ISBN: 9971-50-564-9

                    PERTURBATIVE QUANTUM

ADVANCED SERIES ON DIRECTIONS IN HIGH ENERGY PHYSICS Published Vol. 1 — High Energy Electron-Positron Physics (eds. A. AH and P. Soding) Vol. 2 — Hadronic Multiparticle Production (ed. P. Carruth^rs) Vol.3— CP Violation (ed. C. Jarlskog) Vol. 4— Proton-Antiproton Collider Physics (eds. G. A/tare///and L. Di Leila) Vol. 5— Perturbative QCD (ed. A. H. Mueller) Forthcoming Vol. 6— Quark Gluon Plasma (ed. R. C. Hwa) Vol. 7 — Quantunn Electrodynannics (ed. T. Kinoshita) Vol. 8 — Interactions Between Elementary Particle Physics and Cosmology (ed. E. Kolb) Cover Artwork by courtesy of Los Alamos National Laboratory. "This work was performed by the University of California, Los Alamos National Laboratory, under the auspices of the United States Department of Energy."
Advanced Series on Directions in High Energy Physics—Vol. 5 PERTURBATIVE QUANTUM CHROMODYNAMICS Editor: A. H. Mueller World Scientific Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd., P O Box 128, Farrer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 SDH Library of Congress Cataloging-in-Publication data is available PERTURBATIVE QUANTUM CHROMODYNAMICS Copyright © 1989 by World Scientific Publishing Co Pte Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any forms or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISBN 9971-50-564-9 9971-50-565-7 (pbk) ISSN 0218-0324 Printed in Singapore by Utopia Press.
V FOREWORD With the discovery of asymptotic freedom in 1973 Quantum Chromodynamics (QCD) was born. It was soon realized that the study of nonperturbative effects would be crucial in order to understand color confinement, chiral symmetry breaking and, of course, to achieve a quantitative understanding of bound states and low energy dynamics. It was also realized, right from the start, that the rather rich structure of QCD perturbation theory could be seen in hadronic processes involving a high momentum transfer, that is, in hard processes. QCD is widely viewed to be the correct theory of the strong interactions, mainly because of the success which has been achieved in predicting and describing such hard processes. The road has not been easy. It has been necessary to develop an extensive theoretical apparatus in order to relate properties of the fundamental quarks and gluon of QCD to the observed properties of hadronic interactions. A lot of work has been completed in this direction, but much remains to be done both in giving an even more solid foundation to the formalism which has been developed and in developing new frameworks in which to understand high energy reactions. The articles in this volume aim at describing the formahsm which has been developed in order to relate perturbative QCD to measurable quantities. The emphasis is placed on understanding perturbative QCD and how it relates to physical quantities rather than on detailed fits to data. It is hoped that these contributions will make the rather elaborate formalism of perturbative QCD more accessible to our theoretical colleagues in neighboring disciplines, to graduate students and to the adventurous experimenter who wants to understand exactly where QCD predictions come from and what they really mean. At the basis of most high energy applications of QCD is factorization. Without factorization theorems, the separation of the short distance physics, perturbative QCD, from the long distance physics of observable hadrons would not be possible. The standard factorization theorems for hard processes in QCD, and their proofs, are summarized in the article of Collins, Soper and Sterman. The article by Brodsky and Lepage deals with exclusive processes in QCD. This is a very diverse subject encompassing form factors, wide angle elastic scattering and various reactions involving nuclei. This is also a subject which has important points of contact with nonperturbative QCD and with nuclear physics. The detailed properties of QCD jets, such as particle distributions within a jet or between several jets is given in the contribution of Dokshitzer, Khoze and Troyan. This article also describes the QCD basis for Monte Carlo models of single and multiple jet events. One of the oldest problems in QCD, that of the behavior of small angle high energy scattering, is still not completely solved. Exactly how much of this problem can be solved purely within perturbative QCD is not completely clear at this time.
VI The Pomeron problem remains one of the most challenging questions in QCD. The article by Lipatov describes the present understanding on this topic. Infrared effects and double logarithmic terms in perturbation theory play a crucial role in many processes. For example, the transverse momentum distribution of massive /i-pairs or of W and Z production in hadronic collisions can be predicted only after resuming double-logarithmic terms. Perturbative QCD is applicable to wide angle elastic scattering only because non-hard regions are suppressed, at sufficiently high energy, by the doubling logarithmic Sudakov factors. Understanding particle and multiplicity distributions in QCDjets requires good control over infrared gluon emission. These topics are discussed in the articles by Ciafaloni and by Collins. Of course, the articles which follow are not the final word on any of these subjects. They do, however, furnish soHd and fairly complete discussions as to what is known at present. We can expect factorization theorems to become more rigorous and far reaching in the future. The Pomeron and small-x problems in QCD are perhaps ripe for significant future development, soHdifying our ever growing qualitative understanding of these questions. Infrared and Sudakov behavior in QCD present important technical challenges for the future. We can hope that in 10 — 15 years from now, significant advances and improvements will have been made in all the subjects discussed here. Nevertheless, even at that time, the present articles should remain a good introduction to the subject of perturbative QCD. A. H. Mueller Department of Physics Columbia University New York
CONTENTS Foreword A. H. Mueller Factorization of Hard Processes in QCD /. C Collins, D. E. Soper and G. Sterman S. J. Brodsky and G. P. Lepage Coherence and Physics of QCD Jets Yu L. Dokshitzer, V. A. Khoze and S. I. Troyan M. Gafaloni Sudakov Form Factors /. C Collins VII 1 Exclusive Processes in Quantum Chromodynamics 93 241 Pomeron in Quantum Chromodynamics 411 L. N. Lipatov Infrared Singularities and Coherent States in Gauge Theories 491 573
1 FACTORIZATION OF HARD PROCESSES IN QCD John C. Collins Physics Department Illinois Institute of Technology Chicago, IL 60616, U.S.A. and Institute for Theoretical Physics State University of New York Stony Brook, NY 11794-3840, U.S.A. Davison E. Soper Institute of Theoretical Science University of Oregon Eugene, OR 97403, U.S.A. George Sterman Institute for Theoretical Physics State University of New York Stony Brook, NY 11794-3840, U.S.A. ABSTRACT We summarize the standard factorization theorems for hard processes in QCD, and describe their proofs. 1. INTRODUCTION In this chapter, we discuss the factorization theorems that enable one to apply perturbative calculations to many important processes involving hadrons. In this introductory section we state briefly what the theorems are, and in Sects. 2 to 4, we indicate how they are applied in calculations. In subsequent sections, we present an outline of how the theorems are established, both in the simple but instructive case of scalar field theory and in the more complex and physically interesting case of quantum chromodynamics (QCD). The basic problem addressed by factorization theorems is how to calculate high energy cross sections. Order by order in a renormalizable perturbation series, any physical quantity is a function of three classes of variables with dimensions of mass. These axe the kinematic energy scale(s) of the scattering, Q, the masses, ?7i, and a renormalization scale /i. We can make use of the asymptotic freedom of QCD by choosing the renormalization scale to be large, in which case the effective
2 coupling constant g{n) will be correspondingly small, g{n) ^ l/ln(yu/AQCD)- The renormalization scale, however, will appear in ratios Q/fi and fi/m^ and at high energy at least one of these ratios is large. If we pick n ^ Q, for instance, then at n loops the coupling will generally appear in the combination g^^(Q) \n^^(Q/m), with a = 1 or 2. (See Sect. 7.) As a result, the perturbation series is no longer an expansion in a small parameter. The presence of logarithms involving the masses shows the importance of contributions from long distances, where the precise values of masses (including the vanishing gluon mass!) are relevant. For such contributions we do not expect asymptotic freedom to help, since it is a property of the coupling only at short distances. In summary, a general cross section is a combination of short- and long-distance behavior, and is hence not computable directly in perturbation theory for QCD. There are exceptions to this rule. For reasons which will become clear in Sect. 7, these are inclusive cross sections without hadrons in the initial state, such as the total cross section for e'^e" annihilation into hadrons, or into jets. This leaves over, however, the majority of experimentally studied lepton- hadron and hadron-hadron large momentum transfer cross sections, as well as inclusive cross sections in e'^e" annihilation with detected hadrons. Factorization theorems allow us to derive predictions for these cross sections, by separating (factorizing) long-distance from short-distance behavior in a systematic fashion. Thus almost all applications of perturbative QCD use factorization properties of some kind. In this chapter, we will explicitly treat factorization theorems for inclusive processes in which (1) all Lorentz invariants defining the process are large and comparable, except for particle masses, and (2) one counts all final states that include the specified outgoing particles or jets. The second condition means that we consider such processes as hadron A + hadron B —^ hadron C -f X, where the X denotes "anything else" in addition to the specified hadron C. The first condition means that in this example the specified hadron C should have a transverse momentum comparable to the center-of-mass energy. For such processes, the theorems show how to factorize long distance effects, which are not perturba- tively calculable, into functions describing the distribution of partons in a hadron — or hadrons in a parton in the case of final-state hadrons. Not only can these functions be measured experimentally, but also the same parton distribution and decay functions will be observed in all such processes. The part of the cross section that remains after the parton distribution and decay functions have been factored out is the short distance cross section for the hard scattering of partons. This hard scattering cross section is perturbatively calculable, by a method which we describe below.
3 Some examples of processes for which one expects a factorization theorem of this type to hold include (denoting hadrons by A, jB, C ...) • Deeply inelastic scattering, lepton -f A —> lepton' -f X\ • e+ -f e" -> A + X; • The Drell-Yan process, A + jB ->e++e- +J\:, A + jB->W + X, A + jB-^Z + X; • A + jB-^jet + X; • A 4- jB —> heavy quark -f X. In the last example, the heavy quark mass, which must be large compared to 1 GeV, plays the role of the large momentum transfer. In the Drell-Yan case, the kinematic invariants are the particle masses, the square, s, of the center-of-mass energy, and the invariant mass Q and transverse momentum q± of the lepton pair. The requirement, for the theorems that we discuss, that the invariants all be large and comparable means that not only should Q^ be of order 5, but also that either we integrate over all q± or q± is of order Q. There are applications of QCD to processes in which there is a large momentum scale involved but for which the most straightforward sort of factorization theorem, as discussed in this chapter, must be modified. However, the same style of analysis as we will describe applies to these more general situations. (The Drell- Yan process when q± is much less than Q is an example.) We will summarize these in Sect. 10. Some of the factorization properties, such as those we describe in this chapter, have been proved at a reasonable level of rigor within the context of perturbation theory. But many of the other results have, so far, been proved less completely. The following three subsections give explicit factorization theorems for three basic cross sections from the list above, deeply inelastic scattering, single-particle inclusive annihilation and the Drell-Yan process. These three examples illustrate most of the issues involved in the application and proof of factorization. We close the section by relating factorization to the parton model. 1.1 Deeply Inelastic Scattering Deeply inelastic lepton scattering plays a central role in any discussion of factorization, both because this was the first process in which pointlike partons were "seen" inside the hadron, and because much of the data that determines the par- ton distribution functions comes from measurements in this process. In particular, let us consider the process e -f A —> e -f X, which proceeds via the exchange of
4 a virtual photon with momentum qf^. Prom the measured cross section, one can extract the standard hadronic tensor W^'^{q^,p'^), W" = 1-Jd^e'^y Y^'^A \j 1^{y)\X){X \j"{0)\ A) Fi{x,Q )[-g'' +—^j+F2(x,Q) — , (1) where Q^ = —QfiQ^, x = Q^/^q-p^ p^ is the momentum of the incoming hadron A, and j^{x) is the electromagnetic current. (More generally, j'^{y) can be any electroweak current, and there will be more than two scalar structure functions Fi.) We consider the process in the Bjorken limit, i.e., large Q at fixed x. The factorization theorem is contained in the following expression for W^^^ PF'"'(5^p'') = Y.I T /"MC^'^) ff^(9^ep^M,«s(M)) + remainder. (2) Here fa/A(C^/^) is a parton distribution function, whose precise definition is given in Sect. 4. There, fa/Aiii f^)^C is interpreted as the probability to find a parton of type a {— gluon, u, u, d, d,...) in a hadron of type A carrying a fraction ^ to ^ -f d^ of the hadron's momentum. In the formula, one sums over all the possible types of parton, a. We can prove eq. (2) in perturbation theory, with a remainder down by a power of Q (in this case, the power is Q~^ modulo logarithms, but the precise value depends on the cross section at hand, and has not always been determined). We can project eq. (2) onto individual structure functions: Fi{x,Q^) = ^ / y fa/ Hia (j,—,as{/J,)] + remainder. '-F2{x,Q^) = Y1 I T •^«m(^'^) "■^2a ( 7, —,as(A^) ) + remainder. (3) The extra factors of 1/x and ^jx in the equation for F2 are needed because of the dependence on target momentum of the tensor multiplying ^2- Inspired by the terminology of the operator product expansion for the moments of the structure functions, it is conventional to call the first term on the right of either of eqs. (2) or (3) the leading twist contribution, and to caJl the remainder the higher twist contribution. The same terminology of leading and higher twist is used for the factorization theorems for other processes. It is not so obvious why proving eq. (2) in perturbation theory is useful, given that hadrons are not perturbative objects. But suppose we do decide on
5 a way of computing the matrix elements in eq. (1) perturbatively. For any such formulation for haxiron A, both W^^ and f^/A will depend on phenomena at the scale of hadronic masses (or some other infrared cutoff), and the exact nature of these phenomena will depend on our particular choice of A, as well as on the precise values we pick for both hadronic and partonic masses. The content of the factorization theorem is that this dependence of W^^ on low mass phenomena is (mtirely contained in the factor of fa/A- The remaining function, the hard scattering coefficient H^'^, has two important properties. First, it depends only on the parton type a, and not directly on our choice of hadron A. Secondly, it is ultraviolet dominated, that is, it receives important contributions only from momenta of order Q. The first property allows us to calculate H^^ from eq. (2) with the simplest choice of external hadron, A = 6, 6 being a parton. (We will see an example of this in our calculations for the Drell-Yan process in Sect. 2.) The second property ensures that when we do this calculation, H^'^ will be a power series in aa(Q), with finite coefficients. We now assume that nonperturbative long-distance effects in the complete theory factorize in the same way as do perturbative long-distance effects. Once this assumption is made, we can interpret our perturbative calculation of H^'^ as a prediction of the theory. Parton model ideas, summarized in Sect 1.4, give motivation that the assumption is valid. Note that our definition of the parton distributions, which we will give in Sect. 4, is an operator definition, which can be applied beyond perturbation theory. This ability to calculate the H^^ results in great predictive power for factorization theorems. For instance, if we measure F2{x,Q'^) for a particular hadron A, eq. (3) will enable us to determine fa/a- We then derive a prediction Fi(x, Q^) for the same hadron A, in terms of the observed F2 and the calculated functions Hia- This is the simplest example of the universality of parton distributions. The functions Hia may be thought of as hard-scattering structure functions for parton targets, but this interpretation should not be taken too literally. In any case, methods for putting this procedure into practice, including definitions for the parton distributions are the subjects of Sects. 2 to 4. Originally, eq. (2) was primarily discussed in terms of the moments of the structure functions, such as Fi(n,Q^)= / —x"F,(x,Q'), (4) n-1 F2(n,Q')= / — x"-*F2(x,g^).
6 With this notation, eq. (3) becomes Fiin,Q^) = Y^ fa/A{n,m) hJu, 9.,a^ifi)) . (5) a \ M / In this form of the factorization theorem, when n is an integer, the fj/y^{n,fi) are hadron matrix elements of certain local operators, evaluated at a renormalization scale fi. On the other hand, the structure function moment F2{n,Q ) can be expressed in terms of the hadron matrix element of a product of two electromagnetic current operators evaluated at two nearby space-time points. Equation (5) thus appears as an application of the operator product expansion^'"^'^J. The product of the two operators is expressed in terms of local operators and some perturbatively calculable coefficients Hia{n, Q/n, as(fj,)), called Wilson coefficients. It was using this scheme that the Hia{n,Q/iJ.,as{iJ>)) were first calculated^J. 1.2 Single Particle Inclusive Annihilation In this subsection, we consider the process 7* —> A -f X, where 7* is an ofF- shell photon. The relevant tensor for the process, for which structure functions analogous to those in eq. (1) may be derived, is D>"'{x,Q) = l-Jd*ye'''yJ2{(^\j>'{y)\AX){AX\r{0)\0), (6) where q^^ is now a time-like momentum and Q^ = Q^- The sum is over all final- states that contain a particle A of defined momentum and type. We define a scaling variable hy z = 2p'q/Q^, where p'* is the momentum of A, and we will consider the appropriate generalization of the Bjorken limit, that is, Q large with z fixed. The factorization theorem here is quite analogous to eq. (2), but incorporates the slightly different kinematics, D^'^iz, Q) = Y.fT ^a%^K. Ql\^. «s(/i)) cf^/a(C), (7) with corrections down by a power of Q, as usual. We have used the same notation for the hard functions as in deeply inelastic scattering, and as in that case they are perturbatively calculable functions. Here it is the fragmentation functions dj^jJ^Qj which are observed from experiment, and which occur in any similar inclusive cross section with a particular observed hadron in the final state. For example, single- particle inclusive cross sections in deeply inelastic scattering cross sections require the factorization both of parton distributions iajA-, with A the initial hadron, and of distributions c^b/oj with B the observed hadron in the final state. We shall not go into the details of such cross sections here^^.
7 1,S Drell'Yan Our final example to illustrate the important issues of factorization is the Drell-Yan process: A + jB-./i+-f/i-+X (8) at lowest order in quantum electrodynamics but, in principle, at any order in quantum chromodynamics. q^^ is now the momentum of the muon pair. We shall be concerned with the cross section dcr/dQ^dy^ where Q^ is the square of the muon pair mass, Q'=9%, (9) and y is the rapidity of the muon pair. We imagine letting Q^ and the center of mass energy y/s become very large, while Q^/s remains fixed. The relevant factorization theorem, accurate up to corrections suppressed by a power of Q^, is do- dQ^dy rs-* (11) Here a and b label paxton types and we denote XA = G^y—, XB = e ^J—. (12) The function Hat is the ultraviolet-dominated hard scattering cross section, computable in perturbation theory. It plays the role of a parton level cross section and is often written as when it is not necessary to display the functional dependence of Hah on the kine- matical variables. The parton distribution functions, /, are the same as in deeply inelastic scattering. Thus, for instance, one can measure the parton distribution functions in deeply inelastic scattering experiments and apply them to predict the Drell-Yan cross section. As before, the parameter /i is a renormalization scale used in the calculation of Hah-
8 1.4 Factorization in the Parton Model Having introduced the basic factorization theorems, we will now try to give them an intuitive basis. Here we shall appeal to Feynman's parton model^K In fact, we shall see that factorization theorems may be thought of as field theoretic realizations of the parton model. In the parton model, we imagine hadrons as extended objects, made up of constituents (partons) held together by their mutual interactions. Of course, these partons will be quarks and gluons in the real world, as described by QCD, but we do not use this fact yet. At the level of the parton model, we assume that the hadrons can be described in terms of virtual partonic states, but that we are not in a position to calculate the structure of these states. On the other hand, we assume that we do know how to compute the scattering of a free parton by, say, an electron. By "free", we simply mean that we neglect parton-paxton interactions. This dichotomy of ignorance and knowledge corresponds to our inability to compute perturbatively at long distances in QCD, while having asymptotic freedom at short distances. To be specific, consider inclusive electron-hadron scattering by virtual photon exchange at high energy and momentum transfer. Consider how this scattering looks in the center-of-mass frame, where two important things happen to the hadron. It is Lorentz contracted in the direction of the collision, and its internal interactions are time dilated. So, as the center-of-mass energy increases the lifetime of any virtual partonic state is lengthened, while the time it takes the electron to traverse the hadron is shortened. When the latter is much shorter than the former the hadron will be in a single virtual state characterized by a definite number of partons during the entire time the electron takes to cross it. Since the partons do not interact during this time, each one may be thought of as carrying a definite fraction x of the hadron's momentum in the center of mass frame. We expect x to satisfy 0 < a: < 1, since otherwise one or more partons would have to move in the opposite direction to the hadron, an unlikely configuration. It now makes sense to talk about the electron interacting with partons of definite momentum, rather than with the hadron as a whole. In addition, when the momentum transfer is very high, the virtual photon which mediates electron-parton scattering cannot travel fax. Then, if the density of partons is not too high, the electron will be able to interact with only a single parton. Also, interactions which occur in the final state, after the hard scattering, are assumed to occur on time scales too long to interfere with it. With these assumptions, the high energy scattering process becomes essentially classical and incoherent. That is, the interactions of the partons among themselves, which occur at time-dilated time scales before or after the hard scattering, cannot interfere with the interaction of a paxton with the electron. The
9 cross section for hadron scattering may thus be computed by combining probabilities, rather than amplitudes. We define a parton distribution /a///(0 ^^ ^^^ probability that the electron will encounter a "frozen", noninteracting parton of species a with fraction ^ of the hadron's momentum. We take the cross section for the electron to scatter from such a parton with momentum transfer Q^ as the Born cross section crBiQ^^C)- Straightforward kinematics shows that for free partons ( > X = 2p'q/Q'^, and the total cross section for deeply inelastic scattering of a hadron by an electron is CTeHix, Q2) = ^ r de fa/HiO CrB^x/i, Q^). (14) This is the parton model cross section for deeply inelastic scattering. It is precisely of the form of eq. (2), and is the model for all the factorization theorems which we discuss in this chapter. Essentially the same reasoning may be applied to single-particle-inclusive cross sections and to the Drell-Yan cross section. For example, in the parton model the latter process is given by the direct annihilation of a parton and anti-paxton pair, one from each hadron, in the Born approximation, cr'siQ^^y)- The interactions which produce the distributions of each such parton occur on a scale which is again much longer than the time scale of the annihilation and, in addition, final- state interactions between the remaining partons take place too late to affect the annihilation. We thus generalize (14) to the parton model Drell-Yan cross section d(7 dQ'^dy J2 [ ^^^1 ^^B fa/AiU) h/B{iB) cr's{Q\y\ (15) where xa,b are defined in (12). Equation (15) is of the same form as the full factorization formula, (11), except that there is only a single sum over parton species, since the hard process here consists of a simple quark-antiquark annihilation. In the parton model, the functions fa/A{(.A) iii Drell Yan must be the same as in deeply inelastic scattering, eq. (14), since they describe the internal structure of the hadron, which has been decoupled kinematically from the annihilation and from the other hadron. It is important to notice that the Lorentz contraction of the hadrons in the center of mass system is indispensable for this universality of parton distributions. Without it the partons from different hadrons would overlap a finite time before the scattering, and initial-state interactions would then modify the distributions. We now turn to the technical discussion of factorization theorems in QCD, but it is important not to loose sight of their intuitive basis in the kinematics of high energy scattering. In fact, when we return to proofs of factorization theorems in gauge theories (Sects. 8 and 9) these considerations will play a central role.
10 2. CALCULATION OF THE HARD SCATTERING CROSS SECTION In this and the following two sections, we discuss the explicit calculation of the hard scattering functions for the Drell Yan cross section. In doing so, we will cover most of the technical points which are encountered in applying factorization in other realistic cases as well. At order zero in as for the Drell-Yan cross section, the hard process described by Hab is quark-antiquaxk annihilation, as illustrated in fig. 1. One can simply compute this paxton level cross section from the Feynman diagram and insert it into the factorization formula (11). The resulting cross section is not itself a prediction of QCD, although it is a prediction of the parton model. The factorization theorem will malce the connection between the two. At the Born level, it is natural to define fa/a{0 = ^{1 - 0- We then find ^a,b ^a gQ4 (16) where the factor S^ i indicates that parton a must be the antiparticle to parton b. Here C(e) is 1 if we work in 4 space-time dimensions. However, when one wants to calculate higher order contributions, it will turn out to be useful to perform the entire calculation in 4 — 2€ dimensions. Then (1 - ef r(l - e) ^^^^ (l-2c/3)(l-2€)r(l-2c)' The € dependence here arises from three sources. First, the Dirac trace algebra gives an angular dependence 1 -f cos"^ 0 — 2e. Secondly, one introduces a factor (/i^/(47r) e'^y so as to keep the cross section at a constant overall dimensionality* of M~^. Finally, the integration over the lepton angles in 4 — 2e dimensions gives the remaining e dependence. Actually, it is quite permissible to perform the lepton trace calculation and the integration over lepton angles in 4 dimensions instead of 4 — 2€ dimensions. This procedure results in multiplying the Born cross section and the higher order cross section by a common, e-dependent factor. As we will see below, such a factor will drop out in the physical cross section. Now let us calculate H at one loop. At first order in ag, the cross section gets contributions from the graphs shown in fig. 2, along with their mirror diagrams. In this figure, we show contributions to both the amplitude and its complex conjugate. * We use {/j,'^/(Att) e'^Y rather than (/i^)^ in anticipation of our use of MS renormalization.
11 Fig. 1. Born amplitude for the Drell-Yan process. separated by a vertical line which represents the final state. We will use this notation frequently below, and refer to diagrams of this sort as "cut diagrams". The situation now is not so simple, because a straightforward calculation of the cross section for quark -f antiquark —> fj.'^ -f /i~ -f X according to the diagrams shown above yields an infinite result when we use massless, on-shell quarks as the incoming particles. Fig. 2. Order as contributions to the Drell-Yan cross section. Following Sect. 1.1, we use the factorization formula (11) applied to incoming
12 partons instead of incoming hadrons. Since the details associated with parton masses axe going to factorize, we can choose to calculate the cross section for paxton a -f parton 6—>/i'^-f/i~-f-X' with the partons having zero mass and transverse momentum. Let us call this cross section Gah'- 5g2d^ = Gath^.XB^Q; ^;a,;e] (18) In this calculation there are both ultraviolet and infrared divergences. Dimensional regularization is used to regulate them both. The factorization formula is then Gab[ XA,XB,Q\ ^5<^s;€ C' • Cat (19) Both factors in the formula depend on /i, which is the scale factor introduced in the dimensional regularization and subsequent MS renormalization^J of Green functions of ultraviolet divergent operators. One introduces a factor (20) for each integration J di^~^^k in order to keep the dimensionality of the result independent of e. Ultraviolet divergences then appear as poles in the variable e, which are subtracted away, as explained in Ref. 7. The factor e'^/{Air) that comes along with the fj. is the difference between MS renormalization and minimal subtraction (MS) renormalization. Here 7 = 0.5.77... is Euler's constant. Let us suppose that we have calculated Gab to two orders in perturbation theory. We denote the perturbative coefficients by G.6 = Gi:> + ^GlV+C?K^). (21) (0) :. .-u^ o .:^^ : /i^\ ^(1) Thus G^j is the Born cross section in eq. (16). G^^ is the first correction. The first correction G^^ will generally have ultraviolet divergences at e = 0, coming from virtual graphs, and these divergences will appear as 1/e poles. Following the minimal subtraction prescription, we remove these ultraviolet poles
13 as necessary.* In general 1/e poles of infrared origin will remain in G^j , and we shall discuss these infrared poles presently. Let us similarly denote the perturbative coefficients of the hard scattering cross section Hah by Ha, = H^:^ + ^ Hil^ + Dial). (22) It is these coefficients that we would like to calculate. All we need to know to calculate H from G is the perturbative expansion of the functions fa/h(^y^)j which, according to the factorization theorem, contain all of the sensitivity to small momenta, and are interpreted as the distribution of parton a in paxton b. These functions can be calculated in a simple fashion using their definitions (Sect. 4) as matrix elements (here in parton states) of certain operators. When the ultraviolet divergences of the operators are also renormalized using minimal subtraction, one finds simply where P^/lix) is the lowest order Altarelli-Parisi^J kernel that gives the evolution with fi of the parton distribution functions. We will discuss the computations that lead to eq. (23) in Sect. 4. For now, let us assume the result. When we insert these perturbative expansions (23) into the factorization formula, we obtain G^ahi^A.XB.Q-.-^^ej -h^ G^^^(xA,XB,Q]^\e 2 + 0{at). (24) * In the particular case of the Drell-Yan cross section (or, more generally, a cross section for which the Born graph represents an electroweak interaction), the first QCD correction G^j is not in fact ultraviolet divergent, provided that we include the propagator corrections for the incoming quark lines This follows from (1) the Ward identity expressing the conservation of the electromagnetic current and (2) the fact that the photon propagator does not get strong interaction corrections, at lowest order in QED. It can also be verified easily by explicit computation.
14 We can now solve for Hah- At the Born level, we find H^ai ixA,XB,Q;^;e]= G^^^ f x^, xa, Q; |; e ] . (25) Then at the one loop level we obtain hH^ (xa, xb, Q; ^; e j = gIV ixA,XB, Q; ;^; (26) ^E/d|.PSl(WGirfex.,Q;|; + 2e c + ^E/<i^B^i»^«)^i"i(--if.Q; : 6 Thus the prescription is quite simple. One should calculate the cross section at the parton level, G^j , and subtract from it certain terms consisting of a divergent factor 1/e, the Altarelli-Parisi kernel, and the Born cross section (with e ^ 0). The result is guaranteed to be finite as e —> 0. Recall that the Born cross section G^^^ consists of an e dependent factor C(e) times the Born cross section in 4 dimensions, where C(e) arises from such sources as the integration over the lepton angles in the Drell-Yan process. A convenient way to manage the calculation is to factor C(e) out of the first order cross section also. Then the prescription is to remove the 1/e pole in G^^^(e)/C(e), set e = 0, and multiply by C(0) = 1. Thus we see that a function of e that is a common factor to G^j and G\^ cancels in the physical hard scattering cross section, as was claimed after eq. (17). When calculating G^^\ it should be noted that there are contributions involving self energy graphs on the external lines, as in fig. 2. The total of all external line corrections gives a factor of y/z2 for each external quark (or antiquark) line and y/zs for each external gluon line. Here Z2 and z^ are the residues of the poles in the renormalized quark and gluon propagators. In the massless theory these have infrared divergences. For example the value of Z2 in massless QCD in Feynman gauge is z, = l+^^+Oial). (27) Then the contribution of the self energy graphs to G^^^ is a factor 2as/37re times the Born cross section.
15 3. RELATION TO THE RENORMALIZATION GROUP The prescription (26) for removing infrared poles is intimately related to the ^ dependence of H\i^ — that is, to the behavior of H)^^ under the renormalization group. In this section, we display this connection and show how it leads to the approximate invariance of the computed cross section under changes of /x. (Of course, the complete cross section, to all orders of perturbation theory, is exactly invariant under changes of /x. What we are now concerned with is the behavior of a finite-order approximation.) We recall that the Born cross section G\^ = H^^ contains some /i dependence from the factor C(fi/Qy e), as specified in eq. (16). The one loop cross section G^^ contains this same factor, and we can simply factor it out of eq. (26) and set it to 1 when we set e = 0 at the end. In addition, G^j contains a factor fi^^ from the loop integration. 4-2€ k. (28) The (e'^/J,'^ /AttY factor multiplies the 1/e poles in G^j . Writing -//2' = -+2^ ln(M) + 0(€), (29) and reading off the value of A from eq. (26), we find the /x dependence of G^^ - and thus of ^^j : Hil^(xA,XB,Q; ^) = Hil\xA,XB,Q; 1) (30) Q Ef^UPSliU)HiT(^,xs,Q Here we have set e = 0 and have suppressed the notation indicating e dependence; we have also noted that H^^^ does not depend on /x when e = 0, so we have suppressed the notation indicating /x dependence in H^^\ We see that H^^^ contains logarithms of /x/Q. If ^ is fixed while Q becomes very large, then these logarithms spoil the usefulness of perturbation theory, since the large logarithms can cancel the small coupling a;s(/x) that multiplies H^^\ For this reason, one chooses /x such that ln(/x/Q) is not large. For example, one chooses fx = Q OT perhaps /x = 2Q or /x = Q/2.
16 The freedom to choose /x results from the renormalization group equations obeyed by H and fa/AiO- '^^^ renormalization group equation for Hah is /x— Hah(xA,XB,Q]^,as(fi)] (31) -X^ / ^CsPd/hiCB^Oisil^)) Had(xA,-^,Q;Q,as(fJ')] • Here Pc/a(^5<^s(A*)) is the all orders Altaxelli-Parisi kernel. It has a perturbative expansion Pc/aii,asifi)) = ^ Pl}liO + ... (32) where P^/l(0 is the function that appears in eq. (23). Thus at lowest order the renormalization group equation (31) is a simple consequence of differentiating eq. (23). Parton distribution functions also have a fi dependence, which arises from the renormalization of the ultraviolet divergences in the products of quark and gluon operators in the definitions of these functions, given in eqs. (43) and (44) below. The renormalization group equation for the distribution functions is The physical cross section does not, of course, depend on /x, since p, is not one of the parameters of the Lagrangian, but is rather an artifact of the calculation. Nevertheless, the cross section calculated at a finite order of perturbation theory will acquire some jj, dependence arising from the approximation of throwing away higher order contributions. To see how this comes about, we differentiate eq. (11) with respect to // and use eqs. (31) and (33). This gives d dcr dfj, dQMy E f ^^^ f '-t f ^^- a,6,c *^^^ -^^^ ^^ *^^fi L A \ f ^ A ^ Xi IJ xPa/c{U,0'M) fc/A[-T-,fJ-jHah{y-,J-,Q;Q,as{fJ.)\ fb/B{(B,ti) a,h,c ^^^ J^a/Ia Jxb
17 ^A XB ^ 1^ X fa/A{U,l^)Pc/a(CA,Oisifi)) //"eft ( T-y, J", 0; ^ , Q^s(/^) ) fb/B^^B^fJ') 4- B terms. (34) Here the two terms shown relate to the evolution of the partons in hadron A. As indicated, two similar terms relate to the evolution of the partons in hadron B. We now change the order of integration in the second term to put the (a integration inside the (a integration, then change the integration variable from ^a to (a = CaCaj and finally reverse the order of integrations again. This gives d dcr d/x dQ2dy a,b,c -^^^ X Pa/c{CA,Ois{fi)) /c/aI ^,iW l^aftl 7^,7^,Q;^,Q^s(iw) 1 fb/BUB,IJ') X fa/A\ ■T^.fJ'jPc/aiCAyOisilJ')) f^cftf 7^,7^,0; Q,Q^s(/^)l fb/B{^B,IJ') -h B terms. (35) We see that the two terms cancel exactly as long as Pa/b si-nd Hab obey the renor- malization group equations exactly. Now, when Hab is calculated only to order a^, it only obeys the renormalization group equation (31) to the same order. In this case, we will have when the parton distribution functions obey the renormalization group equation with the Altarelli-Parisi kernel calculated to order a^ or better. One thus finds that the result of a Born level calculation can be strongly /x dependent, but by including the next order the jj, dependence is reduced. We have argued that one should choose fi to be on the order of the large momentum scale in the problem, which is Q in the case of the Drell-Yan cross section. We have the right to choose fi as we wish because the result would be independent of fi if the calculation were done exactly. The choice ^i ^ Q eliminates the potentially large logarithms in eq. (30). Another choice is often used. One substitutes for /x in eq. (11) the value y/s = y/^A^B^- We now have a value of fi that depends on the integration variables in the factorization.
18 Let us examine whether this is valid, assuming that Pa/h and Hah are calculated exactly. We replace /x by t^iKU.ia) = ^l\-^ (yUi^y, 0<A<1. (37) At A = 0 we have a valid starting point. When we get to A = 1 we have the desired ending point. The question is whether the derivative of the cross section with respect to A is zero. Applying the same calculation as before, we obtain instead of eq. (34) the result d da ^ [' [' AU [\, 1 , (UiBS a,6,c "^^^ ^^a/^a ^xb ^ \ H'O X fa/A {Uy K^y U, (b)) Pc/a (Ca, Ois(fi(X, U,(b))) VqaU ^b Q ) -\- B terms. Now making the same change of variables as before, we obtain d darn dA dg2dy (a(bs ^ f '^^ f 77 f '^' I \ .3 X ''"(li'if'**^^^'^^'"*')''"'*^'''''*'^-"*"" -E/'«^/T/'^.i \ (Af-tl In X fa/A f ^,MA,a/CA,^B)j ■Pc/a(a,as(MA,a/G,<jB))) "" ^'' lil' if' ^' ^^^'%^^'^^\ <f^)) f^/BUB.^(A,a/a, ^b)) 4- jB terms. (39)
19 We see that the cancellation between the two terms has been spoiled, first by the differences in the values of /x(A,...) in the two terms, but more importantly by the differences in the arguments of the logarithm in the two terms. We conclude that the substitution of s for /x^ results in an error of order ag no matter how accurately the hard scattering cross section is calculated. This is not a problem If the hard scattering cross section is calculated only at the Bom level, which is, in fact, commonly the case. However, it is wrong to substitute s for ^^ when a calculation beyond the Born level is used. 4. THE PARTON DISTRIBUTION FUNCTIONS The parton distribution functions are indispensable ingredients in the factorization formula (11). We need to know the distribution of paxtons in a hadron, based on experimental data, in order to obtain predictions from the formula. In addition, we need to know the distribution of paxtons in a parton in order to calculate the hard scattering cross section Hah- The hard scattering cross section is obtained by factoring the parton distribution functions out of the physical cross section. Evidently, the result depends on exactly what it is that one factors out. 4.1 Operator Definitions In this section, we describe the definition for the parton distribution functions that we use elsewhere in this chapter. A more complete discussion can be found in Ref. 9. In this definition, the distribution functions are matrix elements in a hadron state of certain operators that act to count the number of quarks or gluons carrying a fraction ( of the hadron's momentum. We state the definition in a reference frame in which the hadron carries momentum P^ with a plus component P'^', a minus component P~ = m^ /2P'^, and transverse components equal to zero. (We use P± = (PO ± P^)/V2). The definition may be motivated by looking at the theory quantized on the plane x^" = 0 in the light-cone gauge A'^ = 0, since it is in this picture that field theory has its closest connection with the parton model^^K In this gauge, ^ = 1, where ^ is a path-ordered exponential of the gluon field that appears in the definition of the parton distributions. The light-cone gauge tends to be rather pathological if one goes beyond low order perturbation theory, and covaxi- ant gauges are preferred for a complete treatment. However quantization on a null plane in the light-cone gauge provides a useful motivation for the complete treatment. In this approach the quark field has two components that represent the independent degrees of freedom; 'y'^il^(x) contains these components and not the other two. One can expand the two independent components in tenns of quark
20 destruction operators b(k'^,k±,s) and antiquark creation operators d(k'^jk±,sy as follows: j+i;{0,x-,xj,) = (27r)-'Y.J "^ dk+ . dk± 2k+ 3 X {y+U(k,s)e-'''''bik+,kjL,s) + -y+V{k,s)e+'''''d(k-^,k±,sy}. (40) The quark distribution function is just the hadron matrix element of the operator that counts the number of quarks. f,/AiO d^ = (2t)-' J2 ifprj / '^^^ (PI KC^+. *x, s)^b{^P+,kjL,s) IP). (41) In terms of t/^(x), this is f,/Aii) = ^j da;-e-«^^^" (P 1^(0, X-, Ox) 7+ ^-(0,0,0±)\P). We can keep this same definition, while allowing the possibility of computing in another gauge, by inserting the operator X g P exp I ig J dy-A+(0, y", Ox)te \ , (42) where P denotes an instruction to order the gluon field operators i4j(0,y~,0x) along the path. The operator Q is evidently 1 in the A"^ = 0 gauge. With this operator, the definition is gauge invariant. We thus arrive at the definition^'^^^ /,m(0 = ^ /clx-e-'«^^-"(P|^(0,a;-,0x)7+aV(0,0,0x)|P) (43) For gluons, the definition based on the same physical motivation is /,m(0 = 2;4p+y dx-e-'«^^-" (P |Fa(0, X-,0x)+''a^t ^6(0,0, Ox)/IP), (44) where F^^, is the gluon field strength operator and where in Q we now use the octet representation of the SU(3) generating matrices t^
21 ^.jB Feynman rules and eikonal lines The Feynman rules for parton distributions are derived in a straightforward manner from the standard Feynman rules. Consider, for instance the distribution fg/g of a quark in a quark. To compute this quantity in perturbation theory, we use tlu' following identity satisfied by any ordered exponential, P expjz^ / dXn'A(Xn^)^ [P exp|z^ / d\n'A((X-\-r])n^)\V P expUg f dAn-A(An^)| . (45) Using (45) in eq. (43), for instance, enables us to insert a complete set of states and write /,/,(0 = ^ /dx-e-«''*^" V(P|#(0,a;-,0x)|n)7+(n|$(0,0,0x)|P), (46) 47r " n where we define ^ as the quark field times an associated ordered exponential, ^(x) = tl^ix) P exp|i^ / dA V'A(x + Xv^)\ , (47) where v^ = 9-^ and A^(x) = A^(x)tc. To express the matrix elements in eq. (46) in terms of diagrams, we note that by (47) the gluon fields in the expansion of ^ are time ordered by construction. Expanding the ordered exponentials, and expressing them in momentum space we find oo . oo n P exp{ig J^ dXn- ^(An")} = 1 + P E 11 _/ (2^ Sn-Aiq^ 1 " • Ej=i 97 + «'« ' (48) where we define the operator P on the right-hand side of the equation to order the fields with the lowest value of i to the left. From eq. (48) we can read off the Feynman rules for the expansion of the ordered exponential^'^^^. They are illustrated in fig. 3. The denominators n-^ • qj + ie axe represented by double lines, which we shall refer to as "eikonal" lines. These lines attach to gluon propagators via a vertex proportional to —ign^. Fig. 3(a) shows the formal Feynman rules for eikonal lines and vertices. In fig. 3(b), we show a general contribution to fg/g, as defined by eq. (46). The positions of all the explicit fields in eq. (46) differ only in their plus com- l)onents. As a result, minus and transverse momenta are integrated over. (They may thought of as flowing freely through the eikonal line.) The plus momentum
22 (a) q > q-u + ie igU«tij q J -1 q-u-ie (b) P Fig. 3. (a) Feynman rules for eikonal lines in the amplitude and its complex conjugate, (b) A general contribution to a parton distribution. flowing out of vertex 1 and into vertex 2, however, is fixed to be ^P"*". No plus momentum flows across the cut eikonal line in the figure. Fig. 4 shows the one loop corrections to fq/q{C)-
23 (a) (c) (d) Fig. 4. One loop corrections to quark distribution, eq. (43). To be explicit, fig. 4(b) is given in n dimensions by 14/ d^o ^^ , .^ ^, os « qf + it tr[(j^-rf)(-W"¥7^)](w'n'') e^^ — I (49) u ' q — le where Na^ is the polarization tensor of the gluon. By applying minimal subtraction to eq. (49) and the similar forms for the other diagrams in fig. 4, we easily verify vx{. (23) for fq/q- Gluon distributions are calculated perturbatively in a similar manner. We will need the concept of eikonal lines again, when we discuss the proof of factorization in gauge theories. ^.S Renormalization The operator products in the definitions (43) and (44) require renormalization, as discussed in Ref. 9. We choose to renormalize using the MS scheme. Of course,
24 renormalization introduces a dependence on the renormalization scale /i. The renormalization group equation for the iaIA is the AltareUi-Parisi equation (33). A complete derivation of this result may be found in Ref. 9. The one-loop result, eq. (23), can actually be understood without looking at the details of the calculation. At order ag, one has simple one loop diagrams that contain an ultraviolet divergence that arises from the operator product, but also contain an infrared divergence that arises because we have massless, on-shell partons as incoming particles. The transverse momentum integral is zero, due to a cancellation of infrared and ultraviolet poles, which we may exhibit separately: J (27r)2-2e k^2 - 4^ j ,^^ ,^^ > • (50) In this way, we obtain fa/kii; e) = Sai6il-0+{—-—] — Pi%iO - counterterm + 0(a,^). (51) The coefficient of l/cuv is the * anomalous dimension' that appears in the renormalization group equation, that is, the AltareUi-Parisi kernel. Following the renormalization scheme, we use the counter term to cancel l/cuv term. This leaves the infrared 1/e, which is not removed by renormalization, /a/»(^;e) = 6a, 6(1-0-~ Pi)liO + 0{al). (52) 4-4 Reldtion to Structure Functions Let us now consider the relation of the parton distribution functions to the structure functions measured in deeply inelastic lepton scattering. If we use the definition of parton distribution functions given above, then the structure function F2 is given by the factorization equation (2). At the Born level, the hard scattering function is simply zero for gluons and the quark charge squared, e^, times a delta function for quarks. Thus the formula for F2 takes the form ^-1 X 'di -r •/- « "^ VC /'/ (53) + Oial). The sums over j run over all flavors of quarks and antiquarks. Gluons do not contribute at the Born level, but they do at order Ofg, through virtual quark-aiitiquark pairs. The hard scattering coeflSicients Cjb can be obtained by calculating (at order
25 G(n) deeply inelastic scattering from on-shell massless partons, then removing the Infrared divergences according to the scheme discussed in Sect. 2. The explicit form of the perturbative coefficients Cjt is'*^ Cjk(z,l) Sjk- 11-j-z 2 1 In 1-^ 3 1 3 + Cj,(z,l) = -l^l{z' + il-zf}l^ln 1-z + 1} - 3^(1 - z) (54) where the plus subscript to the bracket in the first equation denotes a subtraction that regulates the z —^ 1 singularity, dz [C{z)]+ h{z) = / dz [C{z)]+ h(z)e(z > x) 0 / dz C{z) ^h(z)e(z >x)- /i(l)|. (55) 4.5 Other Parton Distributions The definitions (43) and (44) are the most natural for many purposes. They are not, however, unique. Indeed, any function gb/Aiv)^ which can be related to fa/A(^) t)y convolution with ultraviolet functions Dab(x/y^Q/lJ') in a form like 9a/A{x) = Y1 (^y/y)^ab{x/y,Q/n,as{fi))fb/A{y) , L J X (56) is an acceptable parton distribution^"^1. The hard scattering functions calculated with the distributions Qb/A will differ from those calculated with fa/A^^)-, but this difference will itself be calculable from the functions Dab as a power series in as(Q)- The most widely used parton distribution of this type is based on deeply inelastic scattering, and may be called the DIS definition. The definition is DIS /i/i?(^,M) de fj,A{x,n) + Y.f ^hlAitt^) ^ Cji (;,!)+ 0{al). i i (57) for quarks or antiquarks of flavor j. Comparing this definition with eq. (53), we see that X -1 F2{x,Q) = Y.^] ffjiix,Q) + 0{al). (58) That is, we adjust the definition so that the order a^ correction to deeply inelastic scattering vanishes when fi = Q. It is not so clear what one should do with the gluon distribution in the DIS scheme. One choice^^J is rDIS J 9/A (^, /^) = fg/Aix, /^) - ^ ^ / r J X 'fh/AiC,f^)^C,Jj,l]+0{al). (59)
26 This has the virtue that it preserves the momentum sum rule that is obeyed by the MS parton distributions^!, a -^0 (60) If one wishes to use parton distribution functions with the DIS definition, then one must modify the hard scattering function for the process under consideration. One should combine eqs. (52) and (59) to get the DIS distributions of a parton in a parton, then use these distributions in the derivation in Sect. 2. It should be noted that there is some confusion in the literature concerning the term +1 that follows the logarithm in Cjg in eq. (54). The form quoted is the original result of Ref. 4, translated from moment-space to z-space. In the calculation with incoming gluons, one normally averages over polarizations of the incoming gluons instead of using a fixed polarization. This means that one sums over polarizations and divides by the number of spin states of a gluon in 4 — 2e dimensions, namely 2 — 2e. If, instead, one divides by 2 only, one obtains the result (54) without the -j-l, which may be found in Ref. 14. This does no harm if, as in the case of Ref. 14, one wants to express the cross section for a second hard process in terms of DIS parton distribution functions and if one consistently divides by 2 instead of 2 — 2e in both processes. However, it is not correct if one wants to relate the DIS structure functions to MS parton distribution functions, defined as hadron matrix elements of the appropriate operators, renormalized by MS subtraction. 5. FACTORIZATION FOR cj)^ THEORY In this and the next section, we study the factorization theorem in a (j>^ theory for n < 6 space-time dimensions. First we show how the factorization theorem comes about for one-loop corrections in deeply inelastic scattering, and compare the field theory to the parton model. In the next section, we will present a reasonably complete but compact derivation of the factorization theorem in deeply inelastic scattering to all orders of perturbation theory. The scalar theory allows us to study these issues in a simplified but highly nontrivial context. As emphasized above, the purpose of the factorization theorems is to separate long-distance behavior in perturbation theory. In the scalar theory, as we shall see, this behavior is associated with partons that are coUinear to the observed hadrons. The organization of such "collinear divergences" is central to factorization in all field theories, but in gauge theories they are joined by "soft" partons, associated with infrared divergences. Indeed, the basic problem in gauge theories is to show how that infrared or "soft" divergences cancel (see Sect. 9). In
27 ^' theory the infrared problem is absent, so that studying this theory allows us to liiidy the basic physics of factorization in the simplest possible setting. The Lagrangian is £ = i {d(i>f - \rn'^<t>^ - ~5^(A*^e^/47r)'/2<?!>^ + counterterms . (61) We will use, where necessary, dimensional regularization, with space-time dimen- nion n = 6 — 2e. It is worth recalling that at n = 6 the theory is renormalizable, while for n < 6 it is superrenormalizable. We shall not concern ourselves with the theory for n > Q where it is nonrenormalizable by power-counting, /i is a mass which enables us to keep g dimensionless as we vary n. We will renormalize the tlicory with the MS prescription. We use the factor (/i^e'''/47r)^/^ rather than the more conventional /i% so that we can implement MS renormalization as pure pole counterterms. (For convenience, we will define the /i<^ counterterm that renormal- Izes the tadpole graphs by requiring the sum of the tadpoles and their counterterm to vanish.) We define /i = /ix/eV^TT. (62) 5.1 Deeply inelastic scattering Our model for deeply inelastic scattering consists of the exchange of a weakly interacting boson, A, not included in the Lagrangian (61). This is illustrated (liagrammatically in the same way as for QCD, in fig. 5. The weak boson couples to the <l> field through an interaction proportional to hAcfP". There is then a single structure function which we define by F{x,Q) = ^ /cl«ye-'"{p|y(y)i(0)|p}, (63) where j = i(j) . The momentum transfer is g^, and the usual scalar variables are defined by Q^ = —q^ and x = Q^/2p-g, with p^ the momentum of the target. We will investigate the structure function in the Bjorken limit of large Q with x fixed, and our calculations will be for the case that the state |p) is a single (j) particle (with non-zero mass, as given in eq. (61)). When Q is large, each graph for the structure function behaves like a polynomial of \n{Q/m) plus corrections that are nonleading by a power of Q. Factorization is possible because only a limited set of momentum regions of the space of loop and final state phase space momenta contribute to the leading power. First we will explain the power counting arguments that determine these "leading regions", and how they are related to the physical arguments of the parton model. The tree graph for the structure function is easy to calculate. It is Fo = Q^S(2p-q -I- ^2) = S(x - 1). (64)
28 Fig. 5. Deeply inelastic scattering. The one-loop "cut diagrams" (as defined in Sect. 2.2 above) which contribute to F are given in fig. 6. Each of these diagrams illustrates a different bit of the physics, so we shall treat them in turn, starting with the "ladder" correction, fig. 6(a). 5.2 Ladder Graph and Us Leading Regions The Feynman integral for the cut diagram fig. 6(a) is (65) Although for nonzero m this integral is finite, it will prove convenient to retain the dimensional regularization, in order to display some very important dimension- dependent features of the Q —> oo limit. Equation (65) is calculated conveniently in terms of light-cone coordinates. Without loss of generality, we may choose the external momenta, q^ and p^ to be pf" = (p+,m2/2p+,0j_) and qf" = (-xp-^,Q^/2xp-^,0±). Notice that this formula for q^ corresponds to a slight change in the definition of or, which we now define by Q^/2p-q = x/{l — xm^/Q'^). At leading power in Q, there is no difference, but at finite energy our formulas will be simplified by this choice. The (^-functions in (65) can be used to perform the k± and k~ integrals. Then if we set ^ = k'^ /p^ ^ we find E 2(a) 9^ f Q' \"' 1 f^_m^x'''-' QA'K^ ye-^p?x(\ - x)J r(2-€) V Q L (66)
29 (a) (b) (c) (d) (e) Fig. 6. One-loop corrections to deeply inelastic scattering. For graphs (b), (d) and (e), we also have the hermitian conjugate graphs. where the limits ^^in and ^max are given by 1 -\- X 1 — X 2 2 4771^ X {1 - x){Q^ i-m^x) ' (67) In this form, we can look for the leading regions of the ladder corrections. To do this, it is simplest to set the mass to zero, find the leading regions, and then check
30 back as to whether we must reincorporate the mass in the actual calculation. So, to lowest order in A = m?' /Q^ ^ {p&) becomes "^^C) 64x3 VeV^(l-^)y r(2 - e) X(i+^) ^ {^ - x[l + x(l - OA]F ' (68) To interpret this expression, we must distinguish between the renormalizable (n = 6, e = 0) and superrenormalizable (n < 6, e > 0) cases. In the super-renormalizable case, (e > 0), the leading-power contribution (Q/fi)^ comes from near the endpoint ^ = a:(l -j- A). The bulk of the integration region, where ^ — x = 0(1) is suppressed by a power of Q. The integral is power divergent when ?7i = 0, and clearly we cannot neglect the mass. Now consider the renormalizable case, n = 6. When we set e = 0, eq. (68) has leading power (Q°) contributions from both the region (, — x near zero, where, as above, the mass may not be neglected, and the region ^ — x = 0(1), where it may. In the former region, the integral is logarithmically divergent for zero mass, but since the nonzero mass acts as a cutoff, the two regions ^ ~ x and ^ — a: = 0(1) should be thought of as giving contributions of essentially equal importance. We now interpret these dimension-dependent leading regions. 5.3 Collinear and Ultraviolet Leading Regions; the Parton Model To see the physical content of the leading regions identified above, it is useful to relate the variable ^ in (66) to the momentum k^^ by the relations k-= -' 2p+(l - x) (69) ,^^2 ^ g^(i - oii - x) _ m^[(i-o^ + e(i-x)] x{l — x) 1 — X Changing variables to k± , we now rewrite the integral eq. (6S) in a form which is accurate to leading power in A for n < 6^ ^'^^^ - 64^f(2^ X ^ (fcx^-fm2(l-.« + x2))2 • ^^^) We emphasize that this expression is accurate to leading power in the region k±^/Q^ = 0(A), which is suflSicient to give the full leading power for ?? < 6, although not for n = 6, where larger k± also contribute. Now let us choose a frame in which p'^ is of order Q. When (^ —> a:, the components of k^ are of order (Q, (^ — x)Q, Q\/(, — x)^ and at its lower limit, (,— x
31 is of order vn?" jQ^. Hence, in the region that gives the sensitivity to ?7i, kx_ is small, and k^ is ultrarelativistic and represents a particle moving nearly collinear to the incoming momentum, p'*. In addition, the on-shell line, of momentum p^ — fc^, is nearly collinear to the incoming line as well. In fact, when m and fcx are both zero, h^ is also on the mass shell. The energy deficit necessary to put both the momenta h^ and p^ — h^ on shell is of order hx_ jQ in this frame. Thus, in this frame, the intermediate state represented by the Feynman diagram lives a time of order Qlh±^^ which diverges in the collinear limit. The space-time picture for such a process is illustrated in fig. 7, and we see a close relation to the parton model, as discussed in Sect. 1.4, which depends on the time dilation of partonic states. Partonic states whose energy deficit is much greater than m in the chosen frame correspond to ^ — x of order unity, and do not contribute at leading twist.. Thus here, as in the parton model, there is a clear separation between long-lived, time-dilated states which contribute to the distribution of partons from which the scattering occurs, and the hard scattering itself, which occurs on a short time scale. t-z-0 Fig. 7. Space-time structure of collinear interaction. From this discussion, the collinear region, which is the only leading region when n is less than 6, is naturally described in parton model language. When n = Q^ the collinear region remains leading. In addition, however, all scales between k_\_ — m and fc^ = Q contribute at leading power, and there is no natural gap between long- and short-distance interactions. When ^ — x is order unity, fc^ is separated from p^ by a finite angle, and corresponds to a short-lived
32 intermediate state, where (p — k)^ ^ m^. This leading region, which is best described as "ultraviolet", is not naturally described by the parton model. But, in an asymptotically free theory (as {<I>^)q is), such short-lived states may still be treated perturbatively. We shall see how to do this below. In summary, the ladder diagram shows two important features: a strong correspondence with the parton model from leading collinear regions for both su- perrenormalizable and renormalizable theories and, for the renormalizable theory only, leading ultraviolet contributions, not present in the parton model. 5.4 Parton distribution functions and parton model We shall now freely generalize the results for the one loop ladder diagram. Indeed, as we shall see in Sect. 7, some of the dominant contributions to the structure function arise from (two-particle-reducible) graphs of the form of fig. 8. A single parton of momentum k^ comes out of the hadron and undergoes a collision in the Born approximation. If we temporarily neglect all other contributions, we find that Fi^,Q) = / 0^Hk,p)H{k,q) + 0(1/Q<"), (71) where $ represents the hadronic factor in the diagram and H the hard scattering (multiplied by the factor of Q^/27r in the definition of the structure function): Hik, q) = QH((k -f Q)2 - m^). (72) For (j)^ with n < 6, as in the parton model, the parton momentum k^ is nearly collinear to the hadron momentum p^. This implies that we can neglect m and the minus and transverse components of k^ in the hard scattering, so that we can write H = Hix/^,Q) = 6a/x-l), (73) and hence F{x,Q)= I d^|/p+^^^$(fc,p) S{ax-1) + 0{1/Q''). (74) Here we define ^ = k"^ jp'^. The limits on the ^ integral are 0 to 1, since the final state must have positive energy. We therefore define the parton distribution function (or number density): , / Ak'A^kx ---^k, p) s{Cp+/ k+-1).
33 Fig. 8. Dominant graphs for deeply inelastic scattering in parton model. With this definition (74) becomes F{x,Q) 4.^ mHix/(,Q) + 0{\/Q'') (76) /(x) + 0(1/Q") (n < 6). As we shall see in the next section, the factorization theorem is also true in the renormalizable theory, F{x, Q) 1^ f{i)H{xli,Q) + 0{llQ'^) (n = 6) , (77) where now H is nontrivial. The dominant processes that contribute are illustrated by fig. 9, which generalizes the parton model only to the extent of having more than just the Born graph for the hard scattering. These processes first involve interactions within the hadron that take place over a long time scale before the interaction with the virtual photon. Then one parton out of the hadron interacts over a relatively short time scale. We now note that eq. (75) can be expressed in operator form as m ip + oo — — 4 i^ n I *i 27r dy-e-*«P ''>|<^(0,y-,Ox)^(0)|p). (78) OO
34 P Fig. 9. Dominant regions for deeply inelastic scattering in {(I>^)q theory. This is the definition which we use for all n < 6. Of course in the renormalizable theory n = 6 renormalization will be necessary^J. The definition (78) is precisely the analog for (f)^ theory of those we gave in Sect. 4 for QCD. It involves an integral over a bilocal operator along a light-like direction. The graphs for /(^) up to one- loop order are shown in fig. 10. Feynman rules are the same as for the gauge theory, but without eikonal lines. It is natural to interpret f{x)dx as the number of partons with fractional momenta between x and x -f- dx. This interpretation is justified by the use of light front quantization^^J, as we saw in Sect. 4. Note that although the definition picks out a particular direction as special to the problem, it is invariant under boosts parallel to this direction. The ladder graph, fig. 10(c) gives /, g'P'' (2t)' -/ a-n.K.V.--i)"'t"-1-r'"^ (79) The ^-functions may be used to perform the k"^ and k integrals, after which we obtain /, g' (eVmi-0 647r3 r(2 - e) oo dk ± (k^y-^ 0 [A:j.'+m2(l-e-he)] 2 ' (80)
35 (a) (b) (c) Fig. 10. Low-order graphs for part on distribution in (j)^ theory. which matches eq. (70) in the k± —^ 0 limit. That is, we have constructed the par- ton distribution to look like the structure function at low transverse momentum. The significance of this fact will become clear below. For n < 6, eq. (80) is the same as the full leading structure function (70), and it exemplifies the validity of the parton model in a super-renormalizable theory. When n = 6^ however, there is a logarithmic ultraviolet divergence from large k± in (80). So, in the renormalizable theory we must renormalize f{C). (Since / is a theoretical construct defined to make treatments of high-energy behavior simple and convenient, we are entitled to change its definition if that is useful; in particular, we are allowed to include renormalization in its definition.) If we use the MS scheme, then the renormalized value of fc for nonzero mass is: R[fc] = - g 647r ai - 0 In m\i-i+e) F (81) while for zero mass it is (compare eq. (23)) R[fc] = - ^^e(i-oi. 647r (82) Now let us see what this means in the calculation of the hard part, as in Sect. 2. To calculate the hard part, we expand eq. (77) in powers of (7^, as in eq. (24), and solve for H^^\x/^^Q). There is some question about what to do with the
36 higher-twist terms, proportional to powers of m/Q. The simplest method is to simply define H^'\x/(, Q) = \f^'\x/(, Q) - M , (83) m=0 Comparison of eqs. (70) and (80) shows that the low k± region, which is the only leading region which is sensitive to the mass, cancels between F^^^ and p^\ at the level of integrands. Thus, for the combination on the right hand side of eq. (83), it is permissible to set the mass to zero. It is thus practical to set the mass to zero at the very beginning. It should be kept in mind, however, that this is a matter of calculational convenience, rather than principle. The factorization theorem allows us to calculate mass-insensitive quantities whatever the masses we choose, since all sensitivity to these masses will be factored into the parton distributions. Now let us return to the remaining diagrams in fig. 6, treating first the "final state" interactions, fig. 6(b) and (c). 5.5 Final state interactions The graphs of fig. 6(b) and (c) have a self-energy correction on the outgoing line, the final state cut either passing through the self energy or not. As we will show, these graphs have contributions that are sensitive to low virtualities and long distances. However, they are not of the parton model form, and do not naturally group themselves into the parton distribution for the incoming hadron. We will see, however, that there is a cancellation between the two graphs such that they are either higher twist (n < 6), or may be absorbed into the one-loop hard part (n = 6). The self energy graphs give simply the lowest order graph, 6{x — 1), times the one-loop contribution to the residue of the propagator pole: i^2(6) = S{X - 1) r r. r., 2, 2 ^(1-^) dk_L'k 1287r3yo J^ - - [k_^'^m^l-z + z'^)]^ (84) -f-counter term]. We may derive this expression in either of two ways. One way is to combine the denominators of the two propagators in the loop by a Feynman parameter before performing the k^ and k~ integrals. Then z is the Feynman parameter. Alternatively, we may first use contour integration to perform the k~ integral. Then we get (84) by writing k'^ = z (p"'" + ^"'"). The integral is the same by either derivation. But the second method shows that we may interpret 2 as a fractional momentum carried by one of the internal lines. Since we will be concerned with the low k±^ region, while the counterterm, if computed with MS renormalization, is governed by the k± —^ 00 behavior of the integrand, we do not write the counterterm explicitly.
37 There is clearly a significant contribution in (84) from small k±, where the mass m is not negligible. The cut self-energy graph, in fig. 6(c), will also contribute in this region. Now the region of low /?_[_ represents the effect of interactions that happen long after the scattering off the virtual photon, and it is reasonable to expect that interactions happening at late times cancel, since the scattering off the virtual photon involves a large momentum transfer Q and therefore should take place over a short time-scale. However, the uncut self-energy graph only contributes when x is exactly equal to 1, while the cut self energy graph has no ^-function and thus contributes at all values of x. This mismatch is resolved when we recognize that we should treat the values of the graphs as distributions rather than as ordinary functions of x. That is, we consider them always to be integrated with a smooth test function. Mathematically, this is necessary to define the ^-function. Physically, the test function corresponds to an averaging with the resolution of the apparatus that measures the momentum of the lepton that is implicitly at the other end of the virtual photon. After this averaging, a measurement of the lepton momentum does not distinguish the situation where a single quark goes into the final state from the situation where the quark splits into two. We therefore consider an average of the structure function F{x) with a smooth function t{x): (t,F) = f dxt{x)F{x). (85) Then the contribution of the self-energy graph is (t,F2(6)) = t(l)F2(6) . (86) Next we compute the cut graph, fig. 6(c). Its value is 2/n2 t{x) 6{P - m^) S{{p + q-k)'^~m?) ^ ' [{p + qf - m2]2 To make this correspond with the form of (84), we define z = k"^/{p^ -f- q^)^ and then use the ^-functions to do the h~ and x integrals. After some algebra, and after the neglect of terms suppressed by a power of Q, we find where the Bjorken variable x satisfies X 2 , _2 - ~^ A;_L + m Q^z{l-z)\ (89)
38 We now axld the two diagrams to obtain: 9' 1287r Jo Jo oo dkA_'^k±'^[t{x)x^ -t{l)] z{l-z) [it±'+m2(l-^ + ^2)]2 4- counter term. (90) In the region A;j_ <C Q, a: is close to one, and there is a cancellation in the integrand of eq. (90). The cancellation fails when z is close to zero or one, but the contribution of that region is suppressed by a power of Q. We are therefore permitted to set m = 0 in the calculations of the graphs, after which a calculation (with dimensional regularization to regulate the infrared divergences that now appear in each individual graph at k± = 0) is much easier. 5.6 Vertex correction Finally, we consider the vertex correction fig. 6(d). It has the value + counterterm 1 [m2 - Jb2] [m2 - (p + ky] [m2 - (p + it + ^)2] -S{x - 1) 9 647r / dai / Jo Jo da2 In m2(l - ai - a2 - {ai + ^2)^) + Q^Q:2«i f^' (91) where we work in c? = 6 — 2e space-time dimensions to regulate the ultraviolet divergence. When Q —> 00, we can clearly neglect the mass, so that we have (at € = 0) E 2(d) 8{x - 1) 9 1287r3 ln^-3 + 0 (92) F2{d) is higher twist for e > 0. The graph fig. 6(e) is related to fig. 6(d) by moving the final state cut so that it cuts the inner lines of the loop. We will not calculate it explicitly. But when that is done, the quark mass can be neglected, just as for the uncut vertex. In summary, the only diagram from fig. 6 which corresponds to the parton distribution is the ladder diagram, fig. 6(a). Non-ladder diagrams are either higher twist, or contribute only to the hard part (renormalizable case). These results are consistent with the structure of fig. 8 and fig. 9, which show the structure of regions which give leading regions for n < Q and n = Q^ respectively. As we shall show in the next section, it is this structure which enables us to prove that the
39 parton distributions eq. (78) absorb the complete long-distance dependence of the structure function. 6. SUBTRACTION METHOD To establish a factorization theorem one must first find the leading regions for a general graph. We will see how to do this in Sect. 7. The result, for deeply inelastic scattering in a nongauge theory, has been summarized by the graphical picture in fig. 9, and it corresponds closely to our detailed examination of the order g^ graphs. It can be converted to a factorization formula if one takes sufficient care to see that overlaps between different leading regions of momentum space do not matter. An approach that makes this process clear is due to Zimmermann'^J. To treat the operator product expansion (OPE), he generalized the methods of Bogoli- ubov, Hepp, Parasiuk, and Zimmermann (BPHZ)'^'^^J that were used to renor- malize Feynman graphs. (Although the original formulation was for completely massive theories with zero momentum subtractions, it can be generalized to use dimensional continuation with minimal subtraction^^J. This allows gauge theories to be treated simply.) In the case of deeply inelastic scattering a very transparent reformulation can be made in a kind of Bethe-Salpeter formalism^'^^, although it is not clear that in the case of a gauge theory the treatments in the literature are complete. In this section, we will explain these ideas in their simplest form. There are two parts to a complete discussion: the first to obtain the factorization, and the second to interface this with the renormalization. We will treat only the first part completely. In (<^^)6 theory, renormalization is a relatively trivial affair. Moreover, if we regulate dimensionally, with e just slightly positive, one can choose to treat as the leading terms not only contributions that are of order Q^ (times logarithms) as Q —> oo, but also those terms that are of order Q to a negative power that is of order e. The remainder terms are down by a full power of Q"^, and can be identified as "higher twist". In this way one has the same structure for the factorization, without the added complications of renormalization. Zimmermann's approach is to subtract out from graphs their leading behavior as Q —> oo. This is a simple generalization of the renormalization procedure that subtracts out the divergences of graphs. From the structure function -F(x, Q) one thereby obtains the remainder -FRem(^5 Q)? which forms the higher twist contributions. The leading twist terms are F — F^^^. It is a simple algebraic proof to show that F — FRem has the factorized form / * i^, with / being the parton distribution we have defined earlier, and with '*' denoting the convolution in eq. (77).
40 6.1 Bethe-Salpeter decomposition In the graphical depiction of a leading region, fig. 9, exactly one line on each side of the final state cut connects the collinear part and the ultraviolet part. So it is useful to decompose amplitudes into two-particle-irreducible components. This will leaxl to a Bethe-Salpeter formalism. Consider, for example, the two-rung ladder graph, fig. 11, for deeply inelastic scattering off a composite particle. We can symbolize it as Fig. 11 = 73 X 7 X 7 X 7/,. (93) Here 7^ represents the graph that is two-particle-irreducible in the vertical channel and is attached to the initial state partide, 7 represents a rung, and 7/^ represents the two-particle-irreducible graph where the virtual photon attaches. It is necessary to specify where the propagators on the sides of the ladder belong. We include them in the component just below. Thus 7^ and 7 have two propagators on their upper external lines. The purpose of having a composite particle for the initial state is to give an example with a non-trivial 7^, as in QCD with a hadronic initial state. The vertex joining the initial particle is a bound-state wave function. Fig. 11. Example of ladder graph with several rungs. We now decompose the complete structure function as 00 F = 2_^ Gs Gj. Gh N=0 (94)
41 Here G3 is the sum of all two-particle-irreducible graphs attached to the initial state particle, Gh is the sum of all two-particle-irreducible graphs coupling to the virtual photon, and Or is the sum of all graphs for a rung of the ladder. Thus Gr is the sum of all two-particle-irreducible graphs with two upper lines and two lower lines, multiplied by full propagators for the upper lines. The second line of eq. (94) has the inverse of 1 — Gr, and it clearly suggests a kind of operator or matrix formalism. Indeed, if we make explicit the external momenta of two ladder graphs, 71 (A;, /) and 72(^5 0? then their product is (7172)(fc, l) = J ^^7i(fc, k')y2{k', I). (95) The rung graphs can thus be treated as matrices whose indices have a continuous instead of a discrete range of values, while G^ and Gh can be treated as row- and column-vectors. In the case that the initial hadron is a single parton, as in the low order examples in Sect. 5, the soft part G^ is trivial: G^ = 1, where '1' represents the unit matrix. 6.2 Extraction of higher twist remainder We can now symbolize the operations used to extract the contribution of a graph to the hard scattering coefficient. Consider the example that lead to eq. (83). We took the original graph and subtracted the contribution of the graph to f^^^H^^\ where H^^^ is the lowest order hard part. Then we took the large Q asymptote of the result, by setting all the masses to zero. Fig. 12. Hard scattering coefficient from fig. 6(a). We represent this in a graphical form in fig. 12. There, the wavy line represents the operation of short circuiting the minus and transverse components of the loop
42 momentum coining up from below, and of setting all masses above the line to zero. Symbolically, we write this as: Contribution of fig. 6(a) to H^^^ = Pjjh - PjPlh = Pj{l-P)^h, where the operator P is defined by P{kJ) = (27r)^-^6(A;+ - /+)<5(A;-) (5^-^(/j.) X (Set masses to zero in the part of the graph above P). (97) In eq. (96) we have ignored the need for renormalization that occurs if e = 0. Either we can assume that we are only making the argument when e is slightly positive, or assume that all necessary renormalization is implicitly performed by minimal subtraction. Fig. 13. Contribution of fig. 6(a) to f^^^H^^K Fig. 6(a) gives two contributions to the factorization: a contribution to the one-loop hard part H^^^ given in eq. (96) or (83), and a contribution to f^^^H^^\ The second of these we picture in fig. 13 and symbolize as ^Plh. (98) Thus we can write the remainder for fig. 6(a), after subtracting its leading twist contribution, as Rem(fig. 6(a)) = 77,, - 7P7,, - P7(l - P)^f, = (1-P)7(1-P)7/.. ^ ^ Clearly the operator 1 — P subtracts out the leading behavior.
43 In general, we can write the remainder for the complete structure function as oo ^=0 (100) This formula is valid without renormalization, even at € = 0. In the first place, renormalization of the interactions can be done inside the 7's. This is because there is nesting but no overlap between, on the one hand, the graphs to which the operation 1 — P is applied and, on the other hand, the vertex and self-energy graphs for which counterterms are needed in the Lagrangian of the theory. Further divergences occur because of the extraction of the asymptotic behavior, and these give rise to the need to renormalize the parton distribution. But the regions that give rise to such divergences are of the form where lines in some lower part of a graph are coUinear relative to lines in the upper part. All such regions are canceled in eq. (100) since to the operator 1 — P they behave just like the regions that give the leading twist behavior of the structure function. 6.S Factorization It is now almost trivial to prove factorization for the leading twist part of the structure function, which is F-i^Rem=G,^^Gh-G,(l-P)^—^i^—— Gft. (101) Simple manipulations give 1-Gr 1-Gr{l-P) (102) We now have an explicit formula for the hard scattering coefficient: ^ = ^i-G.a-p)^-' (^°^) while the paxton distribution / satisfies fxP = G,j-^^P. (104) One somewhat unconventional feature of our procedure is that not only do we define P to set to zero the minus and transverse components of the momenta going
44 into the subgraph above it, but we also define it to set masses to zero. Setting the minus and transverse momenta to zero while preserving the plus component is exactly the appropriate generalization of BPH(Z) zero-momentum subtractions to the present situation. Setting the masses to zero as well is a convenient way of extracting the asymptotic large-Q behavior of a graph, as we saw in our explicit calculations. Moreover, particularly in QCD, it greatly simplifies calculations if one works with a purely zero-mass theory. Of course, setting masses to zero gives infrared divergences in all but purely ultraviolet quantities. The momentum-space regions that give the divergences associated with the structure function all have the same form as the leading regions for large Q, fig. 9, so that the 1 — P factors in eq. (104) kill all these divergences. Note that, just as with Zimmermann's methods, the P operator can be applied at the level of integrands. In practical calculations, dimensional continuation serves as both an infrared and an ultraviolet regulator. In the one-loop example of Sect. 5, the external hadron is a part on, so that in Gs = S(x — 1) in (104). At one loop, Gr corresponds exactly to /c, eq. (80). This expression, and the distribution / as a whole in (104) is still unrenormalized, and contains ultraviolet divergences. These may be removed by minimal subtraction, as in eq. (81) at one loop, or as discussed more generally in Ref. 9. We should mention, however, that it would be advantageous to have a subtraction procedure which combined factorization and renormalization into a single operation. The particular procedure outlined by Zimmermann'^J does this, but is not immediately applicable when all particles are massless. Duncan and Furmanski^^l have discussed some of these issues at length. 6.4 Factorization for Inclusive Annihilation in {(I>^)q It is easy to generalize the general arguments of this and the previous section to other processes, such as those listed in the introduction. An important example, is the cross section in (j>^ theory that is analogous to one-particle inclusive annihilation in e'^e" annihilation, that was discussed in Sect. 1.2. In the scalar theory, the structure function for this process is Dix,Q) = ^ J d'ye"'yJ2{0yiy)\HX){HXy{0)\0), (105) which is exactly analogous to the QCD version, eq. (6). It is relatively easy to check that the leading regions for this process have a form that generalizes fig. 9 for deeply inelastic scattering, that is, they have the form of fig. 14. This was shown in Ref. 19 (for the case of a non-gauge theory). An example is given by the ladder graph of fig. 15. We must integrate over all values of the momentum (k — pY, When [k — pY is coUinear to p'^, the line k has low virtuality. Then in the overall center-of-mass, the remaining particle q~ k
45 Fig. 14. Example of leading region for inclusive annihilation. Regions with more than one jet inside the hard subdiagram H are also leading. has large energy, approximately Q/2, and is moving in the opposite direction to the first two particles. We therefore consider the lines p, k — p and k as forming the jet J A in fig. 14 and q — k together with the vertex where the 'virtual photon' attaches as forming the hard part H. When (k — p)^ has transverse momentum of order Q, we put both k and k — p into the hard part. In a non-gauge theory, these two regions are the only significant ones, together with a region that interpolates between them. As we shall see in Sect. 7, this statement generalizes to all orders of perturbation theory. In a gauge theory, like QCD, all kinds of complication arise because there are also 'leading twist' regions involving soft gluons. 6.5 Factorization, fragmentation function Simple generalizations of the arguments for deeply inelastic scattering give the scalar factorization theorem: Diz,Q) 1^ C H(z/C,Q)d{0 + O{l/Q''). (106) analogous to eq. (7). Here the fragmentation function is defined in exact analogy to the parton distribution. We choose axes so that the momentum p^ of the detected
46 Fig. 15. Ladder graph. particle is in the positive 2:-direction. Then we define: d{z) P + 27rz dy~e'P'^y /^ 53(0 1(^(0, 2/-,0t)| ifX)(ffX 1(^(0)1 0) X P + (107) This is interpreted as the number density of hadrons if in a parton. The formulae are exactly analogous to eqs. (75) and (78) for the parton distribution. Renormal- ization is needed here also. 7. LEADING REGIONS As we saw in Sects. 5 and 6, the first step in constructing a complete proof of a factorization theorem is to derive the leading regions of momentum space for a graph of arbitrary order. This section begins with a brief description of a general approach to the long- and short-distance behavior of Feynman diagrams that results in a derivation of the leading regions. We apply this method to describe the origin of high-energy logarithms in scalar theories, and go on to discuss the cancellation of final state interactions, and the infrared finiteness of jet cross sections.
