Two-loop representations of low-energy pion form factors and pi-pi scattering phases in the presence of isospin breaking

TL;DR

This paper develops two-loop dispersive representations of low-energy pion form factors and pi-pi scattering phases, incorporating isospin-breaking effects due to pion mass differences, providing analytical expressions and studying their parameter dependence.

Contribution

It introduces new two-loop dispersive formulas for pion form factors and scattering phases that include isospin-breaking effects, extending previous isospin-symmetric analyses.

Findings

Derived analytical expressions for pion form factors and scattering amplitudes with isospin breaking.

Identified universal and form-factor dependent parts of the scattering phases.

Analyzed the impact of isospin-breaking parameters on the phases and form factors.

Abstract

Dispersive representations of the pi-pi scattering amplitudes and pion form factors, valid at two-loop accuracy in the low-energy expansion, are constructed in the presence of isospin-breaking effects induced by the difference between the charged and neutral pion masses. Analytical expressions for the corresponding phases of the scalar and vector pion form factors are computed. It is shown that each of these phases consists of the sum of a "universal" part and a form-factor dependent contribution. The first one is entirely determined in terms of the pi-pi scattering amplitudes alone, and reduces to the phase satisfying Watson's theorem in the isospin limit. The second one can be sizeable, although it vanishes in the same limit. The dependence of these isospin corrections with respect to the parameters of the subthreshold expansion of the pi-pi amplitude is studied, and an equivalent representation in terms of the S-wave scattering lengths is also briefly presented and discussed. In addition, partially analytical expressions for the two-loop form factors and pi-pi scattering amplitudes in the presence of isospin breaking are provided.