Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization

Isospin-breaking (IB) effects in the two-pion contribution to hadronic vacuum polarization (HVP) can be resonantly enhanced, if related interference of $\rho(770)$ and $\omega(782)$ resonances. This particular IB pion vector form factor thus line shape $e^+e^-\to \pi^+\pi^-$ described by residue at $\omega$ pole - $\rho$-$\omega$ mixing parameter $\epsilon_\omega$. Here, we argue that while general analyticity requires this real, radiative channels $\pi^0\gamma$, $\pi\pi\gamma$, $\eta\gamma$ induce a small phase, whose size estimate as $\delta_\epsilon=3.5(1.0)^\circ$ using narrow-width approximation for intermediate-state mesons. We then perform fits data base study consequences HVP anomalous magnetic moment muon, its part due mixing, mass resonance. find global fit does prefer non-vanishing value $\delta_\epsilon=4.5(1.2)^\circ$, close narrow-resonance expectation, but with large spread among sets, indicating systematic differences region.