A Dispersive Treatment of $K_{\ell4}$ Decays

TL;DR

This paper develops a dispersive approach to analyze $K_{ ext{ell}4}$ decays, enabling precise extraction of low-energy constants and understanding rescattering effects beyond standard chiral perturbation theory.

Contribution

It introduces a dispersive framework for $K_{ ext{ell}4}$ decays that resums rescattering effects and fits experimental data to determine low-energy constants of ChPT.

Findings

Resummation of $ ext{pi} ext{pi}$- and $ ext{K} ext{pi}$-rescattering effects.

Determination of LECs $L_1^r$, $L_2^r$, $L_3^r$, and $L_9^r$ from data.

Description of form factor curvature as a rescattering effect beyond NNLO.

Abstract

$K_{\ell4}$ decays offer several reasons of interest: they allow an accurate measurement of $\pi\pi$-scattering lengths; they provide the best source for the determination of some low-energy constants of ChPT; one form factor is directly related to the chiral anomaly, which can be measured here. We present a dispersive treatment of $K_{\ell4}$ decays that provides a resummation of $\pi\pi$- and $K\pi$-rescattering effects. The free parameters of the dispersion relation are fitted to the data of the high-statistics experiments E865 and NA48/2. The matching to ChPT at NLO and NNLO enables us to determine the LECs $L_1^r$, $L_2^r$ and $L_3^r$. With recently published data from NA48/2, the LEC $L_9^r$ can be determined as well. In contrast to a pure chiral treatment, the dispersion relation describes the observed curvature of one of the form factors, which we understand as a rescattering effect beyond NNLO.