⚛️ phenomenology

A Novel One-loop Model for Majorana Neutrino Mass and Dark Matter

This paper presents the first complete field-theoretic realization of the finite one-loop T4-3-i topology for Majorana neutrino mass, proposing a minimal singlet-doublet model with Dirac and Majorana fermions and inert scalars that simultaneously explains neutrino oscillations, predicts a massless neutrino, and offers viable fermionic or scalar dark matter candidates stabilized by a $Z_2$ symmetry.

Original authors: Mohamed Belfkir, Mohamed Amin Loualidi, Salah Nasri

Published 2026-08-14

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CC BY 4.0

Original authors: Mohamed Belfkir, Mohamed Amin Loualidi, Salah Nasri

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible puzzle where the pieces we can see—stars, planets, and us—are only a tiny fraction of the picture. Two of the biggest missing pieces are neutrinos and dark matter. Neutrinos are ghostly, nearly massless particles that zip through everything without stopping; for a long time, scientists thought they had no mass at all, but experiments proved they do, just a tiny bit. We still don't know why they are so light or how they get their mass. Then there's dark matter, the invisible "glue" that holds galaxies together. We can't see it, but we know it's there because of how it pulls on stars. It makes up most of the matter in the universe, yet we have no idea what it's made of. The big question driving this research is: Could these two mysteries be connected? Could the same hidden rules that give neutrinos their tiny mass also create the dark matter that fills the cosmos?

This paper is like a detective story where the authors build a new, clever machine to solve both mysteries at once. They focus on a specific, complex blueprint called the T4-3-i topology. Think of this blueprint as a recipe for a "one-loop" machine. In physics, a "loop" is a path particles take where they pop in and out of existence, interacting with each other before disappearing. Usually, if you try to build a machine to make neutrinos heavy enough to explain the universe, you accidentally create a "tree-level" version that makes them way too heavy, breaking the recipe. The authors' trick is to swap out a key ingredient: instead of using a "Majorana" particle (which is its own antiparticle and causes the recipe to go wrong), they use a "Dirac" particle (a distinct pair of particle and antiparticle) to block the unwanted heavy mass. Then, they hide the actual mass-making magic inside a secret loop involving a different, hidden Majorana particle and some invisible "inert" particles that never interact with light.

The result of this clever swap is a model that predicts a very specific outcome: the universe should have one massless neutrino (a neutrino with zero weight) while the other two have tiny masses. This isn't just a guess; the authors ran massive computer simulations to see if this machine works in the real world. They tested it against a mountain of data: how neutrinos change flavors (oscillate), how often they might cause rare particle decays, how the Higgs boson behaves, and how dark matter should act. They found that the machine works perfectly for two different scenarios: one where the dark matter is a heavy, ghostly fermion (a type of particle like an electron), and another where it's a scalar (a type of particle like the Higgs).

Here's the fun part: the two dark matter candidates behave very differently. If the dark matter is the fermion, it's incredibly shy. It doesn't talk to the Higgs boson directly; it only whispers through a complex, loop-induced connection. This means if we try to catch it in a detector, it barely bumps into anything, making it almost invisible to current experiments. It's like a ninja that only leaves a faint footprint. On the other hand, if the dark matter is the scalar, it's much more chatty. It has a direct line to the Higgs boson, so it's much more likely to bump into atoms in a detector. This makes it a prime target for next-generation experiments that are currently being built.

The authors also checked if their machine fits with the "rules of the game," like the laws of physics that keep things stable and the precision measurements of how particles interact. They found that their model passes every test, whether the neutrinos are arranged in a "normal" order or an "inverted" order. The simulations show that for the fermion dark matter, the mass is likely between 92 GeV and 1447 GeV, while for the scalar dark matter, it could be as light as 62.8 GeV (just a bit heavier than the Higgs boson) or as heavy as 896 GeV.

One of the most exciting findings is that this model predicts specific signals for future experiments. If the dark matter is the scalar type, it might be detectable by upcoming liquid xenon experiments, sitting right in the "sweet spot" between current limits and the "neutrino floor" (the point where background noise from cosmic neutrinos makes detection impossible). If it's the fermion type, it's likely too quiet for current detectors but could be revealed by looking for rare particle decays or by studying the Higgs boson's behavior more closely.

In short, this paper doesn't just propose a wild idea; it builds a complete, working model that explains why neutrinos are light, what dark matter might be, and how to find it. It suggests that the universe might be running on a hidden, one-loop engine where the same mechanism that gives neutrinos their tiny mass also creates the invisible scaffolding of the cosmos. While the authors haven't proven this is the true nature of reality, their simulations show it's a very strong candidate that fits all the known data and offers clear paths for scientists to test it in the near future.

Technical Summary: A Novel One-loop Model for Majorana Neutrino Mass and Dark Matter

Problem Statement
The paper addresses the challenge of constructing a complete, finite one-loop realization of the Weinberg operator ( $O_W = LLHH$ ) that generates Majorana neutrino masses while simultaneously providing a stable Dark Matter (DM) candidate. Specifically, the authors focus on the T4-3-i topology, a one-loop diagram where a fermion links two external lepton–Higgs pairs. A critical issue with this topology is that if the internal fermion is a Majorana singlet or triplet, the diagram is accompanied by a tree-level Type-I or Type-III seesaw contribution, respectively. In such cases, the tree-level term typically dominates, rendering the loop subleading and the model incomplete as a radiative explanation for neutrino mass. Furthermore, previous attempts to realize this topology (e.g., Refs. [15–17]) often relied on specific symmetries or supersymmetry to suppress tree-level terms, but a fully symmetry-invariant field-theoretic realization of T4-3-i had not been constructed.

