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

We parametrise the space of all possible flavour non-universal u(1) 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 u(1)’s) of the SM with such semi-simple completions. The resulting space of abelian extensions is a collection of planes of dimensions ≤ 6. Numerically, we find that roughly 2.5% of anomaly-free u(1) extensions of the SM with a maximum charge ratio of ±10 can be embedded in such semi-simple gauge theories. Any vector-like anomaly-free abelian extension embeds (at least) inside g = su(12) ⊕ su(2) ⊕ su(2). We also provide a simple computer program that tests whether a given u(1)X1 ⊕ u(1)X2 ⊕.. charge assignment has a semi-simple completion and, if it does, outputs a set of maximal gauge algebras in which the sm ⊕ u(1)X1 ⊕ u(1)X2 ⊕.. model may be embedded. We hope this is a useful tool in pointing the way from sm ⊕ u(1)X1 ⊕ u(1)X2 ⊕.. models, which have many phenomenological uses, to their unified gauge completions in the ultraviolet.