47 7.1 Mass dependence and leading regions Consider, then, an arbitrary Feynman integral G{p'^/fi^m/fi)^ corresponding to a graph G, which is a function of external momenta pj', mass m (possibly zero), and renormalization scale //. Without loss of generality, we may take G to be dimensionless. We also assume that the invariants formed from different p^ are all large, while the pj' themselves have invariant mass of order m. Thus: Pi'Pj = Q'^Vij^ P] = C»^^ (108) where Q is a high-energy scale, Q ^ tti, and the r^ij and Ci ^^^ numbers of order unity. In the following, it will not be necessary to consider the 77,^ and Ct dependence, and we will write G as G{Q'^I^^^m? j^'^). We will be interested in the leading term in an expansion in powers of \jQ^. (Always we will allow the possibility of a polynomial in In Q multiplying the power of Q, in each order of perturbation theory.) Suppose G is the result of L loop momentum integrations acting on a product of / Feynman propagators, times a function iV, which is a polynomial in the internal and external momenta. For simplicity, we absorb into N the numerator factors associated with the internal propagators, as well as overall kinematic factors, etc. G may then be represented schematically as G(QV/^^mV/^^) = n U^'U N{{k^]Api})f{ \ (109) The line momenta {A;^}, of course, are functions of the ^f and the pf. Any region in £^ space which contributes to G at leading power in Q^ will be called a "leading region". In addition, by a "short-distance" contribution to (109) we will mean that we have a region of loop momenta in which some subset of the line momenta, {A;^}, are ofF-shell by at least C^(Q^); the short-distance contribution is the factor in (109) given by these far ofF-shell lines. Short-distance contributions are independent of masses to the leading power in Q"^, since the integrand can usefully be expanded in powers of m'^ when propagators axe far ofF-shell. A general leading region has both short-and long-distance contributions, the latter associated with lines which are nearer the mass shell. Roughly speaking, factorization is the statement that the cross section is a product of parton distributions, in which all the long-distance contributions are found, and a hard-scattering coefficient, which has purely short- distance contributions. To study factorization, we must characterize all "longdistance" contributions. Our analysis depends on two observations. The first concerns the close relation between the high-energy and zero-mass limits. That is, if the renormalization
48 scale // is chosen to be of 0(Q), then the two limits are equivalent in the function G(l, m^/Q^). Short-distance contributions to the Q —> oo limit are those involving lines for which kj is of order Q^. Long-distance contributions are parts of the if integrations for which kj is much less than Q^. If we scale all momenta down by a factor proportional to Q, then we axe considering the m —^ 0 limit instead. The short-distance contributions now have fixed kj and the long-distance contributions have Feynman denominators kj -f- ie in eq. (109) that vanish in the m —> 0 limit. Note that if G is such that it only has short distance contributions, then the Q —* oo limit is G(1,0), i.e., we can just set m = 0. The QCD coupling, ^s(Q), is an implicit argument for G, and we have already chosen to set the renormalization scale fi equal to Q. Thus in this case the detailed large Q behavior is renormalization-group controlled in a simpJle way. When there are long-distance contributions to G, an expansion in powers of m will often fail. So to find the long-distance contributions to G, one must look for singularities in the m —^ 0 limit. There are apparent exceptions to this rule, exemplified by the integral However, if we factor out the numerator factor m^^ we axe left with an integral that is singular like 1/m^. This singularity is governed by the denominator. So what we axe looking for is singularities in the m dependence in the integral over the denominators of G. Our second observation is that the integrals in (109) are defined in complex ^f-space. As a result, it is not enough for a set of denominators to vanish in the integrand of (109) for the integral to produce a singularity at m = 0 in G. We must have, in addition, a pinch of one or more of the if integrals at the position of the singularity, between coalescing poles. This fact enables us to apply the simple but powerful analysis due originally to Landau^^'^^1 on the relation of singularities in Feynman integranc^^ to the singularities of Feynman integra/^. In the next subsection we explain the application of this argument. 7.2 Pinch surfaces We begin by using Feynman parameterization to combine the denominators of eq.(109) by G(QV/i',mV/i')= (/-l)!n / da,^(l-5^a,)n d^^, N{{kj},{pi}) x[i2a,{k]{if)~m') + ze] \ j=i (111)
49 where we have exhibited the loop-momentum dependence of the hne momenta. There is now a single denominator D{ii^aj)^ which is quadratic in loop momenta and linear in Feynman parameters. Suppose D{ii^aj) vanishes for some value of loop momenta and Feynman parameters. We will now derive necessary conditions for this zero to produce a singularity in G. Then we will apply these conditions to the case m = 0. A pole from D = 0 will not give a singularity in G if D can be changed from zero by a deformation that does not cross a pole in any one oi the momentum or parameter contours. Consider first the parameter integrals. Because D is linear in the {«j}, a deformation of the aj integral will change D away from zero, unless kj = rr? or aj = 0 (112) for each line. In the first case, D is independent of o^j, while in the second we note that aj = 0 is an endpoint of the aj integral, away from which it cannot be deformed. Now suppose (112) is satisfied, and consider the momentum integrals. D will be independent of those loop momenta which flow only through lines whose Feynman parameters are zero. The contours of the remaining loop momenta must be pinched between singularities associated with the vanishing of D. Since D is a quadratic function of the remaining momenta, each momentum component sees only two poles in its complex plane due to the vanishing of D. The condition for a pinch is thus the same as the condition that the two zeros of the quadratic form be equal. That is, in addition to D = 0 we must have dDjdi^ = 0 for all i'^ which flow through one or more on-shell lines. For each such loop momentum, the extra condition is^^'^^1 where the sum goes over all lines through which the loop momentum if flows. (Note that any line which is not on shell has aj =0, by eq. (112), so the condition (113) can be applied to every loop.) Together, (112) and (113) are known as the "Landau equations". We shall refer to any surface in momentum space on which the Landau equations are satisfied as a "pinch surface" of the diagram G. With each pinch surface we associate a "reduced diagram", in which all ofF-shell lines are shrunk to points. By construction, the reduced diagram contains only those loop momenta of the original diagram which satisfy (113) with nonzero a's. 7.3 Physical propagation The Landau equations axe surprisingly restrictive, especially in the zero-mass limit. To see why, let us rederive the observation of Coleman and Norton^^'^^1 that
50 eqs. (112) and (113) have an appealing physical interpretation. Consider a given pinch surface. We rewrite (113) on this surface as T(aj u;j)v^ = 0, (114) where v^ and ljj axe the four-velocity and energy associated with the momentum kj^. The units of the Feynman parameters are arbitrary, so suppose we may, if we wish, interpret aj as the frame-independent ratio of a time to the energy of line j. Then each of the components of the vector (ajiJj)Vj^ has the units of a distance in space-time. It is the distance traversed in time (ajUj) by a free particle moving classically with velocity v^. Now suppose we associate a definite position a;J* to one of the vertices in the reduced diagram associated with the pinch surface. Then, if line j attaches to the vertex at Xj, x^ -\-{aj (j^j)vj^ may be interpreted as the position of the vertex at the other end of line j. Continuing in this manner, we can associate with the reduced diagram a position in space-time for every one of its vertices, and a physical process in which free particles move between these points. Equation (113) ensures that this program can be carried out consistently, by requiring that in going around any closed loop we come back to the same position. We can use this construction as a necessary condition for a pinch surface. Finally, note that, because eqs. (112) and (113) axe homogeneous in the a's, a rescaling of the a's leaves the Landau equations satisfied. Hence the vertices in the physical picture are an indefinite distance apart, and, in particular, this distance may be arbitrarily large. 7.4 Collinear and infrared pinches; power counting For a general diagram with arbitrary masses and external momenta the criterion of physical propagation allows a very rich analytic structure. In the massless limit, however, this structure actually simplifies since multiparticle thresholds become degenerate. The physical processes of which an isolated massless (but not massive) particle is kinematically capable are as follows. First, a massless particle of momentum p^ may split into two (or more) massless particles of momenta ap^ and (1 — Oi)p^, and vice-versa. This is the source of collinear divergences. Second, a particle may emit or absorb one (or more) zero energy particles. This is the source of infrared divergences. We easily check that arbitrary loops involving only collinear and zero-momentum particles can satisfy the Landau equations. Generally, we will describe a subdiagram consisting of mutually collinear particles as a "jet" subdiagram. Lines that have zero momentum in the massless limit we will call "infrared". A jet subdiagram describes the evolution of a set of collinear lines, as they absorb and emit other collinear and infrared lines.
51 As an example, let us consider the vertex correction fig. 16a, in dimensionally regulated (f)^ theory, VM = / ^ 1 [{p' - ky - m2 + ie] [(p + ky - m2 + ie] (F - m2 + ie)' (115) where we assume a production process, Q^ = {p' + pY > 0. This example is used for illustrative purposes only. The term "leading" will refer here only to this diagram, and not to the behavior of the Born diagram. In fact, fig. 16(a) is nonleading compared to the Born process for all n < 6. The Landau equation for fig. 16(a) is «i(p - J^Y + «2(p + ky + a^k^" = 0. (116) For on-shell (p^ = p'^ = m^) scattering with m ^^ 0, eq. (116) has no solutions at all. Note that this is the case even though there is a singular surface illustrated in fig. 16(b), with (p + kf = (p' - kf = m^, jfc^ < 0. (117) This singular surface corresponds to the production of two particles, followed by a subsequent spacelike scattering. Although such a process is kinematically possible, it clearly cannot correspond to physical propagation, because the two particles produced at vertex 1 propagate in different directions, and would therefore not be able to meet at vertex 2 to scatter again. Now let us consider the case that tti = 0. By the same reasoning, (117) does not give a pinch surface, if A; ^ 0. There are nevertheless two sets of solutions. First, there are infrared solutions where one line has zero momentum, k^ = 0, «! = ^2 = 0, (p + A:)'^ = 0, ai =0^3 = 0, (118) (p - ky = 0, ^2 = «3 = 0. Second, there are coUinear solutions, where two of the lines are parallel to one of the outgoing external particles, 2 «i(p - ^Y + ot:ik^ =0, ^2 = 0, k^ = p'k = 0, 2 .. . (119) «2(p' + ky + a^k^ =0, Oil = 0, k^ = p'-k = 0. The physical pictures associated with typical infrared and collinear pinch surfaces are shown in figs. 16(c) and (d), respectively. In each case, there is physically realizable propagation between vertices.
52 P P (b) p + k-0 k-0 (c) (d) P Fig. 16. (a) Vertex correction, (b) Reduced diagram correspond ing to eq. (117), (c) Infrared reduced diagram, (d) collinear re duced diagram. Now we observe that even though solutions to the Landau equations like eqs. (118) and (119) give pinch surfaces, they still do not necessarily produce mass dependence that is relevant to the leading power of Q, and hence are not necessarily leading regions. The Born graph for the vertex behaves like Q^. The contribution to the one- loop graph from the pure short-distance region, from the region, \k^\ = 0(Q), is Qn-6 'pj-^^s ^}^is region is leading when the theory is renormalizable, at n = 6, but is non-leading relative to the Born graph when the theory is super-renormalizable, n < 6. Next we consider the one-loop graph near its singular surfaces. For example, consider the integral (115) near the surface defined by the first of eqs. (118). To be specific, let |A;'^| < Aj^ax? /^ = 0... 3, with A;max being some fixed scale (which
53 must ^ m). In this region the integral behaves as r Ik. < Q^Jk.<k^.. k' Q '"*^-. (120) rs^ in&x Compared to the short-distance region, this infrared region is leading only for n < 4. (The other two infrared regions in (118) require n < 2.) (We remind the reader that, compared to the Born graph, none of these regions contribute to the leading power of Q.) Similarly, near the coUinear pinch surfaces of eq. (118), the integral behaves as max jl2 An-2h. h^-"^ SO that again only for n < 4 do we find collinear contributions from this diagram that are leading compared with the short-distance contribution. In summary, for the scalar theory in six dimensions, only short distance regions are (relatively) leading for fig. 16. This result generalizes to all orders in the vertex correction for this theory^'*^. The process of estimating the strength of a singularity is known as "power counting". We will give more low order examples below, while more general arguments can be found in Ref. 23. We can, however, summarize the basic result of these arguments briefly. Let Z) be a reduced diagram with 5 "infrared loops" and Is infrared lines whose momenta vanish at the corresponding pinch surface, and with C "collinear loops" and Iq collinear lines whose momenta become proportional to an external momentum at the pinch surface. Finally, let A^2 denote the number of two point subdiagrams in R. In (j)^ theory in n space-time dimensions, let us define^^1 the "degree of divergence" by u;(R) = nS + (n/2)C - 2Is - Ic + ^^2- (122) This corresponds to a power law Q{n-6)V j^^iR) ^^23) as Q ^ oo. Here V is the number of loops and A is a small parameter that parameterizes the approach to the singularities in the massless theory. The power law is measured relative to the power for the Born graph. In the renormalizable case, n = 6, we get leading behavior only if lj{R) = 0 and there are no graphs that give negative uj{R), The term N^^ which tends to suppress the behavior of the integral at the singular point, is due to the fact that the renormalized two-point function must vanish on-shell if the particle is to have
54 zero mass. In this sense, the infrared behavior of the theory is dependent on renormalization^^^. In the superrenormalizable case, n < 6, the first factor in (123) gives a negative power of Q that just corresponds to normal ultraviolet power counting; this is the power that comes from a purely short-distance contribution to the graph. A sufficiently strong power law singularity in the massless theory is needed to overcome this if one is to get a leading contribution. 7.5 Leading regions for deeply inelastic scattering in {<I>^)q As an application, we now discuss the general leading regions for the basic inclusive cross sections in {<I>^)q' The criterion of physical propagation shows why the considerations of Sects. 5 and 6 take into account all relevant leading regions. To see this, we must generalize our concept of leading regions to include cut diagrams, of the type discussed in Sects. 5 and 6. There is no problem in doing this, and power counting may be estimated for collinear and infrared cut lines with the same degree of divergence eq. (122) as for virtual lines. It is useful to apply the optical theorem to reexpress the deeply inelastic scattering structure function eq. (63) as the discontinuity of the forward Compton scattering amplitude T(^,p), F(x,Q2)=discr(g,p), O^ f (124) Tiq,p) = ^ d«j/e"«'{p|T(i(y)i(0)|p). This relation holds diagram-by-diagram, once cuts are summed over, so that a necessary condition for a region L to be leading in F is that it be leading in T. We thus need to consider the leading regions of the diagrams illustrated by fig. 17(a), which represents forward Compton scattering. The pinch surfaces are symbolized in fig. 17(b). The incoming hadron can form a jet of lines, and in addition, may interact with any number of soft lines, connected to a subdiagram S consisting of only lines with zero momentum. To see why this is the general form, consider a singular surface not of this kind, cis in fig. 17(c). Here, one or more lines of the jet may scattering with the incoming photon to form a set of on-shell outgoing jets, which then rescatter to emit the outgoing photon and reform the outgoing jet, which eventually evolves into the outgoing hadron. Such a process is certainly consistent with momentum conservation. Surfaces of this type are not pinch surfaces for the amplitude T(^,p), however, because the outgoing jets can never collide again once they have gone a finite distance from the point at which they are produced. As a result, in every pinch surface, the incoming and outgoing photons attach at the same point in space-time, and we derive the
55 (a) (b) (c) Fig. 17. (a) Diagrams for forward Compton scattering, (b) Reduced diagrams for pinch surfaces, (c) Reduced diagram for a singular surface which is not pinched. picture of fig. 17(b). This result shows that all divergences associated with final state interactions cancel in the sum over final states. As indicated above, not every pinch surface will correspond to a leading region. In particular, power counting using eq. (122) shows that there are no infrared divergences in {(t>^)Qi that is, no zero-momentum lines for any leading region^^l. In addition, we can show that only the minimum number of jet lines (two) can attach the jet to the hard part in fig. 17(b). It is a straightforward exercise in counting
56 to prove these results, using (122), the Euler identity (loops = lines - vertices + 1), and the observation that every internal line of a graph begins and ends at a vertex. In summary, we can show that fig. 9 is indeed the reduced diagram of the most general leading region for deeply inelastic scattering in the scalar theory. 1.6 Unitarity and jets: the cancellation of final state interactions The cancellation of final state interactions in deeply inelastic scattering plays an important role in the analysis for deeply inelastic scattering just described. This cancellation is a general feature of inclusive hard scattering cross sections, and is used repeatedly in factorization proofs. The physics behind this cancellation has already been pointed out in Sect. 5: a hard scattering is well localized in space- time, and, as a result, it cannot interfere with long-distance effects which describe the further evolution of the system. Thus, when we sum over final states in an inclusive cross section, we lose information on the details of evolution in the final state, and are left with the constraint that, by unitarity, the sum of probabilities of all final states is unity. As a result, at each order in perturbation theory, long-distance contributions to final states must cancel. It is worth noting that it is not always necessary to sum over all final states to cancel long-distance interactions. There are three kinds of cross sections, among those mentioned in Sect. 1, for which the cancellation of final state interactions is important. In deeply inelastic scattering and Drell-Yan, for instance, we sum over all hadronic final states. In single-paxticle inclusive cross sections, on the other hand, we shall find in Sect. 9 that cancellation requires the use of Ward identities. Finally, in jet cross sections, the cancellation comes about in a sum over all final states which satisfy certain criteria in phase space. Let us hint at how this happens. If all non-forward particles in the final state emerge from a single hard scattering, the criterion of physical propagation requires that the long-distance contributions will come entirely from soft and coUinear interactions. This is because, as in the low order example of fig. 16, jets emerging from a single point and propagating freely cannot meet again to produce a new hard scattering. In this case, once the energy and direction of a set of jets is specified, the sum over only those final states consistent with these jets will also give unity, and their collinear and infrared divergences cancel in the sum. The technical proof of these statements may be given in a number of ways. The simplest is based on a truncation of the hamiltonian to describe only collinear and infrared interactions. Then, since the truncated hamiltonian is hermitian, it generates a unitary evolution operator whose divergences cancel by the "KLN" theorem^^'^^K It is also useful to see that this cancellation is manifested on a diagram-by-diagram basis within each leading region in perturbation theory^^l. Proofs of this type are most easily given in terms
57 of time-ordered or light-cone ordered perturbation theory^^'"^^J. Technicalities aside, the cancellation of final state interactions at the level of jets has a number of important consequences. The simplest of these is the finiteness of jet cross sections in e"^e~-annihilation cross sections"^®J. We have already seen its importance for the analysis of deeply inelastic scattering in {<t>^)Q- It has a similar simplifying effect for single-particle inclusive cross sections, as well as for the Drell- Yan and related cross sections. To illustrate this, we show, in fig. 18 the reduced diagrams for leading regions in the scalar "Drell-Yan" cross section, defined for the scalar theory by analogy to eq. (63) in {(t>^)Q, after the sum over final states. We see that all information about the final state has been absorbed into a single hard part H. Note that this result holds not only for the fully inclusive Drell-Yan cross section, but also for semiinclusive cross sections such as hadron-hadron -^ Drell-Yan pair -f jets. Pa Pb Fig. 18. Leading regions for the Drell-Yan cross section in {(J>^)q. 8. FACTORIZATION AND GAUGE INVARIANCE In this and the following section, we discuss the extension of factorization proofs to gauge theories. We begin with a discussion of the classical Coulomb field of a fast moving charge, an example that anticipates what happens in the full quantum theory.
58 We next summarize the Ward identities we will need. Then we discuss the leading regions of Feynman graphs in a gauge theory. There are great differences from the case of the (j)^ theory discussed in the previous section. With the aid of the example of the vertex graph, we show how, after an appropriate eikonal approximation. Ward identities are applicable that will combine graphs into a fac tori zed form. In the next section, we will show how factorization may be proved for a variety of experimentally important cross sections, which can be measured in deeply inelastic scattering, e"^e~ annihilation and hadron-hadron scattering. We should emphasize at the outset that although we regard existing proofs in all these cases as reasonably satisfying, there is still room for improvement, especially for hadron- hadron cross sections. We will point out the shortcomings of existing arguments in Sect. 10. 8.1 Classical considerations Before getting into a detailed discussion of Feynman diagrams, it is worth noting that insight can be gained into the physical content of factorization theorems from purely classical considerations. This discussion will at once highlight an important difference between gauge and scalar theories, and at the same time show why this difference, important though it is, respects factorization. As we observed in Sect. 1.4, the parton model picture of hadron-hadron scattering rests in part on the Lorentz contraction of colliding hadrons. Now a simplified classical analog of a hadron is a collection of point charges, each acting as a source of a classical scalar field. We would expect that if the parton model, or factorization, is to make sense, these fields ought to be Lorentz contracted themselves, and this is just what happens. Let us see how. Consider first a static classical scalar field </>ci(x), associated with a point particle of charge q at the origin. If we assume that the field obeys Laplace's equation, it is given in the rest frame of the particle by ^ci(x) = -^. (125) Now consider the same field in a frame where the particle is moving at velocity c/S along the z-axis. Then the field at x'^ in this frame is <?^cl(^') = [4 + 7^(^cf-4)2] 1/2 ' (126) where, as usual, 7 = (1 — ^^)~^/^. For an observer at t' — 0 in the primed system, the (f) field decreases as 7"^ as (3 approaches unity, except near x'^ = 0. Thus, the (f) field is indeed Lorentz contracted, and any force proportional to the <j) field is also
59 Lorentz contracted into a small longitudinal distance about x'^ = 0. This means that in the rest frame of a scalar "hadron", the forces due to another such hadron approaching at nearly the speed of light are experienced in a Lorentz-contracted fashion, just as supposed in the paxton model. Now let us apply this reasoning to a classical gauge theory, in this case classical electrodynamics. Here, the field in the rest frame of a point particle of charge q is precisely analogous to eq. (125), ^c^« = #. (127) Because this is a vector field, however, there is a big difference from the scalar case in a frame in which the particle moves with velocity 0^X3. In this frame, we find [x ^^ + jm3ct'- x'.fY'^ A'Jix') = ^^ ^, (128) For large 7, the field in the zero and three directions are actually independent of 7 at fixed times before the collision. It might therefore seem that a vector field is not Lorentz contracted, and would not respect the assumptions of the parton model. If we look, however, at the field strengths rather than the vector potential, we find a different story. The electric field in the three direction, for instance, is given in the primed frame by which shows a 7"-^ falloff. Since the force experienced by a test charge (or parton) in the primed frame is proportional to the field strength rather than the vector potential itself, the physical effects of the moving charge are much smaller than its vector potential at any fixed time before the collision. This in turn may be understood as the fact that, as 7 —>• 00, the vector potential approaches the total derivative qd^\n{l3ct' -x':^). (130) That is, for any fixed time the vector potential becomes gauge equivalent to a zero potential. We can conclude from this excursion into special relativity that factorization will be a more complicated issue in gauge theories than in scalar theories. Only
60 for gauge invariant quantities will the gauge-dependent, large vector potentials of moving charges, which naively break factorization, cancel. So, in particular, we cannot expect factorization to be a property of individual Feynman diagrams, as it was in scalar theories. On the other hand, we should look for the solution to these problems in the same techniques which are used to show the gauge independence of physical quantities. 8.2 Ward identities The Ward-Takahashi identities of QED and the Taylor-Slavnov identities of non- abelian gauge theories ensure the perturbative unitarity of these theories. We shall refer to them collectively as "Ward identities" below. Ward identities may be expressed in various forms, for instance, as identities between renormalization constants (the familiar Zi = Z2 of QED). For our purposes, however, the basic Ward identity is given graphically by the equation {N\T ^^,A'''{x^)x ■■■ X d^^A'^-ixn) \M) =0, (131) where A^{x) is an abelian or nonabelian gauge field, and where M and N are physical states, that is, states involving on-shell fermions and gauge particles, all with physical polarizations. In particular, physical states do not include ghosts. Equation (131) will be represented graphically by fig. 19, in which the scalar operator d^A'^(x) is represented by a dashed line ending in an arrow. In momentum space, this operator is associated with a standard perturbation theory vertex in which one gluon field is contracted into its own momentum. Here and below, we refer to such a gluon as "longitudinally polarized". Note that this is to be taken as referring to the four-momentum. 0 Fig. 19, Ward identity. Proofs of eq. (131) are most easily given in a path integral formulation using BRST invariance, as in, for instance Ref. 29. They can also be proved in a purely
61 graphical form, as in the original proofs of Refs. 30 and 31. Here we need not concern ourselves with the details of these proofs, although it may be worthwhile to exhibit the very simplest example of eq. (131). This is the lowest order contribution to the electron scattering amplitude with a single longitudinally polarized photon. At this order, we have qf'uip + qhMP) = ^(P + ^) [(/ + ^ + "^) - (/ + ^)] ^(P) = 0- (132) The first equality is sometimes referred to as the "Feynman identity", and the overall result is current conservation at lowest order. This is not surprising, since classical current conservation is a consequence of gauge invariance. In the quantum theory, it appears as a matrix element relation, whose validity is ensured by the Ward identity. A helpful exercise is to construct the analog of eq. (132) for the scattering of a physically polarized gluon. The graphical proof consists essentially of repeated applications of identities like eq. (132). Even without going into the details of the proof of eq. (131), we can elucidate its interpretation. First, it is true order-by-order in perturbation theory, although not graph-by-graph in perturbation theory. In addition, we may imagine constructing a path integral in which only certain momenta are included, for instance ultraviolet momenta and/or momenta parallel to a given direction. Then at a given order, the Ward identities hold for both internal and external lines in this restricted portion of momentum space. This heuristic argument may be verified by a close look at the graphical proof of Ward identities in Refs. 30 and 31. So far, we have discussed Ward identities for the S-matrix. As we saw in Sect. 4, however, we will sometimes be interested in matrix elements involving a gauge invariant but nonlocal operator which includes the ordered exponential of the gauge field. Such matrix elements also obey Ward identities, which may be proved by either of the methods mentioned in connection with eq. (131). The simplest generalization of (131) to this case is (N\T {Uid^,A^^{xi) Fexp{ig / dy-^+(0,y-,Ox)}^(0)) | M> = 0, (133) Jo where in the ordered exponential A refers to the gauge field in the representation of field $, which may be a fermion or gauge particle. Equation (133), in various guises, will be useful in our proofs of factorization. 8.3 Singularities in Gauge Theories Discussions of factorization start with a catalog of the pinch surfaces of the relevant Feynman diagrams, as described in Sect. 7. They then proceed, by power counting, to estimate the strength of singularities encountered in each such surface. The
62 same procedure may be carried out for gauge theories, but, as we will now see, many of the regions that are nonleading in (f)^ are now leading. Thus the results are much richer than in (f)^ theory. The one-loop vertex graph illustrates the origin of the infrared and collinear singularities. Ignoring overall factors, including group structure, we find that the graph is given by t^m(p,p') d^k v(p')7"(-/ + t + m)fM + ^ + m)j^u{p) {2ny [{p' - fc)2 - m2 + ie] [{p + fc)2 - m2 + ie] {k'^ + ie)' (134) The singularity structure (with one minor exception) is the same as in <f)^ theory; what changes is the strength of the singularities. To discuss the large Q region, we will consider, as before, the massless limit. Of the solutions (118) and (119) to the Landau equations, the first of the infrared solutions (k^ = 0) and both of the collinear solutions (119) give leading power behavior at large Q, as we will see. These are singularities in the fully massless theory, and, by our discussion in Sect. 7, they correspond to long-distance contributions when Q is large. In addition to these singularities, there is a genuine singularity, at k^ = 0, even when the fermion mass is nonzero. This is an example of the usual infrared divergences of QED and is caused by the masslessness of the gluon. This singularity survives the Q ^ oo limit, of course, and becomes the first of the solutions (118). The methods that we use to treat both the collinear and especially the infrared singularities in the fully massless nonabelian theory axe explicitly motivated by the elegant methods given by Grammer and Yennie^"^J to treat the ordinary infrared problem in QED. In QED, the infrared divergences correspond directly to the real physics of the long range of the Coulomb field and the genuinely massless photon. But in QCD the infrared divergences are cut off by confinement. Since this is a nonperturbative phenomenon, the resulting cutoff is not easily accessible (if at all) in perturbation theory. Perturbative calculations must be restricted to sufficiently short-distance phenomena so that asymptotic freedom is useful. The singularity structure of the massless theory is just a convenient tool to aid in the factorization of long-distance phenomena. 8.4 Infrared divergences in gauge theories: the eikonal approximation We now consider the infrared singularity at k^ = 0. Although our ultimate aim is to treat the large Q limit, our discussion will not need to assume this limit initially. Let us see how the integral (134) behaves near this point. As k^ —*■ 0, it is valid to make the following two approximations in V^(p,p'), (1) Neglect k^ compared to m and p^ in numerator factors, ) -> . . (135) (2) Neglect k^ compared to p'-k and p'-k in denominator factors.
63 Together, these two prescriptions define the "eikonal approximation" for the graph. Simple manipulations show that in the eikonal approximation V^ is given by /d^ib 1 (2^ i-2p'-k + ie)i2p.k + te)ik^ + tey ^'^^^ In this form it is apparent that the k integral is logarithmically divergent from the region near k^ = 0. Notice also that, because the numerator in proportional to //•p, this divergent integral behaves as a constant at as p'-p —> oo, that is, with the same power as the elementary vertex. This is to be contrasted with the situation in (f)'^ theory that was explained in Sect. 7. Infrared behavior with the same power law behavior in Q as the elementary vertex is a characteristic of theories with vector particles. The eikonal approximation is, not surprisingly, closely related to the ordered exponentials of Sect. 4.2, with their eikonal Feynman rules. In fact in making the (eikonal approximation (135), we are precisely replacing fermion propagators by (ukonal propagators of the type shown in eq. (48) for the parton distribution functions. We can anticipate the importance of the eikonal approximation by relating it to the classical discussion given in Sect. 8.1. Consider a gluon of momentum k^ interacting with an eikonal line in the v^ direction. The only component of the gluon momentum k^ on which the eikonal propagator depends is v-k, and the only component of the gluon polarization e^{k) on which the eikonal vertices depends is v-e. So, as far as the eikonal line is concerned, the gluon acts in the same way as a fictitious gluon of momentum (v'k)u^ and polarization {v'e)u^^ where u^ is any vector for which u-v = 1. But this fictitious gluon is longitudinally polarized. That is, any gluon interacts with an eikonal line in the same way as a longitudinally polarized, and therefore unphysical, gluon. But we have argued above that such gluons, although they may be expected to break factorization on a graph-by-graph basis, should be consistent with it in gauge invariant quantities. When Q is large, we consider not just the actual infrared singularity at k^ = 0, but the whole infrared region k^ <C Q- That is, as Q —> oo, we consider the region k^/Q —> 0. It is possible for the different components of k^/Q to go to zero at such different rates that the eikonal approximation fails. Since we will rely on this approximation in proving factorization we will need to evade this failure. To get an idea of what is involved, let us return to eq. (134), and justify the eikonal approximation eq. (135) in this simplest of cases. Failure of eq. (135) is caused by failure of the second of the approximations of which it is comprised: dropping factors of k^ in the numerator is a safe bet, because, as we have seen, the factors p^ combine to form large invariants. So, the issue is whether or not we may neglect k^ compared to p-k and p'-k. This is nontrivial, because it is easy to find vectors k^ for which p-k and p'-k are small, while k'^ remains relatively large.
64 This will be the case whenever its spatial momentum transverse to the p and p' directions is large, k± ~ —k^ ^P'k, p'-k. (137) This region was called the "Glauber" region in Ref. 33. It is easy to check that in this region the k^ integral of eq. (134) is logarithmically divergent. If we were to put in a gluon mass (as is consistent for an abelian theory), the divergence would disappear, but we would still have a contribution from the region (137) to the leading-power behavior at large Q, that is, a contribution of order Q^ times logarithms. Does this mean that the eikonal approximation is wrong? In fact it does not. To see this, we appeal to our freedom to deform momentum space contours. Suppose, for definiteness that p and p' are in the ±z directions, respectively, and that |p| = |p'|. We then change variables from the set {A;o,ki} to the set {/c^ = 2~^''^(ko ± (p-k)/a;j,), k±}. (These become light cone variables in the high energy limit.) Then in the region defined by eq. (137), the A;'^ —^ 0 singularity of (134) is given by •^(-ibj.2 + ze) '^ ^ {2^/^LOpK- - A;j.2 + ie) (-23/2a;;,/c+ - ibj.^ + ie)' (138) where the variables k^ appear in only one denominator each. In this form we see explicitly that the k^ integrals are not trapped in the region k^ <C k±^ since they each encounter only a single pole in this region. As a result, these contours may be deformed away from the origin into the region \k^\ ~ |^±|- But in this region the eikonal approximation is valid, provided only that lA;'^! <C Q- So, we may relax our criteria for the eikonal approximation to include the possibility that, even if it is not valid everywhere along the undeformed contours, these integrals can be deformed in such a way that it holds along the deformed contours. 8.5 Collinear divergences and choice of gauge In addition to infrared divergences, we have to consider collinear divergences in the massless limit. The nature of the collinear contributions to leading regions depends on the gauge, els we will now show. Consider the gluon propagator in a axial gauge n-A = 0. It has the form DAk) = jf!— (a,. - ''"'''' ^""^-^ + ^^r^) , (139) P + ^e \ n-k (n-k) which satisfies 2 KZ),.(.) = M £^ - ^ ) . (140)
65 Such a gauge is "physical" because its propagator has no particle pole when contracted into any vector proportional to its momentum. Another way of putting this is that in such a gauge longitudinal degrees of freedom do not propagate. This is to be contrasted with a covariant gauge like the Feynman gauge, for which k^D^ty(k) = —ikjy/k'^. As a result, leading regions in which longitudinal degrees of freedom propagate are present in covariant gauges but absent in physical gauges. Let us see what this means in practice. To do so, we turn again to the vertex correction, (134). We have already stated the locations of the collinear singularities, in eq. (119). The two possibilities are that A;'^ is proportional to p^ and that it is proportional to p'^. The corresponding reduced diagrams are shown in fig. 20(a). (a) (b) Fig. 20. Leading collinear reduced diagrams at one loop: (a) covariant gauge, (b) physical gauge. By doing the k integral by contour integration, we easily find that in Feyn man gauge the contribution of momenta close to the singularity where k is pro
66 portional to p is given by Mp,p') « (2^ y + -pr^ip'hMp) 9-+ (1 + ''^/p^)jj^ (141) and similarly in the second region. We have exhibited explicitly the numerator of the gluon propagator. For fixed k"^ = —xp"^, the k± integral diverges, and is leading power, that is, independent of Q. This is the coUinear divergence. (There is an additional infrared divergence as k"^ vanishes; this region we have already discussed. This result for the Q dependence is known as a "Sudakov" double logarithm; it is associated with the overlap of coUinear and infrared divergences.) These regions, summarized by the reduced graphs of fig. 20(a) in which two coUinear lines attach to a hard subdiagram, would not be leading in (<^^)6, because of the lack of the numerator factor. Note, however, that in the numerator of the gluon propagator, the term which gives the leading behavior in the coUinear region is g ^. Since the gluon is moving, by assumption, parallel to p^^ which is in the plus direction, this corresponds to an unphysical polarization at the vertex adjacent to the antiquark line. Thus the collinear divergence is associated with a longitudinally polarized gluon, and we might expect it to be absent in a physical gauge — at least in this particular diagram. To verify this, we can compute V^ in an axial gauge. The leading term in eq. (141) is then replaced by v,iP,p')« ^ J_^^ ^vip'hMp) (1 + ^^/p"")^ n-k^ + k-n^ k-k^ 9—{- ; + (142) n-k (n-k) Using eq. (140) (and remembering that k± = A;^), we easily check that the collinear divergence in (141) is absent in (142), and that the vertex diagram therefore lacks the Sudakov double logarithm in axial gauge. Of course, since the theory is gauge invariant, the corresponding physics, and in particular the double logarithm, has to show up somewhere, and in axial gauge it occurs in the one loop fermion self energy. We leave it as a simple but instructive exercise to check that this is indeed the case. Thus, in axial gauge the reduced diagrams of fig. 20(a) do not correspond to a collinear divergence, while that of fig. 20(b) does. We emphasize here the fact that in the Feynman gauge the jets are one-particle irreducible, while in the axial gauge they are reducible. As we shall see below, this result generalizes to all orders. In this sense, the gauge theory in a physical gauge behaves, from the point of view of reduced diagrams for collinear lines, like (<^^)g.
67 This suggests that an axial gauge is the most appropriate one for proving factorization. However, the singularities at n-A; = 0 cause a lot of trouble. In the first place they obstruct^^'^^'^^J the contour deformations that we have already seen are essential to demonstrating factorization; this is equivalent to saying that the singularities violate relativistic causality on a graph-by-graph basis. Furthermore, the singularities have to be defined by some kind of principal value prescription, and it is difficult to ensure that the products of these singularities that occur in higher order graphs can be defined properly.^'^J 8.6 Power counting for gauge theories As in the scalar theory, we must use power counting to identify those pinch surfaces which actually give leading regions. Again, this approach is discussed in detail in Ref. 23. Here, we once again quote the general result. Assuming the eikonal approximation, for any leading region with reduced diagram R^ we compute the infrared degree of divergence, a;(i^), analogous to eq. (122), Lo{R) = 45 + 2C - 2/5 -Ic-hN2-\- iiVa + F. (143) We have assumed a space-time dimension 4. As in (122), 5, Is, C*, and Ic are, respectively the numbers of soft loops and lines, and coUinear loops and lines in R at the associated pinch surface. N2 is the number of two-point functions in R, while ATs the number of three-point functions all of whose external lines are in the same jet. F is derived from the numerator factors where soft lines connect to collinear lines. It is positive except when all soft lines connecting to coUinear lines are gluons. The suppression terms, ^N^ and F, axe the only differences from eq. (122). We note that the N3 term is present diagram-by-diagram in physical gauges^J, but that in Feynman gauge the computation holds in general only when gauge invariant sets of diagrams axe combined for the hard scattering subdiagrams. In fact, in covariant gauges, individual diagrams may be much more divergent in the presence of infrared and collinear interactions than is the cross section, and may even grow with energy'^®^. This is a consequence of the well-known fact that unitaxity bounds on energy growth are only a property of gauge invariant sets of diagrams. 8.7 General leading regions The general leading regions for e'^e"-annihilation processes, for deeply inelastic scattering and for hard inclusive hadron-hadron scattering are quite analogous to those for the (<^^)g theory. The basic difference is that lines which paxticipate in infrared logarithms must be added to the corresponding reduced diagrams.
68 (a) (b) Fig. 21. Typical leading regions for annihilation processes, (a) physical gauge, (b) covariant gauge. The most general leading region has the possibility of extra jets beyond the two shown here. Fig. 21(a) shows a general leading region for a single particle inclusive cross section in e'^e" annihilation for a physical gauge, and fig. 21(b) for a covariant gauge Compared to the leading regions for (<^^)6, summarized in fig. 14, which include only jets of coUinear lines and hard subdiagrams, to get fig. 21(a) we simply
69 add a "soft" subdiagram, consisting of lines whose momenta vanish at the pinch singular surface in question. The soft subdiagram contains in general both soft gluon lines and soft quark loops (as well as ghost loops in covariant gauges); its external lines, however, are always gluons. These external gluons always attach to (energetic) lines and not to the hard subdiagrams. The lines attaching jet subdiagrams to the hard subdiagrams may be either gluons or fermions, but at leading power only a single line from each jet enters a given hard subdiagram, just as in ((t>^)Q' The physical picture is also the same as in {(t>^)Q', several hard particles recede from a hard scattering at the speed of light, forming jets by their self-interactions. These particles can never interact with each other except by transfer of soft momenta lA;'^! <C Q- The presence of vector particles in the gauge theory, however, does give leading power contributions from the exchange of soft particles. Since k^/Q « 0 for each of these particles, they do not affect those of the Landau equations, (112) and (113), which involve only the jet subdiagrams. Of course, it should be kept in mind that the soft lines have zero momentum only at the exact pinch singular surface. Feynman integrals get contributions from an entire region near this surface where the soft momenta are much smaller than a typical energy of a jet, but may approach a nonzero fraction of that energy. For a covariant gauge, the leading regions are essentially the same, except that, just as in the one-loop case, arbitrary numbers of longitudinally polarized gluons may attach the jet to the hard part, as shown in fig. 21(b). Figures 22(a) and 22(b) show general leading regions for inclusive deeply inelastic scattering, and the Drell-Yan cross sections in Feynman gauge. As with ((^^)g, the sum over final states eliminates pinch singularities involving final state jets. The remaining on-shell lines make up the jets associated with the incoming particles including soft exchanges within and between the jets. In covariant gauges, longitudinally polarized gluons may connect the jets to the hard part. 9. FACTORIZATION PROOFS IN GAUGE THEORIES We are now ready to discuss the extension of factorization theorems to gauge theories, for the basic cross sections discussed above: deeply inelastic scattering, single particle inclusive annihilation and Drell-Yan production. Each of these will require new reasoning relative to the scalar case. Compared to the proof in Sect. 6 for (</>^)g, our treatment of factorization in gauge theories will be much more modest. Rather than derive closed expressions for the factorized forms in terms of explicit subtraction operators, we will deal with the cross sections on a region-by-region basis. We will show that an arbitrary leading region either contributes to the factorized form of the cross section, or cancels to leading power when gauge invariant sets of diagrams are combined.
70 (a) P P. (b) P B Fig. 22. Leading regions in Feynman gauge for (a) inclusive deeply inelastic scattering and (b) Drell-Yan cross sections. 9.1 Deeply inelastic scattering and collinear factorization We start with the deeply inelastic scattering cross section, h(p) +7*(^) —> X^ with p^{q^) being the momentum of the incoming hadron h (virtual photon 7*). Here, as we shall see, the question of factorization reduces to a treatment of collinear singularities associated with unphysically polarized gluons.
71 Fig. 22(a) illustrates the leading regions in a gauge theory for diagrams that contribute to the structure function tensor W^^{q^p). There is a single jet J, in the direction of the incoming particle, and a single hard subdiagram H^^ ^ containing the hard scattering. Divergences associated with final state interactions cancel because of unitarity in the sum over different final state cuts of the same Feynman graph. Thus we have not included regions in which soft gluons from the jet J interact with the outgoing particles in the hard part if, even though such regions can give leading contributions to individual cut graphs. In Feynman gauge, the hard subdiagram is connected to the single jet by more than one collinear line. This makes the transition to the convolution form of eq. (2) more complex than in the scalar case. In physical gauges, the reduced diagram corresponding to an arbitrary leading region has the same form as for the scalar theory, fig. 9. This simplification is the reason that most of the original arguments for factorization were given in physical gauges^J, where essentially the same procedure can be used as in Sect. 6. However, it is important to show how the proof may be carried out in the covariant gauges, for two reasons. First, as mentioned above, there are difiiculties associated with the unphysical singularities encountered in physical gauges, which have not been fully understood yet^^J. Although these are presumably of a technical nature, and not associated with the content of factorization, it is surely desirable not to be completely dependent on this presumption. Second, physical gauges, because of their noncovariance, are ill-suited to proofs of factorization in the crucial case of hadron-hadron scattering. So, in the interests of generality, we shall discuss deeply inelastic scattering in the Feynman gauge. These issues were not treated in Ref. 5. Let us consider a typical cut Feynman diagram G^^\ where C labels the cut, in the neighborhood of a leading region L. L is specified completely once we specify how the graph G is to be decomposed into the subgraphs J and H. We shall write the contribution from region L to G^^^ as G^^'^\ Referring to fig. 22(a), we see that our problem is to organize the set of longitudinally polarized lines which attach the jets to the hard parts. Suppose a set of n gluons of momentum /^* attaches to the hard part H to the left of the cut, along with a physically polarized parton of momentum A:'* — ^^- /f. Similarly, suppose a set of n' longitudinally polarized gluons I' -^ attaches to H on the right of the cut, along with a physically polarized parton of momentum k^ — Yli^'^- Each momentum l^ is parallel to the external momentum p^ and flows into the hard part. Each l'^^ is also parallel to p^, but flows out. We sum over all cuts of the original graph G consistent with this leading region, with fixed n and n'. We
72 can now represent the sum over these allowed cuts C of G^^'^^ as c n/(wn/ (144) X ^ J(^^>(p^ A:^ - E/f, {/r }; A:^ - E/7, {/f 1)^^^^^^^^^ where /x, and Vj are polarization indices for the /j and /'j, respectively, and ry and ry' are the polarization indices associated with the physical partons attaching to the hard part on either side of the cut. Of necessity, the sum includes only those cuts which preserve n and n', and we note that it breaks up into independent sums over the cuts of the hard part and of the jet. The integrals in (144) are restricted to the neighborhood of the region L. We implicitly introduce a variable fi^ to set the scale of L. The integration region in (144) is set by requiring, for instance, that lines within H have transverse momenta of order at least //, while those in J have transverse momenta of ji or less, fi will later be identified with the renormalization scale for the part on distribution. Because all lines in H are, by construction, far off the mass shell, we replace the momenta of all its external particles by lightlike momenta in the corresponding jet direction. Then, if we keep only leading polarization components, the extra coUinear gluons which attach the hard part to the jet are exactly longitudinally polarized. Corrections are suppressed by a power of q^. To formalize this approximation, we introduce the vectors A* _ ^/^ ,,At _ ^A* i;''=s!^, t.'*=<?^, (145) and define U'l^ = Ai, wk"" = k, u'l"^ = a;. (146) In terms of these variables, the approximation is y^H<c„)(^q.. K _ szf, {If'}- k" - E/';, {I'f })iy' (147) Ch ^(g'';(fc-SA.>^{Ai^,-■■};(fc-EA;.)^,^{AXO),,,'^«"■ IT""''
73 where H{q''', {k - EAiK, {Ai^"'-}; {k - EA;.)^^ {A^-^^^}),,,. (148) Ch i' j' This replacement is analogous to the operator P introduced for the scalar theory in Sect. 6. We will now show that the unphysical polarizations of the extra gluons can be used to factor them from the hard part. The hard part will become a function of only the total longitudinal momentum flowing between it and the jet, as is appropriate for a factorized form, while the longitudinally polarized gluons will couple to an eikonal line, which we associate with the jets. Let us show this result first for the diagram on the left-hand side of fig. 23(a), with a single longitudinally polarized gluon of momentum Z'*, which attaches to the hard part along with the physically polarized parton of momentum k^ — l^. (We shall refer to particles by their momentum labels.) If we apply the Ward identity, eq. (131) to this set of diagrams, we find the result on the right hand side of fig. 23(a). The left-hand side of fig. 23(a) would vanish, except that the diagram on the right-hand side, in which the gluon Z'* is attached to the physical parton, is not included in H by construction. But now consider the identity shown in fig. 23(b). Here we consider a diagram in which the unphysical line ends in an eikonal line, while H has a single (physically polarized) external line from J, which carries the total jet momentum k. The right hand side of figs. 23(a) and 23(b) are the same, and we derive the identity of fig. 23(c), in which the longitudinally polarized gluon has been factored onto an eikonal line moving in the opposite direction from the A-jet. To be careful, we should note that in each individual cut diagram of fig. 23, the intermediate states are not physical states, but rather states including on shell gluons with unphysical polarizations and ghosts. Once graphs for a given cut are summed over, however, we may replace the unphysical states by physical ones^^J. So we may, without loss of generality, treat the matrix elements as though they were between physical states. The extension of this reasoning to two gluons is straightforward. We use the identity of fig. 24, analogous to fig. 23. On the right hand side of the first equality in fig. 24 we have two diagrams in which only the physical parton attaches to the hard part, and also two diagrams in which one gluon is still attached to the hard part. (Diagrams in which the two gluon lines are interchanged are not indicated explicitly in the figure.) In a covariant gauge, Lorentz invariance requires that the gluon entering the hard part in diagram 4 also be longitudinally polarized (it has
74 (a) (b) (c) Fig. 23. Ward identities for a single gluon. Group sums follow repeated indices, (a) Identity for hard part; (b) Eikonal identity; (c) Factorization of the gluon. no other vectors on which to depend). Thus we can apply the result of fig. 23 for the single gluon entering the hard part in diagrams 3 and 4. The result is shown in the second and third equalities in fig. 24. This inductive approach can clearly be extended to arbitrary order, and we derive fig. 25 for a general leading region.
75 <r \ \ ^) /=¥% ^\T ^ir \ ;^) ;«) \ «^ir / \ Fig. 24. Application of Ward identities to two collinear gluons
76 Fig. 25. Factorization of collinear gluons. This gives the overall replacement Ui X j^^^Xp"; fc" - ui {ir}; fc" - sr;, {/'f D^;,-;.;^., (^49) F(g^fc^>''W^(«.{^.■})*'"'^«,{A;})<''' } X /(^-^(p'';^ - s;r,{/n;K - s/';, {/f j)^;";.,,.,,
77 where S{u,{Xi})^^'^^ is a lightlike eikonal line in the u^ direction, coupled to n gluons /,-, and similarly S*{u, {A'})^"^'^ is an eikonal in the same direction coupled to the n' gluons /'•. It is natural to group the eikonal lines with the jet, and to define (compare eq. (75)) x5(M,{Ai})<'"'rK{A;.})<'''> (150) X ^ j(^-')(p''; k" - E/f, {If}; k" - U";, {I'f })l';:„.y Cj The function J is linked to the remaining hard part H{q^, kv^) through only the variable ^ and the physical polarization indices x] and r]'. Using (150) and (149) in (144), we have p(t)(^) = y" ^ F(g^$p•«t;'•),,,-i(0■''•'', (151) where x = —q^l^p-q as in Sect. 1. Thus in each leading region the cross section factorizes into an ultraviolet contribution times a contribution to the distribution of the physical parton which remains attached to the hard part. It is clear that we get every leading region for the parton distribution in this way. Note that when we sum over all leading regions, the perturbative sums for the hard part H and the factorized jet J are completely independent. Equation (151) contains most of the physics of factorization for deeply inelastic scattering, but a few more steps are required to obtain the result (2). First, one argues using Lorentz invariance that for unpolarized incoming hadrons the hard part are both diagonal in the spin indices 'q^x]'. Thus we can sum over the spin of the partons leaving the jet part and average over the spin of the parton entering the hard part. This decouples the two factors in spin space. We then sum over all graphs and over the leading regions L for each graph. The result is '^""-E/ y^aM(e,M)wr(9^ep•«^'^M)• (152) a Here a sum over parton types a is indicated, J^ is the jet part summed over graphs, leading regions, and spins, and Ti is the hard part summed over graphs and leading regions and averaged over the incoming spins. Both J^ and H depend on the parameter ji that sets the scale for the leading regions. We can relate the functions !F to the MS parton distributions fa/a iii ^he following manner. We note first that we can carry out exactly the same factorization
78 procedure for parton distributions defined as in eqs. (43) and (44) as for the deeply inelastic scattering structure functions above. Then, in pla<:e of (152), we find where gba is some new hard part (a matrix in the space of parton types), while Ta/A is the same jet function as in (152). Here we define the scale of the leading regions to be the same as the renormalization scale in the parton distribution, and we use the same notation for both. Using eqs. (152) and (153), we find the desired result for the structure functions. w••'(9^p'') ~ ^ r ^ h/Aiv,i^) ^^(9^'7P^M,«s(M)). (154) where the hard part H^^ is defined by the relation X V It should also be possible to demonstrate this factorization in the more careful manner outlined for the scalar theory in Sect. 6. From (151), the leading region L may now be represented by fig. 9, the canonical form for deeply inelastic scattering found in the scalar theory and in physical gauges. Since the same construction may be carried out for any leading region, one could define a subtraction procedure for gauge theories analogous to the one for scalar theories. The subtraction operator for a leading region L then makes the replacement (147) for the hard part for the region. 9.2 Single-particle inclusive cross sections and the soft approximation The leading regions for a single-par tide inclusive cross section, e"^4-e~ -^ A(p)H-X, were shown in fig. 21. There is a jet subdiagram J that describes the jet in which hadron A is observed. The hard subdiagram H contains two short distance interactions (one on each side of the final state cut) involving highly virtual particles, from which one or more jets of interacting coUinear particles emerge. Once again, there are extra longitudinally polarized gluons connecting the jet J to the hard subdiagram H. More importantly, in contrast with deeply inelastic scattering, there is a soft subdiagram S that connects J to H. As a result, the factorization property fails on a graph-by-graph basis. We recall that in any given cut Feynman graph for deeply inelastic scattering there could be soft partons connecting to the hard subdiagram, representing soft
79 interactions between on-shell particles as they enter the final state. However, we argued that any leading region containing such soft interactions gives a cancelling contribution when one sums over the possible final state cuts for a given Feynman graph. Unfortunately, this rather trivial cancellation mechanism does not work for single particle inclusive pro duct ion^^J. The reason is that we are observing a particle in the final state rather than summing freely over all final states. Nevertheless, our aim will be to show that any leading region with a soft subdiagram connecting J to H cancels. The only remaining leading regions are analogous to those already encountered in the case of deeply inelastic scattering, in Sect. 9.1, so that the proof of factorization sketched there carries over. We consider the contribution from a leading region L to a cut Feynman diagram. Each such cut diagram is decomposed into subdiagrams J, H, and S. We now sum over all cut graphs containing the same number of lines connecting the parts J, if, and S on each side of the cut and call the result G. Our object is to show that, after summing in addition over where the lines go relative to the final state cut, ^ G can be rewritten in a factorized form in the high energy limit. In order to write the kinematic approximations, we pick lightlike vectors v^ = gH^ in the p^ direction and u^ = gt in the Q^,p^ plane, and define the momentum fraction ^ of the outgoing hadron A by U'k = U'p/^. (156) As in the case of deeply inelastic scattering, we use the longitudinal polarization of the extra collinear gluon lines which attach the J to the hard part H. We once again approximate these lines by dominant momentum and polarization components, so that they appear as longitudinally polarized. Then, we sum over graphs representing different attachments of the collinear gluons to H and use Ward identities to remove them from H and attach them instead to eikonal lines S in the w'* direction. They are then grouped with the jet to form, in this case, a fragmentation function c?^/a(0- ^^ ^^^^ derive a form analogous to (151), but with the extra complication of the soft lines. e xF(0^(M•p/0t;^{g7}) ll'{rj} Here J is analogous to the function J in eq. (150). It includes the original jet subgraph together with eikonal lines attached to the 'extra' longitudinally polarized gluons that formally attached to the hard part. The indices rj,rj' represent the
80 polarization of the physical parton that enters the hard part carrying momentum k^ ~ {u'pI£,)v^. The extra complication compared to deeply inelastic scattering is the soft subgraph 5, which couples to J and H via soft gluons as indicated in (157). It is useful to interpret (157) in the language of Sect. 8.1. It describes the jet A containing the observed hadron A together with the unobserved jets that we have included in H. All of these jets emerge from a hard scattering and evolve independently, except for the exchange of soft partons, which are coupled to the color current of each jet. In the frame of jet A, for instance, all the charges within the other jets are moving at nearly the speed of light. But then, according to the discussion of Sect. 8.1, the Lorentz transformed field due to these jets should be nearly gauge equivalent to zero. Of course, since jet A arises from a quark or gluon, it is not gauge invariant. We might expect, however, that we can exhibit the gauge nature of this interaction. To do so, we need a generalization of the eikonal approximation which we applied in Sect. 8.4 to single parton lines coupled to soft radiation. The relevant generalization of the eikonal approximation has been termed the "soft approximation" ^^'^^'^^'^^J. For the A-jet,it consists of making the replacement •qr}' " v>' lai J /T/T/ in which we define ^" = q-v u^, (159) This approximation replaces each soft gluon entering the A-jet by a fictitious gluon whose momentum and polarization are both in the w-direction. Before justifying the soft approximation, let us see what its consequences are. Once we make the soft approximation, each cut jet diagram is a contribution to a product of matrix elements precisely of the form to which the Ward identity of eq. (133) can be applied, with the field ^{x) now representing the field associated with the physically polarized parton which couples to the hard part. As a result, we have at out disposal a Ward identity, which can be used to factor soft lines from the jet subdiagram, by an iterative argument very similar to the one just used to factor longitudinally polarized collinear gluons from the hard part. The details of the argument are slightly more complex because of the extra eikonal line, and we refer the interested reader to Refs. 35 and 36 for details. Here we simply quote the result, which is illustrated in fig. 26 and may be expressed as {«/} Jab{iAil])nn' '^oc^ " "^ocr.'^ <^l . . . yf'n ^ Jrf<i(0„'^K,9!)ir<^'''^•(«^5S)^^^''^'' ^^^°^
81 where ^ac(v'*5^2) stands for the lightHke eikonal line in the v^ direction, to which have been connected those soft gluons to the left of cut C, with momenta qf^ and polarization indices cr(L). 5* is defined similarly in terms of soft gluons to the right of the cut, with momenta q^ and polarization indices <7(R). Finally, we have made color indices a,6,... explicit, and d(R) is the dimension of the color representation of the physical parton. b Fig. 26. Factorization of soft lines from a jet
82 The complete Green ftinction may now be written in the form xffa6(Q^e«•9f^{97}) T]T] {tj} Notice that the jet function has now been factored from the rest of the process. When we now perform a sum over cuts, we can sum independently over the cuts of J and over the cuts of the rest of the diagram. In the rest of the diagram, we have a hard interaction producing the eikonal line S and the jets in H. These are coupled by final state interactions with the soft gluons in 5. By the general reasoning of Sect. 7.6, these final state interactions cancel. Thus any leading region with soft exchanges cancels, and the factorization reasoning reverts to the arguments which apply in the scalar theory. We shall skip giving these details, and will close this subsection with a justification of the all-important soft approximation, eq. (158). The soft approximation consists of an approximation for the polarizations, and an approximation for soft momenta. The former may be justified by detailed power counting arguments-^^J, but the underlying motivation is simply that gluon polarizations proportional to u^ can couple to the A-jet by contracting into vectors proportional to p^, the momentum of the hadron A. Since w-p = p"^ is a large invariant, it will dominate by a power over invariants formed from the other internal momentum components present in J. Note, by the way, that in Feynman gauge gluon polarizations will have nonvanishing projections onto u^ only if the soft subdiagram couples to other parts of the diagram as well as J. The approximation associated with the gluon momentum is more subtle. Recall that the A-jet is in the plus direction. We claim that one can neglect the transverse momentum of the gluon compared to its minus momentum. As we have seen in Sect. 8.4, this is nontrivial. In fact, regions where the transverse momentum is nonnegligible are leading by power counting. Recalling the one-loop discussion of Sect. 8.4, a typical denominator from the i4-jet on the left of the cut is of the form of U - 9,)' + '■« ~ ^' - 2^^-?." + 2^x-'/x. - kx.f + «« (162) with t^ a typical line momentum in the A-jet. We would like to set q\_^ to zero in all denominators like (162), and all we need for that is ^. > |2^j_-gj_^ - \qu 2 (163)
83 The relevant question is thus whether the q~ momentum contours are trapped at q7 = 0, at the scale of {2i±'q±i — \q±i\'^)/p'^- Note that poles of this type can only come from denominators from the A-jet, and not from the hard part or the soft subdiagram. Now, although every jet line through which q^ flows gives a pole at a position like (162), close to the origin in the q^ plane, all of these poles are on the same side of the real axis. To see this, consider how each soft momentum q~ flows from the vertex where it attaches to J to the parton line that attaches J to H. In general, the q~ pole from any jet line is in the upper half plane ii q~ flows in the opposite sense relative to the large plus momentum carried by that line. But q^^ may always be chosen to flow so that q~ is directed in this sense for each jet line on which it appears. This is evident from fig. 21. Soft gluons to the right of the cut may be treated analogously. As a result, the q'[ contours may all be deformed away from jet poles into a region where q±i may be neglected, and the soft approximation is justified along the deformed contour. By Cauchy's theorem, it is also justified in the original integral. Thus, the factorization program may be carried out in e"*"e~ annihilation. 9,3 The Drell-Yan cross section The thorniest factorization theorems involve two hadrons in the initial state. The Drell-Yan cross section for the process a{pa) + b(pb) -^ e+e-iQ") + x (i64) is the simplest of these, and has therefore received essentially all the attention. Q^ will represent the momentum of the lepton pair t^f. The step from Drell-Yan to more complex processes, involving observed hadrons or jets in the final state is relatively straightforward, as indicated in Sect. 7.6. Factorization for the Drell-Yan cross section has, at times, been the subject of controversy^^'^^'^^1, although more recent work has, we believe, established its validity at all orders^^'^^'^^J. Nevertheless, as we shall observe below, there is plenty of room for improvement in our understanding. The general leading region for the Drell-Yan process is shown in fig. 22(b). After the sum over final states, all nonforward hadron jets are absorbed into the hard subdiagram if, in the same way as in deeply inelastic scattering. In common with the deeply inelastic scattering and one particle inclusive e"*"e~ annihilation processes, we can factor collinear gluons from the hard part. Once this is done, the sum of cut Feynman diagrams for the Drell-Yan cross section is very similar
84 to eq. (157) for e"*"e annihilation, X fl■(Q^a(«•P^)'^MB(^^•PB)«")'"' , where the lightHke vectors v^ = g!l,u^ — g^ have been chosen in the Pj^^Pq directions, respectively and the parton momentum fractions are defined by ^a = kA'u/pA'U and ^b = kB'v/pB'V. Here JaUai {Qi}) ^^^ JBi^sAQif}) ^^^ similar to the parton distribution function J defined for deeply inelastic scattering in eq. (150), except that soft gluons are still attached to them. Connections between the parton distribution and a soft sub diagram were absent in the deeply inelastic scattering cross section, because, after the sum over cuts, there was only one jet, which cannot by itself produce large invariants in numerator factors. In (165) we have a soft subdiagram as in e"*"e~ annihilation, but now interacting with the jets associated with the two incoming hadrons. Our basic problem is the same as in e"*"e~ annihilation, to show that contributions from any leading region with a nontrivial soft subdiagram cancel in the sum over final states and gauge invariant sets of diagrams. Then the remaining leading regions of eq. (165) are just of a form similar to eq. (144), and the arguments for factorization may be given as above for deeply inelastic scattering, eqs. (152) to (155). Naturally, we would like to proceed by analogy to e"*"e~ annihilation. Thus, we would like to apply the soft approximation to the jets, and factor the soft gluons from them. The jets would then contribute to parton distributions as in eq. (153), and, once the remaining soft contributions cancel, we would derive the desired factorized form, eq. (11). The main obstacle to this program is shown in fig. 27, which illustrates a typical low-order example. It shows a single soft gluon, q^, attached to the A-jet. The soft momentum flows through two lines in the A-jet, an "active" jet line i + q that carries positive plus-momentum into the hard part, and a "spectator" line p — i — q that carries positive plus momentum into the final state. We saw in e"^e~ annihilation that the criterion for the applicability of the soft approximation for qf^ is given by |g~^"^| ^ |^± • (2^j_ -f ^±)|, where i^ is any line in the jet along which q^ may flow. But this condition may be satisfied along the entire q~ contour only if the contour is not pinched by poles on opposite sides of the real line. In e"^e~ annihilation they are not, but in Drell-Yan they are. This is illustrated by our example, since the poles in the q~ plane due to the jet propagators are
85 P p-il-q Fig. 27. Example illustrating obstacles to the soft approximation. approximately at - _ (P - 0' + 2(px - i±)-q± - |gxP + ie which are on opposite sides of the contour, both at a distance of order 2i±'q±/p'^ from the origin. Thus, in the Feynman integral associated with fig. 27, the q~ contour is forced to go through a region in which the soft approximation fails, and we are unable to apply immediately the reasoning introduced for e"*"e~ annihilation. The resolution of this problem is rather technical, and may be found in Ref. 36. It may be understood most simply as a result of the Lorentz contraction of the colliding gluon fields, as in Sect. 8.1. In addition, we can give an intuitive picture here, based on semi-classical considerations in the center of mass frame. We consider the A-jet to be passing through a soft color field produced by the B- jet. Consider first the very softest part of the color field, with a spatial extent ~ (1 fm) [p^/(l GeV)]. The self-interactions of the partons in the A-jet are time dilated, but this field extends so far in space that it interacts with the partons on the same time scale as that of their self-interactions. However, on this distance scale, the soft gluons cannot resolve the hadron jet into individual partons. Thus the jet appears as a color singlet until the time of the hard interaction, at which time it acquires color because one parton is annihilated. The result is that the only interactions of the very soft color field occur long after the hard scattering event, and such interactions cancel because of unitarity. Consider now the part of the color field of smaller spatial extent, say 1 fm. The point is that the interactions
86 of this color field with the spectator partons in the A-jet don't really matter. The reason is that the self-interactions of the partons in the A-jet are time dilated, so that the spectator partons do not interact with the active parton on the 1 fm time scale in which they interact with the color field. Since the spectator partons are not observed, unitarity implies that their interactions with the color field will not affect the cross section. As a technical trick, we could as well replace all of the spectator partons with an equivalent color charge located at x± = 0, right on top of the active quark. Then the color field sees a net color singlet in the initial state and a net colored charge in the final state. Again, the only interactions of the color field occur long after the hard scattering event, and such interactions cancel because of unitarity. Finally, note that the arguments given above are asymmetric between the two incoming jets. This is natural, because it is only necessary that one of the two incoming particles move at the speed of light for our arguments to apply. Indeed, factorization should hold in the (hypothetical) scattering of a truly lightlike particle with a massive particle at any center of mass energy. We should note that explicit two-loop calculations which show that infrared divergences cancel at leading power (although not at higher twist), have been carried out for the most part with one massive and one massless (eikonal) line^^'^^^J. 10. OUTLOOK AND CONCLUSION In the foregoing, we have described the systematics of factorization for hard inclusive cross sections in QCD, and have discussed in some detail the nature of factorization proofs, first for (<^^)6, and then for gauge theories. Along the way, we outlined a systematic approach to perturbative processes at high energy, based on the classification of leading regions. As we have indicated above, the proof of factorization theorems in gauge theories is by no means a closed subject. Factorization proofs for inclusive processes is the first item of a whole list of subjects in which progress has been made, but for which important work remains to be done. In the following, we briefly discuss a few other significant topics which relate closely to the methods discussed in this chapter. Of great importance are extensions of the theorems to more general situations. 10.1 Factorization Proofs Factorization proofs in nonabelian gauge theories have reached a certain level of sophistication in Refs. 34, 35 and 36. Comparison with the discussion for (<^^)6, however, shows that there is as yet in the literature no complete and systematic subtraction procedure in QCD of the type explained in Sect. 6, even in the case of
87 deeply inelastic scattering. A subtraction algorithm would eliminate any lingering uncertainty associated with overlaps between leading regions. Perhaps even more importantly, such a procedure should make it possible to develop bounds on corrections to leading power factorization theorems, and to prove factorization theorems for nonleading power corrections, so-called "higher twist". A model for this program is presumably to be found in the BPHZ formalism for deeply inelastic scattering cross sections in scalar and abelian gauge theories developed by Zimmermann^^, suitably modified to treat the extra infrared problems and gauge structure of QCD (see Sect. 6). It should also be noted that the Monte-Carlo event generators^^^ that are so widespread in analyzing data depend on generalizations of the factorization theorems; these generalizations have not yet gone significantly beyond the level of leading logarithms. In addition, we should mention that additional factorization theorems, of different but related forms, are central to the analysis of the elastic scattering of hadrons, which decrease as powers of the energy'^^J. 10.2 Factorization at Higher Twist It has been proposed'^^j that generalized factorization theorems hold beyond the leading twist for a wide variety of cross sections. Most work on this possibility has been carried out for deeply inelastic scattering, where the systematics are best understood as a generalization of the operator product expansion^^''*^'^^^ In particular, it has been shown that multiparton distributions may be defined in a natural way to parameterize soft physics at higher twist^^'^^J. In hadron-hadron scattering, factorization at higher twist is complicated by the infrared structure of perturbative QCD. We have seen in Sect. 9 that leading twist factorization requires the cancellation of infrared divergences. It has been shown by explicit calculation^^'^^j, however, that infrared divergences do not cancel beyond a single loop in hadron-hadron scattering for QCD at higher twist. This is a sharp contrast between the nonabelian and abelian theories. At two loops, noncancelling divergences occur at the level m^ fs^ in the Drell-Yan process. Refering to Sect. 8.1, this is precisely the level suggested by the classical relativistic kinematics of gauge fields. How one should interpret this lack of cancellation is not quite clear to us. The actual situation, including nonperturbative effects, may be better or worse than suggested by perturbative calculations^^J. The fact that perturbation theory respects factorization at m?Is., however, makes it possible that factorization theorems may hold at this level, even for hadron-hadron scattering^^j. 10.3 Factorization at the Boundaries of Phase Space A rich class of perturbative predictions involve the summation of corrections near
88 boundaries of phase space in different processes. Near some of these boundaries, notably small Q± and small x, cross sections increase greatly. Along these lines, perhaps the most attention has been given to the Drell Yan cross section at measured transverse momentum dcr/dQ^d^Qx^^'^^'^^^ with Q± "C Q and the related two-particle inclusive cross section for e"*" + e~ -^ A -\- B -\- X at measured transverse momentum^^'^^'^®^ The complete leading-twist analysis of these cross sections begins with factorized forms of the type of eq. (165), in which soft partons have been factored from jets, but not yet cancelled. At the boundary of phase space the cancellation of soft gluons outlined in Sect. 9 still occurs, but is incomplete. All infrared divergences still cancel at leading power, but finite remainders depend on the small parameter in the problem, for instance the transverse momentum in the cross sections cited above. By developing generalizations of the renormalization group equation for each of the functions in the factorized form eq. (165)'*^^, it is possible to resum systematically higher order corrections to these quantities. This general approach can be applied in a number of other physically important situations. For instance, the r = Q'^/s -^ 1 limit in the inclusive Drell- Yan cross section is related to the normalization of the Drell-Yan cross section, the "K-factor"^^'®^'®^'®^l. It is possible to sum corrections which are singular at r = 1^2,63] ^j^ interesting feature of the result is that it is sensitive to high orders in perturbation theory^^^ through the running coupling. Because of this, it gives a measure of the sensitivity to higher-twist effects of perturbative predictions based on factorization^^j. This sensitivity is found to be nonnegligible in some, but not all, regions of physical interest. Another regime, which is of crucial importance for experiments at the Teva- tron and SSC, is the x —)- 0 limit in hadron-hadron scattering. The cross sections get into the range of tens of millibarns, which is enormous compared to typical cross sections at larger x. So far, much work has concentrated on the behavior of parton distributions at small j;65,66,67,68]^ assuming the validity of the standard factorization formulas (2), (3) and (11). From a more general point of view, factorization has been shown to hold explicitly in leading logarithms in x^^K We would like to suggest, however, that factorization theorems need a more extensive examination in this region. In conclusion, we emphasize that essentially every calculation in perturbative QCD is based on one factorization theorem or another. In view of this, progress toward developing perturbative QCD as a quantitative system requires further understanding of the systematics of factorization.
89 ACKNOWLEDGEMENTS This work was supported in part by the Department of Energy, under contracts DE-AT06-76ER-7004 and DE-FG02-85ER-40235, and by the National Science Foundation under contract PHY-85-07627 and under grant No. PHY82- 17853, supplemented by funds from the National Aeronautics and Space Administration. REFERENCES 1 K. Wilson, Phys. Rev. 179 (1969) 1699. 2 W. Zimmermann, Comm. Math. Phys. 15 (1969) 208, and Ann. Phys. (N.Y.) 77 (1970) 536, 570. 3 N. Christ, B. Hasslacher and A.H. Mueller, Phys. Rev. D6 (1972) 3543. 4 W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Phys. Rev. D18 (1978) 3998. 5 D. Amati, R. Petronzio, and G. Veneziano, Nucl. Phys. B140 (1978) 54 and B146 (1978) 29; R.K. ElHs, H. Georgi, M. Machacek, H.D. Politzer, and G.G. Ross, Nucl. Phys. B152 (1979) 285; A.V. Efremov and A.V. Radyushkin, Teor. Mat. Fiz. 44 (1980) 17 [Eng. transL: Theor. Math. Phys. 44 (1981) 573], Teor. Mat. Fiz. 44 (1980) 157 [Eng. transL: Theor. Math. Phys. 44 (1981) 664], Teor. Mat. Fiz. 44 (1980) 327 [Eng. transL: Theor. Math. Phys. 44 (1981) 774]; S. Libby and G. Sterman, Phys. Rev. D18 (1978) 3252, 4737; A.H. Mueller, Phys. Rev. D18 (1978) 3705. 6 R.P. Feynman, "Photon-Hadron Interactions", (Benjamin, Reading, MA, 1972). 7 J.C. Collins, "Renormalization" (Cambridge University Press, Cambridge, 1984). 8 V.N. Gribov and L.N. Lipatov, Yad. Phys. 15 (1972) 781 [Engl, transl: Sov. J. Nucl. Phys. 46 (1972) 438]; L.N. Lipatov, Yad. Phys. 20 (1974) 181 [Engl, transl: Sov. J. Nucl. Phys. 20 (1975) 95]; G. AltareUi and G, Parisi, Nucl. Phys. B126 (1977) 298. 9 J.C. CoUins and D.E. Soper, Nucl. Phys. B194 (1982) 445. 10 J. Kogut and D.E. Soper, Phys. Rev. Dl (1970) 2901; J.D. Bjorken, J. Kogut and D.E. Soper, Phys. Rev. D3 (1971) 1382. 11 B. Curci, W. Furmanski and R. Petronzio, Nucl. Phys. B175 (1980) 27; L. Baulieu, E.G. Floratos and C. Kounnas, Nucl Phys. B166 (1980) 321. 12 G. Sterman, Nucl. Phys. B281 (1987) 310. 13 M. Diemoz, F. Ferroni, E. Longo and G. Martinelli, Z. Phys. C39 (1988) 21. 14 G. AltareUi, R.K. Ellis and G. Martinelli, Nucl. Phys. B157 (1979) 461. 15 N.N. Bogoliubov and O. Parasiuk, Acta Math. 97 (1957) 227; K. Hepp, Comm. Math. Phys. 2 (1966) 301.
