Pion-nucleon scattering in chiral perturbation theory I: Isospin-symmetric case

Abstract

We construct the complete effective chiral pion-nucleon Lagrangian to third order in small momenta based on relativistic chiral perturbation theory. We then perform the so-called heavy baryon limit and construct all terms up-to-and-including order $1/m^2$ with fixed and free coefficients. As an application, we discuss in detail pion-nucleon scattering. In particular, we show that the $1/m$ expansion of the Born graphs calculated relativistically can be recovered exactly in the heavy baryon approach without any additional momentum-dependent wave function renormalization. We fit various empirical phase shifts for pion laboratory momenta between 50 and 100 MeV. This leads to a satisfactory description of the phase shifts up to momenta of about 200 MeV. We also predict the threshold parameters, which turn out to be in good agreement with the dispersive analysis. In particular, we can sharpen the prediction for the isovector S-wave scattering length, $0.083 M_\pi^{-1} \leq a_{0+}^- \leq 0.093 M_\pi^{-1}$. We also consider the subthreshold parameters and give a short comparison to other calculations of $\pi N$ scattering in chiral perturbation theory or modifications thereof.