Axion scales and couplings with St\"uckelberg mixing

Understand

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.