Axion-electrodynamics: a quantum field calculation

An axion background field induces tiny oscillating electric and magnetic fields in an external static magnetic field. This signature is used to search for axion dark matter. We use standard quantum field theory techniques to obtain an expression for a transition amplitude, from which we identify the classical electromagnetic fields induced by the background axion field. We confirm previous results, that if the spatial size $R$ of the applied static magnetic field is small compared to the axion Compton wavelength $\lambda$, the induced electric and magnetic fields are parametrically suppressed by the small numbers $(R\mathrm{/}\lambda {)}^2$ and $R\mathrm{/}\lambda$, respectively, relative to the case when $R$ is larger than $\lambda$. Our approach allows an intuitive interpretation in terms of 4-momentum conservation and momentum exchange via the photon propagator.