Topology in the SU(Nf) chiral symmetry restored phase of unquenched QCD and axion cosmology

TL;DR

This paper examines the topological properties of unquenched QCD at high temperatures, challenging existing analytical predictions and suggesting the vanishing of topological susceptibility and axial anomaly effects above the critical temperature.

Contribution

It provides a critical comparison between numerical results and analytical predictions, proposing that the vacuum energy density becomes $ heta$-independent at high temperatures, implying the disappearance of topological effects.

Findings

Numerical results are inconsistent with the saddle point expansion predictions.

Vacuum energy density appears $ heta$-independent at temperatures around 2$T_c$.

Topological susceptibility and axial anomaly effects vanish above $T_c$.

Abstract

We investigate the topological properties of unquenched $QCD$ on the basis of numerical results of simulations at fixed topological charge, recently reported by Borsanyi et al.. We demonstrate that their results for the mean value of the chiral condensate at fixed topological charge are inconsistent with the analytical prediction of the large volume expansion around the saddle point, and argue that the most plausible explanation for the failure of the saddle point expansion is a vacuum energy density $\theta$-independent at high temperatures, but surprisingly not too high $(T\sim 2T_c)$, a result which would imply a vanishing topological susceptibility, and the absence of all physical effects of the $U(1)$ axial anomaly at these temperatures. We also show that under a general assumption concerning the high temperature phase of $QCD$, where the $SU(N_f)_A$ symmetry is restored, the analytical prediction for the chiral condensate at fixed topological charge is in very good agreement with the numerical results of Borsanyi et al., all effects of the axial anomaly should disappear, the topological susceptibility and all the $\theta$-derivatives of the vacuum energy density vanish and the theory becomes $\theta$-independent at any $T>T_c$ in the infinite volume limit.