,pTsm(p,pz), (H.l)
with E = y/m2 +p2, pr = psin#, and pz = pcos6, where p = \p\ and 6 is
the polar angle. The Lorentz-invariant phase space element is
d3p . . dpz _ _ dpz . . /TT _
— = dpxdpy — = pTdpT d<p—-= pTdpT dip dy, (H.2)
where the last form will be defined below, pr, <p, and dpz/E are all invariant
under longitudinal Lorentz boosts (i.e., boosts along the ±z direction).
p*1 can be rewritten as
p^ = (rrtT cosh y, pr cos ip,pr sin (p,rriT smb. y), (H.3)
where
= yjm2+P2T- (H.4)
This quantity is sometimes referred to as the transverse mass or transverse
energy, but we will reserve these terms for different quantitites. The rapidity
*Much more common are the soft minimum bias events, characterized by low transverse
momenta and (typically) low multiplicity, associated with long distance effects.
567

568
The Standard Model and Beyond
y is defined as
y=iln£±^=ln^=tanh-'f. (H.5)
2 E — pz ttit E '
Eq. H.3 follows immediately by taking the cosh and sinh of (H.5). The rapidity
is especially useful for expressing the effects of longitudinal boosts. From
(H.3), (H.5), and the expression in Problem 1.4, the momentum of the particle
as seen in another Lorentz frame moving with velocity (3qz is of the same form,
except
y^y-Vo where y0 = -\n- — = tanh"1 0O. (H.6)
z 1 — po
dy is therefore invariant under longitudinal boosts, with dy = dpz/E from
(H.3), leading to (H.2). There is no longitudinal momentum in the frame
with /?0 = pz/E, where p^ = (ijit.Pt cos tp.pr sin (p, 0).
The measurement of y requires a measurements of both pz and E, so it is
usually easier to work with the closely related pseudorapidity
1. l+cos0 / 0\
^2lnr^-lHtan2J' (H-7)
which is measured directly from the direction of the particle, with the range
0 < 6 < 7r corresponding to +oo > rj > — oo. Eq. H.7 is equivalent to
cos# = tanh r?, sin# = —-—. (H.8)
cosh r? <
Rapidity is identical to pseudorapidity for m = 0 and reduces to it for p ^> m
except for very small angles (6 < m/p). pr, ip, and rj are therefore
useful variables for describing the momenta of the final particles in a high pr
event*. The angular separation between two tracks of momentum p\ and pi
is conveniently expressed as the (longitudinal boost-invariant) distance
AR = V(At7)2 + (Ap)2 = V(V2-m)2 + (V2-Vi)2 (H.9)
between them in the rj-ip plane, and jets can be defined operationally as
consisting of the cone of particles separated from the center of the jet by a distance
AR smaller than some reference value (e.g., 0.7). Events are often displayed
on Lego plots, which indicate the the total transverse energy
ET = EsinO = E/cosh 77. (H.10)
deposited in the electromagnetic and/or hadron calorimeters in cells in the
77 — ip plane. If the masses of the particles in the jet can be ignored then Et
is just the sum of their m^.
tThe various components of the ATLAS and CMS detectors at the LHC will have coverage
up to \rj\ ~ 2.5 - 5.

Collider Kinematics
569
Parton-Parton Scattering
As described in Section 5.6, the cross section for the short distance process
HaHb —> F + X can be expressed in terms of the parton-parton cross
sections (Jij-+F{xA,XB), where i and j label the parton types and xa,b are their
respective momentum fractions in hadrons Ha,b- The total momentum of
the subprocess ij —> F is therefore pp = xaPa + xbPb, where the hadron
momenta in the CM are pa,b = {Ea,b, ±p), with Ea ~ EB ~ p= \p\ in the
(relevant) high energy regime. The invariant mass-square for the subprocess
is
p2F = s ~ xa%bs = ts, (H.ll)
where s = (pa +Pb)2 ~ 4p2. One can express the individual xa,b as
xA = Vre+y, xB = Vre~y, (H.12)
where
y = l^ (H.i3)
2 xb
is the rapidity of the CM of the subsystem F. This interpretation follows
from pf = {xa + %b, 0,0, xa — %b)p and the definition (H.3). The velocity of
the system in the hadron CM is
(3 = ttmhy=XA~XB. (H.14)
xa +xb
It is sometimes useful to trade the variables xa,b for t = s/s and y. In
practical applications, the cross section expression (5.89) may involve the
factor f dxAdxsQ{s — so), where sq is a physical or experimental threshold in
the invariant mass of the system. But Q(s — 3o) = 0(r — To) = @(xaXb — ^o),
where To = so/s. Thus,
pl pi pi pi pi p — \\nr
I dxA I dxB®(s-s0) = / dxA / dxB = dr dy. (H.15)
./O «/0 J to Jto/xa J to J1\t\t
The relation of the kinematic variables is shown in Figure H.l.
Transverse Variables and Missing Momenta
The high pr particles relevant to a short distance process often involve one
or more weakly interacting neutral particles that are stable or long-lived and
therefore do not interact or decay in the detector. These could be neutrinos
produced in weak decays of other particles, or they could be new "beyond the
standard model" particles, such as the lightest supersymmetric partner in
imparity conserving versions of supersymmetry. In principle one can determine
the sum of the four momenta of the missing particles by subtracting the
total four momentum of the observed particles from the initial momentum
(y/s, 0). In practice this is impossible at a hadron collider because many

The Standard Model and Beyond
FIGURE H.l
Relation of xa,b to the rapidity y (lines) and fractional invariant mass yfr =
y/s/s of the hard subprocess. xa corresponds to the x variable for a given y
and y/r, while xb corresponds to the — y line. The maximum rapidity for a
given r is — ^ lnr, which follows from the requirement that r < xa,b < 1.
undetected beam fragments exit the detector close to the beam direction*.
This problem can be evaded by working in terms of the transverse components
of the momenta px- In a high pt process it is a reasonable first approximation
to ignore the transverse momenta of the scattering partons, and the transverse
momenta of the beam fragments is small. Therefore, the total transverse
momentum of the subprocess can be approximated by zero, and the transverse
momentum of the invisible high pr particles is given by the missing transverse
energy vector (which is really a momentum)
£™- = -£piT, (H.16)
i
where the sum is over the observed particles. As a simple example, consider
pp —> W~ + X, where W~ —> e~ve and all of the high pr particles in X are
measured. In that case, one can determine the transverse momentum of the
unobserved ve
PpeT = -Per - ^PiT- (H.17)
iex
*Of course, in the real world there are other complications ignored here, including large
numbers of low energy particles, resolution and threshold effects, inefficiencies in particle
identification, cracks in the detector, etc.

Collider Kinematics
571
Since ppeZ is not known, one cannot directly reconstruct the mass of the W.
However, one can determine its transverse mass
Mwt= [(rnPeT + rneT)2-(ppeT+peT)2]1 2 ~ 2p*eTPeT(l-cos(/?ei/), (H.18)
where the rnr are defined in (H.4). In the last form <pei/ = ipe — (pPe and
we have neglected the lepton masses. It is straightforward to show that 0 <
Mwt < Mw> From a single event one cannot determine Mw However, from
the endpoint and shape of the Mwt distribution for a large number of events
one can obtain an excellent value. Of course, backgrounds and the finite
W width must be taken into account. For more complicated examples, e.g.,
involving two missing particles, see (Barger and Phillips, 1997; Han, 2005).

I
Quantum Mechanical Analogs of Symmetry
Breaking
Many of the field theoretic possibilities for symmetry breaking and realization
described in Section 3.3 have simple analogs in ordinary non-relativistic
quantum mechanics. Consider a single particle of mass \x moving in the potential
V{x) = \^2x2 + ^-, (1.1)
illustrated by the dashed curve in Figure I.l. (We take A > 0 in all of the
examples.) The minimum is at x = 0, and for sufficiently small A the energy
eigenvalues are approximately those of a simple harmonic oscillator, En =
(n + \)w. V exhibits a reflection symmetry, V(x) = V(—x), so the energy
eigenstates ipn{x) will have either even or odd parity under x —> —x. This
will be true even when the effects of the Ax4 term are non-negligible.
One can break the reflection symmetry explicitly, i.e., in the equation of
motion, by adding a small linear term to the potential
1 At4
V{x)^^2x2-ex+^-. (1.2)
The minimum is now at xo ~ e//j,u)2 ^ 0. Expanding around xo,
V(x) ~ \^2{x - x0)2 + Xx0(x - x0f + ^'^ + 0(e2). (1.3)
The energy eigenvalues (in this example) are unchanged to 0(e), but the
reflection symmetry (whether around the origin or around Xo) is broken. The
energy eigenfunctions will no longer have a definite parity.
Another possibility is to maintain the exact reflection symmetry of the
potential, but to break it spontaneously, i.e., in the solutions to the Schrodinger
equation. For
V(x) = -l-^2x2 + ^, (1.4)
(the solid curve in Figure I.l) the origin x = 0 is unstable, but there are
1 /9
degenerate minima at ±xo, where xo = (/xo;2/A) . One can expand around
the minimum, e.g., at +xo, to obtain
V(x) - +/iu;2 (x - x0)2 + Ax0(x - x0)3 + A(X~Xo) , (1.5)
573

574
The Standard Model and Beyond
where we have dropped an irrelevant constant V(xo) = —/j?u4/4\. Assuming
that the barrier is sufficiently high that one can neglect tunneling, we can
quantize around Xo. For small enough cubic and quartic terms, the energy
eigenvalues are ~ (n + \)y/2uj. Again, the reflection symmetry is lost. One
can also combine explicit and spontaneous breaking by adding a perturbation
—ex to (1.4) (the dotted curve in Figure 1.1). This breaks the degeneracy
between the minima at ±#o, and also shifts their position slightly. For e > 0
the minimum near +#o, known as the true or global minimum, is deeper. It
is straightforward to calculate the (9(e) shift in energy eigenvalues. For small
enough e there remains a shallower metastable minimum near — #o, known as
a local or false minimum.
V(x,y)
FIGURE LI
Left: one-dimensional potential in (LI) (dashed), (1.4) (solid), or (1.4) with an
additional —ex term (dotted). Right: two-dimensional Mexican hat potential
in (L7), which leads to spontaneous symmetry breaking.
Now consider a particle moving in the two-dimensional potential
V{x,y) = \^\x* + y>) + A(*2^2)2 = ^V + ^, (1.6)
where r2 = x2 + y2. The potential is independent of the polar angle 6 —
tan-1 y/x, and is therefore invariant under the 50(2) group of rotations on
x and y, as well as under the reflections (sign changes) of x or y. The energy
eigenstates may be classified by the IRREPs of 50(2), i.e., as eigenstates of Lz
with eigenvalue ra = 0,±l,±2,---. For small enough A the energy eigenvalues
are given by the harmonic oscillator formula En = (n+1) a;, where n = 2/c+|ra|
with k = 0 • • • oc the radial quantum number. Equivalently, n = nx + ny in
a rectangular basis. The degeneracy of n is 2n + 1. Adding a perturbation
ey2/2 to V(x,y) in (1.6) breaks the 0(2) symmetry down to discrete sign
changes (i.e., rotations by tt and sign changes in x or y). The degeneracy

Quantum Mechanical Analogs of Symmetry Breaking 575
of the eigenvalues is broken, with EUxyUy = (nx + \)u + (ny + |)u/, where
J ~ u) + e/(2ficj).
The Mexican hat potential, illustrated in Figure 1.1,
V(x,y) = -i^(x2 +y2) + \(x2+V2)2 = _l^2r2 + A£ (J ?)
differs from (1.6) by the sign of the quadratic term. The origin is unstable, and
1 /9
there are a circle of degenerate minima at radius ro = (/xo;2/A) . Similar to
(1.5) one can expand V(r, 6) around ro,
V(r,0) ~ W (r - r0)2 + Ar0(r - r0)3 + ^-^°L, (L8)
where we have again dropped a constant. For suitable parameter values,
V(r, 9) can be approximated by a harmonic oscillator in the radial direction,
centered at ro and with frequency y/2u, along with a flat potential in the 6
direction, leading to non-degenerate eigenvalues
1 m2
Enr,m - (nr + -)V2u> + ^—5. (1.9)
The second term represents the zero-frequency "rolling" of the particle along
the flat 6 direction, and corresponds to quantized momentum A;m = m/ro-
The m = 0 mode has no energy in the angular variable and is analogous to
the Nambu-Goldstone bosons encountered in Section 3.3.3. For large ro the
excitations are closely spaced.
One can add a small explicit breaking term ey2/2 to (1.7). If the
corresponding frequency \JT[\x is large compared to the excitation energy ~ (2/ir2))-1 of
the rolling but still small compared to u, the energy eigenvalues become
Enr,ne - (nr + \)V2u + (ne + \)J^, (1.10)
i.e., the Goldstone modes acquire a minimum vibrational energy.

