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

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.