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93 EXCLUSIVE PROCESSES IN QUANTUM CHROMODYNAMICS* Stanley J. Brodsky Stanford Linear Accelerator Center Stanford University, Stanford, California 94S09, USA G. Peter Lepage Laboratory of Nuclear Studies Cornell University, Ithaca, New York 1^853, USA 1. INTRODUCTION What is a hadron? In practice, the answer to this question depends upon the energy scale of interest. At the atomic scale a hadit)n can be treated as an elementary pointlike particle. The proton's electromagnetic interactions, for example, are well described by the simple Hamiltonian for a point-like particle: This Hamiltonian describes a wide range of low-energy phenomena—e.g. proton- electron elastic scattering (ep —* ep), Compton scattering of protons (7p —> 7p), atomic structure...—and it can be made arbitrarily accurate by adding interactions involving the magnetic moment, charge radius, etc. of the proton. The description of the proton becomes much more complicated as the energy is increased up to the strong interaction scale (~ 1 GeV). In proton-electron elastic scattering, for example, one must introduce phenomenological form factors F(Q^) to correct the predictions from the point-like theory: in effect, T(ep) — F{Q^) ^(epjpoint-like where Q is the momentum transfer and F{Q') ~ ' ^ Q2 + A2 (2) One might try to modify the proton-photon interaction in the point-like Hamiltonian to reproduce the phenomenological form factors, but the resulting interaction * Work supported in part by the Department of Energy under contract number DE-AC03- 76SF00515 and the National Science Foundation.
94 would be very complicated and nonlocal. Furthermore such a modification would not suffice to account for the changes in the Compton amplitude of the proton at high energies. In fact, new terms would have to be added to the Hamiltonian for every process imaginable, resulting in a horrendously complicated theory with little predictive power. The tremendous complexity of the high-energy phenomenology of hadrons stalled the development of strong interaction theory for a couple of decades. The breakthrough to a fundamental description came with the realization that the rich structure evident in the data was a consequence of the fact that hadrons are themselves composite particles. The constituents, the quarks and gluons, are again described by a very simple theory. Quantum Chromodynamics (QCD). The complexity of the strong interactions comes not from the fundamental interactions, but rather from the structure of the hadrons. The key to the properties of the form factors and other aspects of the phenomenology of the proton thus lies in an understanding of the wavefunctions describing the proton in terms of its quark and gluon constituents. In this article we shall discuss the relationship between the high-energy behavior of wide-angle exclusive scattering processes and the underlying structure of hadrons. Exclusive processes are those in which all of the final state particles are observed: e.g. ep —^ ep, 7p —>> 7p, pp —>> pp.... As we shall demonstrate, the highly varied behavior exhibited by such processes at large momentum transfer be understood in terms of simple perturbative interactions between hadronic constituents. ' Large momentum transfer exclusive processes are sensitive to coherent hard scattering quark-gluon amplitudes and the quark and gluon composition of hadrons themselves. The key result which separates the hard scattering am- 4 2 plitude from the bound state dynamics is a factorization formula: ' To leading order in 1/Q a hard exclusive scattering amplitude in QCD has the form 1 M = JTH{xj,Q)l[<i>HX^j.Q)[dx] . (3) 0 H. Here Th is the hard-scattering probability amplitude to scatter quarks with fractional momenta 0 < x^ < 1 collinear with the incident hadrons to fractional momenta collinear to the final hadron directions. The distribution amplitude </>//, is the process-independent probability amplitude to find quarks in the wavefunc- tion of hadron Hi collinear up to the scale Q, and n, / n [dx] = '[[dxj6(l-Y.Xk)- (4) 1=1 ^ k=i
95 Remarkably, this factorization is gauge invariant and only requires that the momentum transfers in Tfj be large compared to the intrinsic mass scales of QCD. Since the distribution amplitude and the hard scattering amplitude are defined without reference to the perturbation theory, the factorization is valid to leading order in l/Q, independent of the convergence of perturbative expansions. Factorization at large momentum transfer leads immediately to a number of 5 important phenomenological consequences including dimensional counting rules, (\ 7 hadron helicity conservation, and a novel phenomenon called "color transparency", which follows from the predicted absence of initial and final state interactions at high momentum transfer. In some cases, the perturbation expansion may be poorly convergent, so that the normalization predicted in lowest order perturbative QCD may easily be wrong by factors of two or more. Despite the possible lack of convergence of perturbation theory, the predictions of the spin, angular, and energy structure of the amplitudes may still be valid predictions of the complete theory. This article falls into two large parts. In the first part, we introduce the general perturbative theory of high-energy wide-angle exclusive processes. Our discussion begins in Section 2 with a discussion of hadronic form factors for mesons composed of heavy quarks. This simple analysis, based upon nonrel- ativistic Schrodinger theory, illustrates many of the key ideas in the relativistic analysis that follows. In Section 3 we introduce a formalism for describing hadrons in terms of their constituents, and discuss general properties of the hadronic wave- functions that arise in this formalism. In Section 4 we give a detailed description of the perturbative analysis of wide-angle exclusive scattering. In the second part of the article we present a survey of the extensive phenomenology of these processes. In Sections 5 and 6 we review the general predictions of QCD for exclusive reactions and the methods used to calculate the hard scattering amplitude. Various applications to electromagnetic form factors, electron-positron annihilation processes and exclusive charmonium decays are also discussed. One of the most important testing grounds for exclusive reactions in QCD are the photon-photon annihilation reactions. These reactions and related Compton processes are discussed in Section 7. In Section 8, the QCD analysis is extended to nuclear reactions. The reduced amplitude formalism allows an extension of the QCD predictions to exclusive reactions involving light nuclei. Quasi-elastic scattering processes inside of nuclei allow one to filter hard and soft contributions to exclusive processes and to study color transparency. The most difficult challenges to the perturbative QCD description of exclusive
96 reactions are the data on spin-spin correlations in proton scattering. We review this area and a possible explanation for the anomalies in the spin correlations and color transparency test in Section 9. General conclusions on the status of exclusive reactions are given in Section 10. The appendices provide a guide to the main features of baryon form factor and evolution equations; a review of light-cone quantization and perturbation theory; 8 and a discussion of a possible method to calculate the hadronic wavefunctions by directly diagonalizing the Hamiltonian in QCD. -k Figure 1. Nonrelativistic form factor for a heavy-quark meson 2. NONRELATIVISTIC FORM FACTORS FOR HEAVY-QUARK MESONS The simplest hadronic form factor is the electromagnetic form factor of a heavy-quark meson such as the T. In this section we show how perturbative QCD can be used to analyze such a form factor for momentum transfers that are large compared with the momentum internal to the meson, but small compared with the meson's mass. The analysis for relativistic momentum transfers is presented in subsequent sections. Heavy-quark mesons are the simplest hadrons to analyze insofar as they are well described by a nonrelativistic quark-antiquark wavefunction. The amplitude that describes the elastic scattering of such a meson off a virtual photon is, by definition of the form factor, the amplitude for scattering a point-like particle multiplied by the electromagnetic form factor. The form factor is given by a standard formula from nonrelativistic quantum mechanics (see Fig. 1): j^r(k+q/2)Hk). (5) (Note that the wavefunction's argument is 1/2 of the relative momentum between the quark and antiquark.) At first sight it seems that we require full knowledge of
97 the meson wavefunction in order to proceed, but in fact we need know very little about the wavefunction if q is sufficiently large. To see why we must determine which regions of A:-space dominate the integral in Eq. (5) when q is large. When f ~ 0 the integral in Eq. (5) is just the normalization integral for the wavefunction, and F[q ) ~ 1—the meson looks like a point-like particle to long-wavelength probes. As q becomes large, large momentum flows through one or the other or both of the wavefunctions in Eq. (5). Since nonrelativistic wavefunctions are strongly peaked at low momentum, the form factor is then suppressed. The dominant region of A:-space is that which minimizes the suppression due to stressed wavefunctions. There are three regions that might dominate: —♦ —♦ —♦ 1) |A: I <C If I, where il)*[k -f q 12) is small but il)[k ) is large; 2) 1^ -I- q I2\ <C If I, where il){k) is small but xl)*{k -\- q/2) is large; 3) \k -f q/2\ ^ \k\ ^ If/4|, where both ^(^) and 'ip*{k -f q/2) are small, but not as small as the stressed wavefunction in either of the other two regions. The ^-dependence of the contributions to F(q ) from each of these regions is readily related to the high-momentum behavior of the wavefunction. In region 1), k can be neglected relative to q/2 in the first wavefunction and so the form factor has ^-dependence 2 F{q')^r{q/2). (6) The contribution from region 2) is essentially identical, as is clear if one makes the variable change k -^ k ~k -f f/2. In region 3), the phase space contributes a factor of q^ while each wavefunction goes like tp{q/4) so that ■2 /'(r)~</iV'(974)r- (7) The dominant region is clearly a function of the high-momentum behavior of the wavefunction. In fact wavefunctions for heavy-quark mesons, like those for QED atoms, fall off as inverse powers of the momentum when it becomes large. As we show below, the ground state wavefunction falls off like 1/q^ up to factors of log(^^). Then the form factor is dominated by regions 1) and 2) for large (nonrelativistic) q , and falls off as ^(f/2) ~ (1/f )^- The contribution from region 3) is suppressed by an additional factor of l/|f j, and so can be neglected when q is sufficiently large. Note that this behavior is characteristic of wavefunctions that vanish as powers of the momentum. With a Gaussian wavefunction, for example, region 3) dominates and the form factor is exponentially damped for high momentum transfers.
98 Neglecting k relative to q/2^ the contribution to F(q ) coming from region 1) has the simple form ^ <fk —^V(A;) = V'(972)0(r=O) (8) where ip{f = 0) is the wavefunction evaluated at the origin (in coordinate space). We can further simplify this equation using the Schrodinger equation for ip{q/2) (Fig. 2): 4>(q/2) = e — (i/2) 2M, 21 -1 {2^)- V{k -q/2)x{>{k) (9) where e is the nonrelativistic binding energy, Mr = Mq/2 is the reduced mass of the quark and antiquark, and V is the interaction potential between them. The ¥(r = 0) Figure 2. Momentum-space Schrodinger equation for the meson wavefunction potential V{q) can be computed using perturbation theory when the momentum transfer q is large; to leading order it is just the Coulomb interaction modified by a running coupling constant: V{q) = - ATras{q'^)CF (10) Here Cf = 4/3 is the value of the Casimir operator for the fundamental repre-
99 sentation of SU3 (i.e. the quark's representation), and a,{Q') = ^'^ ^0 log(QVA^Ci3) is the running coupling constant of QCD, with scale parameter Aqcd ~ 200 MeV, and ^0 = 11 — 2ny/3 where Uf is the number of active quark flavors (ny = 4 for the T). Given this behavior for V we can show that the region |.A: | <C If/2| dominates the integral in Eq. (9) by using arguments similar to those just applied to the form factor (Eq. (5)). Thus when q is large Eq. (9) becomes 4>(q/2) rs^ (i/2) 2Mr 21-1 Vi-q/2)^ir=0), (11) and the form factor takes the form F(r)« ^'(-0) |v(-,72)3(^r^^ + r(-7^^(-^72)} t(r=o) « ^^^- --(ff) ^'^^^ l,(.=o)P (12) where we have now included the contributions from both regions (1) and (2). So all we really need to know about the meson is its wavefunction evaluated at the origin. The high-f form factor is completely determined by perturbation theory up to an overall multiplicative constant! Equation (12) has a simple, intuitive interpretation that generalizes easily to the relativistic case and to other processes. The quantity 2,_-., ^.-, 1 1 (13) 12S7ras(q^/i)MQCF that appears in the first expression of Eq. (12) is just the nonrelativistic meson form factor but with each of the initial and final state mesons replaced by an on-shell quark-antiquark pair. The quark and antiquark share the meson's three- momentum equally. Our analysis shows that momenta internal to the mesons
100 can be neglected relative to q in this "hard-scattering amplitude"—i.e. that Th is roughly independent of the relative momenta of the quark and antiquark when q is large. In coordinate space this means that the separation between the quark and antiquark in this process (~ l/l9*l) ^^ much smaller than the size of the mesons. Thus Eq. (12) for the asymptotic form factor can be recast in the highly suggestive form (Fig, 3) Fif) = ^(f = 0)rif(9')t/'(r =0) (14) where tp{r = 0) is the amplitude for finding the quark and antiquark on top of each other in the initial meson, Tff is the amplitude for scattering the quark- antiquark pair from the initial direction to the final direction, and il)*{f = 0) is the amplitude for transforming the resulting quark-antiquark pair into the final meson. ¥(0) q/2 + q/2 ¥*(0) Figure 3. The asymptotic form factor in terms of the hard scattering amplitude Th and the meson's wavefunction at the origin tl)(f = 0). Notice that we are justified in using perturbation theory to compute Tjj only because the hard-scattering subprocess occurs over short distances. This highlights an important distinction between the perturbative analysis of form factors and that of other processes like deep inelastic scattering. Perturbative QCD is reliable only for phenomena that occur over short distances (or near the light cone). In processes like deep inelastic scattering the short distances arise for largely kinematical reasons: the cross section for deep inelastic scattering is given by a matrix element of two currents separated by z^ ~ ^/Q^- By contrast, we find short distances in our form factor analysis only by looking inside the process. Short distances arise as a result of the properties of the hard-scattering amplitude Th—i.e. as a result of the dynamics of the theory. As a consequence the validity of a perturbative analysis of form factors is perhaps not as well established as it is for, say, deep inelastic scattering. By the same token the analysis is perhaps more interesting because of the critical role played by the dynamics and by hadron structure.
101 Finally we should comment briefly upon the principal limitation of our per- turbative analysis: it is valid only over a limited range of momentum transfer. It is clear from our analysis that q (2 must be larger than the root-mean-square momentum in the wavefunction. This is evident from the form factor for ground- state positronium, which can be computed analytically: '''■""'=' ?Tib I <'^' where 7 = ame/2 is the rms momentum. Here q must be of order 4 times the rms momentum before the form factor begins to fall off like the asymptotic form factor. In the QCD case q 12 must also be sufficiently large that the perturbative part of V(q/2) dominates the nonperturbative part. At the high end, q is limited by the fact that our analysis is nonrelativistic. Also radiative corrections to the form factor (Eq. (5)) and to the quark potential (Eq. (10)) contribute corrections of order q /Mq that become important for relativistic q. These limitations make it unlikely that our results can be used for the ^ or even for the T; neither meson is sufficiently nonrelativistic. So we must develop a relativistic analysis if we are to treat these mesons or, more generally, light-quark hadrons properly. 3. HADRONIC WAVEFUNCTIONS The relativistic analysis of hadronic form factors and other large-px processes is conceptually similar to the nonrelativistic analysis. The only significant difference is in the formalism used to describe hadronic structure in terms of its constituents. To proceed we require a relativistic formulation of the bound state problem. The conventional formalism for relativistic bound states is the Bethe-Salpeter formalism. In this formalism a meson is described by a covariant wavefunction BS ylf^\kuk2) = (0 ITxl;{ki)ij(k2) I M) (16) that depends upon the four momenta of its quark and antiquark constituents. Although formally correct, this formalism is of little use in the description of such simple systems. The problem is that the couplings between different channels— e.g. between quark-antiquark and quark-antiquark-gluon channels—is usually large in highly relativistic systems, and the energy available is more than ample for particle creation. Thus the physics of such systems tends to depend upon the
102 interplay between a large number of channels. A meson for example is a superposition of states involving a quark-antiquark pair, a quark-antiquark pair plus a gluon, a quark-antiquark pair plus two gluons, two quark-antiquark pairs, and so on. In the Bethe-Salpeter formalism this interplay between channels is implicit since the meson is described entirely by a quark-antiquark wavefunction. Reference to all other channels is buried inside the potential and irreducible scattering amplitudes used in analyzing hadronic processes, and as a result these potentials and scattering amplitudes become largely intractable. Even in situations where a single channel dominates, the formalism is still quite complicated and very nonin- tuitive. For example the Bethe-Salpeter wavefunction has no simple probabilistic interpretation analogous to that for nonrelativistic wavefunctions. Because of such complexity the Bethe-Salpeter formalism has been largely abandoned, even in state-of-the-art calculations pertaining to such highly nonrelativistic systems as positronium or the hydrogen atom. Intuitively one would like to describe hadrons in terms of a series of wave- functions, one for each channel, just as one would in nonrelativistic quantum mechanics: e.g. Tt) = ^ \qq) 4^qq/n + X^ \qQ9) i^qqg/ir + ' " ' • (17) Q<1 (1(19 Formally this can be done by quantizing QCD at a particular time, say t — 0, and using the creation and annihilation operators from the fields to define the basis states for such a "Fock-state" representation. The problem with this approach is that the zero-particle state in this basis is not an eigenstate of the Hamiltonian. An interaction term in the Hamiltonian like gip^f^A^ip contains contributions such as b^a^d^ that create particles from the zero-particle state. As a result not all of the bare quanta in an hadronic Fock state need be associated with the hadroii; some may be disconnected and possibly quite remote elements of the vacuum (Fig. 4). This greatly compHcates the interpretation of the hadronic wavefunctions. Also Lorentz transformations are very complicated in this formalism; boost operators tend to create all sorts of additional quanta. This is because the quantization surface ^ = 0 is not invariant under boosts, and thus boosting a state inevitably involves the dynamical evolution (in t) of parts of that state. This is a serious problem for our analysis of large-p_L processes since the initial and final state hadrons necessarily have very different momenta. Fortunately there is a convenient and intuitive formalism, originally due to 9 Dirac, that avoids these problems. This is based upon the "light-cone quantization" of QCD, where the theory is quantized at a particular value of light- cone time T = t -\- z rather than at a particular time t. In this formalism the
103 (a) (b) Figure 4. Perturbative contributions to the pion's qqqqg wavefunction. Contributions of type b) correspond to creation of a qqg from the vacuum, and have nothing to do with the hadron. These latter contributions do not arise in light-cone quantization. hadronic wavefunctions describe the hadron's composition at a particular r, and the temporal evolution of the state is generated by the light-cone Hamiltonian: Hic = P~ = P^ — P^, conjugate to r. Remarkably a simple kinematical argument shows that the zero-particle state in the light-cone Fock basis is an exact eigenstate of the full Hamiltonian Hic- Therefore all bare quanta in an hadronic Fock state are part of the hadron. Furthermore Lorentz boosts are greatly simplified in this framework since the quantization surface r = 0 is invariant under longitudinal boosts. It is also convenient to use r-ordered light-cone perturbation theory (LCPTh), in place of covariant perturbation theory, for much of our analysis of exclusive processes. LCPTh provides the natural perturbative framework for computing amplitudes in terms of the light-cone wavefunctions that describe hadrons, the resulting formalism being conceptually very similar to ordinary time- dependent perturbation theory in nonrelativistic quantum mechanics. LCPTh is also very convenient for analyzing other light-cone dominated processes, such as deep inelastic scattering. Unlike ^-ordered perturbation theory, r-ordered perturbation theory does not suffer from an explosion in the number of diagrams relative to covariant perturbation theory. The advantages of light-cone quantization do not come for free. The quantization surface r = 0 is not invariant under arbitrary rotations or even under parity inversions. As a consequence the operators that generate these transformations are as complicated as the light-cone Hamiltonian, making it difficult, for example, to specify the spin of a particular hadronic state. However the simplicity of the vacuum and of boosts is more important for our applications than is rotation symmetry. Light-cone quantization and perturbation theory are briefly reviewed for QCD in Appendix III. In the following sections we describe the Fock state basis and
104 wavefunctions in greater detail, emphasizing those features important to our analysis of form factors. 3.1. Definitions It is convenient when quantizing on the light-cone to rewrite four-vectors in terms of their H-, —, and ± components: - _ oO P- = P--P' (18) _ / ol r)2 P^ = {P\P') These components transform very simply under boosts along the ^-direction: P^ —^ exp(±Q;)P^ and Pj_ —^ P ±. In this notation dot-products have the form P'P = P^P- - P± P'q = —^ — - P±- q±- (19) If r = x"^ = t -\- z is to play the role of time in our light-cone formalism then P~, the momentum conjugate to r, plays the role of the Hamiltonian, and P = (P"^, P±) is the three-momentum that specifies the state of a particle. The light-cone energy of a noninteracting particle with mass M is just p-^zl+^' p+ 9 (20) and the particle's phase space is given by (27r)4 ^ ' 2P+(27r)3- ^-^' Thus a properly normalized momentum eigenstate satisfies {P\P!) = 2P+(27r)^ 6\E - P!). (22) Note that the longitudinal momentum P"^ for a particle is always positive.
105 To quantize QCD on the light-cone one defines commutators for the independent fields at a particular Hght-cone time r. (See Appendix III). Particle creation and annihilation operators are obtained by Fourier transforming the unrenor- malized field operators. These create and destroy bare quarks and gluons that have specific three-momenta and helicities. Using the creation and annihilation operators we can define a set of basis states for the quantum theory: |0) \qq:k>^i)=b^kiXi)d\k2X2) |0) Iqqg : fciA,} = b\k,Xi)d\k2X2)a\k3\i) |0) *-^' where 6^, d^ and a^ create bare quarks, antiquarks and gluons having three- momenta fcj and helicities Aj. Of course these "Fock states" are generally not eigenstates of the full Hamiltonian Hic- However the zero-particle state is the only one with zero total P"^, since all quanta must have positive fc"^, and thus this state cannot mix with the other states in the basis. It is an exact eigenstate of HiC' Although they do not diagonalize the Hamiltonian, the Fock states form a very useful basis for studying the physical states of the theory. For example, a pion with momentum P_ = (P"^, P ±) is described by state (24) #1 The restriction k"^ > 0 is a key difference between light-cone quantization and ordinary equal-time quantization. In equal-time quantization the state of a parton is specified by its ordinary three-momentum k = (k^^k^^k^). Since each component of k can be either positive or negative, it is easy to make zero-momentum Fock states that contain particles, and these will mix with the zero-particle state to build up the ground state. In light-cone quantization each of the particles forming a zero-momentum state must have vanishingly small k"^. Such a configuration represents a point of measure zero in the phase space, and therefore such states can usually be neglected. Actually some care must be taken here since there are operators in the theory that are singular at k"^ = 0—e.g. the kinetic energy {kj^-\-M^)/k'^. In certain circumstances states containing fc"*" —+ 0 quanta can significantly alter the ground state of the theory. One such circumstance is when there is spontaneous symmetry breaking. However such effects play little role in the sort analysis we deal with in this article, since we are concerned with high-energy, short-distance phenomena. Note also that the space of states that play a role in the vacuum structure is much smaller for light-cone quantization than for equal-time quantization; the state of each parton is specified by a two-momentum rather than a three-momentum since k"^ — 0. This suggests that vacuum structure may be far simpler to analyze using the light-cone formulation.
106 where the sum is over all Fock states and helicities, and where jTc/a;j = TT dxi 6 I 1 — Y^ Xj (25) Y[d^kJi^ = Y[d^k^^l67^^ 6 The wavefunction ^^/^(xj, k±i^ Aj) is the amplitude for finding partons with momenta [xiP'^^Xi P ± -f- k±i) in the pion. It does not depend upon the pion's momentum. This special feature of light-cone wavefunctions is not too surprising since Xi is the longitudinal momentum fraction carried by the z^^-parton (0 < Xi < 1), and k±i its momentum "transverse" to the direction of the meson. Both of these are frame independent quantities. Throughout our analysis we employ the light-cone gauge, rj - A = A"^ = 0, for the gluon field. The use of this gauge results in well known simplifications in the perturbative analysis of light-cone dominated processes such as high-momentum hadronic form factors. Furthermore it is indispensable if one desires a simple, intuitive Fock-state basis, for there are neither negative-norm gauge boson states nor ghost states in A"^ = 0 gauge. Thus each term in the normalization condition E/n ^i0^i^"M^"^^-^"^')i'=1 (26) n,A, I is positive. This equation follows immediately from the normalization condition for the full pion-state. 3.2. Light-Cone Bound-State Equations Any hadron state, such as |7r) for the pion, must be an eigenstate of the light- cone Hamiltonian. Consequently, when working in the frame where P^ = (1,0) and P~ = M^, the state |7r) satisfies an equation (M2 - Hlc) |t> = 0. (27) Projecting this onto the various Fock states (^^1, {qqg\. •. results in an infinite
107 number of coupled integral eigenvalue equations, M E 2. kli H- mf Xi {qqg] v Iqq) {<iq9\ v Im) • • • • ^ggg/w (28) where V is the interaction part of Hic- Diagrammatically, V involves completely irreducible interactions—i.e. diagrams having no internal propagators—coupling Fock states (Fig. 5). These equations determine the hadronic spectrum and 3> 0 • • • 0 'V/V* • • • • • • • • • Figure 5. Coupled eigenvalue equations for the light-cone wavefunctions of pion. a wave functions. Although the potential is essentially trivial, the many channels required to describe an hadronic state make these equations very difficult to solve. Nevertheless the first attempts at a direct solution have been made. The bulk of the probability for a nonrelativistic system is in a single Fock state—e.g. |ee) for positronium, or |66) for the T meson. For such systems it is useful to replace the full set of multi-channel eigenvalue equations by a single equation for the dominant wavefunction. To see how this can be done, note that the bound state equation, say for positronium, can be rewritten as two equations using the projection operator V onto the subspace spanned by ee states, and its complement 2 = 1—7^: Hvv |Ps)^ H- HvQ |Ps>g = M^ |Ps)^ (29) where H-pQ = VHQ...^ and |Ps}^ = ^ |Ps) Solving the second of these equations for |Ps)q and substituting the result into the first equation, we obtain a single equation for the ee or valence part of the positronium state: Heff |Ps)p = M^ IPs) (30)
108 where the effective ee Hamiltonian is Hef[ = H-p-p -\r H-pQ 1 M2 - Hqq Hqv- (31) The second term of /fgfF includes all effects from nonvalence Fock states; in light- cone perturbation theory it is given by the sum of all diagrams for ee —^ ee having no ee intermediate states (i.e. it is "ee-irreducible"). Thus we have (Fig. 6) M k±^ H- ml x{l — x) 1 i^ee(x,k±) = dy —^ V;fF(a;,fcj.;j/,/j.;M^)V'ee(?/,/j.)- 0 (32) where \4fF is given by V,K = Tirr(ee —)• ee) [x(l -x)y(l-y)] 1/2 (33) and Tirr(ee —^ ee) is the ee-irreducible amplitude for elastic ee scattering. The helicity dependence is implicit in this equation. (a) "Cw^ s • • • (b) Figure 6. a) Bound state equation for the ee wavefunction of positronium. b) The ee-irreducible potential. One might wonder whether or not this simple equation is also useful for rel- ativistic states like light-quark hadrons. For positronium the effective potential, Kff ~ M:^oulomb? is little modified by nonvalence Fock states and so this reduction to a valence equation is well warranted. However nonvalence states are most likely quite important for a light-quark hadron, and therefore Veff cannot help but be very complex in this case. For example, retardation effects must become significant when non-valence states become important, as is evident from the
109 normalization condition for the valence wavefunction: ?/n ^^^3 '|V^val(a^t,fcj.»,At)|^ = 1 - (V'vajl ^^ l^vaj) (34) A, " i —the expectation value of dV^f^/dM^^ a measure of the retardation, equals the probability carried by nonvalence Fock states. So usually one is forced to use the full coupled-channel equations when analyzing ordinary hadrons. However, as we shall see, the valence state plays a special role in high-momentum form factors, and so the valence-state equation will be useful in our analysis. 3.3. General Properties of Light-Cone Wavefunctions One major advantage of the Fock-state description of a hadron is that much intuition exists about the behavior of bound state wavefunctions. So, while the task of solving Eq. (28) remains formidable, there is nevertheless much we can say about the hadronic wavefunctions. An important feature that is immediately evident from Eq. (28) is that all wavefunctions have the general form M^i, hu A,) = i (V^). (35) Consequently tpn tends to vanish when 5 = M2-vMi±!!i Xi t oo. (36) This is intuitively plausible. In the Fock state expansion we think of the bare quanta as being on mass shell but off (light-cone) energy shell: i.e. each parton comprising a state with P_ = (P"^, P ±) has » " ^~P+ ki =m^, (37) but the sum over all k~ is not equal to P~. In fact the difference is just Parameter £ is a boost-invariant measure of how far off energy shell a Fock state is. Thus Eq. (35) impHes that a physical particle has little probability of being
110 in a Fock state far off shell. In general S is large when k\i or Xi is small—i.e. the wavefunction should vanish as k\i —>• oo or a;^ —> 0. Formally such constraints appear as boundary conditions on the wavefunctions and are important if the Hamiltonian is to be well defined (e.g. self-adjoint). These are subtle issues that we will not discuss here. Suffice it to note that all wavefunctions must satisfy the conditions ^It V'nl'Si, fcj.i, A,) —► 0 as fc|i —► oo (39) V'nla^t, ^it, Aj) —> 0 as Xi —^ 0. if the free-particle Hamiltonian is to have a finite expectation value. Perturbation theory is a useful source of intuition concerning wavefunctions and Fock-state expansions. The electron's Fock-state expansion, for example, can be computed perturbatively. To lowest and first order there are only electron and electron-photon components in the physical electron state: e.g. an electron with momentum P = (1,0) and positive helicity is described by [physical e|) = |e|) yfz^ + dfX d k \ (I -♦ \ "♦ 7r3(xfl-x)lV2irt^t'^-^-^/ ^er7r/er(^'^-l-)+ (40) 167r3(x(l -x)) e|7| :x,kA il^em/e^i^^^l)-^ "' where the electron in 67 : x,fcj_) has momentum k^ = {x^k±) and the photon has momentum k_^ = (1 — x^—k±). The e7-component of this state is readily computed from the light-cone Hamiltonian using ordinary first-order Rayleigh- Schrodinger perturbation theory. Schematically this term is given by the expression E ey mj - Pry which is identical in form to the LCPTh amplitude for the diagram in Fig. 7. Thus the 67-wavefunctions follow directly from LCPTh: e.g. mj — [kj_ H- xmj)/x[l — x) kj_-\- x^m; Having computed these wavefunctions, the renormalization constant Z2 is fixed by the normalization condition for the full electron state; obviously Z2 is the
Ill probability for finding a bare electron in a physical electron. The wavefunctions for an elementary particle like the electron can be used in much the same way as the wavefunctions for a composite particle; given the wavefunctions, there is little distinction between composite and elementary particles in this formalism. Notice that the 67-wavefunctions do not satisfy the boundary conditions discussed above, and as a result Z2 is not finite. This is of course just the usual ultraviolet divergence in QED. As we discuss in the next section, neither of these boundary conditions is generally satisfied in the absence of ultraviolet {k± —^ 00) and infrared (a; —^ 0) regulators. ( (1.0) (1-X,-kj_) Figure 7. LCPTh amplitude corresponding to the e7-wavefunction for a physical electron. More generally perturbation theory can be used to compute the high-momentum behavior of light-cone wavefunctions. The basic ansatz of perturbative QCD is that the short distance behavior of the theory is perturbative; only perturbative interactions are sufficiently singular to contribute at short distances. Consequently wavefunctions behave in much the same way as perturbative amplitudes (in LCPTh) when k± —^ 00. This is evident from our analysis of the non- relativistic wavefunction for heavy-quark mesons: the large-^ dependence of the wavefunction is obtained by replacing the meson with an on-shell quark-antiquark pair and computing in perturbation theory. A similar analysis in the relativis- tic case shows that the pion's qq wavefunction falls off roughly as l/kj_ when kj_ —► 00, just like the LCPTh amplitude for qq —*■ q*q* that is shown in Fig. 8a. Similarly one expects the qqg wavefunction to fall like the perturbative amplitude in Fig. 8b—i.e. tpqqg ^ l/|^j.| as \k±\ —^ 00. In addition to determining the large-fcj^ behavior of wavefunctions, perturbation theory also serves as a guide to modelling such things as the helicity dependence of wavefunctions. Normally one can say little about the angular-momentum #2 This connection can be made precise using the operator product expansion, as we illustrate in later sections.
112 (a) (b) '^K^ =<^ Figure 8. LCPTh diagrams having behavior similar to that of wavefunctions for k^ large. content of a model wavefunction, since the angular momentum operators are very complicated in light-cone quantization. However perturbation theory can be used to produce examples of wavefunctions having particular spin quantum numbers, and these can be used to motivate non-perturbative models. For example, to see what a pion's qq wavefunction might look like, we can treat the pion as an elementary particle that couples to the quarks through elementary couplings like ?/j75 TT • Til) or ^^757 • ^TT • f?/). The wavefunction can then be computed pertur- batively in much the same way we compute ^e^/e above. This wavefunction has the correct quantum numbers in the limit where the quark-antiquark interactions are negligible, and so it can serve as the starting point for the design of empirical wavefunctions to model the pion. Note that such a wavefunction is more singular at large momenta than the pion's true wavefunction; this is the essential difference between an elementary particle and a composite particle. Further intuition about wavefunctions comes from the physics of nonrela- tivistic bound states. In the rest frame, where P^ = P~ = M and P j. = 0, time t and light-cone time t = t -\- zjc are almost identical for a nonrelativistic system since the speed of light c is effectively infinite. Consequently the usual Schrodinger wavefunction defined at a particular i should be almost the same as the light-cone wavefunction defined at r « ^. To make the connection notice that the v^ constituent has longitudinal momentum A:+ = x,M = k\ H- k\ « m, + 0{m^v^) -f k\ (43) where the constituent's energy fc|^ is just its mass m, plus small corrections (due to kinetic and potential energies) of (9(mjt;^) <^k\ ^ rriiv. Thus the quantity XiM —
113 TTii is effectively equal to A:f, and a Schrodinger wavefunction can be converted to a light-cone wavefunction simply by the replacement: kf —^ XiM — rrii. This is also evident when we note that all energy denominators have the form «^2 Sr^kli-\-mj o»>rli:^ y--^ kli -\- {xiM - mi) (44) when \xiM — mj <C Tni- This correspondence indicates that nonrelativistic light- cone wavefunctions are sharply peaked at ^^ = -n ^-Li = 0, (45) just as Schrodinger wavefunctions are peaked at low ki (<C mi). This is well illustrated by the wavefunction for ground state positronium (or hydrogen) which 1 is given by ,, . , ,2MY\"^ 87r7 TT ' - ^2 (46) A:|-f (a:eM-me)2-f 72 w hen A:^, {xeM — m^)^ <C ml. Here 7 = amr where m,r is the reduced mass. 3.4. Renormalization As we discuss in earlier sections, perturbation theory indicates that hadronic wavefunctions do not fall off sufficiently quickly as ^^ —^ 00. This leads to infinities in the unitarity sum (Eq. (26)), energy expectation values, and in the wavefunctions themselves. Of course this is not unexpected given that the wavefunctions and the theory are as yet unrenormalized. To make the theory finite we must truncate the Fock space by in effect discarding all Fock states with light-cone energy \S\ > A^. This ultraviolet cutoff can be introduced by using Pauli-Villars and related regulators or, equivalently, dimensional regularization. These regulators preserve the Poincare and gauge symmetries of the theory. For our purposes, however, it is simpler and more intuitive to simply truncate the Fock space, excluding all states with \S\ or kj_ greater than some A^. This procedure causes no problems in "leading-log" analyses of the sort we are concerned with here. The end result is that all loop integrations in LCPTh are finite, and the wavefunctions all vanish at large k±.
114 Usually one takes A —> oo when computing. However the key physical characteristic of renormalizable theories is that this cutoff has no effect on the results for any process provided only that A is much larger than all mass scales, energies, and so on relevant to the process of interest. So we can compute with finite A. This is not to say that states with \S\ > A^ are unimportant—the existence of ultraviolet divergences is dramatic evidence to the contrary. Rather it means that all low-energy effects due to these very high-energy states can be accounted for by redefining the coupling constants, masses, etc. appearing in the effective Lagrangian (or Hamiltonian) for the truncated theory—e.g. £(^) = i^{id"y-g{A)A"y-m{A))iP -f 1/4^^ -f O ("^^-p^-\-■ - ■] . (47) These bare parameters vary with A in the usual way, as more or less of the high-energy Fock space is absorbed: "^^ ^ ^ ■ (48) In general nonrenormalizable interactions appear as well, but these are suppressed by powers of 1/A, as is suggested by simple dimensional arguments. Also the effective Lagrangian can change radically as A passes thresholds for new heavy quarks, or say for observing quark substructure (if there is any). Working with a finite cutoff, the couplings, masses, and wavefunctions of the theory are both well defined and well behaved. Furthermore they have a simple interpretation. The bare parameters—5f(A), m(A)...—are the effective couplings and masses of the theory at energies of order A (i.e. at distances of ^ 1/A). Indeed as we shall see, a process or quantity in which only a single scale Q is relevant is most naturally expressed in terms of the couplings, masses, wavefunctions, etc. of the theory with cutoff A ~ Q. Of course one must compute with A ^ Q, but the dominant effect of vertex and self-energy corrections is to replace 5f(A), m(A), ip^^^... by g{Q)y m((5), ip^^^ Thus as Q is increased, ever finer structure is unveiled in the wavefunctions and in the theory. The wavefunction V^n (^i,^±i,'^t) has a multipHcative dependence upon A when xi and k±i are held fixed, and when kj_i <C A^: 2(A) \ '"
115 where Z)- ^ is the usual wavefunction renormalization constant for the j^^ parton. This formula is easily understood by recalling that ZJ ^ is the probability for finding a "bare" parton in a "dressed" parton. Also it follows that 0 < Zj < 1. Furthermore, ZJ ^ generally decreases with increasing A since the effective phase space, and therefore the probability, for the multi-parton Fock states in a dressed parton increases with A. Although the probability shifts from Fock state to Fock state with varying A, the total probability is always conserved: E/n ^^0^ \i>i''\x.,hi,X,)\' = i+0{j). (50) One final modification of theory is required. The polarization sum for a gluon is singular as the gluon's longitudinal momentum k'^ vanishes: J2',{k,X)el(k,X) = -g,.+ "kh^JhbL. (51) As a result wavefunctions for states with gluons diverge as A:^ —* 0, again contrary to the boundary conditions Eq. (39). This singularity is to some extent an artifact of light-cone gauge. For our purposes it can be regulated by making the replacement: 1 x'' 1 r 1 1 + TT^-^T^ • (52) k+ J 2 {{k"^ -\-i8Y {k+ ~i8)^ Physical amplitudes or cross sections are independent of 6 provided it is sufficiently small. This implies that gluons decouple when k^ < 8 for some small 6. Thus we can use this regulator with a small but non-zero 6 to obtain wave- functions that are well behaved when gluons have vanishingly small longitudinal momenta. Typically the cutoff point must be 6 < {k±)/Q^ where (A:_l) is some average of the gluon's k±y and Q is the momentum scale of the probe. Therefore as Q increases, so does the number of "wee" gluons. Notice finally that (A:_l) can never vanish for physical states since very long wavelength gluons cannot couple to a color-singlet state. Thus, with finite 6 .and A cutoffs, all Fock-state wavefunctions are well behaved, both as x^ —> 0 and k±t —^ oo.
116 3.5. Calculating In principle the hadronic wavefunctions determine all properties of a hadron. Here we illustrate the relation between the wavefunctions and measurable quantities by briefly examining a number of processes. These examples also demonstrate the calculational rule for using wavefunctions: i.e. an amplitude involving wave- function il^n \ describing Fock state n in a hadron with P_ = (P*^, P ±)^ has the general form where Tn is the irreducible scattering amplitude in LCPTh with the hadron replaced by Fock state n. If only the valence wavefunction is to be used, Tn is irreducible with respect to the valence Fock state only: e.g. Tn for a pion has no qq intermediate states. Otherwise contributions from all Fock states must be summed, and Tn ^ is completely irreducible. TT —* UU H The leptonic width of the tt^ is one of the simplest processes because it involves only the qq Fock state. The sole contribution to 7r~ decay is from (0| V„7+(l - 75)V'<( k~) = -y/2P-^f, (54) where nc = 3 is the number of colors, /,r ^ 93 MeV, and where only the Lz = Sz = 0 component of the general qq wavefunction contributes. Thus we have 167r3 'f'^^^^''^>-2Vs' V'i^'(^.^i) = :r^- (55) This result must be independent of the cutoff A provided A is large compared with typical hadronic scales. This equation is an important constraint upon the normalization of the du wavefunction, indicating among other things that there is a finite probability for finding a tt" in a pure du Fock state.
117 Hadronic form factor The electromagnetic form factor of a pion is defined by the relation (t : £'| JL |t : £) = 2(P + P'r F {-(P' - Pf) (56) where Jj^rn is the electromagnetic-current operator for the quarks. The form factor is easily expressed in terms of the pion's Fock-state wavefunctions by examining the /i = -f component of this equation in a frame where P_ = (1,0) and P' = (1,^_l)- Then the spinor algebra is trivial since u (i)7"*-w(/) = 2\/FF, and the form factor is just a sum of overlap integrals that is quite analogous to the nonrelativistic result (Fig. 9a): n,A, a dxi (Pk±i (A) / n leJ^' V^i^'^'l^*' '^»' ^0 '^n'^(^^^ ^^»' ^0. (57) Here Ca is the charge of the struck quark, A^ >> qj^^ and lu = k±i — Xiq± 4- q± for the struck quark k±i - Xiq_i for all other partons (58) Notice that the transverse momenta appearing as arguments of the first wave- function correspond not to the actual momenta carried by the partons but to the actual momenta minus Xiq±^ to account for the motion of the final hadron. Notice also that l± and k± become equal as q± —^ 0, and that Ftt —> 1 in this limit as a consequence of the unitarity condition Eq. (50). The behavior at large (f^ is discussed at length in subsequent sections. (a) (b) • • • • • • Figure 9. Diagrams contributing to the electromagnetic form factor of a hadron: a) only terms for /i = -f; b) additional terms for /i 5^ +.
118 It is interesting to note that a very different expression is obtained for the form factor if one examines some other component of the current, for example the /i = — component. Not only does the momentum dependence of the quark- photon become more complicated, but the vertex no longer conserves particle number since there are now terms involving transitions ^ -f 7* —^ Q -\- 9 and Q -\- 9 -\- y* —> ^, as illustrated in Fig. 9b. These various expressions for the form factor must all be equal, and yet there is no simple way of demonstrating this fact. The problem is that rotations must be used to relate one expression to another, and the rotation operators are complicated in our formalism. The equality of these expressions implies a nontrivial relationship between different Fock states, a relationship that ought to be incorporated as much as possible into empirical models for the pion wavefunctions. Note finally that our expression for the pion form factor is actually far more general. The helicity-conserving electromagnetic form factor of any hadron has precisely the same form. Deep inelastic scattering The proton's structure functions are determined to leading order in as{Q^) by the r-ordered diagrams in Fig. 10. Furthermore the only region to contribute in this order is kj^ <C Q^ where Q^ = qj^. This is because the hadronic wavefunctions are peaked at low k±. This has two important consequences: first, we can neglect k± relative to qj^ to leading order; and second, we can set the ultraviolet cutoff A equal to Q since only those Fock states with k^ <C Q^ are important. The structure functions are then 2MFi{x,Q) = ^^^^^^»'£elG,/^(x,Q) (59) a where, from Fig. 10, n,A, i 6=a is the number density of partons of type a with longitudinal momentum fraction X in the proton. (The ^^ is over all partons of type a in Fock state n.)
119 This equation leads immediately to a very useful interpretation of the structure (b) (c) Figure 10. LCPTh diagrams contributing to the proton's structure functions for deep inelastic scattering. function moments: 1 ^+ n+l dxx^^'G,,j,{x,Q) = {p\^al+{iD r+Valp)''^' 0 (2Pp^) + ^n+2 (61) where the matrix element is between proton states and is evaluated with ultraviolet cutoff A = (5, and where the gauge-covariant derivative is m light-cone gauge. The Q-dependence of the moments is determined simply by the cutoff dependence of matrix elements of (twist-two) local operators! 4. A PERTURBATIVE ANALYSIS In this section we develop the techniques needed to understand exclusive processes with large momentum transfer. This relativistic analysis is very similar to the nonrelativistic analysis given in Section 2, and, as in the nonrelativistic case, the result is both simple and intuitive. Generally one finds that the amplitudes for such processes can be written as a convolution of quark distribution amplitudes 4>{^iiQ)i one for each hadron involved in the amplitude, with a hard-scattering amplitude Tjj- 4,2 The pion's electromagnetic form factor, for example, can be
120 •,, 3,4,2 written as 1 1 FAQ^) = Jdx Jdy<l>;{y,Q)TH(x,y,Q)M^.Q) (^ "^ ^ (^)) ' ^^^^ 0 0 Here Tjj is the scattering amplitude for the form factor but with the pions replaced by collinear qq pairs—i.e. the pions are replaced by their valence partons. The process-independent distribution amplitude <t>ir{x, Q) is just the probability amplitude for finding the qq pair in the pion with Xg = x and x-g = 1 — x: M-.Q) = j^^,^f,^-rki) (63) = P.^/^e-^^^-/MO|?(0)^^Wk)(^) ^ .(64) + = fi. = 0 The A:_L integration in Eq. (63) is cut off by the ultraviolet cutoff \ = Q implicit in the wavefunction; only Fock states with energies \S\ < Q^ are important. The structure of Eq. (62) is very reminiscent of that for the nonrelativistic form factor (Eq. (14)). The major difference is that here there is a convolution over the longitudinal momenta of the partons. In a nonrelativistic meson the longitudinal momentum is sharply peaked about x = 1/2, and thus the x-y dependence of T// plays no role. One can set a: = y = 1/2 in Tjj, and factor it out of the integral in Eq. (62). Then one needs only J dx (p, which is just the wavefunction evaluated at the origin, to compute the form factor. As far as the nonrelativistic meson is concerned the hard subprocess occurs over very short distances. The situation is different for a relativistic meson, which is sensitive to the fact that the hard subprocess is not really a short-distance reaction. Although the volume within which the subprocess occurs is small in the transverse direction (|(5£_l| ~ 1/Q), it can extend over large longitudinal distances: 6z~ ~ ^/P^ = l/m^r in the pion's rest frame. A relativistic meson has structure over such distances, and therefore the asymptotic form factor is given by a convolution over #3 The distribution amplitude is gauge invariant. In gauges other than light-cone gauge, a path-ordered "string operator" Fexp(jQ dsig A{sz) ■ z) must be included between the \p and tp. The line integral vanishes in light-cone gauge because A • z =^ A^z~(^ — 0 and so the factor can be omitted in that gauge. This (non-perturbative) definition of 4> uniquely fixes the definition o( Th which must itself then be gauge invariant.
121 longitudinal momentum. Note that the subprocess is still restricted to a region very near the light-cone—i.e. bz^ = 6z'^6z~ — 8zJ^ ^ —1/Q^. Such "light-cone dominated" processes can still be analyzed perturbatively. The distribution amplitude is only weakly dependent on Q, as is evident from the evolution equation ' (which we derive below): 1 QqqMx,Q) = J dyV{x,y,a,{Q^))My,Q} (65) 0 ,2\\ _ V(x,y, a,{Q')) = a,{Q') Vi{x,y) +ai{Q')V2(x,y) + ■ ■ ■. (66) The bulk of the Q dependence comes from Th- To leading order in a3((5^), Tjj is obtained directly from the form factor for 7* -f ^^ —* qq^, where the mesons have been replaced by collinear qq pairs: Th{x, y, Q) = ; ^«(^'^'g) (leading order). (67) [x(l - x)y{l - j,)]'/2 Beyond leading order only the "collinear-irreducible" part of Fgg is retained: all mass singularities are systematically subtracted out since contributions from low momenta are already included in the distribution amplitudes. Therefore we can neglect all quark and meson masses in T//, leaving Q as the only scale. The amplitude must then have the general form TH{x,y,Q) = -^f{x,y,a,(Q^)) (68) where n = 2 from simple dimensional arguments. This means that the pion form factor falls as 1/(3^, up to logarithms of Q. In general the dimension of an amplitude is [energy]"" where n is the total number of quarks, gluons, and leptons in the initial and final states of the process: e.g. n = 6 — 4 for the pion form factor since the process en —^ eir involves four partons and two leptons. This "dimensional-counting rule" implies that the nucleon form factor falls off roughly like l/Q^ with increasing Q, since there is one additional parton in each of the initial and final states of Tjj relative to the pion case and thus n = 8 — 4. Generally the more partons that must be scattered from the initial to the final direction, the more powers of 1/Q there are in the form factor.
122 1 -X, -kjL 1 y.^i+yqj. -y. \ + (1 -y)qi^ (a) + + XjfvVi + ••• Z-5 • • • (b) Figure 11. The ^g-irreducible diagrams contributing to the qq form factor A second consequence of neglecting masses in Tjf is that total quark helicity is conserved since the vector couplings with gluons cannot flip the helicity of massless quarks. By its definition (f) carries no helicity, and so the helicity of the hadron equals the sum of the helicities of its valence quarks in Th- Thus, for example, hadronic helicity is conserved in high-Q^ form factors—i.e. helicity- flip form factors such as the nucleon form factor F2 are suppressed by additional powers of m/Q. In the following sections we derive these results for the pion's electromagnetic form factor; the techniques generalize readily to other large-px processes. We discuss how the distribution amplitudes might be computed nonperturba- tively. We examine problems that arise in certain processes due to singularities in Th' Finally, we address the critical question of how large Q must be for these asymptotic results to hold. We do this by examining competing mechanisms and by investigating the self-consistency of perturbation theory. 4.1. Factorization—Leading Order Analysis The pion's form factor can be written in terms of its qq wavefunction alone: R iQ') = / dxcPk ± 167r3 ^ ^^'Hy, L) /^^'^^^'^'^"'-^^^"^l i>^'\x, h). 167r3 [x{l - x)y{l - y)f' (69) Here T is the sum of all ^^-irreducible LCPTh amplitudes contributing to the qq form factor for 'y* -\- qq —^ qq (Fig. 11). The ultraviolet cutoff is A >> Q. #4 The helicity-projection operators for massless quarks are just 1 ± 75. Noting that, for example, that the vertex t77^(l — 75)1/ equals u^{l — 75)^7^7^^, we see that the vector coupling of the gluons with the quarks preserves quark helicity. This would not be the case if the gluon was a scalar where, for example, the coupling might be t7(l — 75)w which equals u^{l 4-75)^7°^ and flips the quark's helicity. This same sort of argument can also be used to explain why massless neutrinos are always left-handed.
123 Consider first the disconnected part of T (Fig. 11a). For the moment we ignore renormalization diagrams, and consider only terms where the photon attaches to the quark line. The disconnected part then gives a contribution eg ' ' ' 0 jdxjj^ V-'^^*!^, fcx + (1 - x)9-x) ^("Xx, fcx) (70) to Ftt, where tq is the quark's electric charge. The analysis of this contribution follows closely that of the nonrelativistic form factor. The integral is dominated by two regions of phase space when Q^ is large since the wavefunctions are sharply peaked at low transverse momentum: 1) l^xl <^ (1 — x)Q^ where '4)^^\x^k±) is large; 2) |A;j. + (1 - x)^xl < (1 - x)Q, where i/;(^)*(x, ^x + (1 - a:)^j.) is large. In region 1), ^x can be neglected in il)^^^*{x^k_[_-\-[\ — x)q±) until |^_lI ^ (\~x)Q'> at which point tl)^^' begins to cut off the k^ integration. Thus in region 1) we can approximate Eq. (70) by 1 (i-x)Q eg |dx0(^)*(x,(l-x)9-l) j l^V'^^)(x,fcx). (71) 0 The bulk of the integral comes from \k^\ ^ (1 — x)Q. Similarly we obtain the following contribution from region 2): 1 { (i-x)Q d?k e ' ' ' 9 0 Jdxl J i^rl,W*{x,h)U^'^\s,-{l-x)qj_). (72) One can easily show that these approximations are valid to "leading-log" order— i.e. up to corrections of 0{l/ \og{Q^))—given that tp falls off roughly as l/k]_ in QCD. Again as in the nonrelativistic case, we can use the bound-state equation for the valence wavefunction {c.f. , Eq. (32)) to further simplify these expressions by isolating the q± dependence of the stressed wavefunctions. The equation for
124 i/;(^)(x,(l-x)^±)is ' SI (73) where we have neglected masses in the energy denominator. As above the dominant contribution here is from |/x| <C (1 — 2/)Q, and so we can approximate this equation to leading-log order by 1 (i-y)Q 0 It is readily demonstrated that V^ff (a:, (1 —a:)^xj 2/? 0) is free of mass singularities in light-cone gauge. Consequently all loop momenta are of order Q or larger, and perturbation theory can be used to compute \4ff. To leading order V^ff involves the exchange of a single gluon between the quark and antiquark. Combining Eq. (74) with Eqs. (71) and (72) we arrive at a simple expression for the contribution to F^ coming from the disconnected part of T (Eq. (70)): 1 1 j dx j dy<t>l{y,[l-y)Q)e,T^^\x,y,Q)<t>ii{x,{l-x)Q). (75) 0 0 Here the unrenormalized quark distribution amplitude <^o is defined by t dp M^,Q)=y Yg^V'^^H^,fc±), (76) #5 Mass singularities do occur in 14ff(^, (1 — ^)9l; 2/, 0) when using covariant gauges. They arise because the external quarks that carry no transverse momentun:i in this amplitude are effectively on energy-shell. In most covariant gauges such a quark couples strongly to a nearly collinear gluon, resulting in an integral over the gluon's transverse momentum that is logarithmically sensitive to masses and other low-momentum scales: e.g. f dl]_/{lj_ -\- 0(m^)). In light-cone gauge the coupling between a gluon and an on-shell quark vanishes as the gluon becomes collinear with the quark. This means there is an extra factor l^/Q in the integral over the gluon's momentum /x, and thus the logarithmic dependence upon masses is removed. Indeed all contributions from |/x| -C Q are strongly suppressed. The only diagrams that lead to collinear singularities in light-cone gauge are ones in which a gluon is exchanged between two nearly on-shell quarks (or gluons) that are collinear with each other. Such diagrams do not contribute to 1/efT since they are not two particle irreducible.
125 and the hard-scattering amplitude T^^ is given by T^H^ = Vei^(x,{l-x)qr^y,0) 1 ^1(1 -x)/x + (x <-> 2/) (a) (77) Note that Tjj' comes from part of the LCPTh amplitude for 7* + ^^ —> qq (Fig. 12a). (a) X. Ox (a) H 7 1-x. Oj. y. yq z 1-y.1-yq (b) (b) H Figure 12. factor. The unrenormalized hard-scattering amplitude for the pion form In addition to the disconnected parts, the connected part Tc of T contributes to Eq. (69) as Q —> 00 (Fig. lib). By the same reasoning used above, we can neglect /^ and k± relative to q± in Tc to obtain a formula that is identical to Eq. (75) but with egT^^^ replaced by (Fig. 12b) g ^(6) ^ Tc(x,0;y,0;gl) '' "" [x{l~x)y{l-y)f'' (78) Again Tc is free of mass singularities (in A'^ = 0 gauge) and can be computed
126 perturbatively. Still ignoring renormalization, the otherwise complete result is therefore 1 1 KiQ') ^ Jdxjdyl 4>l{y. (1 - y)Q) eg Tl[x, 2/, Q) Hx, (1 - x)Q) 0 0 (79) 0 + <?^o(2/.2/Q)egTjy(l -x,l -y,Q)M^^^Q) where we have now included contributions for the photon attaching to each of the quark and the antiquark. The unrenormalized hard-scattering amplitude in lowest order is given by rpOf ^x rp{a) .rp{b) 1 GtT CjT Qg (A^) r^(x,y,Q) = rV + rV = ^3-^^^^-^^ (so) which is just the Born amplitude for a coUinear qq pair to scatter with the virtual photon (divided by [x(l — x)y{l — y)] ' ). Finally we must consider the effects of vertex and propagator corrections in T^E (Fig. 13). Each of these corrections involves propagators off energy shell (Q) f (A) ^ 1 \ ^Z 7(A) -.(Q) ^1/^3 Figure 13. Vertex and propagator corrections to the hard-scattering amplitude. by (P(Q^) and therefore all loop momenta are of order Q or larger (in A^ = 0 gauge). It is then a straightforward consequence of renormalization theory that the propagators and vertices are modified only by the factors Z^ IZ\^ for propagators (81) Z\ ' IZ\ ' for vertices up to corrections oi 0{as{Q'^))^ where Z\ ' is the usual renormalization constant
127 w ith ultraviolet cutoff A. Thus in leading order T% is multiplied by (Fig. 13) H where Z^p renormalizes quark-gluon vertices, and Z2 and Z\ ^ renormalize the quark and gluon propagators. Here we use the fact that a^ is renormalized by Z^{Z2lZiF?—i.e. that as{h?)zl^\z[^^iz[^^f is independent of A. Also the photon-quark vertex correction in this amplitude cancels the quark-propagator correction by the QED Ward identity. So Eq. (79) is corrected to give 1 1 F^[Q'^) ^ JdxJ dy {0*(2/, (1 - y)Q) e, Th{x, y, Q) <t>[x, (1 - x)Q) -V [q ^ q)} 0 0 (83) where now the leading-order hard-scattering amplitude is ^^^"'^'^^=(l-.)(l-y)Q2 (84) and the distribution amplitude is given by 7^^) f dp Since the bulk of the integral in Eq. (85) comes from k\ <C Q^, we can use Eq. (49) to redefine ^^^^\x,k^) (86) where now the fc^ cutoff at |/:x| ~ 0 is implicit in the definition of the wavefunc- tion. Our equations now have the general form proposed in the introduction to this section. #6 For example, the full unrenormalized quark propagator has the form d/'(A/Q, Qr,(A^))/(q' • 7) as Q^ = —(^ —*• 00. Since the quark is far off energy shell dp is independent of masses in this limit. Furthermore the A dependence can be removed by dividing with the renormalization constant Z\ \ Thus the quantity dF{K/Q,ocs{S?'))lZ2 must equal di^(l, a,(Q^))/Z^^\ up to corrections of 0(ckj(Q)) due to the fact that A/Q is not large in the second case. Since d/'(l, a,(Q^)) = 1 + 0(as{Q'^))^ the final result is dF{A/Q,ot,(A^)) = Z^^V4^\ again up to corrections of 0(c^,(Q'^)).
128 The major effect of the renormalization corrections is to replace as(A^) by c^s{Q^) in the hard-scattering amplitude, and 0^^^ by ip^^^ in the distribution amplitude. This is exactly what is expected on the basis of our earlier discussion of renormalization. The only physical scale in Tjj is Q and so as{Q^) is the natural expansion parameter. Furthermore Tff only probes structure in the wavefunctions down to distances of 0{l/Q). Thus the wavefunction 0(^\ defined in a theory with cutoff Q, incorporates hadronic structure over all distance scales relevant to the physical process. Structure at distances smaller than \/Q is irrelevant except insofar as it determines cts{Q^), '^(Q) The leading order result for Tji is consistent with the dimensional-counting prediction for the pion form factor: i.e. Tff ~ 1/Q^ up to logarithms of Q. This rule also shows why it is that only the valence Fock state is relevant for large Q. For example, the hard-scattering amplitude for scattering a collinear qqqq state has four additional partons and so must fall as l/Q^] this ampHtude has many more far off-shell (^ Q^) internal propagators than does the qq amplitude. The same is true of states with additional gluons provided that one is working in light-cone gauge. 4.2. The Quark Distribution Amplitude Everything one needs to know about the pion in order to compute the asymp- A O totic form factor is lumped into the quark distribution amplitude <^(x, Q). ' Obviously (j) is intrinsically nonperturbative. However its variation with Q can be studied in perturbation theory. To see this we differentiate Eq. (85) with respect to Q to obtain (Q) (87) #7 A hard-scattering amplitude with additional gluons can contribute to leading order in 1/Q when covariant gauges are used. For example, adding a single gluon to the qq hard scattering amplitude introduces one additional denominator of O(Q^). In addition there is typically a numerator factor of 0{e ■ q), where e is the gluon's polarization vector. So such an amplitude is suppressed by e • q/Q"^ ~ l/Q in light-cone gauge where e'^ = 0. However other gauges can have e • q ^ ^"^9~ ~ Q^, in which case the amplitude with an additional gluon is not suppressed at all.
129 where 7/- is the anomalous dimension associated with Z2 d C?^4'^^ = -7.K(Q^))^f' c.MQ-)y,/ + (i-y)V^(,2)U(Q) (88) 0 (The singularity at y = 0 in this equation cancels in the final result because the meson is a color singlet.) The first term in Eq. (87) represents the change in the probability amplitude cj) due to the addition of more qq states as the cutoff Q is increcLsed, while the second term represents the loss of probability from those already present, as Z2 decreases. By using the bound-state equation as in Eq. (74), we can express 0^^^(x,^x) ^^ terms of (j){x^Q). To leading order we need only consider one-gluon exchange between the quark and antiquark, and this gives (Fig. 14) Zi"^ V '■--/ ^-2 J ''y(l-y) where again as(A^) is converted to as{Q^) by propagator and vertex corrections. 4 Substituting into Eq. (87) we obtain finally the leading-order evolution equation 7(A) d? (y. Q) Figure 14. The qq wavefunction for q]_ = Q"^ large. for <^: ^^^("' ^) = ^ I / '^^ ^^) ^(^' '^^ - '^(^' '^^ I (^°)
130 where the evolution potential is V(x,y) = 4Cf |x(l -j,)%-x) (^^_^j-+ -A_^ + (^ il I _ J I = ^(S''^)- (91) Operator A in the potential is defined by 2/(1-2/) 2/(1-2/) x(l-x)' Also h and h are the helicities of the quark and antiquark (<^_/^^ = 1 for pions). The evolution equation completely specifies the Q dependence of (j){x^Q)\ given <^(x, Qo)? <t>{x^ Q) is determined for any other Q by integrating this equation, numerically or otherwise. Still it is instructive to exhibit explicitly the most general Q dependence. Using the symmetry V{x^y) = V{y,x) to diagonalize V^ the general solution of Eq. (90) is easily shown to be oo 3/2 (t>{x,Q) = x{l-x)Y,<^nCr{2x-\) log-f— (93) n=0 -7n/2^o where ,. = .C,|,.4gj-;-^^^}>0^ ,94, By combining the orthogonality condition for the Gegenbauer polynomials and the operator definition of (p (Eq. (64)), we obtain an interpretation for the ex- #8 The evolution potential V(x,y) can be treated as an integral operator. Being symmetric it has real eigenvalues % and eigensolutions <pn{y) that satisfy J dy V{x, y) w{y) <f>n(y) = 7n<f>n{x) where integration weight w{y) = 1/(2/(1 — y))- The eigensolutions must be orthogonal with respect to weight w(x), from which it immediately follows that <f)n{x) oa 3/2 3/2 x(l — x)Cn (2x — 1) where Cn is a Gegenbauer polynomial. It is a straightforward exercise to now extract analytic expressions for the eigenvalues. Given the eigenvalues a general solution of the evolution equation can be written down as an expansion on the complete set of eigensolutions, as we do here. #9 Note that Aqcd is the scale appearing in the running coupling constant; it has nothing to do with the ultraviolet cutoff A. Recall also that Cr = 4/3 and f3o = 11 — 2n//3 where Uf is the number of quark flavors.
131 pansion constants in Eq. (93): 2 , -7n/2^o . ,. ^. 1 ^ 0 (95) (2 + n)(l+n) Za/ztxc —the an's are just matrix elements of local operators. This analysis shows that the distribution amplitude can be expressed as a 11 12 sum of matrix elements of local (twist-two) operators. ' This sum is just the operator-product expansion of the operator 0(0)7"^75i/'(z) in Eq. (64). Such an expansion is warranted since the separation between the fields is very nearly on the light cone: z^ = z'^z~ — zj_ = 0{l/Q^). The Gegenbauer polynomials also appear very naturally in this context, as a consequence of the residual conformal symmetry of QCD at short distances. All of the dimensionful couplings in the QCD lagrangian can be dropped at very short distances, and so the classical theory (i.e. tree order in perturbation theory) becomes invariant under conformal mappings of the space-time coordinates. This conformal symmetry is destroyed in the quantum field theory by renormalization, which necessarily introduces a dimensionful parameter such as the cutoff A. However the evolution potential for (f> is given by tree diagrams in leading order, and so the leading-order potential ought still to be consistent with the requirements of conformal symmetry. One such requirement is that local operators that are multiplicatively renormalizable must transform irreducibly under conformal transformations. In the case of meson operators conformal symmetry is enough by itself to uniquely specify the structure of the these local operators. As these are the operators that appear in the operator-product expansion, conformal symmetry completely specifies the structure of the expansion for (j). These ideas do not easily generalize beyond 13 leading order. The operator-product analysis of the distribution amplitude suggests an important constraint on (f>. The n = 0 Gegenbauer moment of the distribution amplitude is proportional to the amplitude for pion decay {c.f. Eq. (55)): 1 j dx<j>{x,Q) = -^. (96) 0 Given the shape of (t)[x^Q) this equation normalizes it for any Q. Note that the
132 value of this moment is Q independent. This is because the n = 0 operator is just the axial-vector current operator. As far as its ultraviolet behavior is concerned, this operator is conserved and so its anomalous dimension vanishes: 7n=o = 0. Notice also that 7n > 0 for all other n. Thus only the n = 0 term in the expansion of (j){x^Q) survives when Q becomes infinite: <^(x, Q) —» -—^ x(l — x) as Q —> oo. (97) rir So (j){x,Q) is completely determined for pions when Q is very, very large. Notice finally from Eq. (89) that ip^^\x^q±) does in fact fall as l/qj_y up to logarithms, as q± grows. The high-momentum or short-distance behavior of the Fock-state wavefunctions is perturbative in nature, and as a general rule is crudely that of simple Born amplitudes in light-cone perturbation theory. In particular wavefunctions are not exponentially damped at large fj_, as is frequently assumed in phenomenological studies. 4.3. Determination of Distribution Amplitudes Large-px exclusive processes, like most other high-energy processes, involve physics both at short distances and at long distances. A special feature of the large-px processes is that we are able to separate short from long distance physics in a relatively simple fashion. This allows us to analyze each regime separately, using the tools best suited to that regime. The hard-scattering amplitudes and the evolution potentials for distribution amplitudes embody the short-distance physics; they are most effectively analyzed using perturbation theory. However perturbation theory is largely useless for determining anything about the distribution amplitudes beyond their Q-dependence. The distribution amplitudes contain the long-distance physics of a large-px process, and as such require some sort of nonperturbative treatment. Given that the distribution amplitude is intrinsically nonperturbative one might wonder whether it isn't just as well to treat the entire process nonpertur- batively. This is generally a very bad idea. Any nonperturbative analysis of a large-px process would have to deal accurately with QCD dynamics over a huge range of momentum scales—e.g. a vast grid would be required in lattice QCD if one wanted to accommodate both the relatively small momenta that characterize hadronic structure and the very large momenta transferred in the process. Such an analysis would be very inefficient. Instead we can use our renormalization- group analysis to "divide and conquer" the problem in pieces. First we compute
133 the distribution amplitude (j){x^Qo) for some small Qo, of order a few GeV, using a nonperturbative technique. The range of relevant momentum scales is quite modest for this part of the analysis. Then we use the perturbative evolution equations to evolve <^(x, Q) out to the large values of Q characteristic of the process. The evolution equations build up the short-distance structure of the hadronic wavefunction and are trivial to apply. Finally we combine the distribution amplitudes with the hard-scattering amplitude, which incorporates (perturbatively) the short-distance structure particular to the process. We can illustrate the nonperturbative analysis of distribution amplitudes 14 15 with a brief discussion of two such analyses, one using lattice QCD ' and 1 fi the other QCD sum rules. Both methods are based upon the behavior of matrix elements of the form (0| T Ti{0) Tj{t) |0>^^°^ where each Ti{t) is the spatial average of a local operator like those in Eq. (95): T^{t) = ^ Jd^xT^{x,t). (98) V By inserting a complete set {|^)} of hadronic eigenstates between the two operators it is easy to see that (O|ri(0rj(0)|0)^^°) = ^(0|ri(0)|n)(^°^ (n\Tj{0)\0)^^°^ e-*^"* (99) n when t > 0. The matrix elements multiplying the exponential in the sum are precisely those that determine the moments of the distribution amplitude for state \n). In the lattice analysis ordinary time is analytically continued to euclidean time so that it -^ t, and the cutoff Qo is determined by the lattice spacing. The matrix element in Eq. (99) is computed for large t. The sum is then dominated by the lowest mass state |no) that couples both to F, and Tj—e.g. the pion for operators taken from Eq. (95)—and so for sufficiently large t the expectation value has the form (0| Ti{t) Tj{0) |0)(^°^ -. (0| Fi(0) |no)^^°^ (no| Fj(0) |0)^^°^ e"^^"* (100) where Mq is the mass of state |no). The moments of the distribution amplitude for the lowest-lying state can be read off directly from the large-^ behavior of the Fj Fj-amplitude.
134 QCD sum rules can be derived for the Fourier transform of the matrix element, /.,(<?') = /c/te'"(o|r.(t)r,(o)|o>(«°), (101) 2 ^ n A,^^i:4..,j« T..(J2. analytically continued deep into the euclidean region q < 0. Amplitude Iij[q ) can be computed in two ways as q^ —> — oo. First, since the two operators are forced together in this limit, the operator product expansion can be used to relate the amplitude to vacuum expectation values of such local operators as as F^^ and yVa^uu. These matrix elements are universal and their values are usually inferred from other processes. On the other hand, the spectral decomposition Eq. (99) can be used to relate Iij{q^) to the moments of the distribution amplitudes for hadronic states \n). In practice the sum over hadronic states is replaced by a sum over a few low lying hadrons together with a continuum contribution approximated by the formula for free quarks, the threshold being a tunable parameter of the model. The moments are extracted by fitting the spectral formula for Iij{q^) to its operator product expansion. Each of these methods currently suffers from large systematic uncertainties and so one must be cautious in accepting results derived using them. Nevertheless such results form a reasonable starting point for phenomenological studies. Furthermore these methods have played an important role in alerting us to the potential complexity of hadronic distribution amplitudes. For example, one might have expected a relatively smooth distribution amplitude for the pion, not too different perhaps from its asymptotic form x{l — x). However the sum rules, for example, seem to imply a double-humped distribution x{l — x)(2x — l)'^. The sum rule predictions for baryons are even more remarkable—e.g. 65% of the proton momentum is carried by the w-quark with helicity parallel to the proton, while the remaining quarks split the remainder in this model. It is unclear how seriously one should take such predictions, but it is clear that unusual x-dependence is a distinct possibility for hadronic distribution amplitudes. It is also clear that the reliability of the these nonperturbative techniques, particularly the lattice analysis, will improve substantially in the not-too-distant future. Note finally that it was essential for our nonperturbative calculations that the distribution amplitude have a nonperturbative definition—i.e. in terms of operator matrix elements in a cut off field theory. Had the distribution amplitude been #10 In actual practice this procedure is modified to employ a Borel transform so as to de- emphasize the high-mass region.
