Z. Phys.C - Particles and Fields40, 293-297 (1988) Zeitschnft P a r t i c l e s ~r Physik C and F ds 9 Springer-Verlag 1988 The chiral phase transition in QCD T.S. Evans 1 and R.J. Rivers 2 1 Institutefor TheoreticalPhysics,Universityof Alberta, Edmonton,Alberta T6G 2J1, Canada 2 BlackettLaboratory,ImperialCollege, London, SW7 2BZ,UK Received22 February 1988 Abstract. The Chiral Phase Transition in QCD is studied analytically by looking at truncations of the Schwinger-Dyson equation for the quark self-mass. We find that the usual implementation of the gluon propagator at non-zero temperature is far too simple. When the gluons are given the correct qualitative non-zero temperature behaviour, the calculation of the critical temperature changes significantly. QCD is now firmly established as the theory of quarks and gluons from which hadronic matter is created. Recently, the possibility that heavy-ion collisions can create quark-gluon plasma has focused attention on its phase structure. The non-abelian nature of the col- our symmetry necessarily complicates an analytic study of (non-perturbative) phase transitions, and it has been common practice [1 5] to treat the quarks as the fundamental entities, and try to add on the (difficult) gluons iteratively. This continues the early pre-QCD approaches to the strong interactions of the '60's which necessarily concentrated on the quarks and their flavour symmetries. These quark-orientated studies are based on the observation that the pions are very light on the QCD scale, and can be regarded as approximate Goldstone bosons associated with the breaking of the chiral sym- metry q-*exP (2 ~a ~r"75) q ~ exp (~ ~aa"75) (1) in the limit of zero quark mass. (q denotes the SU (2) doublet of up, down quarks and o-" the Pauli matrices.) The idea is that the chiral phase transition in this idealised massless QCD is close to, if not ident- ical to, the deconfining transition (as lattice studies [-6] have suggested). The competing lattice calculations begin from the other end, taking the non-abelian coloured gauge fields as the main ingredients, and grafting on the (difficult) lattice quarks later. The role of gluons in the two methods is so different that there seems no point of contact. The purpose of this paper is to at- tempt a less cursory approach towards gluons in the quark-based scheme for chiral symmetry breaking than has been adopted hitherto. Our conclusions, spelled out below, are that the proper inclusion of gluons will have a significant effect on predictions for critical temperatures, etc. From this viewpoint the prior numerical success of quark-based calculations [1, 2, 3] at zero temperature looks to be felicitous. They seem very unstable to 'realistic' modifications at non-zero temperature and should not be taken as providing a solid base for calculation of phase transi- tions. Chiral symmetry breaking, and its restoration, is inferred from the Schwinger-Dyson equation for the quark self-mass S(p u) given in Fig. 1. The kernel K, Fig. 2, contains all the parts with as yet undetermined dependence on S (as well as on the three-point vertex). At non-zero temperature, we wish to maintain real and continuous energy P0, most conveniently effected in the real-time formalism. This requires a doubling Fig. 1. Schwinger-Dyson equationfor quark self-mass