Proposals for the test of the isospin invariance in the pion-nucleon interaction at low energy
This work elaborates on former remarks of ours regarding two sets of predictions for the observables of the charge-exchange reaction $\pi^- p \to \pi^0 n$ at low energy (pion laboratory kinetic energy $T \leq 100$ MeV). The first prediction is obtained via the triangle identity from the results of fits to low-energy $\pi^\pm p$ elastic-scattering data, whereas the second is based on the same analysis of the combined low-energy $\pi^+ p$ and charge-exchange databases. Assuming the integrity of the data used in our fits (i.e., the absence of significant systematic effects in the determination of the absolute normalisation of the datasets) and the insignificance of residual effects in the electromagnetic corrections, a significant difference between these two sets of predictions may be interpreted as departure from the isospin invariance in the low-energy $\pi N$ interaction. We examine the sensitivity of the standard low-energy observables to the effect, and identify the kinematical regions which are promising for experimental survey. Accurate experiments under the suggested conditions will differentiate between the predictions, and thus provide an independent test of the isospin invariance in the low-energy $\pi N$ interaction.