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Web Sites
• Particle physics
PDG: The Review of Particle Physics, http://pdg.lbl.gov/.
Particle properties and many useful review articles in particle physics,
astrophysics, and cosmology.
SPIRES: http://www.slac.stanford.edu/spires/hep/. Extensive
data base of high energy physics publications.
Archive: http://arXiv.org/. Archive of preprints in physics and
related fields.
Online: Online Particle Physics,
http://www.slac.Stanford.edu/library/pdg/hepinfo.html. List
of particle physics links.
SMB: http://www.sns.ias.edu/~pgl/SMB/. Supplements and
corrections to this book.
• Tevatron and LHC general purpose detectors
CDF: CDF results, http://www-cdf.fnal.gov/physics/physics.html.
DO: DO results, http://www-dO.fnal.gov/results/.
ATLAS: http://atlas.ch/.
CMS: http://cms.cern.ch/.
• QCD and structure functions
CTEQ: The Coordinated Theoretical-Experimental Project on QCD,
http://www.phys.psu.edu/~cteq/.
MRST: http: //durpdg. dur. ac. uk/HEPDATA/PDF.
LHAPDF: Les Houches Accord PDFs (Whalley et al., 2005),
http://hepforge.cedar.ac.uk/lhapdf/.
• Electroweak Physics
LEPEWWG: LEP Electroweak Working Group,
http://lepewwg.web.cern.ch/LEPEWWG/. Major source for LEP
and SLC results.
631

632
The Standard Model and Beyond
ZFITTER: http: //www-zeuthen. desy. de/theory/research/zf itter/.
Electroweak radiative corrections and fits (Arbuzov et al., 2006).
GAPP: Global Analysis of Particle Properties (Erler, 1999),
http://www.fisica.unam.mx/erler/GAPPP.html.
Gfitter: A Generic Fitter Project for HEP Model Testing (Flacher
et al., 2009),
http://proj ect-gfitter.web.cern.ch/proj ect-gfitter/.
• Higgs physics
LEPHIGGS: LEP Higgs Working Group,
http://lephiggs.web.cern.ch/LEPHIGGS/www/Welcome.html.
TeV4LHC: Higgs production cross sections (SM and MSSM),
http://maltoni.home.cern.ch/maltoni/TeV4LHC/.
HDECAY: Higgs boson decays in the SM and MSSM (Djouadi et al.,
1998), http://home.cern.ch/~mspira/proglist.html/.
CPsuperH: MSSM Higgs with CP violation (Lee et al., 2004),
http://www.hep.man.ac.uk/u/j slee/CPsuperH.html.
FeynHiggs: MSSM Higgs.
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HiggsBounds: LEP and Tevatron bounds for arbitrary model (Bechtle
et al., 2008). http://www.ippp.dur.ac.uk/HiggsBounds.
• Supersymmetry: spectrum, renormalization group evolution
SPA: Supersymmetry Parameter Analysis (Aguilar-Saavedra et al., 2006),
http: //spa. desy. de/spa/. Links to supersymmetry analysis
programs.
SuSpect: http://www.lpta.univ-montp2.fr/users/kneur/Suspect/
(Djouadi et al., 2007).
Spheno: http://ific.uv.es/~porod/SPheno.html (Porod, 2003).
SoftSUSY: http://projects.hepforge.org/softsusy/ (Allanach, 2002).
• Event generators (for a general description, see Mangano and Stelzer,
2005) and analysis programs.
Pythia: http://home.thep.lu.se/~torbjorn/ (Sjostrandet al., 2008).
Herwig: http: //hepwww. rl. ac. uk/theory/seymour/herwig/ (Corcella
et al., 2001).
Isajet: http://www.hep.fsu.edu/~isajet/ (Paige et al., 2003).
Sherpa: http://projects.hepforge.org/sherpa/ (Gleisberg et al.,
2009).

Web Sites
633
ROOT: http: //root. cern. ch/root/. Analysis package.
MARMOSET: Mass And Rate Matching In On-Shell Effective
Theories (Arkani-Hamed et al., 2007) http://www.marmoset-mc.net/.
PGS4: Pretty Good Simulation of particle collisions (generic collider
detector) http://www.physics.ucdavis.edu/~conway/
research/software/pgs/pgs4-general.htm.
• Matrix elements
MadGraph: http: //madgraph. hep. uiuc. edu/.
CalcHEP: http: //theory. sinp.msu. ru/~pukhov/calchep.html.
LanHEP: Feynman rules from Lagrangian density (Semenov, 2009).
http://theory.sinp.msu.ru/~semenov/lanhep.html.
CompHEP: http: //comphep. sinp.msu. ru/.
FeynArts: Generation and calculation of Feynman diagrams (Hahn,
2001), http://www.feynarts.de/.
FeynRules: Calculation of Feynman rules (Christensen and Duhr, 2008),
http://ieynrules.phys.ucl.ac.be/.
• K and B physics, CKM matrix
FlaviaNet: Working group on K decays (Antonelli et al., 2008),
http://www.lnf.infn.it/wg/vus/.
CKMfitter: http://ckmfitter.in2p3.fr/ (Charles et al., 2005).
UTfit: http://www.utfit.org/ (Ciuchini et al, 2001).
HFAG: Heavy Flavor Averaging Group,
http: //www. slac. Stanford. edu/xorg/hfag/ (Barberio et al., 2008).
• Dark matter abundance
MicrOMEGAs: http: //lappweb. in2p3. fr/lapth/micromegas/
(Belanger et al., 2007).
DarkSUSY: http://www.physto.se/~edsjo/darksusy/ (Gondoloet al.,
2004).
• Neutrinos
Neutrino oscillation industry: Neutrino links,
http://www.hep.anl.gov/ndk/hypertext.

Index
action, 10
active/passive transformation, 470
Ademollo-Gatto theorem, see form
factor
adjoint, see representation
Adler-Bell-Jackiw, see anomalies
Adler-Weisberger relation, see form
factor
AdS/CFT correspondence, 236, 248
AGS, 387
ALEPH, 340, 422
ALICE, 236
a, see gauge coupling constant: QED
as, see gauge coupling constant: QCD
Altarelli-Parisi equations, see DGLAP
equations
analyticity, 188, 209
anapole moment, 92, 248, 323, 564
annihilation operator, see creation
operator
anomalies, 177-179
U{1)A, 222, 229
tt° -> 27, 193, 258
£7(1)', 514
gauge symmetry, 178
global symmetry, 178
standard model, 283, 448, 454
strong CP, 458
anomalous dimension, 217
ANTARES, 447
anthropic principle, 457, 460
antiscreening, 81
antisymmetric tensor, 3, 42, 129, 152
Argand plot, 548
astrophysics, xi
and neutrinos, 404, 447
Solar i/, 430-435
asymmetric B factories, 392
asymmetries
CP, 388, 394
Z-pole, 340, 346-350
Born, 346
decay, 254, 272, 274, 279
forward-backward, 321, 325, 327,
328, 346, 450
mixed, 348
polarization, 321, 322, 325, 348
asymptotic freedom, 81, 184, 185, 189,
199, 200
ATLAS, 568
atomic parity violation (APV), 304,
321-323
axial baryon number (U(1)a), 131,
187, 222, 223
problem, 229-230
axion, 422, 458, 462, 506
azimuthal angle, 19
B meson, 219, 245
B° - B° mixing, 277, 391-394
oscillations, 393
Bis, 392
CP violation, 394-397
standard model, 395
time-dependent asymmetry, 394
Amd)S, 393-394
decay
B°d -> 7T7T, 396
B°d -> JI^K_s,l, 396
charmless (b —> sqq), 397
tagging, 393
BaBar, 375, 391, 392, 396
Baker-Campbell-Hausdorff
construction, 104, 149, 479
635

636
The Standard Model and Beyond
Baksan, 433
baryogenesis, 71, 301, 460
CPT violation, 461
Affleck-Dine, 461
electroweak, 460, 507, 519
GUT, 533
Sakharov conditions, 460, 533
baryogenesis, see also neutrino: lep-
togenesis
baryon number, 121, 187, 222, 283
violation, 222, 402, 505, 530
neutron oscillations, 404, 505
non-perturbative, 222, 460, 534
baryon number, see also nucleon:
proton decay, baryogenesis
baryon/hyperon, 122
5£/(3), 122
decays, 241
nonleptonic, 99, 150
octet, 122, 128
quark model, 128, 185, 189
Bates, 203, 321
BBN, see cosmology
BEBC, 313
Becquerel, H., 239
Belle, 276, 375, 377, 391, 392, 396
beta beam, 447
0 decay, see nucleon: /? decay
/? function, see renormalization group
equations
/?/?0i/, see neutrino: double beta
decay
Bethe, H. A., 74
beyond the standard model (BSM),
462-537
and neutrinos, 404
new ingredients, 460-462
top down vs bottom up, 462, 508
Bhabha scattering, 54, 99, 195, 325
bilinear form, 5, 48-50
CP, 67
adjoint, 48
charge conjugation, 65
Fierz identity, 49
space reflection, 60
spinor notation, 473
time reversal, 70
two-component, 72-73
Bjorken scaling, see deep inelastic
scattering
Bohr magneton, 78
bootstrap theory, 188
Borexino, 434
Bose-Einstein statistics, 8, 117, 160
bottom (b) quark, 183
SU(2) x (7(1) assignment, 326
Boulder, 321
box normalization, 7
brane world, 455
Breit frame, 17, 93, 210
Breit-Wigner resonance, 90, 100, 326,
341, 550, 559-561
brick wall frame, see Breit frame
Brookhaven, 236, 239, 244, 254, 387,
436
bubble chamber, 202
C, see charge conjugation
C, see charge conjugation: Dirac
matrix
Cabibbo rotation, see quark mixing
Cabibbo-Kobayashi-Maskawa (CKM)
matrix, see quark mixing
Callan-Gross relation, 188, 210, 315
canonical commutation rules, 10, 69,
115, 149, 541-544
in quantum mechanics, 69
Cartan classification, 108, 110
cascade decay, 399, 501, 504, 539,
550
Casimir invariant, 109
quadratic, 110, 111, 199
CCFR, 319
CDF, 218, 321, 328, 340, 348, 350,
361, 370, 376, 391, 392, 519
CDHS, 319
CDM, see cosmology: dark matter
center of mass, 15, 19, 567
CERN, 84, 203, 244, 291, 304, 312,
321, 350, 375, 388, 436

