### abstract

- We extend earlier constructions of the effective action for neutrino-photon scattering, using the connection between low-energy neutrino-photon and photon-photon scattering together with the known effective Lagrangian describing low-energy photon scattering in QED. We use this effective action to calculate analytic expressions for the low-energy cross section for the (unpolarised) processes $\nu \nubar \to 5 \gamma$, $\nu \gamma \to \nu + 4 \gamma$ and $\gamma \gamma \to \nu \nubar + 3 \gamma$. As a byproduct we derive compact expressions for the $N$-body phase-space integrals for massless particles, including those having non-trivial tensor-structure.