Unitarity in axion coupling to photons I have been investigating about axion coupling to EM field, and I have found that it is usually described through the interaction Lagrangian:
$$ \mathcal{L} = \frac{1}{4 M } a \tilde{F}_{\mu \nu} F^{\mu \nu}.$$
What confuses me is the fact that the dimensionality of the product of the fields is [E$^5$]. I have learnt that having a dimensionality greater than [E$^{4}$] causes unitarity issues (although I might be biased by what I learnt from Fermi theory). Is this not the case for axion-photon interaction?
 A: What having operators with mass dimension greater than four causes is renormalizability problems.  A theory with something like the axion-photon coupling in the question is not renormalizable, but it can be interpreted as an effective quantum field theory—which means that we impose a cutoff for loop corrections.  In a theory without gauge symmetry, this would mean cutting off all loop integrals at some momentum $\Lambda\sim M$, while in a gauge theory like axion QED, the cutoff has to be implemented in a gauge-invariant way, which is a little more subtle but still doable.  All quantum corrections are then calculated using finite loop integrals, and so long as all physical momenta are small compared with $M$, the theory gives orderly predictions.
This has become the standard approach in quantum theory in the past few decades, and it has applications everywhere, including in the standard model itself.  The Majorana mass for light neutrinos that is kept small via the seesaw mechanism appears as a dimension-five operator, suppressed by a large $M$.
