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 − → π + 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 − → 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 − → 3π data base, subject to constraints from analyticity, unitarity, and the chiral anomaly, gives a μ 3 π ≤ 1.8 GeV = 45.91 53 × 10 − 10 $$ {\left.{a}_{\mu}^{3\pi}\right|}_{\le 1.8\ \textrm{GeV}}=45.91(53)\times {10}^{-10} $$ for the total 3π contribution to the anomalous magnetic moment of the muon, of which a μ FSR 3 π = 0.51 1 × 10 − 10 $$ {a}_{\mu}^{\textrm{FSR}}\left[3\pi \right]=0.51(1)\times {10}^{-10} $$ and a μ ρ − ω 3 π = − 2.68 70 × 10 − 10 $$ {a}_{\mu}^{\rho -\omega}\left[3\pi \right]=-2.68(70)\times {10}^{-10} $$ can be ascribed to IB. We argue that the resulting cancellation with ρ–ω mixing in e + e − → 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) MeV, M ϕ = 1019.21(2) MeV, Γ ω = 8.71(3) MeV, and Γ ϕ = 4.27(1) MeV