The extraction of hadronic parameters from experiments on pionium

We show how experimental values of the lifetime of the 1s level of pionium and of the difference between the energies of the 2s and 2p levels yield values of the elements a_oc and a_cc respectively of the s-wave scattering matrix for the two-channel (π⁺π^-, π₀π₀) system at the π⁺π^- threshold. We then develop a method, using energy independent hadronic potentials which reproduce the best available pion-pion phase shifts up to 500 MeV total energy in the c.m. frame, for obtaining the values of the isospin invariant quantities a²₀-a⁰₀ and 2a⁰₀ + a²₀ from a_oc and a_cc respectively. We emphasise that the isospin invariant scattering lengths a⁰₀ and a²₀ universally used in the literature cannot be considered to be purely hadronic quantities.