Isospin breaking in pion and $K_{e4}$ form factors

TL;DR

This paper develops a two-loop dispersion relation framework to analyze isospin breaking effects in $K_{e4}$ form factors caused by pion mass differences, connecting experimental phase shifts with theoretical scattering lengths.

Contribution

It introduces a novel two-loop dispersion approach incorporating isospin breaking, enabling more precise extraction of $ ext{pi} ext{pi}$ scattering lengths from experimental data.

Findings

Analytical expressions for two-loop form factor phases are provided.

Reanalysis of NA48/2 data with isospin-breaking corrections yields updated scattering lengths.

The method links experimental phase shifts to theoretical scattering parameters more accurately.

Abstract

Isospin breaking in the $K_{\ell 4}$ form factors induced by the difference between charged and neutral pion masses is discussed within a framework built on suitably subtracted dispersion representations. The $K_{\ell 4}$ form factors are constructed in an iterative way up to two loops in the low-energy expansion by implementing analyticity, crossing, and unitarity due to two-meson intermediate states. Analytical expressions for the phases of the two-loop form factors of the $K^\pm\to\pi^+\pi^- e^\pm \nu_e$ channel are presented, allowing one to connect the difference of form-factor phase shifts measured experimentally (out of the isospin limit) and the difference of $S$- and $P$-wave $\pi\pi$ phase shifts studied theoretically (in the isospin limit). The dependence with respect to the two $S$-wave scattering lengths $a_0^0$ and $a_0^2$ in the isospin limit is worked out in a general way, in contrast to previous analyses based on one-loop chiral perturbation theory. The results on the phases of the $K^\pm\to\pi^+\pi^- e^\pm \nu_e$ form factors obtained by the NA48/2 collaboration at the CERN SPS are reanalysed including isospin-breaking correction to extract values for the scattering lengths $a_0^0$ and $a_0^2$.