| Summary: | We consider the general form of the axion coupling to photons in the axion-Maxwell theory. On general grounds this coupling takes the form of a monodromic function of the axion, which we call g(a), multiplying the Chern-Pontryagin density FF~ of the photon. We show that the non-linearity of g(a) is a spurion for the shift symmetry of the axion. In this context, when g(a) ≠ ℤa, the linearized coupling of the axion g′(a) is not quantized and there is a correlated mass term for the axion. Singularities in g(a) due to the fast rearrangement of degrees of freedom are shown to have corresponding cusps and singularities in the axion potential. We derive the general form of g(a) for the QCD axion, axions with perturbatively broken shift symmetries and axions descending from extra dimensions. In all cases, we show that there is a uniform general form of the monodromic function g(a) and it is connected to the axion potential.
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