Methodology
The authors propose a minimal extension of the Standard Model (SM) to realize the T4-3-i topology without tree-level seesaw contamination. The methodology involves:

Field Content and Symmetries: The model introduces a Dirac fermion $N$ (an $SU(2)_L$ singlet) to replace the would-be Majorana fermion linking the external legs, thereby forbidding the tree-level Type-I seesaw term. Lepton number violation (LNV) is instead placed in a separate Majorana fermion $\psi$ inside the loop. An exact $Z_2$ symmetry is imposed to keep new scalars inert (no vacuum expectation values) and stabilize the lightest $Z_2$ -odd state as a DM candidate.
Model Classification: The authors classify possible electroweak charge assignments for the loop particles based on the hypercharge parameter $\alpha$ $α$ . This yields four variants (T4-3-i-A, -B, -C, -D). The study focuses on the T4-3-i-B1 model, the minimal singlet-doublet realization containing:
- One Dirac fermion $N$ (singlet).
- One Majorana fermion $\psi$ (singlet).
- One inert scalar doublet $\phi$ .
- One inert scalar singlet $\phi'$ .
Analytical Derivation: The authors derive the one-loop neutrino mass matrix, showing it takes the form $M_\nu = \Lambda (Y y^T + y Y^T)$ , where $Y$ and $y$ are flavor vectors. This structure implies a rank-two mass matrix, predicting one massless neutrino. They also derive expressions for Charged Lepton Flavor Violation (cLFV) observables (radiative decays, three-body decays, and $\mu-e$ conversion), Higgs diphoton decay rates ( $h \to \gamma\gamma$ ), and electroweak oblique parameters ( $S, T$ ).
Numerical Analysis: A global scan of the parameter space is performed using a likelihood-based approach. The analysis tests two DM scenarios:
- Fermionic DM: $\psi$ is the lightest $Z_2$ -odd state.
- Scalar DM: The neutral CP-even inert scalar $H^0_1$ is the lightest $Z_2$ -odd state.
  Both scenarios are tested against Normal Ordering (NO) and Inverted Ordering (IO) of neutrino masses. Constraints include neutrino oscillation data (NuFIT 6.0), cosmological bounds on neutrino mass sum, Planck relic abundance, direct detection (DD) limits, cLFV limits, and theoretical consistency (perturbativity, unitarity, vacuum stability).

Key Contributions

First Complete Realization: This work presents the first complete field-theoretic realization of the T4-3-i topology that successfully suppresses tree-level seesaw contributions while maintaining a finite one-loop neutrino mass generation mechanism.
Minimal Particle Content: The T4-3-i-B1 model achieves this with a minimal particle content (one Dirac fermion, one Majorana fermion, one doublet, one singlet) and no large electroweak multiplets.
Dual DM Candidates: The model naturally accommodates two distinct DM pictures: a fermionic candidate ( $\psi$ ) and a scalar candidate ( $H^0_1$ ), both arising from the same minimal field content.
Rank-Two Prediction: The model predicts a rank-two neutrino mass matrix, leading to a massless neutrino at leading order, which strongly constrains the absolute mass scale and the effective Majorana mass.

Results
The numerical analysis demonstrates that the model is viable for both NO and IO neutrino mass orderings and for both DM assignments:

Neutrino Sector: The model successfully reproduces oscillation data. For NO, the mass sum is $\sim 0.058\text{--}0.059$ eV; for IO, it is $\sim 0.098\text{--}0.100$ eV. The effective Majorana mass $m_{\beta\beta}$ is predicted to be in the few-meV range for NO and $0.016\text{--}0.048$ eV for IO.
Dark Matter:
- Fermionic DM: The spin-independent (SI) scattering cross-section is strongly suppressed because the Higgs coupling is loop-induced. Consequently, all viable points lie below current LZ/XENONnT limits and the neutrino floor. The relic abundance is achieved primarily through coannihilation with inert scalars, requiring a compressed spectrum.
- Scalar DM: The SI cross-section is tree-level mediated via the Higgs portal. Viable points exist both below current limits and in the region between current limits and the neutrino floor, making them accessible to next-generation experiments (e.g., DARWIN). Relic density is achieved via the Higgs-funnel region ( $m_{H^0_1} \approx m_h/2$ ) and coannihilation.
Flavor and Precision: The model satisfies all current cLFV limits (e.g., $\mu \to e\gamma$ , $\mu-e$ conversion) and electroweak precision observables ( $S, T$ ). The Higgs diphoton rate ( $R_{\gamma\gamma}$ ) remains consistent with ATLAS measurements, constraining the charged scalar mass and the coupling $\kappa_1$ .

Significance and Claims
The authors claim that this work establishes a compact and predictive framework connecting radiative neutrino mass, Dark Matter, and precision flavor phenomenology.

The model distinguishes itself by avoiding the tree-level seesaw dominance inherent in the T4-3-i topology through a specific field assignment (Dirac $N$ vs. Majorana $\psi$ ) rather than relying on ad-hoc symmetries or supersymmetry.
The rank-two mass matrix provides a sharp prediction (one massless neutrino) that correlates neutrino oscillation data with cLFV processes and DM properties.
The dual nature of the DM candidate allows for distinct experimental signatures: fermionic DM is "stealthy" (below the neutrino floor), while scalar DM is "testable" (potentially above the neutrino floor but below current limits).
The paper concludes that the T4-3-i-B1 model is a robust, minimal realization that satisfies all current theoretical and experimental constraints, offering a concrete target for future cLFV searches, direct detection experiments, and collider studies of the inert scalar sector.

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