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 $\ddot{I}(770)$ and $\ddot{I}‰(782)$ resonances. This particular IB contribution to the pion vector form factor and thus the line shape in $e^{+}{e}^{-}\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}$ can be described by the residue at the $\ddot{I}‰$ pole - the $\ddot{I}$-$\ddot{I}‰$ mixing parameter $\hat{I}{\mu }_{\ddot{I}}‰$. Here, we argue that while in general analyticity requires this parameter to be real, the radiative channels $\ddot{I}{€}^0{\hat{I}}^3$, $\ddot{I}€\ddot{I}€{\hat{I}}^3$, $\hat{I}\cdot {\hat{I}}^3$ can induce a small phase, whose size we estimate as $\hat{I}{´}_{\hat{I}}\mu =3.5(1.0{)}^{\circ }$ by using a narrow-width approximation for the intermediate-state vector mesons. We then perform fits to the $e^{+}{e}^{-}\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}$ 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 $\ddot{I}$-$\ddot{I}‰$ mixing, and the mass of the $\ddot{I}‰$ resonance. We find that the global fit does prefer a non-vanishing value of $\hat{I}{´}_{\hat{I}}\mu =4.5(1.2{)}^{\circ }$, close to the narrow-resonance expectation, but with a large spread among the data sets, indicating systematic differences in the $\ddot{I}$-$\ddot{I}‰$ region.

18 pages; journal version