Dark Grand Unification

• Dark Grand Unification is a theoretical framework that embeds both visible and dark sectors within unified gauge groups like E6, ensuring intrinsic matter parity.
• The unified model naturally predicts stable dark matter candidates through residual discrete symmetries emerging from high-scale symmetry breaking.
• A canonical Type-I seesaw mechanism within the framework explains light neutrino masses by linking electroweak dynamics with grand unification scales.

Dark Grand Unification refers to a family of theoretical frameworks in which the unification of Standard Model (SM) forces at high scales is extended to nontrivially include the dark sector—encompassing dark matter, dark interactions, and often neutrino mass generation—in a single, comprehensive gauge-theoretic structure. The core concept is that dark and visible matter are not only coexistent but are intricately related through their embedding in unified gauge multiplets at a grand unification scale, typically via exceptional or extended groups such as E₆. This paradigm predicts specific low-energy phenomena: a stable dark matter candidate protected by residual symmetries (such as matter parity), the natural emergence of a seesaw mechanism for neutrino mass generation, and distinctive phenomenology including a new variant of the 3-3-1-1 gauge model (Dong et al., 2 Aug 2025). The resulting framework tightly connects dark matter stability and baryogenesis, often through universal family assignments and tightly correlated particle spectra.

1. Unified High-Scale Frameworks and Symmetry Embeddings

Dark grand unification places both visible and dark matter fields within high-dimensional representations of unified gauge groups, exemplified by E₆ for unification and its trinification subgroup SU(3)₍C₎ × SU(3)₍L₎ × SU(3)₍R₎. In such theories, all matter—including conventional quarks and leptons, right-handed neutrinos, and dark sector fields—arises as components of the same E₆ representations. The SU(3) trinification structure ensures left-right symmetry, while the decomposition of the E₆ 27-dimensional representation leads naturally to “dark” fields alongside conventional particles. Upon symmetry breaking, this results in residual discrete symmetries that govern low-energy phenomenology, including dark sector stability.

Two chain patterns are central:

• E₆ → SU(3)₍C₎ × SU(3)₍L₎ × SU(3)₍R₎: The trinification scenario features all SM and dark fields in unified multiplets, and the matter parity is an automatic remnant of the symmetry breaking sequence.
• Trinification → SU(3)₍C₎ × SU(3)₍L₎ × U(1)₍X₎ × U(1)₍N₎: Here, SU(3)₍R₎ is further reduced to two abelian factors, yielding a family-universal “3-3-1-1” model where leptons (e_R and ν_R) are linked in the same triplet, and matter parity persists.

The electric charge and baryon-minus-lepton number operators are then determined by specific combinations of SU(3) generators and the new abelian charges: $$Q = T_3^L + \frac{1}{\sqrt{3}}T_8^L + X,\qquad N = \frac{2}{\sqrt{3}} T_8^R$$ which organizes both dark and visible sector charges.

2. Dark Matter Candidates and Stability Mechanisms

Stability of dark matter arises not by ad hoc imposed global symmetries, but from the spontaneous breakdown of unified symmetries, with a residual discrete symmetry—matter parity—encoded as: $M_p = (-1)^{3(B-L)+2s}$ where $B-L$ is baryon minus lepton number, and $s$ is spin. This matter parity is an intrinsic remnant of E₆ → trinification → 3-3-1-1 symmetry breaking. All ordinary matter fields (quarks, charged leptons, left-handed neutrinos) are even under $M_p$, while exotic (dark) fields are odd. Consequently, the lightest parity-odd state is absolutely stable, qualifying naturally as a dark matter candidate. The framework admits several types of dark matter:

• Scalar, fermion, and vector candidates—specifically, non-degenerate doublets and singlets whose mass splittings are enforced by the underlying symmetry breaking and matter parity (Dong et al., 2 Aug 2025).
• The explicit assignment of matter parity forbids rapid decays into SM particles and suppresses dangerous mixings.

This mechanism is structurally distinct from the imposition of external Z₂ or ad hoc global symmetries in traditional dark matter theories.

3. Neutrino Mass Generation via Seesaw Mechanism

Light neutrino masses are generated naturally via a canonical (Type-I) seesaw mechanism. In the universal 3-3-1-1 model derived from trinification, heavy right-handed neutrinos (or similar “sterile” states within the multiplet) receive large Majorana masses as a consequence of high-scale symmetry breaking. In turn, SM neutrinos couple via Dirac terms to these heavy singlets. The mass matrix takes the schematic form: $$\mathcal{M}_{\nu} = \begin{pmatrix} 0 & m_D\ m_D^T & M_R \end{pmatrix}$$ leading to light eigenvalues $m_\nu \approx m_D^2/M_R$. The parameters $m_D$ (Dirac masses) are set by electroweak-scale VEVs, while $M_R$ is at the scale associated with matter parity or unified symmetry breaking. An explicit realization in (Dong et al., 2 Aug 2025) is: $m_\nu \approx \frac{(h^\nu)^2 u^2}{2\sqrt{2} x}$ where $h^\nu$ is the neutrino Yukawa coupling, $u$ is a weak-scale VEV, and $x$ is a large scale connected to the VEV breaking U(1)₍N₎, typically $x \sim 10^{10} \,\mathrm{GeV}$. This construction makes small neutrino masses a direct prediction of the same underlying unification responsible for dark matter stability.

