Dirac neutrinos and gauged lepton number

This paper proposes the first scotogenic model with gauged lepton number $U(1)_L$ that is spontaneously broken to a residual $\mathbb{Z}_6$ symmetry, thereby generating tiny Dirac neutrino masses at one loop while simultaneously providing a stable scalar dark matter candidate consistent with current experimental constraints and predicting observable charged lepton flavor violating decays.

The Gist

Imagine the Standard Model of physics as a giant, incredibly successful recipe book for the universe. It tells us how to make stars, planets, and even you. But, like any good recipe, it has a few missing ingredients and a couple of unexplained mysteries.

Two of the biggest mysteries are:

1. Why do neutrinos (tiny, ghostly particles) have mass? In the original recipe, they were supposed to be weightless, but experiments show they have a tiny bit of weight.
2. What is Dark Matter? We know it's there because it holds galaxies together with gravity, but we can't see it or touch it. The recipe book has no ingredient for it.

This paper proposes a new "secret ingredient" to fix both problems at once. Here is the story of their solution, explained simply.

1. The New Rule: "Lepton Number" as a Law

In the current recipe, the conservation of "Lepton Number" (a count of certain particles like electrons and neutrinos) is just a lucky accident. It happens to work, but there's no law forcing it to.

The authors propose making this a strict law of the universe, enforced by a new force (a "gauge symmetry"). Think of it like a strict bouncer at a club. In the Standard Model, the bouncer is asleep; in this new model, the bouncer is wide awake and checking IDs.

2. The "Broken" Law and the Magic Residue

Here is where it gets tricky but fun. The authors suggest that this strict "Lepton Number" law gets broken, but not completely.

Imagine you have a round cake (the symmetry). You cut it into 6 equal slices. If you take away 3 slices (breaking the symmetry by 3 units), you are left with a half-cake. But in this quantum world, taking away 3 slices leaves behind a specific, stable 6-slice pattern (a $Z_6$ symmetry).

This leftover pattern is the key. It acts like a quantum lock. It says, "You can change things, but only in ways that respect this 6-slice pattern."

3. Solving Mystery #1: The Ghostly Neutrinos

How do neutrinos get their tiny mass?

• The Old Way: Usually, scientists think neutrinos get mass directly from a heavy particle, like a heavy stone dropping on a spring.
• The New Way (Scotogenic): In this paper, the neutrinos get their mass through a one-loop "scotogenic" mechanism.
  • Analogy: Imagine you want to bake a cake (neutrino mass), but you don't have the ingredients directly. Instead, you have to send a messenger (a new particle) on a round trip to a distant shop, buy the ingredients, and bring them back.
  • Because the messenger has to go on a detour (a "loop"), the process is slow and inefficient. This inefficiency is why the neutrino mass is so incredibly tiny.
  • The "6-slice lock" mentioned earlier ensures that this detour must happen. It prevents the neutrino from getting mass the easy way, forcing it to take the slow, looped path.

4. Solving Mystery #2: The Dark Matter Candidate

Every time you send that messenger on a detour to get the neutrino ingredients, you need a "safe house" where the ingredients are stored.

• The paper introduces a new, invisible particle (a scalar particle) that lives in this safe house.
• Because of the "6-slice lock," this particle cannot decay or disappear. It is absolutely stable.
• Since it's stable, invisible, and interacts very weakly with normal matter, it is the perfect candidate for Dark Matter.
• It's like a ghost that is trapped in a room by an unbreakable lock; it can't leave, so it just hangs around the universe, holding galaxies together.

5. The "Leptophilic" Force

The new force introduced in this model is "leptophilic," meaning it loves leptons (electrons, neutrinos) but ignores quarks (the stuff inside protons and neutrons).

• Analogy: Imagine a new type of radio signal that only plays music for your headphones (leptons) but is completely silent for your speakers (quarks).
• This makes the model very hard to detect in standard particle colliders (which smash protons), but it gives us specific places to look, like in experiments that study how neutrinos bounce off electrons.

