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

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)(mu), 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 gamma*gamma* -> pi pi . 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 (mu) (pi-box) = -15: 9(2) x 10(-11). As an application of the partial- wave formalism, we present a first calculation of pi pi-rescattering effects in HLbL scattering, with gamma*gamma* -> pi pi helicity partial waves constructed dispersively using pi pi 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) (mu). 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(mu)(pi-box) + a(mu,J=0)(pi pi,pi-pole LHC)= -24(1) x 10(-11)