Heavy neutral bosons and dark matter in the 3-3-1 model with axionlike particle

This paper investigates the 3-3-1 model extended with axionlike particles to predict heavy neutral boson masses, analyze their LHC signals, and identify dark matter candidates by linking their relic density to the axion breaking scale.

The Gist

Imagine the universe as a giant, complex machine. For decades, physicists have been trying to understand how this machine works using a rulebook called the Standard Model. It's a great rulebook, but it has some missing pages. It doesn't explain why the universe is expanding so fast (dark energy), what the invisible "dark matter" holding galaxies together is, or why certain particles seem to break the rules of flavor (like a muon turning into an electron).

This paper proposes a new, expanded rulebook called the 3-3-1 Model with Axion-like Particles (331ALP). Think of this as adding a new, secret wing to the Standard Model's house. The authors are exploring what lives in this new wing and how it might solve the universe's biggest mysteries.

Here is a simple breakdown of their findings:

1. The New Residents: Heavy Bosons

In the Standard Model, we have a "Higgs boson" (the particle that gives other particles mass) and a "Z boson" (a messenger particle).

• The Analogy: Imagine the Standard Model is a family with a father (Z) and a mother (Higgs). This new model suggests there are older, heavier cousins living in the attic: a heavier Higgs ($h_2$) and a super-heavy Z cousin ($Z'$).
• The Discovery: The authors calculated how heavy these cousins must be to avoid being caught by the world's biggest particle detectors (the LHC at CERN).
  • The heavy Higgs cousin ($h_2$) must weigh at least 600 GeV (about 6 times heavier than the known Higgs).
  • The super-heavy Z cousin ($Z'$) must be a giant, weighing at least 5.1 TeV (over 50 times heavier than the known Z).
  • Why it matters: If we build bigger, stronger microscopes (future colliders), we might finally spot these heavy cousins, proving this new wing of the house exists.

2. The "Flavor" Mix-Up (Lepton Flavor Violation)

In physics, particles have "flavors" (like electron, muon, tau). Usually, they are very picky and don't change flavors easily. However, in this new model, they can swap flavors, but it's a rare event.

• The Analogy: Imagine a strict bouncer at a club. Usually, an "electron" guest can't turn into a "muon" guest. But in this new model, there's a secret backdoor. Sometimes, a muon sneaks in and turns into an electron, or a tau turns into a muon.
• The Check: The authors checked the "security logs" (experimental data from the LHC). They found that while these flavor swaps can happen in their model, they happen rarely enough to not break the rules set by current experiments.
• The Prediction: They predict that if we look closely at the Higgs boson, we might see it decay into a muon and a tau particle about 1 in 100,000 times. This is a tiny number, but it's a "smoking gun" signal that could confirm their theory.

3. The Invisible Guardian: Dark Matter

Dark matter is the invisible stuff that makes up 85% of the matter in the universe, but we can't see it.

• The Analogy: Imagine the universe is a party. Most particles are the loud dancers. Dark matter is the quiet person in the corner who never leaves the room. In this model, there is a special rule called $Z_2$ symmetry. Think of this as a "Good/Bad" badge system.
  • Particles with a "Good" badge interact with us.
  • Particles with a "Bad" badge are invisible to us and can't decay into "Good" particles.
• The Candidate: The authors identified a specific particle (a heavy right-handed neutrino, $N_{1R}$) that wears the "Bad" badge. Because it's the lightest "Bad" particle, it can't decay into anything else. It just hangs around forever. This is the Dark Matter candidate.
• The Connection: They found a mathematical link between how heavy this Dark Matter particle is and the energy scale where the "Axion" (a particle proposed to solve another mystery) is created. It's like saying, "If the Dark Matter weighs X, then the Axion energy scale must be Y."

4. The "Axion" and the Universe's Expansion

The model includes an Axion-like particle (ALP).

• The Analogy: Think of the early universe as a balloon being inflated. The Axion is like the air pump that helped inflate it (a process called inflation).
• The Mechanism: In this model, the Axion is naturally created because of the specific way the new particles are arranged. It solves the problem of how the universe got its initial "push" to expand.

