AnS-matrix theory of the vertex $$\rho \to {\mathcal{N}}\bar {\mathcal{N}}$$ based on the strip approximationbased on the strip approximation

Abstract

The\(\pi \pi \to {\mathcal{N}}{\mathcal{N}}\) amplitudes are studied in the neighborhood of the ρ-meson mass and a simple dynamical calculation based on a consistent use of the strip approximation is proposed to study the vertex\(\rho \to {\mathcal{N}}\bar {\mathcal{N}}\). In particular, the vertex\(\rho \to {\mathcal{N}}\bar {\mathcal{N}}\) is expressed in terms of a « potential » (assumed to get its strongest contributions from low-mass baryon states exchanged in the\(\pi {\mathcal{N}}\) channels) and theJ=1,I=1 ππ phase shifts. We can therefore generate the vertex\(\rho \to {\mathcal{N}}{\mathcal{N}}\) and compare this prediction with experimental information about the nucleon electromagnetic (EM) structure. If there is agreement, this will provide a posteriori justification of the strip approximation in bootstrap calculations in the\(\pi {\mathcal{N}}\) and\({\mathcal{N}}{\mathcal{N}}\) systems. In both cases the vertex\(\rho \to {\mathcal{N}}\bar {\mathcal{N}}\) plays a nontrivial role. The ρ-meson can couple to the\({\mathcal{N}}{\mathcal{N}}\) system in two ways: first ρ can couple to the charge of the nucleon (call this coupling\(\gamma _{\mathcal{N}} \)) and second to the magnetic moment (call this coupling\(\gamma '_{\mathcal{N}}\)). If we denote the coupling of ρ → ππ asγπ the results of this analysis can be summarized as follows:\(\gamma '_{\mathcal{N}} /\gamma _\pi = + (6/\gamma _\pi ^2 ) (2.3)\),\(\gamma _{\mathcal{N}} /\gamma _\pi = + (6/\gamma _\pi ^2 ) (0.85)\),\(\gamma '_{\mathcal{N}} /\gamma _{\mathcal{N}} = + 2.7\). The first two numbers depend on the details of the strip approximation used for ππ scattering. The ratio\(\gamma '_{\mathcal{N}} /\gamma _{\mathcal{N}} \) is relatively independent of these details. It is also possible to calculate\(\gamma '_{\mathcal{N}} /\gamma _\pi , \gamma '_{\mathcal{N}} /\gamma _\pi \) and\(\gamma '_{\mathcal{N}} /\gamma _\pi \) from the nucleon EM structure. The results are\(\gamma '_{\mathcal{N}} /\gamma _\pi = + (6/\gamma _\pi ^2 ) (2.6)\),\(\gamma _{\mathcal{N}} /\gamma _\pi = + (6/\gamma _\pi ^2 ) (0.7)\),\(\gamma '_{\mathcal{N}} /\gamma _{\mathcal{N}} = + 3.7\). Again the first two numbers depend on the details of the strip approximation used for ππ scattering and the third is relatively independent of these details.