Toward a Special \textbf{$E_6\to G(2) \times SU(3)_A$} Embedding for Standard Model and Dark Matter and an $E_7$ Completion Proposal

Abstract

We developed a unified framework based on a special (non-regular) embedding of the exceptional group $E_6$ in which the main stage of symmetry breaking chain realizes $E_6\to G(2) \times SU(3)_A$. The exceptional factor $G(2)$ plays the role of a hidden strong sector, while $SU(3)_A$ acts as an ancestor of the electroweak gauge group. A minimal scalar sector is organized around a \(\mathbf{650}\)-based \(E_6\) Higgs sector. Its components $(\mathbf7,\mathbf1)$ and $(\mathbf1,\mathbf8)$ implement the subsequent breaking steps $G(2)\to SU(3)_C$ and $SU(3)_A\to SU(2)_L\times U(1)_A$. The \emph{speciality} of this symmetry breaking establishes the feature of \textit{darkness}. Defining an hypercharge from the $t_{8}$ generator of $SU(3)_A$ is not sufficient to recover the exact Standard Model hypercharges, leading to the necessity of an $E_7$ uplift which introduces a proper additional $U(1)_X$ factor. The special embedding naturally suppresses tree-level leptoquark couplings that typically mediate proton decay in regular GUTs. The scalar potential for the Higgs sectors has been constructed, deriving the heavy gauge-bosons spectrum and presenting a consistent one-loop running of the gauge couplings across the intermediate scales, which is shown to satisfy an $E_6$ unification. The exotic states are organized in \(E_7\)-derived vectorlike pairs and are made ultraheavy. The $G(2)$ gluons ensemble confines into heavy dark glueballs with parametrically suppressed communication with $SU(3)_A$ and $U(1)_X$ sectors. Cosmological history is analyzed, including topological defects, inflation and reheating, demonstrating that monopole relics are naturally diluted. The resulting framework provides a minimal and internally consistent exceptional apparatus which includes the Standard Model and a dark matter sector which is secluded by the group-theoretic orthogonality.