Neutrino-photon scattering and its crossed processes in a background magnetic field

We study the neutrino-photon processes such as $\gamma\gamma\to \nu\bar{\nu}$ and $\nu\gamma\to \nu\gamma$ in a background magnetic field smaller than the critical magnetic field $B_c\equiv m_e^2/e$. Using Schwinger's proper-time method, we extract leading magnetic-field contributions to the above processes. Our result is valid throughout the kinematic regime where both neutrino and photon energies are significantly smaller than $m_W$. We briefly discuss the astrophysical implications of our result.