Deriving experimental constraints on the scalar form factor in the second-class $τ\toηπν$ mode

Abstract

The rare second-class decay mode of the $τ$ into $ηπν$ 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 $τ$ decay can be related to photon-photon scattering and radiative $ϕ$ decay amplitudes for which precise experimental measurements have been performed by the Belle and KLOE collaborations.