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Radiative corrections to neutral pion-pair production

We calculate the one-photon loop radiative corrections to the neutral pion-pair photoproduction process $π^-γ\to π^-π^0π^0$. At leading order this reaction is governed by the chiral pion-pion interaction. Since the chiral $π^+π^-\toπ^0π^0$ contact-vertex depends only on the final-state invariant-mass it factors out of all photon-loop diagrams. We give analytical expressions for the multiplicative correction factor $R\sim α/2π$ arising from eight classes of contributing one-photon loop diagrams. An electromagnetic counterterm has to be included in order to cancel the ultraviolet divergences generated by the photon-loops. Infrared finiteness of the virtual radiative corrections is achieved (in the standard way) by including soft photon radiation below an energy cut-off $λ$. The radiative corrections to the total cross section vary between $+2\%$ and $-2\%$ for center-of-mass energies from threshold up to $7m_π$. The finite part of the electromagnetic counterterm gives an additional constant contribution of about $1\%$, however with a large uncertainty.