The pion nucleon coupling constant and the Goldberger-Treiman relation
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 $\Delta ^0 \to \gamma n$.