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D-69121 Heidelberg, Germany FriederLenz HaraldGrietghammer
DieterStoll (Eds.) Lectures on QCD
Applications ~ Springer Editors Frieder Lenz
Institut ffir Theoretische Physik III
D-91o58 Erlangen, Germany Cataloging-in-Publication Data applied for.
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
L e c t u r e s o n Q C D / Frieder Lenz ... (ed.). - Berlin ; Heidelberg ;
New York ; Barcelona ; Budapest ; H o n g Kong ; London ; Milan ;
Paris ; Santa Clara ; Singapore ; Tokyo : Springer
Applications. - 1997
(Lecture notes in physics ; 496)
ISBN 3-540-63442-8 ISSN 0075-8450
ISBN 3-540-63442-8 Springer-Verlag Berlin Heidelberg New York
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55/3144-543210 - Printed on acid-free paper Preface The two volume set "Lectures on QCD" provides an introductory overview of
Quantum Chromodynamics, the theory of strong interactions. In a series of articles, the fundamentals of QCD are discussed and significant areas of application are described. Emphasis is put on recent developments. The field-theoretic
basis of QCD is the focus of the first volume. The topics discussed include
lattice gauge theories, anomalies, finite temperature field theories, sum-rules,
the Skyrme model, and supersymmetric QCD. Applications of QCD to the
phenomenology of strong interactions form the subject of the second volume.
There, investigations of deep inelastic lepton-nucleon scattering, of high energy
hadronic reactions and studies of the quark-glnon plasma in relativistic heavy
ion collisions are presented.
These articles are based on lectures delivered by internationally well known
experts on the occasion of a series of workshops organised by the "Graduiertenkolleg on Strong Interaction Physics" of the Universities of Erlangen-N/irnberg
and Regensburg in the years 1992-1995. The workshops were held at "Kloster
Banz". Kloster Banz is a former monastery overlooking the valley of the river
Main and still serves, for some days of the year, as the stage where certain canons
and orthodoxies are vigorously formulated.
Inspired by the atmosphere of the site, the workshops were set up with the
aim of introducing novices in the field to the basics of QCD. Accordingly, the
character of the lectures was pedagogical rather than technical. With the organisation of these workshops we have attempted to establish a new form in
graduate education. Graduate students of the "Graduiertenkolleg" constituted
a large fraction of the audience. They have worked out these articles on QCD in
collaboration with the lecturers.
Thanks are due to Jutta Geithner and Achim Oppelt for help in the preparation of these proceedings. The support of the "Graduiertenkolleg" by the
Deutsche Forschungsgemeinschaft was instrumental in this endeavor and is gratefully acknowledged.
Erlangen, August 1997 F. Lenz
H. W. Grieflhammer
D. Stoll Contents High Energy Collisions and Nonperturbative QCD
0 . Nachtmann ................................. Perturbative QCD (and Beyond)
Yu. L . Dokshitzer; Notes in collaboration with R. Scheibl and C. Slotta 87 Quark Matter and High Energy Nuclear Collisions
H. Satz; Notes by S. Leupold . . . . . . . . . . . . . . . . . . . . . . . . . 136 Spin~ Twist and Hadron Structure in Deep Inelastic Processes
R. L. Jaffe; Notes by H. Meyer and G. Piller 178 ................ Quark-Gluon Structure of the Nucleon
K. Rith 250 ..................................... Low-x Physics at H E R A
A. Levy; Notes in cooperation with M. Ferstl and A. Gute
Subject Index ................................. ........ 347
478 High Energy Collisions
and Nonperturbative QCD*
Institut fiir Theoretische Physik, Universitgt Heidelberg, Philosophenweg 16,
D-69120 Heidelberg, Germany We discuss various ideas on the nonperturbative vacuum structure in QCD.
