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^0, a^2 from the elements a_cc, a_0c of the s-wave scattering matrix for the (\pi^+ \pi^-, \pi^0 \pi^0) system at the \pi^+ \pi^- threshold. These elements can be extracted from experiments on pionium. Our calculation uses energy independent hadronic pion-pion potentials that satisfactorily reproduce the low-energy phase shifts given by two-loop chiral pertur- bation theory. We also take into account an important relativistic effect whose inclusion influences the corrections considerably.