Cumulants of the QCD topological charge distribution

The distribution of the QCD topological charge can be described by cumulants, with the lowest one being the topological susceptibility. The vacuum energy density in a θ-vacuum is the generating function for these cumulants. In this paper, we derive the vacuum energy density in SU(2) chiral perturbation theory up to next-to-leading order keeping different up and down quark masses, which can be used to calculate any cumulant of the topological charge distribution. We also give the expression for the case of SU(N) with degenerate quark masses. In this case, all cumulants depend on the same linear combination of low-energy constants and chiral logarithm, and thus there are sum rules between the N-flavor quark condensate and the cumulants free of next-to-leading order corrections.