Radiative neutrino mass generation and dark matter through vectorlike leptons

This paper proposes a radiative three-loop model utilizing vectorlike leptons and asymmetric Yukawa couplings between two scalar doublets to simultaneously generate neutrino masses, provide a dark matter candidate, and remain consistent with current experimental constraints on flavor violation and mixing data.

The Gist

Imagine the universe as a giant, complex machine. For a long time, physicists thought they had the instruction manual for this machine (the Standard Model), but they noticed two major glitches that the manual couldn't explain:

1. The Ghosts: Tiny particles called neutrinos were supposed to be weightless, but experiments showed they actually have a tiny bit of mass.
2. The Invisible Stuff: Most of the universe is made of Dark Matter, an invisible substance that holds galaxies together, but we have no idea what it is made of.

This paper proposes a clever new "patch" to fix both glitches at the same time. Here is the story of how they did it, explained without the heavy math.

The Cast of Characters

To fix the machine, the authors introduced three new types of "actors" to the stage:

1. Vector-Like Leptons (VLLs): Think of these as heavy-duty construction workers. They are heavy particles that don't exist in our normal world but are needed to build the solution.
2. Two New Scalar Doublets (S1 and S2): Imagine these as specialized toolboxes. One toolbox is "light" (CP-even) and the other is "heavy" (CP-odd). They are twins, but they have different weights.
3. The Dark Matter Candidate: This is the invisible guardian. In this model, it's one of the particles from the toolboxes (the lighter one) that never interacts with light, making it perfect for Dark Matter.

The Big Idea: A Three-Loop Recipe

In the world of particle physics, getting a particle to have mass is like trying to bake a cake. Sometimes, you can just mix ingredients (tree-level), but sometimes, you need a complex recipe involving many steps (loops).

The authors propose a three-loop recipe. This is like baking a cake that requires three separate ovens working in a circle, passing ingredients back and forth.

• The Secret Ingredient: The key to this recipe is asymmetry. The authors use two different "flavors" of connections (Yukawa couplings) between the heavy workers and the toolboxes.
• The Result: Because of this asymmetry, the recipe produces a specific result: Two neutrinos get a tiny bit of mass, but the third one remains massless. This matches what we see in nature perfectly, without needing to invent a whole new army of particles.

How the Pieces Fit Together

1. The Mass Split (The "Weight Difference" Trick)

For the recipe to work, the two toolboxes (S1 and S2) cannot weigh exactly the same.

• Imagine two identical twins. If they weigh exactly the same, nothing interesting happens. But if one twin gains a few pounds, the balance shifts.
• In this model, the difference in weight between the "light" and "heavy" versions of the toolboxes creates the conditions necessary to generate neutrino mass. If they were identical, the neutrinos would stay weightless.

2. The Dark Matter Connection

The same "invisible guardian" (the Dark Matter particle) that runs around inside the universe is also the one running through the recipe in the particle collider.

• The Analogy: Think of the Dark Matter particle as a ghostly courier. It carries the "mass" from the heavy construction workers to the neutrinos.
• Because this courier is heavy and interacts very weakly with normal matter, it fits the description of Dark Matter perfectly. It doesn't bounce off atoms (which is why we can't see it), but it does the heavy lifting to give neutrinos their mass.

3. The Safety Check (Flavor Violation)

When you introduce new particles, you have to make sure they don't break other rules. One big rule is that a muon (a heavy cousin of the electron) shouldn't spontaneously turn into an electron and a flash of light ($\mu \to e\gamma$).

• The authors ran the numbers and found that their recipe is safe. The new particles are heavy enough and the connections are just right so that this forbidden transformation happens so rarely that our current detectors won't see it yet. This makes the theory "safe" for now but testable in the future.

The Verdict: Does it Work?

The authors built a computer simulation to test their idea against real-world data:

• Neutrino Masses: It matches the observed masses and mixing angles perfectly.
• Dark Matter: It predicts a Dark Matter particle that has the right amount of abundance to explain the universe and is light enough to be found by future experiments, but heavy enough to avoid being caught by current detectors.
• Predictions: The model predicts that if we build bigger, more sensitive detectors (like the next generation of neutrino experiments or Dark Matter hunters), we might finally see the "ghostly courier" or the specific way neutrinos oscillate.

Summary in a Nutshell

This paper suggests that the reason neutrinos have mass and the reason Dark Matter exists are two sides of the same coin. By introducing a few heavy particles and two slightly different "toolboxes," nature runs a complex, three-step dance that gives neutrinos a tiny weight while hiding a massive, invisible guardian in the shadows. It's a minimal, elegant solution that solves two of physics' biggest mysteries with a single, unified mechanism.

