What has been learnt from the analysis of the low-energy pion-nucleon data during the past three decades?

Over twenty-five years ago, two analyses of the pion-nucleon ($\pi N$) data at low energy (i.e., for pion laboratory kinetic energy $T \leq 100$ MeV) reported on the departure of the extracted scattering amplitudes, corresponding to the two elastic-scattering reactions $\pi^\pm p \to \pi^\pm p$ and to the $\pi^- p$ charge-exchange reaction $\pi^- p \to \pi^0 n$, from the triangle identity, which these amplitudes fulfil if the isospin invariance holds in the hadronic part of the $\pi N$ interaction. This discrepancy indicates that \emph{at least one} of the following assumptions is not valid: first, that the absolute normalisation of the bulk of the low-energy $\pi N$ datasets is correct; second, that any residual contributions to the corrections, which aim at the removal of the effects of electromagnetic (EM) origin from the measurements, are not significant; and third, that the isospin invariance holds in the hadronic part of the $\pi N$ interaction. In view of the incompatibility of the results of the various schemes of removal of the so-called trivial EM effects at the $\pi N$ threshold ($T = 0$ MeV), the likelihood of residual effects of EM origin in the extracted $\pi N$ scattering amplitudes (from the data in the scattering region, i.e., above the $\pi N$ threshold) must be reassessed. This work emphasises the importance of the development of a unified scheme for the determination of reliable EM corrections, \emph{applicable at the $\pi N$ threshold and in the scattering region}, in providing a resolution to the established discrepancy at low energy.