Puppi-Stanghellini Discrepancy

In order to establish the extent of the disagreement between the pion-nucleon forward-scattering-amplitude dispersion relations and experiment in a statistical sense, the uncertainty in the dispersion integrals and $S$-wave scattering lengths is systematically included in the analysis. To accomplish this, we fit the total cross sections below 335 Mev by a Chew-Low $P$-wave resonance, phenomenologically modified, and calculate the error matrix for the parameters. Our fit to the total cross sections is statistically at least as good as the Anderson parameterization used in previous work. Ignoring forward scattering amplitudes above 220 Mev because of $D$-wave uncertainties, we still find that there is less than a 4½% probability that the published data are compatible with a unique value for the pion-nucleon coupling constant $f^2$, and that no adjustment of the $S$-wave scattering lengths can remove the discrepancy. However, if the ${\pi }^{-}$ forward-scattering amplitudes measured by Ashkin et al. at 150 and 170 Mev are abandoned in favor of the values recently obtained by Kruse and Arnold at 130 and 152 Mev, the probability rises to 47.2% for our parameterization, or 8.6% for the Anderson parameterization.