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.
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ArXiv ePrint: 1702.07347
On leave from the University of Vienna (Massimiliano Procura).
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Colangelo, G., Hoferichter, M., Procura, M. et al. Dispersion relation for hadronic light-by-light scattering: two-pion contributions. J. High Energ. Phys. 2017, 161 (2017). https://doi.org/10.1007/JHEP04(2017)161
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DOI: https://doi.org/10.1007/JHEP04(2017)161