Virtual photons in SU(2) chiral perturbation theory and electromagnetic corrections to $ππ$ scattering

We construct the generating functional of two flavor chiral perturbation theory including the effects of virtual photons in the one loop approximation. As an application, we calculate the electromagnetic corrections to the elastic $\pi\pi$ scattering amplitude, in particular to its S--wave threshold parameters. Numerical estimates are given for the reaction $\pi^0 \pi^0 \to \pi^0 \pi^0$. These electromagnetic effects are found to be much smaller than the hadronic two-loop corrections, with the exception of the effective range $b_0^0$, which increases by 15% due to the unitary cusp.