QCD $\theta$-vacuum energy and axion properties

At low energies, the strong interaction is governed by the Goldstone bosons associated with the spontaneous chiral symmetry breaking, which can be systematically described by chiral perturbation theory. In this paper, we apply this theory to study the $\theta$-vacuum energy density and hence the QCD axion potential up to next-to-leading order with $N$ non-degenerate quark masses. By setting $N=3$, we then derive the axion mass, self-coupling, topological susceptibility and the normalized fourth cumulant both analytically and numerically, taking the strong isospin breaking effects into account. In addition, the model-independent part of the axion-photon coupling, which is important for axion search experiments, is also extracted from the chiral Lagrangian supplemented with the anomalous terms up to $\mathcal{O}(p^6)$.