Precision determination of the $\pi N$ scattering lengths and the charged $\pi NN$ coupling constant

We critically evaluate the isovector GMO sumrule for the charged $\pi N N$ coupling constant using recent precision data from $\pi ^-$p and $\pi^-$d atoms and with careful attention to systematic errors. From the $\pi ^-$d scattering length we deduce the pion-proton scattering lengths ${1/2}(a_{\pi ^-p}+a_{\pi ^-n})=(-20\pm 6$(statistic)$ \pm 10$ (systematic))~$\cdot 10^{-4}m_{\pi_c}^{-1}$ and ${1/2}(a_{\pi ^-p}-a_{\pi ^-n})=(903 \pm 14)\cdot 10^{-4}m_{\pi_c}^{-1}$. From this a direct evaluation gives $g^2_c(GMO) =14.20\pm 0.07$(statistic)$\pm 0.13$(systematic) or $f^2_c= 0.0786\pm 0.0008$.