6. The "Flavor" Problem (The Flavor Police)

The paper also checks if this new model causes "flavor violations." In the particle world, "flavor" is like a particle's personality (electron vs. muon). Usually, an electron stays an electron.

• The authors calculated that their model does allow particles to change flavors (e.g., a muon turning into an electron and a photon), but only at a rate that is just barely detectable by current or very near-future experiments.
• This is good news! It means the model isn't broken (it doesn't predict impossible things), but it also means we might be able to prove it's true in the next few years with experiments like MEG II or Mu3e.

Summary

The authors have built a new "recipe" for the universe where:

1. They made a new law (Gauged Lepton Number) that gets partially broken.
2. The leftover "lock" from that break forces neutrinos to get their tiny mass through a slow, round-trip process.
3. That same process creates a stable, invisible particle that acts as Dark Matter.
4. The model predicts specific, testable signals that we might see in labs very soon.

It's a clever solution that ties the smallest particles (neutrinos) and the biggest mystery (Dark Matter) together with a single, elegant mathematical key.

Technical Summary

1. Problem Statement

The Standard Model (SM) fails to explain two fundamental phenomena:

1. Neutrino Masses: The origin of neutrino masses and their extreme smallness.
2. Dark Matter (DM): The lack of a viable, stable particle candidate for Dark Matter.

While the scotogenic model (proposed by E. Ma) links neutrino mass generation to DM via a radiative one-loop mechanism, it typically relies on an ad hoc global $Z_2$ symmetry to stabilize the DM candidate. The authors aim to construct a more natural framework where the stabilizing symmetry arises dynamically from a gauged symmetry connected to neutrino physics, specifically gauged Lepton Number ($U(1)_L$). Additionally, they seek to generate Dirac neutrino masses (rather than Majorana) within this scotogenic framework.

2. Methodology and Model Construction

A. Gauge Symmetry and Breaking Pattern

The model extends the SM gauge group to:
$$G \equiv SU(3)_C \otimes SU(2)_W \otimes U(1)_Y \otimes U(1)_L$$
The $U(1)_L$ symmetry is spontaneously broken by three units ($\Delta L = 3$) by the vacuum expectation value (VEV) of a scalar field $\phi$. This specific breaking pattern, combined with the presence of fields carrying fractional lepton numbers, results in a residual discrete gauge symmetry:
$$Z_6 \subset U(1)_L$$
This $Z_6$ symmetry is crucial as it is not imposed by hand but arises naturally from the gauge structure.

B. Particle Content

To ensure anomaly cancellation and facilitate the scotogenic mechanism, the model introduces:

• Scalars:
  • An inert doublet $\eta$ (Lepton number $L = -1/2$).
  • A singlet $\sigma$ ($L = -7/2$) to close the neutrino mass loop.
  • A singlet $\phi$ ($L = 3$) responsible for breaking $U(1)_L$.
• Fermions:
  • Three right-handed neutrinos ($\nu_{aR}$) with specific $L$ charges ($4, 4, -5$).
  • A vector-like family of leptons ($L', e', n$) with arbitrary $L$ charges $\ell$ and $\ell-3$.
  • Vector-like neutral leptons $S_{iL,R}$ ($i=1,2,3$) with fractional lepton number $L = 1/2$. These act as the mediators in the scotogenic loop.

C. Neutrino Mass Mechanism

The model generates Dirac neutrino masses at the one-loop level.

• The effective dimension-5 operator responsible for neutrino masses is realized radiatively.
• The loop involves the inert scalar $\eta$, the singlet $\sigma$, and the vector-like fermions $S_i$.
• The $Z_6$ symmetry forbids tree-level Dirac masses and ensures the stability of the lightest neutral particle.
• The resulting mass matrix is finite and has rank 2, predicting two massive active neutrinos and one massless state (consistent with oscillation data requiring at least two massive states).

D. Dark Matter Candidate

The lightest electrically neutral particle transforming non-trivially under the residual $Z_6$ symmetry is automatically stable. The authors focus on the complex scalar $\phi^0_2$ (a mixture of $\eta^0$ and $\sigma$) as the Weakly Interacting Massive Particle (WIMP) candidate.

