Thermodynamic Geometry of Yang-Mills Vacua

We study vacuum fluctuation properties of an ensemble of $SU(N)$ gauge theory configurations, in the limit of large number of colors, \textit{viz.} $N_c \rightarrow \infty$, and explore statistical nature of the topological susceptibility by analyzing its critical behavior at a nonzero vacuum parameter $\theta$ and temperature $T$. We find that the system undergoes a vacuum phase transition at the chiral symmetry restoration temperature as well as at an absolute value of $\theta$. On the other hand, the long range correlation length solely depends on $\theta$ for the theories having critical exponent $e=2$ or $T=T_d+1$, where $T_d$ is the decoherence temperature. Further, it is worth noticing that the unit critical exponent vacuum configuration corresponds to a noninteracting statistical basis pertaining to a constant mass of $\eta^{\prime}$.