Critical couplings for chiral symmetry breaking via instantons.pdf

a r X i v : h e p - p h / 0 0 0 3 0 5 9 v 2 2 3 M a r 2 0 0 0 Critical couplings for chiral symmetry breaking via instantons F. S. Roux Department of Physics, University of Toronto Toronto M5S1A7, CANADA Using an instanton effective action formalism, we compute the critical coupling for the nonperturbative formation of a dynamical mass via instantons in non-Abelian gauge theories with Nf massless fermions. Only continuous phase transitions are considered. For large values of Nf the critical couplings are found to be much smaller than the equivalent critical couplings obtained from gauge exchange calculations in the ladder approximation. I. INTRODUCTION Nonperturbative gauge dynamics present many challenges that are yet to be fully understood. One of these is the phenomenon of chiral symmetry breaking. A theory with an SU(Nc) gauge symmetry and Nf fermions in the fundamental representation has a global chiral symmetry in the massless limit. Provided that Nf 5.5Nc this theory is asymptotically free. Close to this threshold the coupling is bounded above due to an infrared fixed point in the beta function. As Nf becomes smaller the fixed point value increases. Below a critical value for Nf the coupling can grow strong enough to break the chiral symmetry down to the vector symmetry: SU(Nf )R × SU(Nf )L × U(1)V → SU(Nf)V × U(1)V . (1) As a result the fermions acquire a dynamical mass, which acts as an order parameter for the spontaneous breaking of the chiral symmetry. This much is well understood, but the respective parts played by gauge exchanges and instantons in this process still needs clarification. It has been known for some time that gauge exchanges can generate a dynamical mass. To leading order (ladder approximation) in Landau gauge the critical coupling associated with gauge exchange dynamics is αgec = 2πNc 3(N2c ? 1) . (2) Assuming a two-loop beta function, one can show that the critical number of flavors, below which dynamical chiral symmetry breaking o