The influence of the hadronic interaction on the pionium wave functions

The influence of the hadronic ππ interaction on the wave functions of pionium (the Coulomb bound π+π− system) at distances ⩽10 fm is calculated for s-states with principal quantum numbers n⩽4. Hadronic ππ potentials are used that reproduce the scattering phase shifts of two-loop chiral perturbation theory. The pionium wave functions ψn(r) are obtained by integrating the coupled Schrödinger equations for the (π+π−,π0π0) system. We find that, for r⩽10 fm, n3/2ψn(r) is practically independent of n. From this we conclude that the production rates of the s-states of pionium are proportional to n−3, a result needed for the interpretation of the DIRAC experiment currently running at CERN.