Higgs Sector and Flavour Structure in an Algebraic Three-Generation Model with S3 Family Symmetry

TL;DR

This paper develops an algebraic model incorporating the Higgs sector with S3 family symmetry, resulting in multiple Higgs doublets and a natural separation of Yukawa interactions, while maintaining flavor structure at tree level.

Contribution

It introduces a novel algebraic construction of Higgs components within an S3-symmetric framework, producing multiple Higgs doublets and preserving Type-II Yukawa structure.

Findings

Constructed Higgs components as right-action operators with correct electroweak quantum numbers.

Produced six Higgs doublets organized into S3-orbits with preserved Type-II Yukawa structure.

Tree-level flavor-changing neutral currents are not expected in the exact S3-invariant limit.

Abstract

We extend our previous algebraic construction of three fermion generations in the complex Clifford algebra $\mathbb{C}\ell(10)$ by incorporating the Higgs sector. Using the $S_3$ family symmetry that permutes three algebraically distinguished fermion sectors, we construct Higgs components as right-action operators and extract the corresponding Yukawa coefficients by means of a trace pairing. This yields two first-generation Higgs doublets with the correct electroweak quantum numbers and a natural Type-II-like separation between down-type and up-type Yukawa channels. The $S_3$ action triplicates this Higgs sector, producing six Higgs doublets organised into $S_3$-orbits. In the exact $S_3$-invariant limit, the Yukawa sector retains its Type-II structure, while the generation-space Yukawa matrices are not diagonal in the algebraic generation basis. If electroweak symmetry breaking is implemented in the usual way, tree-level flavour-changing neutral currents are not expected in this limit.