Quadratic Coupling of the Axion to Photons

We show that the QCD axion couples to the electromagnetic kinetic term at one loop. The result is that if axions make up dark matter, they induce temporal variation of the fine structure constant $\alpha$, which is severely constrained. We recast these constraints on the QCD axion parameter space. We also discuss how to generalise our finding to axion-like particles, and the resulting constraints.