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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>α</a:mi></a:math>, which is severely constrained. We recast these constraints on the QCD axion parameter space. We also discuss how to generalize our finding to axionlike particles, and the resulting constraints.