Topological susceptibility and QCD axion mass: QED and NNLO corrections

We improve the precision of the topological susceptibility of QCD, and therefore of the QCD axion mass, by including O(α_em) and NNLO corrections in the chiral expansion, which amount to 0.65(21)% and -0.71(29)% respectively. Both corrections are one order of magnitude smaller than the known NLO ones, confirming the very good convergence of the chiral expansion and its reliability. Using the latest estimates for the light quark masses the current uncertainty is dominated by the one of the low-energy constant ℓ₇. When combined with possible improvements on the light quark mass ratio and ℓ₇ from lattice QCD, our computation could allow to determine the QCD axion mass with per-mille accuracy.