The stochastic vacuum model of Dosch and Simonov is presented in some detail. We
show how this model produces confinement. The model incorporates the idea of the
QCD vacuum acting like a dual superconductor due to an effective chromomagnetic
monopole condensate. We turn then to high energy, small momentum transfer hadronhadron scattering. A field-theoretic formalism to treat these reactions is developed,
where the basic quantities governing the scattering amplitudes are correlation functions of light-like Wegner-Wilson lines and loops. The evaluation of these correlation
functions with the help of the Minkowskian version of the stochastic vacuum model
is discussed. A further surprising manifestation of the nontrivial vacuum structure in
QCD may be the production of anomalous soft photons in hadron-hadron collisions.
We interpret these photons as being due to "synchrotron radiation from the vacuum".
A duality argument leads us from there to the expectation of anomalous pieces proportional to (Q2)I/B in the electric form factors of the nucleons for small Q2. Finally
we sketch the idea that in the Drell-Yan reaction, where a quark-antiquark pair annihilates with the production of a lepton pair, a "chromodynamic Sokolov-Ternov effect"
may be at work. This leads to a spin correlation of the qq pair, observable through the
angular distribution of the lepton pair.
Abstract. 1 Introduction In these lectures I would like to review some ideas on the way nonperturbative
QCD may manifest itself in high energy collisions. Thus we will be concerned
with strong interactions where we claim to know the fundamental Lagrangian
for a long time now : cQco(x) = + - (1.1) q Here q(x) are the quark fields for the various quark flavours (q = u,d, s, c, b, t)
with masses mq. We denote the gluon potentials by G~(x) (a = 1, ..., 8) and the
gluon field strengths by a ~ , (x) = o ~ a ; (x) - o~a~ (x) - gAbcG~ ( x ) a ; (x), (1.2) * Grown out of lectures presented at the workshop "Topics in Field Theory" organised
by the Graduiertenkolleg Erlangen-P~egensburg, held on October 12th-14thi 1993 in
Kloster Banz, Germany 2 O. Nachtmann where g is the strong coupling constant and fabc are the structure constants of
SU(3). The covariant derivative of the quark fields is Dxq(x) = (Oh + igG~ + )q(x), (1.3) with ha the Gell-Mann matrizes of the SU(3) group. The gluon potential and
field strength matrizes are defined as c~(x) : = c ~ , ( ~ ) ~ ,
G.xt,(x) : = Gip(x)~. (1.4) The Lagrangian (1.1) is invariant under SU(3) gauge transformations. Let x -+ U(x) be an arbitrary matrix function, where for fixed x the U(x) are SU(3)
matrices: U(x)ut(x)= 1,
detU(x) 1. (1.5) q(~) + u(~)q(x), (1.6) = With the transformation laws: cA (~) + g(~)c~ (x)v*(x) - y U(x)Oj, Ut (1.7) we find U~p(x) ~ U(x)G~p(x)Ut(x),
and invariance of •QCD: £QCD(X) -* Z:QCD(X). (1.8) If we want to derive results from the Lagraugian (1.1), we face problems, the
most notable being that £QCD is expressed in terms of quark and gluon fields
whose quanta have not been observed as free particles. In the real world we
observe only hadrons, namely colourless objects; quark and gluons are permanently confined. Nevertheless it has been possible in some cases to derive first
principle results which can be compared with experiment, starting from/:QCD
(1.1). These are in essence the following:
(1) Pure short-distance phenomena: Due to asymptotic freedom  the QCD
coupling constant becomes small in this regime and one can make reliable perturbative calculations. Examples of pure short distance processes are for instance
the total cross section for electron-positron annihilation into hadrons and the
total hadronic decay rate of the Z-bosom
(2) Pure long-distance phenomena: Here one is in the nonperturbative regime
of QCD and one has to use numerical methods to obtain first principle results
from £QCD, or rather from the lattice version of £:QCD introduced by Wilson High Energy Collisions and Nonperturbative QCD 3 . Today, Monte Carlo simulations of lattice QCD are a big industry among
theorists. Typical quantities one can calculate in this way are hadron masses and
other low energy hadron properties. (For an up-to-date account of these methods cf. ).
There is a third regime of hadronic phenomena, hadron-hadron collisions,
which are - apart from very low energy collisions- neither pure long-distance nor
pure short-distance phenomena. Thus, none of the above-mentioned theoretical
methods apply directly. Traditionally one classifies high-energy hadron-hadron
collisions as "hard" and "soft" ones:
(3) High energy hadron-hadron collisions:
(a) hard collisions,
(b) soft collisions.
