The spectral density of the QCD Dirac operator and patterns of chiral symmetry breaking

Abstract

We study the spectrum of the QCD Dirac operator for two colors with fermions in the fundamental representation and for two or more colors with adjoint fermions. For N f flavors, the chiral flavor symmetry of these theories is spontaneously broken according to SU (2 N f → Sp (2 N f ) and SU ( N f → O ( N f ), respectively, rather than the symmetry breaking pattern SU ( N f ) × SU ( N f ) → SU ( N f ) for QCD with three or more colors and fundamental fermions. In this paper we study the Dirac spectrum for the first two symmetry breaking patterns. Following previous work for the third case we find the Dirac spectrum in the domain λ ⪡ Λ QCD by means of partially quenched chiral perturbation theory. In particular, this result allows us to calculate the slope of the Dirac spectrum at λ = 0. We also show that for λ ⪡ 1/ L 2 Λ QCD (wing L the linear size fo the system) the Dirac spectrum is given by a chiral Random Matrix Theory with the symmetries of the Dirac operator.