Low-energy parameterisations of the pion-nucleon phase shifts

Compared in this work are a few sets of results, obtained from the pion-nucleon ($\pi N$) data at low energy (pion laboratory kinetic energy up to $100$ MeV) on the basis of the modelling of the $s$- and $p$-wave $K$-matrix elements (or of their reciprocal) via simple polynomials. The fitted values and uncertainties of the model parameters, as well as the corresponding Hessian (covariance) matrices, for three of these parameterisations are given in tabular form for two types of joint fits: to the measurements of the two elastic-scattering processes ${\pi }^{\pm }p\to {\pi }^{\pm }p$, and to those of the ${\pi }^{+}p$ reaction and of the ${\pi }^{-}p$ charge-exchange reaction $\pi^- p \to \pi^0 n$. From these results, reliable and (largely) data-driven (hence model-independent) predictions, accompanied by uncertainties which reflect the statistical and systematic fluctuation of the input data, can be obtained for the low-energy constants of the $\pi N$ interaction (scattering lengths/volumes and range parameters), for the phase shifts, for the $K$-matrix elements, and for the partial-wave amplitudes in the (dominant at low energy) $s$ and $p$ waves. After the addition of the $d$- and $f$-wave contributions, and the inclusion of the electromagnetic effects, corresponding predictions can be obtained for the usual low-energy $\pi N$ observables, i.e., for the differential cross section and for the analysing power.