Aspects of the ETH model of the pion–nucleon interaction

Abstract

The ETH model of the pion-nucleon (πN ) interaction contains t-channel σ-and ρ-exchange graphs, as well as the s-and u-channel contributions with the wellestablished s and p baryon states with masses below 2 GeV as virtual particles; the model amplitudes obey crossing symmetry and isospin invariance. In the present work, we give the analytical expressions for the model contributions to the Kmatrix elements up to (and including) the f waves. We also extract a new phaseshift solution after performing a partial-wave analysis of meson-factory π ± p elasticscattering data below 100 MeV; included in our results now are also the effects of the variation of the σ-meson mass in the interval which is currently recommended by the Particle-Data Group. Finally, we revisit the subject of the πN Σ term and, using the model amplitudes, obtain Σ = 72.4 ± 3.1 MeV. Our prediction agrees well with the result extracted with Olsson's method, after a few flaws in his paper [Phys. Lett. B 482 (2000) 50] were corrected.1 The first evidence for the violation of the isospin invariance in the hadronic part of the πN interaction was presented in Ref.[8] and appeared two years prior to Ref.[9]; the two research programmes were independent.

Roy–Steiner-equation analysis of pion–nucleon scattering

“…In more detail, the discrepancy could be attributed to about equal parts to the πN coupling constant, a + 0+ , and the dispersive integrals. 14 A similar picture emerged from the subthreshold parameters extracted with dispersive techniques from PWAs in [213][214][215][297][298][299], where the input from the GWU group [296,[300][301][302] consistently produced a larger σ-term than the PWAs from [76][77][78] (see also [303][304][305][306][307] in this context). Similarly, extractions of the σ-term from ChPT [20,308] tend to reproduce the value corresponding to the PWA that is used to determine the LECs in the physical region.…”

Systematic effects in the low-energy behavior of the current SAID solution for the pion-nucleon system

“…(3) has been obtained from Ref. [6], see K 1+ in Eqs. (39) and the corresponding K 3/2 1+ element (after the isospin decomposition of K 1+ is taken into account), as well as footnote 10 therein.…”

“…At the second step of our approach, the ETH model of the πN interaction (a complete description of this model and details on its development may be found in Ref. [6]) is fitted to the truncated DBs of the two ES reactions, thus leading to the determination of the values of the model parameters (coupling constants and vertex factors), which account optimally for the measurements of these two reactions. At a third step, we investigate the violation of the isospin invariance in the πN interaction, by comparing the model predictions for the CX reaction (obtained from the fitted values, as well as the correlation matrix of the fit to the two ES DBs) to the measurements of the CX reaction.…”

“…This is one important difference between our approach and theirs. The second difference relates to the procedure: we choose to analyze the two ES processes (this analysis fixes the I = 3/2 and I = 1/2 amplitudes) and subsequently investigate whether isospin invariance is fulfilled in the πN interaction, on the basis of comparisons with the measurements of the third reaction (triangle identity 6 ). On the contrary, the SAID group assume the fulfillment of the isospin invariance in the πN interaction at all energies, and analyze the three πN reactions in a joint optimization scheme.…”

“…Both approaches rely on the correctness of the bulk of the measurements (ours is significantly more sensitive to the presence of outliers in the DB) and both approaches rely on the correctness of the EM corrections, applied to the hadronic phase shifts and to the partial-wave amplitudes on the way to fitting the observables. 6 Assuming that the isospin invariance holds in the πN interaction, only two (of course, complex) amplitudes enter the physical description of the three πN reactions: the I = 3/2 amplitude (f 3 ) and the I = 1/2 amplitude (f 1 ). Disregarding the distortions which are due to the EM effects, the π + p reaction is described by f 3 , the π − p ES reaction by (2f 1 + f 3 )/3, and the CX reaction by √ 2(f 3 − f 1 )/3.…”