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

Abstract

Abstract We parametrise the space of all possible flavour non-universal $$ \mathfrak{u} $$ 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 $$ \mathfrak{u} $$ 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 $$ \mathfrak{u} $$ 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 $$ \mathfrak{g} $$ g = $$ \mathfrak{su} $$ su (12) ⊕ $$ \mathfrak{su} $$ su (2)L ⊕ $$ \mathfrak{su} $$ su (2)R. We also provide a simple computer program that tests whether a given $$ \mathfrak{u}{(1)}_{X^1} $$ u 1 X 1 ⊕ $$ \mathfrak{u}{(1)}_{X^2} $$ 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 $$ \mathfrak{sm} $$ sm ⊕ $$ \mathfrak{u}{(1)}_{X^1} $$ u 1 X 1 ⊕ $$ \mathfrak{u}{(1)}_{X^2} $$ u 1 X 2 ⊕ . . . model may be embedded. We hope this is a useful tool in pointing the way from $$ \mathfrak{sm} $$ sm ⊕ $$ \mathfrak{u}{(1)}_{X^1} $$ u 1 X 1 ⊕ $$ \mathfrak{u}{(1)}_{X^2} $$ u 1 X 2 ⊕ . . . models, which have many phenomenological uses, to their unified gauge completions in the ultraviolet.