Connection between low-energy scattering parameters and energy shifts for pionic hydrogen

Abstract

The usual connection between the “complex scattering lengths” and the level shifts and widths of a hadronic atom is derived using a many-channel formalism without assuming a potential model for the strong interactions. The effect of radiative decay channels is included. The formalism applies in a phenomenological way to hadronic atoms in general, though it is of practical importance only for simple systems for which we have a reasonable understanding of the hadronic interaction between the constituents. A detailed numerical study is made of the 2p → 1s transition in the π −p atom and it is concluded that a measurement of this energy, with an accuracy which might well be attainable, would provide a valuable independent check on the present extrapolations of low energy πN scattering data to threshold.