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Previous representations of pion pair production amplitudes by two realphotons at low energy, which combine dispersion theoretical constraints withelastic unitariy, chiral symmetry and soft photon constraints are generalizedto the case where one photon is virtual. The constructed amplitudes displayexplicitly the dependence on the $\pi\pi$ phase-shifts, on pion form factorsand on pion polarizabilities. They apply both for space-like and time-likevirtualities despite the apparent overlap of the left and right-hand cuts, byimplementing a definition of resonance exchange amplitudes complying withanalyticity and consistent limiting prescriptions for the energy variables.Applications are made to the pion generalized polarizabilies, to vector mesonradiative decays, and to the $\sigma\gamma$ electromagnetic form factor.Finally, and evaluation of the contribution of $\gamma\pi\pi$ states in thehadronic vacuum polarization to the muon $g-2$ is given, which should be lessmodel dependent than previous estimates.

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