Index
637
CESR, 327, 391
channel (s, t,u), 15
charge conjugation, 58, 63-67, 132
Dirac matrix, 64
electromagnetic current, 65
form factors, 564
two-component, 71, 474
violation, 58, 92, 100, 245
charged current, 249
neutral current, 303
charge conservation, see Noether
theorem
charge quantization, 454, 526
CHARM, 319
charm (c) quark, 126, 183, 248
mass prediction, 244, 378, 381
Cherenkov detector, 433
chiral fermion, 130
chiral perturbation theory, 220, 236,
266, 268, 372, 385, 391
effective theory, 306
chiral projection operators, 44
chiral representation, 41, 97
37 CI experiment, see Homestake
Chooz, 436
chromoelectric dipole moment, 399
Cijk, see structure constants
classical electrodynamics, 74, 157, 220
classical mechanics, 103
Clebsch-Gordan coefficients, 124
5£/(3), 125
CLEO, 276, 327, 375, 391
CM, see center of mass
CMB, see cosmology
CMS, 568
CNGS, 436
coherent state, see field: classical
Coleman's theorem, 143
Coleman-Mandula theorem, 475
collinear frames, 19
color, see quantum chromodynamics
combinatoric factors, 11
complex scalar field, see field:
complex
composite Higgs, 457
composite operator
meson, 224
vacuum condensate, 224, 371, 492
compositeness, 357, 374, 455
Compton scattering, 55, 197
computational methods, 193, 236
confinement, see quantum
chromodynamics
conjugate momentum, 10, 541
constituent mass, see quark: masses
conventions, 1
Cornell, 245
cosmic strings, 137, 179
cosmological constant, see gravity
cosmology, xi, 136
acceleration, 461
and neutrinos, 404, 419, 422
baryon density, 420
dark energy, xi, 461
dark matter, xi, 461, 463, 467,
505
cold (CDM), hot, warm, 422
domains, 137
energy density, 137
closure (fi), 422, 461
galaxies and clusters, 461
Hubble parameter, 419
inflation, 137
isotropy, 137
microwave radiation (CMB), 420,
421, 461
nucleosynthesis (BBN), 419-420,
461
4He abundance, 420
quark gluon plasma, 236
quintessence, 461
reheating, 137
relic particles, 237, 421, 452
cosmology, see also axion, baryoge-
nesis, gravity, inflation, su-
persymmetry: LSP, WIMP
Coulomb gauge, 30
covariant normalization, 7, 19, 101,
383
CP, 58, 67-68, 100

638
The Standard Model and Beyond
strong, 187, 230, 372, 400, 457
two-component, 71, 474
violation, 58, 67, 68, 241, 245,
250
B system, 394-397
KL decay, 380, 387
M° - M° mixing, 277
baryogenesis, 301, 372, 461
beyond standard model, 372,
377, 455
CKM matrix, 300, 371
direct, 387, 391
extended Higgs, 371, 372
indirect, 387, 391
PMNS matrix, 414, 427
spontaneous, 458
strong phase, 395, 398
supersymmetry, 468
superweak model, 372, 389
weak phase, 395, 398
CP, see also quark mixing, kaon, B
meson
CPLEAR, 398
CPT theorem, 58, 68-71, 83, 391
two-component, 72, 474
violation, 71, 379, 461
creation operator, 7-9
spin-0, 7
spin-1, 8
Cronin, J., 387
cross section, 18-21, 550
total, elastic, inelastic, 547
crossing symmetry, 38, 51, 188, 563
CTEQ, 217
Cuoricino, 423
current
V + A, 255, 515, 520
V-A,66
axial, 49, 61, 62, 178, 223, 564
(3 decay, 270
pseudoscalar meson, 224, 258
charged, 245-250, 298
mass basis, 300
conserved vector current (CVC),
261, 263, 265, 270
electromagnetic, 50, 65, 87, 92,
100, 206, 302
hadronic, 245
and SU{S)L x SU{3)r
generators, 261, 263
leptonic, 245
neutral, 303
partially conserved axial current
(PCAC), 227, 234, 261
pseudoscalar, 49, 61, 120
scalar, 49, 61
second class, 566
spinor notation, 473
tensor, 49, 61
vector, 49, 61, 62, 65, 223, 564
(3 decay, 269
K£3,Ke3, 262
current algebra, 220, 223, 266-268,
270-271
current mass, see quark: masses
CVC, see current
D meson, 276
D° - D° mixing, 277, 392
DO, 218, 321, 328, 340, 350, 370, 376,
391, 519
Dalitz plot, 397, 553
dark energy, see cosmology
dark matter, see cosmology
Davis, R., 432
Daya Bay, 447
decay
outside detector, 505, 506
rate, 23-24, 550
deconstruction, 457
decoupling theories, 357
decuplet, 125
deep inelastic scattering, 199, 202-
217
fiN, 189
i/N, 189, 203, 207, 212, 313-321
eN, 185, 203-217, 312
charged current, 313-316
(5/18)*h rule, 315
c threshold, 315

Index
639
total cross section, 316
CKM matrix, 312, 316
electroweak theory, 203, 312
hadronic/leptonic tensors, 206,
313
kinematics, 203-204
neutral current, 312
sin2 6W , 317, 319
nuclear/neutron target, 202, 212,
213, 313
isoscalar, 313
NuTeV result, 320
opposite sign dimuons, 316
Paschos-Wolfenstein ratio, 319
polarized beam/target, 203, 207,
321
QCD, 203, 312
quark model, 185, 188, 203, 209-
213, 312, 318
scaling, 208, 314
QCD corrections, 209, 213-217,
316, 329
small x, 217
structure functions, 208, 216
i/N, 314
quark model, 209, 314
transverse momentum, 217
WNC/WCC ratio, 318
isospin, 319
DELPHI, 340, 375
S(x), see Dirac delta function
5ij, see Kronecker delta function
AI = | rule, 150
ArriK, see kaon: rriKL —™>ks
A resonance, 150
^ahadi see SauSe coupling constant:
QED
density matrix, 386
DESY, 84, 189, 203, 215, 312, 324,
350
DGLAP equations, 215, 218, 219
dijk, see symmetry: SU(3)
dilepton, 505
dimensional regularization, see regu-
larization
dimensional transmutation, 202
diquark, 402, 505, 517, 529
Dirac adjoint, 39
Dirac delta function, 4
Dirac equation, 74
Dirac field, see field: Dirac
Dirac matrices, 40-43
Dirac spinor, see spinor: Dirac
direct product, 103, 107, 108, 111,
129
5£/(3), 123
DIS, see deep inelastic scattering
dispersion relation, 81, 85, 220
displaced vertex, 277, 505
divergence
infrared, 36, 76, 191
chiral symmetry, 268
linear, 178
logarithmic, 79
mass singularity, 191
quadratic, 80, 455, 463
ultraviolet, 24, 74, 76, 78
new physics scale, 25, 79, 456
zero-point energy, 542
domain wall, 134, 136-137, 153, 507
DONUT, 240
DORIS, 324
Double Chooz, 447
down (d) quark, 127, 183
Drell-Yan process, 192, 218, 504, 518
asymmetries, 328
dynamical mass, see quark: constituent
mass
dynamical symmetry breaking (DSB),
xii, 358, 362, 367, 371, 457
Dynkin diagram, 108, 111
Dynkin index, 105, 110, 111, 163, 200
Dyson, F. J., 74
early universe, see cosmology
EDM, see electric dipole moment
effective potential, 135, 145, 365
standard model, 288

640
The Standard Model and Beyond
effective theory, 11, 142, 232, 304-
306
eight-fold way, 106, 121
electric charge
conservation, 89, 207
in standard model, 288
generator, 121
nonconservation, 84, 266-267, 534
extended Higgs, 288, 371, 448,
449, 496
supersymmetry, 495
nonrenormalization, 89, 92
quark, 127
electric dipole moment, 40, 92, 100,
399-401, 462
experimental limits, 399
neutron, 188, 458
new physics, 400
Schiff theorem, 400
standard model, 400
supersymmetry, 400, 468, 505
time reversal, 270, 565
transition, 102
electric field, 30
electromagnetic field, see field: gauge
elect roweak interaction, xii, 11, 105,
164
SPTVA analysis, 245, 254, 260,
272
SU(2) x £7(1), 243, 281-308
W°,B, 281
W±,Z,A, 289
breaking, 281, 337
custodial symmetry, 290, 354,
371
FCNC, 244
Fermi theory and QED, 281,
298, 301
fermion, 282-284
fermion masses, 295
Higgs, 284
Higgs mechanism, 243, 287-
289
neutrino, 404
renormalizable, 243
uniqueness, 326, 329
V-A, 241, 245, 248
7T£2 decay, 260
1-Z,W-Z interference, 304,
308, 310, 321-328
charged current, 239-241, 245-
250, 298-301
AS = 0, 246
AS = 1, 246
AS = AQ rule, 246
c quark, 248
charge retention form, 250, 256
quarks, 246
discrete symmetry breaking, 58,
92, 130, 245, 249, 298, 303
energy spectrum, 277
Fermi theory, 239, 240, 245-250
processes, 241
unitarity violation, 241
form factor corrections, 92
gauge self-interactions, 350, 358-
361
gauge symmetries, 105, 130, 243
global analysis, 328, 332, 351,
356 •
hadronic uncertainties, 82, 330
HERA, 350
intermediate vector boson
theory, 242
isospin breaking, 119
LEP 2, 350
leptonic, 247, 250-258, 275
neutral current, 203, 302-304, SOS-
SIS, 316-329
5, P, T, 308, 321
sm29w, 291, 303
confusion theorem, 321
discovery, 244, 304
effective four-fermi, 304
four-fermi parameters, 309, 311,
312, 317, 318, 320, 327
implications, 328
Majorana i/, 452
model-independent analyses, 308
329

