Three-pion contribution to hadronic vacuum polarization

A bstract We address the contribution of the 3 π channel to hadronic vacuum polarization (HVP) using a dispersive representation of the e + e − → 3 π 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 ∗ → 3 π 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 − → 3 π data sets. Overall, we obtain a_^3 a μ 3 π | ≤1. 8 GeV = 46. 2 (6) (6) × 10 −10 as our best estimate for the total 3 π contribution consistent with all (low-energy) constraints from QCD. In combination with a recent dispersive analysis imposing the same constraints on the 2 π channel below 1 GeV, this covers nearly 80% of the total HVP contribution, leading to a_^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. 4 σ. 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 2 π channel.