4. Universal 3-3-1-1 Model: Family Structure and Phenomenology

The “universal” 3-3-1-1 model emerging from dark grand unification differs fundamentally from the standard 3-3-1(-1) model paradigm:

• Family universality: All three generations of leptons and quarks have identical gauge charge assignments, ensuring cancellation of anomalies without the need for family-dependent quantum numbers.
• The lepton triplet, with both right-handed electron ($e_R$) and neutrino ($\nu_R$) at the bottom, is a unique structural prediction, reminiscent yet distinct from both minimal and right-handed neutrino 3-3-1 models.
• Matter parity protection: The dark spectrum possesses mass splittings and charge assignments such that the lightest parity-odd field is a viable WIMP, with expectations of suppressed direct detection rates due to splitting and suppressed mixings.

The gauge embedding yields additional U(1) factors (X, N) and a well-defined charge relation, for instance,

$$X = T_3^R + \frac{1}{\sqrt{3}} T_8^R,\qquad N = \frac{2}{\sqrt{3}} T_8^R$$

which play an important role in the spectrum and phenomenology of new gauge bosons and possible flavor-changing neutral current constraints.

5. Comparison to Alternative Unification Approaches

Relative to normal 3-3-1 or minimal SU(5)/SO(10) GUTs:

• Dark matter candidates are not supplementary fields but are embedded from the outset in the grand unified representations.
• Matter parity emerges as an automatic consequence of the high-scale symmetry breaking rather than being imposed externally.
• The universal model eliminates family-dependent gauge couplings, circumventing flavor-violating neutral currents endemic to many 3-3-1(-1) variants.
• The unified construction yields a spectrum with naturally split vector-like dark states, a necessary feature for consistency with direct detection bounds and successful phenomenology.
• The seesaw mechanism for neutrino mass is not radiative (as in “scotogenic” models) but canonical, arising directly from high-scale couplings.

6. Mathematical Formulations and Testable Predictions

The theoretical structure is encapsulated in the following representative equations (Dong et al., 2 Aug 2025):

• Electric charge:

$Q = T_3^L + \frac{1}{\sqrt{3}} T_8^L + X$

• Matter parity:

$M_p = (-1)^{3(B-L)+2s}$

• Seesaw neutrino mass (Type I):

$m_\nu \simeq \frac{(h^\nu)^2 u^2}{2\sqrt{2} x}$

• Electromagnetic coupling matching:

$$\frac{1}{e^2} = \frac{1}{g^2} + \frac{1}{3g^2} + \frac{1}{g_X^2}$$

Phenomenological consequences include:

• Dark matter relic abundance: The mass splitting within parity-odd dark sector multiplets produces a viable thermal relic density while simultaneously suppressing direct detection rates via inelastic scattering kinematics.
• Collider signatures: The model predicts heavy new gauge bosons (Z′-like states), new scalar bi-triplets and sextets, and possible doublet vector dark matter candidates with masses and couplings determined by the underlying symmetry breaking structure.
• Neutrino physics: Light neutrino masses are generically unavoidable, and the detailed mass spectra and mixing angles tie directly to the parameters of the high-scale unified model.

7. Future Directions and Open Phenomenological Windows

Potential research avenues motivated by the dark grand unification framework include:

• Detailed collider phenomenology of new gauge bosons and exotic scalars in the universal 3-3-1-1 sector.
• Precision studies of the relic density and direct detection for split vector, scalar, or fermion dark matter candidates.
• Analysis of flavor physics and lepton flavor violation resulting from the universal assignment and extended Higgs sector.
• Exploration of alternative gauge completions enabling radiative neutrino mass generation (scotogenic mechanisms) within unified settings.
• Systematic renormalization group running of couplings from the E₆ scale to the TeV scale, incorporating threshold corrections.

A plausible implication is that the observed properties of dark matter and the neutrino sector may provide direct clues to grand unification, particularly if linked through residual symmetries and universal couplings deriving from trinification or related exceptional gauge structures. This framework provides a predictive, highly constrained class of models with significant discovery potential in collider, neutrino, and dark matter experiments.