Dispersive evaluation of the second-class amplitude $τ\toηπν_τ$ in the standard model

We reevaluate the two form factors relevant for the $ηπ$ second-class $τ$ 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 $ρ$ resonance peak which is a background-free signature of a second-class current. Its dispersive construction requires the $ηπ\toππ$ scattering amplitude which we derive from a family of Khuri-Treiman equations solutions constrained with accurate recent results on the $η\to3π$ Dalitz plot.

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