Unified dispersive approach to real and virtual photon-photon scattering at low energy

Abstract

Previous representations of pion pair production amplitudes by two real photons at low energy, which combine dispersion theoretical constraints with elastic unitarity, chiral symmetry and soft photon constraints are generalized to the case where one photon is virtual. The constructed amplitudes display explicitly the dependence on the ππ phase-shifts, on pion form factors and on pion polarizabilities. They apply both for space-like and time-like virtualities despite the apparent overlap of the left and right-hand cuts, by implementing a definition of resonance exchange amplitudes complying with analyticity and consistent limiting prescriptions for the energy variables. Applications are made to the pion generalized polarizabilies, to vector meson radiative decays, and to the σγ electromagnetic form factor. Finally, an evaluation of the contribution of γππ states in the hadronic vacuum polarization to the muon g − 2 is given, which should be less model dependent than previous estimates.

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

“…In performing this analysis the partial waves for γ * γ * → ππ are treated as known, given quantities, which unfortunately they are not. The lack of experimental information can be partly compensated by theory constraints, in particular by dispersion relations in the form of Roy-Steiner equations [54][55][56][57]. A simplified, S-wave variant of these will be solved in section 4.…”

“…This input will be in the form of helicity partial waves which, in principle, could be measured or, in the absence of data on the doubly-virtual process, have to be reconstructed dispersively [54][55][56][57]. The partial-wave expansion turns, however, the amplitude into a polynomial in the crossed-channel Mandelstam variables, i.e.…”

“…However, approaches that exploit more comprehensively the analytic properties of the amplitude, see [54,55,118] for on-shell photons, can be extended towards the off-shell case with limited data input required to determine parameters, as demonstrated for the singly-virtual process in [56]. The essential features of the generalization towards the doubly-virtual case, i.e.…”

“…The essential features of the generalization towards the doubly-virtual case, i.e. the appearance of anomalous thresholds for time-like kinematics [57] and the modifications to tensor basis and kernel functions [28,31], have already been laid out in previous work, but the practical implementation involves a number of challenges: due to the strong coupling between the ππ/KK channels in the isospin-0 S-wave a single-channel analysis is limited to rather low energies [54,56,117,118], assumptions for the LHC and number of subtractions need to be carefully studied to reliably assess the sensitivity to the high-energy input in the dispersive integrals [55], a full analysis of the generalized Roy-Steiner equations [28,31,55] involves solving coupled S-and D-wave systems of various helicity projections, and last but not least constraints on the γ * γ * → ππ amplitudes from asymptotic behavior and the sum rules derived in section 2.4.3 need to be incorporated. A full analysis along these lines will be left for future work.…”

Dispersion relations for hadronic light-by-light scattering and the muon g – 2

“…We then present a first numerical evaluation of S -wave ππ-rescattering effects, which unitarize the pion-pole contribution to γ * γ * → ππ. This constitutes the first step towards a full treatment of the γ * γ * → ππ partial waves [20][21][22]. In particular, our calculation settles the role of the pion polarizability, which enters at next-to-leading order in the chiral expansion of the HLbL amplitude [23][24][25] and has been suspected to produce sizable corrections in [24].…”