~~KSR~POWS-(Se&m C of Physics Letters) 36, No. 3 (1978) 137- 276 NORTH-HOLLAND PUBL1[SHICNG (;Oli4PANY QUANTUM CHROMODYNAMICS’ William MARCIANO and Heinz PAGELS The Rockejdler Univer arty, New York, N.Y. 10021, U.S.A. Recetved 6 June 1977 Contents 1. Introduction 13’) 5. C alor confinement and perturbation theory 212 I.$. What is Q2D? 140 5.1. The infrared structure of QED 213 1.2. The QCD phase transttlons 142 5.2 Infrared divergences m QCD 216 1.3. Properttes of QCD 145 5.3. The Kinoshtta-Lee-Nauenberg theorem and con- 1 4. What IS m :hm review 148 fint ment 221 2 Renormaliza tron of gauge theortes 149 5 4 Beyond perturbation theory a posstble con- i 1. The pat11 integral 151 f lnemen t scheme 224 2.2. Gaub Gelds and the Faddeev-Popov trtck 154 5 5 Non-perturbattle approaches to QCD 227 2.3. Proof of renormahzabihty for gauge fields IS8 6 Topologtcal soliton\ 230 2 4 Ceynrwa rules for QCD 167 6 1 lntroductton 230 2.5 Slavno v -Taylor Identities 170 6 2 Topologrcal sohtons m D = 1. 2 and 3 dtmen$tons 236 2.6 The Sxwmger-Dyson equattons for QCD 173 6 3 The pseudopartrcte or tnstanton for D = 4 245 3 The renormallzatton group for gauge theortc$ 175 7. Recent developments 256 3 1 Renormahzatlon group equations 17x 7 1 Dtlute gases of topological soIlton\ and quark 3 2 Scaling pt operttes of QCD 1x7 continemcnr 256 3 3 Further dppkatlons of the QCD renormaludtwn 7 2 Qc*D dmphtudtv d\ d functton of the gauge group 195 couphng ‘SC, 4 Two dtmenslonal gdUge theone\ 203 8 Conclustons 265 4.1 The Schwmger model 204 References 268 4.2. 2-D QCD 210 -- Stnyle orders for thlr !nue ---1 I PHYC!CC REPORT’: (Section C of PHYS!CS LL _i-TFRc;r _ -_ __, _Ih _, No I11197x1 (__ _, 117-- -276 I 1 Copies ofthIs Issue may be obtained at the prtce gtven below All orders should be sent dtrecily to the ?~rb!;,iler Orden mu\t be dccompdnled by check. Single Issue price Dfl. 59 -, postage Included, * Work supported in part by the U.S. Energy and Development Admmistratron under Contract Grant No. EY--76 -C -02-2232B.*OOO. NORTH-HOLLAND PUBLlSWlNG COMPANY - AMSI ERDAM IV. MarcJanb and H. Pagels, Quantum chromodynamies 1~o Abstract: Quantum Chromodynamics (QCD) is a quantum field theory of the strong interaction with non-abeh~m gauge fields mediating the interactions between quarks The ex perime~tally observed strong interactions ~.re to be epiphenomena o5 these fundamental rater- actions The experimental successes of QCD form the basis for our present optimism that a theory of the strong force has b~:r~ ~ouno However,~QCdD~isstill .very:vu~able'todefati!~ '6n :both.experimental and theoretical grounds. In this article we offer the ~eader a review of the prope~ies and attempted solutions of QCD. Each section of this review can be read independently of the others In the introduction We describe the' pr0perties of QCD and the hoped for confinement and PCAC phase transitions. This ~s follov,ed by a secti0n on' the renormali~tion of non-abelian gaage field theories: the functional methods and the path integral, the BPHZ program. the BRS transformation and proof of renormalizability, the Slavnov-Taylor identities and Schwinger-Dyson equation.~ The renormal- ization group equations are derived and applied to physical proeesses~ Two dimensional prototypes of QCD, the abehan Schwmger model and 2-D QCD, are reviewed. An extensive review of the perturbative development of QCD ms given with empha~s on infrared divergences, exponentiation of leading Iogarithm~ the CornwaU-Tiktopoulos equation and a non-perturbative approach to QCD A self contained section on tor~ologieal solitons follows with discussions of homotopy theory, voruccs, monopoles and especmlly instantons, and the periodic vacuum. Recent results, the attempt to study phase transitmns m QCD using the &lute gas approxlmatmn and Borel resummations in QCD, are ~amine¢l. Most of the major areas of interest in QCD are covered m this review: the prime excerJtibns are lattice gauge theories and phenomenological QCD like the parton and potentml models 1. Introduction In the last four years there has been a growing conviction among many theoretical physicists that they have found a theory of the strong interaction. This theory is called quantum chromo- dynamics (QCD) [1.1]. The 1. 