Large-Ncgauge theory and chiral random matrix theory

Abstract

We discuss how the $1/{N}_{c}$ expansion and the chiral random matrix theory ($\ensuremath{\chi}\mathrm{RMT}$) 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 $\ensuremath{\chi}\mathrm{RMT}$. 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 $\ensuremath{\chi}\mathrm{RMT}$ with some care. 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 overlap-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 an interpretation of the gap and unexpected ${N}_{c}$ scaling, both of which are observed in the Dirac spectrum.