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

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 η3π 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 a3π | = 45.91(53) × 10−10 for the total 3π contribution μ ≤1.8GeV to the anomalous magnetic moment of the muon, of which aFSR[3π] = 0.51(1) × 10−10 and μ aρ–ω[3π] = −2.68(70)×10−10 can be ascribed to IB. We argue that the resulting cancellation μ 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 Mω = 782.70(3) MeV, Mφ = 1019.21(2) MeV, Γω = 8.71(3) MeV, and Γφ = 4.27(1) MeV.