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A Special $E_6\to G(2) \times SU(3)_A$ Embedding for Standard Model and Dark Matter

Nicolò Masi

Abstract

I developed a grand unified framework based on a special (non-regular) embedding of the exceptional group $E_6$ in which the first 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 a progenitor of the electroweak gauge group. Subsequent breaking steps, $G(2)\to SU(3)_C$ and $SU(3)_A\to SU(2)_L\times U(1)_Y$, recover the Standard Model at low energies while preserving a confining dark gauge sector. Hypercharge is defined from the $t_{8}$ generator of $SU(3)_A$, fixing the electroweak coupling normalization and identifying the Standard Model Higgs doublet within a larger exceptional representation. The special embedding naturally suppresses tree-level leptoquark couplings that typically mediate proton decay in regular grand unified theories. The scalar potential for the three 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 $E_6$ unification. The $E_6$ $\mathbf{27}$ representation neatly packages one SM family, while exotic states are lifted vectorlike and made ultraheavy. All non-Standard Model Higgs fields and \textit{broken} massive vectors are found to be invisible to colliders searches. The $G(2)$ gluons ensemble confines into heavy dark glueballs without communications with $SU(3)_A$ and $E_6$ sectors. Cosmological history is analyzed in details, including topological defects, inflation and reheating, demonstrating that monopole relics are naturally diluted. The resulting framework provides a minimal unified, self-consistent exceptional apparatus which includes the Standard Model and a dark matter sector which is secluded by the group-theoretic orthogonality.

A Special $E_6\to G(2) \times SU(3)_A$ Embedding for Standard Model and Dark Matter

Abstract

I developed a grand unified framework based on a special (non-regular) embedding of the exceptional group in which the first stage of symmetry breaking chain realizes . The exceptional factor plays the role of a hidden strong sector, while acts as a progenitor of the electroweak gauge group. Subsequent breaking steps, and , recover the Standard Model at low energies while preserving a confining dark gauge sector. Hypercharge is defined from the generator of , fixing the electroweak coupling normalization and identifying the Standard Model Higgs doublet within a larger exceptional representation. The special embedding naturally suppresses tree-level leptoquark couplings that typically mediate proton decay in regular grand unified theories. The scalar potential for the three 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 unification. The representation neatly packages one SM family, while exotic states are lifted vectorlike and made ultraheavy. All non-Standard Model Higgs fields and \textit{broken} massive vectors are found to be invisible to colliders searches. The gluons ensemble confines into heavy dark glueballs without communications with and sectors. Cosmological history is analyzed in details, including topological defects, inflation and reheating, demonstrating that monopole relics are naturally diluted. The resulting framework provides a minimal unified, self-consistent exceptional apparatus which includes the Standard Model and a dark matter sector which is secluded by the group-theoretic orthogonality.
Paper Structure (86 sections, 156 equations, 1 figure)

This paper contains 86 sections, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Alternative $E_6$ Grand Unification Historical Approaches
  3. The special $E_6$ Grand Unification Bricks
  4. Representation Decompositions Along the Breaking Chain and Vectorlike Completion
  5. Step 0: $E_6$ representations
  6. Step 1: $E_6 \to G(2) \times SU(3)_A$ special embedding
  7. Step 2: $G(2) \to SU(3)_C$ and the DM appearance
  8. Step 3: $SU(3)_A \to SU(2)_L\times U(1)_Y$ electroweak emergence
  9. Hypercharge embedding inside $SU(3)_A$
  10. Identification of the SM Higgs
  11. Fermion decomposition to the SM gauge group
  12. Vectorlike completion: principle and minimal implementation
  13. Completion.
  14. Vectorlike mass generation.
  15. Explicit pairing of exotic multiplets.
  16. ...and 71 more sections

Figures (1)

  • Figure 1: One-loop running of gauge couplings with exact piecewise solutions. The legend labels the inverse gauge couplings in the SM and in the additional $G(2)$ and $SU(3)_A$ sectors. The blue dot marks the $E_6$ meeting point (values shown). Vertical dashed lines indicate the matching thresholds $M_\Sigma$ and $M_\chi$ for the gauge phase transitions, where the conditions are imposed as $\tilde{\alpha}_1=\alpha_A=\alpha_2$ at $M_\Sigma$ and $\alpha_{G(2)}=\alpha_3$ at $M_\chi$.