135 defined in terms of perturbative constructs, it would have been almost impossible to carry that definition over into a nonperturbative framework such as that provided by lattice QCD. In general it is important to provide a nonperturbative characterization for the contributions omitted from the perturbative analysis of a process. 4.4. Higher Order Analysis The leading-order formula for the asymptotic pion form factor results from a series of approximations. One can systematically undo these approximations to obtain ' 0{as{Q^)) corrections to F:^(Q^). For example in our leading-order analysis of the disconnected contribution eg • ' - 0 / c/x / ^ ^l,^^>{x, ^x + (1 - x)qi.) rk^''\x, h) (102) we assumed that large transverse momentum flows through one or the other wave- function. We ignored the contribution from the region where large momentum flows through both wavefunctions: k± ~ ^i. 4- (1 — 3:)q± ~ (1 — x)q±. The contribution from the latter region is easily estimated. We can use the bound state equation to replace both wavefunctions by a convolution of the perturbative potential with the distribution amplitude (Eq. (74)) to obtain a contribution 1 1 jdyjdz <i>l{z, (1 - z)Q) T2{y, z, Q) Mv. (1 " y)Q) (103) 0 0 where 1 ^^ , d'^ks. Vef[{z,0\x,ks.-\-{l-x)qs.) _ Viff (x, ^_l; ?/,0) T2{y,Z,Q) = ax I ~—^ rr— ^ ^^-^ 6 16t:^ -(^k_i-\-{l-x)q_i)yx{l-x) -kl/x{l-x) The k± integration in this expression must be restricted to the region where both k± and k±-{-{l — x)q± are large, because the contributions from the regions where one or the other vector is small are already included in the leading-order result. One way to restrict the range of k^ is to introduce collinear subtractions that
136 remove precisely the contribution included in the leading-order analysis. The region where k± is small is removed by subtracting 1 (i-^)Q -. To'\y,z,Q) = Jdx e Wt' -{{l-x)q±)yx{l-x) " -k^/x{l-x) 0 -L' V / (104) where we neglect k± relative to (1 — x)q± and integrate over \k^\ < (1 — x)Q, just as in the leading-order analysis (c.f. , Eq. (71)). Similarly the region where k^ -{■ (1 — z)q^ IS small is removed by 1 (i-^)Q -. rps2f n\ - f ^ f ^^i- KK{z,0\x,k_i) V^f^{x,-{1 -x)q_i;y,0) 0 -L' V / (105) where we have changed variables so that k±-{- {1 — x)q± —^ k±. The subtracted amplitude (Fig. 15a) contains only large momenta when Q is large, and thus it can be computed perturbatively and gives an 0{al{Q^)) contribution to the hard-scattering amplitude Tjj. All masses can be neglected, and no logarithms of Q can arise from the A:^-integration since Q is the only scale left after the subtractions. A similar analysis can be applied to the bound state equation to obtain higher order corrections to the formula relating the high-f^ wavefunction and the distribution amplitude (Eq. (74)). These corrections lead to additional 0{al) contributions to Tff (Fig. 15b), and to 0{al) contributions to the evolution potential V. In addition to these higher-order corrections, there are corrections coming from the one-loop (and higher) qq-'iTTedncible diagrams, both for Tjj (Fig. 15c) and for V. As discussed in earlier sections, these irreducible amplitudes have no sensitivity to low momenta when they are computed in light-cone gauge, and thus they are perturbative when Q is large. This procedure can iterated to produce still higher-order corrections to the hard-scattering amplitude and to the evolution potential. In this way one establishes the self-consistency of the factored perturbative result to all orders in perturbation theory. The only complication arises when endpoint and/or pinch singularities appear in the hard-scattering amplitude, and these we discuss in the next section. A systematic analysis of higher order corrections, based upon Mueller's cut- vertex formalism, has been given in Ref. 19. Using this method, the validity of the perturbative expression for the meson form factor has been established to all
137 (a) ki (1 -x)Q d^kj. k i (1-x)Q (b) (1-x)Q d2ki k i (c) + ••• Figure 15. Diagrams contributing to the second-order hard-scattering amplitude for the pion form factor. orders in perturbation theory. The one-loop corrections have also been calculated 17 18 for the meson form factor. ' 4.5. Complications The perturbative analysis of large-px processes relies upon the fact that the hard subprocess is confined to a small volume near the light-cone. This is a consequence not of the kinematics of the process but rather of the dynamical behavior of the hard-scattering amplitude Tjj, all of whose internal propagators are typically far off shell {\S\ ~ Q^). Unfortuna»tely the x integrations in the perturbative formula can include points where internal lines in Tjj go on shell. In form factors these points show up as singularities in Tjj at the endpoints of the integration—i.e. a; = 0 or a; = 1—and so they are referred to as endpoint singularities. Singularities can also occur at intermediate values of x in hard- 20 scattering amplitudes for hadronic scattering amplitudes; these are referred to as pinch singularities. Perturbation theory breaks down in the vicinity of such #11 In the covariant calculation of a Feynman amplitude every internal propagator has singular points. Usually these singularities are avoided by deforming the integration contours into the complex momentum plane. A singularity that occurs at the endpoint of a con-
138 singularities, and so our perturbative results are jeopardized if large contributions come from such regions. 19 4 . .21 Remarkably it is just in the endpoint ' and pinch regions that Sudakov form factors appear. In these regions individual quarks (or gluons) tend to scatter independently of the other partons comprising the hadrons. An isolated, nearly on-shell quark wants to radiate gluons when it scatters, the amount of radiation increasing as the change in the quark's state of motion becomes more drastic. In an exclusive process such hremsstrahlung is prohibited, and as a result the amplitude is suppressed. This phenomenon is apparent in perturbation theory. For example, in computing the electromagnetic form factor of a single quark one obtains double logarithms of Q^ coming from the radiative corrections to the quark-photon vertex. These exponentiate when summed to all orders to give a quark form factor that ultimately falls faster than any power of 1/Q. This is the Sudakov form factor. Such form factors tend to suppress contributions coming from the endpoint and pinch regions. Note that double logarithms of Q and Sudakov form factors only appear in the vicinity of singularities in Th- In other regions all of the constituents of each hadron are involved in the same hard subprocess. The collinear bunches of partons representing each hadron in Th carry no color charge, and thus the soft gluons that normally build up Sudakov form factors decouple. In this section we examine the contributions coming from the endpoint and pinch regions. We show where these contributions come from and why Sudakov suppression is expected. Endpoint Singularities Our analysis of the qq contribution to F;,r(Q^) for large Q^ depends upon the assumption that either k\^ ox k^-\- [\ — x)q± is 0(q±) in the overlap integral 1 cgjdxj^ i^^^'^'ix, ^x 4- (1 - x)q^) V.(^)(x, k^) (106) 0 —i.e. that large momentum flows through one or the other of the wavefunctions. This is certainly the case except in the infinitesimal region where 1-x-A/Q (107) if A is the typical transverse momentum in the wavefunction. Within this "end- tour obviously cannot be avoided in this fashion; this is how endpoint singularities arise in exclusive amplitudes. In addition it is possible for a contour to be trapped or pinched between two singularities. This is how pinch singularities arise.
139 point region" both wavefunctions carry small transverse momentum ('^ A). The meson form factor receives a contribution from this region of order 1 Fep(Q^) r\^ dx \il;^^\x, \)\ r\^ ^ Q ^) \ 1+26 (108) when ip^^\x,X) vanishes Uke (1 — x) as a; —> 1. This mechanism, in which spectator quarks are stopped rather than turned, was actually the first parton model suggested for hadronic form factors. To assess its importance here we require information about the qq wavefunction as a; —>- 1. The qq state in the pion is far off shell in the endpoint region— 1^1 A2 r>^ x{l — x) r>^ XQ (109) —suggesting that perturbation theory might be a reasonable guide to the behavior of the wavefunction (Fig. 16). Perturbation theory implies 6=1 and thus the (b) -1 -X -1 -X Figure 16. Born amplitudes whose behavior might be similar to that of the hadronic wavefunctions as x —^ 1. endpoint contributions fall as (A/Q)"^, down by a full power of X/Q relative to the hard-scattering contributions. #12 We consider only the valence Fock state here since the phase space in the case of n spectator partons goes like (A/Q)"—small numbers of spectators are favored.
140 The analysis is similar for baryon form factors where 1 ^/« 2+2« FEPiQn 2)~ / dxi [ dx2\rP^''Kxi,X)\^~(^\ (110) 1-A 0 Perturbation theory again gives 5 = 1, but here the endpoint contribution seems to be suppressed by only two powers of as(XQ) relative to the hard scattering prediction: FEP^^^^^--al{XQ)FHS- (111) Endpoint singularities are far more severe in the nucleon form factor than they are in the meson form factor. In general they are equally severe in more complicated process, such as hadron-hadron scattering. In fact the suppression of the endpoint region is probably a good deal stronger than these equations indicate. As far as the photon is concerned the struck quark is very nearly on shell in the endpoint region since \S\ ~ XQ <C Q^- Furthermore only the struck quark participates in the hard subprocess in this region; it behaves as though isolated from the other quarks over time scales of 0(l/\/XQ). Consequently the endpoint contribution to the amplitude is suppressed by a Sudakov form factor, and most likely is negligible when Q is sufficiently large. Pinch Singularities The pinch singularity ' ' is most serious in hadron-hadron scattering. As an illustration consider the diagram in Fig. 17a, which contributes to tt-tt scattering. Three-momentum conservation requires k±a + k_[.b - ^Ic - ^Ld = (^c - ^a)r\, -f" [Xd - Xa)q± where k±a • • • ^id are the transverse momenta appearing in the wavefunctions for each of the pions, Xa - - -x^i are the longitudinal momenta, and where the relativistic invariants for the process are 3 = r I + ql t = -ql (113) u = —f^ with r±'q± = 0. At high energies and wide angles, f^ and qj^ are both large, and
141 so at least one of k±a • • • ^id must be large for most values of Xa ... a:^. Then, as in our analysis of the meson form factor, the wavefunction with large k± is replaced by a gluon exchange to give a hard-scattering amplitude, as depicted in Fig. 17b (where k^a is large). Dimensional counting then implies Th a r>^ f{6cM',Xa...Xi) (114) for this contribution. Also the energy denominator in D in Fig. 17a, D = (a:c-Xa)r|-f (a:d-a:a)^X+2(^i.d~^i.a)-fi.-f-2(^_Lc-^±a)-rL-H- • .+2e, (115) is of 0{s) indicating that the two quark-quark scatterings occur within a very short time of each other. V^(^i+''i)+'^i a Xq. Xjjrj_+kj_^ (a) ^•V^i Xu, k b''^lb 1.0 / D=^-^i "d-^d^i^^d D (b) Figure 17. a) Diagram contributing to tt-tt scatteting. b) Hard scattering amplitude coming from a). Notice however that in the pinch region. A Xc — Xa\ ^ f^L ^d *^a A ^\^ Wx\' (116) all wavefunction momenta A^ia • • • ^id can be small (~ A). Furthermore the denominator D is 0{\y/s) or less, and can even vanish. Thus the two quark-quark
142 scatterings can occur more or less independently, at widely separated points. The scattering process is no longer localized, and factorization does not occur. The s dependence of the contribution from this region can be readily estimated: a) the quark-quark scattering amplitudes each give (l/^)^, by dimensional counting; b) phase space as restricted by in Eq. (116) gives a factor {X/y/s)^\ c) the energy denominator gives a factor 1/D ^ l/Xy/s. Thus the pinch region contributes Tps-^-^f(ecM;xa) (117) which apparently dominates the hard scattering contribution by a factor y/s. Two things work to suppress this pinch contribution. First the number of hard scattering amplitudes is much larger than the number of pinch singularity diagrams. More importantly, perhaps, radiative corrections to the individual quark-quark amplitudes build up Sudakov form factors that increase the effective power of 1/5 to something like ! + i|£ log log ( ra ) (118) which grows infinitely large as |^| ^ 6 —^ 00. These corrections do not cancel here because the quarks and antiquarks scatter separately here, and not together as color singlets. So the pinch region is probably completely suppressed by Sudakov effects when s is sufficiently large. It turns out that a contribution still remains from a region intermediate between the pinch region and the hard-scattering 21 . . region. This results in a small correction to the power-law predicted by dimensional counting. For example, pp elastic scattering at wide angles should fall off roughly Hke 3~^'^, rather than s~^^ as predicted by dimensional counting. Considerable progress has been made recently towards a complete analysis of such effects. Pinch singularities always show up as singularities in the hard scattering amplitude TH{xa-,X},..., Q) at points Xa^xi,... away from the endpoints 0 and 1. The integrals over Xa, x^... are then singular. Not every midpoint singularity in Tjj actually corresponds to a pinch. For example, singularities that are linear— e.g. l/{x — c + ie)—do not involve pinches. These cause no problems when integrating over x: the real part of the amplitude is obtained using a principal value prescription, while an imaginary part is generated by making the replacement l/{x — c-{-ie) —> —27riS(x — c). When the singularities are more severe they must be cut off by explicitly including Sudakov form factors in the pinch region. The dimensional-counting rule is modified only in these very singular situations.
143 4.6. How Large is Asymptotic Q? The perturbative formalism we have described is only valid at large momentum transfers. A critical question then is, How large is large? Here as in any application of perturbative QCD there are really two issues: 1) the convergence of perturbation theory; and 2) the relative importance of competing nonpertur- bative mechanisms. We examine each in term. The perturbative expansion describing a short-distance process in QCD—e.g. ao+ai as(Qlf^)/7r-\-...—converges quickly if the characteristic momentum Qeflf for the process is large compared with the QCD scale parameter Aqcd ^ 200 Mev. To determine Qeflf for large-pj^exclusive processes we can examine the momentum flow in the hard-scattering amplitude. The pion's form factor, for example, is given by 1 1 F.{Q^) ^ Jdxjdy {<t>*{y, (1 - y)Q) e, Th{x, y, Q) <t>{x, (1 - x)Q) + {q ^ q)} 0 0 (119) where the hard-scattering amplitude is ^^("'^''^)=(l-x)(l-v)Q2- (120) The running coupling in Tu is associated with gluon-exchange between the quark and the antiquark as they scatter from the initial to the final direction. Thus it is natural to set the scale of this coupling equal to the square of the gluon's four momentum: a^ —> as((l — a:)(l — y)Q ) in Tu. The defining relation for Qgff then is obviously 1 ' 2 ^ ' 0 0 0 0 A small complication is that the usual perturbative formula for as{Q^) has an unphysical singularity at Q = Aqcd^ and so the integral on the left-hand-side #13 In earlier sections we set the scale equal to Q^. The changes that result from the replacement Q^ —»• (1 — x)(l — y)Q^ are higher order in a, and so are irrelevant at very large Q^. However we are now concerned with how small Q^ can be made before perturbation theory fails, and for this purpose it is important to use the more physical scale in a,.
144 of this equation is ill-defined. This is easily remedied by redefining the running coupling so that where c is a constant (~ 1-3). This is a rather ad hoc remedy, but the ratio Qef[/Q that results is fairly insensitive to both c and Q unless Q is very small. The ratio Qef[/Q is clearly quite sensitive to the x-dependence of the distribution amplitudes, with broader amplitudes giving more emphasis to the region x,i/ ~ 1 and thus lower Qeflf's. Assuming the asymptotic dependence x(l — x), one finds that Qef[/Q ~ 0.2. In this case a form factor with momentum transfer of say 2 Gev actually probes QCD at scales of order only 400 MeV. The effective momentum transfer is smaller still with the broader distribution amplitudes suggested by QCD sum rules (Qeff/Q ~ 0.1). The running coupling constant is of order unity for such small Qeflf's and so perturbation theory is not likely to converge very well, if at all. Some perturbative properties, such as the dimensional-counting and helicity-conservation rules, are valid to all orders in perturbation theory; these might well be applicable even for such Qeflf's- However it should not be surprising if predictions for things like the magnitude of the form factor are off by factors of 2 or more. (Note, for example, that replacing as{Q^) by cts(Qlf^) more than doubles the perturbative prediction for the form factor at Q = 2 GeV.) It has proven difficult to measure meson form factors for Q's much above a couple of GeV. However the proton form factor has been measured out beyond 5 Gev. Unfortunately the hard-scattering amplitudes for baryon form factors tend to be more singular in the low-momentum region than meson amplitudes resulting in smaller ratios of Qeff/0* ^S- ^^^ finds that Qeflf/Q ^0.1 for the asymptotic distribution amplitude X1X2X3, and the ratio is smaller by another factor of a half to a third for the broader distribution amplitudes predicted by sum rules. So existing data for the proton form factor, although more accurate, still probes much the same region in effective momentum as does the data for the pion form factor. The ratio Qeff/Q is also relevant to the second important issue—the relative importance of nonperturbative contributions. We expect the quark-antiquark interaction in Th to evolve smoothly from nonperturbative to perturbative behavior as Qeff increases, with the crossover occurring around a few hundred MeV. Consequently the pion form factor, for example, could be predominantly perturbative by Q = 2 GeV since QgfF is then of order a few hundred MeV. This is despite the fact that perturbative interactions bring in factors of a^: the coupling ct3(Q^^)
145 is not particularly small when Qeflf is small, and thus it does not suppress such #14 interactions much. With protons, perturbative behavior might set in at 3 GeV or higher, depending upon the distribution amplitude. For larger Q's one must also worry about nonperturbative contributions coming from the endpoint region, particularly in the case of baryon form factors and scattering amplitudes. Perturbative arguments indicate that such contributions are suppressed by Sudakov form factors, but the extent of this suppression at accessible Q's is uncertain. The importance of this region also depends sensitively upon the behavior of the hadronic wavefunctions in the endpoint region: it is easy to make model wavefunctions in which there is little contribution from the endpoint region for Q's greater than a few GeV; ' ' it is also easy to make models in which the region is important even at several GeV (ignoring Sudakov effects). The situation is further complicated in the case of hadronic scattering amplitudes by our incomplete understanding of the Sudakov suppression of pinch singularities. In the light of these uncertainties the best one can do is to assume the validity of the perturbative analysis, at least as a qualitative or semi-quantitative guide to large-p_i^ exclusive processes. This model is quite plausibly correct, and in any case there is currently no other comprehensive theory of these processes. The validity of the perturbative model can then be judged by the extent to which it is capable of accounting for the broad range of available data. #14 Of course perturbation theory will not converge well if a, is large. When we speak of "perturbative behavior" here we are again thinking of behavior that is true to all orders— factorization, dimensional counting, helicity conservation.... It is important to realize that the validity of the factorized form for a large momentum transfer amplitude is not necessarily contingent on the applicability of perturbation theory. Indeed there is likely to be a region of momentum transfer where factorization, dimensional counting... are valid but where perturbation theory does not converge at all.
146 5. APPLICATIONS OF QCD TO THE PHENOMENOLOGY OF EXCLUSIVE REACTIONS In the following sections we will discuss the phenomenology of exclusive reactions as tests of QCD and the structure of hadrons. The primary processes of interest are those in which all final particles are measured at large invariant masses compared to each other: i.e. large momentum transfer exclusive reactions. This includes form factors of hadrons and nuclei at large momentum transfer Q and large angle scattering reactions. Specific examples are reactions such as e~p —* e~p, e'^e" —> pp which determine the proton form factor, two-body scattering reactions at large angles and energies such as ir'^p —> ir'^p and pp —> pp, two-photon annihilation processes such as 77 —> K'^K~ or pp —> 77, exclusive nuclear processes such as deuteron photo-disintegration "yd —> np, and exclusive decays such as tt"^ —> fi'^i/ or J/^|^ —> 7r"''7r~7r^. In this section we will summarize the main features of the QCD predictions developed in the previous sections. QCD has two essential properties which make calculations of processes at short distance or high-momentum transfer tractable and systematic. The critical feature is asymptotic freedom: the effective coupling constant as(Q^) which controls the interactions of quarks and gluons at momentum transfer Q vanishes logarithmically at large Q^ since it allows perturbative expansions in as(Q^). Complementary to asymptotic freedom is the existence of factorization theorems for both exclusive and inclusive processes at large momentum transfer. In the case of "hard" exclusive processes (in which the kinematics of all the final state hadrons are fixed at large invariant mass), the hadronic amplitude can be represented as the product of a process-dependent hard-scattering amplitude ^//(x,, Q) for the scattering of the constituent quarks convoluted with a process-independent dis- tribution amplitude </>(x, Q) for each incoming or outgoing hadron. When Q^ is large, Tjj is computable in perturbation theory as is the Q-dependence of <^(x, Q). We have discussed the development of factorization for exclusive processes in detail in Section 4. Quantum chromodynamics has now been extensively tested in high momentum transfer inclusive reactions where the factorization theorems, perturbation theory, and jet evolution algorithms provide semi-quantitative predictions. Tests of the confining nonperturbative aspects of the theory are, however, either qualitative or at best indirect. In fact QCD is a theory of relatively low mass scales (^M5 ^ 200 ± 100 MeV, < k\ >^/2 ^ 300 MeV), and eventually its most critical test as a viable theory of strong and nuclear interactions will involve relatively low energies and momentum transfer at the interface of the perturbative and nonperturbative domain.
147 The understanding of hadronization and the computation of hadron matrix elements clearly requires knowledge of the hadron wavefunctions. In Table I we give a summary of the main scaling laws and properties of large momentum transfer exclusive and inclusive cross sections which are derivable starting from the light-cone Fock space basis and the perturbative expansion for QCD. As we have discussed in Section 3, a convenient relativistic description of hadron wavefunctions is given by the set of n-body momentum space amplitudes, ^n(3^t5 ^i.,5 A,), i = 1,2, ...n, defined on the free quark and gluon Fock basis at equal "light-cone time" t = t -\- z/c in the physical "light-cone" gauge A^ = A^ + i4^ = 0. (Here x,- = kf jp^^ Z)t^« ^ ^-t ^^ ^^^ light-cone momentum fraction of quark or gluon i in the n — particle Fock state; k^^^ with ^^ k^,^ = 0, is its transverse momentum relative to the total momentum p'*; and A, is its helic- ity.) The quark and gluon structure functions Gg^ff(x,Q) and Gg^ff(x^Q) which control hard inclusive reactions and the hadron distribution amplitudes (I>h(^^ Q) which control hard exclusive reactions are simply related to these wavefunctions: Q Gq/ni^^Q) ^Yl nf/^^i-. / nC?X,- \'(pn{xi,k±^)\'^6{Xq - x) , n and Q ^.(x.,Q)oc/n.^...^.._(..,.x,) . In the case of inclusive reactions, such as deep inelastic lepton scattering, two basic aspects of QCD are relevant: (1) the scale invariance of the underlying lepton-quark subprocess cross section, and (2) the form and evolution of the structure functions. A structure function is a sum of squares of the light-cone wavefunctions. The logarithmic evolution of Gq{x^ Q^) is controlled by the wave- functions which fall off as \'4^{x^k±)y' ^ as {k\)lk\ at large k\. This for m is a consequence of the pointlike q —> gq^ g —> gg^ and g —^ qq splittings. By taking the logarithmic derivative of G with respect to Q one derives the evolution equations of the structure function. All of the hadron's Fock states generally participate; the necessity for taking into account the (non-valence) higher-particle Fock states in the proton is apparent from two facts: (1) the proton's large gluon momentum 28 fraction and (2) the recent results from the EMC collaboration suggesting that, 29 on the average, little of the proton's helicity is carried by the light quarks.
148 Table I Comparison of Exclusive and Inclusive Cross Sections Exclusive Amplitudes Inclusive Cross Sections M^U(f>{xi,Q)^TH(xi,Q) Q Measure <^ in 7 —> MM da ^ UG(xa,Q) (S> da{xa,Q) Q n -^ Measure G in ip—^ iX d<f>(x,Q) d log g2 = 0^3 [dy] V(x,y)<f){y) lim (l>{x,Q) = n ^t'Cfi Q—00 , avor Evolution dG(x,Q) d log g2 = a,Jdy P{x/y) G(y) lim G{x,Q) = 6(x)C Th : expansion in Qs{Q^) Power Law Behavior da (fpjE {AB-.CX)^Y.^\n27n!Xf^'^--) iQ') flact = Tla ■]- nj, ■]- Tic + rid da : expansion in as{Q^) End point singularities Pinch singularities High Fock states Complications Multiple scales Phase-space limits on evolution Heavy quark thresholds Higher twist multiparticle processes Initial and final state interactions
149 In the case of exclusive electroproduction reactions such as the baryon form factor, again two basic aspects of QCD are relevant: (1) the scaling of the underlying hard scattering amplitude (such as / + qqq —> / + qqq)-, and (2) the form and evolution of the hadron distribution amplitudes. The distribution amplitude is defined as an integral over the lowest (valence) light-cone Fock state. The logarithmic variation of <^(x,(3^) is derived from the integration at large A:^^, i.e. wavefunctions which behave as ^(x, kx) ^ <^s(^j_)/^i ^^ large k^ This behavior follows from the simple one-gluon exchange contribution to the tail of the valence wavefunction. By taking the logarithmic derivative, one then obtains the evolution equation for the hadron distribution amplitude. As we showed in Section 3, the form factor of a hadron at any momentum transfer can be computed exactly in terms of a convolution of initial and final light-cone Fock state wavefunctions. In general, all of the Fock states contribute. In contrast, exclusive reactions with high momentum transfer Q, perturbative QCD predicts that only the lowest particle number (valence) Fock state is required to compute the contribution to the amplitude to leading order in l/Q. For example, in the light-cone Fock expansion the proton is represented as a column vector of states il^qqq, i^ggggi ^ggm • • •• III ^he light-cone gauge, A^ -\- A^ = 0, only the minimal "valence" three-quark Fock state needs to be considered at large momentum transfer since any additional quark or gluon forced to absorb large momentum transfer yields a power-law suppressed contribution to the hadronic amplitude. Thus at large Q^, the baryon form factor can be systematically computed by iterating the equation of motion for its valence Fock state wherever large relative momentum occurs. To leading order the kernel is effectively one-gluon exchange. The sum of the hard gluon exchange contributions can be arranged as the gauge invariant amplitude T//, the final form factor having the form 1 1 Fb{Q^) = J[dy] J[dx] <t>Uyj^Q)TH{xi,yj.Q)<t>B{^t.Q) . 0 0 The essential gauge-invariant input for hard exclusive processes is the distribution amplitude <I>h{x^Q)' For example (j>Tr(x,Q) is the amplitude for finding a quark and antiquark in the pion carrying momentum fractions x and I — x at impact (transverse space) separations less than b± < l/Q. The distribution amplitude thus plays the role of the "wavefunction at the origin" in analogous non-relativistic calculations of form factors. In the relativistic theory, its dependence on log Q is controlled by evolution equations derivable from perturbation
150 theory or the operator product expansion. A detailed discussion of the light-cone Fock state wavefunctions and their relation to observables is given in Section 3 and in Ref. 30. The distribution amplitude contains all of the bound-state dynamics and specifies the momentum distribution of the quarks in the hadron. The hard- scattering amplitude for a given exclusive process can be calculated perturba- tively as a function of as{Q^). Similar analyses can be applied to form factors, exclusive photon-photon reactions, and with increasing degrees of complication, to photoproduction, fixed-angle scattering, etc. In the case of the simplest processes, 77 —> MM and the meson form factors, the leading order analysis can be readily extended to all-orders in perturbation theory. Figure 18. QCD factorization for two-body amplitudes at large momentum transfer. In the case of exclusive processes such as photo-production, Compton scattering, meson-baryon scattering, etc., the leading hard scattering QCD contribution at large momentum transfer Q^ = tu/s has the form (helicity labels and suppressed) (see Fig. 18) r Ma+B-^C+d{Q ,^c.m.) = / [dx]<l>c{xc,Q) <l>D{xd,Q) Tnixi^Q ,^c.m.) X (t>A{Xa,Q) <l>B{xb,Q) In general the distribution amplitude is evaluated at the characteristic scale Q set by the effective virtuality of the quark propagators.
151 By definition, the hard scattering amplitude Th for a given exclusive process is constructed by replacing each external hadron with its mgissless, collinear valence partons, each carrying a finite fraction x, of the hadron's momentum. Thus Th is the scattering amplitude for the constituents. The essential behavior of the amplitude is determined by T//, computed where each hadron is replaced by its (collinear) quark constituents. We note that Th is "collinear irreducible," i.e. the transverse momentum integrations of all reducible loop integration are restricted to k\^ > O(Q^) since the small k± region is already contained in <j>. If the internal propagators in Th are all far-off-shell 0{Q^), then a perturbative expansion in as{Q^) can be carried out. Higher twist corrections to the quark and gluon propagator due to mass terms and intrinsic transverse momenta of a few hundred MeV give nominal corrections of higher order in l/Q^. These finite mass corrections combine with the leading twist results to give a smooth approach to small Q^. It is thus reasonable that PQCD scaling laws become valid at relatively low momentum transfer of order of a few GeV. 5.1. General Features of Exclusive Processes in QCD The factorization theorem for large-momentum-transfer exclusive reactions separates the dynamics of hard-scattering quark and gluon amplitudes T// from process-independent distribution amplitudes <^//(x,Q) which isolates all of the bound state dynamics. However, as seen from Table I, even without complete information on the hadronic wave functions, it is still possible to make predictions at large momentum transfer directly from QCD. Although detailed calculations of the hard-scattering amplitude have not been carried out in all of the hadron-hadron scattering cases, one can abstract some general features of QCD common to all exclusive processes at large momentum transfer: 1. Since the distribution amplitude <J)h is the L^ = 0 orbital-angular-momentum projection of the hadron wave function, the sum of the interacting constituents' spin along the hadron's momentum equals the hadron spin: z z S: = 5 In contrast, there are any number of non-interacting spectator constituents in inclusive reactions, and the spin of the active quarks or gluons is only statistically related to the hadron spin (except at the edge of phase space X —^ 1).
152 2. Since all loop integrations in Tjj are of order Q, the quark and hadron masses can be neglected at large Q up to corrections of order ^ m/Q. The vector-gluon coupling conserves quark helicity when all masses are neglected-i.e. ui^y^u-^ = 0. Thus total quark helicity is conserved in T//. In addition, because of (2), each hadron's helicity is the sum of the helicities of its valence quarks in Tjj. We thus have the selection rule initial final i.e. total hadronic helicity is conserved up to corrections of order m/Q or higher. Only (flavor-singlet) mesons in the 0 ^ nonet can have a two-gluon valence component and thus even for these states the quark helicity equals the hadronic helicity. Consequently hadronic-helicity conservation applies 31 for all amplitudes involving light meson and baryons. Exclusive reactions which involve hadrons with quarks or gluons in higher orbital angular states are suppressed by powers. 3. The nominal power-law behavior of an exclusive amplitude at fixed Oc.m. is (l/Q)^"^, where n is the number of external elementary particles (quarks, 5 gluons, leptons, photons, ...) in Th- This dimensional-counting rule is modified by the Q^ dependence of the factors of as{Q^) in Tjj, by the Q^ evolution of the distribution amplitudes, and possibly by a small power correction associated with the Sudakov suppression of pinch singularities in hadron-hadron scattering. The dimensional-counting rules for the power-law falloff appear to be experimentally well established for a wide variety of processes. ' The helicity- conservation rule is also one of the most characteristic features of QCD, being a direct consequence of the gluon's spin. A scalar-or tensor-gluon-quark coupling flips the quark's helicity. Thus, for such theories, helicity may or may not be conserved in any given diagram contribution to Tjj depending upon the number of interactions involved. Only for a vector theory, such as QCD, can one have a helicity selection rule valid to all orders in perturbation theory.
153 (o) (b) + + + + (c) + + + Figure 19. (a) Factorization of the nucleon form factor at large Q^ in QCD. (b) The leading order diagrams for the hard scattering amplitude T//. The dots indicate insertions which enter the renormalization of the coupling constant, (c) The leading order diagrams which determine the Q^ dependence of the distribution amplitude 5.2. Electromagnetic Form Factors Any helicity conserving baryon form factor at large Q^ has the form: [see Fig 19(a)] 1 1 Fb{Q^) = J[dy] j[dx] <t>^s{y,,Q)TH{x,,y,,Q)<^B{x^,Q) , 0 0 where to leading order in as[Q^), Tjj is computed from 3^ + 7 amplitudes: [Fig. 19(b).] 3^ tree graph 2M Th = Q fi^i.yj)
154 and M^i,Q) = [[d^k^,] i>v{xi,h,)S{kl, < Q^) is the valence three-quark wavefunction [Fig. 19(c)] evaluated at quark impact separation 6j_ ~ 0{Q~^). More detailed formulae for the baryon form factor are presented in Appendix I. Since (/)£ only depends logarithmically on Q^ in QCD, the main dynamical dependence of Fb{Q^) is the power behavior (Q^)~^ derived from scaling of the elementary propagators in Th- More explicitly, the proton's magnetic form factor has the form: Gm{Q^) = ■i2 nytn nm Q 2\ -7n-7 A2 m X l + 0(as{Q)) + 0 The first factor, in agreement with the quark counting rule, is due to the hard scattering of the three valence quarks from the initial to final nucleon direction. Higher Fock states lead to form factor contributions of successively higher order in l/Q^. The logarithmic corrections derive from an evolution equation for the nucleon distribution amplitude. The 7„ are the computed anomalous dimensions, reflecting the short distance scaling of three-quark composite operators. The results hold for any baryon to baryon vector or axial vector transition amplitude that conserves the baryon helicity. Helicity non-conserving form factors should fall as an additional power of l/Q^. Measurements of the transition form factor to the J = 3/2 A'^(1520) nucleon resonance are consistent with Jz = ±1/2 dominance, as predicted by the helicity conservation rule. A review of the data on spin effects in electron nucleon scattering in the resonance region is given in Ref. 34. It is important to expHcitly verify that F2{Q^)/Fi{Q^) decreases at large Q^. The angular distribution decay of the J ftp —*■ pp is consistent with the QCD prediction Ap + Ap = 0. Thus, modulo logarithmic factors, one obtains a dimensional counting rule for any hadronic or nuclear form factor at large Q^ (A = A' = 0 or 1/2) F{Q') rs^ F/^~ 1 Q4 ' n-1 F. 1 ^\^ Q2 ' Fd 1 ^\^ 0 10
I c 00 O 0 0 10' 10 -I 10 -2 10 rZ 0 ,-3 0 ,-4 0 Pion, n = 2 Proton, n = 3 Neutron, n = 3 Deuteron, n=6 Helium 3, n=9 Helium 4, n = l2 xO. q2 (GeV2) 155 ''V> 'V *%/ 35 Figure 20. Comparison of experiment rule (Q^)"~^F(Q^) ^ const for form factors. GeV2. with the QCD dimensional counting The proton data extends beyond 30 where n is the minimum number of fields in the hadron. Since quark helicity is conserved in Th and <j>{xi^Q) is the Lz = 0 projection of the wavefunction, total hadronic helicity is conserved at large momentum transfer for any QCD exclusive reaction. The dominant nucleon form factor thus corresponds to Fi(Q^) or Gm{Q^)\ the Pauli form factor F2(Q^) is suppressed by an extra power of Q^. Similarly, in the case of the deuteron, the dominant form factor has helicity A = A' = 0, corresponding to yjA{Q'^). The comparison of experimental form factors with the predicted nominal power-law behavior is shown in Fig. 20. We will discuss predictions for the normalization of the leading power terms in Section 5.6. As we have discussed in Section 4, the general form of the logarithmic corrections to the leading power contributions form factors can be derived from the operator product expansion at short distance ' or by solving an evolution equation for the distribution amplitude computed from gluon exchange [Fig. 19(c)], the only QCD contribution which falls sufficiently small at large transverse momentum to effect the large Q^ dependence. The comparison of the proton form factor data with the QCD prediction arbitrarily normalized is shown in Fig. 21. The fall-off of (Q^)^Gm(Q^) with Q^
156 is consistent with the logarithmic fall-off of the square of QCD running coupling 1 fi constant. As we shall discuss below, the QCD sum rule model form for the nucleon distribution amplitude together with the QCD factorization formulae, predicts the correct sign and magnitude as well as scaling behavior of the proton and neutron form factors. 0.6 O 0.5 a; I ^1 Q. 0.4 Previous Data This Experiment CZ BL 6 0 0.2 o o 8 0 0 20 Q2 [(GeV/c)2] 30 Figure 21. Comparison of the scaling behavior of the proton magnetic form factor with the theoretical predictions of Refs. 4 and 16. The CZ predictions are normalized in sign and magnitude. The data are from Ref. 36. 5.3. Comparison of QCD Scaling with Experiment Phenomenologically the dimensional counting power laws appear consistent with measurements of form factors, photon-induced amplitudes, and elastic hadron- 33 hadron scattering at large angles and momentum transfer. The successes of the quark counting rules can be taken as strong evidence for QCD since the derivation of the counting rules require scale invariant tree graphs, soft corrections from higher loop corrections to the hard scattering amplitude, and strong suppression of pinch singularities. QCD is the only field theory of spin ^ fields that has all of these properties. As shown in Fig. 22, the data for jp TT'^n cross section at 7r/2 are consistent with the normalization and scaling da/dt {^p [lnb/(5/10 GeV)^] f{t/s). rsy
157 00 o 0^ - 0^ - 0^ - 0^ - a \J 5 4 3 2 1 1 0 -1 1 -2 1 ' 1 «< (a) \X \ V \ X s-'^ 1 1 1 1 ' ' ' ' 1 Xp-^7r"'"n (^♦^90°) - • SLAC o MIT X CIT -n_ \ — \^ — 1 1 1 1 1 1 \ 0^ - 0' - 0 0 10 10 (Gev2) 20 Figure 22. Comparison of photoproduction data with the dimensional counting power-law prediction. The data are summarized in Ref. 37. The check of fixed angle scaling in proton-proton elastic scattering is shown in Figs. 23. Extensive measurements of the pp —> pp cross section have been made at ANL, BNL and other laboratories. The scaling law s^^da/dt{pp —> pp) c^ const. predicted by QCD seems to work quite well over a large range of energy and angle. The best fit gives the power A^ = 9.7 ±0.5 compared to the dimensional counting prediction N=10. There are, however, measurable deviations from fixed power dependence which are not readily apparent on the log-log plot. As emphasized by Hendry the s^^da/dt cross section exhibits oscillatory behavior with px (see Section 9). Even more serious is the fact that polarization measurements show significant spin-spin correlations (A^vA^), and the single spin asymmetry {A])/) is not consistent with predictions based on hadron helicity conservation (see Section 6) which is expected to be valid for the leading power behavior. Recent discussions of these effects have been given by Farrar and Lipkin. We discuss a new explanation of all of these effects in Section 9. As emphasized by Landshoff, the ISR data for high energy elastic pp scattering at small \t\/s can be parameterized in the form da/dt ^ const/t^ for 2 GeV^ < \t\ < 10 GeV^. This suggests a role for triple gluon exchange pinch contributions at large energies where multiple vector exchange diagrams could
158 10 -30 _ 10 -31 10 -32 _ > 10 -33 CVJ -1-. 10-^0 10 -31 10 -32 10 -33 10 -34 1 I I I I I i_i_l 1 1 I I I I J L J I I I I I I I I I I J L 10 -30 10 -31 10 -32 10 -33 10 -34 J I I I ' I 10 -31 10 -32 10 -33 S-^15 20 30 40 60 80 s-^15 20 30 40 60 80 S-^15 20 30 40 60 80 10 -34 Figure 23. Test of fixed 6cm scaling for elastic pp scattering. The data compilation is from Landshoff and Polkinghorne. 21 dominate. However, from Mueller's analysis one expects stronger fall-off in t due to the Sudakov form factor suppression. This paradox implies that the role of the pinch singularity in large momentum transfer exclusive reactions is 43 . not well understood and deserve further attention. As discussed in Section 4.5, pinch singularities are also expected to modify the dimensional counting scaling laws for wide-angle scattering, but the change in the exponent of s is small and hard to detect experimentally. However, Ralston and Pire have suggested that the oscillatory behavior in the wide-angle pp scattering amplitude results from interference between the pinch contributions and the ordinary hard-scattering contributions to the pp amplitude. Thus pp scattering may provide a experimental handle on pinch contribution. However it is possible that the oscillations are specific to particular channels, in which case an alternative explanation is necessary. We discuss this further in Section 9. Pinch singularities do not arise in 16 4 form factors, or such photon-induced processes as 77 —> MM, 7* -|- 7 —> M, 7 44 M\...Mp/ at fixed angle, 77 —> BB, ^B —> 7^, etc. 45,46
159 5.4. Exclusive Anti-Proton Proton Annihilation Processes Anti-proton annihilation has a number of important advantages as a probe of QCD in the low energy domain. Exclusive reaction in which complete annihilation of the valance quarks occur {pp —> ii^ 77, (^<^, etc.) necessarily involve impact distances 6_|_ smaller than l/Mp = 5 fm~ since baryon number is exchanged in the ^-channel. There are a number of exclusive and inclusive p reactions which can provide useful constraints on hadron wavefunctions or test novel features of QCD involving both perturbative and nonperturbative dynamics. In several cases {pp —> ?£, pp —> J/t/^, pp -^ 77), complete leading twist (leading power law) predictions are available. These reactions not only probe the subprocesses qqq qqq —> 77, etc., but they also are sensitive to the normalization and shape of the proton distribution amplitude </)p(xi,X2,X3; Q), the basic measure of the proton's three-quark valance wavefunction. The fixed angle scaling laws for the pp channels are: — {pp -> e+e") 2^ -y— /^ ^ {cosOJnpr) — {^pp _, ^^) ^ .^-^ f^ {cos 0, inpr) -^ {pp -> 7M) - -^ p^{cosejnpT) da . . .-^. 1 d^ {PP-.MM) 2. -J- /^^(cos^,£npT) KPt) %{pp^BB)^^^f^\osO,inpT) The angular dependence reflects the structure of the hard-scattering perturbative T// amplitude, which in turn follows from the flavor pattern of the contributing duality diagrams. It is important to note that the leading power-law behavior originates in the minimum three-particle Fock state of the p and p, at least in physical gauge, such as A'^ = 0. Higher Fock states give contributions higher order in 1/s. For pp —> £l this means that initial-state interaction such as one gluon exchange are dynamically suppressed (see Fig. 24). Soft-gluon exchange is suppressed since the incident p or p color neutral wavefunction in the three-parton state with impact
160 operation 6_|_ ^ 0{l/y/s). Hard-gluon exchange is suppressed by powers of cts{s). The absence of a soft initial-state interaction in these reactions is a remarkable consequence of gauge theory, and is quite contrary to normal treatments of initial interactions based on Glauber theory. Figure 24. Analysis of initial-state interactions in PQCD. We will discuss in Section 8.1 another class of exclusive reactions in QCD involving light nuclei, such as pd —> 772 and pd —> n~p which can probe quark and gluon degrees of freedom of the nucleus at surprisingly low energy. We will also discuss the "color transparency" of nuclei in quasi-elastic processes like pA -> U{A - 1).
161 5.5. Additional Tests of Gluon Spin in Exclusive Processes The spin of the gluon can be tested in a wide variety of exclusive processes: (a) 77 —^ /9/9, A'*A'*,.... These cross sections can be measured using e"^e~ colliding beams. At large energies (5^2 — iGeV'^) and wide angles, the final- state helicities must be equal and opposite. These processes can also be used as 1 /? a sensitive probe of the structure of the quark distribution amplitudes. (b) Electroweak form factors of baryons. Relations, valid to all order in a^, can be found among the various electromagnetic and weak-interaction for factors of the nucleons and other baryons. These relations depend crucially upon quark- helicity conservation and as such test the vector nature of the gluon. Current data for the axial-vector and electromagnetic form factors of the nucleons is in excellent agreement with these QCD predictions, although a definitive test requires higher energies. (c) 7rp —^ T^P^PP —* PP^'-" QCD predicts that total hadronic helicity is conserved from the initial state to the final state in all high-energy, wide-angle, elastic, and quasi-elastic hadronic amplitudes. One immediate consequence of this is the suppression of the backward peak relative to the forward peak in scalar-meson- baryon scattering. This follows because angular momentum cannot be conserved along the beam axis if only the baryons carry helicity, helicity is 32 conserved, and the baryons scatter through 180^. Data for irp and Kp scattering is consistent with this observation. However the hard-scattering amplitudes for these processes must be computed before a detailed interpretation of the data is possible. In the case of pp —> pp scattering, there are in general five independent parity- conserving and time-reversal-invariant amplitudes M{-\--\- -^ ++),A^(H— —> +-), A^(-+ -^ +-),A^(++ -> +-), and M{ ^ ++). Total-hadron- helicity conservation implies that M,{-\--\—>• H—) and M,{ >• ++) are power- law suppressed. The vanishing of the double-flip amplitude implies Af^p^ = Ass-) and 2Ann -All = 1 {Oc.m. = 90"). Here A^v^v is the spin asymmetry for incident nucleons polarized normal (x) to the scattering plane. An refers to initial spins polarized along the laboratory beam direction (I) and Ass refers to initial spin polarized (sideways) along y. 48 Data at piab = 11.75 GeV/c from Argonne appears to be consistent with this prediction. (d) Zeros of meson form factors. Asymptotically, the electromagnetic form factors of charged tt's, iC's, and p{\ = 0)'s have a positive sign in QCD. In a theory
162 of scalar gluons, these form factors become negative for Q^ large, and thus must vanish at some finite Q^ since F[Q^ = 0) = 1 by definition. Consequently the absence of zeros in Ft^{Q^^ is further evidence for a vector gluon. We discuss this in detail in the next section. 5.6. Hadronic Wavefunction Phenomenology Let us now return to the question of the normalization of exclusive amplitudes in QCD. It should be emphasized that because of the uncertain magnitude of corrections of higher order in a5((5^), comparisons with the normalization of experiment with model predictions could be misleading. Nevertheless, it this section we shall assume that the leading order normalization is at least approximately accurate. If the higher order corrections are indeed small, then the normalization of the proton form factor at large Q^ is a non-trivial test of the distribution amplitude shape; for example, if the proton wave function has a non-relativistic shape peaked at Xi ~ 1/3 then one obtains the wrong sign for the nucleon form factor. Furthermore symmetrical distribution amplitudes predict a very small magnitude for Q^G^^[Q^) at large Q^. The phenomenology of hadron wavefunctions in QCD is now just beginning. Constraints on the baryon and meson distribution amplitudes have been recently obtained using QCD sum rules and lattice gauge theory. The results are expressed in terms of gauge-invariant moments < x^ >= J Udxi x^ (j){xi^ fi) of the hadron's distribution amplitude. A particularly important challenge is the construction of the baryon distribution amplitude.In the case of the proton form factor, the constants anm in the QCD prediction for Gm must be computed from moments of the nucleon's distribution amplitude (^(x,-, Q). There are now extensive theoretical efforts to compute this nonperturbative input directly from QCD. The QCD sum rule analysis of Chernyak et al. ' provides constraints on the first 12 moments of (t){x,Q). Using as a basis the polynomials which are eigenstates of the nucleon evolution equation, one gets a model representation of the nucleon distribution amplitude, as well as its evolution with the momentum transfer scale. The moments of the proton distribution amplitude computed by Chernyak et al.^ 50 have now been confirmed in an independent analysis by Sachrajda and King. A three-dimensional "snapshot" of the proton's uud wavefunction at equal light-cone time as deduced from QCD sum rules at /i ~ 1 GeV by Chernyak 49 . .50 et al. and King and Sachrajda is shown in Fig. 25. The QCD sum rule analysis predicts a surprising feature: strong flavor asymmetry in the nucleon's momentum distribution. The computed moments of the distribution amplitude imply that 65% of the proton's momentum in its 3-quark valence state is carried by the u-quark which has the same helicity as the parent hadron.
163 A^ Figure 25. The proton distribution amplitude (/>p(a:,-,^) determined at the scale 1 GeV from QCD sum rules. 27 Dziembowski and Mankiewicz have recently shown that the asymmetric form of the CZ distribution amplitude can result from a rotationally-invariant CM wave function transformed to the light cone using free quark dynamics. They find that one can simultaneously fit low energy phenomena (charge radii, magnetic
164 moments, etc.), the measured high momentum transfer hadron form factors, and the CZ distribution amplitudes with a self-consistent ansatz for the quark wave functions. Thus for the first time one has a somewhat complete model for the relativistic three-quark structure of the hadrons. In the model the transverse size of the valence wave function is not found to be significantly smaller than the mean radius of the proton-averaged over all Fock states as argued in Ref. 51. Dziembowski et al. also find that the perturbative QCD contribution to the form factors in their model dominates over the soft contribution (obtained by convoluting the non-perturbative wave functions) at a scale Q jN ^ 1 GeV, where A^ is the number of valence constituents. (This criterion was also derived in Ref. 52.) 53 Gari and Stefanis have developed a model for the nucleon form factors which incorporates the CZ distribution amplitude predictions at high Q^ together with VMD constraints at low Q^. Their analysis predicts sizeable values for the neutron electric form factor at intermediate values of Q^. A detailed phenomenological analysis of the nucleon form factors for different shapes of the distribution amplitudes has been given by Ji, Sill, and Lombard- 54 . Nelsen. Their results show that the CZ wave function is consistent with the sign and magnitude of the proton form factor at large Q^ as recently measured by the American University/SLAC collaboration (see Fig. 26). o 1.6 ^ 1.0 or 0.6 o Previous Data •SLAG E-136 c ?.tVir4^ rWD I i CZ o Q O KD a. Inside Integral mg^=0.3 (GeV/c^)^ 0 10 20 30 q2 [(GeV/c)^J Figure 26. Predictions for the normalization and sign of the proton form factor at high Q^ using perturbative QCD factorization and QCD sum rule predictions for the proton distribution amplitude (from Ref. 54.) The predictions use forms given by Chernyak and Zhitnitsky, King and Sachrajda, and Gari and Stefanis.
165 It should be stressed that the magnitude of the proton forni factor is sensitive to the X ^ 1 dependence of the proton distribution aniplitude, where non- 55 perturbative effects could be important. The asymmetry of the distribution amplitude emphasizes contributions from the large x region. Since non-leading corrections are expected when the quark propagator scale Q^{1 — x) is small, in principle relatively large momentum transfer is required to clearly test the pertur- 49 bative QCD predictions. Chernyak et al. have studied this effect in some detail and claim that their QCD sum rule predictions are not significantly changed when higher moments of the distribution amplitude are included. The moments of distribution amplitudes can also be computed using lattice gauge theory. In the case of the pion distribution amplitudes, there is good agreement of the lattice gauge theory computations of Martinelli and Sachra- 15 jda with the QCD sum rule results. This check has strengthened confidence in the reliability of the QCD sum rule method, although the shape of the meson distribution amplitudes are unexpectedly structured: the pion distribution amplitude is broad and has a dip at x = 1/2. The QCD sum rule meson distributions, combined with the perturbative QCD factorization predictions, account well for the scaling, normalization of the pion form factor and 77 —> M'^M~ cross sections. In the case of the baryon, the asymmetric three-quark distributions are consistent with the normalization of the baryon form factor at large Q^ and also the branching ratio for J/tj; —> pp. The data for large angle Compton scattering IP ""*■ IP ^r^ ^^so well described. However, a very recent lattice calculation of 15 the lowest two moments by Martinelli and Sachrajda does not show skewing of the average fraction of momentum of the valence quarks in the proton. This lattice result is in contradiction to the predictions of the QCD sum rules and does cast some doubt on the validity of the model of the proton distribution proposed by Chernyak et al. The lattice calculation is performed in the quenched approximation with Wilson fermions and requires an extrapolation to the chiral limit. The contribution of soft momentum exchange to the hadron form factors is a potentially serious complication when one uses the QCD sum rule model distribution amplitudes. In the analysis of Ref. 24 it was argued that only about 1% of the proton form factor comes from regions of integration in which 57 all the propagators are hard. A new analysis by Dziembowski et al. shows ■jo 1 z? that the QCD sum rule distribution amplitudes of Chernyak et al. together with the perturbative QCD prediction gives contributions to the form factors which agree with the measured normalization of the pion form factor at Q^ >
166 4 GeV^ and proton form factor Q^ > 20 GeV^ to within a factor of two. In the calculation the virtuality of the exchanged gluon is restricted to |/:^| > 0.25 GeV^. The authors assume as = 0.3 and that the underlying wavefunctions fall off exponentially at the x c^ 1 endpoints. Another model of the proton distribution 58 amplitude with diquark clustering chosen to satisfy the QCD sum rule moments come even closer. Considering the uncertainty in the magnitude of the higher order corrections, one really cannot expect better agreement between the QCD predictions and experiment. The relative importance of non-perturbative contributions to form factors is also an issue. Unfortunately, there is little that can be said until we have a deeper understanding of the end-point behavior of hadronic wavefunctions, and of the role played by Sudakov form factors in the end-point region. Models have been 24 constructed in which non-perturbative effects persist to high Q. Other models have been constructed in which such effects vanish rapidly as Q increases. ' ' If the QCD sum rule results are correct then, the light hadrons are highly structured oscillating momentum-space valence wavefunctions. In the case of mesons, the results from both the lattice calculations and QCD sum rules show that the light quarks are highly relativistic. This gives further indication that while nonrelativistic potential models are useful for enumerating the spectrum of hadrons (because they express the relevant degrees of freedom), they may not be reliable in predicting wave function structure. 5.7. Calculating Th The calculation of hard-scattering diagrams for exclusive processes in QCD becomes increasingly arduous as the number of incident and final parton lines increases. The tree-graph calculations of Tff have been completed for the meson and baryon form factors, as well as for many exclusive two-photon processes such as 77 —> pp for both real and virtual photons and various Compton scattering reactions. Further discussion of the two-photon predictions is given in Section 7. The most efficient computational methods involve two-component spinor techniques where the amplitude itself can be converted to a trace. This method was 59 . . first used by Bjorken and Chen for their calculation of the QED "trident" amplitudes for fiZ —> fi/i/i. It was further developed by the CALKUL group and applied to exclusive processes by Farrar and Gunion and their co-workers. The large number of PQCD tree graph (300,000 for pp scattering) may help to explain the relatively large normalization of the pp amplitude at large momentum transfer. For example the nominal one-gluon exchange amplitude 4i7rCf{s/t)as{t)[Ff{t)]^ gives a contribution only about 10~^ of that required by
167 the large angle pp scattering data. It is clearly necessary to develop highly efficient and autoniatic niethods for evaluating niulti-particle hard scattering amplitudes Tf{ for reactions such as pp scattering. The light-cone quantization method could prove highly effective. In this method one expands the S-matrix in the r-ordered perturbation theory. For numerical computations one can use a discrete basis, such that in each intermediate state one sums over a complete set of discretized Fock states, defined using periodic or anti-periodic boundary conditions. The matrix elements of the light-cone Hamiltonian //"^^e^^ctton ^^^^ simple to compute. In the expansion all Feynman diagrams and all time-orderings are automatically summed. In principle the perturbative QCD predictions can be calculated systematically in powers oi as{Q^). In practice the calculations are formidable, and thus far only the next-to-leading correction to the pion form factor and the 77 —>• tttt amplitude have been systematically studied. The two-photon amplitude analysis is given by Nizic and is discussed further in Section 7. The complete analysis of the meson form factor to this order requires evaluating the one-loop corrections to the hard-scattering amplitude for ^yqq —> qq^ plus a corresponding correction to the kernel for the meson distribution amplitude. The one-loop corrections to T// for the meson form factor have been evaluated by several groups. Because of different conventions the results differ in detail; however Braaten and Tse have resolved the discrepancies between the three previous calculations. An important feature is the presence of correction terms of order j^{^-Ca — |)log[(l ~ ^)(1 ~ y)Q^] which sets the scale of the running coupling constant in the leading order contribution at Qlff = (1 — x)(l — y)Q^. This is consistent with the expectation that the running coupling constant scale is set by the virtuality of the exchanged gluon propagator, just as 'in Abelian QED. This is also consistent with the automatic scale-fixing scheme of Ref. 63. Thus a significant part of the PQCD higher order corrections can be absorbed by taking the natural choice for the argument of the running coupling constant. The next-to-leading correction to the kernel for the meson distribution amplitude has also been evaluated by several groups. A surprising feature of this analysis is the fact that conformal symmetry cannot be used as a guide to predict the form the results even when the /?-function is set to zero. This is discussed in further detail in Section 4.2.
168 5.8. The Pre-QCD Development of Exclusive Reactions The study of exclusive processes in terms of underlying quark subprocesses in fact began before the discovery of QCD. The advent of the parton model and Bjorken scaling for deep inelastic structure functions in the late 1960's brought a new focus to the structure of form factors and exclusive processes at large momentum transfer. The underlying theme of the parton model was the concept that quarks carried the electromagnetic current within hadrons. The use of time-ordered perturbation theory in an "infinite momentum frame", or equiva- lently, quantization on the light cone, provided a natural language for hadrons 64 . as composites of relativistic partons, i.e. point-like constituents. As discussed in Section 3, Drell and Yan introduced Eq. (57) for current matrix elements in terms of a Fock state expansion at infinite momentum. (Later this result was shown to be an exact result using light-cone quantization.) Drell and Yan suggested that the form factor is dominated by the end-point region x ~ 1. Then it is clear from the Drell-Yan formula that the form factor fall-off at large Q^ is closely related to the x —>• 1 behavior of the hadron structure function. The relation found by Drell and Yan was ^(<3')~77T^ if F2(x,Q^)~(l-x) 65 Gribov and Lipatov extended this relationship to fragmentation functions D{z,Q^) at 2: —> 1, taking into account cancellations due to quark spin. Feyn- man noted that the Drell-Yan relationship was also true in gauge theory models in which the endpoint behavior of structure functions is suppressed due to the emission of soft or "wee" partons by charged Hnes. However, as discussed in Section 4, the endpoint region is suppressed in QCD relative to the leading per- turbative contributions. The parton model was extended to exclusive processes such as hadron-hadron scattering and photoproduction by Blankenbecler, Brodsky, and Gunion and fifi by Landshoff and Polkinghorne. It was recognized that independent of specific dynamics, hadrons could interact and scatter simply by exchanging their common constituents. These authors showed that the amplitude due to quark interchange (or rearrangement) could be written in closed form as an overlap of the light- cone wavefunctions of the incident and final hadrons. In order to make definite predictions, model wavefunctions were chosen to reproduce the fall-off of the form factors obtained from the Drell-Yan formula. Two-body exclusive amplitudes in
169 the "constituent interchange niodel" then take the forni of "fixed-angle" scahng laws where the power N reflects the power-law fall-off of the elastic form factors of the scattered hadrons. The form of the angular dependence f(Ocm) reflects the number of interchanged quarks. Even though the constituent interchange is model was motivated in part by the Drell-Yan endpoint analysis of form factors, many of the predictions and systematics of quark interchange remain applicable in the QCD analysis. A comprehensive series of measurements of elastic meson nucleon scattering reac- 69 tions has recently been carried out by Bailer et al. at BNL. Empirically, the quark interchange amplitudes gives a reasonable account of the scaling, angular dependence, and relative magnitudes of the various channels. For example, the strong differences between K^p and K~p scattering is accounted for by u quark interchange in the K^p amplitude. It is inconsistent with gluon exchange as the dominant amplitude since this produces equal scattering for the two channels. The dominance of quark interchange over gluon exchange is a surprising result which eventually needs to be understood in the context of QCD. The prediction of fixed angle scaling laws laid the groundwork for the derivation of the "dimensional counting rules." As discussed in Ref. 5, it is natural to assume that at large momentum transfer, an exclusive amplitude factorize as a convolution of hadron wavefunctions which couple the hadrons to their quark constituents with a hard scattering amplitude T// which scatters the quarks from the initial to final direction. Since the hadron wavefunction is maximal when the quarks are nearly collinear with each parent hadron, the large momentum transfer occurs in T}j. The pre-QCD argument went as follows: the dimension of T// is j^n-4j ^j^ere ^^ — ^^^ _|. ^^^ _|. ^^^ _j. t^^ jg ^^j^^ total number of fields entering T//. In a renormalizable theory where the coupling constant is dimensionless and masses can be neglected at large momentum transfer, all connected tree-graphs for T// then scale as [^l\fs\^~^ at fixed tjs. This immediately gives the dimensional counting law dl^ ' ^nA-fns-fnc-fWD —2' In the case of incident or final photons or leptons n = 1. Specializing to elastic lepton-hadron scattering, this also implies F{Q^) ~ l/(Q^)^"~^ for the spin averaged form factor, where Ufj is the number of constituents in hadron H. These
170 5 results were obtained independently by Matveev et al. on the basis of an "auto- niodality" principle, that the underlying constituent interactions are scale free. As we have seen, the diniensional counting scaling laws will generally be niodified by the accuniulation of logarithnis froni higher loop corrections to the hard scattering aniplitude T//; the phenonienological success of the counting rules in their simplest form thus implies that the loop corrections be somewhat mild. As we have seen, it is the asymptotic freedom property of QCD which in fact makes higher order corrections an exponentiation of a log log Q"^ series, thus preserving the form of the dimensional counting rules modulo only logarithmic corrections. 6. EXCLUSIVE e+e- ANNIHILATION PROCESSES The study of time-like hadronic form factors using e'^e" colliding beams can provide very sensitive tests of the QCD helicity selection rule. This follows because the virtual photon in e"'"e~ —> 7* —>• Ha^b always has spin ±1 along the beam axis at high energies. Angular-momentum conservation impHes that the virtual photon can "decay" with one of only two possible angular distributions in the center-of-momentum frame: (l-f-cos^^) for | \a ~ ^B |— 1, and sin^^ for ^A ~ ^B |— 0, where \a,b are the helicities of hadron Ha^b- Hadronic-helicity conservation, Eq. (7), as required by QCD greatly restricts the possibilities. It implies that \a-{- \b = 2A^ = —2A5, Consequently, angular-momentum conservation requires | A^ | = | A5 |= ^ for baryons and | A>i | = | A5 |= 0 for mesons; and the angular distributions are now completely determined: (e"'"e -^ BB) oc 1 -|- cos^ ^(baryons). dcosO d(J 4. _ — . 2 ^(e e —► MM) oc sin ^(mesons). a cos 0 It should be emphasized that these predictions are far from trivial for vector mesons and for all baryons. For example, one expects distributions like sin 6 for baryon pairs in theories with a scalar or tensor gluon. Simply verifying these angular distributions would give strong evidence in favor of a vector gluon. #15 This follows from helicity conservation as well, which is a well-known property of QED at high energies. The electron and positron must have opposite helicities; i.e. je+lJ = 0, since it is the total helicity carried by fermions (alone) which is conserved, and there are no fermions in the intermediate state. In the laboratory frame (—* Pe = *■ Pf), their spins must be parallel, resulting in a virtual photon with spin ±1 along the beam.
171 The power-law dependence on s of these cross sections is also predicted in QCD, using the dimensional-counting rule. Such "all-orders" predictions for QCD allowed processes are summarized in Table II. ' Processes suppressed in QCD are also listed there; these all violate hadronic-helicity conservation, and are suppressed by powers of m?/s in QCD. This would not necessarily be the case in scalar or tensor theories. Table II Exclusive channels in e'^e" annihilation. The hj\hBY couplings in allowed processes are -ie{pj\ — pfl)''F(s) for mesons, -iev{pB)'r'*G{s)u{pA) for baryons, and -ie^tft„pp'j^e^p^FMy{s) for meson-photon final states. Similar predictions apply to decays of heavy-quark vector states, such as ip,ip\..., produced in e'^e~ collisions. Allowed in QCD Suppressed in QCD e+e -^ /i^(A^)/ifl(Afl) e'^e" —* TT+TT",/{'■'"/<'" e+e--»^+^-(0),/r+/C- e'^e" —* 7r^7(±l),;/7,;/'7 e+e" -^p{±^)p{T^),nn,... c+e" -^ p(±5 JA{t\ ),nA... e+e--.A(±i)A(T^),yV,--- e+e- -^p+{0)p-{±l),ic+p-,K-^ir-,... e+e- -^p+{±l)p-{±l),... e+e- -» p(±i )p(±i),pA, AA,... e+e- -»p(±i)A(±|),AA,... e+e--.A(±|)A(±|),... Angular distribution sin^^ sin^^ 1 + cos^ e 1 + cos^ e 1 + cos^ e 1 + cos^ e 1 + cos^ e sin^e sin2e 1 + cos^ 0 sin^e i|F(5)p^c/52 i |F(.)|2 ~ c/s^ (7ra/2)5 |Fa/^(5)|2 - c/s \G{s)\'^^c/s* \Gis)\^^c/s* |G(s)|2~c/54 < c/s^ < c/s^ <c/s^ < c/s^ < c/s^ All of these perturbative predictions assume that 5 is sufficiently far from resonance contributions. Notice the e"^e 7r/9, TTCJ,/<'/<'*,..., are all suppressed in QCD. This oc- Xp 1= 1 in e"'"e collisions. curs because the 7 — tt — /? can couple through only a single form factor - g/ii.r(T^(7)^(^p)^^7r)^(p)^^^^^^ — and this requires Hadronic-helicity conservation requires A = 0 for mesons, and thus these amplitudes are suppressed in QCD (although, again, not in scalar or tensor theories). Notice however that the processes e"'"e~ —*■ 77r,777,77;' are allowed by the helicity selection rule; helicity conservation applies only to the hadrons. Unfortunately the form factors governing these last processes are not expected to be large, e.g.
172 These form factors can also tell us about the quark distribution amplitudes 4>H{^iy Q)' For example sum rules require (to all orders in as) that tt'^tt", A'"^A'~, and p'^p~ (helicity-zero) pairs are produced in the ratio of ft '• ft: '- ^fp ~ 1:2:7, respectively if the tt, K, and p distribution amplitudes are of similar shape. These ratios must apply at very large energies, where all distribution amplitudes tend to (f) oc x{l — x). On the other hand, the kaon's distribution amplitude may be quite asymmetric about x = 2 ^^ low energies due to the large difference between 5 and u,c? quark masses. This could enhance K^K~ production. (Distribution amplitudes for tt's and /9's must be symmetric due to isospin.) The process e^e~ —> KlI^S is only possible if the kaon distribution amplitude is asymmetric; the presence or absence of KlKs pairs relative to K^K~ pairs is thus a sensitive indicator of asymmetry in the wave function. 6.1. J I'll) Decay to Hadron Pairs The exclusive decays of heavy-quark atoms (J/V', V'', •••) into light hadrons can also be analyzed in QCD. The decay V' -^ PP-, for example, proceeds via diagrams such as those in Fig. 27. Since 0's produced in e"^e~ collisions must also have spin ±1 along the beam direction and since they can only couple to light quarks via gluons, all the properties listed in Table II apply to 0, ?/>', T, T',... decays as well. Already there is considerable experimental data for the V' and il)' , 72,73 decays. Figure 27. Quark-gluon subprocesses for V* —^ BB. ij^\% For example, this amplitude vanishes under the (stronger) assumption of exact flavor- 5C/(3) symmetry. This is easily seen by defining Gu parity, in analogy to G parity: Gu — Cexp(i7rC/2), where the Ui are the isospin-like generators of 5C/(3)/ which connect the Kq and Kq. The final state in e'^e" —»• KlKs has positive Gu parity, while the intermediate photon has negative Gu parity. Gu parity is conserved if SU(Z)j is exact, and e'^e" —»^ K^Ks then vanishes.
173 Perhaps the most significant are the decays t/', i/'' —^ pp-, nn,.... The predicted angular distribution 1+ cos^ is consistent with published data. This is important evidence favoring a vector gluon, since scalar- or tensor-gluon theories would predict a distribution of sin^^H-0(a5). Dimensional-counting rules can be checked by comparing the V' and tp' rates into pp, normalized by the total rates into light-quark hadrons so as to remove dependence upon the heavy-quark wave functions. Theory predicts that the ratio of branching fractions for the pp decays of the 0 and xj;' is B{4>' ^ pp) Q (m^ " B{^ - PP) ~ ^'''" V % where Qe+e- is the ratio of branching fractions into e'^e~\ Q-'-- = -^nTi ^ = 0.135 ±0.023 . n(J/xp —> e+e ) Existing data suggest a ratio [M^ijM^Y ^^^^ n = 6 ± 3, in good agreement with QCD. One can also use the data for xj) —> pp, AA,E!E!,..., to estimate the relative magnitudes of the quark distribution amplitudes for baryons. Correcting for phase space, one obtains (/>p ^ 1.04(13) (/>n "^ 0.82(5) (j)'= ^ 1.08(8) (j)^. "^ 1.14(5) <Pa by assuming similar functional dependence on the quark momentum fractions Xi for each case. As is well known, the decay 0 —> tt'^'tt" must be electromagnetic if G-parity is conserved by the strong interactions. To leading order in as, the decay is through a virtual photon (i.e. 0 —> 7* —► tt'^'tt") and the rate is determined by the pion's electromagnetic form factor: where s = (3.1GeV)^. Taking FttIs) c^ (1 —s/m^p)~^ gives a rate r(0 —> 7r"^7r~) ^ 0.0011 T{ip —> ^"^/z"), which compares well with the measured ratio 0.0015(7). This indicates that there is indeed little asymmetry in the pion's wave function. The same analysis applied to ip -^ K^K~ suggests that the kaon's wave function is nearly symmetric about x — ^. The ratio r(0 —► K'^K~)/r{ip —> 7r"''7r~) is 2 ± 1, which agrees with the ratio [fKlfi^Y ^ ^ expected if tt and K have similar quark distribution amplitudes. This conclusion is further supported by measurements of 0 —> Kl^^S which vanishes completely if the K distribution amplitudes are symmetric; experimentally the limit is r(0 —► KiKs)/^{'^ —^
174 6.2. The -K-p Puzzle We have emphasized that a central prediction of perturbative QCD for exclusive processes is hadron helicity conservation: to leading order in l/Q, the total helicity of hadrons in the initial state must equal the total helicity of hadrons in the final state. This selection rule is independent of any photon or lepton spin appearing in the process. The result follows from (a) neglecting quark mass terms, (b) the vector coupling of gauge particles, and (c) the dominance of valence Fock states with zero angular momentum projection. The result is true in each order of perturbation theory in OLg. Hadron helicity conservation appears relevant to a puzzling anomaly in the exclusive decays Jj^l) and xj)' —^ pir^ K*K and possibly other Vector-Pseudoscalar (VP) combinations. One expects the J/xl^ and 0' mesons to decay to hadrons via three gluons or, occasionally, via a single direct photon. In either case the decay proceeds via |^(0)|^, where ^(0) is the wave function at the origin in the nonrelativistic quark model for cc. Thus it is reasonable to expect on the basis of perturbative QCD that for any final hadronic state h that the branching fractions scale like the branching fractions into e'^e~: Usually this is true, as is well documented in Ref. 74 for pp7r^, 27r"''27r tt^, tt'^'tt'o;, and Stt'^'Stt'tt^, hadronic channels. The startling exceptions occur for pir — . . . 74 and K*K where the present experimental limits are Qp^ < 0.0063 and Qj^'*j(< 0.0027. Perturbative QCD quark helicity conservation implies Qp^ = [B(ip' —> p-K)IB[JI%!) —> p'k)] < Qe+ e-[^j/ii>/^tj;']^ This result includes a form factor suppression proportional to [Mji^jM^i]^ and an additional two powers of the mass ratio due to helicity flip. However, this suppression is not nearly large enough to account for the data. From the standpoint of perturbative QCD, the observed suppression of ij;' —> 75 V P is to be expected; it is the JIxjp that is anomalous. The xjp' obeys the perturbative QCD theorem that total hadron helicity is conserved in high-momentum #17 There is the possibility is the these form factors are dominated by end-point contributions for which quark masses may be less relevant. Such terms are expected to be strongly suppressed by quickly falling Sudakov form factors. This could also explain the rapid falloff of the ^ — tt — /> form factor with increasing Mj.
175 transfer exclusive processes. The general validity of the QCD helicity conservation theorem at charnrionium energies is of course open to question. An alternative 7fi model based on nonperturbative exponential vertex functions, has recently been proposed to account for the anomalous exclusive decays of the J/^. However, helicity conservation has received important confirmation in J/0 —> pp where the angular distribution is known experimentally to follow [1 -|- cos^ 0] rather than sin 0 for helicity flip, so the decays J/'tp —> 7r/9, and KK seem truly exceptional. The helicity conservation theorem follows from the assumption of short-range point-like interactions among the constituents in a hard subprocess. One way in which the theorem might fail for J/xj^ —> gluons —> Trp is if the intermediate gluons resonate to form a gluonium state O. If such a state exists, has a mass near that of the J/ipy and is relatively stable, then the subprocess for J/ip —> Trp occurs over large distances and the helicity conservation theorem need no longer apply. This would also explain why the J/ip decays into Trp and not the 0'. 75 . .77 Tuan et al. have thus proposed, following Hou and Soni, that the enhancement of J/ip —> K*K and J/'tp —> pTr decay modes is caused by a quantum mechanical mixing of the J/ip with a J^^ = 1 vector gluonium state O which causes the breakdown of the QCD helicity theorem. The decay width for J/ip —> pTr{K*K) via the sequence J/0 —> O —> pTr{K*K) must be substantially larger than the decay width for the (non-pole) continuum process J/0 —> 3 gluons —> pTr{K*K). In the other channels (such as pp,pp7r^,27r"''27r~7r^, etc.), the branching ratios of the O must be so small that the continuum contribution governed by the QCD theorem dominates over that of the O pole. For the case of the 0' the contribution of the O pole must always be inappreciable in comparison with the continuum process where the QCD theorem holds. The experimental limits on Qpjr and Qj^^j^ are now substantially more stringent than when Hou and Soni made their estimates of M(p, Tc-^pv and ^c)_j(*~^ in 1982. 78 A gluonium state of this type was first postulated by Freund and Nambu based on OZI dynamics soon after the discovery of the J/0 and 0' mesons. In fact, Freund and Nambu predicted that the O would decay copiously precisely into pTT and K*K with severe suppression of decays into other modes like e'^e~ as required for the solution of the puzzle. Branching fractions for final states h which can proceed only through the intermediate gluonium state have the ratio: ^' - ^'''- {Mr - MoY+\n' It is assumed that the coupling of the J/0 and 0' to the gluonium state scales
176 as the e^e coupling. The value of Q^ is small if the O is close in mass to the J/0. Thus one requires [Mjj^ — Mc))^-\-\ F^ ;S 2.6 Qh GeV . The experimental 1 In hmit for Q^-*^ then implies [(Mj/^ - Mq)^ + \ F^] ;S 80 MeV. This implies Mji^ — Mo \< 80 MeV and To < 160 MeV. Typical allowed values are Mo = 3.0 GeV, To = 140 MeV or Mo = 3.15 GeV, To = 140 MeV. Notice that the gluonium state could be either lighter or heavier than the J/0. The branching ratio of the O into a given channel must exceed that of the J/0. It is not necessarily obvious that a J^^ — 1 gluonium state with these parameters would necessarily have been found in experiments to date. One must remember that though O -^ p-K and O —> K*K are important modes of decay, at a mass of order 3.1 GeV many other modes (albeit less important) are available. Hence, a total width To — 100 to 150 MeV is quite conceivable. Because of the proximity of Mo to Mjj^^ the most important signatures for an O search via exclusive modes J/0 —> K*Kh, J/0 —> p7rh\ h — 7nr,r),r)\ are no longer available by phase-space considerations. However, the search could still be carried out using 0' —> K*Kh^ 0' —> /97r/i; with h — tttt, and rj. Another way to search for O in particular, and the three-gluon bound states in general, is via the inclusive reaction 0' —> (tttt) -|- X, where the tttt pair is an isosinglet. The three-gluon bound states such as O should show up as peaks in the missing mass (i.e. mass of X) distribution. The most direct way to search for the O is to scan pp or e'^e~ annihilation at y/s within ~ 100 MeV of the J/0, triggering on vector/pseudoscalar decays such as TTp or KK*. The fact that the pTT and K*K channels are strongly suppressed in 0' decays but not in J/0 decays clearly implies dynamics beyond the standard charmonium analysis. The hypothesis of a three-gluon state O with mass within = 100 MeV of the J/0 mass provides a natural, perhaps even compelling, explanation of this anomaly. If this description is correct, then the ip' and J/0 hadronic decays not only confirm hadron helicity conservation (at the 0' momentum scale), but they also provide a signal for bound gluonic matter in QCD. 6.3. Form Factor Zeros in QCD The exclusive pair production of heavy hadrons |(5i(52)' \Q1Q2Q3) consisting of higher generation quarks {Qi = t, b,c, and possibly s) can be reliably predicted within the framework of perturbative QCD, since the required wavefunction input is essentially determined from nonrelativistic considerations. The results can be applied to e'^e~ annihilation, 77 annihilation, and W and Z decay into higher generation pairs. The normalization, angular dependence and helicity structure
177 can be predicted away from threshold, allowing a detailed study of the basic elements of heavy quark hadronization. A particularly striking feature of the QCD predictions is the existence of a zero in the form factor and e'^e~ annihilation cross section for zero-helicity hadron pair production close to the specific timelike value q^/iMJj — m}i/2m£ where rrih and m£ are the heavier and lighter quark masses, respectively. This zero reflects the destructive interference between the spin-dependent and spin-independent (Coulomb exchange) couplings of the gluon in QCD. In fact, all pseudoscalar meson form factors are predicted in QCD to reverse sign from spacelike to timelike asymptotic momentum transfer because of their essentially monopole form. For ruh > 2mi the form factor zero occurs in the physical region. To leading order in 1/?^, the production amplitude for hadron pair production is given by the factorized form where [dxi] = 6 (X^^=i ^k ~ 0 nib=i ^'^k ^^^ n = 2,3 is the number of quarks in the valence Fock state. The scale q^ is set from higher order calculations, but it reflects the minimum momentum transfer in the process. The main dynamical dependence of the form factor is controlled by the hard scattering amplitude Tjj which is computed by replacing each hadron by coUinear constituents P-^ — XiPj^. Since the coUinear divergences are summed in <^/f, T// can be systematically computed as a perturbation expansion in as{q ). The distribution amplitude required for heavy hadron production (t>H{^iiQ^) is computed as an integral of the valence light-cone Fock wavefunction up to the scale Q^. For the case of heavy quark bound states, one can assume that the constituents are sufficiently non-relativistic that gluon emission, higher Fock states, and retardation of the effective potential can be neglected. The analysis of Section 2 is thus relevant. The quark distributions are then controlled by a simple nonrelativistic wavefunction, which can be taken in the model form: tpM{xi,k_ii) = C r2r2 A/f2 ^Jl + ^1 ^±2 + ^2 ^1^2 ^^^H ~ Xi ~ X2 This form is chosen since it coincides with the usual Schrodinger- Coulomb wave- function in the nonrelativistic limit for hydrogenic atoms and has the correct
178 large momentum behavior induced from the spin- independent gluon couplings. The wavefunction is peaked at the mass ratio x, = m,7M//: mi V C'l) where (^kl) is evaluated in the rest frame. Normalizing the wavefunction to unit probability gives C^ = 1287r {{v^)f^ml(mi + ms) where (y^ is the mean square relative velocity and rrir — mi7712/(mi -|- m2) is the reduced mass. The corresponding distribution amplitude is (i)[xi) = r^ IGtt^ \x\X2M'^ — X2m\ — xim^ 1 7^^^ six - ^^ \/27r mJ/^ V ^ mi -I- m2 ti It is easy to see from the structure of Th for e^e~ —> MM that the spectator quark pair is produced with momentum transfer squared q^Xsljs — 4m^. Thus heavy hadron pair production is dominated by diagrams in which the primary coupling of the virtual photon is to the heavier quark pair. The perturbative predictions are thus expected to be accurate even near threshold to leading order in a5(4m|) where mi is the mass of lighter quark in the meson. The leading order t^t~ production helicity amplitudes for higher generation meson (A = 0,±1) and baryon (A = ±1/2, ±3/2) pairs are computed in Ref. 79 as a function of (^ and the quark masses. The analysis is simplified by using the peaked form of the distribution amplitude, Eq. (6). In the case of meson pairs the (unpolarized) e"^e~ annihilation cross section has the general form #18 Fy^{(^) is the form factor for the production of two mesons which have both spin and helicity (Z-component of spin) as A and A respectively. There are two Lorentz and gauge invariant form factors of vector pair production. However, one of them turns out to be the same as the form factor of pseudoscalar plus vector production multiplied by M//. Therefore the differential cross section for the production of two mesons with spin 0 or 1 can be represented in terms of three independent form factors.
