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Starting from hyperbolic dispersion relations, we present a system ofRoy--Steiner equations for pion Compton scattering that respects analyticityand unitarity requirements, gauge invariance, as well as crossing symmetry, andthus all symmetries of the underlying quantum field theory. To suppress thedependence on the high-energy region, we also consider once- andtwice-subtracted versions of the equations, where the subtraction constants areidentified with dipole and quadrupole pion polarizabilities. We consider theresolution of the $\gamma\gamma\to\pi\pi$ partial waves by aMuskhelishvili-Omn\`es representation with finite matching point, and discussthe consequences for the two-photon coupling of the $\sigma$ resonance as wellas its relation to pion polarizabilities.

Roy-Steiner equations for $\gamma\gamma\to\pi\pi$