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

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.