Pion mass-difference sum rules beyond the chiral limit

A bstract The difference between the charged and neutral pion masses can be predicted from a well-known dispersion relation involving a weighted integral of the vector-minus-axial spectral function over all energies. This relation, however, holds only in the chiral limit, while phenomenological applications involve physical light-quark masses and spectral input known only up to a finite scale. Here we combine dispersion theory, the operator product expansion and chiral perturbation theory with dynamical photons to determine the form of the corresponding physical finite-energy sum rule, including all corrections linear in the light-quark masses. We test this generalized relation using hadronic tau-decay data.