Dispersion relation for hadronic light-by-light scattering: two-pion contributions

Abstract

In this third paper of a series dedicated to a dispersive treatment of the hadronic light-by-light (HLbL) tensor, we derive a partial-wave formulation for two-pion intermediate states in the HLbL contribution to the anomalous magnetic moment of the muon (g − 2) µ , including a detailed discussion of the unitarity relation for arbitrary partial waves. We show that obtaining a final expression free from unphysical helicity partial waves is a subtle issue, which we thoroughly clarify. As a by-product, we obtain a set of sum rules that could be used to constrain future calculations of γ * γ * → ππ. We validate the formalism extensively using the pion-box contribution, defined by two-pion intermediate states with a pion-pole left-hand cut, and demonstrate how the full known result is reproduced when resumming the partial waves. Using dispersive fits to high-statistics data for the pion vector form factor, we provide an evaluation of the full pion box, a π-box µ = −15.9(2)×10 −11 . As an application of the partial-wave formalism, we present a first calculation of ππ-rescattering effects in HLbL scattering, with γ * γ * → ππ helicity partial waves constructed dispersively using ππ phase shifts derived from the inverse-amplitude method. In this way, the isospin-0 part of our calculation can be interpreted as the contribution of the f 0 (500) to HLbL scattering in (g − 2) µ . We argue that the contribution due to charged-pion rescattering implements corrections related to the corresponding pion polarizability and show that these are moderate. Our final result for the sum of pion-box contribution and its S-wave rescattering corrections reads a π-box µ + a ππ,π-pole LHC µ,J=0 = −24(1) × 10 −11 .

Kaon-box contribution to the anomalous magnetic moment of the muon

“…where again the first error is due to the variation of the model parameter and the second reflects the slightly increased numerical precision as compared to [29]. As a result of the adjustment of the charged pion mass, the central value of our result moves closer to the result of the data-driven dispersive one a π ± −box µ = −15.9 (2) × 10 −11 [24] and agrees very well within error bars. Furthermore we now accurately reproduce their control-calculation with a VMD-type form factor; we obtain a π ± −box-VMD µ = −16.4 (2) × 10 −11 .…”

Revised and improved value of the QED tenth-order electron anomalous magnetic moment

“…It is a very promising method though it requires long and difficult theoretical works [92,93]. The contribution from the two-π intermediate states has been recently determined [94], which is more accurate than that determined using the hadronic models.…”

Consistency of hadronic vacuum polarization between lattice QCD and the R ratio

“…For the HLbL contribution, new analytic approaches [39][40][41][42][43] as well as the first ab-initio lattice QCD calculation [32] building on multi-year methodology development [44][45][46][47][48][49] so far show consistent results and rule out the HLbL contribution as an explanation for the current * christoph.lehner@ur.de † ameyer@quark.phy.bnl.gov tension between theory and experiment. For the HVP contribution, however, tensions exist within lattice QCD calculations [50] as well as between lattice QCD calculations and R-ratio results [27,50].…”