Index
641
observables, 351
standard model test, 308
non-leptonic, 99, 150, 247
AI = ± /octet rules, 248
AI = |/octet rules, 389
precision tests, 330-350
beyond standard model, 353
implications, 350-353
input parameters, 330
scale (i/), 290, 299, 330
semi-hadronic, 247, 275
semi-leptonic, 247, 258-274
Tevatron, 350, 359
weak interaction history, 239-245
electroweak interaction, see also Higgs
boson, neutrino, quark
mixing, lepton mixing
electroweak phase transition, 460, 507
ep elastic scattering, 90-94, 204-207
e~e+ annihilation, 53-54, 56-58, 73,
99, 191, 340
W~W+, 243, 338
triple gauge, 350, 359
Z-pole, 340
color, 190
gluons, 189
neutral current, 304, 321, 324-
328
polarization, 340
quark model, 189
thresholds, 191
e~e+ annihilation, see also Z boson
^ivpa ^ see antisymmetric tensor
Cijk, see Levi-Civita tensor
equivalence theorem, 367, 451
77, 122, 132
7] - rjf mixing, 229, 231
7} -> 3tt, 227
r] problem, see axial baryon number
r/, 122, 128, 223
Euler-Lagrange equation, 10, 115, 478
Euler-Mascheroni constant, 331
exceptional group, see symmetry
exclusive process, 185, 277
exotic fermions, 255, 273, 310, 329,
374, 376, 449, 514
quasi-chiral, 516, 536
extra dimensions, xii, 371, 455, 504
large, 457
universal, 457, 462
ATAT-parity, 457
warped, 457
F and D couplings
hyperon decay, 273
meson-baryon, 122
factorization theorems, 199, 218
false vacuum, see potential: local
minimum
familon, 419
family
fourth, 357, 374
mirror, 327, 354, 357
sequential, 326, 354
topless, 327
vector, 327, 354
family, see also electroweak
interaction: SU(2)xU(l),
symmetry
FCNC, see flavor changing neutral
currents
Fermi constant, 2, 85, 240, 330
\x decay, 253
from SU(2) x £/(l), 290, 299
Fermi, E., 239
Fermi-Dirac statistics, 8
Fermilab, 240, 244, 304, 312, 319,
321, 375, 388, 391, 436
fermion mass (matrix), 406
diagonalization by Al, Ar, 295-
298, 411
Dirac, 74, 296, 413
active and sterile i/, 406
as conserved L, 416
as two Majorana, 411, 423, 424
as two Weyl, 406
variant i/, 406, 416, 417
Majorana, 296, 407, 411, 413,
423

642
The Standard Model and Beyond
active i/, 407
dipole moments, 424
sterile i/, 408
symmetric matrix, 411
mixed, 410, 412, 415
Dirac limit, 411
pseudo-Dirac, 412, 451
seesaw limit, 412, 415
non-Hermitian, 95, 119, 131, 154,
170, 285, 295, 449
non-square, 285
parameter counting, 300, 414
phases
CP-violating, 300
conventions, 297, 409
Majorana, 410, 414, 423
strong CP, 458
unobservable, 297, 300
spinor notation, 473, 475
fermion mass (matrix), see also
neutrino, quark mixing, lepton
mixing, textures
fermion number, 121, 131
ferromagnet, 114
ferromagnetic
domains, 137, 139
Feynman diagram, 7, 173
amplitude calculation, 56-58, 197-
198, 264
two-component, 73
derivation, 545
Feynman gauge, see 't Hooft-Feynman
gauge
Feynman integrals, 13, 83, 555
Feynman parametrization, 556
Feynman propagator, see propagator
Feynman rules, 12
electroweak standard model, 286,
292-294, 299
gauge
abelian, 158, 159
ghost, 171
non-abelian, 161, 162, 195
Hermitian scalar field, 12-15
pion electrodynamics, 35, 36
QED, 50, 556
signs for fermions, 51
Feynman, R. P., 74
field
chiral, 46-48, 73, 130, 179
classical, 133
complex scalar, 9, 25-29, 33, 97,
146-149, 542
Rt gauge, 174-177
free field, 27
Hermitian basis, 26, 118-119,
143, 146, 166
Lagrangian, 25
symmetry breaking, 137-141
Dirac, 39-41, 98, 406, 543
angular momentum, 98
electron, 76
free field, 40
Lagrangian, 39
spinor notation, 473
gauge, 30, 33, 61, 108, 157, 544
abelian, 158
free field, 30
Lagrangian, 30
mass, 33, 164-166, 168, 170,
181
non-abelian, 160, 163
ghost, 32, 171, 173, 197
standard model, 306
Hermitian scalar, 8-15, 143, 166,
169, 542
free field, 12
Lagrangian, 10
symmetry breaking, 133-137
Higgs, 284
Majorana, 41, 64, 70, 355, 407,
538
vl — vcR asymmetry, 407
free field, 409
Lagrangian, 407
spinor notation, 474
mass eigenstate, 95, 116, 144,
246, 282
massive vector, 32-33
free field, 32

Index
643
Lagrangian, 32
matrix notation, 112, 120, 123,
149, 187, 233, 528
symmetry transformation, 112,
114
antiparticle, 113
complex scalar, 115-117
Dirac, 119
Hermitian scalar, 117
unphysical scalar, 173, 175
weak eigenstate, 246, 282
Weyl, 41, 47, 71, 405
free field, 48
independent, 73, 179
Lagrangian, 47
spinor notation, 471-475
field redefinition, 11, 67, 68, 94, 141-
142, 153, 154, 165, 166, 371
soft pions, 235
strong CP, 458
field strength tensor, 30, 158
dual, 188, 229, 458
non-abelian, 163
QCD, 186
standard model, 282
field theory, 7, 103
finite temperature, 136, 220, 236
Lagrangian, 9
Fierz identity
SU{m), 111
SU{2), 5, 58
and charge conjugation, 66
Dirac, 49-50, 98, 250
spinor notation, 474
two-component, 72
fijk, see symmetry: SU(3)
fine structure constant, see gauge
coupling constant: QED
finite energy sum rules, 209
Fitch, V., 387
/k, see kaon: decay constant
flavor, see symmetry
flavor changing neutral currents, 244,
246, 377, 401-404, 462
imparity violation, 505
/x —> 3e, fi —> e7, /i —> e, 401,
402, 446
t decays, 404
Z' mediated, 350, 397, 401, 519
3 quark model, 303, 381
constituent interchange, 402
exotic fermion mixing, 304, 350,
401, 449
extended technicolor, 402
family symmetry, 510
Higgs mediated, 298, 371, 397,
402
higher-dimensional operator, 403
Kaluza-Klein mediated, 401
lepton flavor, 102, 275
leptoquark/diquark mediated, 402,
517
loop induced, 401, 402
nonstandard third family, 327
rare decays, 402
supersymmetry, 468, 502
flavor changing neutral currents, see
also GIM mechanism
form factor, 62, 77, 87-94
i/, 424-425
isoscalar, 93
isovector, 93
Ademollo-Gatto theorem, 265, 267-
268, 270
chiral limit, 268
Adler-Weisberger relation, 270-
271
discrete symmetries, 361, 563-
566
electric, 93
electric dipole moment, 399
induced pseudoscalar, 225
pion pole, 226, 270
neutron, 93-94
pion, 87-90
proton, 90-93, 207
Sachs, 93
triple gauge, 360
weak, 220, 226
/? decay, 269-271

644
The Standard Model and Beyond
7r£3,K£3, 262,265
hyperon decay, 273
four-component notation, 47, 405
four-fermi interaction, 11
four-vector, see vectors
/tt, see pion: decay constant
fragmentation function, see hadroniza-
tion
Prascati, 375
Proggatt-Nielsen mechanism, see
symmetry: family
Proissart bound, 548
fundamental, see representation
^-factor, 78
G-parity, see symmetry
GALLEX, 433
gallium experiments, see GALLEX,
GNO, SAGE
75, see Dirac matrices
7M, see Dirac matrices
GAPP, 332, 351
Gargamelle bubble chamber, 304
gauge coupling constant, 157
electroweak {g,g',gz), 281, 330
GUT normalized (#i), 363
QCD, 183, 213, 330
experimental determination, 201,
352
running, 183, 184, 191, 237
QED, 2, 35, 76, 82
from SU{2) x £7(1), 302
running, 78-82, 184, 190, 330,
337, 351
rescaling, 159
running, 77, 198-202
screening/anti-screening, 183
unification, 329, 358, 463, 507,
526, 533, 534
gauge coupling constant, see also weak
angle
gauge covariant derivative, 34, 158,
159, 180, 181, 187
non-abelian, 160
standard model, 284
gauge field, see field: gauge
gauge invariance, 30, 34, 84, 158, 173
non-abelian, 161
gauge self-interactions, see field: gauge
gauge theory, 157-158
abelian, 30-32
supersymmetry, 485-488
non-abelian, xi, 108
asymptotic freedom, 199
supersymmetry, 488-490
gauge transformation
abelian, 30
classical electrodynamics, 157
non-abelian, 161, 181
infinitesimal, 163, 181
spin-0, 34
spin-^, 50
Gaussian integral, 5
Gell-Mann matrices, 106
Gell-Mann, M., 121
Gell-Mann-Nishijima relation, 121
Gell-Mann-Okubo
ansatz, 124, 128
relation, 125, 152, 230, 238
generator, see group: generator
Georgi-Glashow model, see grand
unification: SU{5)
Gp, see Fermi constant
ghost, see field
GIM mechanism, 244, 303, 401
Glashow, S., 243
Glashow-Iliopoulos-Maiani mechanism,
see GIM mechanism
glueball, see gluon: gluonium
gluon, 183, 186-188
evidence, 188-189
fictitious mass, 191
gluonium, 220, 231
sea, 210
soft, 190
GNO, 433
Goldberger-Treiman (GT) relation, 226,
236
Goldstone alternative, 143, 226, 234

Index
645
Goldstone boson, see Nambu-Goldstone
boson
Gordon identities, 77, 97, 100
Gran Sasso, 423, 433, 434, 436
grand unification, 73, 329, 454, 455,
525-537
£6, 536
exotic fermions, 536
trinification, 536
50(10), 286, 535
v masses, 536
extended Higgs sector, 536
fermion mass relations, 536
Yukawa unification, 536
5£/(5), 287, 526-534
breaking patterns, 530
doublet-triplet problem, 531
fermions, 528
gauge fields, 527
gauge interactions, 529
Higgs, 529
monopole problem, 534
Yukawa couplings, 531
q — £ mass relations, 532
flipped 5£/(5), 535
GUT hierarchy problem, 456, 531
GUT scale, 531
orbifold, 537
Pati-Salam model, 525, 535, 537
supersymmetric, 329, 507, 534
gauge unification, 534
proton decay (d = 4, 5,6), 535
grand unification, see also gauge
coupling constant, nucleon:
proton decay
gravity, xi, 178
cosmological constant, 135, 292,
459, 461, 542
general relativity, 458
quantization, 458
trace anomaly, 178
unification, 459, 463
ground state, see vacuum
group, 103-105
abelian, 103, 104
compact, 104, 105, 108, 470
continuous, 104
discrete, 104
finite, 103
generator, 104, 112, 157
fermionic, 475
Noether charge, 115, 157
Lie, 103-112, 149
non-abelian, 103, 104
rank, 106, 110
simple, 108
subgroup, 103, 108
diagonal, 223
unbroken, 116, 144
9v,A, see nucleon: /? decay
gz, see gauge coupling constant
hadron, 75, 87, 88, 91, 93, 185, 220
spectroscopy, 236
hadron collider
short distance processes, 217-219
transverse momentum, 218
hadronic vacuum polarization, 85
hadronization, 185, 189, 190, 199, 219
cluster, 219
color flux tube, 219
fragmentation function, 219
jets, 189, 219, 504
string fragmentation, 219
Hahn, O., 239
Hamiltonian, 10, 97, 107, 541
heavy ion collisions, 236
heavy quark
decay constants, 276
decays, 241, 275-277
parton model, 275
spectator model, 276
topological diagrams, 276
heavy quark effective theory (HQET),
219, 277, 306, 372
Heidelberg-Moscow, 423
Heisenberg picture, 10, 87
helicity, 9, 31, 56, 98
negative, 44
positive, 44

