SciPost Phys. Proc. 16, 009 (2025) ·
published 14 July 2025
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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.
