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 a⁰, a² from the elements a_cc, a_0c of the s-wave scattering matrix for the (π⁺π^-,π⁰π⁰) 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 V⁰ and V² 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.