179 d(7 . t _ 3 MxMx) = -Ba 4 e+e~ + «- /i-r/i X 1 l^o,o(9')P + -i^r^ \ (3 - 2/?' + 3/?^)|Fi,i(9')P 2m2 -4(l+r)Re(Fi,i(9^)Fo*,i(<7'))+4|Fo,i(9^)| + 3/? 2m2 2(1 - /?2) (l+cos^^)|Fo,i(9^)| 2t:2 where ^-^ = 5 = AMf^q^ and the meson velocity is /? = 1 2^. The production form factors have the general form F _ _ {-') XX ^-2)2 (^XX + f B,j) where A and B reflect the Coulomb-like and transverse gluon couplings, respectively. The results to leading order in as are given in Ref. 79. In general A and B have a slow logarithmic dependence due to the ^^-evolution of the distribution amplitudes. The form factor zero for the case of pseudoscalar pair production reflects the numerator structure of the Tjj amplitude. Numerator "^ ei [ q m 1 4M? 1 m XI H ^22/1 4M? H ^22/2 For the peaked wavefunction, Fo%') 1 oc if)' ei I q'^ mi 2rn' -^e2[f 1712 2m 1 771 771 1 If 7771 is much greater than 7712 then the ei is dominant and changes sign at q^/4:MJj = m,i/2m,2. The contribution of the 62 term and higher order contributions are small and nearly constant in the region where the ei term changes sign; such contributions can displace slightly but not remove the form factor zero.
180 These results also hold in quantum electrodynamics; e.g. pair production of muonium (/x — e) atoms in e_).e_ annihilation. Gauge theory predicts a zero at ^2 — m^/2me. These explicit results for form factors also show that the onset of the leading power-law scaling of a form factor is controlled by the ratio of the A and B terms; i.e. when the transverse contributions exceed the Coulomb mass-dominated contributions. The Coulomb contribution to the form factor can also be computed directly from the convolution of the initial and final wavefunctions. Thus, contrary to the claim of Ref. 24 there are no extra factors of Ois{q^) which suppress the "hard" versus nonperturbative contributions. The form factors for the heavy hadrons are normalized by the constraint that the Coulomb contribution to the form factor equals the total hadronic charge at q = 0. Further, by the correspondence principle, the form factor should agree with the standard non-relativistic calculation at small momentum transfer. All of these constraints are satisfied by the form M/.2X I67' (Mj^Vf. q^ 2m2 At large q^ the form factor can also be written as 1 0 where /m = (67^/7rM//)^/^ is the meson decay constant. Detailed results for FF and BcBc production are give in Ref. 79. At low relative velocity of the hadron pair one also expects resonance contributions to the form factors. For these heavy systems such resonances could be related to qqqq bound states. From Watson's theorem, one expects any resonance structure to introduce a final-state phase factor, but not destroy the zero of the underlying QCD prediction. Analogous calculations of the baryon form factor, retaining the constituent mass structure have also been done. The numerator structure for spin 1/2 baryons has the form A -\- Bq^ -I- cq^ . Thus it is possible to have two form factor zeros; e.g. at spacelike and timelike values of q^.
181 Although the measurements are difficult and require large luminosity, the observation of the striking zero structure predicted by QCD would provide a unique test of the theory and its applicability to exclusive processes. The onset of leading power behavior is controlled simply by the mass parameters of the theory. 7. EXCLUSIVE 77 REACTIONS Two-photon reactions have a number of unique features which are especially important for testing QCD, especially in exclusive channels: 1. Any even charge conjugation hadronic state can be created in the annihilation of two photons—an initial state of minimum complexity. Because 77 annihilation is complete, there are no spectator hadrons to confuse resonance analyses. Thus, one hcis a clean environment for identifying the exotic color-singlet even C composites of quarks and gluons \qq >, l^'^' >, \999 >^ \^^9 >? k^^ >,... which are expected to be present in the few GeV mass range. (Because of mixing, the actual mass eigenstates of QCD may be complicated admixtures of the various Fock components.) 2. The mass and polarization of each of the incident virtual photons can be continuously varied, allowing highly detailed tests of theory. Because a spin-one state cannot couple to two on-shell photons, a J = 1 resonance can be uniquely identified by the onset of its production with increasing photon mass. 3. Two-photon physics plays an especially important role in probing dynamical mechanisms. In the low momentum transfer domain, 77 reactions such as the total annihilation cross section and exclusive vector meson pair production can give important insights into the nature of diffractive reactions in QCD. Photons in QCD couple directly to the quark currents at any resolution scale (see Fig. 28). Predictions for high momentum transfer 77 reactions, including the photon structure functions, ^2^(^,(5^) and F2{x^Q )^ high pT jet production, and exclusive channels are thus much more specific than corresponding hadron-induced reactions. The pointlike coupling of the annihilating photons leads to a host of special features which differ markedly with predictions based on vector meson dominance models. 4. Exclusive 77 processes provide a window for viewing the wavefunctions of hadrons in terms of their quark and gluon degrees of freedom. In the case of 77 annihilation into hadron pairs, the angular distribution of the production cross section directly reflects the shape of the distribution amplitude (valence wavefunction) of each hadron.
182 hadrons q - y* Figure 28. Photon-photon annihilation in QCD. The photons couple directly to one or two quark currents. Thus far experiment has not been sufficiently precise to measure the logarithmic modification of dimensional counting rules predicted by QCD. Perturbative QCD predictions for 77 exclusive processes at high momentum transfer and high invariant pair mass provide some of the most severe tests of the theory. A simple, 4.0 but still very important example is the Q -dependence of the reaction 7*7 —> M where M is a pseudoscalar meson such as the rj. The invariant amplitude contains only one form factor: It is easy to see from power counting at large Q^ that the dominant amplitude (in light-cone gauge) gives F-ytjiQ^) ~ ^/Q^ ^-nd arises from diagrams (see Fig. 29) which have the minimum path carrying Q^: i.e. diagrams in which there is only a single quark propagator between the two photons. The coefficient of l/Q^ involves only the two-particle qq distribution amplitude <t>{x,Q), which evolves logarithmically on Q. Higher particle number Fock states give higher power-law falloff contributions to the exclusive amplitude. 83 The TPC/77 data shown in Fig. 30 are in striking agreement with the predicted QCD power: a fit to the data gives Fjrf(Q^) ^ [XjQ^Y ^^^^ ^ — 1.05 ±0.15. Data for the t]' from Pluto and the TPC/77 experiments give similar results, consistent with scale-free behavior of the QCD quark propagator and the point coupling to the quark current for both the real and virtual photons. In the case of deep inelastic lepton scattering, the observation of Bjorken scaling tests these properties when both photons are virtual. The QCD power law prediction, F^^((5^) ~ l/Q^? ^^ consistent with dimen- 5 sional counting and also emerges from current algebra arguments (when both
183 1/Q 11 1/Q 11 Figure 29. Calculation of the 7 - r; transition form factor in QCD from the valence qq and qqg Fock states. p Form Factor (|> Form Factor PQCD prediction lO^P 12 3 4 Q2 (GeV2/c2) 5 0 -. CM 1 2 CM U- 0 12 3 4 Q2 (GeV2/c2) 5 83 Figure 30. Comparison of TPC/77 data"" for the 7 - 77 and y - rj' transition form factors with the QCD leading twist prediction of Ref. 82. The VMD predictions are also shown. See S. Yellin, this meeting. 84 photons are very virtual). On the other hand, the 1/Q^ fallofF is also expected in vector meson dominance models. The QCD and VDM predictions can be readily discriminated by studying 7*7* -> rj. In VMD one expects a product of form factors; in QCD the fallofF of the amplitude is still 1/Q^ where Q^ is a linear combination of Ql and Ql. It is clearly very important to test this essential feature of QCD. Exclusive two-body processes 77 —> HH at large s = W^^ = (^1 + ^2)^ and fixed 62m provide a particularly important laboratory for testing QCD, since the
184 large momentum-transfer behavior, helicity structure, and often even the absolute normalization can be rigorously predicted. ' The angular dependence of some of the 77 —> HH cross sections reflects the shape of the hadron distribution amplitudes <f>H{^ii Q)- The 7a7A' -^ HH amplitude can be written as a factorized form 1 Mxx'iW^-r.Ocm) = [dyi]<f>*H{xi,Q)^{yi,Q)Txx'{x,y]Wy^,9cm) 0 where Txx' is the hard scattering helicity amplitude. To leading order T oc a(as/W^^)^ and da/dt ^ W^-/ " /(^cm) where n = 1 for meson and n = 2 for baryon pairs. Lowest order predictions for pseudo-scalar and vector-meson pairs for each helicity amplitude are given in Ref. 82. In each case the helicities of the hadron pairs are equal and opposite to leading order in 1/W^. The normalization and angular dependence of the leading order predictions for 77 annihilation into charged meson pairs are almost model independent; i.e. they are insensitive to the precise form of the meson distribution amplitude. If the meson distribution amplitudes is symmetric in x and (1 — x), then the same quantity 1 dx (l-x) 0 controls the ^-integration for both FTr{Q ) and to high accuracy M(77 —> 7r"*"7r ). Thus for charged pion pairs one obtains the relation: Note that in the case of charged kaon pairs, the asymmetry of the distribution amplitude may give a small correction to this relation. The scaling behavior, angular behavior, and normalization of the 77 exclusive pair production reactions are nontrivial predictions of QCD. Recent Mark II 85 meson pair data and PEP4/PEP9 data for separated 7r"*"7r~ and K^K~ production in the range 1.6 < VK-y-y < 3.2 GeV near 90° are in satisfactory agreement with the normalization and energy dependence predicted by QCD (see Fig. 31). In the case of ttOttO production, the cos ^cm dependence of the cross section can be inverted to determine the x-dependence of the pion distribution amplitude.
185 The wavefunction of hadrons containing light and heavy quarks such as the K, D-meson are likely to be asymmetric due to the disparity of the quark masses. In a gauge theory one expects that the wavefunction is maximum when the quarks have zero relative velocity; this corresponds to x, oc m^x where m^ = k\^-{- rn?. An explicit model for the skewing of the meson distribution amplitudes based on QCD sum rules is given by Benyayoun and Chernyak. These authors also apply their model to two-photon exclusive processes such as 77 —> K'^K~ and obtain some modification compared to the strictly symmetric distribution amplitudes. If the same conventions are used to label the quark lines, the calculations of Benyayoun and Chernyak are in complete agreement with those of Ref. 82. The one-loop corrections to the hard scattering amplitude for meson pairs have been calculated by Nizic. The QCD predictions for mesons containing 56 admixtures of the \gg) Fock state is given by Atkinson, Sucher, and Tsokos. The perturbative QCD analysis has been extended to baryon-pair production in comprehensive analyses by Farrar et al. ' and by Gunion et al. ' Predictions are given for the "sideways" Compton process 77 —> pp, AA pair production, and the entire decuplet set of baryon pair states. The arduous calculation of 280 77 —> QQQW^ diagrams in T// required for calculating 77 —> BB is greatly simplified by using two-component spinor techniques. The doubly charged A pair is predicted to have a fairly small normalization. Experimentally such resonance pairs may be difficult to identify under the continuum background. The normalization and angular distribution of the QCD predictions for proton- antiproton production shown in Fig. 32 depend in detail on the form of the nucleon distribution amplitude, and thus provide severe tests of the model form 49 derived by Chernyak, Ogloblin, and Zhitnitsky from QCD sum rules. An important check of the QCD predictions can be obtained by combining data from 77 —> pp and the annihilation reaction, pp —> 77, with large angle 87 Compton scattering 7p —► 7p. The available data for large angle Compton scattering (see Fig. 33). for 5 GeV^ < 5 < 10 GeV^ are consistent with the dimensional counting scaling prediction, s^da/dt = f{Ocm)- In general, comparisons between channels related by crossing of the Mandelstam variables place a severe constraint on the angular dependence and analytic form of the underlying QCD exclusive amplitude. Furthermore in pp collisions one can study timelike photon production into e"*"e~ and examine the virtual photon mass dependence of the Compton amplitude. Predictions for the q^ dependence of the pp —► 77* amplitude can be obtained by crossing the results of Gunion and Millers. The region of applicability of the leading power-law predictions for 77 —>
186 I I t 10 10 10 0 -2 10 10 0 10 -I T T 1 r = tfc TT+TT" data : 1 |cos0l^O.3 ' 1 — Brodsky a Lepage I0-' y 0^ ^ K^'K" data cos^l^ 0.6 Brodsky ft Lepage .5 2.0 2.5 M (GeV/c2) 3.0 3.5 Figure 31. Comparison of 77 —> tt+tt" and 77 —* K^K~ meson pair production data with the parameter-free perturbative QCD prediction of Ref. 82. The theory predicts the normalization and scaUng of the cross sections. The data are from the S5 TPC/77 collaboration. pp requires that one be beyond resonance or threshold effects. It presumably is set by the scale where Q^Gm{Q^) is roughly constant, i.e. Q^ > 3 GeV^. 88 Present measurements may thus be too close to threshold for meaningful tests. It should be noted that unlike the case for charged meson pair production, the QCD predictions for baryons are sensitive to the form of the running coupling constant and the endpoint behavior of the wavefunctions. The QCD predictions for 77 —> HH can be extended to the case of one or two virtual photons, for measurements in which one or both electrons are tagged. Because of the direct coupling of the photons to the quarks, the Q\ and Q2 dependence of the 77 —> HH amplitude for transversely polarized photons is minimal at W'^ large and fixed ^cm, since the off-shell quark and gluon propagators
187 1.4 - 1.0 - 0.6 - 0.2 = 1.4 - © 1.0 •g|^ 0.6 V) 0.2 - 1.4 - 1.0 0.6 - 0.2 = 0 0.2 0.4 cose 0.6 0.8 Figure 32. Perturbative QCD predictions by Farrar and Zhang for the cos(^cm) dependence of the 77 —>■ pp cross section assuming the King-Sachrajda (KS), Chernyak, Ogloblin, and Zhitnitsky (COZ) , and original Chernyak and Zhitnitsky (CZ) forms for the proton distribution amplitude, <f)p(xi^Q).
188 10 10 10 to 10^ = 1 0 < i ^ O 2GeV ■ 3GeV o 4GeV V 5GeV • 6GeV ^ *<^ 0 cose -1 Figure 33. Test of dimensional counting for Compton scattering for 2 < E"/^^ < 6 GeV.^' in Th already transfer hard momenta; i.e. the 27 coupling is effectively local for Qh Ql "^ PT' ^^^ ^*^* ~^ ^^ ^^^ ^^ amplitudes for off-shell photons have been calculated by Millers and Gunion. In each case, the predictions show strong sensitivity to the form of the respective baryon and meson distribution amplitudes. We also note that photon-photon collisions provide a way to measure the running coupling constant in an exclusive channel, independent of the form of 82 hadronic distribution amplitudes. The photon-meson transition form factors F^-,m{Q^)^ ^ = '^^•>Vl^', fi ^tc, are measurable in tagged 67 —i QCD predicts e'M reactions. a.(Q2) = 1 i^.(Q') 47r Q2|ir^^(Q2)|2 where to leading order the pion distribution amplitude enters both numerator and denominator in the same manner. The complete calculations of the tree-graph structure (see Figs. 34, 35, 36) of both 77 —)> MM and 77 —► BB amplitudes has now been completed. One can use crossing to compute Th for pp -^ 77 to leading order in asijpj^) from the
189 calculations reported by Farrar, Maina and Neri and Gunion and Millers. Examples of the predicted angular distributions are shown in Figs. 37 and 38. TT TT y = . 89 Figure 34. Application of QCD to two-photon production of meson pairs. Figure 35. Next-to-leading perturbative contribution to Th for the process 77 — S9 MM. The calculation has been done by Nizic. As discussed in Section 2, a model form for the proton distribution amplitude has been proposedby Chernyak and Zhitnitsky based on QCD sum rules which leads to normalization and sign consistent with the measured proton form factor (see Fig. 21). The CZ sum rule analysis has been confirmed and extended by 50 King and Sachrajda. The CZ proton distribution ampHtude yields predictions for 77 —► pp in rough agreement with the experimental normalization, although the production energy is too low for a clear test. It should be noted that unlike
190 r\^ NAAA/j r\jy naaa; NAAAlj /\yy, VAAA* naaaT I NAAA NAAA ruy /vn NAAAl AA/VWNA) OuTs /vry VVSM AAA/AAA(| vaaa; AAAAtvAA^ NAAAJ- AAA/'^AA/ NAAAl AAA/ A/SA NAA/^' 0^/\ (/\/> A/W A/NA WW/' rsy> /VW V>A >^k/N \/S/\A/ r\/\ /VS/S/" A/VSj Figure 36. Leading diagrams for 7 + 7 —^ p + p calculated in Ref. 56. 89 meson pair production the QCD predictions for baryons are highly sensitive to the form of the running coupling constant and the endpoint behavior of the wavefunctions. It is possible that data from pp collisions at energies up to 10 GeV could greatly clarify the question of whether the perturbative QCD predictions are reliable at moderate momentum transfer. As emphasized in Section 4, an important check of the QCD predictions can be obtained by combining data from pp —► 77, 77 ~^ PP with large angle Compton scattering 7p —^ 7p. This comparison checks in detail the angular dependence and crossing behavior expected from the theory. Furthermore, in pp collisions one can even study time-hke photon production into e'^'e" and examine the virtual photon mass dependence of the Compton amplitude. Predictions for the q^ dependence of the pp —^ 77* amplitude can be obtained by crossing the results of Gunion and Millers. '
191 8 7 - 6 - 0) 1^ 4 - t b 3 - 2 - 0 0 0.2 0.4 cos9 0.6 0.8 Figure 37. QCD prediction for the scaling and angular distribution for 7 + 7 —»^ 56 «i p-{-p calculated by Farrar et ai The dashed-dot curve corresponds to Ahr/s = 0.0016 and a maximum running coupling constant a^^'^ = 0.8. The solid curve corresponds to Ah? 1$ = 0.016 and a maximum running coupling constant aj*"^ = 0.5. The dashed curve corresponds to a fixed a, = 0.3. The results are very sensitive to the endpoint behavior of the proton distribution amplitude. The CZ form is assumed. 8. QCD PROCESSES IN NUCLEI The least-understood process in QCD is hadronization — the mechanism which converts quark and gluon quanta to color-singlet integrally-charged hadrons. One way to study hadronization is to perturb the environment by introducing a nuclear medium surrounding the hard-scattering short distance reaction. This is obviously impractical in the theoretically simplest processes — hilation. However, for large momentum transfer reactions occurring in a nuclear target, such as deep inelastic lepton scattering or massive lepton pair production. e'^e or 77 anni-
192 I b X3 10 t/> 10^ c 0 10 0' 10 -I T—a •/ - • • = Running a • — ? as=0.27 0 0.2 0.4 0.6 COS^cm 0.8 .0 Figure 38. QCD prediction for the scaling and angular distribution for 7 + 7 —> c z* fi 1 p-{-p calculated by Gunion, Sparks and Millers. ' CZ distribution amplitudes are assumed. The solid and running curves are for real photon annihilation. The dashed and dot-dashed curves correspond to one photon space-like, with Ql/s = 0.1. the nuclear medium provides a nontrivial perturbation to jet evolution through the influence of initial- and/or final-state interactions. In the case of large momentum transfer quasiexclusive reactions, one can use a nuclear target to filter and influence the evolution and structure of the hadron wavefunctions themselves. The physics of such nuclear reactions is surprisingly interesting and subtle — involving concepts and novel effects quite orthogonal to usual expectations. The nucleus thus plays two complimentary roles in quantum chromodynamics: 1. A nuclear target can be used as a control medium or background field to modify or probe quark and gluon subprocesses. Some novel examples are color transparency, the predicted transparency of the nucleus to hadrons participating in high-momentum transfer exclusive reactions, and formation zone phenomena, the absence of hard, collinear, target-induced radiation by a quark or gluon interacting in a high-momentum transfer inclusive reaction if its energy is large compared to a scale proportional to the length of the target. (Soft radiation and elastic initial-state interactions in the nucleus still occur.) Coalescence with co-moving spectators has been discussed as a mechanism which can lead to increased open charm hadroproduction, but which also suppresses forward charmonium production (relative to lepton pairs) in heavy ion collisions. There are also interesting special features of nuclear diffractive amplitudes — high energy hadronic or electromagnetic
193 reactions which leave the entire nucleus intact and give nonadditive contributions to the nuclear structure function at low xbj- The Q^ dependence of diffractive 7*p —> p^p is found to have a slope in the ^—dependence exp bt where b = b{Q^) is of order 1^2 GeV~^, much smaller than expected on the basis of vector meson dominance and ^—channel factorization. 2. Conversely, the nucleus can be studied as a QCD structure. At short distances nuclear wavefunctions and nuclear interactions necessarily involve hidden color, degrees of freedom orthogonal to the channels described by the usual nucleon or isobar degrees of freedom. At asymptotic momentum transfer, the deuteron form factor and distribution amplitude are rigorously calculable. One can also derive new types of testable scaling laws for exclusive nuclear amplitudes in terms of the reduced amplitude formalism. 8.1. Exclusive Nuclear Reactions — Reduced Amplitudes An ultimate goal of QCD phenomenology is to describe the nuclear force and the structure of nuclei in terms of quark and gluon degrees of freedom. Explicit signals of QCD in nuclei have been elusive, in part because of the fact that an effective Lagrangian containing meson and nucleon degrees of freedom must be in some sense equivalent to QCD if one is limited to low-energy probes. On the other hand, an effective local field theory of nucleon and meson fields cannot correctly describe the observed off-shell falloff of form factors, vertex amplitudes, Z-graph diagrams, etc. because hadron compositeness is not taken into account. We have already mentioned the prediction F^(Q^) ~ l/Q^^ which comes from simple quark counting rules, as well as perturbative QCD. One cannot expect this asymptotic prediction to become accurate until very large Q^ is reached since the momentum transfer has to be shared by at least six constituents. However there is a simple way to isolate the QCD physics due to the compositeness of the nucleus, not the nucleons. The deuteron form factor is the probability amplitude for the deuteron to scatter from p to p -\- q but remain intact. Note that for vanishing nuclear binding energy Cd —^ 0, the deuteron can be regarded as two nucleons sharing the deuteron four-momentum (see Fig. 39). The momentum i is limited by the binding and can thus be neglected. To first approximation the proton and neutron share the deuteron's momentum equally. Since the deuteron form factor contains the probability amplitudes for the proton and neutron to scatter from 92 93 p/2 to p/2 -f q/2; it is natural to define the reduced deuteron form factor ' ^'^^ ^=Fr. m F.. m •
194 The effect of nucleon compositeness is removed from the reduced form factor. QCD then predicts the scaling UQ') ^ rsj Q i.e. the same scaHng law as a meson form factor. Diagrammatically, the extra power of l/Q^ comes from the propagator of the struck quark line, the one propagator not contained in the nucleon form factors. Because of hadron he- licity conservation, the prediction is for the leading helicity-conserving deuteron form factor (A = A' = 0.) As shown in Fig. 40, this scaling is consistent with 94 experiment for Q = pj- ^ 1 GeV. /%/ p+q=p' Figure 39. Application of the reduced amplitude formalism to the deuteron form factor at large momentum transfer. The distinction between the QCD and other treatments of nuclear amplitudes is particularly clear in the reaction 7c? —> np; i.e. photodisintegration of the deuteron at fixed center of mass angle. Using dimensional counting, the leading power-law prediction from QCD is simply ^(7^? —^ np) ^ gnF{6cm)' Again we note that the virtual momenta are partitioned among many quarks and gluons, so that finite mass corrections will be significant at low to medium energies. Nevertheless, one can test the basic QCD dynamics in these reactions taking into
195 account much of the finite-mass, higher-twist corrections by using the "reduced amplitude" formalism. ' Thus the photodisintegration amplitude contains the probability ampHtude (i.e. nucleon form factors) for the proton and neutron to each remain intact after absorbing momentum transfers pp — l/2pd and pn — l/2pd, respectively (see Fig. 41). After the form factors are removed, the remaining "reduced" amplitude should scale as F{Ocm)/PT- The single inverse power of transverse momentum pr is the slowest conceivable in any theory, but it is the unique power predicted by PQCD. 6.0 b 4.0 CM o 2.0 - 0 CO O 0.2 CO O E 0.1 - 0 0 2 3 4 Q2 (Gev2) Figure 40. Scaling of the deuteron reduced form factor. The data are summarized in Ref. 92. Figure 41. Construction of the reduced nuclear amplitude for two-body inelastic . 92 deuteron reactions. The prediction that /(^cm) is energy dependent at high-momentum transfer is compared with experiment in Fig. 42. It is particularly striking to see the QCD
196 prediction verified at incident photon lab energies as low as 1 GeV. A comparison with a standard nuclear physics model with exchange currents is also shown for comparison as the solid curve in Fig. 42(a). The fact that this prediction falls less fast than the data suggests that meson and nucleon compositeness are not taken to into account correctly. An extension of these data to other angles and higher energy would clearly be very valuable. An important question is whether the normalization of the 7c? —► pn ampli- 98 tude is correctly predicted by perturbative QCD. A recent analysis by Fujita shows that mass corrections to the leading QCD prediction are not significant in the region in which the data show scaling. However Fujita also finds that in a model based on simple one-gluon plus quark-interchange mechanism, normalized to the nucleon-nucleon scattering amplitude, gives a photo-disintegration amplitude with a normalization an order of magnitude below the data. However this model only allows for diagrams in which the photon insertion acts only on the quark lines which couple to the exchanged gluon. It is expected that including other diagrams in which the photon couples to the current of the other four quarks will increase the photo-disintegration amplitude by a large factor. w CM E * o CD 1 C\J 0 1 1 n Previous Work • This Expt. ec.m=90^ _^^^' ^^j^pcoff^ 1 "¥r- ■V (a) a1 - 1 0 500 1000 (MeV) 1500 (/) C\J CM » 0.6 0 1.2 0.6 0 1.8 1.2 0.6 0 1.8 1.2 0.6 t 0 « F"i~n[ T ec.mf36.9 r-r-7 1—^1 Z 101.5 113.5 I 126.9 1.2 0.6 b 0 0 143.1* J U*K (b) d .^\ ^ >4H^*v^11 k - 0.6 0 1.8 1.2 0.6 0 1.8 1.2 0.6 0 0.4 0.8 Photon Lab Energy (GeV) Figure 42. Comparison of deuteron photodisintegration data with the scaling prediction which requires P(9cm) to be at most logarithmically dependent on energy at large momentum transfer. The data in (a) are from the recent experiment of Ref. 95. The nuclear physics prediction shown in (a) is from Ref. 96. The data in (b) are from Ref. 97.
197 The derivation of the evolution equation for the deuteron and other multi- quark states is given in Refs. 99 and 93. In the case of the deuteron, the evolution equation couples five different color singlet states composed of the six quarks. The leading anomalous dimension for the deuteron distribution amplitude and the helicity-conserving deuteron form factor at asymptotic Q is given in Ref. 99. There are a number of related tests of QCD and reduced amplitudes which require p beams such as pc? —>- ^n and pc? —>- 7r~p in the fixed ^cm region. These reactions are particularly interesting tests of QCD in nuclei. Dimensional counting rules predict the asymptotic behavior ^ (pc? —► 7r~p) ~ pryir /(^cm) since there are 14 initial and final quanta involved. Again one notes that the ^d —► 7r~p amplitude contains a factor representing the probability amplitude (i.e. form factor) for the proton to remain intact after absorbing momentum transfer squared f = (p — l/2p^)^ and the NN time-like form factor at s = (p + l/2p(f )^. Thus A^p(f_^;r-p ~ Fii^{t) Fii^{s) Mri where Mr has the same QCD scaling properties as quark meson scattering. One thus predicts fe {pd ^ x-p) fjil) The reduced amplitude scaling for 7c? —^ pn at large angles and pT ^ 1 GeV (see Fig. 42). One thus expects similar precocious scaling behavior to hold for pd —^ 7r~p and other pd exclusive reduced amplitudes. Recent analyses by Kondratyuk and Sapozhnikov show that standard nuclear physics wavefunc- tions and interactions cannot explain the magnitude of the data for two-body anti-proton annihilation reactions such as pd —>- 7r~p. 8.2. Color Transparency A striking feature of the QCD description of exclusive processes is "color transparency:"The only part of the hadronic wavefunction that scatters at large momentum transfer is its valence Fock state where the quarks are at small relative impact separation. Such a fluctuation has a small color-dipole moment and thus has negligible interactions with other hadrons. Since such a state stays small over a distance proportional to its energy, this implies that quasi-elastic hadron- nucleon scattering at large momentum transfer as illustrated in Fig. 43 can occur additively on all of the nucleons in a nucleus with minimal attenuation due to elastic or inelastic final state interactions in the nucleus, i.e. the nucleus becomes "transparent." By contrast, in conventional Glauber scattering,
198 A-1 Figure 43. Quasi-elastic pp scattering inside a nuclear target. Normally one expects such processes to be attenuated by elastic and inelastic interactions of the incident proton and the final state interaction of the scattered proton. Perturbative QCD predicts minimal attenuation; i.e. "color transparency," at large momentum transfer. one predicts strong, nearly energy-independent initial and final state attenuation. A detailed discussion of the time and energy scales required for the validity of the PQCD prediction is given in by Farrar et al. and Mueller in Ref. 7. A recent experiment at BNL measuring quasi-elastic pp —► pp scattering at ^cm = 90° in various nuclei appears to confirm the color transparency prediction—at least for piah up to 10 GeV/c (see Fig. 44). Descriptions of elastic scattering which involve soft hadronic wavefunctions cannot account for the data. However, at higher energies, piah ^ 12 GeV/c, normal attenuation is observed in the BNL experiment. This is the same kinematical region Ecm ^ 5 GeV where the large spin correlation in AjviV are observed. Both features may be signaling 103 new s-channel physics associated with the onset of charmed hadron production 43 or interference with LandshofF pinch singularity diagrams. We will discuss these possible solutions in Section 9. Clearly, much more testing of the color transparency phenomena is required, particularly in quasi-elastic lepton-proton scattering, Compton scattering, antiproton-proton scattering, etc. The cleanest test of the PQCD prediction is to check for minimal attenuation in large momentum transfer lepton-proton scattering in nuclei since there are no complications from pinch singularities or resonance interference effects. In Section 5.4 we emphasized the fact that soft initial-state interactions pp —► li are suppressed at high lepton pair mass. This is a remarkable consequence of gauge theory and is quite contrary to normal treatments of initial interactions based on Glauber theory. This novel effect can be studied in quasielastic pA —^ l£ {A — 1) reaction, in which there are no extra hadrons produced and the
. o 0.5 - X u UJ cr < a z 0.2 < q: I- 0.1 199 T T T T T 6 GeV/c Aluminum lOGeV/c 12 GeV/c 1 i 1 i 1 0 5 10 15 INCIDENT MOMENTUM (GeV/c) Figure 44. Measurements of the transparency ratio ^=¥=> 101 P(A-W^\PA-*PP] near 90° on Aluminum. Conventional theory predicts that T should be small and 7 roughly constant in energy. Perturbative QCD predicts a monotonic rise to T = 1. produced leptons are coplanar with the beam. (The nucleus {A — 1) can be left excited). Since PQCD predicts the absence of initial-state elastic and inelastic interactions, the number of such events should be strictly additive in the number Z of protons in the nucleus, every proton in the nucleus is equally available for short-distance annihilation. In traditional Glauber theory only the surface protons can participate because of the strong absorption of the p as it traverses the nucleus. The above description is the ideal result for large s. QCD predicts that additivity is approached monotonically with increasing energy, corresponding to two effects: a) the effective transverse size of the p wavefunction is 6j_ ~ l/\/5, and b) the formation time for the p is sufficiently long, such that the Fock state stays small during transit of the nucleus. The color transparency phenomena is also important to test in purely hadronic quasiexclusive antiproton-nuclear reactions. For large pr one predicts
200 {pA -^ TT+TT + {A - 1)) C:^ Y^ Gp/Aiv) "^ {PP "^ ^T+TT ) where Gpu{y) is the probability distribution to find the proton in the nucleus with light-cone momentum fraction j/ = (p^ + p')l{p\ + P>i)' ^^^ ~r(pv -^ tt'^tt ) :^ The distribution Gpjj^{y) can also be measured in eA —^ ep{A — 1) quasiexclusive reactions. A remarkable feature of the above prediction is that there are no corrections required from initial-state absorption of the p as it traverses the nucleus, nor final-state interactions of the outgoing pions. Again the basic point is that the only part of hadron wavefunctions which is involved in the large pT reaction is iPh{^± ^ ^(1/pt))- J-e. the amplitude where all the valence quarks are at small relative impact parameter. These configurations correspond to small color singlet states which, because of color cancellations, have negligible hadronic interactions in the target. Measurements of these reactions thus test a fundamental feature of the Fock state description of large pr exclusive reactions. Another interesting feature which can be probed in such reactions is the behavior of Gp/j^^{y) for y well away from the Fermi distribution peak at ?/ ~ m^/MA' For y —^ I spectator counting rules predict Gpi^iv) ~ (1—2/)^ — (1 — J/)^"^"^ where A^^ = 3(^4 — 1) is the number of quark spectators required to "stop" [yi —► 0) as J/ —► 1. This simple formula has been quite successful in accounting for distributions measured in the forward fragmentation of nuclei at the BEVALAC. Color transparency can also be studied by measuring quasiexclusive J/i/' production by anti-protons in a nuclear target pA —>- J/ip{A — 1) where the nucleus is left in a ground or excited state, but extra hadrons are not created (see Fig. 45). The cross section involves a convolution of the PP —^ J/ip subprocess cross section with the distribution Gp^^iv) where y = {p^ + P^)/{p\ + P>i) is the boost-invariant light-cone fraction for protons in the nucleus. This distribution can be determined from quasiexclusive lepton-nucleon scattering iA —> £p{A — 1). In first approximation pp —>- J/ip involves qqq + ^qq annihilation into three charmed quarks. The transverse momentum integrations are controlled by the charm mass scale and thus only the Fock state of the incident antiproton which contains three antiquarks at small impact separation can annihilate. Again it follows that this state has a relatively small color dipole moment, and thus it
201 Figure 45. Schematic representation of quasielastic charmonium production in pA reactions. should have a longer than usual mean-free path in nuclear matter; i.e. color transparency. Unlike traditional expectations, QCD predicts that the pp annihilation into charmonium is not restricted to the front surface of the nucleus. The exact nuclear dependence depends on the formation time for the physical p to couple to the small "ggq configuration, Tf oc Ep. It may be possible to study the effect of finite formation time by varying the beam energy, Ep, and using the Fermi-motion of the nucleon to stay at the J/ip resonance. Since the J ftp is produced at nonrelativistic velocities in this low energy experiment, it is formed inside the nucleus. The A-dependence of the quasiexclusive reaction can thus be used to determine the J/V'-nucleon cross section at low energies. For a normal hadronic reaction pA —► HX, we expect Aeff ~ .4^/^, corresponding to absorption in the initial and final state. In the case of pA —>- J/ip X one expects A^f^ much closer to A^ if color transparency is fully effective and o-(J/ip N) is small. 9. SPIN CORRELATIONS IN PROTON-PROTON SCATTERING One of the most serious challenges to quantum chromodynamics is the behavior of the spin-spin correlation asymmetry Aj\fjs[ = \dVt\\y+dat\\)\ ^^^sured in large momentum transfer pp elastic scattering (see Fig. 46). At piab = 11.75 GeV/c and ^cm = 7r/2, Aj\i^ rises to ~ 60%, corresponding to four times more probability for protons to scatter with their incident spins both normal to the scattering plane and parallel, rather than normal and opposite. The polarized cross section shows a striking energy and angular dependence not expected from the slowly-changing perturbative QCD predictions. However,
202 • ACS BROWN tt 01. COURT tt ol. o3G«V/c MICCCR ttol. o C G«V/c MILCER tt ol FERNOW ft ol RATNCR ft ol CINN ft ol. OII.79 6fV/c ABEftol. MCTTlNCN 01 ol OFALCON tl OL CRAdB ft ol. 4 MT LIN ot ol CROSBie ft ol lob (6«V/c) fOi p/(6«V*/c*) Figure 46. The spin-spin correlation Ann for elastic pp scattering with beam 1 nfi and target protons polarized normal to the scattering plane. Ann — 60% implies that it is four times more probable for the protons to scatter with spins parallel rather than antiparallel. the unpolarized data is in first approximation consistent with the fixed angle scaling law s^^da/dt{pp —*■ pp) = /{Ocm) expected from the perturbative analysis (see Fig. 23). The onset of new structure at s c^ 23 GeV is a sign of new degrees of freedom in the two-baryon system. In this section, we will discuss a possible explanation for (1) the observed spin correlations, (2) the deviations from fixed-angle scaling laws, and (3) the anomalous energy dependence of absorptive corrections to quasielastic pp scattering in nuclear targets, in terms of a simple model based on two J = L = S = 1 broad resonances (or threshold enhancements) interfering with a perturbative QCD quark-interchange background amplitude. The structures in the pp —>- pp amplitude may be associated with the onset of strange and charmed thresholds. If this view is correct, large angle pp elastic scattering would have been virtually featureless for piab ^ 5 GeV/c, had it not been for the onset of heavy flavor production. As a further illustration of the threshold effect, one can see the effect in Aj\fi^ due to a narrow "^^3 pp resonance at y/s = 2.17 GeV [piab = 1-26 GeV/c) associated with the pA threshold. .2 The perturbative QCD analysis of exclusive amplitudes assumes that large momentum transfer exclusive scattering reactions are controlled by short distance
203 quark-gluon subprocesses, and that corrections from quark masses and intrinsic transverse momenta can be ignored. The main predictions are fixed-angle scaling 5 . laws (with small corrections due to evolution of the distribution amplitudes, the running coupling constant, and pinch singularities), hadron helicity conservation, and the novel phenomenon, "color transparency." As discussed in Section 8.2, a test of color transparency in large momentum transfer quasielastic pp scattering at ^cm — 7r/2 has recently been carried out at BNL using several nuclear targets (C, Al, Pb). The attenuation at piab = 10 GeV/c in the various nuclear targets w£ls observed to be in fact much less than that predicted by traditional Glauber theory (see Fig. 44). This appears to support the color transparency prediction. The expectation from perturbative QCD is that the transparency effect should become even more apparent as the momentum transfer rises. Nevertheless, at Plab = 12 GeV/c, normal attenuation was observed. One can explain this surprising result if the scattering at piab = 12 GeV/c {y/s = 4.93 GeV), is dominated by an s-channel B=2 resonance (or resonance-like structure) with mass near 5 GeV, since unlike a hard-scattering reaction, a resonance couples to the fully-interacting large-scale structure of the proton. If the resonance has spin 5 = 1, this can also explain the large spin correlation Af^f^ measured nearly at the same momentum, piab = 11-75 GeV/c. Conversely, in the momentum range Plab = 5 to 10 GeV/c one predicts that the perturbative hard-scattering amplitude is dominant at large angles. The experimental observation of diminished attenuation at piab = 10 GeV/c thus provides support for the QCD description of exclusive reactions and color transparency. What could cause a resonance aX y/s = b GeV, more than 3 GeV beyond the pp threshold? There are a number of possibilities: (a) a multigluonic excitation 108 such as \qqqqqqggg)^ (b) a "hidden color" color singlet \qqqqqq) excitation, or (c) a "hidden flavor" qqqqqqQ^ excitation, which is the most interesting possibility, since it is so predictive. As in QED, where final state interactions give large enhancement factors for attractive channels in which Za/v^ei is large, one expects resonances or threshold enhancements in QCD in color-singlet channels at heavy quark production thresholds since all the produced quarks have similar velocities. One thus can expect resonant behavior at M* = 2.55 GeV and M* = 5.08 GeV, corresponding to the threshold values for open strangeness: pp —> AK'^p^ and open charm: pp —> AcD^p^ respectively. In any case, the structure at 5 GeV is highly inelastic: its branching ratio to the proton-proton channel is B^p ~ 1.5%. A model for this phenomenon is given in Ref. 103 In order not to over com-
204 plicate the phenomenology; the simplest Breit-Wigner parameterization of the resonances was used. There has not been an attempt to optimize the parameters of the model to obtain a best fit. It is possible that what is identified a single resonance is actually a cluster of resonances. The background component of the model is the perturbative QCD amplitude. Although complete calculations are not yet available, many features of the QCD predictions are understood, including the approximate s~^ scaling of the pp —> pp amplitude at fixed ^cm ^i^id the dominance of those ampli- tudes that conserve hadron helicity. Furthermore, recent data comparing dif- 33 ferent exclusive two-body scattering channels from BKL show that quark in« terchange amplitudes dominate quark annihilation or gluon exchange contributions. Assuming the usual symmetries, there are ^vq independent pp helicity amplitudes: 0i = M(++,+-f), 02 = M( —,++), h = M( + -,+-), (f>A = M( —-|-,-f —), <j>s = M(-f-|-,H—). The helicity amplitudes for quark interchange have a definite relationship: (^,(PQCD) = 2(^3(PQCD) - -2(^4(PQCD) u — mj The hadron helicity nonconserving amplitudes, (^2(PQCD) and <^5(PQCD) are zero. This form is consistent with the nominal power-law dependence predicted by perturbative QCD and also gives a good representation of the angular distribution over a broad range of energies. Here F{t) is the helicity conserving proton form factor, taken as the standard dipole form: F{i) = (1 — t/fn^)~^^ with mj = 0.71 GeV . As shown in Ref. 40, the PQCD-quaxk-interchangc structure alone predicts Ajvjv — 1-/3, nearly independent of energy and angle. Because of the rapid fixed-angle s""^ fallofF of the perturbative QCD arnpH- tude, even a very weakly-coupled resonance can have a sizeable effect at large momentum transfer. The large empirical values for Aj^j^ suggest a resonant pp —> pp amplitude with J = L = 5 = 1 since this gives A^pj = 1 (in absence of background) and a smooth angular distribution. Because of the Pauli principle, an 5 = 1 di-proton resonances must have odd parity and thus odd orbital angular momentum. The the two non-zero helicity amplitudes for a J = L — 5 = 1 resonance can be parameterized in Breit-Wigner form: <^3(resonance) = 127r-^^<i} i(^cra) ^ Pcm '•'' ^'"'M* - Ecrn ~ hT 2
205 (^4 (resonance) = — 127r ^-i,i(^cm) Pcm "*•■' ^""M'-£cm-jr cm 2 (The ^Fz resonance amplitudes have the same form with cf^j j replacing d\.-^ j.) As in the case of a narrow resonance like the Z^, the partial width into nu- cleon pairs is proportional to the square of the time-like proton form factor: rPP(5)/r = 5PP|F(s)|Vl^(^^*^)l^ corresponding to the formation of two protons at this invariant energy. The resonant amplitudes then die away by one inverse power of (^cm — M*) relative to the dominant PQCD amplitudes. (In this sense, they are higher twist contributions relative to the leading twist per- turbative QCD amplitudes.) The model is thus very simple: each pp helic- ity amplitude (f>i is the coherent sum of PQCD plus resonance components: (j) = <^(PQCD) + E<^(resonance). Because of pinch singularities and higher-order 43 corrections, the hard QCD amplitudes are expected to have a nontrivial phase; the model allows for a constant phase 8 in (^(PQCD). Because of the absence of the 4>s hehcity-flip amplitude, the model predicts zero single spin asymmetry 112 Ai^. This is consistent with the large angle data at p/^ft = 11.75 GeW/c. At low transverse momentum, pr < 1.5 GeV, the power-law fall-off of (?>(PQCD) in s disagrees with the more slowly falling large-angle data, and one has little guidance from basic theory. The main interest in this low-energy region is to illustrate the effects of resonances and threshold effects on /Ia^//. In order to keep the model tractable, one can extend the background quark interchange and the resonance amplitudes at low energies using the same forms as above but replacing the dipole form factor by a phenomcnological form F{t) oc e~*'^^vl'l. A kinematic factor of \/sf2pcm is included in the background ampHtude. The value /? = 0.85 GeV 113 then gives a good (it to da/dt at ^cm = ^/2 for pi^b S 5.5 GeV/c. The normalizations are chosen to maintain continuity of the amplitudes. The predictions of the model and comparison with experiment are shown in Figs. 47-52. The following parameters are chosen: C = 2.9 x 10"^, ^ = -1 for the normalization and phase of <^(PQCD). The mass, width and pp branching ratio for the three resonances are M^ = 2.17 GeV\ V^ — 0,04 GeV, B^ = I; M; =. 2.55 GeV, P^ = 1.6 GeV, ^f = 0.65; and M,* = 5.08 GeV, Fc = 1,0 GeV, B^^ = 0.0155, respectively. As shown in Figs. 47 and 48, the deviations from the simple scaling predicted by the PQCD amplitudes are readily accounted for by the resonance structures. The cusp which appears in Fig. 48 marks the change in regime below p^^fc = ^-^ GeV/c where PQCD becomes inapplicable. It is interesting to note that in this energy region normal attenuation of quasiclastic pp scattering is observed. The angular distribution (normalized to the data -1
206 at ^cm = ^/2) is predicted to broaden relative to the steeper perturbative QCD form, when the resonance dominates. As shown in Fig. 49 this is consistent with experiment, comparing data at pi^b = 7.1 and 12,1 GeV/c. CM CD E o o ■D 10 10 10 10 -2 -3 -4 -5 ^ r- - 1 1 1 Si- ^T)t*> 1 1 1 : ^= ^5^^. 1 1 ■•-> 6 P 8 lab 10 (GeV/c) 12 14 Figure 47. Prediction (solid curve) for d(T/dt(pp -* pp) at 0cm = ''■/2 compared 113 with the data of Akerlof ei al. The dotted line is the background PQCD prediction. o CL o < Q 2 T5 c:) T3 1 ■o 5 10 P lab 15 (GeV/c) 20 Figure 48. Ratio of dc/dt{pp -* pp) at ^cm = ?r/2 to the PQCD prediction. The 113 data are from Akerlof et ai (open triangles), Allaby ei ai (solid dots) and Cocconi ei al. (open square). The cusp at piab = 5.5 GeV/c indicates the change of regime from PQCD. The most striking test of the model is its prediction for the spin correlation Af^j^ shown in Fig. 50. The rise of A^pj to ~ 60% at piah = 11.75 GeV/c is correctly reproduced by the high energy J=l resonance interfering with <;^(PQCD). The narrow peak which appears in the data of Fig. 50 corresponds to the onset
20 15 10 5 0 0 0.4 z=cos 6 207 0.8 cm. Figure 49. The pp — pp angular distribution normalized at ^cm = ^/2 The data are from the compilation given in Si vers ei a/., Ref. 32. The solid and dotted lines are predictions for ptab = 12.1 and 7.1 GeV/c, respectively, showing the broadening near resonance. of the pp —> pA(1232) channel which can be interpreted as a uuuuddqq resonant state. Because of spin-color statistics one expects in this case a higher orbital momentum state^ such as a pp '^F-^ resonance. The model is ailso consistent with the recent high-energy data point for Ai^n at piah = 18,5 GeV/c and pj^ = 4.7 GeV (see Fig. 51). The data show a dramatic decrease of /4^jV to zero or negative values. This is explained in the model by the destructive interference effects above the resonance region. The same effect accounts for the depression of A^^v for Plab ^ 6 GeV/c shown in Fig. 50. The comparison of the angular dependence of Ai^i^ with data at p/afe = 11.75 GeV/c is shown in Fig. 52. The agreement with the data for the longitudinal spin correlation An at the same p^^fc ^^ somewhat worse. The simple model discussed here shows that many features can be naturally explained with only a few ingredients: a perturbative QCD background plus resonant amplitudes associated with rapid changes of the inelastic pp cross section. The model provides a good description of the s and t dependence of the differential cross section, including its "oscillatory'' dependence in 5 at fixed ^cmi and the broadening of the angular distribution near the resonances. Most important, it gives a consistent explanation for the striking behavior of both the spin-spin correlations and the anomalous energy dependence of the attenuation of quasielastic pp scattering in nuclei. It is predicted that color transparency should reappear at higher energies [piah > 16 GeV/c), and also at smaller angles (^cm ^ 60°) at plab = 12 GeV/c where the perturbative QCD amplitude dominates. If the J = l resonance structures in Aj\fj^ are indeed associated with heavy
208 0.8 0.6 - 0.4 A NN 0.2 0 -0.2 0 5 10 P,ab (^^V^^) 113 Figure 50. /I/va' as a function of piab at Ocm = ^/2. The dala^*'' are from Crosbie ei ai (solid dots), Lin ei ai (open squares) and Bhatia ei ai (open triangles). The peak at piab = 1-26 GeV/c corresponds to the pA threshold. The data are well reproduced by the interference of the broad resonant structures at the strange [piab = 2.35 GeV/c) and charm {piab = 12.8 GeV/c) thresholds, interfering with a PQCD background. The value of Aj^^ from PQCD alone is 1/3. 0.8 0.6 0.4 A NN T (b) p^=4.7(GeV/c)^ 0.2 0 -0.2 12 P iab 14 16 (GeV/c) 18 Figure 51. A{s/!\i at fixed p^- = (4.7 GeV/c)^. The data point''"' at piab = 18.5 GeV/c is from Court et ai quark degrees of freedom, then the model predicts inelastic pp cross sections of the order of 1 mb and 1/ib for the production of strange and charmed hadrons lift near their respective thresholds. Thus a crucial test of the heavy quark hypothesis for explaining Apj^, rather than hidden color or gluonic excitations, is the observation of significant charm hadron production at piab > 12 GeV/c. Recently Halston and Pire have proposed that the oscillations of the pp elastic cross section and the apparent breakdown of color transparency are associated
209 0.8 0.6 0.4 A NN 0.2 0 -0.2 3 4 p| [(GeV/c)2] 5 106 Figure 52. A^vyv as a function of transverse momentum. The data are from Crabb ei ai (open circles) and O'Fallon ei ai (open squares). Diffractive contributions should be included for p|. < 3 GeV^. with the dominance of the LandshofF pinch contributions ai y/s ^ 5 GeV. The oscillating behavior of dcr/dt is due to the energy dependence of the relative phase between the pinch and hard-scattering contributions. Color transparency will disappear whenever the pinch contributions are dominant since such contributions could couple to wavefunctions of large transverse size. The large spin correlation in AjVTV is not readily explained in the Ralston-Pire model. Clearly more data and analysis are needed to discriminate between the pinch and resonance models. 10. CONCLUSIONS The understanding of exclusive processes is a crucial challenge to QCD. The analysis of these reactions is more complex than that of inclusive reactions since the detailed predictions necessarily depend on the form of the hadronic wavefunctions, the behavior of the running coupling constant, and analytically complex contributions from pinch and endpoint singularities. Unlike inclusive reactions, where the leading power contributions can be computed from an incoherent probabilistic form, exclusive reactions require the understanding of the phase and spin structure of hadronic amplitudes. These complications are also a virtue of exclusive reactions, since they allow a window on basic features of the theory which are extremely difficult to obtain in any other way. The perturbative QCD analysis is based on a factorization theorem so that only one distribution amplitude is required to describe the interaction of a given hadron in any large momentum transfer exclusive reaction. In some cases the prcdlctlony for exclusive processes in PQCD are completely rigorous in the sense that the results can be derived ro
210 all orders in perturbation theory. In particular the PQCD results for the pion form factor, the transition form factor FyTr{Q^)^ and the 77 —> tttt amplitudes are theorems of QCD and are as rigorous £ls the predictions for R^-¥e-{s), the evolution equations for the structure functions, etc. Although the perturbative QCD analysis is complex, it is hard to imagine that any other viable description would be simpler. At this point there is no other theoretical approach which provides as comprehensive a description of exclusive phenomena. The application of perturbative QCD to exclusive processes has in fact been quite successful. The power laws predicted for form factors and fixed angle scattering amplitudes have been confirmed by experiment, ranging from the theoretically simplest reactions 7*7 —> 77 to the most complicated reactions such as pp —> pp. The application to nuclear exclusive amplitudes such as the deuteron form factor and 7c/ —♦ np have also been surprisingly successful. Taken together with input from distribution amphtudes predicted by QCD sum rules, the sign and magnitude of the meson form factors, the 77 —>- 7r"^7r~, K'^K'^ ^ the Compton amplitude 7p —> jp and the proton form factor are all apparent, though model dependent, successes of the theory. The fact that PQCD scaling laws appear to hold even at momentum transfer as low as 1 GcV/c suggests that the QCD running coupling constant is rather slowly changing even at momentum transfers of order 200 MeV. Barring a conspiracy between non-perturbative and perturbative contributions, the evidence from exclusive reactions is that A^^ is of order 100 MeV or even smaller, MS Alternatively the running coupling constant may "freeze" at the low effective momenta characteristic of exclusive processes. Thus the analysis of exclusive reactions provides important information on the basic parameters of QCD. As we discussed in Section 8.2, recent BNL data for pp quasi-clastic scattering in nuclei at 0cm = f shows that the number of effective protons in the nucleus rises with the momentum transfer as predicted by color transparency at least up to piab = 10 GeV/c. This remarkable empirical result clearly rules out any description of exclusive reactions based on soft wavefunctions. The observation of the onset of color transparency in quasi-elastic pp —^ pp scattering appears to be an outstanding validation of a fundamental feature of perturbative QCD phenomenology. The tests of color transparency address directly the central dynamical assumption of the perturbative analysis, that exclusive reactions at high momentum transfer are controlled by Fock components of the hadron wavefunction with small transverse size. However, in direct contradiction to PQCD expectations, the BNL data at higher momentum, piab ~ 12 GcVjc, indicates normal Glauber attenuation. He-
211 cause of the importance of this and other anomalies and the challenges they pose to the theory, we have devoted several sections of this article to these topics and their possible resolution. The successes of fixed-angle scaling laws could of course be illusory, perhaps due to soft hadronic mechanisms which temporarily simulate the dimensional counting rules at a range of intermediate momentum transfer. If such a description is correct, then the perturbative contributions become dominant only at very large momentum transfer. Quantities such as Q^Fir(Q^) would drop from the present plateau to the PQCD prediction, but at a high value of (J^, much higher than the natural scales of the theory. An important question is whether a soft hadronic model can also account for the normalization of the cross sections for other exclusive processes besides form factor measurements. For example, consider hadronic Compton amplitudes such as 7J3 —> 7p or 77 —♦ 7r"*';r~, As wc have shown in Section 7, the data appeax to scale in momentum transfer according to the perturbative QCD predictions. One can consider a simple model where the hadronic Compton amplitude is given by the product of a point-like Compton amplitude multiplied by the corresponding hadronic form factor. This mode] predicts d(T/dt('yp -► 7p) :^ 5 pb/GeV^ at 5 = 8 CeK^, 0 cm = ^/2 compared to the experimental value of 300 pb/GeV'^ (see Fig. 33). The same simple model predicts (7(77 —*■ tt'^'tt") :x O.i 726 at 5 = 5 GcV^ compared to the experimental value of 2 nb (see Fig. 31). The above estimates are also characteristic of the soft-scattering models in which the end-point large x regime dominates so that the Compton amplitude is given by the sum of coherent point-like quark Compton amplitudes with a:^ c^ 1 multiplied by the electromagnetic form factor. Again one has the problem that the normalization of data for large angle Compton scattering is one to two orders of magnitude larger than predicted. In contrast, in the perturbative QCD description there are many more contributing coherent hard scattering amplitudes for Compton scattering than lepton-proton scattering, so the large relative magnitude of the proton Compton cross section can be accounted for. In the case of large angle pp scattering, the large normalization of the data relative to that obtained by simply multiplying form factors can be understood as a consequence of the many coherent contributions to T// for this process. We also emphasize that the observation of color transparency in the BNL experiment implies minimal attenuation of the incident and outgoing protons and thus appears to exclude any model in which the full size of the hadron participates in the hard scattering reaction. Questions have been raised recently on a number of questions concerning the application of perturbative QCD to exclusive reactions in the momentum
212 transfer range presently accessible to experiment. The issues involved are very important for understanding the basis of virtually all perturbative QCD predictions. The debate is not on the validity of the predictions but on the appropriate range of their applicability because of possible complications such as nonperturba- tive effects. The questions raised highlight the importance of further experimental tests of exclusive processes. As we have discussed in this article, there are, in addition to the numerous successes of the theory, a number of major conflicts between perturbative QCD predictions for exclusive processes and experiment which can not be readily blamed on higher contributions in as(Q'^). For example, the helicity selection rule appears to be broken in irp —>• p^p scattering at large angles, the J/'tp —> Trp and J/^ —> KK* decays. The strong spin correlations seen in large angle pp scattering at >/5 = 5 GeV are not explained by PQCD mechanisms. Color transparency appears to fail at the same energy. Small but systematic deviations or oscillations are observed relative to the PQCD power-law behavior. In each case, the data seems to indicate the intrusion of soft non-perturbative QCD mechanisms such as resonances perhaps due to gluonic or color excitations or heavy quark threshold effects. The presence of contributions from Landshoff pinch singularities may also be indicated. Thus exclusive reactions still remain a challenge to theory. A crucial requirement for future progress is the computation of hadron light-cone wavefunctions directly from QCD. Unfortunately it appears very difficult to obtain much more than the leading moments of the distribution amplitude from either lattice gauge theory or QCD sum rules. The discretized light-cone quantization method reviewed in Appendix III shows promise, but so far solutions have been limited to QCD in one space and one time dimension. The computation of hadronic structure functions, magnetic moments, and electroweak decay amplitudes also require this non-perturbative input. The detailed understanding of the relative role of perturbative and non-perturbative contributions to exclusive amplitudes will unquestionably require a fuller understanding of the hadronic wavefunctions. Much more theoretical work is also required to compute the hard scattering amplitudes for experimentally accessible exclusive processes, and to understand in detail how to integrate over the pinch and endpoint singularities, taking into account Sudakov suppression in the non-Abelian theory. The computerized algebraic methods now available can be used to compute the hard-scattering quark- gluon amplitude Tjj for processes as complicated as pp —> pp and the deuteron form factor. Each Feynman diagram which contributes to T// represents a particular overlap of the participating hadron wavefunctions. Considering the uncertainties in the wavefunctions and the myriad number of diagrams contributing
213 to pp scattering, even getting the correct order of magnitude of the large angle cross section would be a triumph of the theory. Computations of the higher order corrections to high momentum transfer exclusive reactions will eventually also be needed. More precise predictions for color transparency is needed, particularly ep quasi-clastic scattering in nuclei. The analysis requires computing the detailed parameters which control the color transparency effect due to smallness of the participating Fock state amplitude, and by uncertainties involving the role of formation zone physics, which controls the length of time the hadron can stay small as it traverses the nucleus. The experimental study of exclusive reactions is also in its infancy. Much more experimental input is required paxticulajly from ep, 7p, pp, and 77 initial states. Ratios of processes such as 77 —^ pp and A'^'^A can isolate important features of the baryon wavefunctions. The ratio of the square transition form factor for 7*7 —^ tt^ to the pion form factor provides a wave-function independent determination of Qs{Q^)- ^t is important to confirm the color transparency phenomena, particularly in the simplest channels such as ep quasi-elastic scattering. It is important to verify that both elastic and inelastic initial and final state interactions are suppressed in the nucleus. Once this phenomena is validated it can be used as a "color filter" to separate soft and hard contributions to a large range of exclusive reactions. We have emphasized in this article that the correctness of the PQCD description of exclusive processes is by no means settled. There is now a strong challenge to design decisive experimental and theoretical tests of the theory. If the theory survives, the reward is high: through exclusive reactions we can explore both the behavior of QCD and the structure of hadrons. APPENDIX I BARYON FORM FACTORS AND EVOLUTION EQUATIONS The meson form factor analysis given in Section 3 is the prototype for the calculation of the QCD hard scattering contribution for the whole range of exclusive processes at large momentum transfer. Away from possible special points in the X, integrations a general hadronic amplitude can be written to leading order in l/Q^ as a convolution of a connected hard-scattering amplitude Tjj convoluted with the meson and baryon distribution amplitudes: <!>m{^.Q)= / :^^'^(a:,^x)
214 and m<Q'' M^iy Q)= / ld^f^±\i^qgq{^t, ^±t) • The hard scattering amplitude Tjj is computed by replacing each external hadron line by massless valence quarks each collinear with the hadron's iriomen- A fi turn pj* = 3^%?^' For example the baryon form factor at large Q^ has the form where T/f is the 3^ + 7 —► Zq' amplitude. For the proton and neutron we have to leading order [Cb = 2/3] where X3(l -a:i)2 7/3(1 - J/i)^ a^2(l - X\)^ 2/2(1 -2/1)^ «5(aJ2j/2Q^) Of5(x3y3(5^) 0:2X3(1 - X3) 2/2^/3(1 -l/i) and xiX3(l -xi) 1/12/3(1 - j/3) 7'] corresponds to the amplitude where the photon interacts with the quarks (1) and (2) which have helicity parallel to the nucleon helicity, and T2 corresponds to the amplitude where the quark with opposite hehcity is struck. The running coupling constants have arguments Q corresponding to the gluon momentum transfer of each diagram. Only the large Q^ behavior is predicted by the theory; we utilize the parameter Mq to represent the effect of power-law suppressed terms from mass insertions, higher Fock states, etc.
215 The Q^-evolution of the baryon distribution amplitude can be derived from the operator product expansion of three quark fields or from the gluon exchange kernel, in parallel with derivation of Eq. (90). The baryon evolution equation to leading order in a^ is 1 {U ~ 3 Cy F " 1 C B f 0 Here (j> = xiT2a;3<^,C = log(log(3VA^), Cf = (n? - l)/2nc = 4/3, Cb = [ric + \)/2nc — 2/3, /9 = 11 — (2/3)n/, and V(xi,yi) is computed to leading order in as from the single-gluon-exchange kernel [see Fig, 19(b)]: Vj I ^f^^hj A V{^t.yi) = 2xtX2X^y^ $(yt - Xi)6(xk -yk)— ~- + Xj \ Xt T X7 yx •*'» = V'(y.,^«) • The infrared singularity at Xj = yi is cancelled because the baryon is a color singlet. The evolution eqiiation automatically sums to leading order in aj(Q^) all of the contributions from multiple gluon exchange which determine the tail of the valence wavefunction and thus the (J^-dependence of the distribution amplitude. The general solution of this equation is <^(a:i,Q) = 0:10:2x3 ^ a„ Un-J ) <^„(xi) , n=0 ^ where the anomalous dimensions 7n and the eigenfunctions ^n(xt) satisfy the characteristic equation: 1 0 A useful technique for obtaining the solution to the evolution equations is to construct completely antisymmetric representations as a polynomial orthonormal
216 basis for the distribution amplitude of multiquark bound states. In this way one obtain a distinctive classification of nucleon (N) and delta (A) wave functions and the corresponding Q^ dependence which discriminates N and A form factors. This technique is developed in detail in Ref, 117. Taking into account the evolution of the baryon distribution amplitude, the nucleon magnetic form factors at large Q^, has the form «?(0')V-. A QW''" [, , ^/ ,^2^ m 2 Gm{Q') - ^ E ^""^ ( >°g72 ) 1 + ^ ( -'(«')' Q where the 7„ are computable anomalous dimensions of the baryon three-quark wave function at short distance and the bmn are determined from the value of the distribution amplitude <^b(^i Qq) a.t a given point Qq and the normalization of T//. Asymptotically^ the dominant term has the minimum anomalous dimension. The dominant part of the form factor comes from the region of the x, integration where each quark has a finite fraction of the light cone momentum. The integrations over Xi and y, have potential endpoint singularities. However, it is easily seen that any anomalous contribution [e.g. from the region X2,X3 ^ 0(m/Q).x\ ~ 1 — 0{Tn/Q)] is asymptotically suppressed at large Q^ by a Sudakov form factor arising from the virtual correction to the q'yq vertex when the quark legs are fi 1 0 near-on- shell [p2 - 0{mQ)]:' This Sudakov suppression of the endpoint region requires an all orders resummation of perturbative contributions, and thus the derivation of the baryon form factors is not as rigorous as for the meson form 19 factor, which ha^ no such endpoint singularity. One can also use PQCD to predict ratios of various baryon and isobar form factors assuming isospin or 56^(3)-flavor symmetry for the basic wave function structure. Results for the neutral weak and charged weak form factors assuming standard SU(2) x ^(1) symmetry are given in Ref. 47. APPENDIX II LIGHT CONE QUANTIZATION AND PERTURBATION THEORY In this Appendix, we outline the canonical quantization of QCD in A'^ = 0 gauge. The discussion follows that given in Refs. 4 and 51. This proceeds in several steps. First we identify the independent dynamical degrees of freedom in the Lagrangian. The theory is quantized by defining commutation relations for these dynamical fields at a given light-cone time t — t -\- z (we choose r — 0). These commutation relations lead immediately to the definition of the Fock state ba^is. Expressing dependent fields in terms of the independent fields, we then
217 derive a light-cone Hamiltonian, which determines the evolution of the state space with changing r. Finally we derive the rules for r-ordered perturbation theory. The major purpose of this exercise is to illustrate the origins and nature of the Fock state expansion, and of light-cone perturbation theory. We will ignore subtleties due to the large scale structure of non-Abelian gauge fields {e.g. *instan- tons'), chiral symmetry breaking, and the like. Although these have a profound effect on the structure of the vacuum, the theory can still be described with a Fock state basis and some sort of effective Hamiltonian. Furthermore, the short distance interactions of the theory are unaffected by this structure, or at least this is the central ansatz of perturbative QCD, Quantization The Lagrangian (density) for QCD can be written C = 1 --Tr(F'^^F^,) + 0(^^-m)T/; where F*""" = d^'A'' - d'^A^ + iglA^", A""] and iD*' = id^ ~ gA^. Here the gauge field A^ is a traceless 3x3 color matrix {A*" = J2a ^"^y'^ Tr(r'^T'') = l/2(5'^^ jy-a j>6j _ ^^abcrpc^ ^^ ^^^j ^^^ Quark field 0 is a color triplet spinor (for simplicity, we include only one flavor). At a given light-cone time, say r = 0, the t independent dynamical fields are ?/>± = A-i-0 and ^4^^ with conjugate fields 2^| and d'^A*j_, where A± = 7^7^/2 are projection operators (A+.\_ = 0, Aj. = A±i A+ + A_ — 1) and ^* = C^^ i: d"^. Using the equations of motion, the remaining fields in C can be expressed in terms of ^-|., A^j_: tl}- = A_^ = T 1 id+ —# iD± • a_L + /?m ^+ = p-- 1 id+ /l+ = 0 , A- = 2 2<9+ 2.9 idi • ii + T-^ {[id^A^^. A\] ^ 24 r ^'^ T /.t T« „. T^ = A- + 2g (t5+) 2 Uid-^A\,A\]^2'^\T''xp+T with P = ^^ and qx — 7^7-
218 To quantize, we expand the fields at r = 0 in terms of creation and annihilation operators, ^^(^)= / %Te^J:{'(!^''^-^(^''^''''- ifc+>0 + d^{LX) v+ik,X) e'^''\ , T = x-^ = 0 dk-^dH ^'^^'^^ / ^^E{«(^'^)^iW^"'''' + ^'^} ' ^ = ^^-0, Ar+>0 ^ with commutation relations {h= (^"'",^_l))' {b{k,x), b^p,x)]^[d{k,x), </t(p,Y)} = [a(LX), a\p,X^) 3 L-\- a = ieiT^k^6%k-p)Sxx' , {6,^} = {d,d} = ... = 0 , where A is the quark or gluon helicity. These definitions imply canonical coni- mutation relations for the fields with their conjugates (r = j;"*" — i/"^ = 0, x_ = (x-,xj_),...): The creation and annihilation operators define the Fock state basis for tlic theory at r = 0, with a vacuum |0} defined such that 6|0} — d\Q) — a |0) = 0. The evolution of these states with r is governed by the light-cone Hainiltonian. Hic = P~-, conjugate to r. The Hamiltonian can be readily expressed in terms * of ip^ and .4j_:
219 where //n = / d^x Tr {d]^A{diA'^) + 4 (id± ' a± + /?m) j^ {id^ • a^ + 0m) 0+1 JA:+c/2jt _ V f 1 h^. ^k+ a^k, A) a(^, A)-^ + 6^ (i, A) b{k, A) /b+ colors X k^. + 1 m /b+ + ^t(^,A)6(&,A)^ii^ + constant is the free Hamiltonian and V the interaction: 1 V= I d^x [2glY{id*'A''\At,,AS\- ^grp /A^l;-^9^'i\[ id^A\A^ 2 1 Tr A^,A {id^) ^a+A^A A 9- 7+ ~ o^ , / 1 2id+ (^a+) ^aM^^^ ■> + Y ^7+ T> -^ ,/.7+ TV !■ , ^j^i^l j 0 with 0 = V- + 0+ (—* 0 as ^r —► 0) and A'* = (0, A~, /1*l) (—^ .4'' as ^ Fock states are obviously eigenstates of IIq with Hq \n : k;^,k_i^) = ^ f ibU 1 m A:+ /I • K^ ) /Cj^ ^ ) « t 0). The It is equally obvious that they are not eigenstates of V, though any matrix element of V between Fock states is trivially evaluated. The first three terms in V correspond to the familiar three and four gluon vertices, and the gluon-quark vertex [Fig. 53(a)]. The remaining terms represent new four-quanta interactions containing instantaneous fcririion and gluon propagators [Fig. 53(b)]. All terms conserve total three-momentum ^ = {k^,k±), because of the integral over x in V. iMirthermore, all Fock states other than the vacuum have total /j"*" > 0, since each individual bare quantum has k'^ > 0. Consequently the Fock state vacuum
220 (o) (b) 'Vrx/y-^ •^yv^ 3-83 4507A26 Figure 53. Diagrams which appear in the interaction Hamiltonian for QCD on the light cone. The propagators with horizontal bars represent "instantaneous" gluon and quark exchange which arise from reduction of the dependent fields in A"^ = 0 gauge, (a) Basic interaction vertices in QCD. (b) "Instantaneous" contributions must be an eigenstate of V and therefore an eigcnstate of the full light-cone Hamiltonicui. Light-Cone Perturbation Theory We define light-cone Green's functions to be the probability amplitudes that a state starting in Fock state \i} ends up in Fock state |/} a (light-cone) time r later (f\i)G{f,i;r) = {f\e-'^'^'^'\i) __ f dt ,^ = t / -— e J 27r -i€r/2 G(f.z;e){f\i), where Fourier transform G(/,r, e) can be written {mG{f,i;t)=(f 1 f - Mlc + iO+ = f 1 + 1 V 1 e - ^LC + iO+ c - //o -f ^0+ c - Hq-{- iO+ + 1 V 1 V 1 t - i/o + tO+ €~Ho-\-iO+ e- Jh + ^0+ + I The rules for r-ordered perturbation theory follow immediately when (e — Ho) is replaced by its spectral decomposition. -1 ' = y dkfd^k J Al 167r3Jb+ e~ n : ^, A,) {n : 4,-, A, S(A;2 + m2)./A;+ + iO+
221 The sum becomes a sum over all states n intermediate between two interactions. To calculate G[f,i\t) perturbatively then, all r-ordered diagrcims must be considered, the contribution from each graph computed according to the following rules: 1. Assign a momentum k^ to each line such that the total A:"^, ^x ^^^ conserved at each vertex, and such that A:^ = m^, i.e. k' =^ {k^ -\- Tn^)/k'^. With fermions associate an on-shell spinor. x(T) A-T Vk+ V" ' -"" ' ^^ •^-/ 1 x(i) A =i u(i, A) = -j== (^k-^ 4- /5m + ax or ^(i> ^) - ~/= (k^- i3m i- d±'k^ A=T A=i where x(T) = l/\/2(l,0,1,0) and x{l) = l/vf (0, LO,-1)^. For gluon lines, assign a polarization vector t*^ = (0, 2e± • k±/k'^^ e±) where ex(T) — -l/x/2(l,0 and eKi) - l/\/2(l, -i). 2. include a factor 0(k'^)/k^ for each internal line. 3. For each vertex include factors as illustrated in Fig. 54. To convert incoming into outgoing lines or vice versa replace u <r-^ V , w <-♦ —v , f <-+ f* in any of these vertices. 4. For each intermediate state there is a factor 1 interin where c is the incident P , and the sum is over all particles in the intermediate state. 5. Integrate J dk'^cfik^/I^tt'^ over each independent k, and sum over internal helicities and colors. 6. Include a factor -1 for each closed fermion loop, for each fermion line that both begins and ends in the initial state {i.e. v., .u), and for each diagram in which fermion lines arc interchanged in either of the initial or final states.
