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TLDR: In this article, the available two-loop expressions in the framework of chiral perturbation theory were recalculated in order to check the pion polarizabilities, using the chiral Lagrangian at order p 6.

Abstract: In view of ongoing experimental activities to determine the pion polarizabilities, we have started to recalculate the available two-loop expressions in the framework of chiral perturbation theory, because they have never been checked before. We make use of the chiral Lagrangian at order p 6 now available, and of improved techniques to evaluate the two-loop diagrams. Here, we present the result for the neutral pions. The cross section for the reaction γγ → π 0 π 0 agrees with the earlier calculation within a fraction of a percent. We present analytic results for the dipole and quadrupole polarizabilities, and compare the latter with a recent evaluation from data on γγ → π 0 π 0 .