Isospin-breaking in the $\pi \pi$ scattering amplitude I: Effects due to the pion mass difference

TL;DR

This paper investigates how the mass difference between charged and neutral pions affects the $cetaeta$ scattering amplitude, deriving modified Roy equations and providing numerical solutions within a chiral framework.

Contribution

It introduces the first detailed analysis of isospin-breaking effects in $cetaeta$ scattering due to pion mass differences, modifying Roy equations accordingly.

Findings

Derived modified Roy equations accounting for pion mass difference

Numerically solved the equations within a chiral representation

Provided detailed discussion of numerical results

Abstract

This is the first of a series of papers devoted to a detailed analysis of isospin-breaking effects in the $\pi \pi$ scattering amplitude and the vector form factor of the pion. Isospin breaking originates from the mass difference between up and down quarks and from electromagnetic effects. The latter can be further split into effects due to the pion-mass difference and the remaining virtual and real photonic effects. In this paper, we derive the modifications to the Roy equations for $\pi\pi$ scattering due to the mass difference between the charged and the neutral pion. We solve the equations numerically after matching them to the chiral representation of the $\pi \pi$ scattering amplitude evaluated in the same approximation, which is also provided. Numerical results are presented and discussed in detail.