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

Abstract

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

18 pages; journal version

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