New light on electromagnetic corrections to the scattering parameters obtained from experiments on pionium

We calculate the electromagnetic corrections needed to obtain isospin invariant hadronic pion-pion s-wave scattering lengths a, a from the elements acc, a0c of the s-wave scattering matrix for the (π+π−, ππ) system at the π+π− threshold. The latter can be extracted from experiments on the π+π− atom (pionium). Our calculation uses energy independent hadronic pion-pion potentials V , V 2 that satisfactorily reproduce the low-energy phase shifts given by two-loop chiral perturbation theory, with the hadronic mass of the pion taken first as the charged pion mass and then as the neutral pion mass. We also take into account an important relativistic effect whose inclusion influences the corrections considerably. PACS: 13.75.Gx,25.80.Dj