Summary: What's the Big Picture?

The authors have built a theoretical "house extension" to the Standard Model. They checked the blueprints and found:

1. Heavy Cousins: There are new, very heavy particles ($Z'$ and $h_2$) that are currently hiding but might be found soon if we look hard enough.
2. Rare Swaps: These new particles allow for rare flavor swaps between leptons, which we might detect in future experiments.
3. Dark Matter: The model naturally produces a stable, invisible particle that fits the description of Dark Matter.
4. Consistency: All of this fits within the current rules of physics and experimental limits.

In short: This paper says, "If you look for these specific heavy particles and these rare flavor swaps, you might just find the missing pieces of the universe's puzzle, including the identity of Dark Matter." It's a roadmap for future experiments to test if this new, expanded version of reality is true.

Technical Summary

1. Problem Statement

The Standard Model (SM) fails to explain several phenomena, including the origin of neutrino masses, the nature of dark matter (DM), and the existence of lepton flavor violation (LFV). While the 3-3-1 model extends the SM gauge group to $SU(3)_C \otimes SU(3)_L \otimes U(1)_X$, previous variations have not fully integrated an axion-like particle (ALP) mechanism, addressed heavy neutral boson phenomenology at the Large Hadron Collider (LHC), or identified stable dark matter candidates within a unified framework.

This paper investigates the 3-3-1 model with Axion-Like Particles (331ALP). The specific goals are to:

• Analyze the spectrum of heavy neutral bosons ($Z'$ and new Higgs bosons).
• Calculate and constrain LFV processes (charged lepton decays $l_a \to l_b \gamma$ and Higgs decays $H \to l_a l_b$) against current experimental limits.
• Identify viable dark matter candidates based on residual symmetries.
• Predict the mass ranges of new particles consistent with LHC data (ATLAS and CMS).

2. Methodology

Model Construction

The authors construct a model based on the gauge group $SU(3)_C \otimes SU(3)_L \otimes U(1)_X$ augmented by discrete symmetries $Z_{11}$ and $Z_2$.

• Particle Content: Includes three generations of fermions, exotic quarks ($U_3, D_n$), right-handed neutrinos ($N_{aR}$), and scalar sectors consisting of three $SU(3)_L$ triplets ($\eta, \rho, \chi$) and one singlet ($\phi$).
• Symmetry Breaking: The model undergoes a three-stage spontaneous symmetry breaking (SSB):
  1. $v_\phi$ breaks $Z_{11}$, generating the axion-like particle and an inflation field.
  2. $v_\chi$ breaks $SU(3)_L \to SU(2)_L$.
  3. $v_\eta, v_\rho$ break $SU(2)_L \to U(1)_Q$.
• Residual Symmetry: The $Z_2$ symmetry remains unbroken after SSB. Particles odd under $Z_2$ are stable, providing a dark matter candidate.

Theoretical Calculations

• Feynman Rules & Amplitudes: The authors derived the Feynman rules for the model, specifically focusing on lepton-flavor-violating couplings. They calculated one-loop amplitudes for:
  • Charged lepton decays: $l_a \to l_b \gamma$.
  • Higgs decays: $h_{1,2} \to l_a l_b$ (where $h_1$ is SM-like and $h_2$ is a heavy Higgs).
• Loop Contributions: The calculations utilized Passarino-Veltman (PV) functions. Key loop particles included neutrinos ($\nu, N$), charged gauge bosons ($W, V$), and charged scalars ($H^\pm$).
• Relic Density: The dark matter annihilation cross-section $\langle \sigma v \rangle$ was calculated for the process $N_{1R} N_{1R} \to \text{SM particles}$ (mediated by the heavy scalar $\Phi$) to determine the relic abundance $\Omega_{DM} h^2$.

