Current-Algebra Determination of Low-Energy Pion-Nucleon Scattering

Abstract

The $P$-wave pion-nucleon scattering lengths and $S$-wave effective ranges are studied by means of the current-commutator algebra and the hypothesis of partially conserved axial-vector current (PCAC). This requires a model for the weak axial-vector production; but it is shown that the production of baryon resonances with $J={\frac{3}{2}}^{-}$ and $J>\mathrm{}\frac{3}{2}$ do not contribute to these threshold quantities, so that only the nucleon and $N^{*}$ contributions are included in the model for the weak amplitude. The predicted $P$-wave scattering lengths are in reasonable agreement with experiment, while a discrepancy for the isotopic even effective range implies a possible $T=0\mathrm{}$ exchange in the weak amplitude. The other $S$-wave effective range gives a determination of the induced pseudoscalar coupling which is a factor 2 smaller than the PCAC value.