1abashed hope is that QCD will provide a complete description of all strong interaction physics just as quantum electrodynamics (QED) has in the past prowded a high precision account of the interactions of photons and electrons. Should this hope become realized it would be a major triumph for theoretical physics. At this time we are fa~" from realizing this hope. QCD is a Lagrangian field theory ~n search of a solution. Nothing comparable to the Feynman rules mad the perturbation approximation series in QED exists for the bound state physics of QCD. There may even be difficulties in precisely defining the theory. According to QCD all the strongly interactmg particles are bound state~ of permanently confined constituents called quarks. Yet no one has ever proven the existence of a single bound state let alone the confinement property fin any relativistic, 3 + 1 dimensional. quantum field theory. The outsider may then well wonder why this theory has attracted the attention of theoretr=al physicists. First, QCD has not been shown to b" in conflict with an2~ existing phenomenology of the strong interaction. In part the compatability of QCD with experiment is due to our ignorance of its solution. But the symmetries that we can extract fi'om QCD with confidence are precisely the observed symmetries ol the strong interactions and no more. Where dynamical consequences of the theory can be extracted with some confidence, ~,:ch as scaling behavior, they are in accord witt" ~,xperiment. Other examples of successful contact between QCD and experimental observa- ,~,,v,o...... ,,,,, m be given I"-'t,~t,,v, .... In principle,"- ' " as far as we know today, QCD could be the complete theory of all strong interaction physics. Secondly, QCD is the realization that the simple principle of a ]ocal gauge group S U~(3) and the formal existence of fermions called quarks transforming as the fundamental representation of this group essentially specifies the complete theory. So QCD could fulfill our hope that the plethora of material phenomena, in spite of their apparent c 9mplexity, are the consequence of elementary physical principles. 140 W. Marciano and H. Pagels, O.uant ,m chromodynam~cs We thought that it would be useful to outline the properties of QCD and review recent develop- ments in the attempts to solve the theory. This review could well be premature. Our perspective on the theory could alter dramatically in the near furore. QCD could be proven wrong, irrelevant or incomplete. This review is written for the theoretical physics student. Its main purpose is to outline ~ome of the major developments that have occured in QCD, so that the student can have an overview before being swallowed by a jungle of technical details. This review is to serve as an introduction to the literature and nor as a substi)ute for it. 1. L What is QCD? QCD is a renormalizable Lagrangian quantum field theory of the strong interactions. Its fun- damental constituents are spin ½Fermi fields called quarks which carry fractional electr;c charge and non-abelian spin 1 gauge fields called gluons which interac) with the quarks as well as among themseh,es. To begin oar description of this theory, we introduce the Dirac spinor quark field qa~(x) (the 4 spinor components are implicit) where A and a label internal degrees of freedom. These indices distinguish the members of the fundamental representations of simple Lie groups to which the quark field belongs. The index A refers to Havor and corresponds to the observed degrees offreedom of existing hadrons such as isotopic spin, strangeness, charm and very possibly more. At present 4 flavor,~ are known and the quarks corresponding to these are commonly called: ql = u (up quark), q2 = d (down quark), q3 ._ S (strange quark), q4 = c (charmed qv :rk). In QCD the flavor index has no dynamical role; it just goes along for the ride. The we~k ip'~eraction current will couple to flavor, q he index ct refers to the color degrees of freedom. It is as, umed that there are exactly three colors and the usual nomenclature is: q~ = red quark, qg = ¢vh.'te quark, q~ = blue quark. (So for example, q~ rs a blue up quark.) The reason for fixing the number of colors at 3 is so that the statistics of the baryons come out right. This and other phenomenotogical support for 3 colors will be explained in more detail subsequently. Because of this constraint, the only candidates for a simple Lie group of color are SU(3) and SO(3). With ,~ 0(3) one loses the much desired property of asymptotic treedom when the number of flavors ex,'eeds 2, while for SU(3) one can have up to 16 flavors [1,.2]. Therefore, since 4 flavors are already known, SU(3) is the only viable possibility for the color group of QCD. Further with SO(3) a diquark system could be a color singlet and such objects are unobserved. The main idea of QCD is to make the SU¢(3) color symmetry a local, rather than just a global symmetry [1.3]. As is well known, to implement this local SU,(3) symmetry, one must introduce a compensating vector gauge field [1.4] A~(x), a = 1, 2..