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

Abstract

Isospin-breaking (IB) effects are required for an evaluation of hadronic vacuum polarization at subpercent precision. While the dominant contributions arise from the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ channel, also IB in the subleading channels can become relevant for a detailed understanding, e.g., of the comparison to lattice QCD. Here, we provide such an analysis for $e^{+}{e}^{-}\to 3\pi$ by extending our dispersive description of the process, including estimates of final-state radiation (FSR) and $\rho$-$\omega$ mixing. In particular, we develop a formalism to capture the leading infrared-enhanced effects in terms of a correction factor ${\eta }_{3\pi }$ that generalizes the analog treatment of virtual and final-state photons in the $2\pi$ case. The global fit to the $e^{+}{e}^{-}\to 3\pi$ data base, subject to constraints from analyticity, unitarity, and the chiral anomaly, gives $a_{\mu }^{3\pi }{|}_{\le 1.8\text{GeV}}=45.91(53)\times 10^{-10}$ for the total $3\pi$ contribution to the anomalous magnetic moment of the muon, of which $a_{\mu }^{\text{FSR}}[3\pi ]=0.51(1)\times 10^{-10}$ and $a_{\mu }^{\rho \text{-}\omega }[3\pi ]=-2.68(70)\times 10^{-10}$ can be ascribed to IB. We argue that the resulting cancellation with $\rho$-$\omega$ mixing in $e^{+}{e}^{-}\to 2\pi$ can be understood from a narrow-resonance picture, and provide updated values for the vacuum-polarization-subtracted vector-meson parameters $M_{\omega }=782.70(3)\text{MeV}$, $M_{\varphi }=1019.21(2)\text{MeV}$, ${\Gamma }_{\omega }=8.71(3)\text{MeV}$, and ${\Gamma }_{\varphi }=4.27(1)\text{MeV}$.

Comments: 29 pages, 4 figures, result for ${\bar{\eta }}_{3\pi }$ included as supplementary material; journal version