Aspects of the QCD $θ$-vacuum
This paper addresses two aspects concerning the $θ$-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, $χ_\mathrm{top}$, 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 $χ_\mathrm{top}=\mathcal{O}(N_c^0)$, 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 $θ\simπ$ 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 $θ\simπ$ two vacuum states coexist, which become degenerate at $θ=π$. The wall tension of the energy barrier between these degenerate vacua is determined as well as the decay rate of a false vacuum.