222 a—> :p^ Vertex Factor <ju{c) jl^ u{n) Color Factor r + cyclic permutations} iC ^abc ^^abe ^(^cde a 2 ,-, d r ^(a) ;^6 7 + 2[p7 - Pj) ^r W(c) (Pc^ + p;~) ''''' ^Wvt^d "' "'"' ^^"<^' ,+ _r.+ (p?-pI) .. Cj • e, {P?+Pj)^ '■ b c 2: d" {Pr. - pi r^ fj^h rpd zC°^^ iC"^^ ^QcAe jc C T'C rpej rKr\/\ /\/\,^K/\/xr /\/\A^\/vr -f < Figure 54, Graphical rules for QCD in light-cone perturbation theory. As an illustration, the second diagram in Fig. 54 contributes 1 Oik: - K) i—b,d + X .9' E ^W ^\ka-kb. A) u{a) u(d) Aka - h. A) ti(c) A 1 t=a,c (times a color factor) to the qq —> qq Green's function. (The vertices for quarks and gluons of definite helicity have very simple expressions in terms of the mo- iiicritaof the particles.) The same rules apply for scattering amplitudes, but with
223 propagators omitted for external lines, and with e = P of the initial (and final) states. Finally, notice that this quantization procedure and perturbation theory (graph by graph) are manifestly invariant under a large class of Lorentz transformations: 1. boosts along the 3-direction - - i.e. p^ —^ ^^P^j P" —* K~^p~, p± -+ p± for each momentum; 2. transverse boosts — i.e. p"*" ^ p"*", p~ —► p~ + 2p± ♦ Q± + p'^Q\y Pi —^ p_l_ + p'^Q± for each momentum {Q± like A' is dimensionlcss); 3. rotations about the 3-direction. It is these invariances which lead to the frame independence of the Fock state wave functions. APPENDIX III A NONPERTURBATIVE ANALYSIS OF EXCLUSIVE REACTIONS- DISCRETIZED LIGHT-CONE QUANTIZATION Only a small fraction of exclusive processes can be addressed by pcrturba- tive QCD analyses. Despite the simplicity of the c'*'e~ and 77 initial state, the full complexity of hadron dynamics is involved in understanding resonance production, exclusive channels near threshold, jet hadronization, the hadronic contribution to the photon structure function, and the total e"^e~ or 77 annihilation cross section. A primary question is whether we can ever hope to confront QCD directly in its nonperturbative domain. Lattice gauge theory and effective Lagrangian methods such as the Skyrme model offer some hope in understanding the low-lying hadron spectrum but dynamical computations relevant to 77 an- 1 fi nihilation appear intractable. Considerable information on the spectrum and the moments of hadron valence wavefunctions has been obtained using the ITFP QCD sum rule method, but the region of applicability of this method to dynamical problems appears limited. Recently a new method for analysing QCD in the nonperturbative domain 118 has been developed: discretized light-cone quantization (DLCQ). The method has the potential for providing detailed information on all the hadron's Fock light-cone components. DLCQ has been used to obtain the complete spectrum of neutral states in QED and QCD in one space and one time for any mass and coupling constant. The QED results agree with the Schwinger solution at infinite coupling. We will review the QCDjl-fl] results bolow. Studies of QED in 120 3-1-1 dimensions are now underway. Thus one can envision a nonperturbative
224 Table III. Comparison Between Tinne-Ordered and r-Ordered Perturbation Theory Equal t Equal T = t -{- z k^ = V f^ + m- (particle ma^s shell) j^_ ^ ^j + m2 Y+— (particle mass shell) ^ k conserved Y^ kxy k"^ conserved Mab = Kl6 + E Kc y l^0_j. j^O _|_ ^^ Vac A^«6 = Vab-^Z Vac 1 Ea^ -Z.^'-^i^ ^Vcb n\ time-ordered contributions A:+ > 0 only Fock states T/-'n(^i) Fock states 07i(^*it*^i) n y: k^ = p^o (0< xi < 1) n f = P" - E ^r n 1=1 V
225 method which in principle could allow a quantitative confrontation of QCD with the data even at low energies and momentum transfer. The basic idea of DLCQ is as follows: QCD dynamics takes a rather simple form when quantized at equal light-cone "time" r = t -\- z/c. In light-cone gauge A-^ = A^ + A' ~ 0, the QCD light-cone Hamiltonian //qcd =Ho+9lh-\-g'^H2 contains the usual 3-point and 4-point interactions plus induced terms from instantaneous gluon exchange and instantaneous quark exchange diagrams. The perturbative vacuum is an eigenstate of //qcd ^ind serves as the lowest state in constructing a complete basis set of color singlet Fock states of //q in momentum space. Solving QCD is then equivalent to solving the eigenvalue problem: as a matrix equation on the free Fock basis. The set of eigenvalues {M } represents the spectrum of the color-singlet states in QCD. The Fock projections of the eigenfunction corresponding to each hadron eigenvalue gives the quark and gluon Fock state wavefunctions tpni^i-, ^±t5 ^%) required to compute structure functions, distribution amplitudes, decay amplitudes, etc. For example, as shown by Drell and Yan, the form-factor of a hadron can be computed at any momentum transfer Q from an overlap integral of the ?/>« summed over particle number n. The e'^e" annihilation cross section into a given J ~ \ hadronic channel can be computed directly from its ^g^ Fock state wavefunction. The hght'Cone momentum space Fock basis becomes discrete and amenable to computer representation if one chooses {anti-)periodic boundary conditions for the quark and gluon fields along the z~ = z — ct and zj_ directions. In the case of renormalizable theories, a covariant ultraviolet cutoff A is introduced which limits the maximum invariant mass of the particles in any Fock state. One thus obtains a finite matrix representation of //qcj) which has a straightforward continuum limit. The entire analysis is frame independent, and fermions present no special difficulties. Since HiCy ^^> ^l-> ^^id the conserved charges all commute, Hic iy block diagonal. By choosing periodic (or antiperiodic) boundary conditions for the basis states along the negative light-cone ip(z~ — -\-L) — ±il}{z~ = —L), the Fock basis becomes restricted to finite dimensional representations. The eigenvalue problem thus reduces to the diagonalization of a finite Hcrmitian matrix. To see this,
226 note ihat periodicity in z~ requires P*^ = ^7^ , kf = ^ ni , Yll^i ^^» — ^^• The dimension of the representation corresponds to the number of partitions of the integer K as a sum of positive integers n. For a finite resolution K, the wavefunction is sampled at the discrete points kf n, { \ 2 K -I •^t ~ r^^ — T^ ~ ^ T^ y r^t P+ K \K' X ' *" K The continuum limit is clearly K -+ cx>. One can easily show that P~ scales as L. One thus defines P~ ~ ^H . The eigenstates with P^ :^ ^2 ^^ f^^^j p4- ^^d Pj. = 0 thus satisfy i/ic l^> ^ KH 1^) — M^ 1^), independent of L (which corresponds to a Lorentz boost factor). The basis of the DLCQ method is thus conceptually simple: one quantizes the independent fields at equal light-cone time r and requires them to be periodic or antiperiodic in light-cone space with period 2/y. The commuting operators, the light-cone momentum P^ = ^/\ and the light cone energy P~ ~ -^H are constructed explicitly in a Fock space representation and diagonalized simultaneously. The eigenvalues give the physical spectrum: the invariant mass squared M^ — P^Pj,. The eigenfunctions give the wavefunctions at equal r and allow one to compute the current matrix elements, structure functions, and distribution amplitudes required for physical processes. All of these quantities are manifestly independent of L, since M^ = P'^P~ = HK. Lorentz-invariance is violated by periodicity, but re-established at the end of the calculation by going to the continuum limit: L -^ oo, K —^ oo with F"*" finite. In the case of gauge theory, the use of the light-cone gauge A'^ = 0 eliminates negative metric stales in both Abelian and non-Abelian theories. Since continuum as well as single hadron color singlet hadronic wavefunctions are obtained by the diagonalization of Hic^ ^^^ can also calculate scattering amplitudes as well as decay rates from overlap matrix elements of the interaction Hamiltonian for the weak or electromagnetic interactions. An important point is that all higher Fock amplitudes including spectator gluons are kept in the light- cone quantization approach; such contributions cannot generally be neglected in decay amplitudes involving light quarks. The simplest application of DLCQ to local gauge theory is QED in one-space and one-time dimensions. Since A'^ = 0 is a physical gauge there are no photon degrees of freedom. Explicit forms for the matrix representation of Hqed are given in Ref. 8.
227 The basic interactions which occur in Hic{QCT)) are illustrated in Fig. 53. 119 Recently Hornbostel has used DLCQ to obtain the complete color-singlet spectrum of QCD in one space and one time dimension for Nq = 2,3,4. The hadronic spectra are obtained as a function of quark mass and QCD coupling constant (see Fig. 55). Where they are available, the spectra agree with results obtained earlier; 8 CT> 0 a> SU(2) SU(3) SU(4) Homer: SU(a) Lattice T T (q) Baryon Mass (b) Meson Mass 0 0.5 .0 L5 m/g Figure 55. The baryon and meson spectrum in QCD [l + l] computed in DLCQ 119 for Nc = 2,3,4 6is a function of quark mass and coupling constant. in particular, the lowest meson mass in SU(2) agrees within errors with lattice 121 Plamiltonian results. The meson mass at Nc = 4 is close to the value obtained in the large Nc limit. The method also provides the first results for the baryon spectrum in a non-Abelian gauge theory. The lowest baryon mass is shown in
228 Fig. 55 as a function of coupling constant. The ratio of meson to baryon mass as a function of Nc also agrees at strong coupling with results obtained by Frishman 122 and Sonnenschein.*'^* Precise values for the mass eigenvalue can be obtained by extrapolation to large K since the functional dependence in IjK is understood. ' ' T ' I T—'—I—r 1—•—r 51 in I K - 13/2 0 0.2 0.4 0.6 0.8 |/(l.7rmVq2)'/2 .0 Figure 56. Representative baryon spectrum for QCD in one-space and one-time ,. 119 dimension. As emphasized above, when the light-cone Hamiltonian is diagonalized for a finite resolution A', one gets a complete set of eigenvalues corresponding to the total dimension of the Fock state basis. A representative example of the spectrum is shown in Fig. 56 for baryon states (B = 1) as a function of the dimensionlcss variable A = 1/(1 + 7rm^/g^). Antiperiodic boundary conditions are used. Note that spectrum automatically includes continuum states with B = 1 . 0.5 0.4 I- •f""T 1 ' 0.3 - -O 0.2 0. o -©-.-• i 0 *—i-Mi^i- 1— 1 T—I—I I '—I—r"'^ I—till $U<3) MESON o m/B« 1 w r -0' — • — * —O — ♦ — • — O— + — « . - • *• • J i- 0 0.? 0.4 0.6 . 1 , 0.8 •I -I .0 Figure 57. The meson quark momentum distribution in QCD[l-f I] computed using DLCQ.
229 1.5 1.0 0,5 0. —r " * 1—L O- Jfl ' - - I_ ' 1 ' 1 - ♦ . 1 . "T 1 1 1 ' * ' X J 1 1 I 1 1 * ' 1 t» 1 -r— X r 1 I ■ r ■' r ■ SU(3) BARYON X m/g = I 6 <> iii/g= 1 *<> i ly 1 L^ I ■1 -I — — c 0.2 0.4 0.5 0.8 X -- k/K 0 Figure 58. The baryon quark momentum distribution in QCD[1 + 1] computed using DLCQ. r—r [—!■ r--t ' 1 * 1—| i i ■ I T—\—'—T—I—I—^—v-'v -I—1—r SU(3) BAHYON "=• m/g= I (-10*) 1 ■ J -I .*H «- ^ • .X Q li I J 1 J I . 1 0 0.2 J L _L J- i— 1 -3C 1 0.4 0.6 ^ 1 0.8 i.O Figure 59. Contribution to the baryon quark momentum distribution from qqqqq states for QCD[1+1].^^^ The structure functions for the lowest meson and baryon states in SU(3) at two different coupling strengths mjg = 1.6 and mjg = 0.1 arc shown in Figs. 57 and 58. Higher Fock states have a very small probability; representative contributions to the baryon structure functions are shown in Figs. 59 and 60. For comparison, the valence wavefunction of a higher mass state which can be identified as a composite of meson pairs (analogous to a nucleus) is shown in Fig. 61. The interactions of the quarks in the pair state produce Fermi motion beyond X = 0.5. Although these results are for one time one space theory they do suggest that the sea quark distributions in physical hadrons may be highly structured. 4 In the case of gauge theory in 3+1 dimensions, one also takes the k^^ —
230 1.5 I I .0 •- 0.5 0 i.. 0 1111 r—T" O- <^ 1—r S{]{'\) RAHYON >* ni/g=l 6 (HO') 0 m/g- 1 (xlO"*) ___| 1 |_j^J 1 L-i- i L L0 I 0.2 0.^ 0.6 0.8 1.0 Figure 60. Contribution to the baryon quark momentum distribution from qqqqqqq states for QCD[H-1]. .5 c I 1.0 « 1—I—I—I—r—r—^-——I—•—I—I—I I r—i—1~ SOU) MSbON y- vuj^Nrr wrrs or ucsok p«ir « S'MGlf We50N 4-QK HIGFrR TOOK WFN («13') H •t hr I I- 0.5 -r C ^ ■ \ * 61 . «( >? 0 \ - _!. J I I J. I A . 0.2 0/f 0.6 0.8 .0 X = k/K Figure 61. Comparison of the meson quark di.stributions in the qqqq Fock sate with that of a continuum meson pair state. The structure in the former may be due 119 to the fact that these four-particle wavefunctions are orthogonal. (27r///_L)nj^ as discrete variables on a finite cartesian basis. The theory is covari ariLly regulated if one restricts states by the condition E ^i, + mj < A Xi where A is the ultraviolet cutoff. In effect, states with total light-cone kinetic energy beyond A^ arc cut off. In a renormalizable theory physical quantities are independent of physics beyond the ultraviolet regulator; the only dependence on A appears in the coupling constant and mass parameters of the llamiltonian,
231 123 consistent with the renormalization group. The resolution parameters need to be taken sufficiently large such that the theory is controlled by the continuuni regulator A, rather than the discrete scales of the rnomcntuiii space basis. There are a number of important advantages of the DLCQ method which have emerged from this study of two-dimensional field theories. They are as follows: 1. The Fock space is denumerable and finite in particle number for any fixed resolution K. In the case of gauge theory in 3+1 dimensions, one expects that photon or gluon quanta with zero four-momentum decouple from neutral or color-singlet bound states, and thus need not be included in the Fock basis. 2. Because one is using a discrete momentum space representation, rather than a space-time lattice, there arc no special difficulties with fermions: e.g. no fermion doubling, fermion determinants, or necessity for a quenched approximation. Furthermore, the discretized theory has basically the same ultraviolet structure as the continuum theory. It should be emphasized that unlike lattice calculations, there is no constraint or relationship between the physical size of the bound state and the length scale L, 3. The DLCQ method has the remarkable feature of generating the complete spectrum of the theory; bound states and continuum states alike. These can be separated by tracing their minimum Fock state content down to small coupling constant since the continuum states have higher particle number content. In lattice gauge theory it appears intractable to obtain information on excited or scattering states or their correlations. The wavefunctions generated at equal light cone time have the immediate form required for rel- ativistic scattering problems. In particular one can calculate the rclativistic form fcictor from the matrix element of currents. 4. DLCQ is basically a relativistic many-body theory, including particle number creation and destruction, eind is thus a basis for relativistic nuclear and atomic problems. In the nonrelativistic limit'the theory is equivalent to the many-body Schrodinger theory. Whether QCD can be solved using DLCQ — considering its large number of degrees of freedom is unclear. The studies for Abelian and non-Abelian gauge theory carried out so far in 1-f-l dimensions give grounds for optimism. ACKNOWLEDGEMENTS We wish to thank the following: G. de Teramond, J, F. Gunion, J. R. lliller, K. Hornbostel, C. R. Ji, A. H. Mueller, H. C. Pauli, D. E. Soper, A. Tang and S. F. Tuan.
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COHERENCE AND PHYSICS OP QCD JETS Yu.L.Dokshitzer, V.A.IQioze, S.I^Troyan Leningrad Nuclear Physics Institute Gatchina, Leningrad 188350, USSR 241 This paper presents a review of euaalytical perturbative approach to QCD jet physics. The role of coherent phenomena reflecting the collective character of multiple hadroproduction is emphasized
242 TABLE OP CONTENTS 1• INTRODUCTION 1.1. Perturbative Approach to Hard Processes and Jets 1.2. Perturbation Theory and Shower Picture 1.3« Leading Logs, Coherence and Hadronization Schemes 2. SPACE-TIME PICTURE OP QCD BREMSSTRAHLUNG AND LOCAL PARTON-HADRON DUALITY 2.1. Radiation of Partons 2.2. Porraation and Hadronization Times 2.3« Gluons and 'Gluers': Soft Confinement Scenario 2.4# Angular Ordering and 'Partonic Gas* 2.5* LPHD Concept 3. ESSENCE OP QCD COHERENCE 3*1• Angular Ordering of Successive Parton Branching 3«2. Hump-Backed QCD Plateau in Particle Spectra 3«3» Soft Gluon Emission from Colourless 'Quark- -Ant iquark An t enna * 3*4. Physical Origin of Drag Effect 4. DOUBLE LOG APPROXIMATION 4.1. Tree Multigluon Amplitudes for e'*"e"" —»► qq + Ng 4«1.1# Two gluon emission off a quark p_ 4.1.2. Angular ordering (N = 2) 4#2. Proof of Angular Ordering 4»3# Virtual Corrections A*A* Cross Section. Method of Generating Punctionals 4.5« Multiplicity Distributions in DLA 4*6. Inclusive Particle Spectra in DLA
243 4.7* J$ -Scaling 5. MODIFIED LEADING LOG APPROXIMATION 5.I. Exact Angular Ordering 5•2* MLLA Master Equation 5«3« V-Scheme for Gluon Cascades 5.3.1• Conditional probability V and'interference remainder' 5.3.2. Test of V-scheme in higher orders 5.4. Jet Polarizability and Colour Monsters 5.5. Magnitude of Dipole Corrections to Jet Characteristics 6. MLLA RESULTS FOR JET CHARACTERISTICS 6.1. Correlators of Jet Multiplicity 6.2. Inclusive Energy Spectrum of Partons in MLLA 6.3. Developed Cascade and LPHD Concept 6.4. On Infrared Stability of Limiting Parton Spectrum 7. CHROMODYNAMICS OP HADRONIC JETS 7.1. On Experimental Selection Procedures 7.2. On Structure of Particle Plows in Multiset Events 7.3. QCD Portrait of Individual Jet 7.3.1. Colliraation of energy in jet 7.3.2. Energy spectrum of particles within given cone 7.3.3. Collimation of multiplicity inside jet 7.3.4. Angular distribution of multiplicity inside jet 8. RADIOPHYSICS OP PARTICLE PLOWS 8.1. Inclusive QCD Portrait of qqg Events of e"*'e"" Annihilation 8.1.1. Spatial distribution of multiplicity flow
244 8.1.2# On total particle multiplicity In qqg events 8•^•3. Drag effect in three-jet events 8•2. Drag Phenomena in High p^ Hadronic Reactions 8.3• Prompt J Production at Large p 8#4« Two Jet Production at Large p^^ 8#5« Correlations of Internet Particle Plows 8.6« Azimuthal Asymmetry of QCD Jets 9. COHBHMCB AND PINAL PINAL STATES IN DEEPLY INELASTIC SCATTERING ^.^• The structure of Soft Radiation Associated with DIS 9•2. Angular Ordering for Space-Like Cascades 9*3» The Structure of Inclusive Specti^im in Target Pragmen tat i on 9.4. On the QCD Solution of Peynman-Gribov Puzzle 10. CONCLUSIONS REPERENCES
245 1• INTRODUCTION The aim of this paper is to review the perturbative approach to multiple hadroproductioQ which we consider to be an important tool for gaining information about the colour confinement. By the word * confinement' we meam here not the formal proof of the desired property of gauge theory with unbroken non-Abelian symmetry, but the meeining of concrete knowledge about how the 'device* transforming colour fields into white hadrons operates in real processes* 1.1. Perturbative Approach to Hard Processes and Jets Clearly it is the Hard Process (HP) to serve as the base for scrutinizing this know-how. Here one can unambiguously use the language of quarks and gluons to give a detailed description of the small-distance stage of the 1 —8^ evolution by means of the Perturbation Theory (PT) ''. 9-1S^ Recent experiments ^ ^' have presented the solid evidence for the jet structure of final states in HPs. These jets are now being intensively studied both at e"^e" and hadronic machines. Hadronic jet physics will be one of the central problems of investigation for the e"*'e", pp fiind ep colliders of the future. Detailed studies of jets is of importance for better understanding and testing both PT and non-PT QCD, for designing experiments of the present and of the future, and for finding manifestations of new physics. It would be impossible to be complete in covering the field, announced in the title, so we must apologize in advance for being selective in topics discussed and references cited. One of the main objects of this paper was to give an introduction to the analytical PT approach which has not
246 been presented so far in English systematically ''6-20) ^.2^ Petrurbation Theory and Shower Picture In cxirrent high energy accelerators a collision between two particles may typically lead to the production of ten to hundred offsprings* Exact calculations of QCD matrix elements for multipartonic systems are difficult to use even in the cases where they can be obtained* Therefore, one meets the problem of developing an appropriate PT technique to describe, at least approximately, the properties of such systems* The desired PT approximation has to be: 1* complete in accounting for main physical ingredients (colour dynamics of parton multiplication processes, asymptotic freedom, energy-momentum conservation etc*), 2* asymptotically exact, 3* powerful in giving testable quantitative predictions with controllable accuracy, 4* physically transparent and 5* systematic in improving the accuracy. The key idea is to invoke the parton shower picture 4,u,21-2d; ^iiepe qjjq yiews the evolution, say, of a jet as a sequence of parton branchings* Generally speaking, using a shower picture does not necessarily lead to a loss of accuracy in describing multiparton phenomena. The main idea of the shower picture is to reorganize the perturbative expansion in such a way that its zero order approximation is systematic and involves an arbitrary number of produced particles* This zero order approximation can be achieved through an iteration of basic, A —^ B+C, parton branchings. In principle, it should be possible to include higher correction to
247 the basic braschlng along with higher point branching Tertioes A -^ B^-C-fD*** in order to systematically improTe the accuracy of a calculation. This procedure '^ is closely related to a renormalization group 27) approach '^ where the branchings are not so visible and where higher order corrections can be systematically calculated* 1«3« Leading Logs, Coherence and Hadronization Schemes The above mentioned zero order approximation is the Modified Leading Logarithmic Approximation (MLLA) taking care of both double logarithmic (DL) and single logarithmic which proves to be necessary to predict quantitatively properties of multi- particle systems with reasonable accuracy* Constructing the MLLA we shall pay special attention to maintaining the probabilistic interpretation of jet evolution* An existence of such interpretation is far from trivial in the problems connected with description of soft pai*ticle distributions (x «1)* Here interference contributions play an important role and prove to be unavoidable, unlike the case of the fcuniliar PT problems dealt with x ^ 1 particle spectra (deep inelastic scattering and e'^e" annihilation structure functions, Brell- -Yan and related semiinclusive processes, etc*, see 1,5) and refs* therein) where it was the matter of skillful choice of a gauge to approve the straightforward probabilistic picture* Nevertheless it appears to be possible, by choosing an appropriate evolution parameter (jet opening angle) and accounting for specific angular dependence of soft emission probabilities to maintain probabilistic interpretation of the dynamics of soft partonic cascades 24,25»17;^
248 This not only helps 02ie>s physical intuition but provides the base for Monte Carlo simulations of jet physics* The most elaborated MC schemes ^^"•^S)^ basing on the concept of well-developed QCD cascade, becoming better and better at building in realistic freigmentation and proper QCD evolution, successfully describe the wealth of experimental data* It is important to notice, however, that the use of MC generators for describing the development of multipartonic systems in terms of classical Markov chains proves to be of limited value, in principal* For example, the collective QCD phenomena in multiset ensembles could be reproduced by MC simulations only in the large-N^ limit* We shall focus specially on manifestations of coherent phenomena* The rediscovez^ of coherence in QCD context in early eighties •^^»-^'' has led to dreunatic revision of theoretical expectations about the structure of soft particle distributions *"°^. Thus the coherent effects in the intrajet partonic cascades, resulting, on average, in the angular ordering of sequential branching, gave rise to the hump-backed plateau - one of the most striking PT QCD predictions 38-42,28) # Due to the internet coherence, responsible for the dxag effects In nmltldet eTents *3.7.85, ^^^ ^^^ ^^^^. ant physical phenomenon can be said to be experimentally Terified, namely the fact that it is the dynamics of the colour which governs the particle production in accordance with the QCD radiophysics of hadron flows* Surely, the main lesson of the observations is not the proof of coherence: it would be inexcusable to check quantum mechanics at modem accelerators* Of real importance is that the PT-coherence has revealed itself in
249 hadron spectra, i.e* confinemexit has not disturbed the PT -picture of particle generation. The fact that non-PT effects do not radically rearrange a parton system at the confinement stage provides evidence in favour of locality of parton blanching and hadronization processes in configuration space 4*>^^', thus, supporting the hypothesis f Local Parton-Hadron Duality (LPHD) ^^7,41)^ The phenomenological fragmentation schemes reflections way or another, the coherent phenomena* Thus the cluster Webber-Marches in i model ^^""-^^ ^ naturally incorporates the angular ordering; the Lund string picture ^^^, on the other hand, appears to be suitable for a qualitative reproducing the drag phenomena. Moreover, in the modem versions of MC algorithms both types of QCD coherence might be built in ""^ ^« Such schemes seem to be well subtle for reproducing the bulk of interference phenomena, excluding some suitable effects. This paper is organized as follows. In Sec. 2 we shall sketch the space-time picture of the partonic system evolution, the role of PT bremsstrah- lung in the soft confinement scenario for foundation the concept of the Local Parton-Hadron Duality (LPHD). In Sec. 3 the physical origin of QCD coherence phenomena are discussed. One can find here the qualitative guide to the Angular Ordering (AO) in both time-like and 9pace-like partonic cascades, of the qq//qqg drag effect. In Sec. 4 we briefly discuss the main steps of the formal proof of AO in the Double Logarithmic Approximation (DLA) and present the most important DLA results. In Sec. 5 Logarithmic Approximation (MLLA) - asymptotically exact zero order approxima tion of PT, which corectly keeps leading eind next to
250 leading logarithms, is constructed* Specially emphasized is the possibility to provide the probabilistic interpretation of the partonic system evolution in MLLA and the failure of a naive classical branching picture beyond the BOiLA due to soft 'colour monster* contributions. In Sec* 6 the MLLA evolution equations for generating fianctionals are used to predict the shape of inclusive energy spectra of particles and to calculate the corrections to asymptotic KNO multiplicity distributions. In Sec* 7 we consider the gross features of the QCD event portrait, describe the angular collimn.tion of energr and multiplicity flows with the increase of jet energy* The problem of experimental selection rules is also discussed here* It is pointed out that forcing each hard scattering event to correspond to a definite number of jets is seemingly not a good procedure* We emphasize the use of infrared stable criteria for jets and suggest that the purely inclusive determinations of jet characteristics are probably the best way to make sharp connections between theory and experiment* In Sec* 8 the interference drag phenomena in the interjet soft particle distributions in e'^'e"" —^hadrons, higb-p^ /(E^WjH,***) and other jet production processes are studied* The azimuthal jet asymmetries prove to be of interest for checking subtle QCD effects* In Sec* 9 we consider the manifestations of QCD coherence in the structure of final states in Deep Inelastic Scattering (DIS) processes in the small-x region The QCD solution of the old-famous Feynman-Gribov paradox is discussed*
251 2. SPACB-TIMB PICTURE OP QCD BREMSSTRAHLUNG AND LOCAL PARfDON-HADRON DQALITY ^ ^ ^ 2*1 • Radiation of Partons The wealmess of colour isiteractlozi at small space-time distances does not imply a poverty of dynamics* Indeed, each HP is followed by cascades of parton (gluons and qq pairs) production* For example, at, say, W s 20+20 GeV e'^'e" annihilation energy parton population runs to a dozen of bremsstrahlung gluons accompanying the parent qq system* Radiation of a secondary parton does not lead normally to appearance of an additional resolvable jet since this quantum is quasi-collinear to the direction of original q and prefer to have relatively small energy* This is the characteristic propeorty of the bremsstrahlung gluon spectrum ^ ^s^ ^i^ ^^1 ^'< dy3' ^c -tr" —t ~r (2-^^ which is referred to as Double Logarithmic (DL)*The broad distribution over transverse momentum Ki ('collinear' or transverse' logs) occurs logarlthmi spectrum 'infrared' or 'longitudinal' logs)* The DL spectrum (2*1) corresponds to the wide region of gluon momenta % «> kj^« k « Q , (2.2) which results in the large total emission probability increasing logarithmically with the 'hardness' of the process Q*
252 [otlce that eq« (2*1} describes gluon radiation in the roved Bom approximation: the account of high order >cts makes the effective coupling o^^ run with kj_^ (2.3) %(^\) - ^ ,^^^,^ f ^ « 11/3-N^ - 2/3-n^ b In This can explain, formally, the appearance of the collinear cut-off Q^^ in (2.2): splitting of a parton into two with the small relative transverse momenttun k. ;^ R~^ -^ few hundred MeV proves to be beyond the scope of PT (large oC )• 2.2. formation and Hadronization Times To approach the problem of coexisting PT and non-PT physics we have to look at the space-time picture of the qq system evolution in e'*"e~ annihilation. Prom QED experience two phenomena are well known to be closely connected with each other, namely, the bremsstrah- l\xn& of real quanta and the regeneration of the classic field attached to a free charged particle. Just after the hard interaction one meets an accelerated charge as a bare particle which will be accompanied by the normal Lorentz contracted disk of Coulomb field only at large distance from the interaction point. The time needed to regenerate the field component with fixed longitudinal and transverse momentum projections can be estimated as *regen. ^ ^^^l ' (2.4) same formati bremsstrahlung , Applying the uncertainty relation to the virtual state p in the Peynman amplitude for photon radiation (Pig. 1) one has f - -^ = PQ ~ -^ =: ^ (2.5)
253 P P Pig«1« BremsstraJiluiig from a charged particle. The fact that in relativistic situation formation (regeneration) time may become macroscopic leads to a number of well-knomi peculiar QED phenomena* These considerations in the QCD context clear up the fortune of quarks created in e'^e" —^ qq« In the rest frame of a hadron the gluon field confining quarks has typical momenta Per the case of relativistic quarks with p ^W/2 such confining field has the momentum components k^ -ky^R-S k^ ~ R"^-(pR) = p €U3d due to eq« (2*4) needs some time *regen ^ P»^ (2.6) to be regenerated. Thus, starting from the annihilation time t^^ />J^/p^ 10"""^ fm up to hadronization time "^hadr'^^ ^ ^^ ^^* the 200 GeV-quark behaves as a true colour particle radiating gluons pex*turbatively« An instructive lesson comes from the case of ultraheayy quark Q with mass mQ> 100 GeV '*^^K Due to the semiweak decay (Q -^ W + q)its lifetime "^q^ becomes shorter than the hadronization time C^-1 fin-(—) . p^/mq < t^^^1 fm • p^/mq (2,7)
254 and all the bremsstrahlxmg processes prove to be tmder liie jurisdiction of PT QCD. One cem say that such quark in all aspects behaves as if it were a free coloured object« Applying the same arguments to a secondary part on k one concludes that its lifetime as a coloured object is restricted by t < thadr.'^^^* ^2*^^ Clearly one can say that an additional gluon is emitted really only if its formation time (2#5} is smaller than the hadronization time (2«8) k/k^ < t < kR^. (2.9) We come to the conclusion that the requirement kj^R > 1 (2.10) not only approves the applicability of PT, keeping o^^ formally small, but justifies the very opportunity to use the quark-gluon language for describing the process. The parameter (k^R) regulates, so to say, the lifetime of a secondary part on. The gluons with momenta satisfying the strong inequality kj^R » 1, (2.11) which are the main personages of the DL kinematics (2.2), will live for long, radiating, in their turn, new offsprings thus leading to the cascade multiplication of partons. 2.3* Gluons and 'Gluers': Soft Confinement Scenario What will be the final hadronic state for such a complex partonic system ? In attemping to answer this question let us turn before to the role of more dangerous - from the PT point of view - kinematic region, netmely, radiation at the lower edge of PT phase space - with finite transverse momenta.
255 It should be 'Something' which is radiated strongly ( c*^^ '^ 1) and even could be hardly treated as a gluon since due to (2«9) this 'Something' is forced to 'hadro- nize' just immediately after being formed. For a sake of definiteness let us call such an object Gluer : k^R ~ 1, (2.12) stressing the point that it is the prerogative of Gluers and not of Gluons to glue. Appearance of gluers is a signal of switching on the real strong interactions in a coloured system. According to (2.5) first gluers (with k ~k^~R ) are formed at t ^R after annihilation. It is the moment when the distance between q and q starts to exceed 1 fm. What a non-PT phenomenon has to happen at this moment ? The answer should be the separation of two jets as globally blanched subsystems. Such a blanching is neecbd (pragmatically) to dump the unstability which meuiifests itself in the PT framework as the catastrophic increase of interaction strength and, thus, restrains the gluer radiation probability. Though up to now we have no qiiantitative description of blanching process, the plausible picture of what is going on can be extracted from the Gribov's confinement scenario f where the cixicial restructuring of the Dirac sea of light quarks is forced by the strong external colour field created by outgoing quarks and leads to phenomenon qualitatively similar to the Q£D physics of supercharged ions with Z > 137* With time increasing gluers with larger and larger energies are formed. Their resulting energy spectrum, according to (2.1), , .
256 can be said to represent the famous imiform hadronlc plateau of the old partonlc picture* Thus, the plateau of relatively soft hadrons k « p appears due to sequential blancing of spreading colour fields in the qq system How will additional PT gluons (2.11) contribute to the hadronic yield ? We have met already with two characteristic time scales in the evolution of a secondary gluon: t^^^^ (2.5) and t^^^ (2.8). Let us introduce one more scale, namely, the moment when the bremsstrahlung pairticle and its parent will be separated by the critical 'confinement' dlataxioe R in the transverse plane Notice that the three time scales characterizing the gluon's being are parametrically ordered due to (2.11) ^hadr./'^separ. ^ ^separ.^'^form. ^ ^ »1- (2.15) At "^separ. ®^®® ^^^ specific non-PT interaction must take place to blcmch the total colour charge of the outgoing gluon (e.g., with the help of light qq pair in the octet state). Our qualitative estimates do support this need: at this moment appropriate gluers \<^ are formed which follow the gluon 6 »S , k --^ 1/R© , k. ^R""'* (2.16) *form ^ "^hadr. ^ ^^^l " ^© =" ^separ. Starting from t « t„^_^^ the gluon becomes an in- dependent source of hadrons with energies 1/R© < ^hadr ^ ^ • ^^'^^^ This additional plateau, from the PT point of view, seems to be C^/Cp a 9/4 ^^2 times higher than that of original quark ( qf. eq.(2.1)). It looks like a jet produced in
257 some HP with the effective hardness Q ~k^ but boosted with the Lorentz-factor Jf « ^/Q -^ 1. At first glance it might seem strange that the new subjet did not contribute to the yield of softest hadrons with ^"^ - ^hadr. "^ ^""^/^ ^^^ ^"^^- ^2.18) This is, in fact, oxi interesting phenomenon which stems from the very nature of QCD as gauge theory• The reason for this is the conservation of colour current plus coherence: soft hadrons in the energy interval (2«18) are formed early (t < t_^_^^ ), when the quark and the gluon k appear to be close to each other in the transverse plane. Therefore, they act with respect to gluers ( i^ €N/ R) as a single emitter with total colour charge equal to that of the original q (for more details see Subsec* 3*2}« The same arguments work when one considers the first blanching (and hadronization) acts at t ^^ R« In the previous discussion we spoke about the colour field in the qq system. In fact, the quarks here are not in solo flight, being accompcuiied by narrow bunches of secondary M partons (gluons) with t^^^ < R. Indeed, the probability to meet two original quarks without any accompaniment can be evaluated as Po ^ exp (- 2W ) , (2.19) where (2.20) is the total qtiark radiation probability (2.1) restricted
258 by ^fQjpm^ < "t. Here £, stands for the energy resolution, Q^j^a . i^Z) A o^ (2.21) A* Substituting t'^.R'^A"" one obtains which leads to the power form factor dumping the bare qq state 8 C HviZ P ^(W) b . (2.22) This means that normally the multipartonic * coats* haye already appeared nearly q and q at t < R* However, it is the coherence that makes the long-wave field insensitive to the internal structure of fraying parton jets and, thus, maJces the yield of hadrons with finite energies ^^hadr. ^^ ^ ) independent of W. The latter conclusion is among the brightest predictions of the PT approach and will be discussed in details later in Sec. 6« It was based on qualitative considerations with use of semiclassical space-time description of radiative processes. The coherent phenomena which can be emalysed in the PT framework strongly affect the evolution of pure partonic systems as well. 2.4. Angular Ordering and 'Partonic Gas' Later on we shall sketch the formal proof and give the qualitative explanation of the Angular Ordering (AO) discovered by Fadin, Ermolayev and Mueller -^ *"^''. AO states that the stmcture of partonic system representing the jet development can be treated (in the leading DL approximation) as a tree of independent soft gluon emissions Into sequentially shrinking angular cones. Now let us use this fact to justify an important
259 property of the AO partonic sceletons, namely, that the partons produced in QCD cascades have to hadronlze independently* To do this consider a pair of partons with the aaiae hadronization time t. = eR 2, k (see Fig. 2). ^0 z e,R Z (2.23) E Pig.2 Partonic branching in QCD cascade. 2 Spatial separation between partons at this moment can be estimated as ^? t^i0,-.6,) n (2.24) C^l viv-< A k I »©. Z »i\ (2.25) where we have substituted t^^ instead of the total evolution time of two partonic branches as the largest contributor due to strong time ordering (2.15)# The relative angle 6 can be expressed through the time scale of the decay of their common 'grandpa* 2 k 2 J- (2.26) mix) For the longitudinal distance (2.25) one easily derives U0 1= ■E^'I ^ " ^1 rr , (2.27) E i E where Zi(2) O -^1 ~I. •—Vy\iv\ B-|(2)/Bo " energy (longitudinal momenttim)
260 fractions in the decay. Comijiiilug eqs. (2.24)-(2.27). we have -i i, 1 Jx h ^ Spill. >*»*>' ^ Hulii.' ^2„ i^.^/i!viJ>^j_R(7:^ (2.28) t<J,4, Evniw ^ ^^ '^i^ii!. where z-=i„ = -4l:'t:} (2.29) ^ characterizes the 'softeness' of the partons £:^ ^ £. relatively to their parents (B^tE2)« Now *h > *fS. ^ ^/e©^^ , (2.30) where &^ stands for the emission angle of the part on with respect to its subjet ; 2 Vi 7- *^-^^ - ^0^ . (2.31) Therefore, it is the AO restriction S^ « 6 that forces the (Z* \/'^Q^m^ parameter in (2.28) to be large and thus providing the partons to be involved in strong interactions (non-PT hadronization) at relative distances larger than the typical size of a relativistic hadron: Ap^»R , AZ,^ »y; -R . (2.32) We conclude that in spite of intensive multiplication of partons in the main DL kinematical region their density in the configuration space appears to remain small: PT cascading produces 'the partonic gas'. Noteworthy to mention, in the deep inelastic scattering very
261 different sltixatloxi takes place. There one faces with the partonic systems which become dense with decreasing Bjorken x and look like liquid rather than gas -''. In this Section we tried to advocate the view that the PT radiative processes (including gluon emissions and g —> qq splittings) are likely to prepare comfortable conditions for the so-called soft confinement. Some non- -PT physics must surely be there but to our feeling it reduces to nothing but soft independent hadronization of partons already prepared (in a controllable way ) at the PT stage. 2.5. LPHD Concept ^^7,41) First evidence in favour of the role of PT bremsstrah- lung in hadroproduction, based on the QCD form factor dumping the colour correlations between partons, has become the cornerstone for the *preconfinement' idea ^^. If colour confinement acts, indeed, locally in the phase space, providing the global blanching of separating pieces of a partonic system, then there remains no place for long-range uncontrollable strong interaction effects which might be pictured as long strings or colour tubes, lightening bolts etc. Therefore confinement would have nothing to do with multiplicities, energy and angular distributions of particles produced in HPs. Obviously, ^/K or, say, Meson/Baryon ratio lies beyond the scope of PT. It is the challenge to future qualitative theory of hadrofo3rmation to describe the mass and the quantum number dependencies of hadron yield. However, in the soft confinement pict\ire distributions of different hadrons must be similar and proportional to the calculable spectrum of PT partons (at least asymptotically, outside the domain of influence of phase space boundaries and kinematical mass effects). Moreover, these similarity
262 coefficients have to be universal constants independent of the kind of the process, of particle energies and the total hardness Q* This is the essence of the Local Part on Hadron Duality (LIHD) concept. The LPHD approach attempts to describe the general features of the hadronic systems produced in HPs, such as the mean multiplicities and multiplicity distributions, angular patterns of energy and multiplicity flow, inclusive energy spectra and correlations of particles etc* without invoking any hadronization scheme at all. This makes predictions very restrictive and, therefore, simply testable since there are few parameters to vary in connecting PT QCD results to experiment. One of the main purposes of the LPHD approach is to look for phenomena where PT disagrees with experiment, in order to deduce some actual knowledge about the physics of confinement• 3. ESSENCE OP QCD COHERENCE '^6,18,8) This Section is intended to an elementary introduction to the basic ideas of coherence phenomena. The purpose is to provide the reader with the essential qualitative background helpful for better understanding the material covered below. Roughly speaking, there are two types of coherence effects which occur. The first manifestation of coheraice is the angular ordering (AO) of the sequential parton decays. Coherence of the second type deals with the angular structure of particle flows when three or more partons are involved in a hard process. Here theparticle angular distributions depend on the geometry and colour topology of the whole jet ensemble (radiophysics of particle flows, see Sec. 8).
263 3.1 • Angular Ordering of Successive Petrton Branching To elucidate the physical origin of AO let us consider a simple model of the jet cascade, namely, the radiation pattern of soft photons produced hy a relati- vistic e"^e"" pair in a QED shower. (See Pig#3). The question is to what extent the e"^ and e" independently emit Y^s* To answer this question one has to estimate the formation time, the time interval needed for the y - -quantum to be radiated from, say, e"" leg. According to eq.(2«3) one has form ^ . (3-1) where Gy^ is the 6tngle between the emitted photon and the electron. Now ^^-61^* ^x " ^x with Xj^ the transverse wavelength of the radiated photon# Thus, *form * ^l/^ye / fc Pig. 3. Emission of soft photon, k, after e"^e"" pair production. During this period of time the e'*"e"" pair separate, trans versely, a distance 9 e'e e -t ^ \. -W^ . (3.2)
264 One concludes that for large angle photon emissions, the separation of the two emitters, e*^ and e", is smaller than Xj^ • In this case the emitted photon cannot resolve the internal structure of the e'^'e" pair and probes only its total electric charge, which is zero. Thus for 0wg^» ®e*e" ^® expect 48) strongly suppressed. This is the Chudakov effect ^ ',well known in the physics of the Q£D shower ^, The e and e can he said to emit photons independently only when ff^' » X^ , that is when 0^^^. , &^^ < ^eT ^ ' The same discussion can be given for QCD cascades whexe soft gluon radiation is governed by the conserved (colour) currents. The only difference is that the coherent radiation of soft gluons by an unresolved pair of gluons, or quarks, is no longer zero but the radiation ^^■^s g^s i^ i't were emitted from the parent gluon imagined to be on shell, as is illustrated in Pig.4a. The remarkable fact is that one gets all double log and single logarithmic effects correctly, for angular averaged observables, by emitting the gluon, independently, off line 1 when 9»^^^.^9 off line 2 when 6).<0 and off the parent, line g, when In the general case of large emgle soft gluon emission off the raultiparton jet (see Fig.4b) the intrajet partons p^ can be considered as collimated. As the result this gluon can be treated as the classical probe testing the total colour charge of the jet, i.e. of the original parton p. ' In the middle of fifties in cosmic ray physics it was observed that in high energy Jf-pair conversion the ionization is diminished as long as the e'^e^ spatial separation is below the diameter of an atom.
265 k (a) It II k 2 1 Fig, 4« An illustration of coherence were wide angle emission of soft gluon, k» acts as if the emission came off the parent parton: a) gluon conversion into the quark-antiqtrnrk pair; b) parton jet emission off a hard part on p • The AO occurs not only for the time-like jet evoluti hut also for the space-like partonic cascades. Consider» firstly, the soft gluon radiation in the case of high-p scattering of an energetic parton, when in the t-chcuanel colour is not transferred (e«g«, electroweak quark scattering}* As well known one should observe here two bremsstrahlung cones with opening angles uG c^ centered in the directions of incoming ejid scattered <e a < ©c • a , &r -scattering euagle, see partons (©,^ - .^ , ^^.^ - _^ , .^ Pigs. 5 a, b)# Soft emission at larger angles 6.'^Q > ©c is absent since during the time t^^j^ the ti^ansverse displacement of the charge proves to be small: ^Pi ^ A and the situation looks like there were no current change
266 at all 1 2 Fig* 3* Soft gluon radiation in the process of parton our transfer However^ in the case of scattering with colour treinsfer (that is of importance, e«g#, for deeply inelastic scattering, see Sec* 9) the additional bremsstrahlxuig contribution appears, which corresponds effectively to the emission off the t-channel gluon g, when ©. ^0 ^ > ©c > see Fig* 5c}. 3•2. Hump-Backed QCD Plateau in Particle Spectra *^ The depletion of emission of soft particles inside a jet (htunp-backed plateau) in the inclusive energy spectrum, remains one of the most striking predictions of perturbative QCD# The suppression of soft radiation follows from the angular ordering of partonic cascade in going from greater to lesser virtuality and is a direct manifestation of coherence in QCD. This can be inader- stood on kinematical ground as the result of two conflicting tendencies: on one hand due to the restriction k^Vf^ a slow particle is *forced out* at large emission angle 6 > ''/j<f^ , and on the other hand the allowed decaying single, after a few successive branching, is shrunk to small values* Let us illustrate the influence of the colour coherance
267 on particle spectra with the help of the toy model for part on branching, based on the first order QCD. We start with an old-fashioned plateau of particles with limited transverse momenta k0= k^^ R"" for a qiiark jet with energy £• Here the structure of energy (In k) and angular (In 1/0 ) spectra appeared roughly the same dia , , dh dy\ (3.3) ^^^'=-dl^ ' ?<^' -1 ., ,. .. -n. /«T,N-1 for R"'« k <:<B, (ER) '« e « 1 (see shaded a Pig.6 a-c). I33 eq. (3.3), Vi. is t^ie rapidity p is the component of momentum of an outgoing particle, measured along the parent quark direction* Accounting for a gluon with energy £ and emission angle 0^ , let us use a double-log expression for the radiation probability The step function iT restricts here the treinsverse momentum p^£8^> R"" to ensure the gluon's existence How does the gluon contribute to the particle yield ? From the standpoint of the orthodox parton model one might expect the gluon to give rise to a sub jet of ^ —1 hadrons with k© ^ R (e» being the angle between the registered hadron axxd the gluon) sind an energy plateau as wide as R""^ < k < £ . (3-5) The reduction of this additional plateau to (R-Go)"'^ < k < e (3.6)
268 8nkR PnG -1 Enkj^R Pig« 6. The effect of coherence on energy, angular and k^ distributions. Dotted areas correspond to the contribution which is removed when turning from the incoherent model (dashed) to the coherent one (solid curve). Shaded areas show the old-fashioned plateau (without taking accoimt of bremsstrahlung). proves to be the major consequence of the coherence. To verify that the restriction (3«6) is intimately connected with QCD coherence, let us represent the plateau
269 distribution of particles from a gluon jet symbolically as follows: ^ dn = -^ ^S(Ke'-R-') . (3.7) This expression can be thought of as a DL spectrum of bremsstrahlung (k,60 from the gluon (t^©^ ),'projected' onto the domain of the most intensive radiation (ol^(k©)/or ^i ) . As it follows from the AO in cascade, the off-spring particles are independently emitted by the gluon only inside the cone with the opening angle Applying this inequality to eq. (3«7)»one obtains restriction (3«6) at once* The condition (3.6) reflects the fact that the particle yield from bremsstrahlung depends rather on Pi^ g,- © of a parent gluon than on its energy. Finally,the particle multiplicity can be written schematically as follows: E , ^1 (3.8) VI o Here the first term stands for the background quark plateau, the second one is constructed from the gluon emission (3*4) and fragmentation (3#7). The difference between the coherent euad incoherent approaches has been encoded in eq#(3«8) with the help of 6 ®i?S^ = 1 for the incoherent case max (3^gj Qmf-^ "= ^r^ ^or the coherent one. max o Now we are ready to deduce various differential
270 distributions p , inserting appropriate 6^ function to fix K , ©=-© -4-©^ or K. -»c© of the registered particle. Por the density of the energy »plateau» p(K) our naive model gives: ^'^-'^(K) = ^ + ^*(^^ER -^^R) (3.10a) The additional multiplicity Jd In k (j)(k)-1) appears to be twice as large for the incoherent case (this factor 36 •2' exponentiates with account of multiple branching "^ * •^'^0» Expressions (3«10) illustrate qvialitatively the well-known fact that the coherence substantially depletes the soft part of the energy spectrum giving rise to a CO hump (see Pig.6a). For the rapidity (angular) distribution one obtains j)^*''''^-(6)) = ^ + u.^lyi^(E9R) (3.11a) Thus the rapidity spect3?a happen to be qualitatively similar, both demonstrating maxima at y ^ 0 (see Fig. 6b). Higher order analysis maintains this conclusion. Therefore for a purpose of finding the clear manifestations of QCD coherence y^ is not a good variable. The essential difference in the structure of particle distributions over In k and the rapidity y^^ turns out to be an important lesson. The understanding of this fact will help to overcome prejudices originated from the old theory of strong interactions with limited transverse momenta.
271 3.3* Soft Gluon Emission from Colourless ' Qimrk-Ant iquark Ant enna' ^-^ > ^^ ^ Let us examine soft emission associated with qq pair produced in a colour-singlet state in some hard subproceas, see Pig# ?• This radiation pattern is interesting in its own right, e.g., in connection with two-jet physics in the process e'*"e'" —^ qq. Furthermore, neglecting the terms of order 1/N^ , one can represent the radiation pattern in the case of complex hard partonic system as a sum of terms in which each external quark line is uniquely connected to 6U3 external antiquark line of the same colour (colourless 'qq-antennae'). e[,^i I H J Pig.7# Soft gluon emission from a hard colourless qqpair. In the lowest order the soft gluon distribution takes the familiar form, cf eq. (2.1)2 (Zrr z,K oi,-8r-C otc-2C 4 5r^ Here A ID CP^Pj) (p.xHPjk) £ . (4) 5s a^ •/a^ a •; a id (3.12) (1 - »i^j); a^ = (1 - nn^), (3.13)
272 ]J^,Ei. denote the directions of q,q momenta respectively, ? - direction of the emitted gluon. A Let us call the distribution ij , describing the 3?adiation pattern of the colourless qq pair, 'qq-euitenna'* Antenna ij may be represented in the foiro A id = P^j + Pji, where ^±i * V2-[l/ai ♦ -fli-1^1 . (3*14) ^±^i The point about splitting the radiation pattern into two terms P^^ and P.j» is that only the former (latter) has the pole at 0. s 0 ( 8. = 0), so this term can be treated as 'belonging to' qtiark i (j). Notice that after averaging with respect to the azimuthal angle <f • around the direction n^ we get <p > H Ci^^ p.. =-i-l/(a,-6>.) '■i J 23r ^} CLi ^ tj t^ • (3.15) In other words, <^^i-j/^ is just the incoherent 3?adiation from quark i, confined to the cone 8^ < ^^y Similarly P^^ azirauthally averaged around n^ describes the radiation from j into the cone 8^. < 8^^« This result allows one to incorporate some of the soft gluon interference effects into the MC programs in a probabilistic fashion. 3^ restricting the phase space for soft emission using the angular ordering criterion interference effects are included - on the average - as a sum probabilities* 3.4. Physical Origin of Drag Effect ^^ The drag (or string) effect in the qqg events of e'^e"* annihilation is the best example of QCD coherence of the second kind. In Sec. 8 we shall have much more to say
273 about this coherence, however, our purpose here Is simply to explain the basic idea. So far, the most striking experimental test (see Refs. ^>'*"»'-^'and references there< in) of this idea is the comparison of associated hadron production in qqg three jet events with that of q5^ events with the g and Y having similar kinematics ^-^ . In the plane of the three jets, counting the photon as a Jet, one finds a suppression of associated hadrons in the region between the q and q in qqg events as compared to qqV events. To illustrate the physical origin of the destructive interference one can use the simple QED model with the q and q replaced by e"«s and the gluon by a collinear e+e* pair, 'recharging' electrons, see Fig. da. 3 Ca) 3 (b) "g"=:(e-^e^) JJy"=(e-^e-) 1 2 Fig.8. QED model for illustrating drag effect. The *glu- on t having double electric charge compared to the electron. The 'photon' is replaced by a collinear e'^e" pair. The q5/ event is illustrated in this model by Fig.8b. The corresponding radiation pattern is dQ^ T-. (^2.) . Hw k Zli 2 (3.16a)
274 The soft radiation spectrum in the qq'g' case is determined by the standard classical currents and may be written as ^ "'^qq'g') " f ''^K-Tl^ "^^ ^ 23 - U2'^^). (3.16b) Let us pay attention to the negative contribution of the antenna 12 , connected with the'repulsion ' of e 's. It is easy to see that there is no radiation emitted directly opposite the e'*'e* pair in the symmetric configuration^ The depletion of radiation originates from the compensation of electromagnetic fields caused by two Analogously in QCD the opposite colour charges of q and q in the qqjf event are replaced by the effectively eqixal ones in the qqg case, that leads to the destructive interference• Thus, one meets here the colour 'recharging' of the quark pair by a gluon. 4. DOUBLE LOG APPROXIMATION ^4-26) In this Section we shall construct the multigluon amplitudes Mjj corresponding to the most probable brems- strahlung patterns, reformulate the answer for [Mjjl in terms of the classical shower picture (Markov chain) and discuss briefly the main DLA predictions. Being too crude to describe quantitatively the 2 evolution of particle spectra with Q , DLA predicts correctly the asymptotical shape of the KNO distribution, the position of the hump in energy spectra etc. This asymptotics, however, proves to be 'too academic',since the DLA, accounting for gluon cascading and QCD coherence ignores the energy conservation (recoil effects). ' Notice that this pattern mimics the qqg sample at Nq « '^, cf. eq. (8.24).
275 We shall consider the DLA as the base for formulating the true zero order approximation - the MLLA. - in Sec. 5» The DLA analysis splits into three problems: construction of tree amplitudes, proof of the Angular Ordering (AO) and account of virtual corrections. 4.1 • Tree Multigluon Amplitudes for e'^'e"" -^ qq + Ng Consider gluon radiation accompanying production of the quark and antiquark with 4-momenta P..fP^« Let k^^ be the 4-moment\im, e^ - the polarization vector and a^^ - the colour index of the gluon i. Energies of the final gluons are to be strongly ordered ^+(») » ^i ^> ^2, »-.»60^ (4.1) to give the dominant contribution to the cross section: To pick up the angular logs it is useful to choose a physical gauge, where the gluon emission vertex, either q —^ qg or g —^ sSf vanishes at collinear momenta Following Ref. ^^ we use the planar gauge where the gluon propagator reads (4.3) Av^ T^V (KG) It is convenient to take the gauge vector c proportional to the total momentum, c « (1, "o) ; "c = 0 in the e'**e'" cms. In this gauge q and q emit soft gluons independently since the interference between them vanishes due to
276 V V^ ^^^ ^-v "^ ^ (4.4) The resulting amplitude will be explicitly gauge invariant and it is natural to use only two physical polarizations ej,'»^' for the final ('realO gluons: (e,(^^2)^.j 0, (e^. c) = 0 (4.5) The planar gauge has the advantage of diminishing the contributions from two nonphysical polarizations in virtual gluon propagators: 0<M9 ( '<^ 3 (4.6) (0.5) e ' A K^ ± V^- C. (aco(co + fi^)) va 2 where CO« (kc) is the gluon energy (c « 1). 4.1.1. Two gluon emission off a quark P- Q k 2 1 (a) (fe) 1 2 Q Pig. 9. Peynman amplitudes for e e- 2 k 1 Cc) qqg-i S2 Peynman amplitude corresponding to graphs of Pig. 9 can be written as e.P_ x.P- (VXt^P- t^2 t^i (4.7a)
277 % t^'* t^^ . {4.7b) (4.7c) where k as k^ + kg* (4.8) Singular dependence on the directions of gluon momenta contains no angular dependence in denominator), and thus the kinematical regions where both dn^ and dnp integrations are logarithmic caji be easily shown to be the following: «0, « ^ ^ '^ 1 Diagrams of Fig.9 a,b are of the QED type, so let us concentrate on the third eimplitude Fig.9 c: Here 0. (OJ - the angle between ^ and k- (ko); d - the angle between k^» kp* Making use of Eq«(4«9c} one has ^) v/ / .. ..X J<>^)
278 e- ^'::''(^)-?^ - ^^""A^^z^i^» <r - "^ i (0.5) ^s- These inequalities show that nonphysical polarizations Q}^*^hK) prove to be negligible indeed. Therefore, the d^5.(k) factor in eq. (4«7c) can be replaced by the tremsversal tensor domiHEuit term S^y*(2ki+lt2)v = (4.13) « C P ^ ^' ^ r'<) dj..(K) p'^o^ 2 (e^K^) ef ^^^(K) f Finally, one can replace g _ in eq.(4.13) by the unit tensor g„ ^ : _» ^ ^ _» 2, ^ /--*x (e..'<)CKp-) (4.14) where we have used eq. (4»9c) to estimate * e, p_ -- E.0^ » ^ ^^- To; % • The resulting DL expression for the gluonic amplitude (4.7c) of Pig.9 c looks very much alike eqs.(4.7a) and (4.7b): ^ e^k., e-p a-iarjC ^ % - go }. V • ^-?^ • i f ^ ^ t^ . (4.15) 2^1 (k^+k2)p. 4.1.2. Angular ordering (N«2). Further simplification of DL amplitudes (4.7a), (4.7b) and (4.15) is connected with formulation of a shower picture.
279 From the kinematical restrictions (4«9) one concludes that the DL regions a) and c) overlap, and, thus, the corresponding amplitudes interfere. It proves to he possible, however, to avoid the examination of an interference. To do this let us consider the three non- -overlapping angular regions I. ©, » e II. 6^ » e^ (4.16) III. 0,. « 0^ ;^©^ and show that the matrix element in each of these regions takes the form of a product of trivial independent radiat ion fact ors. Indeed, in region I the only contribution comes from the amplitude of Pig. 9a which takes the form (k-jP2>k2P«) 2 ^2^- ®lP- ^2 ®'1 Mj « Sg ' FT" * FT" * "^ * • (4.17) Kinematical inequality II splits into two subregions: and -^ 6) ^» e^^ » e; 60« 1 c ^ In the first case (cf. eq. (4.9b) only graph of fig.SB contributes as (k2P_» k^P_) 2 ®1^- %I * Sg • (4.18) In the second subregion one has to account for both Figs. 9a and 9c. Here, however, Q,^^&^ , ^^-^ ^ 1^- and summing (4.7a) and (4.15), with accoxmt of commutation relation [t^^ "fcU^ifa^c*^ ^^ inequality (4.9a) one simply arrives at eq.(4.18), which, thus, proves to be correct all over the region II (4.16).
280 Finally, the diagram of Fig. 9c dominates in the region III of a quasi-collinear g-jg2 pair# Here k^p_ >> kpP^ and the amplitude reads ^ e^k- e-p a-iarjC . Mill « Ss •-tV-•-rV^-i ^ * • (^•'^9) The tree amplitudes (4#19) in the angular regions (4#lS) represent the final result of the Na2 sample • To make the degeneralization transparent let us formulate the answer in terms of a classical chain of sequential branching processes. Pig. 9a (b): quark with momentum p_ emits first the gluon k^ (kp) and then kp (k^); we attribute the angular region I (II) to this diagram. Pig. 9c: quark emits the gluon k- which, in turn, emits the kg; the region III. Elementary radiation contributes to the matrix element by the classical bremsstraJilung factor; emission angles strongly decrease along each chain; the colour factor corresponds directly to the classical graph describing genealogy of the process. 4»2. Proof of Angular Ordering Now we are ready to foiroulate the rules of constinict- ing the tree multigluon amplitudes in the DLA. (i) Draw a Peynman diagram D without 4-gluon vertices. Strong energy ordering (4«1) makes it possible to group gluon propagators into N 'gluon lines' - sets of virtual g states with (approximately) the same energies (straight lines in Pig. 10). A vertex i —> ±i (j > i) we shall treat as an emission of j by the gluon line i (parent particle). (ii) Now define the region f^ in the space of emission angles, corresponding to D: angles decrease along each
281 path in the tree, starting from Jf* -—> qq vertex. Along the quark (antiquark) line emitting sequentially t, L,..« l^ gluons 1 ^ k. p ^ » ..•>> ^4 ? ; (4.20a) along the gluon line j kjj y>j » k^ k^ »..»kj kj , (4.20b) where j.., d2»***^m <1®^^"^® ^^^ momenta of its offsj^rings, - momentum of its parent (p. or k^ of a harder gluon i < j). Inequalities (4.20) defdLne the strong AO ( r )• Notice, that the sequential offsprings here are not ordered in their energies. (iii) The dieigram D in the angular region VL is described by the matrix element where m is the number of gluons emitted by q, 5^-4- -momentum of the parent. Noteworthy to mention, ^ is the momentum of one of the final ('real') partons and not that of any virtual state. The colour factor G is built up according to the usual Peynman rules for D: t^ « \^/2 for any q(q) q(q)g vertex, i fo^j^ for a 3g-vertex, where a(b) marks the gluon with the lowest (highest) energy. Eq.(4.21) represents the so-called QCD 'soft insertLon' rules (see, e.g., Ref. O. The proof follows, in fact, the logic line we have met with in the N=2 case. Let us enumerate eind briefly discuss the main steps of the proof (for details see Ref. 24)), 1. Simplify the denominators of the virtual propagators:
282 "^UH V g Fig.10. The scheme of a gluon cascade. ^i Slim nm t i' ^—k ^t > (4*22) virtual over all the final decay products k^ of article q^, k^ being the most energetic among Eq* (4.22) follows from the K,( K, -f- K ) ~ a>, (a).e^ + to^efg) » cJ.a) (ef ^-e!;) ^ J 'J J e' ^j ^j'^e^A K-kg (co.»cOj ^co^; . (4.23) 2. For each vertex (V) in the tree impose the following angular restrictions (to pick up all the necessaary G-logs) s St (4.24a) (4.24b) where s marks the eldest 'son' of f ('father'), u - its the eldest uncle (among the young brothers of f); see Pig. 10. 3. Show that due to eq.(4.24) virtual propagators between V and the qq creation point prove to be independent of kg. k^. 4. Prove that the DL contribution may come only from the singular region where the inequalities (4.24) are fulfilled for each vertex (by showing that the violation of
283 this restriction somewhere in the tree will lead to a loss of at least one angular logarithm)* 5# Simplify the ntimerators using the plemar gauge and physical polarizations for the final gluons and proving the dominance of physical polarizations X^b 1, 2 for c each virtual vertex V (Fig#10) (Xj CX^) (Xj) /tVj> zi^jli%)-^p-{^\>^\)), ^ (4.25) where q-Q c^Qpo'^^a* ^30^ ^f Considering then the chain of vertices along the line f and estimating (similarly to the N«2 case) one finally arrives at eq» (4*21)• 6» The last statement to be proved is the possibility to replace the DL angular regions (4*24)t which might coincide partially for different diagrams, by the non- overlapping AO regions P (4*20). One can do this by induction* 4«3» Virtual Corrections In refs. ^^^^^^ it was supposed that the account of virtual corrections results in the multiplication of the matrix element (4»21) by the factor (4.27) F exp[- i[A?^,i) -^ i^fp,^) ^ E^(\ ,^t))} where 9^^ is the angle between gluon i and its parent:
284 (4.28) p e is the Bom probability for a gluon to decay g(p) —3*^ s(p)+ + g(k <<:p ) inside a cone of half-angle 9 . ' (4.29) In eq«(4«27) ixf denotes the quark emission probability: with the DL accuracy. 2 P , thus, is the product of two quark and N gluonic DL SO) formfactors "^ 0 The ansatz (4«27) was based on the low order PT calculations and the physical intuition. The formal proof S1) was given in the recent paper -^ ' where Gribov brems- strahlung theorem based on the dispersional considerations -'^ ' and the Kirshner-Lipatov method of constructing the 53) Bethe-Salpeter-type equations for PT amplitudes "^•^' had been exploited. 4«4« Cross Section. Method of Generating Junctionals 22, 24-26) The cross section of N-gluon production according to eqs. (4.21),(4*27) reads ^ i\ Ut? ^ 2(0, K.9) ^rn Jt^ r ^ -^ ^ (4.31) (where d6]^ denotes the Bom annihilation cross section). It can be written as a product of the generating function- als (GP) describing the evolution of two quark jets as follows:
285 de-im} •«.xcl. (4-32) (4-33) U-0 Generating fiAnctional Z^(p,0) for a Rluon the total momentum p and the opening angle = z (p.e) jet with e (4.34) satisfies the Master equation I (p,e) exp 3 (4.35) d\ el P 00 n(p,©) Differentiating Z a [u(K)Z(K,0j-l] . over the probing fxinction u(kj^) near the ^point' u=0 one can obtain multi-particle exclusive probabilities for the production of any number of soft gluons with momenta k^^ from the g-jet. Using eqs. (4#35), (4«34) one can check directly the correspondence between the exclusive cross section (4«31) and the functional derivative (4«33). When studying inclusive N-particle cross sections one has to apply to d6^fu| the operator (4.36) d6: (N) owcL n 1-1 ^ SruCKp V»\ro s- S-U(lc) )m4 u.^o (n dv. — )«p{Ja'K ^ ' BuiKA Su(k) j • c/e-fzx] Uro s- SuiK.) ae'{u) Xk^l > which proves to be equivalent to the expansion of
286 the functional d6'[u] near the 'point* u=s1 • As an example, (no logs in the total cross section); this follows from eq# (4.35) which gives Z (p0) =1 . (4.38) Note, that the GF similar to eq.(4.32) can be constructed for any other initial state ig) consisting of number of qxiarks and hard gluons with energies of the same order and relative angles 0^* '^ 1# (4*39) where N , N are the numbers of primary q(q) and g - prajets (e**"e"" —> qq, j> « 8/9; e'^'e" -» qqg, ^ = 17/9; T-^ 3g, 5^ = 3, etc.). DLA 26) In this Subsection we apply the GP technique to deri"v^ the asymptotic behaviour of the KNO-multiplicity distributions '^^^ in HPs, starting with a single g-jet. Calculating the multiplicity distribution R=^ ^^ t one has to integrate over the whole phase volume of each gluon emitted, applying to the functional Z^(v) the following operator This procedure however proves equivalent to the simple differentiation of function Z^ (u(k)^ u = const): ^ v^r \3u) ^u ,_; ^ ^ ^ a<-»o wtio = ^ (4.41) 'U.-l that Replacing a probing function u in the Master (4.35) by a constant, one can simply check up solution appears to depend only on the product of jet variables, i.e. Oh the 'transverse momentum* of a jet:
287 Z^(P,0 ) = Z (m p6/Q^ u)s Z(y,u). (4.42) Here Q^ stands for an infrared regulator (gluon 'mass') to fix the starting point for the perturhative development of a parton system: Z(y = 0; u) « 1. (4*43) Equation (4«35) then takes finally the form lnZ(y;u) = ^ay'(y-y') a^(y»)Cu Z(y';u) - l] , (4*44) «■ - Sr 'TiT^XJ ' ^"^A -(4.45) Running coupling ot (which we suppose to depend on the transverse momentum of the offspring parton) should be sufficiently small all over the region tinder consideration^ even at k^,'^ ^o Thus, the condition X ::^ 1 has to he imposed to justify formally the perturbative approach. As it is well known an asymptotic KNO distribution exists with the limit ^^^ lim rn(y) P„(y)] « f(x), x = n/n(y) = fixed. In terras of GF this is equivalent to the existence of the limit lim Z (y;u) = exp(-S/n(y)) = Z( S ). (4.46) Indeed, replacing a sum over n by an integral in the Tailor expeinsion for Z (dominant n /N^n(y) —^oo with y —> oo ) one obtains v\^o -JS S dx[y^^cv)]e and applying 'llua' operation, (4.47)
288 Z(^ ) = ^ dz f(x) e'?^ . (4.48) O This equation shows that the asymptotic KNO distrihution function f(x) can be obtained using an inverse Mellin kx) = J transformation -^. llf) e (4.49) (here Re Jf > Re &^ , where R^ - the position of the rightest singularity of Z(a) in the jj -plane; as we shall see later, ^^ < 0)« Another way of studying the KNO function f(x) is connected with the multiplicity correlator njj.(y)~ <n(n-1)...(n-k+1)> « XH n(n-1 )...(n-k:+1 )Pjj Vt::K = (d/du)l^[i: uXlL-1 = (d/du)^ Z(y;u)L., ^4.50) no«1 ; n^ = n(y). Writing down an alternative Tailor expansion of GF near ^^^ ^ xk z(y;u) « ^ ^^'V "k^y^^ ^«o ^ '^^^ (4-51^ KnO and constructing Z(j^), one obtains Z{B)^llml(^;u-e^ )-amZ-trie -i) = 21 -zr ^"-^ I F—tk J . ^"^-^^^ K-=0 Comparing eqa. (4»52), (4»48) one concludes that the normalized multiplicity correlators n^(y) fv = lim ii-—r (4.53) ^ (H(y))^ are nothing hut the moments of the KNO function fk = ^ dz x^ f(x). (4.54)
289 Let us differentiate eq. (4»44) over y to obtain Z'(y;u) = Z(y;u) ^dy' a^(yO [uZ(y';u) - 1^ . (4.55) o Then substituting expansion (4«51) for Z and making the ansatz nj^(y) ^^ (n) • fj^, collect terras proportional to (u-1)^# The result will be for k=1: (4.56a) (d/dy)n(y) = ^ dy' a^(y)(5(y') -^ 1)^ J dy •a^(y')5(y • ) , o o for k > 1: k ^ -(k-1)/ X d -/ N v-^ ^k-m ^m -, ^"^ ~ f^ n^"^ ''(y) —• n(y) = >^ ^ -— n(y) x ^- ^ ^y ;^ (k-m)! ml • \ dy' a^(y') n"^(y)[l + 0(1/5)] . (4.56b) Eq» (4*56a) describes the energy behaviour of the total multiplicity. For large y one has V , , [765 '(4.57) n(y) ^ exp( \ dy^a^(y') ) ^exp U ^ (y +\). 0 I 0 The rate of multiplicity growth certainly depends on the coupling or. This, however, is not the case for the KNO distributicn, Indeed, estimating the integral terra in eq.(4.56b) as vn i^Y\ and using eq. (4.56a) we come to the reccurency relations or (4.58) K ^ k[ L^.^ ^j<!l ^ iiLMj^ ^K K^-1 fe< Wil(i^.m)! vn ^(k^.^) ^^ vnl(K-irn)! yy^(K-w^) (f^ « f^ » 1). Following the same lines for a general case o ^^ one obtains
290 < *<«- ^K ;^4 (K-w.)!v*»l w, VV^, , (4.59) f cy) , f (p) ^ 1. 0 1 These relations contain no memory about the coupling at all* This means that the QCD KNO function f (x) appears to he insensitive to ol^ being moving or fixed, a being smaller or larger. This amusing phenomenon had been noted by Bassetto et al. in ref. -^'# Account for the abovementioned coherent effects which modifies multiplicity behaviour from refs. ^'^•'^-^^ through 2 2 ^^ ^true "^ (1/2 . a )^i|;jio^t coherence consequently does not affect f(x) (moments fj^ satisfy eq (4.59) which coincides with those obtained by Konishi in 22) ref. ^ In the framework of the old approach). p We caji now simplify the problem keeping a fixed (^^(y') « ot^s const). This allows one to find the first integral of the differential equation (In Z)" « a^(uZ-1); Z(0) = 1, Z'(0) « 0; following from eq. (4«55); namely, Z t2 = 2a^Z^[u(Z-1) - Inz] • (4.60) Writing down the solution . —,. . II = ay (4.61) 2 XNJ 2 fud-l) - In x] one has to substitute u = exp(-&/n(y)) and consider the limit y -^oo . This leads to the connection ( a>0) ^4 - '2 ^ Then, making use of the asymptotic relation n(y)a.'^e^, we finally obtain the following representation for Z(6);
291 8Xld Here we denoted by R > 0 and ^^ < 0 *^® numbers Vz, ^a(a:-<--4,x) dx. I 1 The Tightest singularity of Z(p>) in the complex p -plane lies at p - p>o - -2.552 (^=0 where Z(p) « 1 - p + + 0(p ) is an analyticity point). Expanding Z(^) near the singularity one obtains using eq*(4#63b) '2(p)-^a/(W + ^^./(^-^o)- 1^^ 2 '^'^ /M + (nonsingular terms). Substituting this expansion into the Mellin integral (4«49), we derive an asymptotic formulae for f(x) at x»1 (the tail of the KNO distribution) which has the form ^{x:)=e'^a(j,(p,x+1 -^ ^^ 4.^) (4.64) In the opposite limit x-> 0 (n« n) large f> dominate where due to eq. (4.63a) Z(P \ , ^ exp (- 1/2 In^B/c). Evaluating Mellin integral (4.49) one has roughly ^(2c)^_ "^ 1/x.exp (- 1/2 In^x), (4.65) which reflects the form factor damping of the low multiplicity events. There exists another way of dealing with the KNO distribution which does not appeal to the Mellin
292 representation and might appear to be useful for the future more delicate analysis of the KNO phenomenon with account of nonleading corrections• The idea is to ac- ciunulate an information ahout discrete moments f^ (i«e« normalized multiplicity correlators) into the compact nonlinear equation for the distribution fix) itself. This equation reads ^^ X f(x) = J dy f(x-y) S dt f(t) In t/y. (4.66a) Different methods can be used to derive it. We shall restrict ourselves here by noticing that one can simply S dx X J operation,which results in the known recurrency relations (4«58). Let us remind the reader that f(x) corresponds to the KNO distribution in a single gluon jet. For a general case (5^/1) the function f^^'(x) can be found from the subsequent equation analogous to (4»66a) X f^?\x) = S dy f^^^(x-y/p) S dt f (t)-ln t/y. ^ (4.66b) Concluding the discussion of the QCD KNO phenomena let us emphasize once more that one should not be 'too optimistic' to apply the DL formulae to the direct comparison with experimental data. The point here is that except the standard long-lived problem with non-PT hadro- nization dynamics the purely PT Single Logarithmic(SL) corrections appear to be of crucial importance for the quantitative QCD predictions. As we shall see in the next Sec, the SL contributions coming mainly from an account of the recoil effects give sizable preasymptotic corrections to the KNO distributims f°^(x) = lim f^^ (xJ=6„(q2)' ). (4.67) . s The Q -dependence appears to be too weak to lead to any testable violation of the KNO scaling, however the shai)
293 of real multiplicity distributions in HPs, due to the SL effects, will remain to be far from its true asymptotical DL limit 'forever*. Concluding this Subsec. let us draw the reader attention to the point that the Fadin eqtxation (4«66) (nonlinear integral selfcontained equation for f(x)) seems to be too nice to have no direct clear physical explanation basing on the theory of Markov processes. 4*6. Inclusive Particle Spectra in DLA ^' Applying the operator (4.36) to the product of the jet functionals (4»39) one can obtain and solve the integral equations for inclusive spectra (N«1) and correlations (N :^ 2) of the bremsstrahlung particles, following from the Master eqxiation (4#35)« formulae spectrum of gluons with rapidity y = Inco/Q^, 0 $ y ^ y = In E^et/^o from the q-jet (for g-jet Cp/C^ v-^ 1), keeping ol^ fixed which makes it possible to derive simple analytical expressions (a « 2Z^^ /<s(^ const , I^ stands for the modified Bessel functions). 1) Energy distribution (hump-backed plateau) ^S-^iali « Cp/C^-a^(—2—)^/^-I^ (2a>j(y-y)y ). (4-68) 2) Double-differential distribution can be presented as dn . , ^ ^y©»^e^ —2£ = d/dlne ( ^^ ^ dyd In e ^ = d/dlne ( ^~-2-) , (4-69) where y^ = y - In 1/0 > 0, Y^ = Y - In 1/0 : (4.70) dn dy d m 0 = Cp/C^ • a^-I^ (2a ^|(y-y)yQ )• (4-71) 3) Angular distribution one derives integrating eq.(4.71)
294 over y: dn/dlne = Cp/C^ • a sh (a Y^). (4*72) The total multiplicity of bremsstrahlung gluoias, as it follows trivially from eq.^(4«72), reads This expression solves the differential equation ( d^/dY^ ) n(Y) « a^ (1 + n (Y)) (4.74) with the boundary conditions n(0) - n^(0) = 0 (cf•(4*56q))* 4.7* ^ -Scaling ^^^ It is interesting to notice that certain characteristics of final states exhibit a kind of new scaling behaviour when one arrives at the well-developed partonic cascades with increase of the hardness of HP. Let us illustrate this statement by two examples starting with the study of the shape of the inclusive energy specti^im. The DLA energy distribution of gluons, with account of the running coupling oL(k ) » 2S7blnk^//\ , can be shown to have the following Mellin representation: (4.75) where A^ « iGC^b (= 16/3 for n^= 3), \ = In Q^A • Evaluating the integrals by the steepest descent method and neglecting for the sake of simplicity a preexponential factor one obtains D(y,Y)^exp f(y,Y) = exp 4?(o^^(y,Y), |5^(y,Y), y,Y), (4-76) where 2. 4^(«^,^Vy)-o((Y-y)-^py-^4^/i^f^ -As, ^ (4.7fe) ^o-f (^aVx ^u-p)^ - (^t^) )
295 Functions et^(y,Y), ^oi7,'!i) must be fovmd from ^f _ £SP = O . (4.77) 3-i »P Introducing new convenient variables m ^^ through A ^ . •< a' -A ,V (4.78) -I A •> •'A one can resolve eqs. (4.77) to obtain /- r,n\ In D(y,Y) wf(y,Y) = A {\ Y+X - nTX ) ."^ ~ . , where iU.,V obey the following equations r-T , (4.79a) ^ nyTx It can be easily seen that the maximura of D corresponds to y » 1/2-Y ( >> "^ v^x f^ "" °^* Expanding (4.79) near this point one can reproduce the known formula for the shape of the hump D(y,Y) ^ exp(-4 A -tt! n-rs-), (4.80) A = 4 i^^. It is interesting to note that the general expression (4.79) exhibits a kind of a new asymptotical 'scaling laW' -^ , namely in D(y,Y) ^ In D (y.Y? ^ p^y/y^^ (4^81) In D,^ In n (Y) Y»X Indeed, keeping 5= y/Y fixed we have ■v>«M for Y »X so that P( "^ ) reduces to As we have already mentioned above, the restriction
296 X a In Qq/A^ ^ ^^ ^^ ^^ imposed for the formal applicability of the PT considerations• Por this reason the inequality Y >>X might seem to be too'academic'• However the shape of the inclusive spectrum according to the approximate relation (4*79) turns out to be *infrarel stable* iia a sense, i#e# it has a final limit withX—*> 0 (•^^CQq) -»<» formally] )• Later we shall discuss this property in detail in the context of the MLIjA. Here let us notice that the insensitivity of the spectrum to the Q value at large y makes it reasonable to compare spectra of different hadrons in the t -scaler Por such €in attempt see ref. ^'K The second example illustrating ^-scaling concerns the two-particle energy correlator: 2/ (4.83) IC^ 4 + ZcU(/,-/<i) where JU* = tc (y^,Y) are determined by eq. (4.79a)# The dispersion of the correlator should exhibit therefore ^ - -dependence d2 = Cw 1 in high energy limit Y »X (4.84) 3. MODIPIED LEADING LOG APPROXIMATION ^'^^ This Section is devoted to the description of the PT approach which has been designed to describe quantitatively soft particle spectra (x« 1), following the logic of the famous Gribov-Lipatov-Altarelli-Parisi(GLAP) approach
297 to BIS and e"^e"" structure functions in hard momenta region (x'^ 1) "^^ The standard LLA being equivalent in fact to the renorm-group (RG) approach, is known to maintain a clear probabilistic picture of the Jet development via the chain of elementary part on branchings A B(2) + C(1-Z) Within the IjLA accuracy the evolution parameter which separates sequential partonic decays could be chosen in different wegrs. So, at x '^ 1 (Z. '^ 1) the strong order- p ing of parton virtiialities k^ or of transverse momenta of products kj^j^ or of decay angles ©. worked equally well. As we know now from the DM experience, the angular ordering proves to be correct for soft gluon cascades. Constructing the probabilistic scheme with account of both the DL €Uid the essential SL effects one has to pay for better acc\xracy of the approximation by the tremendous growth of the nxiraber of interference contributions which must be analysed and interpreted. The interference graphs contain soft gluon lines connecting harder partons of quite different generations. Meanwhile, the very idea of a classical shower picture implies that the structure of elementary parton decays, i.e. the blocks for building up the partonic cascade, should depend on just the nearest 'forefathers' of a considered parton. Thus the possibility to absorb all essential interference terms into the local probabilistic scheme is far from being obvious. Even much more striking, therefore, looks the fact that such a scheme not only exists but proves to be (a posteriori !) a simple generalization of the standard LLA scheme. That is the reason to refer to this 'zero order' approximation (at x« 1) as the Modified IjLA (MLLA).