Technical Summary

1. Problem Statement

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

1. Neutrino Masses: Oscillation experiments confirm neutrinos have non-zero masses, implying physics beyond the SM.
2. Dark Matter (DM): Cosmological observations require a non-baryonic DM component, which the SM cannot provide.

While radiative neutrino mass models (where masses are generated via loop corrections) exist, there is a need for comprehensive models that:

• Incorporate Vectorlike Leptons (VLLs).
• Utilize asymmetric Yukawa couplings to generate specific neutrino mass structures.
• Simultaneously address the Dark Matter candidate nature and Charged Lepton Flavor Violation (cLFV) constraints (specifically $\mu \to e\gamma$).
• Fill gaps in the classification of $\Delta L = 2$ operators involving SM gauge fields at the three-loop level.

2. Methodology and Model Framework

A. Particle Content and Symmetries

The authors propose a three-loop radiative model extending the SM with:

• Two Scalar $SU(2)_L$ Doublets: $S_1 \sim (1, 2, 1)$ and $S_2 \sim (1, 2, 3)$ (hypercharges $Y=1$ and $Y=3$).
• One Vectorlike Lepton (VLL) Doublet: $F = F_L + F_R \sim (1, 2, -1)$.
• $Z_2$ Symmetry: All new fields ($S_1, S_2, F$) are odd under a discrete $Z_2$ symmetry, ensuring the lightest $Z_2$-odd particle is stable (a DM candidate) and preventing tree-level neutrino masses.

B. Lagrangian and Mass Generation

The interaction Lagrangian involves asymmetric Yukawa couplings ($g_{ia}$ and $\tilde{g}_{ib}$) connecting the VLLs to SM leptons and the new scalars:
$$\mathcal{L} \supset -g_{ia} \bar{F}_{Li} S_1 \ell_{Ra} - \tilde{g}_{ib} \bar{\ell}^c_{Rb} \tilde{S}_2^\dagger F_{Ri} + \text{h.c.}$$
where $\tilde{S} = i\sigma_2 S^*$.

Neutrino Mass Mechanism:

• Neutrino masses are generated at the three-loop level.
• The process requires Lepton Number Violation (LNV) by two units ($\Delta L = 2$), mediated by:
  1. Mass splitting between CP-even ($H$) and CP-odd ($A$) neutral scalars (controlled by coupling $\lambda_5$).
  2. Mixing between singly charged scalars ($S_1^+$ and $S_2^+$) (controlled by coupling $\lambda_7$).
• The resulting neutrino mass matrix element is:
  $$(m_\nu)_{ab} \propto \frac{\lambda_5 \lambda_7}{(16\pi^2)^3} (g_a \tilde{g}_b + \tilde{g}_a g_b) \times I_{3L}$$
  where $I_{3L}$ is a dimensionless three-loop integral dependent on the masses of the new particles.

Key Structural Feature:
The mass matrix takes the form $m_\nu \propto (g \tilde{g}^T + \tilde{g} g^T)$. With only one generation of VLLs, this structure yields a rank-2 matrix, naturally predicting two non-zero neutrino masses and one massless neutrino.

C. Scalar Potential and Spectrum

The scalar potential includes terms mixing the SM Higgs ($\Phi$) with $S_1$ and $S_2$.

• Inert Doublets: $S_1$ and $S_2$ do not acquire Vacuum Expectation Values (VEVs) due to $Z_2$.
• Physical Spectrum: After Electroweak Symmetry Breaking (EWSB), the model contains:
  • Two CP-even neutral scalars ($h, H$).
  • One CP-odd neutral scalar ($A$).
  • Two singly charged scalars ($H_1^\pm, H_2^\pm$) resulting from mixing angle $\beta$.
  • One doubly charged scalar ($S_2^{\pm\pm}$).

3. Key Contributions

1. Minimal Rank-2 Realization: The model demonstrates that a single generation of VLLs is sufficient to generate a realistic neutrino mass spectrum (two massive, one massless) via asymmetric Yukawa couplings, avoiding the need for multiple VLL generations.
2. Three-Loop Topology with VLLs: It provides a UV-complete realization of a specific class of three-loop operators involving VLLs and scalar doublets with distinct hypercharges, linking neutrino mass generation directly to DM phenomenology.
3. Unified DM and Neutrino Sector: The model identifies the lightest inert scalar ($H$ or $A$) as the DM candidate. The same mass splittings required for the DM relic density ($m_H \neq m_A$) and charged scalar mixing ($m_{H_1^\pm} \neq m_{H_2^\pm}$) are the exact conditions required to generate non-zero neutrino masses.
4. Phenomenological Consistency: The study rigorously tests the model against:
   • Neutrino oscillation data (mass squared differences and mixing angles).
   • Dark Matter relic density ($\Omega_{DM}h^2$).
   • Direct detection limits (XENONnT, LZ).
   • Charged Lepton Flavor Violation ($\mu \to e\gamma$).