646
The Standard Model and Beyond
pseudoscalar, 59
wrong, 98
HERA, 202, 203, 215, 217, 312
Hermitian scalar field, see field: Her-
mitian
Higgs boson, 58, 118, 245, 288, 362-
371
interactions, 108
decays, 368-370
W+W~,ZZ, 368
77, 369
66,r-r+,//, 297, 368
doublet, 284
Hermitian basis, 287
extended models, 176, 255, 288,
329, 353, 354, 370-371, 447-
449
CP violation, 371
baryogenesis, 461
charged, 371
doublets, 370
doubly charged, 371, 522
FCNC, 297
pseudoscalar, 371
singlets, 371, 507
triplets, 371, 521
interactions, 292
mass, 80, 85, 330
experimental limits, 292, 368-
370
precision constraints, 349, 352
quadratic divergence, 455
supersymmetry, 352, 366, 463
production
associated (tiH), 368
associated (Z -> Ah), 371, 499,
538
gluon fusion (GG -> H), 368
Higgstrahlung (Z -> ZH, W ->
WH), 292, 368, 500, 538
vector boson fusion (VV —>
H), 368
strong coupling, 367
theoretical constraints, 292, 362-
368
tree unitarity, 366-367
triviality, 363-364
vacuum stability, 364-366
weak coupling, 364, 367
width, 369
Higgs boson, see also potential,
equivalence theorem,
supersymmetry
Higgs mechanism, see symmetry
breaking: spontaneous
Higgsless models, 457
higher twist, 217
higher-dimensional operator, 11, 146,
151, 329, 353
higher-dimensional operator, see also
effective theory,
supersymmetry
higher-order diagrams, see loop
diagrams, radiative corrections
holomorphic, 480
Homestake, 432, 433
horizontal symmetry, see symmetry:
family
HPW, 304
hyperon, see baryon
IceCube, 447
ideal (magic) mixing, see p — u — <j>
mixing
identical particles, 19
IHEP, 375
1MB, 421, 435
in and out states, 69, 398
inclusive process, 185, 277
inelasticity, 548
infinite momentum frame, 209
infinitesimal transformation, 104, 144
inflation, 461, 534
eternal, 460
infrared slavery, 184, 185, 189
inner product, 109, 168
instanton, see tunneling
interaction picture, 62, 87
International Linear Collider, xi, 370
GigaZ option, 358

Index
647
invariance, see symmetry
invariant subgroup, 108
irreducible tensor operator, 103, 121,
124
IRREP, see representation: irreducible
isoscalar factors, 125, 150
isospin, see symmetry: isospin
Jacobi identity, 149
JADE, 326
Jefferson Lab, 203, 321, 334
jets, see hadronization
J/ip{cc), 191, 244, 276
J/^->e+e", 280
1)' -> ip7r°, 227
K2K, 436
Kaluza-Klein excitations, 510, 515
Kaluza-Klein theory, 458
Kamiokande, 421, 433, 435
Kamland, 434, 436, 437
kaon
CP violation, 380, 387-391
observations, 388
parameters, 387
standard model, 388-391
superweak phase, 389
K° - K° mixing, 244, 262, 277,
378-391
FCNC, 378
oscillations, 385-387
tf°2, KStL, 379
mK+ -mKo, 227, 262, 379
m^L.— tuks ? 380-385
5£/(3), 122
decay, 241
Ks,l, 380, 385-387
K&, 262
Ki3, 262, 265-266, 279, 380
non-leptonic, 248, 249, 380
decay constant, 262, 381
/*//*, 261
properties, 259, 262
regeneration, 381, 386
Wu-Yang convention, 379
KARMEN2, 437
KATRIN, 421
KEK, 84, 245, 324, 391, 436
Kibble transformation, 166, 167, 367
standard model, 288
kinematic edge, 504, 539
kinematics
collider, 567-571
relativistic, 2-3, 15-17, 549
kinetic energy, 142
non-abelian, 163
non-canonical, 94, 154, 167
Klein-Gordon operator, 11, 12
Klein-Nishina formula, 55
^l,s? see kaon: K° — K° mixing
Kronecker delta function, 2
KTEV, 388
L field, see field: chiral
L3, 340
laboratory frame, 17, 21, 97, 203
Lagrangian, 7, 10
5£/(5), 529, 531
V-A, 245
\x decay, 250
v in matter, 428
vee~ —> vee~', 256
a model, 233
SU{2)L x SU{2)R x £7(1), 520,
523
U{1)', 511, 518
chiral, 130, 143
complex scalar, 25
Dirac, 39
electric dipole moment, 399
Fermi theory, 240, 246
gauge, 30, 157
R^ 169, 174, 176
SU{2), 181
abelian, 159, 160
Hermitian scalar, 181
Higgs mechanism, 165, 168
non-abelian, 163
ghost, 171, 174
Hermitian scalar, 10

648
The Standard Model and Beyond
massive vector, 32
meson-baryon, 122, 124
MSSM, 498, 501-503
soft breaking, 494
neutral current
i/N, 317
i/e, 309
eN, 322
pion electrodynamics, 33, 158
pion-nucleon, 120, 130
QCD, 186
flavor, 221
QED, 50, 158
spinor notation, 473
quark, 128
standard model
W, Z decay widths, 339
charged current, 298
effective couplings, 346
fermion, 284
gauge, 282
Higgs, 284
neutral current, 302
QED, 301
Yukawa, 285
standard model with SSB, 453
Rt gauge, 306-308, 382
fermion, 297
gauge, 291, 293
Higgs, 291
unitary gauge, 289-304
Yukawa, 295, 297
supersymmetry, 482-485
gauge, 486-488
Lagrangian density, see Lagrangian
A, see quantum chromodynamics: scale
parameter
A, see potential: Higgs
A, see Gell-Mann matrices
Landau gauge, see renormalizable gauge
Landau pole, 357, 364
landscape, xi
lattice techniques, 88, 91
Higgs, 364
QCD, 220
weak interactions, 248, 266, 276,
372, 385, 391
Lee, T.D., 249
left-handed, see helicity: negative
left-right (LR) symmetry, see SU(2)Lx
SU(2)R x £7(1)
LEP, 84, 244, 254, 292, 324, 328, 340,
368, 391
LEP 2, 84, 321, 324, 338, 350, 352,
359, 361, 368, 371, 375, 502
LEPEWWG, 343, 351
lepton
e — \x — r universality, 260, 275,
340, 345
lepton flavor conservation, 296, 401
lepton mixing, 241
mixing matrix (PMNS), 300, 413
CP violation, 414, 427
013, 427, 436
approximate form, 445
bimaximal, tri-bimaximal
mixing, 445
Major ana phases, 410, 414
non-square, 416
PDG parametrization, 414
lepton mixing, see also fermion mass
(mixing), neutrino
lepton number, 283, 406
as fourth color, 526
neutrino masses, 404
violation, 255, 402, 407, 422, 505,
530
lepton number, see also fermion mass
(mixing), neutrino
leptoquark, 273, 329, 350, 353, 374,
423, 505, 517, 529
Levi-Civita tensor, 2
LHC, xi, 218, 236, 368, 392, 404, 502,
519, 568
LHCb, 397
Lie algebra, 103-112, 149, 157
classical, 108, 111
exceptional, 108
Lie group, see group
light by light diagrams, 85

Index
649
little hierarchy problem, 457
Little Higgs, 358, 456, 462, 504
T-parity, 457
local (gauge) supersymmetry, see su-
pergravity
local symmetry, see symmetry: gauge
local transformation, see gauge
transformation
loop diagrams, 24, 169, 183, 193
M°-M° mixing, 244, 379, 381-
385, 392, 402
CP violation, 391
SU(2) x £7(1), 241
anomalous magnetic moment, 78,
83-87, 555
Fermi theory, 240
Higgs mass, 463
LO, NLO, NLLO, 215, 217, 218
penguin, 276, 384, 391, 396, 397
QED, 74-78
renormalization, 76
running coupling, 78
supersymmetry, 86, 397
loop diagrams, see also radiative
corrections, penguin diagram,
see also radiative corrections
Lorentz boost, 3, 6, 97, 470, 538, 567
Lorentz gauge, 30
Lorentz invariance, see symmetry
Lorentz invariant phase space, see phase
space
Los Alamos, 310, 437
LSND, 310, 412, 415, 437
implications, 439
LSP, see supersymmetry
luminosity, 190
M, see transition amplitude
MACRO, 435
magnetic dipole moment, 40, 74, 78
anomalous, 78
//*, 84-87, 254
i/, 424
e±, 83
baryon, 92, 93, 128, 152, 207
electroweak, 83, 85
hadronic uncertainties, 82, 83,
85
Schwinger contribution, 78, 555-
558
supersymmetry, 86
transition, 102, 152, 424
magnetic field, 30, 78
Mainz, 203, 321, 421
Majorana fermion, see field: Majo-
rana
Majorana mass, see field: Majorana,
fermion mass (matrix)
Majoron, 419, 422
singlet, 443
triplet, 346, 441
Mandelstam variables, 15-17
MARK II, 352
mass insertions, 95-96
matrices, see operators and matrices
Maxwell's equations, 133, 157
Maxwell, J. C, 74
Meitner, L., 239
Mellin transformation, see parton
distribution function: moments
meson, 122
quark model, 185
metastable, see potential: local
minimum
metric, 2-3
MiniBooNE, 437
minimal electromagnetic substitution,
34, 50
classical and quantum
mechanics, 34
minimal flavor violation (MFV), 403
MS, see renormalization: minimal
subtraction scheme
minimum bias events, 567
missing energy, 504, 570
MIT bag model, 220
MIT-SLAC experiments, 208, 212
mixing
kinetic, 94-96, 513
mass, 94-96, 511, 522

