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 ≤ 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 ±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) ⊕ mathfrak{su} (2)L ⊕ mathfrak{su} (2)R. We also provide a simple computer program that tests whether a given mathfrak{u}{(1)}_{X^1} ⊕ mathfrak{u}{(1)}_{X^2} ⊕ . charge assignment has a semi-simple completion and, if it does, outputs a set of maximal gauge algebras in which the mathfrak{sm} ⊕ mathfrak{u}{(1)}_{X^1} ⊕ mathfrak{u}{(1)}_{X^2} ⊕ . model may be embedded. We hope this is a useful tool in pointing the way from mathfrak{sm} ⊕ mathfrak{u}{(1)}_{X^1} ⊕ mathfrak{u}{(1)}_{X^2} ⊕ . models, which have many phenomenological uses, to their unified gauge completions in the ultraviolet.