Periodic axion EFT data and sources

Fix the four-dimensional Einstein-frame scalar EFT conventions of the earlier cosmology EFT datum. A single-axion datum on a stated field patch consists of:

1. A dimensionless angular coordinate $\theta$ with target $\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$. The identification $\theta \sim \theta +2\pi$ is either derived from a specified gauge identification or explicitly adopted as a model input.
2. A positive kinetic coefficient $f^2$ and a declared region on which it is constant, so the local canonical scalar is $\chi =f\theta$ and its kinetic term is $-\frac{1}{2}{g}^{\mu \nu }{\mathrm{\partial }}_{\mu }\chi {\mathrm{\partial }}_{\nu }\chi$. If moduli vary or several axions mix, the full positive-definite kinetic matrix on the joint field patch is supplied; the formula $\chi =f\theta$ is then only a local fixed-background normalization, not a global field redefinition.
3. A specified finite or infinite set of nonperturbative sources indexed by integer charges $n\in \mathbb{Z}\setminus \{0\}$, dimensionless actions $S_n\ge 0$, phases ${\delta }_n\in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$, and scales ${\mathrm{\Lambda }}_n\ge 0$. Their declared periodic potential terms are $V_n(\theta )={\mathrm{\Lambda }}_n^4{e}^{-S_n}[1-\cos (n\theta +{\delta }_n)]\mathrm{.}$ The contribution of this source set is $V_{\mathrm{n}\mathrm{p}}(\theta ):={\sum }_n{V}_n(\theta )$. For an infinite source set, require this series and the termwise series of every derivative in a local angular coordinate to converge uniformly on compact subsets of the stated field patch. Thus $V_{\mathrm{n}\mathrm{p}}$ is smooth, as required of the potential in the earlier scalar EFT datum. A harmonic approximation near a chosen minimum is additional model data and is valid only on its stated field range.
4. If a monodromy branch is used, a separate branch label and its charge/brane data. Such a branch is not identified with a single-valued function on the axion circle and is not one of the periodic terms above.

In natural units $\mathrm{\hslash }=c=1$, $\theta ,n,S_n,{\delta }_n$ are dimensionless, $f$ and ${\mathrm{\Lambda }}_n$ have mass dimension one, $\chi$ has mass dimension one, and $V_n$ has mass dimension four. For multiple axions, the periodicity lattice, kinetic matrix, charge vectors and source actions are all stated in a common basis. No axion mass, instanton action, decay constant, portal, or compactification realization follows from the name axion alone.

Four-dimensional Einstein-frame multi-scalar cosmology EFT data