Determination of the pion-nucleon coupling constant and scattering lengths

We critically evaluate the isovector Goldberger-Miyazawa-Oehme (GMO) sum rule for forward $\pi N$ scattering using the recent precision measurements of ${\pi }^{-}p$ and ${\pi }^{-}d$ scattering lengths from pionic atoms. We deduce the charged-pion-nucleon coupling constant, with careful attention to systematic and statistical uncertainties. This determination gives, directly from data, $g_c^2\mathrm{}\left(\mathrm{GMO}\right)/4\mathrm{}\mathrm{\pi }=14.11\mathrm{}\pm 0.05\left(\mathrm{statistical}\right)\mathrm{}\pm 0.19\left(\mathrm{systematic}\right)\mathrm{}$ or $f_c^2/4\mathrm{}\pi =0.0783\mathrm{}\left(11\right).$ This value is intermediate between that of indirect methods and the direct determination from backward np differential scattering cross sections. We also use the pionic atom data to deduce the coherent symmetric and antisymmetric sums of the pion-proton and pion-neutron scattering lengths with high precision, namely, ${(a}_{{\pi }^{-}\mathrm{p}}{+a}_{{\pi }^{-}n})/2=[-12\pm 2\left(\mathrm{statistical}\right)\pm 8\left(\mathrm{systematic}\right)]\times {10}^{-4}{m}_{\pi }^{-1}$ and ${(a}_{{\pi }^{-}\mathrm{p}}-a_{{\pi }^{-}n})/2=[895\pm 3\left(\mathrm{statistical}\right)\pm 13$ $\left(\mathrm{systematic}\right)]\times {10}^{-4}{m}_{\pi }^{-1}.$ For the need of the present analysis, we improve the theoretical description of the pion-deuteron scattering length.