The role of crossing symmetry for a low energy pion-nucleon scattering from dispersive method

Abstract

The slow convergence of chiral perturbation theory for heavy baryons (HBChPT) suggests that any attempt to unitarize the amplitude following from this method will fail to describe the experimental phase shifts. However, it was possible to obtain a ChPT pion nucleon amplitudes respecting exact unitarity relation by using the Inverse Amplitude Method (IAM), but the resulting total amplitude violates the important property of crossing symmetry [1] . On the other hand, the use of a dispersive calculation, starting directly from a result at second order in the pion momentum, is an alternative approach to get unitarized scattering amplitude. By this method it was possible to fit, with two parameters, the P33 partial wave to the experimental low energy phase shifts, and to present the resulting S and P partial wave phase shifts [2]. This was done with a crossing symmetric amplitude, that respect approximate elastic unitarity relation. In the present exercise, we do not impose crossing symmetry for the amplitude obtained in the previous work, in order to verify the role played by crossing symmetry in the dispersive approach. As in the previous work, our strategy was to perform a fit of the P33 amplitude to the experimental phase shifts and then use the fixed parameters in the S and P partial waves to compare them with the corresponding experimental phase shifts. We conclude that, when we do not impose crossing symmetry for the total amplitude, more parameters are needed in the fitting procedure for P33, moreover the theoretical results for S11, S31, P11, P31 and P13 are quite far from the experimental points.