Photon Neutrino Scattering in Non-Commutative Space

We extend the non-commutative standard model based on the minimal $SU(3)\times SU(2)\times U(1)$ gauge group to include the interaction of photon with neutrino. We show that, in the gauge invariant manner, only the right handed neutrino can directly couple to the photon. Consequently, we obtain the Feynman rule for the $\gamma \nu \bar{\nu }$-vertex which does not exist in the minimal extension of non-commutative standard model (mNCSM). We calculate the amplitude for $\gamma \nu \to \gamma \nu$ in both the nonminimal non-commutative standard model (nmNCSM) and the extended version of mNCSM. The obtained cross section grows in the center of mass frame, respectively, as $({\theta }_{NC}{)}^2{M}_Z^{-4}{E}^6$ and $({\theta }_{NC}{)}^4{E}^6$ which can exceed the cross section for $\gamma \nu \to \gamma \gamma \nu$ and $\gamma \nu \to \gamma \nu$ in the high energy limit in the commutative space.