$K_{l4}$ - Decays Beyond One Loop

TL;DR

This work extends CHPT analyses of $K_{l4}$ decays by computing the $R$ form factor at next-to-leading order and employing a dispersive, unitarized approach to estimate higher-order effects on the $S$-wave amplitude. By fitting to $K_{e4}$ data and $\,\pi\pi$ threshold parameters, it extracts the low-energy constants $L_1^r,L_2^r,L_3$ and tests the large-$N_c$ relation $L_2^r\approx 2L_1^r$, finding reasonable agreement with data and improved control over systematic uncertainties. The paper then makes concrete predictions for the slope of the form factor $G$ and for total decay rates across all $K_{l4}$ channels, demonstrating that accurate measurements of leading partial waves could enable precise determinations of CHPT parameters. Overall, the work integrates one-loop CHPT with a dispersive framework to enhance the reliability of low-energy constants and to provide predictive power for kaon semileptonic decays relevant to hadronic scattering and Standard Model tests.

Abstract

The matrix elements for $K\rightarrow ππłν$ decays are described by four form factors $F,G,H$ and $R$. We complete previous calculations by evaluating $R$ at next-to-leading order in the low-energy expansion. We then estimate higher order contributions using dispersion relations and determine the low-energy constants $L_1,L_2$ and $L_3$ from data on $K_{e4}$ decays and on elastic pion scattering. Finally, we present predictions for the slope of the form factor $G$ and for total decay rates.