  We calculate the one-photon loop radiative corrections to the neutral
pion-pair photoproduction process $\pi^-\gamma \to \pi^-\pi^0\pi^0$. At leading
order this reaction is governed by the chiral pion-pion interaction. Since the
chiral $\pi^+\pi^-\to\pi^0\pi^0$ 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 $R\sim \alpha/2\pi$
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 $\lambda$. The
radiative corrections to the total cross section vary between $+2\%$ and $-2\%$
for center-of-mass energies from threshold up to $7m_\pi$. The finite part of
the electromagnetic counterterm gives an additional constant contribution of
about $1\%$, however with a large uncertainty.

High Energy Physics - Phenomenology