Axion scales and couplings with St\xfcckelberg mixing

Abstract

We study the axion field range and low energy couplings in models with Stückelberg mixing between axions and U(1) gauge bosons. It is noted that the gauge- invariant axion combination ξ in the model is periodic modulo an appropriate shift of gauge-variant axions eaten by the massive U(1) gauge bosons, which in some cases makes the connection between the field range and the low energy couplings less transparent. We derive the field range of ξ for generic forms of the axion kinetic metric and U(1) charges, and identify the field basis for which all non-derivative couplings of ξ are quantized in a manner manifestly consistent with the periodicity of ξ. Generically Stückelberg mixing reduces the axion field range. In particular, the mixings between N axions and (N − 1) U(1) gauge bosons typically result in an exponentially reduced field range {M}_{xi }=mathcal{O}left({k}^{-left(N-1right)}f/sqrt{N!}right) for the residual gauge-invariant axion ξ in the limit N ≫ 1, where f and k denote the typical decay constant and the root mean square of the U(1) gauge charges of the original N axions. Using simple examples, we study also the reparameterization-invariant physical quantities such as the axion effective potential and 1PI couplings to gauge bosons, which are determined by the reparameterization-dependent axion couplings in the model.