$θ$ dependence of $T_c$ in SU(2) Yang-Mills theory

Abstract

We present an exploratory study to determine the confinement-deconfinement transition temperature at finite $θ$, $T_c(θ)$, for the 4d \SU(2) pure Yang-Mills theory. Lattice numerical simulations are performed on three spatial sizes $N_S=24$, $32$, $48$ with a fixed temporal size $N_T=8$. We introduce a non-zero $θ$-angle by the sub-volume method to mitigate the sign problem. By taking advantage of the universality in the second order phase transition and the Binder cumulant of the order parameter, the $θ$-dependence of $T_c$ is determined to be $T_c(θ)/T_c(0)=1-0.16(2)\,(θ/π)^2-0.03(4)\,(θ/π)^4$ up to $θ\sim 0.9\,π$. The reliability of the extrapolations in the sub-volume method is extensively checked. We also point out that the temperature dependence of the topological susceptibility should exhibit a singularity with the exponent for the specific heat.