The Zero Temperature Chiral Phase Transition in SU(N) Gauge Theories

We investigate the zero temperature chiral phase transition in an SU(N) gauge theory as the number of fermions Nf is varied. We argue that there exists a critical number of fermions N c f , above which there is no chiral symmetry breaking or confinement, and below which both chiral symmetry breaking and confinement set in. We estimate N c f and discuss the nature of the phase transition. An SU(N) gauge theory, even at zero temperature, can exist in different phases depending on the number of massless fermions Nf in the theory. The phases are defined by whether or not chiral symmetry breaking takes place. For QCD with two or three light quarks, chiral symmetry breaking and confinement occur at roughly the same scale. By contrast, in any SU(N) gauge theory, asymptotic freedom (and hence chiral symmetry breaking and confinement) is lost if the number of fermions is larger than a certain value (= 11N/2 for fermions in the fundamental representation). 1 If the number of fermions Nf is reduced to just below 11N/2, an infrared fixed point will appear, determined by the first two terms in the beta function. By taking the large N limit or by continuing to non-integer values of Nf [1], the value of the coupling at the fixed point can be made arbitrarily small, making a perturbative analysis reliable. Such a theory with a perturbative fixed point is a massless conformal theory. There is no chiral symmetry breaking and no confinement. As Nf is reduced further, chiral symmetry breaking and confinement will set in. There have been lattice Monte Carlo studies of the Nf dependence of chiral symmetry breaking [2]. For example, Kogut and Sinclair [2] found that for N = 3 and Nf = 12 there is no chiral symmetry breaking, while Brown et. al. [2] have found chiral symmetry breaking for N = 3 and Nf = 8. In this paper we will estimate the critical value N c f at which this transition occurs. We then investigate the properties of the phase transition for Nf ≈ N c f . Our discussion will parallel an analysis of the chiral phase transition in QED3 and QCD3 [3, 4]. In a large Nf expansion it was found that an appropriate effective coupling has an infrared fixed point with strength proportional to 1/Nf , and that as Nf is lowered, the value of the fixed point exceeds the critical value necessary to produce spontaneous chiral symmetry breaking. It was argued that this critical value is large enough to make the 1/Nf expansion reliable. An Nf dependence similar to the one we describe here has been found in N = 1 supersymmetric QCD [5]. This theory is not asymptotically free for large enough Nf , and has an infrared, conformal fixed point for a range of Nf below a certain value.