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The enhancement mechanism due to the resonant properties of the $\rho$ and$\omega$ mesons, which are close in mass, are analysed when such resonancescarry different momenta. Considerations from the particular process where theyappear to the individual resonant features are at play for the appearance ofthe 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 agiven angle of emission of one of the pions, to tune the individual features ofthe two resonances and exhibit how both resonances combine to produce theenhancement. Then, we incorporate the $\rho^\prime$ using the informationobtained from the $ e^+e^- \to \pi^0 \pi^0 \gamma$ total scattering process andshow that, it becomes important thanks to the same enhancement mechanismbetween the $\rho$ and the $\omega$. In a second step, we use a similarapproach to describe a model dependent contribution to the $\tau^- \to\pi^-\pi^0\nu_\tau\gamma$ decay, the so-called $\omega$ channel. We show thatthe dipion invariant mass distribution at particular angles is sensitive to theindividual resonant states. We compute the interference of this channel withthe known dominant model independent contribution, and show how a betterknowledge of the $e^+e^- \to \pi^0 \pi^0 \gamma$ process can help to properlyaccount for such model dependent effects. The implication on the isospinsymmetry breaking correction to tau-based estimates of the muon magnetic dipolemoment is assessed.

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