Matter-induced vertices for photon splitting in a weakly magnetized plasma

We evaluate the three-photon vertex functions at order $B$ and $B^{2}$ in a weak constant magnetic field at finite temperature and density with on shell external lines. Their application to the study of the photon splitting process leads to consider high energy photons whose dispersion relations are not changed significantly by the plasma effects. The absorption coefficient is computed and compared with the perturbative vacuum result. For the values of temperature and density of some astrophysical objects with a weak magnetic field, the matter effects are negligible.