Change of theta dependence in 4D SU(N) gauge theories across the deconfinement transition

TL;DR

The paper examines how 4D SU($N$) gauge theories depend on the topological parameter $\theta$ at finite temperature, focusing on the deconfinement transition. It employs the lattice Wilson formulation and computes the finite-temperature free-energy expansion $F(\theta,T)$ around $\theta=0$ up to $O(\theta^6)$, extracting the topological susceptibility $\chi(T)$ and expansion coefficients $b_2(T)$, $b_4(T)$ for $N=3$ and $N=6$. The results show a sharp change in the $\theta$-dependence across $T_c$: in the low-$T$ confined phase the dependence scales with $\theta/N$, while in the high-$T$ deconfined phase it follows the instanton-gas prediction with $\theta$ as the relevant variable, with $b_2$ tending to $-1/12$ and $b_4$ to $1/360$. The crossover becomes sharper with increasing $N$ and $\chi(T)$ is exponentially suppressed at high temperature, supporting a regime change in topological dynamics and informing large-$N$ and finite-temperature QCD phenomena.

Abstract

We investigate the dependence of four-dimensional SU(N) gauge theories on the topological theta term at finite temperature and, in particular, across the deconfinement transition. For this purpose, we exploit the lattice formulation of the theory and present numerical results for the expansion of the free energy up to O(theta^6), for N=3 and N=6. Our numerical analysis shows that the theta dependence of 4D SU(N) gauge theory experiences a drastic change across the deconfinement transition: the low-temperature phase is characterized by a large-N scaling with theta/N as relevant variable, while in the high-temperature phase the scaling variable is just theta and the free energy is essentially determined by the instanton-gas approximation. The crossover between the two different behaviours gets sharper with increasing N, suggesting that the instanton-gas regime sets in just above Tc at large N.

Figures (3)

• Figure 1: (Color online) 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 \ref{['t0res']}.
• Figure 2: Results for SU($3$).
• Figure 3: Results for SU($6$).