Dispersion Relations for Pion-Proton Scattering

The dispersion relations are used to predict the values at zero kinetic energy, of the derivatives, $\frac{\partial D_{\pm }\mathrm{}\left(0\right)\mathrm{}}{\partial k^2}$, of the real parts, $D_{+}\mathrm{}\left(0\right)\mathrm{}$ and $D_{-}\mathrm{}\left(0\right)\mathrm{}$, of the forward elastic scattering amplitudes for ${\pi }^{+}$ and ${\pi }^{-}$ mesons scattered by protons. The experimental value of $\frac{\partial D_{+}\mathrm{}\left(0\right)\mathrm{}}{\partial k^2}$ is fairly well known, and, when compared with the predicted value, yields a determination of the coupling constant, $f^2=0.104\mathrm{}\pm 0.014$. The predicted value for $\frac{\partial D_{-}\mathrm{}\left(0\right)\mathrm{}}{\partial k^2}$ disagrees badly with experiment, especially with an $f^2$ as large as 0.10.