Dispersion Relations for Pion-Proton Scattering

The dispersion relations are used to predict the values at zero kinetic energy, of the derivatives, ∂D_±(0)∂k², of the real parts, D₊(0) and D_-(0), of the forward elastic scattering amplitudes for π⁺ and π^- mesons scattered by protons. The experimental value of ∂D₊(0)∂k² is fairly well known, and, when compared with the predicted value, yields a determination of the coupling constant, f²=0.104±0.014. The predicted value for ∂D_-(0)∂k² disagrees badly with experiment, especially with an f² as large as 0.10.The dispersion relations are modified by introducing an extra energy denominator in such a way as to contain, as the additional constants, the derivatives ∂D_±(0)∂k². This enables us to check the values of ∂D_±(0)∂k² obtained from the usual dispersion relations as well as the assumption that ω^-2T_±(ω) vanishes at infinity. It is found that as long as the agreement with experiment obtained for the π⁺ relation is retained, no appreciable change in the values of ∂D_±(0)∂k² is possible and that the high-energy behavior of T_±(ω), usually assumed, is correct. The predicted value for ∂D_-(0)∂k² strongly suggests a nonzero effective range for α₁ and a relatively large α₁₁.