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, 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 (with L the linear size of the system) the Dirac spectrum is given by a chiral Random Matrix Theory with the symmetries of the Dirac operator.

Finite volume and partially quenched QCD-like effective field theories

“…The partially quenched expressions reduce to the unquenched results whenever we set the sea mass equal to the valence mass. In addition we reproduce the known results at NLO for the condensate [23] also for the partially quenched case. The finite volume expressions have been checked against the known NLO results and numerically with the earlier known NNLO results, as discussed in section 7.…”

“…Alternative methods of calculation are the replica trick [50] and the quark flow method [30]. The earliest partially quenched work for QCDlike theories used the supersymmetric method [23]. The replica trick has been used in [51].…”

Spectral universality of real chiral random matrix ensembles

“…The equivalence of the effective theory for the Goldstone modes and chiral Random Matrix Theory has been demonstrated nonperturbatively for the chiral Unitary Ensemble [45,60,31,32]. For the the other two values of the Dyson index this has only been shown perturbatively [13].…”

Lattice study of largeNcQCD

“…The actual condensate might be obtained from the zero mass limit of this quantity. We are aware of more modern ways of finding the condensate, from the spectrum of eigenvalues of the Dirac operator [47][48][49][50][51], but these methods seem to us to require smaller quark mass data than we can safely obtain given our simulation volumes.…”