Flatland: abelian extensions of the Standard Model with semi-simple completions

Abstract

We parametrise the space of all possible flavour non-universal $\mathfrak{u}(1{)}_X$ extensions of the Standard Model that embed inside anomaly-free semi-simple gauge theories, including up to three right-handed neutrinos. More generally, we parametrise all abelian extensions (i.e.) by any number of $\mathfrak{u}(1)$'s) of the SM with such semi-simple completions. The resulting space of abelian extensions is a collection of planes of dimensions $\le 6$. Numerically, we find that roughly $2.5\%$ of anomaly-free $\mathfrak{u}(1{)}_X$ extensions of the SM with a maximum charge ratio of $\pm 10$ can be embedded in such semi-simple gauge theories. Any vector-like anomaly-free abelian extension embeds (at least) inside $\mathfrak{g}=\mathfrak{su}(12)\oplus \mathfrak{su}(2{)}_L\oplus \mathfrak{su}(2{)}_R$. We also provide a simple computer program that tests whether a given $\mathfrak{u}(1{)}_{X^1}\oplus \mathfrak{u}(1{)}_{X^2}\oplus \dots$ charge assignment has a semi-simple completion and, if it does, outputs a set of maximal gauge algebras in which the $\mathfrak{sm}\oplus \mathfrak{u}(1{)}_{X^1}\oplus \mathfrak{u}(1{)}_{X^2}\oplus \dots$ model may be embedded. We hope this is a useful tool in pointing the way from $\mathfrak{sm}\oplus \mathfrak{u}(1{)}_{X^1}\oplus \mathfrak{u}(1{)}_{X^2}\oplus \dots$ models, which have many phenomenological uses, to their unified gauge completions in the ultraviolet.

42 pages. v2: Matches published version

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