Radiative corrections in $K_{e4}$ decay

TL;DR

This paper calculates radiative corrections in $K_{e4}$ decay to accurately extract pion-pion scattering lengths from experimental data, considering leading and next-to-leading logarithmic effects.

Contribution

It provides the first detailed expressions for radiative corrections in $K_{e4}$ decay using the lowest order and factorization hypothesis, including the dependence on lepton mass.

Findings

Decay width depends on electron mass via a standard Drell-Yan form.

Radiative corrections are expressed in leading and next-to-leading logarithmic approximations.

Numerical estimates of the corrections are provided.

Abstract

The final state interaction of pions in the decay $K^\pm\to \pi^+\pi^-e^\pm\nu$ allows to obtain the value of the isospin and angular momentum zero pion-pion scattering length $a_0^0$. To extract this quantity from experimental data the radiative corrections (RC) have to be taken into account. Basing on the lowest order results and the factorization hypothesis, we get the expressions for RC in the leading and next-to leading logarithmical approximation. It is shown that the decay width dependence on the lepton mass $m_e$ through the parameter $\sigma=\frac{\alpha}{2\pi}\br{\ln\frac{M^2}{m_e^2}-1}$ has a standard form of the Drell-Yan process and is proportional to the Sommerfeld-Sakharov factor. The numerical estimations are presented.