298 To obtain the soft particle content of a jet within the MLLA one has to use the GLAP chains of two particle decays eq« (5*1) with 1) the LLA splitting ftinctions H^ ii) describing 1) g —^ gg, q —> qg and g —> qq subprocesses *', 2) oL(k^ ) prescription for a decay vertex, and ^i+i«®- exact AO 0. -. ^ 0* (instead of the strong AO we have leaamed from the DLA). Now let us concentrate on the last point. 5.1• Exact Angular Ordering The basic idea of the shower approach is to 'exponentiate* sequential partonic decays separated by an appropriately chosen evolution parameter t Z = C (o6g(t)) • exp[ 5 ^ (o<.s(t')) dt»] (5*2) and to study PT expansions for the 'coefficient function' C(ol^) and for the 'anomalous dimension' ^i^^) in terms of probabilities of elementary partonic processes. Starting from the DLA probabilistic picture, where an account of QCD coherence has led to the strong AO, it seems paramet opening angle: de dt tz ' e (5-3) This means that all the contributions, which are singular in the relative angle between partons, should be attributed to the evolution of a jet and must be absorbed in the exponential factor of eq.(5«2), whereas the regular residue factor C, being free of collinear ('mass') singularities, could be said to describe wide-angle partonic configurations ('multijet' contributions). Successive terms of PT series for V(o^s) correspond to the increasing accuracy in description of
299 elementary partonic decays (6j««1) and thus of the jet evolution; iterating the coefficient function C « 1 + ^ + U ^ + .•• (5.5) one would account for ensembles of increasing number of such jets with 8j^. ^ 1 (some of which could be soft g- -jets). Structure of symbolical series (5»4)i (5«5) might seem strange ( ^fZ^ as an expansion parameter]), but it is inherent in soft particle spectra and, as we shall see below, has clear mathematical basis* The leading term in eq.(5«4) corresponds to the DLA for Jf and originates from g —> gg^ decay as where 1 denotes the 'longitudinal' log coming from the soft gluon emission• Eq« (5«6) shows, thus, that the soft log compensates one power of ^[^^ smallness in the expression for )(C^^) , effectively. Bearing in mind this rule we are ready now to estimate how do different partonic decays contribute to ifi'L^) , in order to extract MLLA effects and analyse higher corrections* a) Hard two-parton decays A -> BC (Z '^(l-Z) ^ 1) ^Y ^ 5S^^ = ^s » (5.7a) {5-7b) b) Z-dependence of coupling in a soft radiation c) Three-^parton soft decays A -^Ag'g" The three contributions (5*7) define the MLLA correction '° ^('^^ '- /mlla = ^^^s + oCc (5.8)
300 erm and can be taken into account by the specification of the z-dependence of splitting functions: 4C^ dz/z -^q^^^^ (z) dz. (5.9) Similarly, the MLLA correjction (5.7c) describes a loss of one 8tngle-log and needs the angular pattern of multiple soft gluon production to be analysed in detail: This problem will be studied in the Subsec^ 5.3 where we shall construct the exact 'angular kernel' V(3^) (an gmalcg of the exact energy kernels ^ Ct) : A BC «^e(K^) Be clQ (5.11) Vf(g)(^ = -S^ f , a^j,^ 1 - (n^nj^), ^sf ^-sg where subscripts are referred to the gluon - 'son' (s), its'father' (f) and'grandpa' (g). Noticing that the integral over S-direction at ®sf^ ®fg? "^^^^= 2/agf is and at large emission angles 6g£:j5> ©^ , V(fl)<^1/eg^ (; — Y(^) ^ \^l^ = Cohst (5.12b) one concludes that the kernel (5.11) reproduces the AO restriction within the DLA accuracy and gives a definite prescription for account of the Q^^ ^ 0^„ ^ 0 region (cf. eq. (5.7c)),To see that this prescription is nothing but the exact AO, the reader is advised to check the following nice property of the V-kemel:
301 zr ^^^^ azimuth ; 2^ <(j) ^sf ^» ^t' average (5.13) where tJ denotes the step fimction^ This means that th decay probability (5.11) integrated over the azimuth of 's' (aroiand 'f*) results in the logarithmic ©-distrib tion (5«12a) inside the parent cone 6^^ ^ ^i^ ^^^ vanishes outside* empha »V-scheme' (5.11) proves to eliminate not only A -^Ag'g" ( aY = 0^3 ) but also A -^ Ag'g'^ g'^' i^){^oL^^ elementary splitting process, factorizing them into the chains of two-i>arton decays completely. The first specific soft contribution, arising only in the 4 loop, corresponds to subtle interferences between a parent and its four offsprings ordered in energy (with all emission angles of the same small value), contributes to A.V'^ol^ , happens to 2 have an anomalous (1/N^ suppressed) colour factor ('colour monster') and can be interpreted physically in terms of the 'colour polarizability' of a jet (Subsec.5.4)» First soft corrections to C( tsl^ ) eq.(5.5) correspond to '4 jet' e'^e'" -»► qq + g'g'^ (/XC = o(^ ), '5-jet' e'^e"' - s 0.5 + g^g'''' &^^^ (AC - oi.|^^ ) events etc. They might be interpreted as multipole interactions between jets: 'colour charge-dipole', 'dipole-dipole' and so on (see Subsec. 5*3)• 5.2. MLLA lyiaster Equation The exact AO makes it possible to construct simple evolution equation for GF. The system of two coupled equations for the quark (Zj,) and gluon (Zq) functionals reads (A.B,C = P.G) ^^ ^^^^ ^^^^^^
302 The first term in r#h«s« corresponds to the (form factor damped) situation when the A-jet (with energy E and opening angle 6 ) consists of the parent parton onlj The integral term describes the first splitting A —^ BC with eingle S between the products. The exponential factor provides this decay being the first one indeed: it is the probability to emit nothing in the angular interval between Q and 6 {^ 0) • The two last factors account for the further evolution of produced subjets having smaller energies and 6 as the opening angle • The MLLA form factors W. are the following (cf.eqs. (4«28) - (4-30)): Q "^f^^l W ifs^^^[it%)-.^j^j (5.15) &~^ 6' I " lie Li'^v^/ v-<f The transverse momentum of partons in eqs* (5#14),(5»15) is bounded from below as usual: k^^^E-z^d-z)©"" > Q^. (5.16) Differentiating the product expW.(E0) • 2»^( 0 ) over 0 and using eqs»(5«15) one can derive the Master equation which is free from the DL form factors (cf# eq.(4.35)): t Z (zE,e) 2^((^z)£,e) - Z^(E,e)] (5.17) The Initial condition for this 'regularized' equation reads
303 ZjE,e;l.(K)) ^_^^^ =^(K-=E) . (5.18) Readers are welcomed to derive the DLA eq» (4«35) from MLTiA. eq.(5»17) bearing in mind slightly different normalization: Z^^-"^ -»u Z^^^. Notice, that the GPs (5»H) have correct normalization of eq*(4.38). All the properties of the parton system produced in e'*"e" annihilation with total energy W = 2E are described in the MLLA by the GP Z + . {u} « (Zp (E, 8; u(k))) 2 (5.19) When expanding beyond the MLLA range some additional correction terms oo z^ ^ Z^ etc# should appear in r#h#s. of the evolution equation (5*15) as well as in eq«(5.19)» 5.3. V-Scheme for Gluon Cascades Our intention now is to analyse the angular pattern of multiple soft gluon production in order to formulate the probabilistic scheme of cascading (V-scheme, see Subsec. 5.1)• Studying the system of gluons with strongly ordered energies (eq.(4*0) one can apply successively the factorization property (soft insertion rules) to build up the tree amplitude /c on) where T^ is an appropriate colour generator. This amplitude gives rise to the N-particle exclusive cross section
304 with the maximum power of energy logs. Looking for the same power of Q -logs in the DLA context (Sec. 4) we have proved that the angular factor ©^(l'^^}) corresponded to prohahilistic cascade with strong AO. Now we have to etnalyse ^{{^Vn^] ) more carefully to show that the V- -scheme (eq.(5«11)) keeps trace of subleading 0-logs as well* The amplitude (5«20) is gauge invariant. However it proves to be convenient to make use of the planar gauge connected with the cms of the process (eq.(4*3)) which kills the interference between '+' (q) and '-* (q) and reduces the number of diagrams contributing to 6"({y\.]) • The number of topologically nonequivalent Feynman graphs for 6" (lV\;\) could be estimated as = 1, 5, 45, ^700... for N=:1,. ..y).... The gluon 'i* connecting two harder lines 'l* and 'm' of A and A"^ introduces the factor /c pp) T ={K.h) ^ —; = for 1 / m (Interference terms), and for 1 = m ('self-energy' graph). 5*3*1* Conditional probability V and 'interference remainder' K = 1. <y-^''^ = Cp(H^^ +H_'') (5.24)
305 can be treated laaturally as a sum of probabilities of independent a^ radiation by q and q. N g 2> Let us list all the contributions related to q-^ radiation by q : (5*25) a b d CpC^ «>H=v''l^^H)*S'T'"l(-^tK'«l"-* =F«!"+ The other part (g- off q) cajai be obtained vxa symmetriza- 2 tion J * + » <—» '""' 3 • ^^° colour structures C^ and C^C, have appeared. The Cp terms (5#25 d, e) describe independent radiation of two gluons by q and "q. p^V The contributions with CpC^ factor are the material for constructing the kernel VCnp) determining the cascade 1 2 Graphs of eq,(5«25) have essentially different angular behaviour. The item (a) is singular both at a 21 0 and at a^^ —-^ ^2+^ ^21» ^1 + cancel, leading then If Bo I 2 _ is emitted at large angle H^ and items (a) and (b) to the AO. Contrary to (a) exxd (b), the item (c) has no singularity at all. Strictly speaking, 2 ._..a._ .„^.^ _ _ _ . ^ -.2 the pole in I^_ exists when, e.g., a 21 0: ^T. ^ sin ^21^^12 1/0 12* But it is reasonable to think of this behaviour as nonsin^ilar because such a pole gives no angle log contribution to the cross section. 2 2 factor) could be made obvious by rewriting -I^_=3} r+iv where Cancellation of the singularity at a-|^ D l|m,n] I Im - I In (5.26) could be called naturally the 'angular dipole'.
306 Such *n033singular' dependence on the angles between any pair of particles (i,d) we call in what follows the 'friability' of a contribution. Less evident is another jproperty of the item (5•25c) - namely the global integrability over directions of all the gluons involved. We shall call this property the 'ideality' of contribution. It means that the term contains no one eingle log and corresponds, therefore, to configuration with all the angles being large: 9. ^ ^ 1. With help of the formula 4 ST tvn a^^ one obtains for the item (c) (5.27) •^ dx e .... tr Z G ' (5.28) The ideality of (5.25c) is a reason to consider this term as the 'remainder' (R), excluding it from the definition of V(n) (i.e. V e (5.25a) + (5.25b)): Finally, C has the following representation: ^a)=p(2) ^r(2)^ (5.30J (2) The contribution p^*^' corresponds to the probabilistic scheme P<2) = C/(H^U hJ)(H^2^h2) + {Cjfljc^v2^^j}gy^(i) (5.31) The remainder reads: (5.32) l F V + -D-i3JsYm(i) This is the soft correction to coefficient function C we have discussed above in Subsec. 5.1. As we shall show below In Subsec."5.5. this dioole term contributes to
307 characteristics as OCoi^) and happen ond the MLLA scope* 2 Vw N is the 'conditional probability* of the emission of a »son* ('2») by a 'father' ('1') up to 'grandpa' (' + ') This saying gets clearness after the azimuthal integration of n'p aroxmd n-: ^) ^ 21 £3r a a+ 1 a, - a I (5.33) Zor 1 (+) a 4- (X..-^ i± a 11 la - 1+ which leads to the exact AO, see eq* (5*13) Test scheme er orders P (3 Let us write down the main probabilistic part accounting for the processes of independent and cascade radiation: p IS) p (5) 1 Z (5.34) P O (3) 3 Z Z c:(h>h:)(h;-hh:)(h; + h:) R (5) cx <H^"-)Xw ^ ( hV . Hixi,) 2.(+) i{-) ^ 5 -x; ( h; + Hi)] SV^Yi (+-) R (5) 2. c c^ 2. 3 Each of P^ emissions: (3) S'yvn(+.-) Pi^^^ c^ C/-^ C/ P (3) ent radiation of '1', '2 I I 3' by q nvunber of gluon is clearly independ P (3) q. 1 describes the contribution of the three similar = H ^ processes (H^ H ^):
308 SYfTn(i) p (3) . 2 includes the independent radiation of '2' and '3' by M', which is restricted by direction of their common • gran dpa' +'• as well as the ordered cascade '+* 1 I t 2 3 I The remainder R of contributions: (3) contains the two different types R (3) « = R (3) R (3) (5*35) (oi)- the 'winding* g-j (3) previous order (R^''^'' = interference structures• round the remainder R (R^^>/). (2) of the (b) - the new irreducible The contributions of Pig.11 contain the collinear divergencies eind start the evolution of each of the four (2) jets forming the dipole R^ '• The interferences Pig.11. Radiation of go by the 'dipole leg* gg* Black circles show other possible singular insertions. I 3 ik between partons of the dipole might be absorbed into as well. These interferences do not spoil the 'friability* and 'ideality' of the remainder and lead to
309 •next-to-next order' correction to coefficient function C« The evolution picture of the dipole can be finished by extraction from the remainder a few other singular terms which are topologically identical to those of Pig.11, but correspond to different order of gluon energies. Indeed, a gluon, dressing '+ »-• and M' partons in the final state, could be not only the softest one among the three (as marked by black points in Fig.11), but might have intermediate or even the largest energy (for the emission off ' + » and '-')• E*g*, recorabining two interference diag3?ams, not matched by p^-^-' (eq. (5«34)), one obtains such a contribution: I 2x + (^P H. C^ c Z 2 ^+^1^-Ci23 (5.36) The first term here is singular at a^2 0 and corresponds to the dressing of the dipole rung (M») by the gluon (*2'), harder than the 'dipole leg* ('S')* The second term in eq.(5«36) has no a-2 singularity and has to be combined with a similar one, arising from (5.37) CpH; C z ^ . .2, ^5 -^-zKK J?r..,
310 to form a new friable structure: l/2-CpC/H| (H^2 jj^3^^^ H+ ^-[+2]^ (5.38) To prove the ideality of this remainder it is convenient to convert eq, (5*38) into the sum of three contributions which might be thought of as dipole interaction of the part on '-' with the AO group ' + ','1* and '2': Z ^fSt^+Xw-^-C^z] (5.39) (5.40) Here we used the identity 1t2 ^+'^+1 = «A(2) (5.41) that is easy to check. Notice, the symmetry of eq.(5.40) allows one to remove the energy order between gluons '1' and '2'. The first term of eq.(5.37) together with (5.42) 2C. _.= C.-^^H'hV. =C„H I F Z "+ + -&-0 P + contributes to the evolution of the dipole (Fig. 11). Dealing analogously with interference diagrams contain- 1 ing the factors H H_ 2 and H %_, 2 one singles out
311 the missing contributions necessary for the total•dreealng' of R^ '. The corresponding remainder (RV ) looks like dipole-dipole interaction between jets (5*43) C Z Z C--^ H.'H^d) 3 p a + - iC.O ^cP + {^ 2) and proves to be 'ideal' as well. An additional series of R (3) comes from the dipole interaction inside jet. It originates after removing the interference terms, necessary for P (3) , from the diagrams of the type of fig.12. This procedure being executed, S^v"A"i -2C F V % W^^l -^^,Wf«' I,.2.3 12 Pig.^2* Ext3?action of dipole interaction inside quark jet one obtains 2rTl /t.2t.3 2n3 ^R = 1/2-Cj.C/H;(h;D-^^2J^ «1 ^;[12] - 2*1 2 3 (5.44) +1^+t12l^* 2 The expression in brackets vanishes at a-i .-^ 0 (I-j^ H-j^), so the 'friability' is evident. To prove the •ideality' of eq.(5«44) let us rewrite it in terms of H 2 cascade: (5.45) aR = i z z z 2c,c:h:(v;i)!--^vm)-- + F V +(1) l[+Z] Z -C C z '^+ % (+) +[1Z1 2 -tnS I'4 l) On the last step we used the identity (5.41).
312 |C,dH'V^X>^ (5.46) A a 1 i 5 ^5.47) - 2, F V ■*- Hi) iC+a] symmetry of eq.(5«47) allows one to remove the energy order between M* and *2* (compare with eq. (5*40))# Let us siimmarize the result for 6^^^(l5^i]) = P^^^+(R^^^/ ^"BSp . (5.48) The set of Feynman graphs splits into 1. the probabilistic part P^*^' describing independent and cascade radiation of z^^*^-^ (V-scheme), 2. the first expansion terra of the product of evolution 2 exponents Zp Zq/ Zq// describing the dipole (5.32) 3. the new irreducible interference remainder (5.49) ^3) = (Eq.(5.39) + Eq.(5.40) + Eq.(5.46) + Eq.(5.47)) . , ^ . Eq. (5.43). ''''^^ The full angle integrals of the items of eq.(5.49) are, the following: R t~< S7>»»(t) 1 r^ d^ ^ a. X M3)=i f^^ The ideality (total integrability) of the remainder shows
313 that the V-scheme kept trace of all suhleading (as well as leading) angular logs. When constnictlng P^ ', we dealt with consequtive emissions of gluons by the seune 'father' as independent probabilities• For example: Cp%jH^%^^, CpC^H|v^^^jV^^^j etc. Keeping in mind further integration over azimuths, one could proceed to replace the energy ordering with the strict angular ordering of sequential emissions. For example, hJh/ = H^V^^) +h/v^J2) (5.51) is a clear identity. Along this way the V-scheme gets natural generalization to incorporate hard partonic decays (a). ~ Cx).) described with use of 9^ (z) splitting functions. The resultant picture gets symbolical expression EXP (o^^Ck^ • 4>' V) -(1 + b!^1^ + R^2^ + R^3) ^ _^) (5.52) where EXP denotes the evolution operator 'propagating' both the major qq configuration (first item in brackets) and the multiset ensembles qqg, qqg'g'' , qqg g g^^^ etc. Let us remark that the prescription <3^s(^x^ which affects essentially soft g emission (see Subsec. 5«1) had been tested by the direct calculation of renormaliza-* tion functions in t: is of 2-loop anomal e.g., ref. ^^). ^6) "^ ' and by the analys The first nonasymptotic exponentiating correction is mected with 3-particle hard decays A —» BCD that r± a single 0-log for the whole group (cO "^ cO^ '^ ^3> )»^"*' could be extracted from the 2-loop analysis of Y (see,e. g., ref. ""^ '). The same order correction originates from soft 'colour monsters' mentioned above in Subsec. 5«1f which we are going to describe now.
314 5»4« Jet Polarizability and Colour Monsters The V-scheme had been checked in the Ns4 case as well. In few words, the procedure of the analysis would be the following. First, the necessary probabilistic part P^^-'is constimcted. Then the remainder is separated r(4) ^ r(4) ^ r(4) ^ „i,^^e r(4) =(r(2)/' ^ (r(3)/ accounts for evolution of previous remainder terms. Finally, the irreducible RV" is 'dipolized' and acquires the 'friability'. However R a happens to be 'nonideal' owing to specific interferences which have topology of a 'gluonic square' (see Pig. 13). 1 ^ /^3 qC^(c^^2C^) I Fig*13» The diagram with nontrivial colour factor (the 'colour monster'). C^=Cp= 4/3 for A=q, C^=CYa3 for A=g. It is only the contributions proportional to C^'Cy which spoil the ideality. The total 'monster' contribution to the cross section is clearly gauge-invariant since it has the unique colour structure. Therefore, the axial gauge n^^ = (^«)a^ can be chosen, where there are only 15 'monster' diagrams (lesser than in pleuiar gauge). Three of them are shown in Pig.14. After the 'dipolization' is fulfilled the remainder is reduced to the AO cascade and the trick of eq.(5»41) is used to convert the interference I^^ to the 1 conditional probability V^(2)t one arrives at the clear result
315 CX hV^D^ D'* -^ (i<^l) (5.53) 3 H 'f^A'^m^iafim (5.54) that could be interpreted as the 'double-dipole' interaction inside a jet. Notice, the notations here are to be corresponded to the gauge n = k_, e.g., ^1^ b'A 1- a •f- a 1 + O 1 Omm ^2+^-2- ^21^*2+ (5-55) V i(H-rf 2 1 -} = a 1-f 1- a +- ^21^2+ ^•21 ^2- ^2+^-2 (5.56) Let us recall that in this gauge there is no soft emission off q_, and so the graphs of eqs.(5.53),(5.54) contain both the q -jet eaid the q^-jet contributions. (a) + 4SS (fc) — c c (C) Pig.l4. Some of 'monster' graphs. Colour factor of the graph (c) depends on the position of '3* in interference between '1* and '2'.
316 Integrating the 'monster' (5»53) over angles one obtains the collinear divergency: (5 £7) e^i fi^ u^v^ iv,^-^ = ^H \-.^Vx + \ -T^^^h a^. L J -l-JC ""-^ ^a -l-x 3<^ 11- ^ c|a,+ ... Z 5C3)i -or: + 6^(^) have since in the main contributing region a-i j.'^Q'pi^ ^31'^^41 « 1 Eqs. (5.55) and (5«56) could be reduced to the planar gauge expressions. In the case of single D-interaction (5«46),(5*47) one would get the first power of log in the square brackets and hence the finite result (5•50). Singular 'monster' contributions of eqs.(5»53)j(5»54) to the evolution exponent could be treated physically as an effect of 'polarizability': a parton interacts with the induced ♦colour dipole moment' of some parton pair* 5*5» Magnitude of Dipole Corrections to Jet Characteristics Let us evaluate the 'dipole contribution'of eq»(5«32) to the mean multiplicity of particles in e"^e""annihilaticin, This contribution arises when the registered particle comes from the softest of 'dipole' jets (the 'dipole leg' gp); other cuts are cancelled by corresponding virtual corrections. Hence one gets /c cq) z. where NQ(k2) denotes the multiplicity originated from ^2* Here the integral over the directions of g^ and gp (see eq. (5*28)) and the large opening angle of g2-3et (©2 '^ 1) were taken into account.
317 Differentiating the DLA eqtiation for multipli city25.26) one gets for the inner integral of eq,(5«58) that leads to ^ (5.60) '^"ev'^l ^" 6-2-Cpl^NiE)(< -0M = - I^T■'^eV The result displays the relative smallness '^ oi of a'dipole' contribution to average characteristics of a process. 6. MLLA RESULTS POR JET CHARACTERISTICS 6,1• Correlators of Jet Multiplicity 17) Let us calculate the multiplicity correlators as the first example of exploiting the MLLA GP-technique# Here we use the correlators normalized to the mean multiplicity of particles in a gluon jet ? = <n>^ ... : g-jeu gjj. =<n(n-1)...(n-k-f1)>g/n^ , g^ = g^ = 1, (6,1a) f^ =<n(n-1)...(n-.k+1)>q/n^, f^^ = 1, (6.1b) where gj^ denotes the correlators of a gluon jet, and fj^ - of a quark jet. For simplicity, all the particle produced are considered to be identical. These quantities define the coefficients in the series expansion of the functionals Z [u] and Z^ [u] on a class of constant probing o HI functions (u » const). For example, oo ,_ ^ xk K-z o qL-J =2^ ^kV ^ * ° * ^k» (6.2) S" \>« ^ f =^(Tu) Zqt^^ U-, ' '^ = ^V^^ u^^ It is convenient to use also the correlators ^^ and4^^ that are given by the expansion of the functionals
318 ffu] s m Zq[u] , 4^[ul = m Zq|u] , (6.3) similar to eq, (6.2)• They describe the true k-particle correlations irreducible to the correlations of smaller groups of particles. The correlators f^iM^^^ ^^^ related to Sv(^k) ^ 9' following way; where the coefficients Qrm-ji ^^ "^^^ polynomial (6,4) are to be extracted from the expression For instance: /^ c\ Notice, the quantity V^ s= f^ = <n> /<n> means the multi^ plicity ratio; nJ % /k = d^ and vfTp" = d^ are the fit A P ^2* (f* normalized multiplicity dispersions for q- and g-jets# Let us write down the system of MLLA equations for the GPs, which follows from eq. (5»17): r" ^ i 'r y ' " v ' ^ ^ o y where '^^d/dy, y = In EC/A (E - jet energy, 0 - angle of a first decay in a jet). Using the symmetry z ^-^(l-z) in (6.6a) one could change ^4^(71)-> ('f-Z)^CZ) to remove the singularity of the kernel at Z=1 : other kernels are regular at Z=1 also: '^(Z) = Z^ + (1-Z)2, (6.8)
319 9p(7.) -i + 2. } (6.9) Thus, eqs. (6.7) - (6.9) imply the mean value of ln(1-Z) 1 « y, and so one may use the approximation 2 (y + In (1-Z)) ^Z(y) (6.10) in eqs, (6.6). In the DLA case the account of the singular parts of the kernels (6.7),(6.9) was quite sufficient. The non- singular parts as well as the entire kernel (6.8) are therefore the source of MLLA corrections. Neglecting the dependence of Z(n/ + ^7; ) integrands, analogously to eq.(6.tO) system (6.6) substantially: and ol*^ on Z in nonsingular , one simplifies the Zm Zp)-]d/4Y)[ip)-i H z ,,K„cv)£/v)(Z(v)-i 4- a h oL 3 f 2sr lAV] (6.11a) ZI1)-2I1)K^^^^ - fto)[Zci)H] .(6.11b) Here ^ {\j\ = "^i^i^ oL (VjAgr' ^® ^^® anomalous dimension that determines the rate of the multiplicity growth in DLA (see Sec. 4). Now let us rewrite the system (6.11) in terms of the GPs ^ and ^ defined by eq. (6.3). Subtracting eq. (6.11b) from eq.(6.11a) one obtains the relation between multiplicity correlators for q- and g-jets: C H^-M') ((^H-3fJ Z, (J^ I \ 3 c Z-1 h h ^«=->i N. ^ -1]). (6.12) Here we used the DLA relation 2 q = z Cp/N s (6.13) in the correlation term of eq.(6.1la)
320 Dividing eq. (6.11a) by Z- and taking the derivative of the result one comes with the account of eq«(6.13) to the following equation /g -^x It is useful to introduce the 'anomalous dimension' ^ - __ V n = exp ^ dy'/ (y) (6.15) and reduce eq. (6.14) with the help of eqs.(6.2)-(6.4) to the series of recurrent relations for <^. : 2...2, ./^ <^,W^■■■■"^t^ ^I^-l''<^ ^"»(f -^) ^ . (6.16) where a « 11/3-N^ + 2/3-nf-(1 - 20^7^^). The r.h.s. of eq.(6.l6) equals zero identically at k ss 1. Hence, ^ Solving this equation by substitution ^'=^ ^o'=-f^ ^0(1^') , (b^rf^,-!*^,) (6.18) with account of smallness of li (X^^ol.) > one gets the ** 28 17) MLIiA-corrected rate of the multiplicity growth * '': ^^^o- ^•i(<^-|) + ou!) . (6.19) Por k > 1 let us insert eqs.(6.18),(6.19) in eq.(6.l6) to obtain ^ (6^20) /^^^ where the symbol Q denotes the series PC \^"^2» Q^-O, $2 = 1, Q3= 2Cp/N^ = |, 0^4= 1/3 +(2Cp/lIc)^=9V81,... ^^^ #-*^ ^r\^
321 The recurrence (6,16) allows one to calculate "f^ consequtl- vely. The limit i^^> 0 corresponds to DLA: 4^= 1/3, ^ = 1/4, etc. For the dispersion of the g-jet one gets the MLLA correction 'A J. _£ \ K 4- Lo J_ I. 5 (6.22) C C Similar to eq.(6.14), the relation (6.12) can be reduced to (6 23) that expresses the q-jet correlators through the g-jet ones. For instance, eq,(6.23) gives the ratio of q-jet to g-jet multiplicities 58,59). For the normalized quantity V^/% °^® gets, therefore, The dispersion of q-jet, e*g#, reads (6#26) Notice that MLLA corrections to the dispersion of q- and g-jets appear to be approximately equal, as the dominating terms in the square brackets of eqs,(6.22),(6.26) are the sajne matter of fact, it follows from eq.(6.25) that the SL correction to the DL relation ^^Kf_\ (^ ^^) (6.27) vanishes at n^=0 and is proportional to the small factor 1 - 2Cp/N^ = ^/^c^ = 1/9. Notice, that the smallness of the colour factor (1 - 2Cp/Nc) allows one to omit in eqs.
322 (6.20) ,(6.23) the terms with 'Q-^ as being proportional to N^""^ and, thus, to simplify considerably the calculations Therefore, the account of the SL effects results in the substantial corrections to 4^^^ and ^^ while the relation (6.27) between them is violated rather weakly (see also ref. ^^^)m Using the numerical values 11^= 3, n^s 3 one obtains approximately dq^ ^ 3/4-(l - n5 >o) ^3/4-(1 - 2.1 \| o^^(E) ). So far we considered the multiplicity correlations for a single jet. The real events contain seveiul jets. For example, in the case of e'**e'" —^ q^^+ + q 2 -f « = 2 ^ e e ^ jet " ^jet which implies that Combining these equations one gets the numerically large correction to the DLA-value of <n>/D, which improves the agreement with experiment. It is of interest to note that last equation was checked experimentally. This gave a clear evidence in favour of the independent evolution and hadronization of the two back-to-back quark jets. Attention would be paid also to the fact that the true parameter of expansion in the recurrent relations (6.20), (6.23) is the quantity ( K-'^ ). It means that MLLA corrections to the higher correlators Cat K :>, c^^ » 1) can not be estimated according to eqs.(6.23), (6.25). But this non-uniformity in k is connected just with the crudeness of the chain of transformations leading to eqs. (6.12),(6.14) and does not come from the initial functional eqs. (6.6). As a matter of fact, the dependence of
323 Z (y + Ind-z)) on z in eqs. (6«6) for high value of k could be considered as weak. That is why the reconstruc- KNO fimct large z. » 1 (highe formulae gjj. and fjj. (or 4^^ ,4^^ ) , which are uniform in (K-^^) MC simulation of events corresponding to the evolution eqs. (6#6) is likely to "be the straight way to predict 29) the shape of KNO distribution at the realistic energies ^; 6.2« Inclusive Energy Spectrum of Partons in MLLA The inclusive gluon spectrum of A-jet (A=q,g) with the given 'opening angle' 6 , ^/^ ^A ~ rr-^ 2a (E, 8; tu(k)]) I , (6.28) ^ 6u(k) ^ Iuh1 is determined by the system of two equations following from the eqs* (5.15), (5.17): sli^.v) - Sit) s! * Ww^'^a-t'my) ''•^" 0 0 ^ Here ^^ stands for the regularized AP kernels ; 1 = In r- = In 1/x, y = In kS/Q^ (1^ = In z/x, y^ = In keJ^Q^)* The running coupling o(. ^ depends on the transverse momentum of soft gluon + _- The energy spectoTum for e e annihilation is the sum of two q-jet contributions: e e ^^ - e m>^) (5:* - d X % e-i (^=-ek.|) (6.31) Notice, the region 0 ~ 1 in the angular integral of eq (6.29) leads to the negligible correction of order/v^ol to eq. (6.31).
324 Eqs. (6.29) can be solved directly with use of double Mellin transformation similar to eq.(4.75)» The resulting integrand proves to have an additional SL factor('^-y as compared to that of eq. (4#75)» We shall follow the way that manifests the relation of eqs. (6.29) to the RG approach. First of all, let us introduce the variable Y s In -M. = y + 1 (v>0) (6.32) that is connected with the largest k available for partons in a jet. Then turn to the moment representation d'(u), y) = [ dl e""^ D(l, Y). (6.33) O Owing to the fact that the evolution parameter 6 appears due to strict AO, exclusively in the upper limit of the integral (6.29), one can write down eqs.(6.29) in a symbolic matrix form as ^ §(a),Y) =Tde e^^Wi^^—©KY-t) . (6.34) o Eq. (6.34) generalizes the RG equation for the LLA (60^1) over the region of parametrically small moments u) ~ 1/Ty" <jC 1. Indeed, neglecting 1 «Y in the arguments of o^w^ and D one could transform eq.(6.34) to /\ Diagonalization of the kernel matrix 4^(a>) results in the two * trajectories' tlriat determine the anomalous dimensions of the two operators arising from mixing of g and q states in a cascade:
325 At X « 1 the trajectory v^(u)) ^ singular at a) =0, ^^M = ^'-a.OM, a=f^/^^|-^ , (6.38) — c gives the main contribution do D(l,y)# The 'nonlocal' in Y eq«(6«34) encloses in a compact form the information about the anomalous dimension, which is not easy to reveal by meeois of the standard RG approach. Indeed, let us trace the chain of ti^insformations of •q. (6.34)= " r <! . ± S(a),Y) =( I'll e ^""^4-v'cp,z,)^Y'»,^y, The diagonalization leads formally to the known LLA trajectories with the differential operator as an argument (see eq* (6.37)}: Using eq. (6*39) one obtains d . d ^+ ,.., ^^ ^+ ./ . d X ot Introducing V as follows : y i)iuyy) - S(tOX) e.xp 5dv ir(a),cl^tV)) ^ (6.42) one gets the equation for ^(od^^^) which clearly possesses the necessary property of locality where PK)=^'^.(Y}cx-J^4(Y)
326 The r.h.s. of eq# (6#43) proves to be the correction 'ol^^^ to the l*h*s* C'^^)* In the DLA case s 5 where ^ = 4a/^ -^ . This result certainly needs neither diagonalizing nor accurate handling with d/dY and follows immediately from eq# (6*39): ^£KY,=f(<o.i)|js=^i2=ir£ . (6.45) Meanwhile, to derive /txt a in the framework of conventional RG approach one has to sujn up the series ZZ ^icv^s'^ / representing in fact the square root with c-^ to he found from the PT expansion of eqs. (6.29)* Thus, the nonlocal eq# (6«34) reduces the RG series, defining /pjA, to the shift of the argument of D, which is inherent to the evolution equation: ^^^'''s) ^(^X; -^^ (^zT^ = (6.46) The SL correction to Jt^ta corresponds in terms of the ordinary RG technique to series ^Ws)-^?C,(^f -H <., . (6.47) This results from the account of the regions ^j'^^^v^ an<i the renormalization of dL^ ; the last term ^<^s ^^ eq# (6«47) is connected with the hard decays k^r^^ ^i+l (^®® eqs#(5*7)). But here eq.(6,34) proves to be very effective once more: it has already accounted for both the angular regions 6^^6/^j, (exact AO) and the renormaliza- tion effect via the shift of the arguments:
327 ici/{>iH.cL,'t''+ ...)] ot^cY)S(y; -y[«i,^Jo(,(Y3S(Y-€} ots(Y-€) ^CY-e) . Therefore, combining eqs. (6,43),(6.44) one gets immediately the SL contribution to V(tO,«iL,) • ^z (6.48) » o ^ The omitted corrections here are small uniformly in cO . Eq.(6.48) coincides with the result of RG calculations '' One can solve the differential eq#(6«41) exactly, reducing it to confluent hypergeometric equation for the function <^^(Y)*'D{Y)• Accounting for the initial conditions fixed by the integral eqs#(6.29) one obtains the result: where V-(V*X)cO =(^^,-^ ' V^^" ^^A ''^ =^'♦ ^ = T * For the Mellin representation of the spectrum in a physical region (I = In 1/x < Y) the second term in eq.(6.49) could be omitted: p/.v B (6.50) Since it decreases exponentially at R^^D-^-foo and so gives zero contribution to the contour integral. Its role consists in the cancellation of the first term near the poles of rf-T^ + ^ ) and at the left cut (Reo) < 0).
328 ingularit The asymptotic behaviour e"" provides zero value of the spectrum D(3c) in a non-physical region xE < Qq(1 > Y)# The integral eqs* (6.29) determine also the relation of q and g-jet momentum functions D q(<*^,Y,^): 21F ^ + ^ itLiL_^(£) oi ^ + o(j) , (6.51) 3- 9+(i)-v)_(3) where j s a)+y(<o,<(Y;) ^<1 (for derivation of the LLA 1 ^ residues C see ref. '). Thus, the asymptotic spectra of q- and g-jets prove to be similar. In the DLA they have the known constant ratio Cp/Cy = 4/9. This similarity appears to be slightly violated by V d^ terras which, as it follows from Subsec. 6.1, are under the MLLA control. For the jet multiplicity, taking 60=0 in eq.(6.49),one gets: where I and K are the standard modified Bessel finictions. The second term in eq.(6.52) decreases with Y increasing and provides the normalization D(a>= 0, E ^ Qq) -^ 1. The asymptotical energy behaviour of the multiplicity reads N -(^^P"^expi^-^'^>^ (6.53) The ratio of parton multiplicities in q- and g-jets follows from eqs. (6.51) (cf. eq. (6.29)):
329 o where 60 = 0 was taken• It is seemingly the simplest way of derivation of the q/g ratio -^^>-^"^, which needs in fact no more than the LLA coefficient functions and /t)ta« 6.3« Developed Cascade and LPHD Concept A rum quantity Q^ which regularizes collinear divergencies.This quantity represents the minimal value of the relative Kj^ of decay products in jet evolution. Q also bounds parton energies E- = x E >,k^/©Q >, Qq^^o' "thus playing the role of effective mass of a parton• The choice of the Q^ value sets a formal boundary between two stages of jet evolution: the one of the parton branching process, which is controlled by PT, and then the stage of non-PT transition into hadrons. In essence, partons and hadrons are the complementary languages. So, if the theory of hadronization would exist, the result would be independent of the formal quantity Q^ separating the two stages. As a matter of fact, for large enough Q (for example, at Qq^3 GeV) the number of partons produced at recent energies is certainly small. So, one is forced to apply for some 'ad hoc' hadronization model describing the multihadron production as the evolution 'below Q ' of a partonic system with large invariant masses of parton pairs. Unfortunately, an experimental verification of such results looks rather like a touning of parameters, which are inevitably present in any phenomenological model,than a test of QCD predictions. But with the intent look at eq.(6.50) an opportunity
330 to make a model independent prediction may be found. If PT partons hadronize independently of each other, then the distribution of a hadron h over the energy fraction Xv^ is given by convolution of the parton x-spectrum with the gluon hadronization function C (x^^/x, Q , %)• Hence, because of the factorized dependence of the integrand in eq. (6*50) on the jet energy B and Q^, the hadron spectrum has the same form of eq.(6.50) but with KCtOjQ^) replaced by the product K^(a), M^) = K(a),QQ)C^(^,Q^,Mj^). (6.55) Here Cg^(^,Qo,Mj^) = S dz z Cg^(z,QotMj,) is a moment of the hadronization fxinction. What can be said about the 60 -dependence of the K factor which influences a shape of the hadronic spectrum ? We are interested in the kinematical region of relatively soft (though relativistic) particles. In such a case essential values of cO under the integral (6.50) are small: cO « 1 (near the maximum of the spectrum they are parametrically small: oO'>^'{oC^ ). Therefore, to umderstand how hadronization affects the spectrum shape one needs to know behaviour of K at cO —> 0. Let us consider the two qualitatively different variants: K^(oO,Mt^) i^ 1/60-C(Mt^) + const + 0(co), (6.56a) K^(a),Mj^) ^ const +0(co). (6.56b) The first (singular) case corresponds to the physical picture where each coloured parton produced hadrons v/ith h a plateau-like energy distribution: C (^) 00 1/z. One o can see that in such a case the dip at small x which is characteristic for partonic spectra never will manifest itself in experimental hadron spectra. The regular behaviour (6.56b) corresponds to local in
331 the phase space blanching and hadronization of partons (see Sec. 2)# It is perhaps surprizing to see the x- -dependence of xD_ being given completely in teiros of the PT evolution• Non-PT effects can smear the hadron distribution over a finite interval in In 1/x# However, such a smearing is, formally, a higher order effect in the framework of MLLA. Thus, the overall normalization factor K is the only phenomenological parameter which remains arbitrary, concerning the PT-LPHD approach. It may be found, e#g«, by fitting the average multiplicity. The parton spectrum (6.50) has another Interesting property: for large energy E, when Y > X^ (recall that y = In E/A , X = In ^ q/A ) $ "tiie sha rum as far as Q^, determin ^ ^ ^^^^ ^^ ^Q -.^^ ^Q stant factor: K(u),Q^)c:. __i-_. (zy2)S Kg (Z^) (6.57) at coX « 1, where T^o^ ^^6'^ XA>' (recall, that essential values of 60 small <60> ^ I/nTy" « !)• So, this situation resembles the hadronization function (6.56b). At asymptotically high jet energy B, when only small cO -region is important, the spectrum (6.50) for any X '^ 1 has the same shape as in the case X =0, differing just in the numerical factor (6.57). Thus, when the bremsstrahlung cascade is developed enough, the shape of resulting energy distribution of particles gets insensitive to the processes occurring at the last steps of evolution (at kj^/^ ^o^* This observation may serve to justify the attempts to provide the developed cascade not with the expensive increase of total energy E, but with the decrease of Q , thus enhancing the responsibility of PT for jet evolution at recent
332 energies• 6.4« On Infrared Stability of Limiting Parton Spectrum The PT formula (6.50) was based on the smallness of z ^si^D , that of strong inequality X = Id Qq//V » 1 even at the last steps of part on cascade. Decreasing X (and so extending the responsibility of PT) one comes to the finite 'limiting' parton spectrum at Qq = A. • Moreover, at X = 0 eq. (6#50) simplifies noticeably since in this case K(u) ,Q ) = 1 at all a) . The fact, that PT formula (6.50) leads to finite result in a somewhat senseless limit, may be explained formally by integrabili- ty of the anomalous dimension over the region of 'infrared pole': ^ j^z ^^^ (6.58) But so fas one is still unaware, in a rigorous sense, of 14 . , r..Z. ^4. runn Furthermore small .e ol^ should disappear from the theory, since new physics turns on and the language of the coloured qxxarks and gluons ceases to be appropriate. Nevertheless, one might dare to consider the perturbative approach, PT formulae, to be plausible, if the predicted results turn to be independent of the concrete function <^c(^<x^ ^ "^^® dangerous region. A characteristic feature of the limiting spectrum is the presence of the maximum. One may find analytically the asymptotic shape of distribution not too far from its peak by saddle-point evolution of the integral (6.50): x»(xy, ^^ exp[-|f (*ii|Pf Y = In E/A . Y (6.59) It has a broad Gaussian shape with a peak at Ep=Eo=zoE,
333 (6,60) that grows rather slowly with the jet energy E E d E^ I T > o - /^ ^1 b K^ = '^' - ^6^ (6.61) The subtracted term here represents a SL correction to DLA, which is asymptotically small. For example, eq# (6.61) gives dlnE^dlnE « 0.4 at E = 30 GeV (A. = 150 MeV). Thus, inclusion of SL terms shifts the maximvim to lower X. This result is quite obvious because these SL terms accoxint for the more complete description of harder parton emissions than the pure DLA does. Therefore, a share of the total jet energy, which is supplied for creation of the soft plateau, somewhat lowers due to the recoil effect, and the spectrum softens. It is noteworthy to emphasize, that the recent data on inclusive energy distribution of hadrons in e"*"e""ajinihila- 10 12) tion (see refs. ^»*^'' and references therein) demonstrate the existence of the hvimp-backed plateau, supporting the concept of LPHD. So far, probably the most convincing evidence for the hump-backed distribution is the growth of the energy, ^^mp' ^^ which the spectrum reaches its maximum as one increases the jet energy. In refs. ^'j^^-' one may find the discussion of the unexpectedly close correspondence between the limiting spectrum and the observed hadron distributions. To sharpen the influence of AO on the parton multiplication process, and in attempt to find the dip in jets produced in hadronic collisions, it proves to be importeoit to look at the spectra of particle restricted to lie within a particular
334 opening angle with respect to the jet (see for detail ref. ^). Here the hump should be observed for the energetic particles and the influence of the kineraatical phase space effects will be reduced. ?• CHROMODYNAMICS OP HADRONIC JETS ''9,61) In this Section we shall discuss the QCD portrait of a ;jet ensemble and the properties of an individual jet in HPs, The emphasis is on the collective QCD phenomena in the jet dynamics. 7.1. On Experimental Selection Procedures Traditionally, the final state structure in a hard collision is interpreted in terms of jets of hadrons with kinematics corresponding to those of the energetic quarks and gluons participating in HP. The standard exclusive procediires for jet finding as well as the reconstruction of the jet parameters (energy, angular direction, mass and so on) differ in some details, but all of them are based on the idea of assigning each hadron in an event to a certain jet. This has been a very fruitful approach especially as regards three jet events in e^e""collisions, where the gluon was found, and two high Pj^ jet events in hadronic collisions, where the point-like nature of quark and gluon interactions has been best measured. However,
335 number is inherently ambiguous, especially as one goes to higher energies• The ambiguity comes from several sources• (i) In a particular part of an even it may be equally correct to identify a set of particles as belonging to one jet, two jets or even more jets# After all a jet often has an identifiable substructure consisting of further jets. ii) Such a procedure completely ignores the collective QCD nature of pa3?ticle production in HPs. In particular, due to colour coherence soft hadrons do not belong to any particular jet, but have emission properties dependent on a jet ensemble. The spectrum of particles associated with 3~jet events in e"^e"" annihilation, especially in the wide etfigle regions, is of this character (see the next Sec). In particular, this collective phenomenon is not the least of the factors explaining the measured in the three -fold-nsymmetric 3-jet events ^ ratio for hadron multiplicities in g- gind q-jets, that turns out to be lower than the famous asymptotic value 9/4, see for details refs. '»'*'^. We emphasize, in addition, that the coherent influence of the colour topology of the overall jet ensemble affects not only the flows of interjet particles, but also the particle distributions inside each jet (azimuthal asymmetry of jets, see Subsec. 8.6). Attempting to force particles to belong to some jet in an event may cause difficulties. This leads, e.g., to the sizable uncertainties in the finding of a 'jet axis', resulting in the bias-effects for the particle distribution relative to such an axis. It is instructive to recall here the appearance of the artificial two-humped distribution in the rapidity spectmra dN/dy observed in experiments (e.g., O. As it has been demonstrated above in the language of the toy
336 model (see Sec. 3 ) the coherence of QCD does not produce a dip in dN/dy at y| = 0. Analysis, accounting for the QCD cascades, maintain this conclusion. So, in the BL expression dM _ ^ . isHc ^J (cf# eq» (4«72) there is no hint of a dip as ^ decreases* An important point is, that one produces a dip at n/ = ss 0 simply from a bias ag&inst choosing the jet axis so that 7^^ = 0 (see for details ref. *^0. The serious shortcoming of some exclusive procedures (e.g., dealing with the sphericity tensor) is the lack of infrared stability of the event characteristics. If the jet algorithms do not use infrared safe quantities, comparison with QCD cannot be carried to higher orders axid the whole procedure, though adequate when only crude data and crude calculations are available, may have limited quantitative significance. Even if the jet finding algorithms are infrared stable, the procedure for assigning particles to jets remains, in principle, unjustified. Especially as higher energies are attained a purely inclusive procedure for quantitatively dealing with hard collisions is preferable to organizing the event according to a certain number of jets. There is in general a rather direct correspondence between jets ajid energy correlations so that any observable which can be described in terms of jets can also be described in terms of energy correlations fiJ,^-)^ As the simplest example consider the angular distribution of the multiplicity flow in two-jet events of e"^e~ annihilation. Its study is accessible through Z 2 axi (energy) multiplicity correlation (E MC) (7.1a) —^ n T [ AE A£ dE ^gE^ c{g"3
337 E, E^ 46: (7.1b) a y where the sura is over all particle types. The energy weighted integrals over E^^ and E^^, at fixed angular directions n and n^^ '^--n^, specify the 'jet' directions about which one has an associated multiplicity distribution at variable angular direction n(£?). The cross section (7«1b), describing the correlation between two back-to-back energy fluxes (EEC), contains the known double logarithmic form factor ^* K This reflects the natural disbalance of the jet directions, caused by the gluon brerasstrahlung. The same angular distribution may be discussed in terms of a more simple double-inclusive correlation between the energy flux and the multiplicity flow (EMC) a^r ^ ^ ^^a^^pT 6;dE,dE^ d^^dQ^ ^ ^^-^-^ £. de: s; ? i ^^o- dlTja: ■ "•^'" The point is that here the main contribution also comes from the two-jet sample whose kinematics is practically fixed by the choice of the direction n • The difference between the distributions (7.1a) and (7#1b) occurs only when the angular direction n is parametrically close to the backward »jet axis*, n ^ -n^# In this case the shape of the distribution (7.2a) near n = -n^ becomes somewhat wider due to the natural for QCD 'shaking* of the non- -registered jet ^5) (0^^^-- (A/W)^, Jf '=^ b/(b+4Cp) (-^0.64 at n^ = 3) ). The drag effect physics becomes accessible through
338 a more complicated correlation, E%C, see the next Section In the general case multiple correlations between the energy eoid multiplicity flows could be referred to as In the discussions that follow we shall refer to measurements involving determinations of jets and jet axes. To see, qualitatively, the effects we shall be considering even crude determinations of jet axes are probably sufficient. However, in making precise quantitative relation to theory the purely inclusive approach (a use of E M'^C ) seems to be the best way* 7»2. On Structure of Particle Plows in Multijet Events As it is discussed in detail in the next Subsec*, the collimation of the QCD cascade around the parent parton becomes stronger as the parton energy increases. Moreover, the collimation of an energy fliix grows much more rapidly as compared to a multiplicity flow. Therefore, at asymptotically high energies each event should possess the clear geometry, that reflects the topology of the partons participating in the hard interaction. Therefore, the best characteristic of the final hadronic system is probably the spatial distribution of the energy flxixes. After some smearing this takes the shape of the closed energy surface with a few comparatively sharp bumps, corresponding to the primary partons. The widths of these bumps determine the angular apertures of each one of the main colour currents (see the next Subsec). Fixing a jet axis with the accuracy higher than the natural angular width of the corresponding energy flux is unreasonable. Therefore, the space-energy portrait of events represents a natural partonometer for registration the kinematics of the energetic partons participating in HP. While the hard component of a hadron system (a few
339 hadrons with the energy fraction % '^^) determines the partonic skeleton of an event, the soft component (the other hadrons with 7» « 1) forms the bulk of multiplicity. Closely following the radiation pattern, associated with the partonic skeleton, the soft component is concentrated inside the bremsstrahlung cones of QCD jets. Theoretically, the opening angle of each cone is bounded by the nearest other jet, since at larger angles particles are emitted coherently by the overall colour charge of both jets. Even though the bremsstrahlung cones of the neighbouring jets strongly overlap, the resulting total multiplicity can be presented as the additive sum of the contributions of the individual jets, see for detail Subsec. 8.1. If one keeps the angle between the two jets fixed, then with increase of the total energy these jets become experimentally distinguishable. This fact stems from the asymptotical collimation of the energy and multiplicity flows within each jet. Shrinkage of the characteristic opening angle permits one to introduce the notions of the 'intrajet' and the 'interjet' hadron flows. 7.3. QCD Portrait of Individual Jet ^^^ Let us consider the general inclusive characteristics which may be called, in some sense, the characteristics of an isolated jet (neglecting the mutual influence of jets in their ensemble). One can study the properties of £01 individual q\iark jet when measuring the different inclusive distributions in the process e'^'e" —> hadrons. The decay into two gluons of the C-even heavy quarkonium states, JCq= QQ> might define, by analogy, the individu al gluon jets. In spite of the high importance of the coherence phenomena the notion of the isolated jet makes sense, if one does not deal with the azimuthal effects.
340 but considers only multiplicities, energy spectra and correlations, etc. In this case all the influence of the jet ensemble on a given jet may be encoded in a single parameter 6^ , the jet 'opening angle', this, in essence, being the angle between the considered jet and the nearest other one. Multiplicity, energy spectra of particles and the other jet characteristics prove to depend not on the jet energy E but on the hardness, Q, of the process producing jet, i.e. on the product of jet energy and its opening angle 0 , , Q = E e^ olliraation o at e << 1. jet. Consider a jet with the energy E and the opening angle ©^ . Let us try to answer the question, what is the angular size 6 (Qq/E ^ 0 <. 8 ) of the cone, where the definite fraction Ti ^ 1 of the jet energy is deposited (see Pig. 15)» The smaller Pig.15. Production of a subjet B being registered by the calorimeter with the angular aperture 9 . is the angle, where the bulk of the energy is concentrated (aperture of the energy flux), the higher is the jet collimation. Experimentally this task would correspond to the calorimetric measurement of the energy flux being deposited within the given cone. Prom the point of view of PT the sequential parton decays in a cascade are ordered in angles (the 'hard' decays due to the LLA kinematics and the 'soft' ones due to the QCD coherence^ Therefore a calorimeter measures
341 an energy of a subjet, initiated by a parton B,produced at that stage of the cascade evolution where the characteristic transverse momenta in the decays, kj^ , are of order zBG, In other words, a calorimeter with an aperture 9 registers the energy spectrum of the intermediate partons at the certain phase of the development of the partonic system. Hence, the probability that the energy fraction z is deposited in a cone with an opening angle 0, should be related to the inclusive spectrum of partons 5,(2,^ ,\)^Y1 ^l{z,\ ^ ) ^'^•^^ where A denotes the incoming parent parton (A = q, g)and ^^= 1/l> Inln (Ee/A) at 0 < 0^ « 1. We assume here that the type of the registered parton B is not identified. To quantify the energy collimation in jet let us suppose that the deposited share of energy is large, z —> 1. Then A 1 ) the 'valence' contribution , D. ', dominates in eq. 'A (7.3) (7.4) »:<^.i.,u=^(^-r*'''-^#^^^^ A t where ^'5~le'"'5© corresponds to the evolution from the incoming parton A to the parton B=A, decaying inside a given cone with opening angle 0 (/jg<^0.5772, C s Cp = = 4/3, Cg= N^ = 3). At fixed value of z this formula describes the distribution in 0 . that has a characteristic maximum at some angle 6= &^ • Indeed, when 0 •—> 0^ , practically the whole energy should be deposited in the cone. The fact, that only a certain energy fraction Z has been registered, means that the fraction (1-z) is carried away by the additional hard partons produced at larger angles (smallness ^d (E0J ; D^^(^ ,^1^ ) -» (1-z) at^^->0).
342 Therefore, the probability D for z ^ 1 should rapidly decrease when &—> 6^ . On the contrary, with 0 decreasing (down to 6>^A/E ) the share of emission outside the cone grow5 . The quantity /i^ increases, and so the probability D, that the energy fraction z is deposited in the cone, decrease^again (the effect similar to the Sudakov form factor). To illustrate the energy dependence of the quantity ^z > v we present the approximate values of V(2L) Q-'t Z=0.9 and Z=0.5 (see Fig. 16) /q(0.9) ^ 0.55, iq(0.5) ^ 0.83; ;g(0.9) ~ 0.30, Jfg(0.5) ^ 0.54. It follows from eq.(7.5) and Pig. 16 that the energy collimation in a quark jet is stronger than in gluon one, and the collimation grows as energy increases. Energy spectrum of particles within given cone. Even more subtle proves to be a spectral characteristic of the energy flux registered by a calorimeter with the angular aperture 6 . Such a quantity represents a correlation between the energy flux and a particle within this flux. This implies, in essence, the double inclusive cross section: a parton B is registered as well as one of its offstrings, a particle h, as shown in Pig.15. The distribution in x, the energy fraction, of hadrons of type h within the registered energy flux may be presented as the convolution In the above D.^ determines the probability to find the parton B, initiating the subjet with the energy E and the opening ooigle S » within a jet A^ and J>q describes
343 eoCdecj^ =50% = 50% = 90% = 25% = 25% £g=50% Pig.16. Shrinkage of the cones, in which the fixed shares of multiplicity (S\ ) or energy (^a ) of a jet A(A=q,g) are concentrated. the distribution in the energy fraction x/^z* of hadrons of type h in a subjet B. Notice, that one can obtain eq. (7#6) by integrating the standard expression for the double-inclusive correla- 1) tion between particles . An integration over z in eq.(7#6) corresponds to the simplest case when it is known, that an energy flux is deposited within a given solid angle, but the corresponding
344 energy share is not measured. If one fixes the value of z, the integration in eq.(7#6) should be omitted. To estimate the integral in eq.(7.6) one can neglect z in the arguments of all logs since only the values of z '^ 1 are essential. This stems from the behaviour of the function B-^ at X « 1 B^^U,E%,q^) = 1/x^j)(1/x, InEB^/A , In Q^/A), (7-7) where P is a slowly changing logarithmic function that describes the hump-backed plateau (see Sec. 6). For better understanding the correlative nature of eq. (7.6) one may consider the two limiting cases. (i) e-*0„ ,»^Z) -^S'd-Z)^*, gr^'^^^^>^a;,E6l„,Q,). (7.8) In this case the whole energy flux of a jet A is deposited in a calorimeter, and the particle spectrum coincides with that in the overall jet. Here a'subjet' reduces only to one hadron h, the energy flux is predetermined by the value of x. Then the correlation disappear, and the expression (7.6) factorizes into X, the energy flux, and D^ , the probability for finding a hadron h with an energy fraction x inside a jet A. The correlation, that in a general case (at x < 1) is described by eq.(7#6), disappears, in fact, also for soft hadrons h, where x « 1. Emission of such particles proves to be less sensitive to the energy balance, and it should be determined by the average 'colour current* of hard partons. Substituting eq.(7.7) to eq.(7»6) at x « 1 one has ¥^ (x,e ; E,e^ ) ^ <CVNc • Dg^(x,E0,Q^), (7.10) where D_ is the known spectrum of particles h in a gluon
345 jet having the energy E and the openting angle 6^ , and the quantity <C>. is (7.11) In the above represents the aveiTage 'colour current' of part on B registered by a calorimeter. Integrating eq. (7#6) over x one immediately obtains the multiplicity of hadrons of type h in a registered flow n/ ( e; E, e^) - <CVN^-Ng^ (E0). (7.12) Thus, as it is easily seen from eqs.(7.10), (7.12),for a registered part of a jet the energy spectrum of particles as well as their multiplicity are proportional to those in eoi isolated g-jet with the different hardness Q = E0. The hardness of a primary A-jet, £0^, determines only a proportionality coefficient, that is the average colour current of the parent parton initiating the registered part of a jet (as measured in the units of a gluon charge, Cy = N^). The value of the mean colour current, originated by a parton A depends on the momentum balance between the quarks and the gluons in evolution of a jet A* The momenta carried away by the quarks (^Z^>a) and the gluons 15 " (< 2 > A) are calculable in the LLA ^» Substituting them to eq#(7.11) one obtains <o>^= <^^c -^pG F where
346 Eqs.(7»13) describe, in some sense, the process of losing by a registered parton B the memory of the colour charge of a parent parton A, as the aperture 6 decreases. In asyraptotics, when E and & ( e > Qq/E), one has <C^> n^= 3 and 2.12 for n^ = 6 = <C>^ = 2.4 for This mean colour current of 00 a parton in a cascade proves to be, naturally, somewhere in-between the gluon and the quark charges. Therefore, for a gluon jet the 'colour-grasp" of an emitter decreases with 6 decreasing, and for a quark jet this quantity increases, as shown in Pig. 17. <c> Pig. 17* The mean colour current of parton <G> in the q- or g-jet with hardness Q=E0o (E is the jet energy, e 0 is its opening angle)as registered by the calorimeter with an angular aperture 0, *$ = = Inln ^d^/A - lnlnE0/A. . It is of interest that, while the ratio of the total multiplicities in g- and q-jets asymptotically equals to C /C = 9/4, for the case of a narrow cone of observation this ratio tends asymptotically to 1.
347 xF g 9 15 10 5 0 e.. (1) 0.05-(2) O0O5-(3) io~enx Pig.18. Energy spectrum of partons (versus In 1/x) in a gluon Jet with an opening angle 9^, registered by the calorimeter with an angular aperture 0: (1) for the whole aperture B/B^ =1, (2)- 6/©^ = = 0.05, and (3) - &/Br. = 0.05. Fig. 18 illustrates the dependence of an energy distribution on the aperture of the registered particle flow, as given by eq. (7#6). The narrower is the registration cone, the harder are the particles within this cone, x > "s-^^ = Q^/E6. On the other hand, eq.(7#6) describes the average energy flux deposited within a cone 0 around the registered hadron h carrying an energy fraction x. As seen in Fig.18, a soft hadron with x« 1 is 'accompanied' by the energy flux only starting from the sufficiently large values of the calorimeter aperture, e > e min = Qq/xE. (7.14)
348 Collimatiop of multiplicity inside .iet> By analogy with the discussion above, one can ask, what is the angular size © of the cone, where the main part of jet multiplicity is concentrated, and what is the energy behaviour of this aperture. To answer this question quantitatively one should solve the equation N/(©^;E,e, )= S'-H/(Ee^) (7-15) and find the value of the angle ©g. . where the share S' of the total multiplicity is concentrated. Then accounting for the DL relation 1T^(E ©^^ ) = C^/N^*N (E©^), from eqs. (7#12),(7»15) one can obtain Using eqs. (7#13) we can rewrite eq.(7.l6) in the form 5^0 (7.17) E© A A\ p^^ ^Oo -^ r ^' 3- -Cvi A where a cr1.8, b <^0.8 and a <=« 0.8, b ^ 0.2. Shown in Fig.l6, for both q- and g~jets, is the dynamics of shrinkage of the cones ©^ , where the shares S= 0.25 and S= 0.5 of the jet multiplicity are concentrated. As it is easily seen, the multiplicity flow in a q-jet is collimated around the direction of the energy flux much stronger than in a g-jet. With the jet energy increasing the collimation of the multiplicity flow grows much slower than that of the energy flux. One can obtain a qualitative estimate for the growth of the multiplicity collimation roughly simplifying eq. (7.17), N (E©5-) = S'-N(E) at ©o -- 1,
349 exp 16_M..Ee b A = S' • exp This leads to e 8- LN (E)J V8Wc-ln 1/6- (7.18) Thus, the solid angle, where a half of jet multiplicity is concentrated, decreases with the increase of jet hard- —1 /A ness (E ©^ '^ E) approximately as W '^ (E), i#e# parametrically much slower than in the case of the energy collimation (see eq# (7#5)). lar distribution of multiplicity inside .iet< Finally, let us consider the angular distribution of a multiplicity flow around the direction of a jet energy flux. This can be obtained by differentiating eq.(7.12) over the variable In ©• u ' - £ <i^^«) - oi^s<>." •"' M mam exponent describing the asymptotical growth of the particle multiplicity in a jet, as given by eqs.(6.52), (6.53) k dM.t4(Ee)) d^9 ■> (7.20)
350 8. RADIOPHYSICS OP PARTICLE PLOWS 7,8,19) In the framework of LPHD the source of multiple hadro- production in HPs is gluon brerasstrahlung, so one should expect that all of the produced hadrons are the consequences of the colour dynamics. Therefore, the properties of the partonic skeleton, such as the flow of colour quantum numbers, should influence the distribution of colour singlet hadrons in the final state. The phenomenon of such a kind has been first observed in the experiments (see refs. •^»"-»'-^' and references therein), studying the angular flows of hadrons in three- -jet (qqg) events from e'^e"' annihilation, the so-called string '' (or drag ^•^^) effect. The data have strongly supported the predicted drag of the internet particles in the direction of the gluon jet (net destructive interference in the region between the q and q), for details see Subsec. 8.2. Detailed studies of the string-like phenomena are of importeince for the high energy HPs. These effects are interesting not only in their own right as tests of QCD. They should be valuable in helping to distinguish new physics signals from the conventional QCD backgrounds. In this Section we examine the distributions of multiplicity flow accompanying reactions with the complex topology, where three or more hard partons are involved in a HP. Our emphasis is on the QCD coherent drag phenomena and their manifestations in different types of hard interactions. Let us begin v/ith the particle distributions in the gold-plated three-Jet events.
351 8«1. Inclusive QCD Portrait of qqg Events of e'^e" Annihilation 8.1.I. Spatial distribution of multiplicity flow 61) In terms of the inclusive approach, discussed in the previous Sec. a proper inclusive characteristic of the spatial distribution of particle flows in the e'^e" qqg events is the E-^MC (8.1a) dH dS2 ^=:ZlJaEdEdEdE E.E^E, ds; a,b,c 1, c ^x dE, dE^dE^ dE^ ^^n^A d5^ (8.1b) <3; = ZZ ^l^E dE^dE 3 ^—' 0 A 1> c a\> c > 3 EaEt^, '^«'3 dE^dEdE.dQ^dQ.clQ, where the sum is over all particle types. This represents an angular correlation between the three registered hard particles (a, b, c), moving in the directions Hq^, n^ and n^ and the multiplicity flow around the direction n, see C V w ft. 0*^1Q^ When all tViT»ftP vftr-hn-nn r\ ^ n, and n^ are in the D C Pig.19• When all three vectors n same plane, the main contribution to dN^ comes from the qqg configuration of the primary parton system. In the leading order in ol ^ the parton kinematics is unambiguously fixed as follows a 6 Fig#19# Angular inclusive correlation between three energetic (a,b,c) and one soft particle (if) in the process of e'^e"" annihilation °+ ^ °a' n n b n 1 n c'
352 (8.2) highe 6J UJ i'J (x^ + x_ + x^ = 2), where x^^ = 2Ej^/W are the normalized part on energies, Q^^ are the angles between partons i and j ( + ,- s qp^; Emphasize here, that owing to coherence, the radiation of a secondary soft gluon S2 ^^9 ^ ^±^ ^^ angles than the characteristic angular size of each parton jet proves to be insensitive to its internal structure: ^2 is emitted by the colour current which is conserved when the jet splits. That is the reason v/hy one may replace each jet by its parent parton with P^"^— 0, as, e.g., in eq. (8.2). Let us turn firstly to the simpler case of two-jet events. Here the particle flow distribution corresponding to the discussed above correlation (7.2a) can be written as where N^(Y^,Y) =(d/dY^)N^(Y^,Y) ; N^(Yi, Y) stands for the multiplicity in a jet A(A = q,g) of particles, concentrated in the cone with an angular aperture B^ around the jet direction n^. In the above a^^s 1 - nn^i^ (i = +, -), n^^ H^, ?r,^n^ = -F^, Y^= In (E/A- >fIT72), Y = In E/A. and E = W/2 is the jet energy. To understand the meaning of the quantity N^(Yj^,Y) it is helpful to represent it as
353 A?=i^^ , Y,-Y,.&z=4(f|f) Yi (see for details Subsec. 7»3)« In the above N-g is the multiplicity initiated by a parton B within the cone 0? , and D** denotes the structure function for parton fragmentation A —^ B (see ref# •')• Eq« (8#4) accounts for the fact that due to coherence the radiation at small angles 6^ «* 1 is governed not by the overall colour current of a jet A, but by that of a sub jet B, developing inside a much narrower cone ©. • This formula has a correct limit at 6/—► ST, Y^ —^ Y, A^ -^ 0, d/ ^g'd-Z) ^/, N^(Yi,Y) ^N^(y). Eq« (8»2) looks, formally, as the sum of two contributions, accounting for the independent evolution of the q and q-jets. However, one can see, that this reflects also the collective character of the soft radiation at large angles• Indeed, in this case neglecting relative corrections of order 0(<^ ) one obtains Nq'(Y^,Y) =>^Nq (Y_,Y) «Nq'(Y)co>{A^-Nq(Y), (8.5) and eq.(8.2) can be transformed to Ei!!M= i£t(;-)/(^I-) , (8.6, where the notations of eqs«(7«20) and (3.13) are used (a -f a_ = ^^J^^)* Eq« (8,6) represents, in fact the radiation pattern for the internet gluon emission by ein antenna (+ -), of eq«(3«12), the factor N_ takes into o account that the final gluon is a part of a cascade For the ao-called radiative two-jet events (e e~—♦ qq/) the emission pattern is given by a qq sample Lorentz
354 boosted from the quark cms to the lab.system (i#e# the cms of qq/), and the corresponding particle multiplicity 4- - — 2 should surely be equal to that in e e —v qq at W^ = 2 ' * (Rv + Px) • It is useful also to introduce here the Lorentz-invariant generalization of the rapidity ^^^ of an emitted gluon K and its momentum component transverse to the + and - directions, kj^, (8 7) By analogy with eq.(8#2) the formula for angular distribution of particle flow in qq/ events can be written as (8.8) where and ^q(q) == '" ^' V = l^(V^-K^).^q-=^^A^^^^ Y = In E/A. , E = W/2, I^^ =(+«)- 1/a^ - 1/a^. (8.10) Por the emission at large angles (a^^'^a^'^ 1) when according to eq.(8.5) all the factors N* are approximately the sajne, eq. (8.8) coincides with eq. (8.6). We are ready now to deal with the three-jet event sample when a hard radiative photon is replaced by a gluon g-j. For a given qqg^ configuration the particle flow can be presented, analogously to eqs. (8.8)-(8.10), as (a ii)
355 with Y^ = In E^/A, Y^^ = In(E^/A-\|^i72 ), Y = In E/A This formula accounts for both types of coherence: the angular ordering inside each of jets and the collective nature of the internet flows. The first three terms in eq. (8.11) are collinear singular as 0. ~>0 and contain the factors N describing the evolution of each of the , The QCD Jet initiated by the hard emitters q^q" and g^ • last term in eq.(8#11) accounts for the interference between these Jets. It has no collinear singularities and contains the common factor N (Y, Y) independent of the direction n# 1000 T 2i^dM 100- 10^ I i I I t I I I ■ ' ■ 1 ■ ' ' ■ I I -120 -60' O' 60 120 9 Pig.20. Particle angular flows in the three-fold symmetric qqg events on the event plane with respect to the q-jet axis at different values of the parameter E/A = 60 (1), 1000 (2). Arrows indicate the mean cone apertures, where the 50% of the energy flow of each Jet is concentrated.