650
The Standard Model and Beyond
M0ller scattering (e_e_), 99, 321, 328
momentum operator, 541
monopoles, 137, 221, 454, 534
Mott scattering, 99
Mp, see Planck constant
MRST, 217
MSSM, see supersymmetry
MSW effect, see neutrino
muon (/i)
H -> e7, 102
decay (/i~ —> e~veVy), 241, 250-
256
SPTVA analysis, 254
Michel parameters, 255, 278
polarized /i, 254, 279
inverse decay {y^e~ —> ii~ve),
241, 247
properties, 250
muon (//), see a/so magnetic dipole
moment
NA48, 388
Nambu-Goldstone boson, 139, 142-
145, 148, 164, 166, 167, 223
Rt gauge, 169-177, 381
bound state, 226
longitudinal vector mode, 145
pseudo, 140, 143
strong interactions, 145, 202,
226, 258
quantum mechanics analog, 575
Nambu-Goldstone theorem, see Nambu-
Goldstone boson
narrow width approximation, 559
TV"1 expansion, 220, 385, 391
Ne'eman, Y., 121
neutrino
vL - i£, 427
vl — vcR asymmetry, 407, 420
i/M(i/M), 239
i/r(Pr), 239, 329
i/e(Pe), 239
active (ordinary, doublet), 405
active-sterile mixing, 412, 415,
439
anarchy models, 444
as a probe, 404
atmospheric, 245, 435
beams, 312, 319
counting, 419-421
CMB, 420
invisible Z width, 340, 346,
347, 419
nucleosynthesis, 419-420
precision electroweak, 419
decay, 418, 424
dipole moments, 424-425
Dirac mass models, 405, 407, 440-
441
Le- LM - Lr, 416
extra dimensions, 441
higher-dimensional operators,
440
loop, 440
non-holomorphic terms, 441
string instantons, 441
Zeldovich-Konopinski-Mahmoud
(ZKM), 416, 444, 451
double beta decay, 407, 410, 414,
418, 422-423, 451
/?/?2„/%,M, 422
alternative mechanisms, 423
heavy i/, 423
nuclear uncertainties, 422
factory, 447
geophysical, 436
heavy, 418, 423
hypothesis, 239
leptogenesis, 372, 443, 534
Majorana mass models, 405, 409,
441-444
imparity violation, 444, 506
54, A4, 445
higher-dimensional operators,
442
loop, 444
seesaw (5O(10)), 443, 536
seesaw {SU(2)L x SU(2)R x
tf(l)), 522
seesaw (extended), 444

Index
651
seesaw (lopsided), 443, 532
seesaw (minimal), 442-443
seesaw (string), 443
seesaw (type II), 442, 536
triplet, 441
mass and mixing, xi, xii, 241,
245, 404-417
/?#>„, 422
q — £ complementarity, 445
Cabibbo haze, 445
cosmological constraints, 422,
452
kinematic constraints, 421
open questions, 446, 460
when negligible, 296, 300, 417
matter (MSW) effect, 427-429
vl vs vcr, 429
adiabatic, 429, 434
level crossing, 429
resonance, 429, 434
mixed mass models, 444
higher-dimensional operators,
444
mirror world, 444
non-orthogonal, 427
observation, 239
oscillations, 425-427, 430-437
L/E dependence, 436
accelerator experiments, 436
appearance, 426
dark side, 426
disappearance, 426
first and second class, 427
general case, 426
long-baseline experiments, 436
oscillation length, 426
reactor experiments, 436
two neutrino, 426
vacuum, 426
relic, 421
detection, 421
scattering, 241
vN, 310
v^e~ —> v^e~, 309-310
vee~ -> i/ce~, 256-258, 279,
310
neutral current, 304, 309-313,
316-321
Solar, 245
pp chain, 430
pp, 7Be, 8B, 432
alternative explanations, 434
astrophysical explanations, 433
astrophysical uncertainties, 432,
433
LMA solution, 434
spectrum, 432
standard Solar model (SSM),
430
spectrum, 438-439
global analysis, 438
normal, inverted, degenerate
hierarchies, 438, 439
sterile (right-handed, singlet), 282,
405, 437, 439
ultra high energy, 447
neutrino, see also deep inelastic
scattering, fermion mass
(matrix), lepton mixing
neutron, see nucleon
neutron star, 236
Noether charge, see Noether current
Noether current, 28, 35, 114, 142,
149, 153, 543
SU(m)L x SU(m)R, 222
(axial) baryon number, 222
chiral, 130
complex scalar, 28, 116
Dirac, 119
divergence, 115
Hermitian scalar, 117
Noether theorem, 27-29, 77, 114-119
non-covariant normalization, 7
non-polynomial interactions, 11
non-relativistic limit, 41, 78, 93, 100
non-relativistic QCD (NRQCD), 219
non-renormalizable operator, see non-
renormalizable theory
non-renormalizable theory, 11, 24, 240

652
The Standard Model and Beyond
effective theory, 305
nonet, see symmetry: SU(3)
normal order, 65, 97, 115, 542
notation, 1-5
NOi/A, 447
NRQM, see quantum mechanics
i/, see electroweak interaction: scale
nuclear (3 decay, 272-273
Fermi transition, 272
Gamow-Teller transition, 272
super allowed, 272
nuclear magneton, 92
nucleon
/? decay (n -> pe~ue), 239-241,
268-273, 279
v mass, 421
energy nonconservation, 239
Fermi integral, 272
inverse (e~p —> ven), 241
weak magnetism, 269, 273
mp — mn, 227
s quark, 203
SU{2), 120
electrodynamics, 90-94, 100
proton decay, 222, 404, 462, 505,
526, 530
experimental limits, 526, 535
spin distribution, 203
number operator, 8, 542
Noether charge, 29
NuMI/MINOS, 436
NuTeV, 320, 333
oblique parameters, see 5, T, U
parameters
octet, see symmetry: SU(3)
0(m), see symmetry
fi, see cosmology: energy density
u resonance, 125, 133
Q-{sss), 125, 128, 189
OPAL, 340
OPERA, 436
operators and matrices
adjoint, 1
anti-commutator, 1, 7-9, 106
commutator, 1, 7-9, 104-106
exponent, 104, 149
Hermitian, 4, 104, 105
orthogonal, 109
special, 105
symplectic, 109
trace, 1
Dirac, 42-43
group generators, 105
transpose, 1
unitary, 105, 107, 109
optical theorem, 547
oscillations, see kaon, B meson,
neutrino
OZI (Zweig) rule, 220, 238
P, see space reflection
Palo Verde, 436
parameter
bare, 76
physical, 76, 77
parastatistics, 190
Paris, 321
parity, see space reflection
parity violating electron scattering,
see asymmetries:
polarization
partial rate asymmetries, 71, 533
partial wave expansion, 548
Particle Data Group, 351
parton distribution function, 210
global analysis, 214, 217, 316
gluon, 212, 217
hadron collider, 218
moments, 217
momentum, 210, 212
non-singlet, 217
quark, 210
singlet, 217
splitting, 213, 215
sum rules, 211
parton model, 17, 188, 199, 202, 208
r decay, 275
deep inelastic scattering, 209-213,
314, 318

Index
653
heavy quark decays, 275
QCD improved, 213, 214
spin-0, 210
spin-±, 210
Pauli matrices, 4, 41, 104, 105
Pauli spinor, see spinor: Pauli
Pauli, W., 239
Pauli-Dirac representation, 41, 97
Pauli-Villars, see regularization
PCAC, see current
PDF, see parton distribution
function
Peccei-Quinn mechanism, see
symmetry
penguin, see loop diagrams
PEP, 84, 321, 324, 391
perturbation theory, 12-15, 185
time dependent, 101
PETRA, 84, 189, 321, 324, 326
phase shift, 389, 548
phase space, 8
n-body, 549
2-body, 20, 24, 549
3-body, 550-553
Kgs, 279
fi, decay, 252, 278
tt*3, 264, 279
energies, 551
factorization, 550
narrow resonance, 551
cylindrical coordinates, 567
phase transition
chiral, 236
deconfinement, 236
4>(ss) resonance, 125, 191
photon (A, 7), see QED, electroweak
interaction
physical constants, 1
PIBETA, 264
pion
5£/(2), 120
as bound state, 88
charge radius, 89
Compton scattering, 38, 55
decay, 241
7T° -> 27, 192
^i-^i^jz), 193
7T£2 helicity suppression, 260
^£2 (7I>2> 7re2), 258~260
tt*3, 262-264, 279
7Te2/7i>2 ratio, 260
decay constant, 193, 202, 225
chiral breaking, 226, 234, 259
measurement, 260
electrodynamics, 33-39, 87-90,
98, 99, 158
field conventions, 26
phase convention, 120, 150, 151
pion-nucleon a term, 237
pion-nucleon interaction, 22, 87,
92, 120, 152, 226
soft pions, 235
properties, 258, 259
pseudoscalar, 61
self-interactions, 87, 120, 151, 158
soft pion theorems, 142, 227, 235,
236
pion beta decay, see pion decay: 71^3
Planck constant, 2, 25, 79
Planck scale, see Planck constant
Pr,l, see chiral projection operators
Poincare invariance, see symmetry
polar angle, 19
polarization, 31, 210
circular, 31
fermion, 43, 56-58, 100
linear, 31
longitudinal, 33, 165, 168
high energy, 33, 243, 366
sum, 31, 33
gluon, 195
transverse, 30, 33
pomeron, 220
potential, 10, 101, 133
SU{5), 530, 531
U{l)f, 513
atomic parity violation, 323, 450
boundedness, 134, 145-146
classical, 134
confinement (linear), 220

654
The Standard Model and Beyond
constant term, 135
Coulomb, 36, 78, 99
effective, 80
extremum, 134
flat direction, 146, 513
four-fermi, 101
Higgs, 285
0(4) symmetry, 287
running coupling, 363
Higgs (with SSB), 291
induced cubic, 135, 136, 140, 168,
170, 174-176, 292
maximum, 140, 147
Mexican hat, 139, 575
minimum, 134, 138, 140, 144,
146-149
global, 136
local, 136, 366
MSSM, 495
saddlepoint, 147
scalar exchange, 101
supergravity, 482
supersymmetry, 475, 476, 482,
487
vector exchange, 101
Yukawa, 22, 102, 220
potential scattering, 21-22
propagator, 7
Breit-Wigner, 326, 341, 559
ghost, 172, 307
gluon, 183
Majorana, 417
mass eigenstates, 95
spin-0, 12, 27
Rt gauge, 172, 307
spin-1, 32, 33, 78
Rt gauge, 171, 307
spin-^, 44
proton, see nucleon
proton decay, see nucleon: proton
decay
pseudo-Goldstone boson, see Nambu-
Goldstone boson
pseudorapidity, 568
pseudoscalar
interactions, 122
octet, 122, 125, 202, 229, 263
dual interpretation, 231
Goldstone bosons, 223
quark model, 128
pseudoscalar decay constant, see pion:
decay constant
pseudoscalar mixing angle, see 77: 77—
7/ mixing
PSI, 255, 264
QCD, see quantum chromodynamics
QED, see quantum electrodynamics
quantum chromodynamics, xii, 81, 106,
119, 183, 186-188
color, 183, 221
average/sum, 194
evidence, 189-193
color factors, 193
color identities, 193-194
confinement, 185, 199, 220
flux tube, 220
evidence, 188-193
parton processes, 193-198, 218,
237, 569
scale parameter, 187, 200, 202
sum rules, 220
quantum electrodynamics, xii, 33, 74-
94, 158
and strong interactions, 75, 87-
94, 206
high energy, 84
Lagrangian, 50
tests, 82-87
quantum mechanics, 7, 74, 101, 103,
107, 120, 121, 545
measurement theory, 386
symmetry breaking, 573-575
quark, 88, 91, 93, 126-129, 133, 183,
185-188
evidence, 188-189
integer charge, 191, 316, 526
masses
mu - ma, 227
mu = 0 and strong CP, 458

