Dark Matter as a Source for Lepton Flavor Violation

This paper explores a dark matter model where a fermionic dark matter particle simultaneously satisfies existing collider and direct detection constraints while serving as a source for observable charged lepton flavor violation signals, such as $\mu\to e \gamma$ and $\mu\to e$ conversion.

The Gist

Imagine the universe as a giant, complex puzzle. For a long time, scientists have had a picture of how the pieces fit together called the "Standard Model." But this picture has two huge holes: it can't explain Dark Matter (the invisible stuff holding galaxies together) and it can't explain why neutrinos (tiny, ghostly particles) have mass.

This paper proposes a new way to fill those holes using a specific blueprint called the 331-LHN model. Think of this model as a new set of rules for the puzzle that introduces a few new, hidden pieces.

Here is the story of what the authors found, explained simply:

1. The New Characters: Dark Matter and the "Heavy Neutrino"

In this new model, the authors introduce a new type of particle that acts as Dark Matter. Let's call him "N1."

• The Costume: N1 is a "heavy neutral fermion." In plain English, it's a heavy, invisible particle that doesn't interact with light, making it perfect for being Dark Matter.
• The Bodyguard: To keep N1 stable (so it doesn't just disappear), the model uses a special "security rule" (called R-parity). Only the lightest particle with this rule survives, and that's our Dark Matter candidate.

2. The Secret Connection: Lepton Flavor Violation

The most exciting part of this paper is a secret handshake between Dark Matter and ordinary matter.

• The Problem: In our normal world, a muon (a heavy cousin of the electron) is supposed to stay a muon. It shouldn't suddenly turn into an electron and a photon (light). This is called "Lepton Flavor Violation" (LFV). We've never seen it happen yet, but if we did, it would prove new physics exists.
• The Connection: In this model, the Dark Matter particle (N1) and a new, heavy force-carrying particle (called W') act as a bridge. They allow a muon to accidentally "leak" into an electron.
• The Analogy: Imagine a muon is a person trying to walk through a locked door. Normally, the door is locked. But in this model, the Dark Matter particle and the W' boson are like a secret tunnel behind the door. If the tunnel exists, the person can slip through and turn into an electron.

3. The Three Tests (The "Detective Work")

The authors looked at three different ways to catch this "leak" happening:

1. The Flash (µ → eγ): A muon turns into an electron and flashes a photon of light. This is the most famous test.
2. The Split (µ → 3e): A muon turns into an electron and a pair of other electrons (like splitting into three).
3. The Swap (µ-e conversion): A muon orbiting an atom's nucleus swaps places with an electron in that nucleus.

The paper calculates exactly how often these events should happen based on the new model. They found that while the "Flash" (µ → eγ) is usually the strongest signal, the other two tests (the Split and the Swap) have a special trick: they are sensitive to exotic quarks (strange, heavy particles predicted by this model) that the "Flash" test doesn't see.

4. The Great Filter: What Actually Works?

The authors ran a massive simulation to see which versions of this model could survive real-world tests. They had to pass three strict exams:

1. The Cosmology Exam: Does the model produce the right amount of Dark Matter to match what we see in the universe?
2. The Direct Detection Exam: Does the Dark Matter bump into normal atoms (like in the LZ experiment) too hard? If it does, we would have seen it by now, so the model is ruled out.
3. The Collider Exam: Have the Large Hadron Collider (LHC) experiments already seen the new heavy particles? If not, the model must predict particles heavy enough to have been missed so far.

The Big Discovery:
When they combined all these rules, they found a very specific "Goldilocks Zone."

• Inside the Zone: In the only area where the model works (where Dark Matter is stable and fits the universe's history), the "leak" is almost entirely driven by the simple "Flash" (dipole) mechanism. The complex, exotic parts of the model don't change the outcome much here.
• Outside the Zone: If you look at areas where Dark Matter wouldn't work (it's too heavy or unstable), the exotic parts (the Z' boson and the box diagrams) take over. In these "forbidden" zones, the "Swap" test (µ-e conversion) becomes the most powerful tool to detect the model, even more so than the Flash.

5. The Conclusion

The paper concludes that this model is a very tight, predictive framework.

• Right Now: The best way to test this model is to look for the "Flash" (µ → eγ). If we find it, it fits the model's predictions for the safe, working version of Dark Matter.
• In the Future: As our detectors get better, the "Swap" test (µ-e conversion) will become the star player. It is the only test that can peek into the "exotic quark" sector of the model, acting like a special lens that reveals parts of the puzzle the other tests miss.

In short: The authors built a model where Dark Matter and strange particle physics are linked. They found that for the model to be real, it must behave in a specific, simple way right now, but future experiments will be able to see the complex, hidden machinery underneath.

Technical Summary

Technical Summary: Dark Matter as a Source for Lepton Flavor Violation

Problem Statement
The Standard Model (SM) fails to explain neutrino masses and the existence of Dark Matter (DM). While Lepton Flavor Violation (LFV) has not been experimentally confirmed, it serves as a sensitive probe for Beyond Standard Model (BSM) physics. The 331-LHN model, an extension of the SM with gauge symmetry $SU(3)_C \otimes SU(3)_L \otimes U(1)_N$, offers a framework where a fermionic DM candidate (the lightest heavy neutral fermion, $N_1$) and LFV processes are intrinsically linked via the same gauge interactions. Previous studies of this model focused primarily on the dipole contribution to LFV, specifically the radiative decay $\mu \to e\gamma$. However, this approach neglects non-dipole contributions (such as $Z'$ penguins and fermionic box diagrams) which become relevant in specific regions of the parameter space, particularly where the DM candidate mass ($M_{N_1}$) approaches the mediator mass ($M_{W'}$). The challenge is to reassess the model's viability by simultaneously accounting for DM relic abundance, direct detection limits, collider bounds, and a complete calculation of LFV observables ($\mu \to e\gamma$, $\mu \to 3e$, and $\mu-e$ conversion in nuclei).

