Universal 3-3-1-1 Model Overview

• Universal 3-3-1-1 model is a gauge extension of the SM defined by SU(3)C × SU(3)L × U(1)X × U(1)N, where U(1)N gauges the B–L symmetry to ensure anomaly cancellation and dark matter stability.
• It employs spontaneous and radiative symmetry breaking via an extended scalar sector, enabling seesaw mechanisms that generate hierarchical SM and exotic fermion masses.
• A residual discrete matter parity emerges after symmetry breaking, automatically stabilizing dark matter candidates while linking collider signals with cosmological and neutrino phenomenology.

The Universal 3-3-1-1 model is an extension of the Standard Model (SM) that embeds the SM gauge group into the product group SU(3)₍C₎ × SU(3)₍L₎ × U(1)₍X₎ × U(1)₍N₎, where the additional U(1)₍N₎ (or, more generally, U(1)₍G₎ in some conventions) is closely linked to a gauged B–L symmetry. This framework has been developed to address several deficiencies of the SM, including the origin of the number of fermion generations, the structure and stability of dark matter, the generation of small neutrino masses, the mechanism underlying baryogenesis via leptogenesis, the naturalness problem, and the hierarchy of fermion masses. The symmetry structure imposes strong constraints on the particle content, dynamics of spontaneous symmetry breaking, and the possible connections between cosmology and collider phenomenology. A key feature is the emergence of discrete matter parity or W-parity as a remnant of gauge symmetry breaking, resulting in the automatic stabilization of certain dark matter candidates.

1. Gauge and Charge Structure

The Universal 3-3-1-1 model is defined by the gauge symmetry: $$SU(3)_C \otimes SU(3)_L \otimes U(1)_X \otimes U(1)_N$$ where:

• SU(3)₍C₎ is the QCD color group,
• SU(3)₍L₎ generalizes the electroweak SU(2)₍L₎,
• U(1)₍X₎ is required for charge assignment closure,
• U(1)₍N₎ (alternatively labeled U(1)₍G₎ or U(1)₍B-L₎ in differing conventions) is introduced to gauge B–L.

The electric charge Q and B–L are embedded as: $Q = T_3 - \frac{1}{\sqrt{3}} T_8 + X$

$B-L = -\frac{2}{\sqrt{3}} T_8 + N$

where $T_3$, $T_8$ are SU(3)ₗ generators and X, N are the charges under the corresponding U(1) groups (Dong et al., 2013).

Gauging B–L at low energy requires extending the SU(2)ₗ doublets and singlets of the SM into SU(3)ₗ triplets (or anti-triplets) and appropriately assigning charges to prevent chiral anomalies. The representation assignments for the ordinary quarks and leptons, together with new exotic fermions, are dictated by anomaly cancellation conditions: $$\operatorname{Tr}[SU(3)_L^2 U(1)_X] = 0,\quad \operatorname{Tr}[SU(3)_L^2 U(1)_N] = 0,\quad \operatorname{Tr}[U(1)_X^3] = 0,\quad \operatorname{Tr}[U(1)_N^3] = 0,\ \ldots$$ For three generations, these constraints are automatically satisfied (Doff et al., 2012).

2. Symmetry Breaking and Scalar Sector

The spontaneous breaking of the 3-3-1-1 gauge group proceeds via vacuum expectation values (VEVs) assigned to an extended scalar sector. The minimal field content usually consists of three SU(3)ₗ triplets (η, ρ, χ) and a scalar singlet φ (to break U(1)₍N₎ at a high scale) (Dong et al., 2014, Huong et al., 2015, Dias et al., 2022). In some models, additional singlets or sextets are included to implement specific seesaw or phenomenological mechanisms (Huong et al., 2016).

The generic breaking sequence follows: $$SU(3)_C \otimes SU(3)_L \otimes U(1)_X \otimes U(1)_N \longrightarrow[VEVs\, of\, (\eta, \rho, \chi)] SU(3)_C \otimes U(1)_Q \otimes U(1)_{B-L} \longrightarrow[VEV\, of\, \phi] SU(3)_C \otimes U(1)_Q \otimes P$$ Here, P is a remnant matter (or W-) parity derived from B–L (Huong et al., 2015).

In several realizations, radiative symmetry breaking via the Coleman-Weinberg mechanism is employed due to classical scale invariance at tree level, with the Gildener-Weinberg method used to analyze the vacuum structure and identify the flat direction of the scalar potential. Under copositivity constraints, the minimal scalar sector is proven to be sufficient for successful and stable breaking (Dias et al., 2022, Dias et al., 29 Jan 2025).

