Unified dispersive approach to real and virtual photon-photon scattering at low energy
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Previous representations of pion pair production amplitudes by two real photons at low energy, which combine dispersion theoretical constraints with elastic unitariy, 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 $\pi\pi$ 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 $\sigma\gamma$ electromagnetic form factor. Finally, and evaluation of the contribution of $\gamma\pi\pi$ states in the hadronic vacuum polarization to the muon $g-2$ is given, which should be less model dependent than previous estimates.
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