Thermodynamic Geometry of Yang–Mills Vacua

We study vacuum fluctuation properties of an ensemble of $SU\left(N\right)$ gauge theory configurations, in the limit of many colors, viz. $N_c\to \infty$, and explore the statistical nature of the topological susceptibility by analyzing its critical behavior at a non-zero-vacuum parameter $\theta$ and temperature html-italic^>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 with critical exponent $e=2$ or $T=T_d+1$, where $T_d$ is the decoherence temperature. Furthermore, it is worth noticing that the unit-critical exponent vacuum configuration corresponds to a non-interacting statistical basis pertaining to a constant mass of ${\eta }^{\prime }$.