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.