Low-energy parameterisations of the pion-nucleon phase shifts

Abstract

Compared in this work are a few sets of results, obtained from the pion-nucleon (πN ) 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 π ± p → π ± p, and to those of the π + p reaction and of the π − p charge-exchange reaction π − p → π 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 πN 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 πN observables, i.e., for the differential cross section and for the analysing power.

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

“…• In the separate fits, measurements of only one πN reaction comprise the input DB. Such fits may be performed using the phenomenological/empirical models of Section 3.1 [32,33,34,35,36]. Due to the sizeable correlations among the model parameters, separate fits of the ETH model to the πN data have not been attempted for a long time.…”

“…As aforementioned, the history of the development of the hadronic model, which is the backbone of the ETH πN project, may be found online [1]. • Several phenomenological/empirical models have appeared, which are based on hadronic potentials [32], on the polynomial parameterisation of the sand p-wave K-matrix elements [33,34], on the parameterisation of the sand p-wave phase shifts [35,36], etc.…”

“…In this respect, of interest in this work are the models of Refs. [1,32,33,34,38]. It is interesting to examine the similarities of and the differences between the aforementioned models in the chronological order in which the first analyses, involving these models, appeared.…”

“…This is due to two reasons: first, the method has been firmly established as the means of identification of the outliers which the low-energy πN DB contains; second, it provides a modelindependent way of analysing the measurements, one which is devoid of theoretical constraints. There are two issues in analyses using similar parameterisation: (1) the number of terms which one retains from the power series (in a suitable variable, e.g., in the pion CM kinetic energy ǫ or, more frequently, in the square of the CM 3-momentum in the initial state [34]), and (2) the forms which one uses in the modelling of the resonant contributions.…”

“…Therefore, the identification of datapoints as outliers in the fits cannot be attributed to the inadequacy of the employed parametric forms to account for the energy dependence of the πN phase shifts; consequently, the outliers are suggestive of experimental shortcomings. In the data analysis, terms up to O(ǫ 2 ) (or up to O(q 4 ) in the parameterisation of the K-matrix elements in powers of q 2 , where q := | q |) are retained [34]. Owing to the current uncertainties of the measurements, the coefficients of higher orders cannot be determined reliably from the data.…”