$\tau^-\to\eta^{(\prime)}\pi^-\nu_\tau \gamma$ decays as backgrounds in the search for second class currents

Observation of $\tau^-\to\eta^{(\prime)}\pi^-\nu_\tau $ decays at Belle-II would indicate either a manifestation of isospin symmetry breaking or genuine second class currents (SCC) effects. The corresponding radiative $\tau^-\to\eta^{(\prime)}\pi^-\nu_\tau \gamma$ decay channels are not suppressed by $G$-parity considerations and may represent a serious background in searches of SCC in the former. We compute the observables associated to these radiative decays using Resonance Chiral Lagrangians and conclude that vetoing photons with $E_\gamma>100$ MeV should get rid of this background in the Belle-II environment searching for the $\tau^-\to\eta \pi^-\nu_\tau $ channel. Similar considerations hold inconclusive for decays involving the $\eta^\prime$ given the theory uncertainties in the prediction of the $\tau^-\to\eta^\prime\pi^-\nu_\tau$ branching ratio. Still, additional kinematics-based cuts should be able to suppress this background in the $\eta^\prime$ case to a negligible level.