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

Abstract

Over twenty-five years ago, two analyses of the pion-nucleon (πN ) data at low energy (i.e., for pion laboratory kinetic energy T ≤ 100 MeV) reported on the departure of the extracted scattering amplitudes, corresponding to the two elastic-scattering reactions π ± p → π ± p and to the π − p charge-exchange reaction π − p → π 0 n, from the triangle identity, which these amplitudes fulfil if the isospin invariance holds in the hadronic part of the πN interaction. This discrepancy indicates that at least one of the following assumptions is not valid: first, that the absolute normalisation of the bulk of the low-energy πN 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 πN interaction. In view of the incompatibility of the results of the various schemes of removal of the so-called trivial EM effects at the πN threshold (T = 0 MeV), the likelihood of residual effects of EM origin in the extracted πN scattering amplitudes (from the data in the scattering region, i.e., above the πN threshold) must be reassessed. This work emphasises the importance of the development of a unified scheme for the determination of reliable EM corrections, applicable at the πN threshold and in the scattering region, in providing a resolution to the established discrepancy at low energy.

Low-energy parameterisations of the pion-nucleon phase shifts

“…The phase-shift analyses, performed within the context of the ETH πN project (as well as those of the SAID group), make use of the minimisation function which Arndt and Roper introduced half a century ago [15]. All details can be found in Section 3.3 of a recent study [16]; there is no need to repeat them here.…”

“…In this work, I will refer to the modelling, based on Eqs. (15,16), as 'ELW parameterisation', despite the fact that it was first suggested by Höhler.…”

“…( 22) is taken from Eqs. (12,16,19,21), according to the selected parameterisation, whereas the singular term has been obtained from Ref. [7], see K 1+ in Eqs.…”

“…( 2) of Ref. [16]), whereas the other is obtained from fits involving the π − p CX data; in Ref. [16], the former amplitude is denoted by f CX , the latter by f extr CX .…”

“…[16]), whereas the other is obtained from fits involving the π − p CX data; in Ref. [16], the former amplitude is denoted by f CX , the latter by f extr CX . For the sake of variety, let me also use the results obtained from the ETH parameterisation.…”