G-parity breaking in $\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ decays induced by the $\eta^{(\prime)}\gamma\gamma$ form factor
Breaking of G-parity or new weak (second class) currents can be responsible for $\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ decays. Forthcoming measurements of $\tau$ lepton properties at the Belle II experiment will be able to measure this decay channel for the first time. Isolating new physics contributions from the measured rates will require a careful evaluation of G-parity breaking contributions. Here we evaluate the one-loop contribution to $\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ decays induced by the emission of two virtual photons and its later conversion into an $\eta^{(\prime)}$ meson. As expected, this contribution is very small and may be relevant only for new physics searches contributing at the $10^{-4}$ level to the decay rate of $\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ .