Chiral dynamics and fermion mass generation in three-dimensional gauge theory.

We examine the possibility of fermion mass generation in 2+1–dimensional gauge theory from the current algebra point of view. In our approach the critical behavior is governed by the fluctuations of pions which are the Goldstone bosons for chiral symmetry breaking. Our analysis supports the existence of an upper critical NF and exhibits the explicit form of the gap equation as well as the form of the critical exponent for the inverse correlation length of the order parameter. This work is supported in part by a grant from the M.U.R.S.T. This work is supported in part by the Natural Sciences and Engineering Research Council of Canada. G.S. acknowledges the hospitality of the Physics Department of the University of Perugia and I.N.F.N., Sezione di Perugia where this work was completed. The possibility of dynamical generation of fermion masses is of fundamental importance to our present view of quantum chromodynamics (QCD) as a theory of the strong interactions. Central to our understanding of this phenomenon is the existence of a critical coupling. When fermions have a sufficiently strong, attractive interaction there is a pairing instability and the resulting condensate breaks some of the flavor symmetries, generates quark masses and represents chiral symmetry in the Nambu-Goldstone mode. This idea dates back to the earliest models for chiral symmetry breaking [1] and is prevalent in the modern literature [2]. Recently the issue of critical coupling has been investigated in 2+1 dimensional gauge theories [3, 4, 5]. These theories provide toy models which exhibit simpler behavior than their 3+1–dimensional relatives [6, 7] and are also of interest as effective field theories for some condensed matter systems [8]. Typically, their dimensionless expansion parameter is 1/NF , where NF is the number of quark flavors [3]. Using Schwinger-Dyson equations in the 1/NF approximation for QED and QCD, [3] have found that there is a critical N crit F such that when NF < N crit F chiral symmetry is broken and when NF > N crit F it is not broken and quarks remain massless. In the case of QED, this result has been the subject of some debate [4, 5, 9, 10]. Either an improved ladder approximation [5, 10] or renormalization group computation [4] find no critical behavior and that chiral symmetry is broken for arbitrarily large NF . There are, however, numerical simulations [11] of 2+1– dimensional QED which find an N crit F remarkably close to that obtained by [3]. In this Letter we shall present further support for the existence of N crit F . We shall advocate a picture which is complementary to that of critical attractive quark-quark interactions and in which the critical behavior of the chiral symmetry breaking phase transition is governed by the fluctuations of the pions which are the Goldstone bosons for broken continuous flavor symmetries. We shall argue that an upper critical NF is natural since the order parameter is renormalized by the N F/2 pions with classical coupling constant ∼ 1/NFNC where NC is the number of quark colors. Their fluctuations are strong enough to destroy the ordered state when NF = N crit F ∼ NC . We are partially motivated by our recent study of strong coupling gauge theory [12] on the lattice. We showed that, in the strong coupling limit, a Hamiltonian lattice gauge theory with NC colors and NF/2 lattice flavors of staggered fermions (because of fermion doubling this corresponds to NF