A typical hard reaction is the Drell-Yan process, e.g.
7r- + N --+ 7 * + X
¢-~ e+e - (1.9) where l = e, #. All energies and m o m e n t u m transfers are assumed to be large.
However, the masses of the n - and N in the initial state stay fixed and thus we
are not dealing with a pure short distance phenomenon. T" N Fig. 1. The lowest order diagram for the Drel1-Yan reaction (1.9) in the QCD improved
patton model. In the reaction (1.9) we claim to see directly the fundamental q u a n t a of the
theory, the patrons, i.e. the quarks and gluons, in action (cf. Fig. 1). In the usual
theoretical framework for hard reactions, the QCD improved p a r t o n model (cf.
e.g. ), one describes the reaction of the partons, in the Drell-Yan case the qq
annihilation into a virtual photon, by perturbation theory. This should be reliable, since the parton process involves only high energies and high m o m e n t u m 4 O. Nachtmann transfers. All the long distance physics due to the bound state nature of the
hadrons is then lumped into patton distribution functions of the participating
hadrons. This is called the factorization hypothesis, which after early investigations of soft initial and final state interactions  was formulated and studied
in low orders of QCD perturbation theory in . Subsequently, great theoretical
effort has gone into proving factorization in the framework of QCD perturbation
theory -. The result seems to be that factorization is most probably correct
there (cf. the discussion in ). However, it is legitimate to ask if factorization
is respected also by nonperturbative effects. To my knowledge this question was
first asked in  -. In Sect. 4 of these lectures I will come back to this question and will argue that there may be evidence for a breakdown of factorization
in the Drell-Yan reaction due to QCD vacuum effects.
Let us consider now soft high energy collisions. A typical reaction in this
class is proton-proton elastic scattering:
p + p ---+ p + p (1.10) at c.m. energies Ecm = X/~ >~ 5GeV say and small momentum transfers ~ / ~ =
Iql <~ 1GeV. Here we have two scales, one staying finite, one going to infinity:
Ecru --4 cx), Iql <~ 1GeV. (1.11) Thus, none of the above mentioned calculational methods is directly applicable.
Indeed, most theoretical papers dealing with reactions in this class develop and
apply models which are partly older than QCD, partly QCD "motivated". Let
me list some models for hadron-hadron elastic scattering at high energies:
additive quark model ,
Regge poles ,
topological expansions and strings ,
leading log summations ,
two-gluon exchange ,
the Donnachie-Landshoff model for the "soft Pomeron" .
It would be a forbidding task to collect all references in this field. The references given above should thus only be considered as representative ones. In
addition I would like to mention the inspiring general field theoretic considerations for high energy scattering and particle production by Heisenberg  and
the impressive work by Cheng and Wu on high energy behaviour in field theories
in the framework of perturbative calculations .
I will now argue that the theoretical description of measurable quantities of
soft high energy reactions like the total cross sections should involve in an essential way nonperturbative QCD. To see this, consider massless pure gluon theory High Energy Collisions and Nonperturbative QCD 5 where all "hadrons" are massive glueballs. Then we know from the renormalization group analysis that the glueball masses must behave as
mglueball (X Me -c/g2(M) (1.12) for M --~ c~, i.e. for g(M) ~ O, due to asymptotic freedom. Here M is the
renormalization scale, g(M) is the QCD coupling strength at this scale and c is
a constant: g2(M)
C~ A: 12
33 ln(M2/A 2) for M ~ c~, 11'
QCD scale parameter. (1.13) Masses in massless Yang-Mills theory are a purely nonperturbative phenomenon, due to "dimensional transmutation". Scattering of glueball-hadrons
in massless pure gluon theory should look very similar to scattering of hadrons
in the real world, with finite total cross sections, amplitudes with analytic t dependence etc. At least, this would be my expectation. If the total cross section
atot has a f...
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- Quantum chromodynamics, Nonperturbative QCD