Neutrino Photon Interaction in a Magnetized Medium II

In the presence of a thermal medium or an external electro-magnetic field, neutrinos can interact with photon, mediated by the corresponding charged leptons (real or virtual). The effect of a medium or an electromagnetic field is two-fold - to induce an effective $\nu \gamma$ vertex and to modify the dispersion relations of all the particles involved to render the processes kinematically viable. It has already been noted that in a medium neutrinos acquire an effective charge, which in the standard model of electroweak interaction comes from the vector type vertex of weak interaction. On the other hand in a magnetized plasma, the axial vector part also start contributing to the effective charge of a neutrino. This contribution corresponding to the axial vector part in the interaction Lagrangian is denoted as the axial polarisation tensor. In an earlier paper we explicitly calculated the form of the axial polarisation tensor to all odd orders in external magnetic field. In this note we complete that investigation by computing the same, to all even orders in external magnetic field. We further show its gauge invarience properties. Finally we infer upon the zero external momentum limit of this axial polarisation tensor.