3. Discrete Remnant Symmetry and Dark Matter Stabilization

After symmetry breaking, a remnant discrete Z₂ symmetry—dubbed matter parity (Pₘ = (–1)^{3(B–L)+2s}) or W-parity—remains unbroken. This symmetry arises because the scalar responsible for B–L breaking (usually φ) is chosen such that its VEV leaves a residual discrete invariance under B–L (Dong et al., 2013, Huong et al., 2015). Explicitly: $P = (-1)^{3(B-L)+2s}$ All SM fields are even under this parity, whereas new states with “wrong” B–L (relative to the SM) are odd. As a result, the lightest parity-odd state is stable, providing robust dark matter candidates without requiring additional ad hoc symmetries (Loi et al., 2020, Dong et al., 2014, Dias et al., 29 Jan 2025).

4. Fermion Masses and Seesaw Mechanisms

In the Universal 3-3-1-1 model, all SM fermion masses as well as those for new exotics are generated via a combination of Yukawa couplings and universal seesaw mechanisms. Extended sectors with vector-like quarks, leptons, and singlet neutrinos are introduced to mediate seesaw suppression (Hernández et al., 2019, Dias et al., 29 Jan 2025). The mass matrices typically possess block structures that, after integrating out the heavy degrees of freedom, yield: $$M_\text{light} \sim \frac{v_{low} \times v_{mid}}{v_{high}}$$ where vₗₒw is the electroweak scale, v₋ₗ𝒾d an intermediate scale (from the VEV of an additional scalar), and vₕᵢgₕ the scale at which extra vector-like fermions attain mass.

Neutrino masses are explained using combined type-I and type-II seesaw mechanisms (with or without scalar sextet extensions), yielding light effective masses: $m_{\text{light}} \simeq M_L - M_D M_R^{-1} M_D^T$ Here $M_L$ arises from type-II contributions (sextet VEV), $M_D$ is the Dirac term, and $M_R$ a heavy Majorana mass (Huong et al., 2016).

These mass textures often account for the hierarchical pattern observed in SM fermions and provide the foundation for generating charged lepton, quark, and neutrino masses, while maintaining consistency with flavor-changing neutral current constraints (Dias et al., 2022, Dias et al., 29 Jan 2025).

5. Dark Matter Candidates and Phenomenology

The spectrum of Pₘ-odd fields contains neutral fermions (e.g., N_R, $\chi_1$), neutral scalars (e.g., H′ ≈ η₃), and, in some realizations, exotic vector bosons (X^0). Viable dark matter candidates are selected based on cosmological, collider, and direct detection limits.

• Fermion DM: In scenarios where the lightest parity-odd state is a neutral fermion (e.g., f_d), allowed mass ranges that satisfy relic abundance ($\Omega_{\textrm{DM}} h^2 \simeq 0.12$) and direct detection constraints are typically $$160~\mathrm{GeV} \lesssim m_{f_d} \lesssim 520~\mathrm{GeV}$$, with the lower bound on the symmetry breaking scale $v_\chi \gtrsim 3.6~\mathrm{TeV}$ (Dias et al., 29 Jan 2025).
• Scalar DM: The Higgs portal coupling enables annihilation mainly into SM Higgs pairs; viable mass windows reside in the multi-hundred GeV to few TeV regime (Loi et al., 2020, Dong et al., 2014).
• Gauge DM: Non-Hermitian neutral gauge bosons (e.g., X^0) typically have annihilation cross-sections too high to constitute the observed relic density (Dong et al., 2013).

Matter parity ensures the stability of these candidates, and, depending on parameters, models can feature multi-component dark matter or superheavy dark matter (for high-scale breaking scenarios or gravitational production) (Huong et al., 2016).

The dominant dark matter portal is generally the Higgs sector via mixing between the SM-like Higgs and the scalar that breaks U(1)₍N₎; in Majorana DM scenarios, direct Z′-mediated interactions are suppressed by p-wave and helicity selection rules (Luong et al., 20 Mar 2024). Direct detection experiments probe most parameter space, with some regions near sensitivity thresholds for future detectors such as XLZD and PandaX-xT (Dias et al., 29 Jan 2025).

6. Collider, Precision, and Cosmological Constraints

Limits on new neutral gauge bosons (Z′, Z_N), extra scalars, and vector-like fermions arise from precision electroweak data (ρ-parameter, Z width), collider searches (e.g. LEPII, LHC dileptons), and flavor observables (K–$\bar{\textrm{K}}$, $B^0_s$–$\bar{B}^0_s$ mixing). Perturbativity, anomaly cancellation, and minimal flavor violation further constrain the viable parameter space.

