The pion nucleon coupling constant and the Goldberger-Treiman relation

TL;DR

This paper re-analyzes pion-nucleon scattering data to precisely determine the pion coupling constant, examines its consistency with the Goldberger-Treiman relation, and investigates the mass and width differences of the Delta resonance charge states.

Contribution

It provides updated values of the pion-nucleon coupling constant and explores the small discrepancy with the Goldberger-Treiman relation, including effects of the pion decay constant and resonance properties.

Findings

Coupling constant g_c^2/4π = 13.74 ± 0.10

Delta mass difference M^0 - M^{++} = 2.0 ± 0.4 MeV

Width difference Γ^0 - Γ^{++} = 3.8 ± 1.0 MeV

Abstract

The latest $\pi N$ elastic scattering data are re-analysed to determine the coupling constant $g_c$ of the charged pion, using the dispersion relation for the invariant amplitude $B^{(+)}$. Depending on the choice of data-base, values $g^2_c/4\pi = 13.80$ to 13.65 are obtained with errors of $\pm 0.12$. We re-examine the well known discrepancy with the Goldberger-Treiman relation. After allowing for the mass dependence of the pion decay constant $f_\pi$, a (2-3)% discrepancy is predicted, hence $g^2_c/4\pi = 13.74 \pm 0.10$ in the prior case. The mass difference between charge states of $\Delta (1232)$ is $M^0 -M^{++} = 2.0 \pm 0.4$ MeV, close to twice the mass difference between neutron and proton. The difference in widths on resonance is $\Gamma ^0 - \Gamma ^{++} =3.8 \pm 1.0$ MeV. One may account for a width difference of 4.5 MeV from phase space for decays and the extra channel $\Delta ^0 \to \gamma n$.