Sizing the double pole resonant enhancement in $e^{+}e^{-} \to \pi^{0} \pi^{0} \gamma$ cross section and $\tau^{-} \to \pi^{-}\pi^{0} \nu_{\tau}\gamma$ decay

The enhancement mechanism due to the resonant properties of the $\rho$ and $\omega$ mesons, which are close in mass, are analysed when such resonances carry different momenta. Considerations from the particular process where they appear to the individual resonant features are at play for the appearance of the global resonant manifestation. In this work, we first consider the $e^+e^- \to \pi^0 \pi^0 \gamma$ process. We use the differential cross section at a given angle of emission of one of the pions, to tune the individual features of the two resonances and exhibit how both resonances combine to produce the enhancement. Then, we incorporate the $\rho^\prime$ using the information obtained from the $ e^+e^- \to \pi^0 \pi^0 \gamma$ total scattering process and show that, it becomes important thanks to the same enhancement mechanism between the $\rho$ and the $\omega$. In a second step, we use a similar approach to describe a model dependent contribution to the $\tau^- \to \pi^-\pi^0\nu_\tau\gamma$ decay, the so-called $\omega$ channel. We show that the dipion invariant mass distribution at particular angles is sensitive to the individual resonant states. We compute the interference of this channel with the known dominant model independent contribution, and show how a better knowledge of the $e^+e^- \to \pi^0 \pi^0 \gamma$ process can help to properly account for such model dependent effects. The implication on the isospin symmetry breaking correction to tau-based estimates of the muon magnetic dipole moment is assessed.