4. Results

A. Dark Matter Phenomenology

• Candidate: The DM is identified as the lightest inert scalar (either CP-even $H$ or CP-odd $A$). The fermionic VLL neutral component is excluded due to unsuppressed $Z$-boson coupling leading to excessive direct detection cross-sections.
• Relic Density: Numerical scans using micrOMEGAs show that viable regions exist for heavy DM masses ($m_{DM} \gtrsim 700$ GeV).
  • Co-annihilation: For specific benchmark points, co-annihilation with charged scalars and other neutral states helps achieve the correct relic abundance.
  • Direct Detection: The spin-independent scattering cross-section is mediated by Higgs exchange. The model successfully satisfies current bounds from XENONnT and LZ, with many points falling below the "neutrino floor."

B. Neutrino Observables

• Fit Quality: Using a $\chi^2$ analysis with the FlavorPy package, the model fits current neutrino oscillation data (NuFIT v6.0) extremely well for both Normal Ordering (NO) and Inverted Ordering (IO).
  • Best-fit $\chi^2$ values are very low ($\sim 0.01$ for NO, $\sim 0.005$ for IO in Benchmark 2).
• Predictions:
  • Massless Neutrino: Consistent with the rank-2 matrix prediction ($m_1=0$ for NO, $m_3=0$ for IO).
  • Mixing Angles: Predictions for $\theta_{12}, \theta_{13}, \theta_{23}$ and the Dirac CP phase $\delta_{CP}$ fall within $1\sigma$ experimental ranges.
  • Effective Masses:
    • NO: Predicts $m_{\beta\beta} \approx 3.7$ meV (below current KamLAND-Zen sensitivity).
    • IO: Predicts $m_{\beta\beta} \approx 48$ meV (within reach of next-generation $0\nu\beta\beta$ experiments like LEGEND-1000).
    • Sum of Masses: $\sum m_i \approx 59$ meV (NO) and $\approx 99$ meV (IO), both well below the Planck cosmological limit ($<120$ meV).

C. Charged Lepton Flavor Violation (cLFV)

• The branching ratio for $\mu \to e\gamma$ is calculated.
• The best-fit points for the Yukawa couplings yield branching ratios well below the current MEG II limit ($1.5 \times 10^{-13}$) and the projected future sensitivity ($6 \times 10^{-14}$), ensuring the model is safe from current cLFV constraints.

D. Benchmark Points

The authors define two benchmark points (BP1 and BP2) representing viable scenarios:

• BP1: DM is the CP-even scalar $H$ ($m_H \approx 842$ GeV). VLL mass $m_E \approx 6.5$ TeV.
• BP2: DM is the CP-odd scalar $A$ ($m_A \approx 998$ GeV). VLL mass $m_E \approx 9.6$ TeV.
  Both points satisfy all theoretical (vacuum stability, perturbativity) and experimental constraints.

5. Significance and Conclusion

This paper presents a minimal, predictive, and testable framework that unifies the origin of neutrino masses and the nature of Dark Matter.

• Theoretical Elegance: It resolves the neutrino mass hierarchy problem (two massive, one massless) naturally through the rank-2 structure of the mass matrix derived from a single VLL generation and asymmetric couplings.
• Experimental Viability: The model is not ruled out by current data. Instead, it makes specific, testable predictions:
  • Neutrinoless Double Beta Decay: A clear signal is predicted for the Inverted Ordering scenario ($m_{\beta\beta} \sim 48$ meV), which is within the reach of upcoming experiments.
  • Direct Detection: While some parameter space is below the neutrino floor, regions exist that could be probed by future direct detection experiments or collider searches for the heavy scalars and VLLs.
  • cLFV: The model predicts $\mu \to e\gamma$ rates that are accessible to future upgrades of the MEG II experiment.

The work highlights that radiative mechanisms at the three-loop level, involving vectorlike fermions and inert scalars, offer a robust solution to the dual problems of neutrino mass and dark matter, providing a clear roadmap for future experimental verification.