Dispersive evaluation of the second-class amplitude $\tau \to \eta \pi {\nu }_{\tau }$ in the standard model

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 $\eta \to 3\pi$ Dalitz plot.