Axion scales and couplings with St\"uckelberg mixing

We study the axion field range and low energy couplings in models with St\"uckelberg mixing between axions and $U(1)$ gauge bosons. It is noted that the gauge-invariant axion combination $\xi$ 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 $\xi$ for generic forms of the axion kinetic metric and $U(1)$ charges, and identify the field basis for which all non-derivative couplings of $\xi$ are quantized in a manner manifestly consistent with the periodicity of $\xi$. Generically St\"uckelberg 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 $M_\xi ={\cal O}\left({k^{-(N-1)} f}/{\sqrt{N!}}\right)$ for the residual gauge-invariant axion $\xi$ in the limit $N\gg 1$, 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.