$\theta$ dependence in $SU(3)$ Yang-Mills theory from analytic continuation

Abstract

We investigate the topological properties of the $SU(3)$ pure gauge theory by performing numerical simulations at imaginary values of the $\theta$ parameter. By monitoring the dependence of various cumulants of the topological charge distribution on the imaginary part of $\theta$ and exploiting analytic continuation, we determine the free energy density up to the sixth order order in $\theta$, $f(\theta ,T)=f(0,T)+\frac{1}{2}\chi (T){\theta }^2(1+b_2(T){\theta }^2+b_4(T){\theta }^4+O({\theta }^6))$. That permits us to achieve determinations with improved accuracy, in particular for the higher order terms, with control over the continuum and the infinite volume extrapolations. We obtain $b_2=-0.0216(15)$ and $|b_4|\lesssim 4\times 10^{-4}$.

Comments: 10 pages, 9 eps figures (minor changes)