Numerical Analysis

The authors scanned the parameter space using:

• Experimental Constraints: Limits on $Br(\mu \to e\gamma)$, $Br(\tau \to e\gamma)$, $Br(\tau \to \mu\gamma)$, and $Br(h \to \mu\tau)$ from MEG, Belle, and LHC.
• LHC Data: Cross-section limits for heavy neutral bosons ($Z'$ and $H$) from ATLAS and CMS (integrated luminosities up to 140 fb$^{-1}$ at $\sqrt{s}=13$ TeV).
• Parameters: Free parameters included mixing angles ($t_\alpha$), masses of new charged scalars ($m_{H^\pm_2}$), and the vacuum expectation value (VEV) of the singlet scalar ($v_\phi$).

3. Key Contributions

1. Unified 331ALP Framework: The paper successfully integrates the axion generation mechanism (via the singlet $\phi$ and $Z_{11}$) into the 3-3-1 model while maintaining a stable dark matter candidate via $Z_2$.
2. LFV Constraints: The study provides a rigorous analysis of the parameter space where both charged lepton LFV and Higgs LFV decays satisfy current experimental bounds.
3. Dark Matter Identification: It identifies the lightest right-handed neutrino ($N_{1R}$) as the sole viable dark matter candidate in this model, as other $Z_2$-odd particles either break the electroweak scale or constitute ordinary matter.
4. Mass Predictions: The paper establishes specific lower bounds for new heavy bosons based on LHC null searches.

4. Key Results

• Lepton Flavor Violation:

  • The model satisfies experimental limits for $Br(\mu \to e\gamma) \le 4.2 \times 10^{-13}$ and $Br(\tau \to \mu\gamma) \le 4.4 \times 10^{-8}$ in specific regions of the $(t_\alpha, m_{H^\pm_2})$ plane.
  • Higgs Decays: In the allowed parameter space, the branching ratio for the SM-like Higgs decaying to mu-tau is predicted to be $Br(h_1 \to \mu\tau) \sim 10^{-6}$, and for the heavy Higgs $Br(h_2 \to \mu\tau) \sim 10^{-5}$. These are well below the current experimental limit of $10^{-3}$ but potentially observable in future high-luminosity runs.

• Heavy Neutral Boson Masses:

  • $Z'$ Boson: Based on the non-observation of high-mass dilepton resonances at ATLAS and CMS, the mass of the new $Z'$ boson is constrained to $m_{Z'} \ge 5.1$ TeV.
  • Heavy Higgs ($h_2$): Analyzing gluon-gluon fusion signals ($pp \to h_2 \to \mu\tau$), the paper predicts a lower mass limit of $m_{h_2} \ge 600$ GeV.

• Dark Matter:

  • The lightest right-handed neutrino ($N_{1R}$) is the dark matter candidate.
  • A direct relationship is established between the dark matter mass ($m_{N_{1R}}$) and the axion breaking scale ($v_\phi$). For relic densities matching Planck data ($0.1 \le \Omega_{DM} h^2 \le 0.3$), the model requires specific correlations between these scales, typically involving $v_\phi$ in the TeV to 100s of TeV range.

5. Significance

This work is significant for several reasons:

• Phenomenological Viability: It demonstrates that the 3-3-1 model can be extended to include axion-like particles and dark matter without contradicting stringent LFV constraints.
• Testability: The paper provides concrete, testable predictions for the LHC. The lower bounds of $m_{Z'} \ge 5.1$ TeV and $m_{h_2} \ge 600$ GeV guide future searches for heavy resonances.
• Dark Matter Mechanism: By linking the stability of dark matter to a residual $Z_2$ symmetry arising naturally from the 3-3-1 breaking chain, it offers a compelling alternative to the standard WIMP paradigm, specifically utilizing right-handed neutrinos.
• Higgs Physics: The prediction of enhanced but sub-limit LFV Higgs decays ($h \to \mu\tau$) suggests that the 331ALP model could be probed via precision Higgs measurements at the High-Luminosity LHC (HL-LHC).

In conclusion, the paper successfully maps the viable parameter space of the 331ALP model, constraining new physics scales and offering a coherent explanation for neutrino mass, dark matter, and the axion problem within a single theoretical framework.