Low-Energy Pion-Nucleon Scattering from Current Commutation Relations and Partial Conservation of the Axial-Vector Current

Abstract

Starting with current commutation relations and the hypothesis of partial conservation of the axial-vector current, relations are obtained for the pion-nucleon scattering amplitude considered to the second order in the energy variable $\nu$. The structure-dependent contributions to the second-order terms in the matrix element involving the axial-vector currents are approximated by the direct and exchanged poles and resonances, particularly the $N^{*}$ resonances with spins ½ and $\frac{3}{2}$. The contribution of the $\rho$ meson to this matrix element is estimated to be small. Expressions are obtained for the $P$-wave scattering lengths and the $S$-wave and $P$-wave phase shifts at low energies. Numerical estimates are made for these and are compared with the available data; it is found that resonable agreement with experiment can be obtained except for the $P$-wave scattering length ${a_{1-}}^{\mathrm{}(-)\mathrm{}}$ and the partial-wave amplitude ${f_{1-}}^{\mathrm{}(-)\mathrm{}}$. Some questions relating to the off-mass-shell extrapolation are discussed. We have also examined the validity of the assumption made, in the usual derivation of the relations for the $S$-wave scattering lengths, that the terms of higher order in $\nu$ may be neglected. We suggest that at least for the isospin-symmetric scattering length this is not a good approximation, so that this scattering length cannot be obtained adequately by using only the current commutation relations and the hypothesis of a partially conserved axial-vector current. The scalar term arising from the commutator [${\mathcal{a}}^0\mathrm{}\left(x\right)\mathrm{}$, $\phi \mathrm{}\left(y\right)\mathrm{}$] is estimated; we suggest that the contribution of this term is important and cannot be ignored.