Further study of the maximally symmetry breaking patterns in an ${\rm SU}(8)$ theory

TL;DR

This paper explores symmetry breaking patterns in an SU(8) grand unified theory, analyzing their implications for fermion masses, mixing, and gauge coupling unification, and proposing a novel interpretation within affine Lie algebra.

Contribution

It extends previous work by examining alternative symmetry breaking sequences and analyzing two-loop RGEs, proposing gauge coupling unification via affine Lie algebra.

Findings

SWW sequence successfully generates fermion mass hierarchies

Alternative sequences do not achieve gauge coupling unification in field theory

Gauge coupling unification may be better understood through affine Lie algebra

Abstract

An ${\rm SU}(8)$ theory was previously found to be the minimal simple gauge group where all three-generational Standard Model (SM) fermions can be nontrivially embedded. It is maximally broken into a subgroup of ${\rm SU}(8)\to {\cal G}_{441}\equiv {\rm SU}(4)_s \otimes {\rm SU}(4)_W \otimes {\rm U}(1)_{X_0}$ at the grand unified theory scale by the ${\rm SU}(8)$ adjoint Higgs field of $\mathbf{63_H}$. Gauge symmetries in the strong and the weak sectors are extended by one and two ranks, respectively. The sequential strong-weak-weak (SWW) symmetry breaking stages were found to generate the observed hierarchical SM quark/lepton masses as well as the Cabibbo-Kobayashi-Maskawa mixing pattern with the precise flavor identifications~[17, 20]. We further study the possible weak-strong-weak and weak-weak-strong symmetry breaking patterns, and compare with the results that we have obtained by following the SWW sequence. The two-loop renormalization group equations following both patterns are analyzed, where we cannot achieve the gauge coupling unification in the field theory framework. Through these analyses, we suggest the gauge coupling unification to be interpreted in the context of the affine Lie algebra.