Methodology
The authors analyze the 331-LHN model, where the leptonic sector is enlarged to include heavy neutral fermions ($N_a$) forming $SU(3)_L$ triplets. The model incorporates an $R$-parity-like symmetry to stabilize the lightest neutral fermion ($N_1$) as a WIMP candidate.

1. Model Setup: The gauge sector includes new heavy bosons ($W'$, $Z'$, $U^0$). The scalar sector consists of three triplets ($\eta, \rho, \chi$) responsible for symmetry breaking. The mass spectrum is determined by the vacuum expectation values (VEVs) of these scalars, with $v_{\chi'} \gg v$ (the electroweak scale).
2. Dark Matter Phenomenology:
   • Relic Abundance: The authors compute the thermal freeze-out of $N_1$ using micrOmegas, considering annihilation channels mediated by $Z'$, the pseudoscalar $P_1$, the CP-even scalar $S_1$, and $t$-channel processes involving $W'$ and charged scalars. Coannihilation with nearly degenerate states $N_2$ and $N_3$ is included.
   • Direct Detection: Spin-independent DM-nucleon scattering is calculated via $t$-channel $Z'$ exchange.
3. LFV Analysis: The study extends previous work by calculating amplitudes for $\mu \to e\gamma$, $\mu \to 3e$, and coherent $\mu-e$ conversion in nuclei (Au and Al) in the Feynman gauge.
   • Contributions: The analysis includes the dipole operator (dominant for $\mu \to e\gamma$), non-dipole photon terms, $Z'$ penguin diagrams, and fermionic box diagrams.
   • Specifics: For $\mu-e$ conversion, the authors explicitly include contributions from the exotic quark sector (specifically the $u-D_1$ transition via box diagrams), which is a distinctive feature of the 331 model.
4. Constraints: The parameter space is constrained by:
   • Planck data for relic density ($\Omega h^2 \approx 0.12$).
   • LZ direct detection limits on spin-independent cross-sections.
   • LHC bounds on $Z'$ resonance production ($M_{Z'} \gtrsim 4.374$ TeV).
   • Current and projected experimental limits on LFV branching ratios and conversion rates.

Key Contributions

• Comprehensive LFV Calculation: Unlike previous studies that approximated LFV solely via dipole operators, this work provides a full calculation including $Z'$ penguins and box diagrams. This is crucial for accurately predicting the hierarchy between $\mu \to e\gamma$, $\mu \to 3e$, and $\mu-e$ conversion.
• Global Parameter Space Analysis: The paper performs a simultaneous scan of the $(M_{Z'}, M_{N_1})$ plane, integrating constraints from DM relic density, direct detection, collider searches, and all three LFV channels.
• Exotic Quark Sector Impact: The authors highlight that $\mu-e$ conversion is uniquely sensitive to the exotic quark sector of the 331 model, a feature absent in simpler models.

Results

• DM Viability and Dipole Dominance: The region of parameter space compatible with the observed DM relic density and direct detection bounds (where $M_{N_1} < M_{W'}$) is found to be dipole-dominated. In this viable region, the LFV observables maintain a linear correlation with $\mu \to e\gamma$, consistent with previous approximations.
• Instability Region Effects: In the region where the DM candidate would be unstable ($M_{N_1} > M_{W'}$), non-dipole contributions (specifically $Z'$ penguins and box diagrams) become significant. Here, the hierarchy of LFV observables changes, and $\mu-e$ conversion emerges as a distinct probe, potentially exceeding the sensitivity of direct detection experiments.
• Current vs. Future Sensitivity:
  • Currently, $\mu \to e\gamma$ is the most restrictive LFV observable within the viable DM region.
  • Future experiments (e.g., $\mu-e$ conversion in Aluminum) are projected to probe deeper into the parameter space, particularly in the region where non-dipole effects are relevant.
• Mass Scales: The combination of constraints pushes the viable DM candidate mass and the new physics scale ($M_{Z'}$) into the multi-TeV regime (roughly $M_{N_1} \sim 1-4$ TeV and $M_{Z'} \gtrsim 4.4$ TeV).

Significance and Claims
The paper claims that while the 331-LHN model remains highly predictive, a complete understanding of its phenomenology requires looking beyond the dipole approximation. The primary finding is that DM viability selects a specific regime where LFV is essentially dipole-dominated, validating previous simplified studies for the physically motivated region. However, the authors emphasize that outside this regime, the contributions from the exotic particle sector (particularly the exotic quarks) can substantially modify the hierarchy of LFV observables.

The work establishes that $\mu-e$ conversion in nuclei is the most promising channel for future discovery, not only due to improved experimental sensitivity but also because it directly tests the exotic quark sector inherent to the 331 framework. The study concludes that the 331-LHN model remains a viable, predictive, and experimentally accessible framework for exploring the connection between Dark Matter and Lepton Flavor Violation, provided all constraints are imposed simultaneously.