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 afaTr(G˜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 afaF˜F, where F is the QED field strenght. I thought a term like F˜F could be expressed as a vanishing total derivative, unlike Tr(G˜G), so why are we talking about axion couplings to photons ?
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