Chiral Lagrangian Calculation of Pion-Nucleon Scattering Lengths

Abstract

We have considered two chiral-invariant Lagrangian models of pion-nucleon scattering: Weinberg's model, which contains a direct pion-nucleon scattering term; and an extension of Schwinger's model in which this direct term is replaced by a $\rho$-mediated term. These models have been extended by including a $\pi N{N}^{*}\mathrm{}\left(1236\right)\mathrm{}$ interaction. This has allowed us to calculate the $N^{*}$ contributions to the $S$-wave pion-nucleon scattering lengths and to compute the $P$-wave scattering lengths. The theoretically obtained values for these scattering lengths are found to be in reasonable agreement with experiment, with the possible exception of the ${a^{-}}_{P\frac{1}{2}}$ scattering length, for which only the direct model gives agreement with the present experimental data. For the ${a_S}^{+}$ scattering length, agreement was obtained only after the introduction of a contact term. The presence of such a term is made more plausible by the study of the asymptotic behavior of the isospin-even amplitudes. We have also compared our analysis with similar current-algebra calculations and obtain over-all agreement, especially with the $\rho$-exchange model, although we differ from these calculations in various details.