One-channel Roy equations revisited

TL;DR

This paper revisits the single-channel Roy equations, analyzing solution neighborhoods, explicit amplitude expressions, and the role of analyticity in ensuring solution uniqueness and removing unphysical singularities.

Contribution

It provides explicit solutions and clarifies the conditions for uniqueness of the Roy equation solutions by incorporating analyticity properties.

Findings

Explicit amplitude solutions with free parameters

Unphysical singularity at the interval's upper end

Analyticity ensures solution uniqueness

Abstract

The Roy equation in the single channel case is a nonlinear, singular integral equation for the phase shift in the low-energy region. We first investigate the infinitesimal neighborhood of a given solution, and then present explicit expressions for amplitudes that satisfy the nonlinear equation exactly. These amplitudes contain free parameters that render the non-uniqueness of the solution manifest. They display, however, an unphysical singularity at the upper end of the interval considered. This singularity disappears and uniqueness is achieved if one uses analyticity properties of the amplitudes that are not encoded in the Roy equation.