Three generations of colored fermions with family symmetry from Cayley–Dickson sedenions

Niels Gresnigt; Liam Gourlay; Abhinav Varma

doi:10.1140/epjc/s10052-023-11923-y

2024 Impact factor 4.8

Particles and Fields

Open Access

Eur. Phys. J. C (2023) 83: 747
https://doi.org/10.1140/epjc/s10052-023-11923-y

Regular Article - Theoretical Physics

Three generations of colored fermions with $S_{3}$ family symmetry from Cayley–Dickson sedenions

Niels Gresnigt^a, Liam Gourlay and Abhinav Varma

Department of Physics, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Rd., Dushu Lake Science and Education Innovation District, Suzhou Industrial Park, 215123, Suzhou, People’s Republic of China

^a niels.gresnigt@xjtlu.edu.cn

Received: 19 June 2023
Accepted: 9 August 2023
Published online: 24 August 2023

Abstract

An algebraic representation of three generations of fermions with $S U {(3)}_{C}$ $$SU(3)_C$$ color symmetry based on the Cayley–Dickson algebra of sedenions $S$ ${\mathbb {S}}$ is constructed. Recent constructions based on division algebras convincingly describe a single generation of leptons and quarks with Standard Model gauge symmetries. Nonetheless, an algebraic origin for the existence of exactly three generations has proven difficult to substantiate. We motivate $S$ $\mathbb {S}$ as a natural algebraic candidate to describe three generations with $S U {(3)}_{C}$ $$SU(3)_C$$ gauge symmetry. We initially represent one generation of leptons and quarks in terms of two minimal left ideals of $C ℓ (6)$ $\mathbb {C}\ell (6)$ , generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the $S_{3}$ $$S_3$$ automorphism of order three, which is an automorphism of $S$ $\mathbb {S}$ but not of $O$ $\mathbb {O}$ , to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner.

© The Author(s) 2023

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