Does the neutrino interact with the photon? I know that the straight answer is no, but in my EFT course, where we're interested in nonrenormalizable operators of the Lagrangian, things aren't so straightforward.
The non-minimal QED Lagrangian is
$$\mathcal{L}=\mathcal{L}_{ren}+\mathcal{L}_{nonren}=\bar{e}(i\hat{D}-m)e-\frac{1}{4}F^2+\frac{1}{M}\bar{\nu}\sigma^{\mu\nu}\nu F_{\mu\nu}$$
does the last term actually exist? If so, is it "naturally" small? If not, why, since it's allowed by symmetries?
To be clear, I'm asking if this interaction can appear in a theory like the one I wrote above (with abelian $F$, so only electron, photon, and neutrino) and if it can happen at the tree level.
 A: Yes, the neutrino may have a magnetic moment at the 1 loop level in vacuum, cf here, e.g.

This is summarized by your unrenormalizable effective (fake tree)
dimension 5 operator, since the loop into which neutrinos can resolve involves charged leptons or gauge bosons.
Highly suppressed by the weak/EM coupling factor (Fujikawa & Schrock, 1980), $m_\nu G_F$.
In practical terms, the neutrino mass is so much smaller than the W mass that the question is "purely academic".$^\natural$    Your dimension 5 term you tacked on needs a dimension -1 coupling in front of it, for dimensional consistency. It is what's provided above.
There is a voluminous literature on the subject, but astrophysicists focus on interesting media beyond vacuum, affording enhancements, like intense magnetic fields, as cited in the comments above.

$^\natural$ You do notice your term requires a right-chiral neutrino. You might be alarmed that the Ws don't couple to it, but the Higgs, in similar diagrams, omitted here, does!