Index
655
and pseudoscalar masses, 227
bare, 187, 281
constituent, 131, 202, 224
current, 131, 187, 226, 295
deep inelastic scattering, 212
heavy quarks, 228
light quarks, 228
pole, 228
running, 228
model of hadrons, 126-129, 188,
231
sea, 210
valence, 210
non-observation, 189
quark mixing, 241
Vcd, Vcs, 375
Vtd, Vta, Vtb, 375
Vub, Vcb, 375
Vcb from vN, 316
from B decay, 277
Vud, 374
from (3 decay, 272
Vus, 375
from Kes, 265
Cabibbo rotation, 246, 301
Cabibbo/weak/CKM
universality, 246, 374, 376, 525, 539
CKM matrix, 248, 275, 300, 371-
376
CP violation, 301
global analysis, 397
Jarlskog invariant, 374, 378,
397
PDG parametrization, 373
Wolfenstein parametrization, 373,
397
in SU{2)L x SU{2)R x £7(1), 523
unitarity, 245, 301, 373
unitarity triangle, 374, 376-378
quark mixing, see also fermion mass
(matrix)
quark-hadron duality, 186
Qw, see atomic parity violation
R field, see field: chiral
radiative corrections
AmK, 384
(3 decay, 273, 279
\x decay, 253, 279
finite in Fermi theory, 253
atomic parity violation, 323
neutral current, 338
precision electroweak, 309, 329,
342
MH, 335, 352
Z -> bb vertex, 333, 335
mu 335, 337
box, 335
mixed QCD-EW, 332, 335
oblique, 333, 354
QCD, 335
reduced QED, 333
vertex, 335
weak universality, 376
radiative hierarchies, 455
radiochemical detector, 432, 433
Randall-Sundrum models, see extra
dimensions: warped
rank, see group
rapidity, 6, 218, 469, 568
real scalar field, see field: Hermitian
reduced matrix element, 124, 125
reduction formula (LSZ), 563
reflection, see space reflection, group:
0(m)
Regge theory, 220
regular representation, see
representation: adjoint
regularization
dimensional, 79, 191, 331
Pauli-Villars, 79, 331
relativity, 2-3, 74
renormalizable gauge, 173
renormalizable theory, 11, 24, 157,
164, 178
spontaneously broken gauge, 169
renormalization, xii, 74, 84
constant, 76
counterterm, 331

656
The Standard Model and Beyond
minimal subtraction scheme, 82,
330, 331
on-shell scheme, 82, 330, 331
parameter, 25, 76
QED, 74
scale, 82, 183, 199, 218
wave function, 25, 76, 79, 481
renormalization group equations, 25,
81, 199-202, 363, 508, 533
v mass, 445
fixed points, 450
hadronic uncertainties, 81
mass, Higgs, Yukawa, soft, 363,
468, 481, 495, 532
thresholds, 81, 200, 535
representation, 103-105, 112
adjoint, 105, 106, 108, 111, 112,
149, 180, 510
anti-fundamental, 129
complex, 108
conjugate, 106, 108, 113
defining, 105-109, 111
double-valued, 109
equivalent, 107
fundamental, 107, 109, 121, 129
inequivalent, 107
irreducible, 103, 107
real, 108, 112
reducible, 107
spinor, 109
supersymmetry, 475-476
tensor, 109
tilde {SU{2)), 285, 402, 405
vector, 109, 180
RGE, see renormalization group
equations
RHIC, 203, 236
p resonance, 88, 90, 125, 133, 188,
202
p parameters
po, 354, 371, 447
radiative corrections, 337
p-u-(f> mixing, 227, 231, 238
Riemann Zeta function, 253
right-handed, see helicity: positive
Rosenbluth cross section, 93, 207
rotation, see symmetry: rotational
Rutherford, 437
Rutherford cross section, 99
Rutherford experiment, 188
Rt gauges, 32, 169-177
standard model, 306-308
S matrix, see transition amplitude
S-matrix theory, 220
5, T, U parameters, 352, 354-357
chiral fermions, 355
extensions, 357
non-degenerate multiplets, 354
SAGE, 433
Salam, A., 243
scattering amplitude, see transition
amplitude
Schrodinger equation, 69
Schur's lemma, 109
Schwinger, J., 74
screening, 81
seagull diagram, 35, 38, 161, 164
seesaw, see fermion mass (mixing),
neutrino
self-energy correction, 75, 76, 78, 79,
183
quadratic divergence, 80
self-energy correction, see also
radiative corrections: oblique
<7, see Pauli matrices
a model
linear, 232-234, 238
nonlinear, 235-236
a^v', see Dirac matrices
similarity transformation, 107
simple parton model (SPM), see par-
ton model
single-particle state, 8
singularity, see divergence
sin2 9w, see weak angle
Skyrme model, 220
SLAC, 84, 188, 199, 203, 244, 245,
321, 322, 324, 328
SLC, 84, 244, 324, 328, 340, 352, 391

Index
657
SLD, 340
SM, see standard model, see
standard model
sM, see spin four-vector
SNO, 433, 435
soft collinear effective theory (SCET),
219, 306, 372
soft pion theorems, see pion, 306
soft radiation, 76
solid angle, 20
SO{m), see symmetry
Soudan, 435, 436
Sp(2m), see symmetry
space reflection, 56, 58-63, 77, 91,
120, 122, 153, 198, 207
antifermion, 60, 71
axial vector, 58, 279
tt^2 K12 decay, 258
form factors, 62, 564
pseudoscalar, 59, 61, 120
scalar, 58
two-component, 71, 474
vector, 58
violation, 58, 92, 100, 207, 241,
245, 398
NN interaction, 248
/? decay, 249
/i/r polarization, 84, 254, 279
r — 0 puzzle, 249
0(a), 221
charged current, 249
deep inelastic scattering, 313
neutral current, 258, 303
SPEAR, 324
sphaleron, 460, 534
spherical coordinates, 19
spin and statistics theorem, 128, 189
spin average/sum, 31, 53
spin effects, 56-58, 100, 189, 198
(3 decay, 272
/i decay, 254
t decay, 450
form factors, 93
neutral current, 304
supersymmetry, 504
two-component, 73
spin four-vector, 43, 59
spinor
chiral, 47, 98
Dirac, 41, 43-48, 56-58
helicity, 44, 46, 57, 98
Pauli, 5, 44
spinor, see also field: Weyl
splitting functions, see parton
distribution function
SppS, 340
spurion, 150, 506
SSB, see symmetry breaking:
spontaneous
SSM, see neutrino: Solar
standard model, xi, 108, 135, 179,
281-308
and neutrinos, 404
anomaly free, 283
problems, 453-462
fermion, 454
gauge symmetry, 454
gravity, 458
Higgs/hierarchy, 455-457, 463
parameters, 454
strong CP, 457
standard model, see also QCD, QED,
electroweak interaction
statistical factor, 19
step function, 5
strange (s) quark, 127, 183
strangeness, 121, 246, 379
violation, 379
string theory, see superstring theory
strong CP problem, see CP: strong
strong hypercharge, 121, 127, 227
strong interaction, 87-94, 106, 183-
184
long distance, 183-185, 199, 220-
221
short distance, 185-186, 199, 202
symmetries, 105, 221-223
breaking, 223, 226
chiral, 130, 142, 145, 193, 202,
220-223, 236, 271

658
The Standard Model and Beyond
discrete, 58, 77, 132, 187, 221
flavor, 106, 119, 131, 183, 185,
187, 188, 221-223
gauge, 106
strangeness, 379
structure constants, 104, 108, 149
structure functions, see deep
inelastic scattering
Stuckelberg mechanism, 33
5£/(2), see symmetry
SU{2)L x SU(2)R x £7(1), 255, 287,
353, 374, 376, 402, 423, 508,
519-525
50(10), 520, 522
W-WR mass/mixing, 255, 273,
376, 522, 539
v seesaw, 522
breaking scale, 522, 524
discrete symmetries, 519, 524
experimental constraints, 525
gauge interactions, 520, 523
Higgs sector, 520
left-right symmetry, 519, 524
manifest/pseudo-manifest, 525
warped extra dimensions, 520
Yukawa couplings, 523
SU(3), see symmetry
SU{S)C x SU(2)L x £/(l)y, see
standard model
SU(3) x SU{2) x £7(1), see symmetry
SU(m), see symmetry
super-renormalizable theory, 25
supergravity, 458, 475, 481, 487
SuperKamiokande, 433, 435, 436, 526,
535
Supernova 1987A, 421, 433
superstring theory, xi, 109, 454, 455,
458, 487
heterotic, 517, 537
intersecting brane, 441, 455, 537
landscape, 459, 460
supersymmetry, xi, 73, 146, 353, 357,
374, 402, 456, 463-468
CP violation/EDM, 468
i?-parity, 467, 505
violation, 255, 402, 423, 505
^-symmetry, 484, 504, 507
\x problem, 461, 464, 468, 506-
507
Giudice-Masiero mechanism, 506
singlet extension of MSSM, 507,
513
g„ - 2, 86, 463
algebra, 475
auxiliary field, 476, 478
breaking/mediation, 403, 466, 488,
490-492
A terms, 494
D-term, 490
D-term mediation, 492
F-term, 490
Z' mediation, 492, 519
anomaly mediation, 492, 504
dynamical, 491
gauge mediation, 492
gaugino condensate, 492
gaugino mediation, 492
gaugino unification, 504
gravitino mass, 492
hidden sector, 467, 491
higher-dimensional operator, 506
minimal supergravity, 494, 504
non-holomorphic terms, 494
radion mediation, 492
soft (parameters), 467, 490, 492
spontaneous, 490
sum rules, 467, 491
super-Higgs mechanism, 491
supergravity mediation, 492
cascade
CP violation, 399
decoupling limit, 467, 500
Fayet-Iliopoulos term, 487
FCNC, 468, 491, 502
gauge kinetic function, 487
generators, 475
Grassman variable, 477, 538
integration, 477
Higgs
MA, 496, 499

Index
659
tan/?, 466, 496
doublets, 465
experimental limits, 466, 500
interactions, 466, 498
mass, 463
maximal mixing/no mixing, 498
precision constraints, 466
radiative electroweak breaking,
468, 497
radiative mass corrections, 497
spectrum, 497
theoretical mass bound, 466,
497, 507
interactions
abelian, 485-488
Dirac fermion, 490
non-abelian, 488-490
superpotential, 482-485
Kahler potential, 482
lightest supersymmetric partner
(LSP), 461, 467, 504
MSSM, 176, 286, 298, 467, 493-
505
decoupling limit, 371
notation, 493
superpotential, 493, 494
Yukawa couplings, 499
next lightest sparticle (NLSP),
505
non-renormalization theorem, 464,
481
O'Raifeartaigh model, 490
parton processes, 195
precision electroweak, 467, 500
singlet extended
U{l)f, NMSSM, nMSSM, 507
split, 459
superfield, 478-481
D-component, 481, 486, 488
F-component, 480, 482
chiral, 479-481
vector, 479, 481, 485
supergauge transformation, 485
supermultiplet
chiral, 476
gravity, 476
massive vector, 488
vector, 476
superpartners, 86, 87, 354, 465-
466
chargino, 466, 502-505
gaugino (G, W, B), 465, 486
gluino, 501
Goldstino, 488, 491
gravitino, 462
Higgsino, 466, 506
left-right mixing, 498, 501
neutralino, 466, 502-505
scalar lepton (slepton), 465, 500
scalar quark (squark), 195, 465,
500
singlino, 539
superpotential, 480
superspace, 477
transformation, 478-480
trileptons, 504
vacuum, 475
charge/color/B/L breaking, 495
metastable, 491
Wess-Zumino gauge, 486
Wess-Zumino model, 483
SU{2) x[/(l), see electroweak
interaction
symmetry
G-parity, 67, 132, 153, 270
form factors, 565
O(ra), 109, 117, 138
SO(m), 109, 149, 180
SU{2), 105, 107, 108, 119, 150
multiplication rule, 149
SU{3), 106-108, 119, 121-129,
185, 186, 189, 273
nonet, 122, 125, 128, 230-232
octet, 121, 152
SU(S) x SU{2) x £7(1), 108
SU{6), 128
SU(m), 107-109, 111, 116, 129,
199
SU(m)L x SU(m)R, 131, 152,
222-228

