Pair production of neutrinos I learned that neutrinos have a much lower energy than electrons. Pair production of electrons occurs when the photon energy is above 2 times the energy of an electron. So I am wondering if pair production of neutrinos wouldn't be even more common and occur at much lower energy levels?
 A: Pair production of electron/positron happens in the electric field of the atoms to satisfy conservation laws and the same will be true for off mass shell Z0 going into a neutrino antineutrino pair, interacting with the weak field of the atoms. 
The equivalent to the photon weak interaction mediator is the Z0 and neutrino antineutrino pairs can be formed that way. The weak interaction is orders of magnitude smaller than the electromagnetic one, and thus the probability of getting pairs of neutrinos-antineutrinos is high only in special situations as in the Big Bang or in a Super Nova explosion  where the density of matter is high and there is energy available.
The weak couplings lower the probability of interaction  drastically so the advantage of a  smaller neutrino  mass with respect to the electrons is lost for experiments possible  in the laboratory.
A: There is no tree level vertex for a neutrino scattering off of a photon, so you have two choices:


*

*Weak Drell-Yan if the progenitors are leptons. That is $$l + \bar{l} \to Z^0 \to \nu + \bar{\nu} \,.$$

*A multi-step process involving a weak radiative correction. Say running a $W^\pm$ across the out-going lepton lines in a normal Drell-Yan diagram. Unlike the above case, however this can happen starting with a on-shell photon, though as Anna says it requires a heavy spectator to conserve momentum.
In either case there are two low energy weak vertexes which suppresses the interaction rate by a factor of circa $10^{5}$-$10^{6}$. The process is possible but low probability.
