Change of $\theta$ Dependence in 4D $\mathrm{SU}(N)$ Gauge Theories Across the Deconfinement Transition

We study the dependence of 4D $\mathrm{SU}(N)$ gauge theories on the topological $\theta$ term at finite temperature $T$. We exploit the lattice formulation of the theory, presenting numerical results for the expansion of the free energy up to $O({\theta }^6)$, for $N=3$ and $N=6$. Our analysis shows that the $\theta$ dependence drastically changes across the deconfinement transition: the low-$T$ phase is characterized by a large-$N$ scaling with $\theta /N$ as relevant variable, while in the high-$T$ phase the scaling variable is just $\theta$ and the free energy is essentially determined by the instanton-gas approximation.

Figure 1

Finite-$T$ results of $b_2$ for $N=3$ and $N=6$, versus the reduced temperature $t\equiv T/T_c-1$. The shadowed regions indicate the $T=0$ estimates of $b_2$ for $N=3$ and $N=6$; see Table .