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We reconsider Muskhelishvili-Omn\`es (MO) dispersive representations ofphoton-photon scattering to two pions, motivated by the very high statisticsresults recently released by the Belle collaboration for charged as well asneutral pion pairs and also by recent progress in the determination of thelow-energy $\pi\pi$ scattering amplitude. Applicability of this formalism isextended beyond 1 GeV by taking into account inelasticity due to $K\bar{K}$ . Amodified MO representation is derived which has the advantage that allpolynomial ambiguities are collected into the subtraction constants and havesimple relations to pion polarizabilities. It is obtained by treatingdifferently the exactly known QED Born term and the other components of theleft-hand cut. These components are approximated by a sum over resonances. Allresonances up to spin two and masses up to $\simeq1.3$ GeV are included. Thetensor contributions to the left-hand cut are found to be numericallyimportant. We perform fits to the data imposing chiral constraints, inparticular, using a model independent sum rule result on the $p^6$ chiralcoupling $c_{34}$. Such theoretical constraints are necessary because theexperimental errors are dominantly systematic. Results on further $p^6$couplings and pion dipole and quadrupole polarizabilities are then derived fromthe fit. The relevance of the new data for distinguishing between two possiblescenarios of isospin breaking in the $f_0(980)$ region is discussed.

MO analysis of the high statistics Belle results on $\gamma\gamma\to \pi^+\pi^-,\pi^0\pi^0$ with chiral constraints