Determination of the s-wave pion-nucleon threshold scattering parameters from the results of experiments on pionic hydrogen

We give the conversion equations which lead from experimental values of the $3p \to 1s$ transition energy in pionic hydrogen and the total width of the $1s$ level to values of the s-wave threshold scattering parameters for the processes $\pi^-p \to \pi^-p$ and $\pi^-p \to \pi^0n$ respectively. Using a three-channel potential model, we then calculate the electromagnetic corrections to these quantities, which remove the effects of the Coulomb interaction, the external mass differences and the presence of the $\gamma n$ channel. We give the s-wave scattering parameters obtained from the present experimental data and these electromagnetic corrections. Finally we discuss the implications for isospin invariance.