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 $e^+e^-\toπ^+π^-$ 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 $e^+e^-\to 3π$ by extending our dispersive description of the process, including estimates of final-state radiation (FSR) and $ρ$-$ω$ mixing. In particular, we develop a formalism to capture the leading infrared-enhanced effects in terms of a correction factor $η_{3π}$ that generalizes the analog treatment of virtual and final-state photons in the $2π$ case. The global fit to the $e^+e^-\to 3π$ data base, subject to constraints from analyticity, unitarity, and the chiral anomaly, gives $a_μ^{3π}|_{\leq 1.8\,\text{GeV}}=45.91(53)\times 10^{-10}$ for the total $3π$ contribution to the anomalous magnetic moment of the muon, of which $a_μ^\text{FSR}[3π]=0.51(1)\times 10^{-10}$ and $a_μ^{ρ\text{-}ω}[3π]=-2.68(70)\times 10^{-10}$ can be ascribed to IB. We argue that the resulting cancellation with $ρ$-$ω$ mixing in $e^+e^-\to 2π$ can be understood from a narrow-resonance picture, and provide updated values for the vacuum-polarization-subtracted vector-meson parameters $M_ω=782.70(3)\,\text{MeV}$, $M_ϕ=1019.21(2)\,\text{MeV}$, $Γ_ω=8.71(3)\,\text{MeV}$, and $Γ_ϕ=4.27(1)\,\text{MeV}$.