Three-pion contribution to hadronic vacuum polarization

Abstract

Abstract We address the contribution of the 3π channel to hadronic vacuum polarization (HVP) using a dispersive representation of the e + e − → 3π amplitude. This channel gives the second-largest individual contribution to the total HVP integral in the anomalous magnetic moment of the muon (g − 2) μ , both to its absolute value and uncertainty. It is largely dominated by the narrow resonances ω and ϕ, but not to the extent that the off-peak regions were negligible, so that at the level of accuracy relevant for (g − 2) μ an analysis of the available data as model independent as possible becomes critical. Here, we provide such an analysis based on a global fit function using analyticity and unitarity of the underlying γ ∗ → 3π amplitude and its normalization from a chiral low-energy theorem, which, in particular, allows us to check the internal consistency of the various e + e − → 3π data sets. Overall, we obtain $$ {a}_{\mu}^{3\pi } $$ a μ 3 π |≤1.8 GeV = 46.2(6)(6) × 10−10 as our best estimate for the total 3π contribution consistent with all (low-energy) constraints from QCD. In combination with a recent dispersive analysis imposing the same constraints on the 2π channel below 1 GeV, this covers nearly 80% of the total HVP contribution, leading to $$ {a}_{\mu}^{\mathrm{HVP}} $$ a μ HVP = 692.3(3.3) × 10−10 when the remainder is taken from the literature, and thus reaffirming the (g−2) μ anomaly at the level of at least 3.4σ. As side products, we find for the vacuum-polarization-subtracted masses M ω = 782.63(3)(1) MeV and M ϕ = 1019.20(2)(1) MeV, confirming the tension to the ω mass as extracted from the 2π channel.