The Spectral Density of the QCD Dirac Operator and Patterns of Chiral Symmetry Breaking

TL;DR

This paper analyzes the spectral density of the QCD Dirac operator for different gauge groups and fermion representations, revealing how symmetry breaking patterns influence the Dirac spectrum and connecting it to chiral Random Matrix Theory.

Contribution

It extends the study of the Dirac spectrum to new symmetry breaking patterns in two-color and adjoint fermion QCD, providing analytical results for the spectral density and its slope at zero.

Findings

Derived the Dirac spectrum in the domain chiral perturbation theory

Calculated the slope of the Dirac spectrum at

Showed the Dirac spectrum matches chiral Random Matrix Theory for small eigenvalues

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 $SU(2N_f)\to Sp(2N_f)$ and $SU(N_f)\to O(N_f)$, respectively, rather than the symmetry breaking pattern $SU(N_f) \times SU(N_f) \to SU(N_f)$ 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 $\lambda \ll \Lambda_{\rm QCD}$ by means of partially quenched chiral perturbation theory. In particular, this result allows us to calculate the slope of the Dirac spectrum at $\lambda = 0$. We also show that for $\lambda \ll 1/L^2 \Lambda_{QCD}$ (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.