Dispersive evaluation of the second-class amplitude $\tau\to\eta\pi\nu_\tau$ in the standard model

TL;DR

This paper reevaluates the form factors for the second-class $ au$ decay into $ au o ext{eta} ext{pi} u_ au$, using dispersive methods, updated chiral inputs, and scattering amplitudes to analyze the $ ho$ resonance shape as a background-free signature.

Contribution

It provides a dispersive analysis of the $ au o ext{eta} ext{pi} u_ au$ decay form factors, incorporating recent data and solving Khuri-Treiman equations for the scattering amplitude.

Findings

Detailed shape of the $ ho$ resonance peak analyzed.

Updated scattering amplitude from Khuri-Treiman equations.

Enhanced understanding of second-class current signatures.

Abstract

We reevaluate the two form factors relevant for the $\eta\pi$ second-class $\tau$ decay mode, making systematic use of analyticity, unitarity, combined with updated inputs to the NLO chiral constraints. We focus, in particular, on the shape of the $\rho$ resonance peak which is a background-free signature of a second-class current. Its dispersive construction requires the $\eta\pi\to\pi\pi$ scattering amplitude which we derive from a family of Khuri-Treiman equations solutions constrained with accurate recent results on the $\eta\to3\pi$ Dalitz plot.