Deriving experimental constraints on the scalar form factor in the second-class $\tau \to\eta \pi \nu$ mode

TL;DR

This paper develops a method to constrain the scalar form factor in the rare $ au o \eta \pi u$ decay mode using analyticity properties and experimental data, aiding in predicting its branching fraction within the Standard Model.

Contribution

It introduces a systematic approach linking the scalar form factor to experimental measurements of photon-photon scattering and radiative $\phi$ decay, improving theoretical predictions.

Findings

Relates the scalar form factor to photon-photon scattering data.

Provides bounds on the branching fraction of $ au o \eta \\pi u$ decay.

Enhances understanding of second-class decay modes in $ au$ decays.

Abstract

The rare second-class decay mode of the $\tau$ into $\eta\pi\nu$ could be observed for the first time at Belle II. It is important to try to derive a reliable evaluation of the branching fraction and of the energy distribution of this mode within the standard-model. Many predictions exist already in the literature which can differ by one to two orders of magnitude. Here, an approach based on a systematic use of the property of analyticity of form factors and scattering amplitudes in QCD is discussed. In particular, we will show that the scalar form factor in the $\tau$ decay can be related to photon-photon scattering and radiative $\phi$ decay amplitudes for which precise experimental measurements have been performed by the Belle and KLOE collaborations.