Large-$N_c$ gauge theory and chiral random matrix theory

TL;DR

This paper explores how large-$N_c$ gauge theories can be studied using the $1/N_c$ expansion and chiral random matrix theory, demonstrating their combined effectiveness in detecting chiral symmetry breaking through numerical analysis.

Contribution

It clarifies the parameter regimes for applying $1/N_c$ expansion and $ ext{chi}$RMT, and shows their combined use can detect chiral symmetry breaking in large-$N_c$ gauge theories.

Findings

Chiral symmetry breaking is confirmed in large-$N_c$ gauge theory with heavy adjoint fermions.

The combined approach accurately reproduces known symmetry breaking patterns.

Observed Dirac spectrum features support the validity of the combined theoretical framework.

Abstract

We discuss how the $1/N_c$ expansion and the chiral random matrix theory ($\chi$RMT) can be used in the study of large-$N_c$ gauge theories. We first clarify the parameter region in which each of these two approaches is valid: while the fermion mass $m$ is fixed in the standard large-$N_c$ arguments ('t Hooft large-$N_c$ limit), $m$ must be scaled appropriately with a certain negative power of $N_c$ in order for the gauge theories to be described by the $\chi$RMT. Then, although these two limits are not compatible in general, we show that the breakdown of chiral symmetry can be detected by combining the large-$N_c$ argument and the $\chi$RMT with some cares. As a concrete example, we numerically study the four dimensional $SU(N_c)$ gauge theory with $N_f=2$ heavy adjoint fermions, introduced as the center symmetry preserver keeping the infrared physics intact, on a $2^4$ lattice. By looking at the low-lying eigenvalues of the Dirac operator for a massless probe fermion in the adjoint representation, we find that the chiral symmetry is indeed broken with the expected breaking pattern. This result reproduces a well-known fact that the chiral symmetry is spontaneously broken in the pure $SU(N_c)$ gauge theory in the large-$N_c$ and the large-volume limit, and therefore supports the validity of the combined approach. We also provide the interpretation of the gap and unexpected $N_c$-scaling, both of which are observed in the Dirac spectrum.