As I understand it, the axion $a$ originates from the spontaenous symmetry breaking of $U(1)_{PQ}$. This symmetry being anomalous, and because of the QCD vacuum structure, a non vanishing term like $\frac{a}{f_a}Tr( G \tilde{G})$ is included in the Lagrangian, where $G$ is the gluon field strenght. This determines the axion couplings to gluons. Talking about a coupling to photons would mean to consider a term like $\frac{a}{f_a} F \tilde{F}$, where $F$ is the QED field strenght. I thought a term like $ F \tilde{F}$ could be expressed as a vanishing total derivative, unlike $Tr( G \tilde{G})$, so why are we talking about axion couplings to photons ?
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