356 Pig* 20 illustrates the predicted distribution of particle flows projected onto the event plsuae for the three-fold symmetric qqg events# As it follows from eqs. (8«8),(8.11) when replacing a radiative photon by a gluon g^, with otherwise identical kinematics, an additional particle flow arises ( SgrdH ) Q^^qqg QSTdN^q^ (8.12) = a/a^.Ng' (Yg^ . Yg) + [I^^ + Ii_ - I^.] Ng (Y). Note that for the case of large radiation angles both cascading factors N become approximately equal and one has ^"-^T^h = ( (1"+) + (l""-) - (+'^-))^n' (Y). (8.13) dS^ S 6 An interesting point is that this expression which seems to look as a 'time' gluon contribution is not positively definite. One can see clearly, e.g., the net destructive interference in the region between the q- eind q-jets. The soft radiation including the gluon jet proves to be less than that in the absence of the gluon jet ^-^ . This drag phenomenon is clearly seen in the recent experiments 9 12 13) (see refs. ^^ >*-^^) thus strongly supporting the LPHD concept. The physics of drag effect shall be discussed in more detail in Subsec. 8.1.3* Finally,let us emphasize that for detailed study of qqg events it would be important to identify quark and gluon jets. This could be made in heavy quark events QQg (Q = = c, b) ^^ . For example, one may utilize the high rates at the Z°-factories. If we take the integrated luminosity JX dt /experiment to be ^ 2»10'^ cm" , we expect ^^0^ ccg, bbg events produced. Possible taggings of the heavy quark jets include, e.g., secondary vertex tag, triggering on high p^leptons or the specific decay modes
357 of the heavy particles, etc. 8>1»2« Od total particle multiplicity in ggg; events. Let us discuss the connection of the particle multiplicities in two-jet etnd three-jet samples of e'*'e"'annihilation. The particle multiplicity in an individual quark (gluon) jet can be defined through that in e'^e" —> qq —> hadrons ( jCg^—^ gg —^ hadrons) as follows N . (W) = 2 N (E).(1 + 0(oL (E))), E = W/2, (8.14a) ^ (M^) = 2 Ng (E)-(1 + 0(o^3(E))), E = M^/2. (8.14b) When three or more part on s are involved in a HP, say, e"^e" —> qqg-i the multiplicity can not be interpreted in a similar simple way. The point is that multiplicity becomes depending on the geometry of the whole jet ensemble. So, the problem arises of describing the multiplicity in three-jet events, N - , in terras of the discussed above characteristics of q- and g-jets. The quantity N -- should depend on the qq^g geometry in a Lorentz- qqg -invariant way and should have a correct limit when the event is transformed to two-jet configuration by decreasing an energy of a gluon g^ or by decreasing its emission angle. Note that when deriving the formula for N^^ accounting for the interjet contribution, one needs to control systematically the relatively small ^'{oU terms. Formally, s the MLLA analysis does not provide such an accuracy, since the change of scale of parameter A, say, A.-^C A. leads to the relative change of N AN/N ^ >joCg(E) • In C. The subleading to the MLLA, d^^ corrections to the ajiomalous dimension l((<^^) (see Sec. 6), that generate ^p!^ terms in the multiplicity, can be calculated if the second
358 loop contribution to the jg function is included. This is interesting, theoretically, in its own right, but for our purposes here it is enough to note, that *o^ corrections to jf can be formally absorbed into the definition of parameter A. , the value of which is determined phenomeno- logically from connection of the MLLA partonic spectra with the measured ones ^ >^ ^, In the following we shall use the MLLA eqs.(6.52), (6.53)~formulae for particle multiplicities with just that very value of TV • This permits one to fix the energy dependence of the hadron multiplicities including the terms ^ ^ot^- N. One can easily check that integration of eq. (8.2) over the total solid angle reproduces eq.(8.14a). Similarly, with an account of eq.(5»27) the angular integral of eq. (8.8) can be written as /g ^c\ aN.j «^- Ki ; Now we can transform this formula to the Lorentz-invariant form. For this let us rewrite ^ (-n as ^q(q) = ^ + 1" ^+(-) ' ^+(-) ^ Sq(-/E (8.I6) and use the expansion Nq(Yq) = N^(Y) + Inx^-N^ (Y) + O(oi^.Nq). (8.17) Then where (8.18)
359 aaid E* is the quark energy in the cms of qq, i.e. the Lorentz-invarifiuit generalization of a true parameter of hardness of q (q)-det. The multiplicity N_-_ is, by H4S analogy (8.20) . [ft, ^-^ . ^^ ^ if - ]. ^; (Y) , where Y (-j « Y + In 3j:^(.)» Y = Y + In x^ , Y = In B/A . Using the expansion analogous to eq. (8.17) for the multi plicity of each of jets one comes to the final Lorentz- -invariant result ^qqg = C2Nq(Y;.) + Ng(y|) ] • (1+0 i^^) ) (8.21) with y/. = in (^(EgjA). Y* = lxi(lIWlS^) = 4l^ ^%?J where P^_l stands for the transverse momentum of g^ in the cms of qq, cf. eq. (8.7). As is easily seen from eqs.(8.l8),(8.21), the replacement of a photon by a gluon g-j leads to an additional multiplicity which depends not on the gluon energy but on its transverse momentum, i.e. on the hardness of the primary process. Eq. (8.2t) reflects the coherent nature of soft emission and has a correct limit when the event is transformed to two-jet configuration. Another form of representing N - . /o 03) where Y^^ = In if^/A. ) = In E^ ./A , deals with the multiplicities of two-jet events at the appropriate invariant pair energies E^^. This formula has also
360 proper limit , 2N_, when qqg configuration is transformed to a quasi-two-jet one, g + qq with the small etngle between q eind q« 8.U Drag effect in three-.iet events 43) Consider now the particle flows at large angles to the jets in +^- ^ «r^ j^g^ ^2,1 the angles between jets^0j^- and the e^e qqg B^ "^ E. E jet energies E-. , be large: ©^ ^&, ^^ /^.^ ^ ~E ^ W/3» As it was discussed above, the angular distribution of soft internet hadrons carries information about the coherent gluon radiation off the colour antenna being formed when three emitters (q, q and g) separate from one another. The angular distribution of a secondary I Pig. 21• Kinematics of internet radiation in three-jet events. soft gluon gp (see Pig. 21) can be written in the notations of eq. (3«13) as sgrdN ~ qqg dQ^ = 1/IVWj:^ («2^*\ ^^m^ = (r^- ) 1/N.^.(A ))-Ng (Y^), (8.24) m cf. eqs. (8.11)-(8.13) and (3.l6b). In the above y„ = s: In Ee^/A ,0^,= min[0^, e_., 0^} with cos m ^i ~ ^p^i* The radiation pattern, corresponding to the case when a photon )( is emitted instead of a gluon g-j is, cf.(8.6).
361 (8.25) 83rd N ^v _^ A ^ = 1/N^.W^.(5*2) ♦ N' (Y„) = 2VN.(+ -)N*(Y„) The dashed line in Pig. 22 displays the corresponding 'directivity diagram', projected onto the qq^ plane: H.(^,) zc cjcosa . A z (+-) =2Cpa^_V(=t,^) , (8.26) where V(o(.P) = z 5r-oc or-p COSoC - cos ^ \ Stv\ol s ^ Here oC = ^, , ^ = 0^.-4^ ^ (see Fig. 22). (8.27) Pig.22. Directivity diagram of soft gluon radiatioia, projected onto the qqjf (dashed line) and qqg (solid line) event plane. The curves corresponding to expressions (8.26) and (8.28) are drawn in polar coordinates: &-% ^ "^ - In 2W(^2,)* I^otted circles show the constant levels of density flow W(H^^) = 1, 2, 4 •
362 Distribution W (Hp) is simply related to the particle distribution in the two-jet events e'*'e"' —^ q(p^) + q(p_)> Lorentz boosted from the quark cms to the lab system (i« e* the cms of qqjf). V Replacing i by g^ chaxiges the directivity diagram essentially because the antenna element g-j now participates in the emission as well. However, contrary to intuition , this change does not only lead to the appearance of an additional particle flow in the g-| direction. Integrating eq.(8.24) over 9^ one obtains (*»^-^2 ^ W±i (^,) = Nc[^^r^^^.^^ ^ a.^-V(^,)f) - i-ja^^jC^p^ N<5 Pig. 22 illustrates that the particle flow in the direction opposite to n-j appears to be considerably lower than in the photon case. So the destructive interference cancels radiation in the region between the quark jets. For example, suppose n^?_ = ^4?1 " ^-^i ®^^ °2 Points in a direction exactly opposite to ^^, that is, midway between the directions n and n_. Then neglecting the weak dependence N^. on 9 one arrives at dM-JdS^ N„^ - 2 ^^S' 2 ^ c ^ j/^^^ ^3^29) 2 Due to the constructive interference effects, there is a surplus of radiation in the q-g and g-q regions. Thus, the analysis of soft bremsstrahlung radiation pattern demonstrates particle 'drag' by the g-j jet. This phenomenon is easy to understand qualitatively. If a term p proportional to 1/N^ is dropped, the two remaining terms in eq.(8.24) may be interpreted as the sum of two independent (1^^+) and(1^-) antenna pattei^is, boosted from their respective rest freimes into the ovei^all qqg cms. The depletion of the q-q region is a direct
363 consequence of these boosts. This scenario literally 67) repeats the explanation given in the Lund string model. ^ So it appears that the latter provides an excellent picture for mimicking the collective QCD effects. Experiment -^* ^t^^^ j^^g presented evidence of the drag effect in three-jet events. The depletion of particles was observed in the q-q valley relative to the q-g and g-q valleys. The strong support of this effect comes from the comparison of qqg eind qqjf events, that provides the test 4-3) of coherence effects in a model independent way •^'. Pig. 12) 23 shows the measured ratio 'of particle density in the q-q region for qqg eind qq/ events. This ratio should be 1 if no coherence effects would be present, since kinematic- al configurations of both event types were similar. Emphasize that eqs. (8.11) and (8.24) provide not only the planar picture, but also the total three-dimensional pattern of the particle flows in three-jet sample. It is worth noting that the destructive interference proves to be so strong, that the particle flow in the region opposite the gluon jet is smaller than that in the most kinematically 'unfavourable' direction, which is a normal to the event plame. In the asymptotics the ratio of these flows in the case of three-fold symmetric qqg events is ^^ ^ ^ . = 17/U. (8.30) N<qq> 2(4Cj.-N^) As we shall see below, in other types of HPs coherence should lead to a rich diversity of the collective drag phenomena. So, let us enumerate the main lessons from studying this phenomenon. 1. The effects of gluon interference do not permit one to formulate, a priori, a probabilistic scheme for the development of partonic cascades. However for each specific
364 icr cr icr PC 0.0 0.0 0.2 0.4 0.6 0.8 1.0 X Fig.23* Ratio of the particle density in qqg and qq/ events as a function of the scaled angle X=04./0+ > where e^. is the angle be- tween jets q and q. Shown are data from JADE, MARK II and inclusive characteristic it is possible to divide the essential coherent effects into two types: (i) accounts for the coherence effects in the intrajet cascades. These are reduced on average to the Markov process of particle multiplication into the sequentially shrinking angular G (ii) gives an account of the inter- cones 9 ^ -u , ference effects in the total amplitude for production of the minimal number of Jets (partons), whose configuration corresponds to the given experiment Just this amplitude reflects the specificity of HP. Each of the produced jets evolves in an universal manner inside a fixed opening angle 0^ , which size depends on the mutual location of jets from the ensemble. The aforementioned division permits, in any concrete case, a classical probabilistic picture of parton branchiqg to be retained, thus allowing event modeling . 2. The experimental evidence of such bright phenomena as the hump-backed plateau (coherence of the first kind) and the drag effect (coherence of the second kind) has shown quite convincingly that these interference effects
365 survive the hadronization stage• Therefore, one can say, that in spite of confinement the hadronic system reflects very delicate features of the colour field dynamics,which in turn stem from the nature of QCD as a gauge theory• 3. The observation of the colour interference between soft hadrons from, say, q- and g-jets reveals the QCD wave properties of hadronic flows. Thus, it can be considered as an experimental proof of the common brems- strahlung nature of the hadroproduction mechanisms for both jets. The properties of drag interference phenomena are deeply rooted in the basic structure of non-Abelian gauge theory. 4« The relative smallness of the interference effects does not diminish their fundamental importance. This consequence of QCD radiophysics is a serious warning against continuing ideas about independently evolving jets. 5. Drag effects lead to a noticeable azimuthal asymmetry of particle flow relative to the 'jet axis'. The character of this asymmetry depends on the geometry of the whole event, see for details Subsec. 8.6. 8.2. Drag Phenomena in High p^ Hadronic Reactions'* °^^-^' Jets or individual particles at high p^ in hadron- -hadron collisions originate from hard parton interaction at small distances. It is the colour dynamics of these quarks and/or gluons that determines the topology of the final state. Because of the presence of coloured constituents in both the initial and final states, the study of high Pj_ HPs has proved more complicated than in the case of e'^e" annihilation. However, the nature of jets, basing on the dominant role of the QCD bremsstrahlung processes, is the seime for both reactions. Therefore, the main physical phenomena and characteristics of final states are very
366 much alike. This puts jet physics on essentially the same footing as e'*'e*" annihilation. Let us enumerate the virtue of high p^ processes. 1. A diversity of hard interactions at small distances; by varying the experimental conditions (triggers) one may extract the dominant subprocess and turn from one subprocess to another. 2. Dependence of length and height of the 'plateau* in the hadron spectra on the different parameters: the length is determined by the total energy of the collision, and the height and the plateau structure - by the process hardness (trigger Ej_ )• Thus^information becomes available that is inaccessible in e"*'e" annihilation, where both the energy and the hardness are given by the same quantity W. +^- 3. Unlike e e annihilation, where about half of the jets are generated by heavy quark pairs, subprocesses with heavy quarks Q are suppressed in hadronic collisions in a standard way. 4. Pinally, there is the purely practical argument that just in high Pj_ hadronic collisions the largest possible energies (hardness) will be reached in the near future. These reactions are also the best source of high energy gluon jets. Detailed studies of such processes are necessary for designing the future experiments and the analysis of their data, for finding new heavy objects. In particular, drag effects could provide a valuable additional tool, helping to extract and to study new physics. The variety and complexity of colour antennae typical for high p , processes, complicate the picture of final hadronic distributions. However at very high energies, when particle multiplicities become large enough.
367 the interesting possibility arises of using a detailed analysis of hadron flows on an event by event basis to extract information about colour transfer at small distances. Of course, the question still persists, whether the colour coherence effects will be visible clearly enough above the normal soft scattering background In course of a hard interaction colour is transferred abruptly from one parton to another. For example, to leading order in 1/N the colliding quarks q- and q^ are simply *recharged': ^2 (8.31) ji mk 2 im jk 2N^ ij mk The parton-parton scattering acts here as a colour antenna Gluon bremsstrahlung associated with the incoming and the outgoing partons leads to the formation of jets of hadrons around the directions of these coloured emitters. It is the colour topology of the partons, participating in the scattering , that determines the radiation pattern. To demonstrate, how the coherence of bremsstrahlung connectes the structure of hadronic accompaniment with the t-channel colour transfer, let us consider high p_j__ scattering of energetic partons A and B (% "^ E-g^\rs)at relatively small angles 0„ '^ P» /E « 1, as shown in Pig.24• The hardness of the process is determined by the momentum transfer Pj_<^ NJ -^ , which naturally restricts the transverse momenta of the accompanying gluon bremsstrahlung Ix < Px and, so the development of partonic cascades.
368 \ e s Pig«24» Soft gluon bremsstrahlxing accompanying small angle scattering of partons A+B the scattering angle)* A'+B' ( ®s « 1 is In the structure of the final hadronic system three regions may be separated. Two of them adjoin the fragmentation regions of the colliding hadrons and occupy the intervals av^ = In Pj^/A (8*32) where I is the pseudorapidity 4 a 1 ■¥ CjOS 6 \ a ^' 1 - cos 0 The hadronic spectrum in each of these intervals ('extended fragmentation regions^ is saturated with the particles from the hremsstrahlung cones of the incoming and the scattered partons, and so results from an sum B height of the distribution is determined, roughly sum and f Cg+C-g respectively* In the central region (8.33) (final particle angles larger than the scattering angle V,) the incoming and the scattered partons radiate
369 coherently, and, as a result, the hadron density is determined by the colour charge C+ of the t-channel , A A . exchange• Since in the given kinematics (-t « s) gluon exchange dominates, we conclude that in the central region (8.33) hadronic spectrum is determined by the ♦colour strength' of the gluon current C^., and what is of importance, it becomes universal, independent of the nature of scattered partons (A, B = q or g). As one can see from a not complicated analysis, the resulting spectrum, accounting for the parton branching effects, is independent of the energy (pseudorapidity) of the particle registered at angle 6 > 6^ at fixed transverse momentum k j^ : R Jv>^.. -^'^c] .Z OCT ^S^W. '^ A (8.34) In this expression x-D^ (x,ln Q/A) is the standard distribution of particles with energy fraction x in a gluon jet, for which the product of energy and opening angle equals Q, see Sec. 6. Integrating over kj^ of hadrons at fixed vj , we obtain 1 E Thus, the hadron yield in the interval lh(e)l<4(6^)5;-tM-p- does not depend on the rapidity, so a flat distribution emerges ('true plateau', see Pig. 25), whose height is determined by the hardness of the scattering process. This is given by the formula, familiar already from the discussion in the previous Subsec, f (In Pj^) <:«n' (In Pi/A). (8.36)
370 i Hri.kO B + B I PLATEAU NOMCOHEREhfT | -7(®s^ Ah. A I MONCOHE RENT 1 Pig.25* Universal rapidity plateau ( Iv^ I < -^ V^c ^' ^^® solid and dashed lines illustrate schematically the difference between qq £ind gg scattering. +_- A similar distribution appears in e'e annihilation 4-2 ) (the so-called truncated plateau ^ ^) when one considers a special kind of events, where at a given hardness of the process, E = W/2, all the hadron transverse momenta are bounded from above p. < p, ^^^^ « E (Pj^»A)* au in high The 'X HPs should coincide with the doubled density of the internet hadron flow in e'*'e'"annihilation at W -^ 2p X* Listed above conclusions about the structure of inelastic hadron scattering processes in the central region ( lAV^l < 1/2 In (s/jtl), It I « s) are valid provided the one-gluon exchange dominates. To guarantee this, one should register in the final state of hadron- -hadron collision at least one particle with transverse momentum Pj^ exceeding some typical value <^PL'>(iiff|^gion' characteristic for hadron processes at high energies. The latter emerges in the framework of QCD analysis of the total cross sections etnd grows very slowly as s
371 increases (see, e.g.^Ref,^'^). Colour interference between jets seems to become a phenomenon of large potential value as a new additional tool for discriminating between HPs For example, reconstruction of antenna pattern by the effects of particle drag may help to visualize the production of new colourless heavy objects - the Higgs boson H, new quarks and leptons, supersymmetric particles, and so on. Most of these objects produce hadronic jets, and the configurations of interjet particles should differ from familiar QCD processes like parton scattering. An instructive example comes from the study of the radiation pattern associated with the hadronic production 7) of a heavy Higgs . If H boson is produced via the g-g fusion of Pig.26a, then the similar plateau f(io,ln %/A ) emerges in hadronic spectrum. However in the case of the W-W mechanism of Pig.26b the central region (8.33) should be (a) (fe) P P P P Pig.26. Hadronic Higgs boson production via a) gg fusion and b) WW fusion. empty and the process looks as the quasi-diffractive one (the gluon emission by initial and final partons at large
372 angles cancel each other coherently because in t-channel colour is not transferred (see the discussion in Subsec. 3.1). Another example is the comparison of the production of a colourless object via gg or qq collisions. Here, if the hard kinematics is the same, the heights of the accompanying plateau should differ approximately by a factor of two (N^/Cp = 9/4). High Pj_ processes are rich in the drag effects. Let us consider, for example, the topology of events, result ing from the quark scattering ' In this case the two crossing processes shown in Pig. 27a and Pig.27b have approximately equal probabilities. However each of them has its own colour topology, and therefore specific particle flows, as seen in Pig.27c and Pig. 27d. Por the subprocess of Pig.27a, the soft particle radiation pattern is 4Srdl^'^^'^2 r A A ^ A A = [(14 + 23) + —'— • (2 (12 + 34) - d Q ^ 2N^Cp A A A A «. ^ ^ , (8.37) - (14 + 23) - (13 + 24)) JCp/N^ • Ng (In E//V), see for notations eqs. (3*13) and (7.20). In full analogy with the discussion of string effect in the previous Subsec, one may say that the leading contribution (the first term in (8.37)) has the structure - A A of the sum of tv/o independent qq~antennae 14 and 23. This fact also can be mimicked by means of the topological picture of the Lund model. ' One can find a comprehensive study of the radiation associated with the different types of parton-parton scattering in ref. 69).
373 P (a) p u (e>) 3 f (c) 1 3 I A (d) 1 Fig.27. Colour antennae for two crossing high Pj_ qq scattering processes and the corresponding particle flows. It is important to emphasize, however, that here,-unlike the Lund approach, to each contribution (single string) there corresponds not a phenoraenological particle spectrum, connected with the standard plateau of the parton model, but a dynamical distribution which takes into account the effects of cascade multiplication.
374 The height of this distribution is determined "by the hardness of the process. Furthermore, PT approach permits one to control not only the leading colour contributions, but also the 0(1/N^) corrections. The interference between the subprocesses of Pigs.27a and 27b also leads to the colour-suppressed effects. Basing on the perturbative prediction for the distributions of the final particles, it may be possible to distinguish- on an event by event basis - the definite fluctuations in the angular structure of particle flows, corresponding to one of the two topologies of Pig. 27c or Pig. 27d. As the first step, one can study the correlations of the multiplicity flov/s, e.g., their angular asymmetry. pii cf. ref. ^ (8.38) ^ = (%R ^ \l - NuL - %r) / (%R -^ \l -" % ^ %R^- Here N. . is the number of the final particles? in the angular region ±i on the scattering plane (i denotes the upper or the lower quadrant, j denotes the right ot the left quadrant). As a more complicated example we shall discuss in Sec. 8.4 the characteristic features of the antenna pattern, associated with a subprocess gq(q) —^ gq(c[)« Finally, let us briefly consider hard gluon-gluon scattering g^ + gg —> go + g^ at small angle 63 « 1, when one-gluon exchemge dominates in t-channel. Here to leading order in 1/N_ the associated soft radiation is ^^ OI OQ A. A. A. A A A 85rdN ~ [13 + 24 + 1/2-(12 + 34 + 14 + 23)]lJ .(8.39) dSS » This formula shows that two colour configurations of the participating partons, as shown in Figs. 28a and 28b, contribute to the radiation pattern.
375 t t Pig.28. Colour antennae for hard scattering g-j+gp "^E-^-^Sa in the case when t-channel gluon exchange dominates (©g « D- 8.3« Prompt ^Production at Large p 8) X Drag effects, very similar to 3-jet events, can be studied in high p^^ processes, such as Y /tj^ pair, W, Z, ••• production, where a colourless object is used 7 71) as a trigger ' • Consider the three jet production '^) P + P H (Pj^) + jet^ + jetg + j et 3* (8.40) The basic graphs describing the process are shown in Fig. 29• We shall argue in a moment that gq yq dominates qq—»Yg« Keeping only the hard Compton scattering process, the cross section for producing a hard photon of transverse momentum p = E, corresponding to 90^ scattering in the centtr of mass system of the parton-parton 72) scattering, is ' The effects of colour coherence in the prompt W,Z production at high p. were studying in detail in refs.7,8,31)-
376 — dp~ d cos 0 = 11- 2 q q Jdy ^qD^(Xq,p^)XgD^(Xg.p^) dS- + (P P). dPj_ (8.41) g CL Pig.29• Hard scattering graphs leading to 2+ jet production. In the above x •x^-s = 4p 2 g „^ _ .^^ and y = y^+y2 with y^ and yp the rapidities of the outgoing jC and gluon jet respectively. As usual e is the electric charge of the quark having flavour q. Then dG/Cdp^dy d cos 6) = P (8.42) P) The corresponding formula for the hard process qq -^Vs IS dSyCdp^dy d cos 8) = = Z q 2 JToC • oL 9P ^ x^Dp<l(x^,p^2) x-Dp^(x.,p/) ^ (P (8.43) P). When x-(x-.) is less than about 0.1 the Compton process 4 o dominates over annihilation and so we shall neglect the contribution given by eq.(8.43)» Now to the main point of this Subsec. In addition to
377 the jet produced in the hard collision there are also soft gluon emissions associated with the incoming qiiark etnd gluon lines and with the final state qiiark which lead to an interference pattern (drag effect) almost exactly as in the process e'^'e" qqg# The picture of the soft gluon emission is schematically illustrated in Pig. 30. + g(2) q Pig•30. Soft gluon emission from the hard scattering graphs of Pig. 29. The formula for radiation in direction n, associated with the hard scattering has the same form as eq.(8.24), 8Sr dN dn = ( (2^3) + (2^1) - -^ C^^^^n' (Ym), (8.44) where n^^ is the direction of the part on i Pig. 31• The variable Y m = In Ee^jj/A , as shown in govemes the evolution of a jet with energy E and opening angle 0j^. In the above 0j^ = min{0^ ^^z>^x} ^^"^^ ^^® ®* "^ ^^±* As discussed in detail in refs. ^'>^^^ (see also Sec sr^ 6) a reasonable phenomenology can be done for production^taking the MLLA formulae for multiplicity of pions in a gluon jet, N^ (see eqs. (6.52),(6.53)). To quantify the drag phenomena we shall evaluate
378 !/ I Fig«31. The kinematics of 90° scattering in the hard process for Z+jet production. the radiation pattern for final state pions projected onto the plane of the hard scattering axid at angles midway the between the parton^ involved in the HP, directions labelled by ABC and D in Fig.31. Thus, dN^/d^ corresponds to the number density of pions in the plane of the hard scattering and midway between the directions determined by the incoming gluon and the outgoing photon (note in Fig.31 that q- and gp are incoming while V and g^ are outgoing lines). Por purposes of illustration we present in Table the values of pion multiplicities in the 30° sectors around the direction A, B, C and D 73) respectively for different values of B_^/A • Table The energy rise of pion multiplicities in the internet 30° sectors VA A B C D 60 0.62 0.52 0.51 1.47 200 1.04 0.88 0.87 2.44 10^ 1.94 1.57 1.61 4.47 io4 4.3 3.4 3.5 9.6 As is easily seen, the particle production is the largest
379 between the directions of the incoming gluon and the outgoing quark, but '^•2.3 times smaller between the directions of the incoming quark and the outgoing quark. So, just as in the reaction e'^'e" —> qqg, the drag of hadrons is predicted in the direction of the gluon jet. This drag effect leads to an azimuthal asymmetry of particles, which can be seen by looking end on at the struck q-jet, see for details Subsec. 8.6. The observation of such an asymmetry may indicate which of the incoming particles in a given event has shaken off the gluon. Finally, notice that one can find in refs. ^^t^J ^-j^q discussion of the observability of the drag effects in prompt Jf. W production at collider energies above the soft scattering background. 8) 8.4« Two Jet Production at Large p^ ^ As it ]:ias been mentioned above coherence effects are more difficult to observe in the spectrum of hadrons associated with two jet production than in Y or W production. Nevertheless, there are some specific effects which should be observable. The only hard scattering process which has a large asymmetry in the spectrum of associated hadrons is gq(q) ->gq(q) for which subprocess the two jet cross section is /o .c> dy-jdygdpj^^ ^ -^ ^ ^-^dpf with dc/d p2 referring to the fundamental hard process. In general, the process gq —^ gq does not dominate the competing hard processes such as gg —^gg, qq --> qq, etc. However, if one chooses x„ very small, say x_. < 0.1, and Xq not too small, then two jet cross sections will be dominated by eq. (8.45)• As an example of a region where Sq "^ gq should be the dominant subprocess we might
380 imagine the production of two jets having p = 90 GeV at the Tevatron pp collider and with x » 0.03 and x = 0.3« o H In what follows we assume an appropriate kinematic region has beeen chosen so that the QCD Compton process is the dominant hard scattering. The spectrum of hadrons associated with the hard process eq.(8.45) is dN -xl-1 8gr^=[H^(a,t,S)] P A A - Cj,[2 (24) + 2 (13) 2/N^-{H^({i,?,S) [Cj,(2'4) + N^(1^)] A (23) (1^2) - (34) - (U)]H^(u,t,£) Cp[(l'2) (3^4) A (23) A - (U)jH2^(u,t,S)3-Ng(Y^), (8,46) where we use the notation of ref • 69) for the hard scattering amplitudes. The incoming gluon and quark lines are labelled by 1 and 2 while the outgoing gluon and quark lines are labelled by 3 and 4 as shown in Pig# 32. Por t = u =s - s/2, 90° scattering in the partonic center of mass system, one finds Pig. 32. The kinematics g 90*^ scattering in the hard process q(2)+g(1) q(4) + g(3). 85v(iN 2 dS A (12) W [Cj,(24) -h N^(13) - ^p[2(2^)+2(1^)-(23) - A A - (34)-(14)] (8.47) 704 ^ [(12)+(34)-(2'3)-(14)]}Ng(Yjjj) .
381 Evaluating eq. (8.47) in the plane of the scattering and midway between the direction of the parton momenta, in the partonic center of mass frame, one finds ^^^ » 2.1 N^' (Y_), (8.48b) dn S ro ^^T- ^ 0.5 Nl (Y ) . (8.48c) dif g m or Por pion production d/dy N can be estimated to be . or S d/dy N!. » 9 at p, = E = 90 GeV. Thus, the asymmetry indicated in eqs. (8.48) may be observable above the normal soft hadronic background. However, the situation is quite different than in the case of hard )( or W production. There one had only a single outgoing jet. In the present situation there are two outgoing jets axid it is not apparent which one is the gluon and which one is the quark jet. In fact, one must use the asymmetry indicated in eqs. (8.48) to decide which jet is the gluon and which is the quark. This, in principle, would give the possibility of comparing high p. quark and gluon jets in the same event. 8.5« Correlations of Interjet Particle Plows A new interesting majiifestation of the QCD wave nature of hadronic flows arises from studying the double- -inclusive correlations of the interjet flows in e"^e~ — qqg events, as shown in Fig. 33• The point is that here, unlike the case of a single-inclusive distributions examined so far, one faces such tiny effects as the mutual influence of different qq antennae. These effects can not be mimicked by the orthodox Lund string model , which deals with the independently fragmenting string segments (antennae). To our understanding, only much more
382 sophisticated algorithm for the Lund Monte Carlo the so-called 'dipole f onnulation * -^^^^-^^ may reproduce them. The double-inclusive correlation of Pig.33 bear the information on the colour shielding of qqg- antenna by a field of gluon ^2* which has been produced in a cascade on the stage, preceding the emission of a soft gluon go* Just the latter initiates the registered flow of particle^ so the energies are ordered E-j « E2 « E^ -- E^ <s^ E^ ^ E = Y//3. (8.49) It is v/orth noting that' for the single-inclusive multiplicity flows dN(Vi )/dQ examined so far, these shielding effects are asymptotically inessential since ^ dQj / J dSS^dSPj ^ J Ej^ 2Tr J Ej 2r (8.50) / iwter jet \ ^ A^ ' ^^ 3- ^ dQ The double-inclusive correlation of the internet flows, d% / d^^d^^ , is of the order of ( fZI )^ N„^. Just as in the case of drag effect, one can neglect the weak dependence of the cascading factors N* on angles and the ratio of the correlation functions will be determined "by the total lowest order amplitude for e'^e" —» qqg^g2g3* The angular distribution of particle flows in the directions of the secondary jets gp and g^, whose energies are ordered according to eq.(8.49), is given by the sum of three terms having different structure: where
383 3 .2 ^ , 3,2 + lI^(2Cp-N^)[A^^A^ _+ A fA^ _] , (8.52) BlI=NcW>1++^1^1- + N^(2Cj.-N^)[A3_Af^+A3^\f+A^3A2_ ^ Bill = (2Cj. - N^)(2Cp-2N^)Aj_A^2_ ^ In the above A^^ = ^^^/i^^^^^^. a^^. = 1 - ni^j (n^ is the direction of a parton i; i = +,-,1,2,3)» The genetic link between the gluons g^ and gg, initiating the flow correlations, is encoded in the tenn Bj« The remaining terms Bjy and Bjjj correspond to the independent emission of these gluons. In the large-N^- limit expressions for By and Bjj simplify and might be reproduced by the mentioned above ''dipole formulation' of the Lund MC. To quantify the correlation effects we shall compare the ratio of the single-inclusive particle flows between jets, dl^ / dN ^■^^^H)/ ^S^^^-) ^ ^^-""^ with the ratio of the double-inclusive ones, r ^ clQ(,^^dQ(,^/ A^^,^^AQ,^,^y (8.53b) Here (ij) denotes the direction midway between the partons i and j. In the picture of the independently emitting antennae (string segments) the additional registration of a particle flow in the region (1-) does not affect the fatio ^ , so one would expect *^5~'^4 • ^^^ mutual influence of antennae leads to the numerically small difference between the ratios Xj and ^^ • For example, the ratios of flows, projected onto the event plane, in
384 the case of three-fold symmetric events are (see eqs (8.24) and (8.51)-(8,53)) t^^ 2.42, ^^<^ 2*06. The relative smallness of the colour shielding effects does not diminish their fundamental importance. This consequence of QCD radiophysics is a serious warning against ideas about independently fragmenting string segments. a JPig03* Double inclusive correlation of two multiplicity flows in qqg events. 6) 8#6. Azimuthal Asymmetry of QCD Jets 7,73) As we have already discussed, the treatment of the structure of final states given by the string picture qualitatively reproduces the QCD radiation pattern only up to 0(1/N^ ) corrections (the large-N_-limit). However, the 1/N_-expansion appears to be, as mathematician would say, nonuniform. Namely, under specific conditions 1/N terms become sizable or even dominating. The simplest example is given by the study of the azimuthal asymmetry of a quark jet in qqg events. The radiation pattern is given here by eq.(8.24)» When all the angles are large the third term in eq.(8.24)(negative colour-suppressed antenna (+ -) leads to a small (^^0%)
385 correction to the Lund interpretation ^'^ of the drag effect. The azimuthal distribution of particles produced inside a cone of the given opening half-angle 6^, may be characterized by an asymmetry parameter (see Fig» 34)* A(ej M^Je<©) - M^a(©<eo) 1 (^M) as ^totf®<eo) ^l •tot , (8.54) qqg 3 (a) 2 (&) 3 Fig.34• The azimuthal asymmetry of the quark jet in e'^e (a) Geometry in the event plane, dashed lines show the topology of colour strings, (b) Scheme of azimuthal separation of particles from quark jet, see eqs. (8.54) and (8.55) . The azimuthal integration for (^N) as IT ■^ <iiQ) f (8.55) can be done explicitly (see ref. '^^). For parametrically small ©^ values the contribution of the (+ i)-antenna (i = -, 1) to (iiJJ) a as IS de+/2^ I e 4- or t Z )\{^^%) -^ (8.56)
386 while the nonsingular antenna (-1) produces negligible correction to (aN^„) ^ 6^ • The resulting asymmetry cLS O parameter reads (8.57) +- 2C ^ 2. 2N,Cp ^ 2, where we have substituted following eq» (7»20) ^ depends on the geometry of the jet ensemble* The first colour-favoured term in (8»57) describes the Lund-motivated asymmetry due to the 'boosted string* connecting q- and g-directions* The corresponding asymme erm enters the game^ forcing the asymmetry to increase anew as shown in Pig.35a. This behaviour might be interpreted as an additional repulsion between particles from two neighbouring q- and q-jets. For symmetric configuration the colour-suppressed term in eq.(8.57) prevails when ^4.. 4 2 arctg (1/N^) » 37°. (8.59) To realize this effect one has to select qqg-events with unnatural kinematics, when the hard gluon moves in opposite direction to the quasi-collinear qq pair. Fig. 35b displays the predicted asymptotical asymmetry of the quark jet at finite values of 6© as a fvmction of
387 Fig.35. QCD (solid) versus 'string' (dashed) predictions for a) G-factor (see eq.(8*57)) and b)asyraraetiy parameter A in the symmetric qqg events. the relative angle between the q-, q-jets. The increase of A with decreasing 0 - discussed above can be seen on3y for very small values of 8^. The reason is that for ©Q >^ 5® the effect of repulsion is masked by the fragments of the neighbouring q-jet which fill partially the e^-cone, leading to drastically increasing negative contribution to A. It would be important to observe the aforementioned phenomenon with the heavy quark (Q=:c,b) jets identified. The azimuthal asymmetry of produced jets is certainly not specific to 3-de"t events. An analogous picture should be observed, e.g., in high p processes. To elucidate the essence of the phenomenon, let us compare the angular pattern of the radiation accompanying high i qq' and qq' scattering (where q and q» have differer
388 flavours) in the quark cms, as shown in Fig. 27a and Fig* 36 respectively. 1 Fig•36. Colour antennae for high p^ qq' qq' scattering process. The corresponding particle distributions are given by eq.(8.37) and by the analogous formula 45rdli^^' dS^ A A (12+34) 1 A A. 2N,Cp A A A A A A^ -, (2.(14-h23)-(12+34)-(13+24))] Cp/N^-N^ (In E/A) (8.60) Similarly to the discussed above qqg example, first colour-favoured terms in eqs.(8.37),(8.6O) correspond topologically to Lund strings, as illustrated in Figs.27a A A and 36. 'Boosted' 14 and 23 antennae for the quark scattering lead to the certain drag phenomena and, in particular, to the azimuthal asymmetry of the jets . Unlike this case, for qq' scattering, following the Lund scenario, one should not expect such asymmetry since both A A 12 and 34 antennae appear to be straight (each strings is in its cms). In the case of qq' scattering the colour-suppressed term in eq.(8.37) leads to some deviation from the Lund- -motivated asymmetry. But for the qq' scattering such a term in eq.(8.60) determines the whole effect.
389 For illustration let us consider the azimuthal distribution of particles inside the jet-'3* • I^e"tennining the asymmetry parameter A(0 ) analogously to eq# (8.54) (see Fig. 37) A(e^) . N 2 (e < e^^) - N^^ (e < e^) N (e < 0^) (8.61) one obtains for small opening half-angle 8 around the det- '3 * axis / H'^ (8.62) z. ^eMCp h[^h % I -4lJM^. w Aiej-^r ^ a a ^ t,,, (at,f ^ 4?)l^«/-r (8.63) <^ F where 9« denotes the scattering angle s E stands for a jet energy in the cms. 6>^5= ©,^ = or - a + 2 (a) 1 2 (6) 1 Pig.37. Definition of the azimuthal asymmetry A(0 ) of the scattered jet-'3'. Pig.38 demonstrates the comparison of event shape factors G (factors in square brackets in eqs.(8.62)^8.63) with their large-N^j-limits. Od/N^^) term in eq.(8.62) dominates for angles e. < nI 33.6°. (8.64)
390 0 Fig.38. G-factor for qq» and qq* scattering. Solid - QCD, dashed - large-N_ ('string*)limit for qq* case ('string' prediction for qq' case is G=0). S The predicted magnitude of the QCD asymmetry in qq* scattering appears to be comparable with that of qq'^case. For example, ^"^^ ^ A^^' at small 0^ and A^^ = 3/8*A^^ at Gg = 90^. The absolute values of the asymmetry can be estimated using eqs. (8.37),(8.60). Then, A^^^ (0 A^^' (0 0 10^, 0 s 90^) 5%, ( ol 10^, 0 0.12). s 90^) 2%, (8.65) For small opening half-angles the effect grows linearly with e 0 To increase the magnitude of the asymmetry one can take 0 larger and use the original eqs.(8.37),(8.60), accounting for the all contributions including terms, non- singular in the direction of the jet-'3'« Notice, that to study the azirauthal properties of jet-'3' the natural restriction 0 > 20 has to be imposed. For the interaction of identical quarks new effects arise leading to the complication of the antenna patterns.
391 (i) Modification of the qq scattering amplitude, (ii) Opening of the annihilation channel for the qq case. The associated soft gluon distribution is now given by for process q(1) + q(2) —^q(3) + q(4) = [(14+23)R^ + (13+24)R^ + (12+34)(1-Rt-R^) + 1 AA AA AAt 1 fa C.C\ ■ (2(12+34) - (14+23) - (13+24))]-Cp/N-'N ^^'^^^ 2N Cj, P'"c g' and for process q(1) + q(2) —> q(3) + q(4) ^^ =[(12+34)R^. + (13+24)Rg + (14+23)(1-R^;-Rg) + i AA AA AA-, , (8.67) j^ (2(14+23) - (12+34) - (13+24))] Cp/N^*Ng. c F Here ff ->' S' d^ A = r^--— . A. = - s A/Z-tu. ' «* N5 St Keeping in mind that the asymmetry of the jet-»3» comes A A mainly from 13 and 23 antennae, one can simply observe the following peculiarities. The inclusive asymmetry givaa by eq.(8.66) vanishes at 0 = 90^ (^t~ ^^ ^^® "^° compensa- tion between the antennae 23 (positive drag) and 13 (negative one). The eq.(8.67) leads to the asymmetry which, unlike the case of distinguishable quarks (see eq. (8.6o)) contains also the colour-favoured term 13*Rs caused by the einnihilation contribution. Noteworthy to mention, this Lund motivated term
392 produces the negative asymmetry (Fig.39), opposite to the positive one due to remaining terms. Note, that the colour-favoured negative asymmetry occurs also for the cases of qq —> q'q* and qq —> 2g subprocesses. P 2 1 Fig.39» String topology for the qq annihilation sub- process. Let us emphasize that just the colour-suppressed 0(1/N_ ) term proves to govern the overall asymmetry owing to the numerical smallness of the annihilation cross section ( 61/(51- £ 1/10 for 6^ ^ 90^). One concludes, thus, that QCD differs here qualitatively from the Lund picture• In this point we at first time meet the situation when QCD and Lund model give opposite predictions. To our understanding, this qualitative divergence should hold on for aoiy string based hadronization scheme, even with QCD cascades built in. The quark-quark scattering can be studied in pp collisions at high p, • Here both qq and qq» scatterings occur with the relative weights 5/9'(2u«2u + d»d) and 4/9*(2u«d + 2d«u) respectively* The resulting QCD asymmetry of the jet-'3' (at 0^ ^ 90^), as depicted in Pig#40, differs noticeably from its large-N^ limit. In the case of pp scattering the azimuthal distribution of particles inside the jet-'3* without identification its
393 A(%) 5 0 -5 Pig.40. e o o 1 - 10 2-20 3- 3 0 S Asymmetry parameter A(0 ) of the scattered tagged) q-jet in pp collision as a function of the cms scattering angle Q„. Solid: QCD prediction. Dashedj large-H„ ('string') limit. species (q or q jet) may "be characterized by an asymmetry pattern shown in Pig#41« The asymmetry (un- km 2 1 - 0 -1 G^ = 20 ^^^^^^^^^^>^ 50^ "string" Pig.41. Different sign ^ asymmetries A(e =20°) of ^ the scattered untagged (q 90 or q) jet-'3* in pp collisions as predicted by QCD (solid) and its large -Nq limit ('string' - dashed)•
394 proves to "be considerably smaller than in pp case, but reveals the same sign of effect in contrast to the string picture. Curves in Fig.41 correspond to n^= 3# Account of the process qq works in the same direction as a rise of n f enhancing the colour-favouring negative asymmetry. To make the qualitative difference between the predictions of QCD and its large-N^ limit (string) more spectacular it is necessary to identify scattered quark jet. The asymmetry, predicted for the case of tagged quark jet, is shown in Pig.42. As is easily seen, in the region of cms scattering angles QCD predicts A = +(4t7)% jet asymmetry at the half-angle 6 = 30° unlike the opposite sign effect A = -(2.5 T 1.5)%, originated from the large-N^ treatment of QCD formulae, here referred to as the »string-motivated* approach. 70° < e„ < 110° 3 A(%) - 5 0 0=30 O '/////////////////////M -5 30 "string" I / / / / / / / Pig.42. Asymmetry parameter A(©q=30°) of the scattered tagged q-jet in pp-collisions.
395 9. COHERENCE MD FINAL STATES IN DEEPLY INELASTIC SCATTERING ^^•'^5) The parton model had its first great successes in explaining the scaling observed in deeply inelastic lepton -hadron scattering (DIS). Later, scaling violations were explained by the approach to asymptotic freedom dictated by QCD and became the first quantitative testing area of that theory. The spectrum of particles associated with a deeply inelastic scattering event has only recently begun * * '-^ >f^-My ^Q receive serious theoretical treatment. In this Section we shall briefly summarize the present situation. ^^^• The Structure of Soft Radiation Associated with DIS 2 2 Hard lepton-hadron interaction with high -q = Q 2 / rN and fixed x = Q /2Pq knocks the quark with longitudinal 2 2 momentum k = xP at virtuality level k^ ^ Q out of the initial partonic fluctuation which was prepared long before scattering. The probability to find the appropriately prepared quark-parton inside target nucleon determines the DIS cross section (structure functions). The structure of final state is governed by two main phenomena: dissociation of the initial parton system whose coherence was destroyed by removing the quark (target fragmentation) and evolution of the struck quark (current fragmentat ion). These fragmentation regions are best separated kinematically in the Breit frame (q^ = 0, 2xP = -q)• Here, similar to e'^e"'-annihilation, the process looks like abrupt spatial spreading of two colour states 3 (the struck q) and 3 (the disturbed proton) moving in the opposite directions.
396 The current jet (time-like cascades) should be identical to that produced in e e annihilation at W =Q*^. One again finds a hump in the energy distribution with the dip occurring for particles with finite energies in the Breit frame. In the target fragmentation region situation proves to be much more complicated, especially for x« 1# The DIS occurs in this case on a 'sea' quark coming from brems- strahlung of soft (qq)^. pairs in colour octet state O (gluon exchange in t-channel)# The dominant structure of the appropriate fluctuation can be characterized in terms of the multirung ladders of Pig# 43 determining the small -X behaviour of D^(x,Q^)# However, in order to calculate the single particle spectrum in the target jet it is not enough to consider those graphs shown in Pig#43 even Pig»43« Ladder graphs, with transverse momentum ordering, dominating in structure functions at small x.
397 allowing that the horizontal parton lines may fragment. Sets of graphs which cancel in the structure fiinction no longer cancel in the inclusive spectrum. In addition to the offsprings of decaying subjects - remnants of the ordered 'ladders* (structural contribution), the collective coherent accompaniment arises, v^hich is determined by the overall colour topology of partonic system (soft t-cheinnel contribution). 9#2# Angular Ordering for Space-Like Cascades Similar to the time-like case, the PT-analysis has proved the AO of the radiation of soft gluons * I » associated with the space-like fluctuations shown in Fig. 43 ^^^ (see also ref. ^^h» The character of 'soft* radiation associated in the hard scattering can be understood, to a large extent, by considering radiation from the elementary vertex p —^ k + k i appearing in Pig.43 7) 1 Suppose an addition gluon, ^ is radiated from this vertex as shown in Pig.44• Call ^^^ h^-^ ^ Pi-^FpF^ . ^i^^^P/^ ' ^"^^^ ^^ suppose the usual strong^ ordering appropriate to small -X processes. Then there are two cases to consider. (i)B t < ?, and (ii) p^> p^^ Ca) Cb) (c) Pig»44» Additional soft gluon,-d , radiated from lower vertex of the graph shown in Pig. 43•
398 (i) When p^ < ^^ p(k^ ) when © there is Pi ( '^[t) ' is 1 of course radiation off / _ in the usual manner. The interesting » ©^ . Now when Zi^:t) < K^^ s than 6 = 6p^' region is ^ X that is when e < ©^ < P p ^ 0 (9.1) € the ^ - line can be emitted off the p and k. lines coherently. When 2(K^*^) > }fC^^ only the graph of Pig.44c is effective and here one covers the angular region p e ■ p ^l \h < e I (9.2) However, the coherent emission of Pigs.44a and 44b give the same answer in the region (9.1) as does the emission corresponding to Pig.44c in region (9.2). (ii) When h > P^, it does not make sense to route the momenta as shown in Pig#44# Rather, we should write the momenta for graphs (a) and (b) as shown in Pig*45# Then so long as -^^« K^j^^* K^. these two graphs add coherently that is in the region 6 t < JI e (9.3) (a) (6,) Pig»45« Graphs which add coherently in region -6 :3S> k 1
399 The graph analogous to Pig#44c is already included in Pig»43 since here k^ is the soft emission off t • 9.3« The Structure of Inclusive Spectrum in Target Fragmentation As a result in the target region there also arises the hump-backed energy particle distribution with predictable 2 shape, evolving with In Q sind In 1/x. The resulting inclusive spectrum of hadron h in the target fragmentation can be represented conveniently as a sum of three terms• The contribution I related to the upper quark cell in Fig.43 comes from the soft emission -^ < OC P off the line k^ , when 9^ , < ©p # , and off the line k^^ when -^) = s'f^ »'(-P• f^x)] . ''■'' where y = lniO//\.\'^ being the energy of hadron h. The contribution II also accovmts for t <xP coming from effective emissions off the vertical gluon lines of Pig.43 up to angles 6^p < 0p^' V^dy/{f (9.5) «iTP>^^)] xP °^^VxP' A where D^ is the sea quark distribution. The contribution III combines relatively soft gluons (xP < I < P) off the lower part of the ladder with fragments of ladder rungs:
400 / <i^\ S'dv . 1 p ^^ lil p ^p z SW) '^? ^""l^p' P '"'' H' ^I(f'«^^-') 2 "^Je.^(«I)f-£,V^,e„«x -e A i ZT J U A (9.6) where % k i*«tn^ Pig.46 demonstrates contributions I, II and III Contributions II and III, being negligible at x ^ with X decreasing. 1 , rise 5- y = en CO A Pig.46« Contributions to the energy particle spectrum in DIS target fragmentation region at InQ/A =5, In 1/x =s 5« I - quark box contribution (dotted line), II - coherent *t-channel' colour radiation (dashed), III - fragmentation of ladder rungs (dash-dotted); solid line - total sum. The relative magnitude of II versus I depends on the characteristic value of the *sea* quark k! emission angle which becomes larger with increase of number of cells in
401 the ladder, i.e# with In 1/x increasing. Angular structure of the basic ladder is responsible for the shape of the curve III as well. The most energetic particles with cO ;C P can originate only from the very bottom of the ladder where emission angles of partons, and thus the opening angles of fragmenting subjets, are comparatively small. This damps the parton cascades, leading to a smooth increase of the particle yield with decreasing cO . 2 Evolution of resulting hadron energy spectra with q and X is shown in Pig.47. The left wings correspond to the current fragmentation, the right ones - to the hadrons from target fragmentation. Similarly to the e'^e" annihilation case, coherence in DIS stiffens energy spectra. The yield of 'slow* particles (IncO/m <, 1) should be independent of Q . 9.4# On the QCD Solution of Peynman-Gribov Puzzle It is important to note that the fragmentation of the ladder rungs (the contribution III, see eq.(9«6) and Pig. 46) does not populate the energy interval cO < xP. This coherent phenomenon has been predicted long ago in the framework of the parton picture suggested by Peynman. Prom the general physical arguments based on quantum- 78 ) -mechanical coherence V.N.Gribov has shown '' that the DIS on a quark with the momentum xP ci -Q/2 (in the Breit system) did not affect the development of the soft part of the partonic fluctuation. The undisturbed upper part of the partonic fluctuation in Pig.48 results just in a single final hadron, thus, leading to the lack of particles within the rapidity y = lnu)/A interval 0 < y <ln Q/A. (the so-called Gribov gap). Later on the experimental observation of the continuous
402 (a) (&) Fig.47. Evolution of DIS energy hadron distribution in the Breit system with a) Q^ : In Q/A = 3, 5, 10 at In 1/x = 5, and b) x: In 1/x = 2, 5, 10 at In Q/A = 5. plateau (without the gap) was interpreted as an evidence in favour of the Peynman conjecture about the identity of partons with fractionally charged quarks. This fact did not remove the puzzle however. Moreover, establishing QCD has only sharpened it, since QCD maintains both the partonic concept and the physics of coherence which provided the base for Gribov conclusion about the rapidity gap.
403 Q P Pig.48. Development of partonic fluctuation in DIS and rapidity distribution of final particles* As is clear now it is the coherent soft bremsstrahlung caused by the t-channel colour transfer that fills partially this gap, being insensitive to the details of the target wave function• 10• CONCLUSIONS The perturbative approach represents a model independent scheme for the quantitative predictions of hadronic properties of hard processes. So far no experimental fact exists which endangers this endeavour• This means that the QCD part on bremsstrsthliing can be thought as the main source of multiple hadropreduction in hard processes• Looking for manifestations of the nonperturbative
404 ma^ conclude that a coloured parton was substituted promptly by a hadron at the large-distance stage of the evolution. Finally, we list the main lessons from the above discussions* 1. It is time to critically revise the experimental approach to the analysis of the Jet structure of hard processes. One should abandon any attempt to attribute each particle in the event to a certain jet. That is, the notion of isolated Jet should be rejected. Purely inclusive studies of jet characteristics (calorimetric and many- -particle E Mr correlation measurements) are probably the best way to make sharp connection between theory and experiment. 2. A theoretically substantiated scheme exists (MLLA + + LPHD) for making quantitative predictions for jet characteristics without invoking any phenomenological hadronization model. Such a scheme maintains the probabilistic picture of parton branching and could be simulated by Monte Carlo technique. 3« The hump in the energy spectra - one of the brightest consequences of the intrajet coherence - evolves with the hardness of the process in a predictable way. Valuable information on the confinement mechanism may be obtained from a comparison of spectra of different hadron species and from the studies of the fine structure of distributions varying independently the pairs of parameters: s 2 and t in large p processes, and x and q in DIS. 4« The drag effects, reflecting the interjet coherence strongly support the LPHD concept. The collective nature of multiple hadropreduction reveals itself here via QCD wave properties of multiplicity flows. The drag phenomena should be valuable in helping to distinguish New Physics
405 signals from the conventional QCD background. 5« Hadroproduction studies within the perturbative approach are far from being exhausted. The MLLA - LPHD approach accomodates the attractive features of the current fragmentation models being free however from their shortcomings. Moreover the evolution of the phenomenological models seems to lead to some convergence between them» In particular the most successful schemes incorporate nowadays the concept of the well developed coherent parton cascade as the basic ingredient necessary to withstand the pressure of the experiment. ACKNOWLEDGMENTS We wish to thank Ya. Azimov, V.Fadin and Al.Mueller for a nice collaboration and sharing with us the belief in power of the perturbative approach to hadron Jet physics. We are indebted to S.Bethke, M.Derrick, W.Hofmann, B.Ioffe, E.Levin, L.Lipatov and T.Sjostrand for fruitful discussions. We would like to express our heartfelt gratitude to V.Gribov for the constructive criticism and stimulating discussions.
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411 POMERON IN QUANTUM CHROMODYNAMICS L#N«Lipatov Leningrad Nuclear Physics Institute Gatchina, Leningrad 188350, USSR Various approaches to the investigation of the high energy asymptotics of scattering amplitudes in QCD are reviewed♦ We begin with the approximation of the two gluon exchange• Then quark scattering amplitudes v/ith the elastic and quasielastic unitarity are constructed. The inelastic amplitude for production of gluons in the multi-Regge kinematics is obtained in the leading logarithmic approximation. The bare pomeron trajectory is calculated at large momentum transfers. In conclusion the problem of unitarization of the scattering amplitudes is discussed. 1. INTRODUCTION Hadron-hadron collisions AB -^ A'B* (see fig. 1) ^' V .—^ yP^' A R ^ _.-* , '^ at high energies ya and fixed momentum transfers q =
412 are of interest for elementary particle physicists as a way to get an important information of hadronic interactions. Many methods were worked out for theoretical interpretation of the scattering processes in the Regge kinematics (1) (see, e.g., ref./1/). The most consistent approach seems to be the complex angular momentiim method. In this method the invariant scattering amplitude A(s,t) is written in region (1) as an integral over the variable j which is the ana continued total angular momentum in the t-channel i\(i,i)=^ ii,s^ i' fa). lly (2) ^_i /-T/ / J Here tA^) is proportional to the t-channel partial wave amplitude with the signature j^ ^-^i (-1) corresponding to the even(odd) physical values of the angular momentum. The integration in eq. (2) goes along a straight line parallel to the imaginary axis to the right of all singularities of f.^. The signature factor ^ equals e (3) ^ SihTj Usually it is assumed that f P(t) has only moving poles in the j-plane-reggeons. The Regge pole trajectory depends in the linear approximation on two parameters o/ and 0/ which are its intercept and slope. The reggeon contribution to A(s,t) (2) is ^o^a) ^ 0C^>^. (5) where /(t) = PaaiC'^) ^OTt("t) is a factorized residue of
413 J?, /f) in the Regge pole# Besides the signature p the Regge poles have other quantum numbers (electric and baryonic charge , isospin and so on)# Masses m_ and spins s of the resonances 8 belonging to a Regge family are related by the equation c^K^) = s , (6) where s are even (odd) for the positive (negative) signature• The reggeon having vacutom quantum numbers (the Pomeran- chuck pole or the pomeron) determines the high energy behaviour of elastic scattering amplitudes and total cross sections . . ' ' ^ (7) The pomeron intercept is assumed to be close to unity \t)^^ d (8) in accordance with experimental data on a slow growth of hadronic cross sections with energy* For /^>0 the power-like increase of o^^fvvlth energy contradicts to the Proissart theorem /2/: 6^ J ^ const (In s)^. (9) This contradiction is resolved by the fact that besides Regge poles the partial waves f-(t) must have the Mandel- J stam branch points /3/» The reggeon field theory which allows to calculate their contributions was constructed by V.N.Gribov /4/« It is known /5/ that in the quantum field theory one must distinguish the bare and renormaliz- ed fields# Hence, the bare pomeron parameters - trajectoriesfl/(t)^residues /(t) and coupling constants ^
414 are renorraalized due to reggeon interactions /4/# But there is such critical value of A ^ £i^ ^ Z^ , that for A > A^ the Regge behaviour (7) of ^^f is transformed into the Proissart-like one /6/: ^ I ^ c (In s)^ . (10) The experimental data on the hadronic interactions at hd^ energies can be described by the logarithmic dependence (10) with a small constant c# So, the real pomeron is assumed to be supercritical one ( A >^c) with rather small parameters A 9 ^c ^^^ <^^ /7/# These facts agree qualitatively with QCD predictions if one assumes that transverse momenta of virtual gluons forming the pomeron are sufficiently large M » /^<?^5 ' ^''^ where A qqt)^ 100 MeV is the QCD constant* The arguments supporting the idea of the anomalously large mass scale in the pure gluonic systems were given in the framework of the QCD sum rules /8/# In the region (11) the pomeron slope is expected to be small «__— ^' ^ //crj"^ ^^ A'f^r^ (12) Furthermore, inequality (11) justifies the use of the QCD perturbation theory when taking into accovint the asymptotic freedom for the QCD coupling constant g /9/ where n^ is the number of light quarks• The Pomeranchuck singularity of f^(t) was investigated in the non-Abelian gauge theory with the Higgs mechanism
415 (14) in refs# /10/# In a general case of SU(N) gauge group the leading singularity turns out to be an immovable square -root branch point at This result was obtained by summing asymptotic contribu- 2 n tions (a In s) of the Peynman diagrams, that is in the leading logarithmic approximation (LLA)# Por colourless particle collisions in QCD (where N=3) %IZZ infrared divergences cancel in the scattering amplitud calculated in LLA /11/# Due to the asymptotic freedom (13) the above discussed cut in the j-plane turns into a set of reggeons, accumulated to the right of the point 0=1 /10,11/» Using conformal invariance of gluon-gluon scattering amplitudes in the space of impact parameters the trajectories of these Regge poles are calculated at large moment\im trajisfers and the lower boiinds on their intercepts are given /12/, Another approach to the pomeron problem in QCD was advocated by A»White /13/# One additional argument for using the perturbation theory is a certain success of the model in which the hadron scattering at high energies is described by the two gluon exchange diagram /14/# For large transverse momenta (11) of these gluons some results of the known additive quark model /I5/ are reproduced in spite of the fact that in this approximation the total cross secticxns turn out to be proportional not to the number of quarks in colliding hadrons but to their squared sizes /16/* In the next section we consider the two gluon approximation in the version close to ideas of the QCD sum rules /l6/# In the third section a more complicated model for high energy hadron interactions is investigated* In this model the quark-quark scattering amplitude satisfies
416 quasielastic unitarity requirements with taking into account an arbitrary number of soft gluons in the intermediate states of the s- and u-channels /17/. In the fourth section we build inelastic amplitudes for gluon-gluon collisions with produced gluons in a multi-Regge kinematics /10/# For construction of the amplitudes in the tree approximation the t-channel unitarity is used. The multiparticle s-channel unitarity is applied to find radiative corrections in LLA# The relation of the obtained expressions with analogous results /18/ in quantum gravity and string theory is also analysed, Gluon production amplitudes calculated in this section are used by other people for estimating the magnitude of the minijet cross sections /19/« In the fifth section an integral equation for t-channel colourless partial waves is builtin LLA /10/. It has the diagrammatic form of an evolution equation /21/. In the sixth section this equation is solved with using its two-dimensional conformal invariance and the parameters of the bare pomeron in QCD are calculated /12/, In the conclusion unsolved problems are discussed, in particular - the odderon in QCD /22/ and the iHiitariza- tion program. 2. HADRON SCATTERING AMPLITUDES IN THE BORN APPROXIMATION In the lowest order of the QCD perturbation theory invariant amplitudes A(s,t) of the colourless particle scattering are determined by the diagram of fig#2 with the two gluon exchange Ca
417 /\ (^,^) = //, / U^k /^-^'/^'-'^^^ ,<' where A^ and Ap are scattering amplitudes of the virtual gluon Compton-effect for corresponding external partidLes In eq# (15) the factor 2 arises as a result of summing over colour indices fS/^1^},^^^)^- -^''^ -? for N = 3y and the factor 1/2 is due to the identity of virtual gluons. The leading asymptotic contribution r^B in eq,(15) is supplied by longitudinal components of the gluonic polarization tensors , l . s Introdvicing Sudakov's parameters A=: V^.AAAr/ y &-^yo.(^.k)f Eq«(15) can be represented in a factorized form (compare /20/) where to q for 3 -^ ^>-=* do not depend on -^ , are real and equal j ^ P ^ P Al^ The integration contour L in the plane •^z is situated
418 between the right and left cuts of the amplitude Ag (see fig.3) in accordance with the Peynman rules. Thus express Pig. 3. by closing it up around the right-hand cut we '^j (a) in terms of the discontinuity of Awgx: ^ia) U r •^4,,. dW } where 'th* means the threshold value of the invariant ^, Note that due to the colour current conservation A/^/? with the use of (17) expression (20) can be rewritten in the form /20/ <Pz Js \? and therefore we obtain h ^x'^^z-'^-i X4 /» 1/ (21) Furthermore, if the total cross section of the gluon- -extemal particle scattering falls with growing energy j/^I then the integral in eq.(21) is convergent at large s As a simple example let us consider in fig,2 the Jfjf-scattering. In this case the amplitudes A- and A2 describe the gluon-photon scattering. In the lowest order of the perturbation theory they correspond to the diagrams of fig.4# Por the nonforward scattering of real photons
419 '^ Pig. 4. through the virtual heavy quark pair production we obtain /20,11/ =- ^f^jL^i)' i^.fo^i)^ ^^i.?;- ^^/,M ^ , /- ^ 'x(i-^)if{d--{f)I^J)(eJ)4f/J^-¥i)^^^^^^^ ^^'^/y- ;?-^ ^ >*^'^ r^D' An analogous expression can be obtained for the virtual photon scattering. We write down it only for q = 0 and for transverse polarized photons ■ t- Ha-^>^u-,) (A ^J rC?J * V ^4 f,i) (i- izli-3)-f}a-yJtgM.. P^- p: - - >'■ (24) In eqs. (23), (24) ^^ is the quark, electric charge measured in units of the electron charge. The strong
420 coupling ^ -. .f^*. is given above (see (13)), and the use of the perturbation theory can be Justified if the quark mass m or the photon virtuality /) are large. Using the optical theorem (see (7)) the total cross section for inclusive production of hadrons built from heavy quarks in the fragmentation regions of each colliding photon can be calculated /20,11/ : ^w ^ ^ 4 -^ -^^ (25) air V ^ ^' ^=/ '^^ Let us consider now the production of two pairs of massless quarks in the high-energy collisions of virtual photons interacting only with the isovector part of the electromagnetic current In this case we use eq# (24) with the substitution By averaging it over the transverse polarizations of photons the corresponding total cross section of the // scattering is ^ 5^ ^ I / i (28) where —/<,■ are the isovector photon virtualities. On the other hand, in the vector dominance model the total cross section corresponding to the diagram of fig.4 is given
421 by the following expression ^ Fig, 4/ (29) where m is the $ meson mass, the constant g determin 4-^ — es the probability of the decay S -> -^"^ because it parametrizes the pole contribution in the photon mass operator /7^/?^ =///.-l^^jM'^- ""' For/q /—> O^^ rj is given in QCD by the quark loop diagram nj^)i, -(i%-sL^.)i-fA€>'''' where the factor 3/2 appears due to summing over colour and isospin (27) indices. In the method of the QCD sum i*ules /8/ after the Borel transformation (32)
422 expressions (30) and (31) are shovm to coincide ^, ^ ^"fr^ ^-■■■ = ^-h-- (33) 2 2 at the value M = mr of the Borel parameter M^where nonresonance and nonperturbative corrections (denoted in eq. (33) hy dots) approximately cancel • This gives / $;^^^^ (34) We assume the analogous cancellation of corrections 2 2 to expressions (28) and (29) at M« = m^ after 9 2 «^ transformation (32) in each photon virtuality q_. = -A- and obtain /16/ Before doing an estimation of O^^ let us examine in (35) the contribution of large and small values of '^ . For m^ the essential region of ^ lies near the boundaiy points ^ =0, 1, which results in where ^^^^ is the qq scattering total cross section in the Bom approximation The factor 2 in eq.(37) has the same origin as
423 the analogous factor in (15)s the fraction (1/3)^ arises as a result of averaging over three possible colour states of quarks, the rest expression coincides with the Rutherford formula for the ^^scattering cross section in quantum chromodynamics. Eq» (36) corresponds to the impulse approximation. In the general case of scattering of two composite objects built up of n-j and np quarks we would have as in the well known additive quark model /15/« (38) In the region of small transverse momenta the leading contribution in (35) is given by the region close to the boundary points V =0.1 : = Jtr r icy^i^i (39) < 4 Mf where C^ ^ 0.577 is the Euler constant. It is obvious from the comparison of eqs. (37) and (39) that the region of small /k ) is emphasized due to additional factorA/tp/^ > which is justified in computation where for J = const the half of the integral contribution appears from the region of small k <0.2;rm^.. Putting in eq. (13) £^ equal to 0.2rm^ for A^^^.^^00 M^ and nx.=3 we obtain the following value for /O ^^ /16/: ^ ^ ^ ^0/m ^ cz ^^ mb. (40) ^^ - S ^^s This should be compares with ^4.^^^^'^ "^^ found from the factorization relation ^^^ ^ X ^^^for ff ^^30 mb which corresponds to the laboratory epergy C^ 70 GeV. For such rough model the above estimate (40) may be
424 considered as rather satisfactory one because the cross section SI has a nonzero dimensionality/^1/m and ^ A therefore depends drastically on the choice of A ^nj) and on the value of the Borel parameter M for which the nonresonant and nonperturbative corrections cancel. To find the stability region of ^^ in the QCD sum rules for n. , these corrections must be calculated. The ^H- 2 n power corrections (1/M ) are also important for finding the mass scale in the vacuum channel. Before passing to more complicated models for high energy scattering in QCD we dwell upon the interpretation of eq. (18) from the point of view of the Wilson operator expansion. Let us transform (18) to the impact-parameter representation: - • '*>i('i,%) "^Mn^^) ^^-^.^.--^^.W, (41) where )II C i^ ^ (44) Here the fictitious gluon mass is introduced for removing the infrared divergence. Note that due to (22) we have \A^'\h)=U%*'\i,)^o, (45) which means that A(s,t) does not depend on A
425 Furthermore, eq,(45) allows us to use for Cils. f-;&/.f«/J another expression (of. (43)): a^X;W.)'-^^^¥^iM^\ Here the complex notations .(46) are used. h= ^^^^^2 , I^J= f^/T^ (47) It is important that expression (46) is invariant vmder the Mobious transformations (48) where a,b,c,d are complex parameters. It is possible to consider expression (43) as a four- -polnt Green function A ,u. ..r , //5>e where ^-^^ (f) ^^^ some free fields. In general an arbitrary field 0 (/*) is transformed under (48) as follows . 7~ _ ^ ''^ -* (wrJ (91 The representation weights h and h determine the conforraal spin n of the operator and its dimension d: n = h - "h, d = h + h. (51) In the free theory for the local operators built bilinear ly from the fields , , f / ^T
426 the weights h, "E are integer numbers. Due to the equation of motion ^p ^=5 0 the complete set of the operators (52)_is the following one . _, _ and the Wilson operator product expajision can be written in the form , p . "t ViVi where non-vanishing structure constants c in the free theory are eqxial to >/» lip ^ <?/ / ViVi The operators 0 /j^jin eq. (54) are normalized in such way that due to the conformal invariance the three-point Green function is (cf. /23/) . ^Ol fpf9, ) f(0 0 /3P„ )lo> if fO The two-point correlator of operators (52) equals^ Ai T ffj - {fc.i (57) By using eqs. (49), (54) and (57) we can represent (46) in the following separable form J - (5_8)
427 Thus, the scattering amplitude A(s,t) (41) can be rewritten in such way ^^aU r 1/ /^4./// 4r./;=.w^^/VVyt.'*^^'l;/.s, ViVi where constants Cwp) ^"^^ proportional to the coefficient functions for the Wilson expansion of the blobs in fig. 2 over the operators (cf. (52)): analytically continued to the unphysical point j=1. Operators (60) enter in/ the momentum (q) representation. Because q has small longitudinal components the correlator of the operators is integrated over its longitudinal coordinates and after that converts into two dimensional one {51)• Eq.(59) seems to be helpful for finding nonperturbative radiative corrections. The conformal invariance and the Wilson operator product expansion are used below when calculating the bare pomeron parameters in QCD. 3. QUARK SCATTERING AMPLITUDES WITH QUASI-ELASTIC UNITARITY Por elastic scattering of composite objects in the region of large momentum transfers q/ ^>/1 QCD (61) the most probable process is such simultaneous interaction of all partons after which the relative momenta of the final partons are small enough to produce again their bound states (see fig. 5)« In particular, the hadron-hadron scattering in region (61) can be described in terms of quark-quark collisions. However, the scattering amplitudes of colour objects in
428 the perturbation theory contain infrared divergences. On the other hand, from the physical point of view it is obvious that the transverse components of the virtual A 8 ^/ Pig. 5. "^^ gluon momenta must be larger than mean values of transverse momenta of quarks inside the hadron because soft gluons would interact with the colourless hadron as a whole object. When calculating the quark-quark scattering amplitudes within the logarithmic accuracy this infrared cut-off can be achieved by introducing the fictious small gluon mass X in the gluon propagator. /A is supposed to be of the order of transverse quark momenta. In region (61) we have for t2?ansverse gluon momenta k^ in the logarithmic approximation } ^^ li^j,]<r<r^ . (62) In this section we calculate the massless quark scattering in the(extended) double logarithmic approximation (DLA) where the effective parameters of the perturbation theo3?y are T^'/-^^^-§^> (63) Here S =: ( /^'■^^) > U =. f P-P'")^^^ ^^® squared energies in the s-cnannel (qq scattiring)and in the u channel (qq scattering) (see fig. 6) .
429 t a relatively small term 'X- i Fig. 6. In the "R^z^Q kinematics (1) we have ^ Cr-S and therefore two parameters (59) differ from each other only in rs^^^i^^.i'TT . Taking this difference into account we are going to conserve the analytic properties of the scattering amplitude and the ^-matrix unitarity at high energies. To sum DL terms Tet us use the bremsstrahlung theorem /24,25,18/ which states that the gluon with the minimal value of its transverse momentum is emitted from external particles which are supposed to lie on their mass shell. For the elastic amplitude this theorem allows us to write down the equation which is drawn diagrammatically in fig.7 i -X Pig. 7. Here blobs correspond to the mass shell quark scattering amplitude with the infrared cut-off shown inside them. After its differentiation the equation takes the form ; 9/ (64) ? where in the right-hand side we neglected the terms of 2 2 ^ the order of g In Q /i^ appearing in particular from
430 the last diagram in fig.7. Eq,(64) is written in a general case of the SU(N) gauge group (N=3 for QCD), A = ^ ^ ^ * ' ^ " Y^ J unifies two xnvariant amplitudes corresponding to the singlet A^ and ajoint A representations of the group: Here i^ , i-, {±09 ip'^ ^-^^ ^^® colour indices of quarks (antiquarks) and ^^ are the Gell-Mann matrices normalized as follows Jp^ ^^)^ ^~ j^ ^^ It is possible also to write down analogous to (64) equations for inelastic amplitudes of producing the gluons with a small transverse momentum (see, e.g#, fig. 8)(cf. /25/) crsC-)s^ V- + (66) -h Pig. 8. The total set of such equations describes the S-matrix evolution with decreasing the infrared cut-off \ (compare similar equations for parton distributions in the ultraviolet region /21/)# The unitarity property of the S- -matrix is conserved during this evolution. So, below we consider a model in which the S-matrix satisfies the requirements of the elastic unitarity in the s- and u- -channels at some fixed value ^ = M. In principle M may depend on energy. For ^ <' M scattering amplitudes have additional imaginary contributions due to the iTp-terms
431 of the matrix in the right hand side of eq. (64). These imaginary parts of A correspond to quasielastic processes of the soft gluon production. As in the last section it is convenient here to use the impact parameter representation ^^ where f^ (f ) are the partial waves in the s(u)-channel. s u If we introduce the S-matrix in the ^ -representation £^T.i4, ^ S,^ I- c-f^^ , I=^fl) (67) then due to (64) it satisfies the equation ^5 / 0 i^^i. ^am (68) where e n u^ (69) iir^ / ^ A general solution of eq. (68) is ^ ' Ci± ^ y (71)
432 In particular at large energies we have a + 1 s. ^i^U / ^^ Li ^ (72) The coefficients c ± /v^y Jh<^ are determined from the initial condition for eq« (68) at ^ = M • In the perturbation theory where S / ^ = I we obtain + _ T y 2 Y^^.^->^ / (73) (^^'-ci'^JCi Below another initial condition for eq. (68) is used. Namely, we assume that the S-matrices of the s and u <? 2 channels at ^ = M : r ^\'-.,' ' ^-^^ IA .^ (74) satisfy to the s- and u-channel unitarity correspondingly (75) 1-41 'j,/ ^*" (76) The scattering amplitude f^ is determined by eq.(66). Further, the analytic continuation of T along the path 'a' in Pig* 9 ^--t—^ (D Pig. 9. with its subsequent complex conjugation leads to the relation i CC'A) - df/^i^'U. ■/(■ ill)
433 Another possibility to get in the u-channel consists in moving along the path 'b' in fig«9» At high energies (1) the l