# Aspects of the QCD $\theta$-vacuum

## TL;DR

This paper uses chiral perturbation theory to analyze the topological susceptibility and vacuum structure of QCD at different theta angles, revealing new scaling behaviors and vacuum degeneracy properties.

## Contribution

It provides the first next-to-leading order calculations of cumulants of the winding number distribution and explores vacuum degeneracy at heta \, \sim \, \pi in QCD.

## Key findings

- Topological susceptibility scales as N_c^0.
- Fourth cumulant scales as N_c^{-3}, correcting previous assumptions.
- Vacuum degeneracy occurs at heta=\pi with calculable wall tension.

## Abstract

This paper addresses two aspects concerning the $\theta$-vacuum of Quantum Chromodynamics. First, large-$N_c$ chiral perturbation theory is used to calculate the first two non-trivial cumulants of the distribution of the winding number, i.\,e. the topological susceptibility, $\chi_\mathrm{top}$, and the fourth cumulant, $c_4$, up to next-to-leading order. Their large-$N_c$ scaling is discussed, and compared to lattice results. It is found that $\chi_\mathrm{top}=\mathcal{O}(N_c^0)$, as known before, and $c_4=\mathcal{O}(N_c^{-3})$, correcting the assumption of $\mathcal{O}(N_c^{-2})$ in the literature. Second, we discuss the properties of QCD at $\theta\sim\pi$ using chiral perturbation theory for the case of $2+1$ light flavors, i.\,e. by taking the strange quark mass heavier than the degenerate up and down quark masses. It is shown that --- in accordance with previous findings for $N_f=2$ and $N_f=3$ mass-degenerate flavors --- in the region $\theta\sim\pi$ two vacuum states coexist, which become degenerate at $\theta=\pi$. The wall tension of the energy barrier between these degenerate vacua is determined as well as the decay rate of a false vacuum.