Axial-vector–vector amplitude and neutrino effective charge in a magnetized medium

To one loop the effective neutrino photon interaction takes place through vector-vector-type and axial-vector\char21{}vector\char21{}type amplitudes. In this work we explicitly write down the form of the axial-vector\char21{}vector amplitude to all orders in the external magnetic field in a medium. We then infer its zero external momentum limit which contributes to the effective charge of the neutrinos inside a magnetized medium. We further show its gauge invariance properties.