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On the nature of the finite-temperature transition in QCD

We discuss the nature of the finite-temperature transition in QCD with N_f massless flavors. Universality arguments show that a continuous (second-order) transition must be related to a 3-D universality class characterized by a complex N_f X N_f matrix order parameter and by the symmetry-breaking pattern [SU(N_f)_L X SU(N_f)_R]/Z(N_f)_V -> SU(N_f)_V/Z(N_f)_V, or [U(N_f)_L X U(N_f)_R]/U(1)_V -> U(N_f)_V/U(1)_V if the U(1)_A symmetry is effectively restored at T_c. The existence of any of these universality classes requires the presence of a stable fixed point in the corresponding 3-D Phi^4 theory with the expected symmetry-breaking pattern. Otherwise, the transition is of first order. In order to search for stable fixed points in these Phi^4 theories, we exploit a 3-D perturbative approach in which physical quantities are expanded in powers of appropriate renormalized quartic couplings. We compute the corresponding Callan-Symanzik beta-functions to six loops. We also determine the large-order behavior to further constrain the analysis. No stable fixed point is found, except for N_f=2, corresponding to the symmetry-breaking pattern [SU(2)_L X SU(2)_R]/Z(2)_V -> SU(2)_V/Z(2)_V equivalent to O(4) -> O(3). Our results confirm and put on a firmer ground earlier analyses performed close to four dimensions, based on first-order calculations in the framework of the epsilon=4-d expansion. These results indicate that the finite-temperature phase transition in QCD is of first order for N_f>2. A continuous transition is allowed only for N_f=2. But, since the theory with symmetry-breaking pattern [U(2)_L X U(2)_R]/U(1)_V -> U(2)_V/U(1)_V does not have stable fixed points, the transition can be continuous only if the effective breaking of the U(1)_A symmetry is sufficiently large.