Theory of the low-energy pion-nucleon interaction

A once-subtracted form of the Low equation for the pion-nucleon scattering amplitude is derived, with partial conservation of axial-vector current used to define the amplitude when one pion is off the mass shell. The static approximation is not made and both the seagull terms and the antinucleon contribution ($z$ graphs) are retained. The theory is applied to calculate the $S$-wave amplitudes in the elastic scattering region. Good agreement is found with the phase shift fits to the data when we use $\left|g_{\pi }\right(4\mathrm{}{M}^2\left)\right|=11.69\mathrm{} \mathrm{and} 25.5$ MeV for the $\pi N\sigma$ commutator. The implications of this work for the analysis of low-energy elastic scattering of pions from nuclei are discussed. In particular, we point out how this work establishes the presence of a Laplacian term in the pion-nucleus optical potential with a magnitude that is fixed from the value of the $\sigma$ commutator.