660
The Standard Model and Beyond
and weak current, 261
5p(2m), 109
£/-spin, 127, 152
£/(l), 26, 27, 30, 33, 105, 107,
116, 137, 141, 158, 199
as rotation, 118, 138
as subgroup, 159
monopoles/st rings, 137
U{m), 105, 109, 116, 152
V-spin, 127
Zn, 103, 132, 180
accidental, 221, 287
P, C, T, CP, 221, 372
baryon number, 222
lepton number/flavor, 403, 462
chiral, 129-131, 178
fermion mass, 131, 143
continuous, 103, 112, 144
custodial, 290, 520
discrete, 58-71, 112, 114, 132-
133
gauge, 180
spinor notation, 474
family, 283, 285, 444, 455, 508
field theory, 112-133
flavor, 106, 183
gauge, 28, 103, 105, 106, 157-
158
abelian, 32, 158-160
chiral, 158, 159
direct product, 164
gauge boson mass, 163
non-abelian, 160-163, 186
global, 28, 103, 157
internal, 103, 112
isospin, 88, 103, 105, 113, 119-
121, 132, 150, 185, 213
Lagrangian, 114
Lorentz, 4, 62, 89, 103, 104, 134,
469-471
violation, 71
Nambu-Goldstone realization, 139,
143
non-chiral, 130, 131
Peccei-Quinn, 458
permutation (54, A4), 445
Poincare, 4, 67, 469-471
rotational, 58, 63, 103, 105, 114,
120, 121, 469
space-time, 103, 112
translation, 3, 28, 62, 103, 134,
470
vacuum, 113
weak isospin, 105
Wigner-Weyl realization, 138, 143
chiral, 223, 233
symmetry, see also QED, QCD, elec-
troweak interaction, U(\)',
SU(2)L x SU(2)R x £7(1),
grand unification, C, P, T,
CP, CPT
symmetry breaking, 133-145
SU{2) x £7(1), 281
5£/(3), 227-228, 230-232
SU{3), 124-126
charge non-conservation, 153
chiral, 131, 140-141
fermion mass, 141, 143, 226,
234, 286
non-analytic, 268
electromagnet ism, 121, 126
explicit, 114, 115, 117, 132-134,
137, 143
Nambu-Goldstone, 140
Wigner-Weyl, 138
isospin, 85, 119, 121, 125, 126,
132, 227-228
in nuclei, 228
leakage from multiplet, 115, 231,
268
quantum mechanics, 573-575
quark masses, 119, 121, 126, 131,
187, 223-228
soft, 138
spontaneous, xi, 33, 94, 114, 133,
135, 139, 143-145, 153
chiral, 140, 193, 202
discrete, 137
Higgs alternatives, 277, 290,
366, 371, 457

Index
661
Higgs mechanism, xi, 33, 142,
145, 164-168, 243, 287-289,
362
polar basis, 141, 142, 148, 153,
167
rectangular basis, 139, 153
T, see time reversal
T, see time reversal: Dirac matrix
't Hooft-Feynman gauge, 173, 381
T matrix, see transition amplitude
T2K, 447
tadpole diagram, 26
tadpole operator, 136
TASSO, 189
r, see Pauli matrices
tau (r)
SU(2) x (7(1) assignment, 327
decay, 239, 241, 274
vT mass, 275
t~ ->7r-i/r, 279
Michel parameters, 275
universality, 275
properties, 274
technically natural parameter, 531
technicolor/ extended technicolor, 356,
457
tensor notation, 111, 129, 187
tensor operator, 102
Tevatron, 218, 245, 328, 338, 340,
361, 368, 392, 502, 519
textures, 137, 444-446
Oakes relation, 452
9(x), see step function
9 vacuum, see tunneling: instantons
9C (Cabibbo angle), see quark mixing
9qcd, see CP: strong
Thomson cross section, 56
't Hooft, G., 243
(3*, 3) + (3, 3*) model, 223-228
three-vector, see vectors
tilde trick, 286, 465, 520
time evolution operator, 545, 547
time reversal, 58, 68-71, 255, 263
antiunitary, 69
Dirac matrix, 70
form factors, 565
quantum mechanics, 69
two-component, 72, 474
violation, 92, 397-401
K° oscillations, 398
pseudo, 398
triple correlations, 398
time reversal, see also electric dipole
moment
time-ordered product, 12, 44
time-varying coupling constants, 461
Tomonaga, S., 74
top (t) quark, 183, 329
decays, 277
discovery, 350
mass prediction, 244, 329, 350,
352
radiative electroweak breaking,
468, 497
top-color, 457
topological defects, 136-137
trace identities, 42-43
transition amplitude, 12, 17, 101, 141,
173, 188, 545, 547
translation invariance, see symmetry:
translation
transverse variables, 567-571
triangle diagram, see anomalies
TRISTAN, 84, 321, 324
TRIUMF, 255, 256
Troitsk, 421
tunneling
instantons
B + L, 222, 460
U{\)A, 222, 230
metastable vacuum, 136, 366
phase transition, 460
Twin-Higgs, 457
TWIST, 256
twistors, 236
two-component notation, 47, 71-73,
405
dotted/undotted indices, 471
raising/lowering indices, 472

662
The Standard Model and Beyond
spinor, 471-475
U(1)a problem, see axial baryon
number
t/(l), see symmetry
U(l)', 286, 310, 329, 350, 353, 374,
440, 507, 510-519
#6 models, 516
Z-pole, 518
Z — Z' mass/mixing, 511
\x problem, 513, 517
v mass, 519
collider, 519
diagnostics, 519
FCNC, 519
gauge interactions, 510
hidden sector couplings, 514, 519
Higgs/neutralino sector, 519, 539
intermediate scale, 513
kinetic mixing, 513
fractional electric charge, 514
left-right models, 515
leptophobic, 514
minimal unification models, 517
motivations, 510
neutral current/LEP 2, 518
sequential, 514
UA1, 291
UA2, 291
ultraviolet cutoff, see divergence:
ultraviolet
t/(ra), see symmetry
unit sphere (surface area), 5, 558
unitarity, 188, 547-548
5-wave, 241, 366
violation
Born approximation, 241
Fermi theory, 241
Higgs, 366
intermediate vector boson
theory, 243
unitary gauge, 33, 166, 167, 173
multiscalar interaction, 173, 181
standard model, 288-304
units, 1 r
unpolarized rate, see spin average/sum
up (u) quark, 127, 183
T(66), 191, 219, 276, 392
V — A, see current
vacuum, 7
vacuum condensate, see composite
operator
vacuum expectation value, 11, 133,
138, 165
vacuum polarization, see self-energy
vacuum polarization functions, 355
vacuum saturation approximation, 383,
384
vacuum stability, 145, 365
vector fermion, 130
vector field, see field: gauge
vector mesons, 125
vector potential, 30
vector symmetry, see symmetry: non-
chiral
vectors, 2-3
Veltman, M., 243
Veneziano (dual resonance), 220
vertex correction, 75, 78, 91, 555
VEV, see vacuum expectation value
W boson, 242, 336-338
W+W- -► W+W~, 243, 366
strong coupling, 367
and color, 192
decay widths, 339-340
discovery, 244, 291
helicity, 361
mass, 290, 291, 350
propagator effects, 350
W boson, see also electroweak
interaction: charged current
Ward-Takahashi identity, 76, 78
weak angle, 85, 331
v^N, 319
i^e, 310
atomic parity violation, 323
defined, 290
global analysis, 329, 351

Index
663
M0ller scattering, 328
renormalized, 331-333
Z-mass (slfz), 331
MS (52MJ, 331
effective (*f), 331, 342
on-shell ( s^), 331
running, 334
weak charged current (WCC), see elec-
troweak interaction
weak hypercharge, 281, 283
weak interaction, see electroweak
interaction
weak neutral current (WNC), see
electroweak interaction
weakly interacting massive particle
(WIMP), 422, 461
weight diagram, 122, 123, 126, 127,
247
Weinberg, S., 243
Weyl spinor, see field: Weyl
Wick ordering theorem, 546
Wick rotation, 382, 464, 557
Wigner-Eckart theorem, 121, 124, 151
SU(S) octets, 125
WMAP, 420
WR, see SU(2)L x SU(2)R x £7(1)
Wu, C.S., 249
Yang, C.-N., 249
Young tableaux, 111
Yukawa interaction, 58, 60, 120, 150,
170, 185
standard model, 285, 297
diagonal for mass eigenstates,
297
running coupling, 363
Yukawa matrices, 285
Yukawa potential, see potential: Yukawa
Z boson
Z-pole, 340-350
r polarization, 348
asymmetries, 340, 346-350, 450
effective couplings, 333
invisible width, 345
lineshape, 340-346, 560
measurements, 304, 344
widths, 340-346
decays, 99, 340
discovery, 244, 291
mass, 290, 291, 331, 336-338
measurement, 343
predicted, 243
propagator effects, 325
Z boson, see also electroweak
interaction: neutral current,
precision tests, asymmetry
zero-point energy, 542
ZFITTER, 332, 351
Zn, see symmetry
Z' boson, see U(X)'

Physics
Si rii in High Energy Physics, Cosmology, and Gravitation
Series Editors: Brian Foster and Edward W. Kolb
The Standard Model and Beyond
aul Langacker
"The Standard Model and Beyond is a state-of-the-art description of what we
know about the particles and forces that build up the world we see. Most books
that cover these topics are quantum field theory books that treat the quarks and
leptons, and the electromagnetic, weak, and strong forces as examples This
is the first treatment with the opposite priorities, focusing on the structure and
applications of the standard model and bringing in the field theory as needed, in
a pedagogically reliable and thorough treatment. Langacker knows well that the
standard model is the platform on which a deeper understanding of the laws of
nature will be constructed, perhaps from clues soon to come from the Large Hadron
Collider, and provides preparation so the reader can participate in that progress."
— Gordon Kane, Victor Weisskopf Collegiate Professor of Physics and Director
of the Michigan Center for Theoretical Physics, University of Michigan,
Ann Arbor, USA
The Standard Model and Beyond presents an advanced introduction to the
physics and formalism of the standard model and other non-abelian gauge
theories. Thoroughly covering gauge field theories, symmetries, and topics beyond
the standard model, this text equips readers with the tools to understand the
structure and phenomenological consequences of the standard model, to construct
extensions, and to perform calculations at tree level. It establishes the necessary
background for readers to carry out more advanced research in particle physics.
Features
• Covers the fundamental interactions
• Describes the construction, experimental tests, and phenomenological
consequences of the standard model
• Presents a self-contained treatment of the complicated technology
needed for tree-level calculations
• Explores applications in astrophysics and cosmology
• Lists many useful reference books, review articles, research papers, and
Web links
• Offers supplementary materials on the author's Web site
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