A brief history of the pion-nucleon coupling constant

Abstract: This work provides a brief history of determinations of the pion-nucleon ($\pi N$) coupling constant from $\pi N$ and $N N$ data. From robust analyses of twenty reported values of the charged-pion coupling constant, exhibiting sizeable fluctuation, the result $f_c^2 = 762.9^{+6.5}_{-6.2} \cdot 10^{-4}$ is obtained. Similar values are extracted for the other two $\pi N$ coupling constants, $f_0^2$ and $f_p^2$, from fewer data. The average values of the various $\pi N$ coupling constants, extracted in this work, suggest no splitting, in agreement with the thesis of the Nijmegen group. Additional analysis of the $f_c^2$ and $f_0^2$ values, both reported in four studies, turned to be inconclusive: one of these studies suggests that $f_0<f_c$, whereas another slightly favours $f_0>f_c$; no significant splitting effects are observed in the other two studies. The analysis of the low-energy $\pi N$ data with the ETH model indicates significant splitting and, under certain conditions, it implies that $f_0>f_c$. Also discussed in the paper are the electromagnetic corrections, which need to be applied to the strong shift and to the total decay width of the ground state of pionic hydrogen in order that estimates for the hadronic $s$-wave $\pi N$ scattering lengths be obtained; this is a relevant subject as $f_c^2$ may be extracted from the isovector scattering length by use of the Goldberger-Miyazawa-Oehme sum rule. Regarding the removal of the electromagnetic effects in the $\pi N$ system at threshold, my opinion is that Theory must find a way to provide reliable and accurate corrections, matching the level of accuracy of the experimental results.