DOI: 10.1140/epjc/s10052-026-16104-1 ISSN: 1434-6052
Toward a special $$E_6\rightarrow G(2) \times SU(3)_A$$ embedding for Standard Model and dark matter and an $$E_7$$ completion proposal
Nicolò Masi Abstract
We developed a unified framework based on a special (non-regular) embedding of the exceptional group
$$E_6$$
E
6
in which the main stage of symmetry breaking chain realizes
$$E_6\rightarrow G(2) \times SU(3)_A$$
E
6
→
G
(
2
)
×
S
U
(
3
)
A
. The exceptional factor
G
(2) plays the role of a hidden strong sector, while
$$SU(3)_A$$
S
U
(
3
)
A
acts as an ancestor of the electroweak gauge group. A minimal scalar sector is organized around a
$$\textbf{650}$$
650
-based
$$E_6$$
E
6
Higgs sector. Its
$$G(2)\times SU(3)_A$$
G
(
2
)
×
S
U
(
3
)
A
components
$$(\textbf{7},\textbf{1})$$
(
7
,
1
)
and
$$(\textbf{1},\textbf{8})$$
(
1
,
8
)
implement the subsequent breaking steps
$$G(2)\rightarrow SU(3)_C$$
G
(
2
)
→
S
U
(
3
)
C
and
$$SU(3)_A\rightarrow SU(2)_L\times U(1)_A$$
S
U
(
3
)
A
→
S
U
(
2
)
L
×
U
(
1
)
A
. The
speciality
of this symmetry breaking establishes the feature of
darkness
. Defining a hypercharge from the
$$t_{8}$$
t
8
generator of
$$SU(3)_A$$
S
U
(
3
)
A
is not sufficient to recover the exact Standard Model hypercharges and the correct Weinberg weak angle, leading to the necessity of an
$$E_7$$
E
7
uplift which introduces only a proper additional
$$U(1)_X$$
U
(
1
)
X
factor to fix the abelian sector. The special embedding naturally suppresses tree-level leptoquark couplings that typically mediate proton decay in regular grand unified theories. 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
$$E_6$$
E
6
unification. The exotic states are organized in
$$E_7$$
E
7
-derived vectorlike pairs and are made ultraheavy, while a chirality-selecting projection keeps only the desired Standard Model zero modes. All non-Standard Model Higgs fields and
broken
massive vectors are found to be invisible to colliders searches. The
G
(2) gluons ensemble confines into heavy dark glueballs with parametrically suppressed communication with
$$SU(3)_A$$
S
U
(
3
)
A
and
$$U(1)_X$$
U
(
1
)
X
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 and internally consistent exceptional apparatus which includes the Standard Model and a dark matter sector which is secluded by the group-theoretic orthogonality.