• Stability: Guaranteed by the $Z_6$ symmetry.
• Nature: The model also admits a fermionic DM candidate ($S_1$), but the scalar scenario is analyzed in detail.

3. Key Contributions

1. First Gauged $U(1)_L$ Scotogenic Model: This is the first proposal of a scotogenic model where the stabilizing symmetry is a residual discrete gauge symmetry ($Z_6$) arising from the spontaneous breaking of a gauged lepton number.
2. Dirac Neutrino Masses: It successfully generates tiny Dirac neutrino masses radiatively, avoiding the need for heavy Majorana scales.
3. Natural DM Stabilization: The DM stability is a consequence of the gauge structure, removing the need for an arbitrary global $Z_2$ symmetry.
4. Fractional Charges: The model utilizes fields with fractional lepton numbers ($1/2$) to achieve the specific $Z_6$ remnant and close the mass loop.

4. Results and Phenomenology

A. Dark Matter Viability

• Relic Abundance: The authors performed a scan of the parameter space (varying Higgs portal couplings and masses) to reproduce the observed relic density ($\Omega h^2 \approx 0.12$).
• Direct Detection: The model features a "singlet-dominated" DM candidate (small mixing angle $\theta$ between $\eta^0$ and $\sigma$). This suppresses the coupling to the SM $Z$ boson, evading current constraints from LUX-ZEPLIN (LZ).
• Co-annihilation: The viable parameter space is significantly widened by co-annihilation/rescattering effects when the masses of the dark scalars are nearly degenerate ($\Delta m / m_{DM} \lesssim 0.1-0.2$).
• Fermionic DM: The model also supports a fermionic DM candidate ($S_1$) which can annihilate via $Z'$ exchange, offering an alternative to the scalar scenario.

B. Charged Lepton Flavor Violation (cLFV)

The model predicts cLFV processes such as $\mu \to e\gamma$ and $\mu \to 3e$ at the one-loop level, mediated by the charged inert scalars ($\eta^\pm$) and heavy neutral leptons ($S_i$).

• Constraints: The branching ratios were calculated and compared against current limits (MEG II for $\mu \to e\gamma$) and future sensitivities (Mu3e for $\mu \to 3e$).
• Findings:
  • For small Yukawa couplings, the dipole contribution ($\mu \to e\gamma$) dominates.
  • For large Yukawa couplings, box diagrams ($\mu \to 3e$) become significant.
  • The model is compatible with all current experimental bounds.
  • To satisfy constraints, the charged scalar masses ($m_{\eta^\pm}$) typically need to be in the multi-TeV range (near 10 TeV) or the Yukawa couplings must be small.
  • The predicted rates are within the reach of upcoming experiments (MEG II, Mu2e, COMET, Mu3e).

C. Gauge Bosons

• The model predicts a new neutral gauge boson $Z'$ associated with $U(1)_L$.
• In the "leptophilic" limit (negligible kinetic mixing), the $Z'$ does not couple to quarks, suppressing hadron collider production.
• Constraints from LEP-II imply a lower bound on the symmetry breaking scale: $w \gtrsim 1.7$ TeV.

5. Significance

This work provides a compelling, theoretically motivated extension of the Standard Model that simultaneously addresses the origin of neutrino masses and the nature of Dark Matter. By promoting lepton number to a gauge symmetry, the model:

• Naturalizes the stability of Dark Matter (no ad hoc symmetries).
• Explains the smallness of neutrino masses via a radiative mechanism protected by the gauge structure.
• Predicts distinct signatures: A stable WIMP DM candidate, a heavy $Z'$ boson, and observable charged lepton flavor violation rates that are testable in the near future.
• Demonstrates that fractional charge assignments can be consistently implemented in anomaly-free gauge theories to generate residual discrete symmetries.

The model remains viable across a broad parameter space, particularly where the dark sector scalars are nearly degenerate, offering a robust framework for future experimental verification at colliders and direct detection experiments.