Roy-Steiner equations for pion-nucleon scattering

TL;DR

This paper derives and analyzes Roy-Steiner equations for pion-nucleon scattering based on dispersion relations, providing a framework that respects fundamental symmetries and exploring their solutions and implications for nucleon form factors.

Contribution

The paper formulates Roy-Steiner equations for pion-nucleon scattering from hyperbolic dispersion relations, including kernel functions and unitarity relations, and solves a related Muskhelishvili-Omnès problem.

Findings

Derived Roy-Steiner equations respecting analyticity, unitarity, crossing symmetry.

Solved a Muskhelishvili-Omnès problem for partial waves.

Discussed implications for nucleon electromagnetic form factors.

Abstract

Starting from hyperbolic dispersion relations, we derive a closed system of Roy-Steiner equations for pion-nucleon scattering that respects analyticity, unitarity, and crossing symmetry. We work out analytically all kernel functions and unitarity relations required for the lowest partial waves. In order to suppress the dependence on the high-energy regime we also consider once- and twice-subtracted versions of the equations, where we identify the subtraction constants with subthreshold parameters. Assuming Mandelstam analyticity we determine the maximal range of validity of these equations. As a first step towards the solution of the full system we cast the equations for the $\pi\pi\to\bar NN$ partial waves into the form of a Muskhelishvili-Omn\`es problem with finite matching point, which we solve numerically in the single-channel approximation. We investigate in detail the role of individual contributions to our solutions and discuss some consequences for the spectral functions of the nucleon electromagnetic form factors.