Key experimental bounds include:

• Gauge boson masses: $m_{Z′}, m_{Z_N} \gtrsim 2.5–3~\mathrm{TeV}$ (Dong et al., 2014, Rodríguez et al., 2016).
• Lower bound on the symmetry breaking scale (from the ρ-parameter): $v_\chi \gtrsim 3.6~\mathrm{TeV}$ (Dias et al., 29 Jan 2025).
• Preferably small Z–Z′ mixing ( $\lesssim 10^{-3}$ radians) (Rodríguez et al., 2016).
• Lepton flavor violating rates and Higgs diphoton decays compatible with SM (possible in specific low-scale seesaw variations) (Hernández et al., 2019).
• Predicted new particles near the TeV scale, potentially accessible at high-luminosity LHC (Dong et al., 2014, Dias et al., 2022).

In cosmology, inflation and baryogenesis are realized by identifying the high-scale singlet scalar breaking U(1)₍N₎ as an inflaton, with radiative corrections producing a viable slow-roll potential. Both thermal and nonthermal leptogenesis scenarios are available, with heavy right-handed neutrinos playing the central role in generating lepton asymmetry (Huong et al., 2015, Huong et al., 2016). The parameter space for inflation, neutrino masses, and baryon asymmetry is shown to be compatible with Planck, WMAP, and BICEP experimental data (Huong et al., 2015).

7. Variants, Flipped Embeddings, and Extended Features

Variants of the Universal 3-3-1-1 model arise through different assignments of SU(3)ₗ triplets and anti-triplets, and also in the context of [SU(3)]³ trinification, which allows "flipped" models (weak-I, U, V spin assignments) (Rodríguez et al., 2016). The notable finding is that, despite different embeddings of SM fermions, the effective Z′ boson couplings to SM fermions remain identical across all flipped versions, and the same electroweak/collider constraints apply universally.

Kinetic mixing between U(1)₍X₎ and U(1)₍N₎ introduces corrections to Z couplings, ρ-parameter, and can induce extra sources of Z–Z′ mixing, though these must remain small ($\lesssim 10^{-3}$) to satisfy precision constraints (Dong et al., 2015).

Models featuring radiative or dynamical symmetry breaking mechanisms, such as the scale-invariant 3-3-1-1 with B–L symmetry, employ the Coleman-Weinberg and Gildener-Weinberg mechanisms to generate the hierarchy of VEVs. This approach links the mass of the "scalon" or dilaton to beyond-SM physics and maintains a minimal scalar sector (Dias et al., 2022, Dias et al., 29 Jan 2025). In scenarios where the symmetry breaking and inflation occur at the same scale, nonthermal production of superheavy dark matter emerges, stabilized by matter parity (Huong et al., 2016).

Summary Table of Core Features in Universal 3-3-1-1 Models

Feature	Principle or Formula	Typical Value / Implication
Gauge Group	$SU(3)_C \times SU(3)_L \times U(1)_X \times U(1)_N$	Extension of SM gauge symmetry
Electric Charge	$Q = T_3 - \frac{1}{\sqrt{3}}T_8 + X$	Fixes new particle charges
B–L Charge	$B-L = -\frac{2}{\sqrt{3}}T_8 + N$	Required for anomaly cancellation
Remnant Parity	$P = (-1)^{3(B-L) + 2s}$	Stabilizes DM
DM Mass Range	Depends on candidate (e.g., $$160~\mathrm{GeV} \lesssim m_{f_d} \lesssim 520~\mathrm{GeV}$$)	Direct detection bounds (Dias et al., 29 Jan 2025)
Z′ Mass Bounds	$m_{Z′} \gtrsim 2.5~\mathrm{TeV}$	LHC/dilepton search constraints
Seesaw Mass Terms	$m_{\text{light}} \simeq M_L - M_D M_R^{-1} M_D^T$	Sub-eV neutrino masses
3-3-1 Breaking Scale	$v_\chi \gtrsim 3.6~\mathrm{TeV}$	ρ-parameter/LEP constraints

• Gauge structure, charge embeddings, and W-parity: (Dong et al., 2013, Dong et al., 2014, Dong et al., 2015, Dias et al., 2022).
• Neutrino masses and seesaw: (Huong et al., 2015, Huong et al., 2016, Hernández et al., 2019).
• Dark matter candidates, relic density, and direct detection: (Loi et al., 2020, Dias et al., 29 Jan 2025, Dong et al., 2014, Dong et al., 2013, Luong et al., 20 Mar 2024).
• Collider constraints and Z′ search limits: (Cao et al., 2016, Rodríguez et al., 2016, Dong et al., 2015).
• Radiative breaking and scale invariance: (Dias et al., 2022, Dias et al., 29 Jan 2025).
• Flipped versions and trinification: (Rodríguez et al., 2016).

The Universal 3-3-1-1 model provides a predictive and testable framework, where the interplay of anomaly cancellation, symmetry breaking (both dynamical and radiative), matter parity, and the generic structure of the gauge and scalar sectors leads to correlated predictions for collider signals, cosmology, and dark matter phenomenology.