Floccinaucinihilipilification: Semisimple extensions of the Standard Model gauge algebra

TL;DR

This paper classifies all semisimple gauge algebras containing the Standard Model symmetry, revealing possible extensions and unification schemes, especially with multiple generations of matter, and provides a comprehensive catalog for up to three generations.

Contribution

It introduces a systematic method to classify semisimple algebras extending the Standard Model gauge symmetry, including exotic symmetries with multiple generations.

Findings

Only known gauge algebras for one generation: SU(5), SO(10), and SU(4)×SU(2)×SU(2).

Multiple generations allow exotic symmetries mixing flavor, color, and electroweak sectors.

Complete catalog provided for three or fewer generations.

Abstract

We show how one may classify all semisimple algebras containing the $\mathfrak{su}(3)\oplus \mathfrak{su}(2) \oplus \mathfrak{u}(1)$ symmetry of the Standard Model and acting on some given matter sector, enabling theories beyond the Standard Model with unification (partial or total) of symmetries (gauge or global) to be catalogued. With just a single generation of Standard Model fermions plus a singlet neutrino, the only {gauge} symmetries correspond to the well-known algebras $\mathfrak{su}(5),\mathfrak{so}(10),$ and $\mathfrak{su}(4)\oplus \mathfrak{su}(2) \oplus \mathfrak{su}(2)$, but with two or more generations a limited number of exotic symmetries mixing flavour, colour, and electroweak degrees of freedom become possible. We provide a complete catalogue in the case of 3 generations or fewer and outline how our method generalizes to cases with additional matter.