Roy-Steiner equations for $\gamma\gamma\to\pi\pi$

TL;DR

This paper develops Roy--Steiner equations for gamma-gamma to pion-pion scattering, incorporating fundamental symmetries and exploring implications for pion polarizabilities and the sigma resonance.

Contribution

It introduces a system of Roy--Steiner equations for gamma-gamma to pion-pion scattering that respects all fundamental symmetries and connects to pion polarizabilities and resonance properties.

Findings

Derived Roy--Steiner equations for gamma-gamma to pion-pion scattering.

Analyzed the impact of subtraction constants on pion polarizabilities.

Discussed the relation between the sigma resonance and two-photon couplings.

Abstract

Starting from hyperbolic dispersion relations, we present a system of Roy--Steiner equations for pion Compton scattering that respects analyticity and unitarity requirements, gauge invariance, as well as crossing symmetry, and thus all symmetries of the underlying quantum field theory. To suppress the dependence on the high-energy region, we also consider once- and twice-subtracted versions of the equations, where the subtraction constants are identified with dipole and quadrupole pion polarizabilities. We consider the resolution of the $\gamma\gamma\to\pi\pi$ partial waves by a Muskhelishvili-Omn\`es representation with finite matching point, and discuss the consequences for the two-photon coupling of the $\sigma$ resonance as well as its relation to pion polarizabilities.