Two-loop neutrino mass model with modular $S_4$ symmetry

This paper proposes a two-loop radiative neutrino mass model based on modular $S_4$ and $Z_3$ symmetries that successfully explains neutrino oscillation data and charged lepton masses while predicting observable lepton flavor violation and providing viable scalar and fermionic dark matter candidates consistent with cosmological and experimental constraints.

The Gist

The Big Picture: Solving Three Mysteries at Once

Imagine the Standard Model of physics as a very successful recipe book for the universe. It tells us how to make particles like electrons and quarks. However, this recipe book has three glaring holes:

1. The Ghostly Particles: It doesn't explain why neutrinos (tiny, ghost-like particles) have mass, even though the recipe says they should be weightless.
2. The Invisible Stuff: It doesn't explain "Dark Matter," the invisible glue holding galaxies together.
3. The Family Tree: It doesn't explain why some particles are heavy and others are light, or why they mix in specific ways.

This paper proposes a new "master recipe" that fixes all three problems at once. It suggests that neutrinos get their tiny mass not from a direct ingredient, but through a complex, multi-step cooking process that happens twice (a "two-loop" process).

The Secret Ingredient: A Modular "Flavor" Symmetry

To organize this recipe, the authors use a mathematical concept called Modular $S_4$ symmetry.

• The Analogy: Imagine a dance troupe. In the Standard Model, the dancers (particles) move somewhat randomly. In this new model, the dancers must follow a strict, geometric choreography based on the shape of a square (the $S_4$ group).
• The "Modular" Twist: The choreography isn't static; it changes based on a hidden dial called the modulus ($\tau$). When the universe cooled down, this dial was set to a specific position. This setting dictated exactly how the particles interact, determining their masses and how they mix. It's like the dial setting the "flavor" of the universe.

The Kitchen: How Neutrinos Get Their Mass

In many old recipes, neutrinos get mass through a simple, one-step interaction. But this paper argues that if neutrinos got mass that easily, they would be too heavy.

• The Two-Loop Mechanism: Instead of a direct path, the authors propose a "detour." Neutrinos get their mass through a complex, two-step loop involving heavy, invisible particles and new types of Higgs-like fields.
• The "Scotogenic" Effect: Think of this like a secret recipe that only works in the dark. The paper introduces a "Z2 symmetry" (a kind of cosmic "odd/even" rule).
  • Particles with an "odd" number cannot turn into normal "even" particles easily.
  • This rule forces the neutrino mass generation to happen only through the complex, two-step loop.
  • The Result: Because the process is so complicated and indirect, the resulting neutrino mass is naturally tiny, explaining why we haven't noticed it before.

The Bonus: A Double-Duty Dark Matter Candidate

Here is the clever part of the recipe: The same "odd" particles that force the neutrinos to get their mass through the complex loop also serve as Dark Matter.

• The Guardian: Because of the "odd/even" rule (Z2 symmetry), the lightest "odd" particle cannot decay into normal matter. It is stable. It lives forever.
• Two Types of Guardians: The model offers two candidates for this invisible guardian:
  1. A Scalar Candidate: A new type of invisible particle that is a mix of a "singlet" (a lone wolf) and a "doublet" (a pair). Depending on the mix, it interacts with the rest of the universe differently.
  2. A Fermionic Candidate: A heavy, invisible cousin of the neutrino.

The Flavor Connection: Why We Can't See Them (Yet)

The paper connects the invisible Dark Matter to something we can test: Charged Lepton Flavor Violation (LFV).

• The Analogy: Imagine a family where the parents (neutrinos) and the children (electrons/muons) share the same secret handshake. If the parents do a secret dance (neutrino mass generation), the children might accidentally mimic a move they shouldn't (an electron turning into a muon and a photon).
• The Prediction: The model predicts that experiments should eventually see an electron turning into a muon and a flash of light ($\mu \to e\gamma$).
• The Catch: The paper calculates that while this event is possible, it is very rare. Current experiments haven't seen it yet, but the model predicts it will be within reach of future, more sensitive detectors (like the MEG II experiment).

The "Split" Mystery

One of the most unique features of this model is how it handles the "mass splitting" of the Dark Matter particles.

• The Tree-Level Problem: In many theories, you have to manually force two particles to have slightly different masses to make the math work.
• The Radiative Solution: In this model, the two particles start with exactly the same mass (they are twins). However, because of the complex quantum loops (the "two-step cooking"), a tiny difference in their mass is generated naturally over time. It's like two identical twins who, after years of different experiences, end up with slightly different weights. The model doesn't need to force this; it happens automatically as a result of the universe's rules.

Summary of Results

The authors ran the numbers on their new recipe and found:

1. It Works: It successfully reproduces the known masses of charged particles (like electrons) and the mixing patterns of neutrinos, but only if the neutrinos follow a "Normal Ordering" (a specific hierarchy of weights).
2. It's Testable: It predicts that future experiments will likely find the "electron-to-muon" decay signal.
3. It's Viable: It identifies specific ranges of particle masses and interaction strengths where the Dark Matter abundance matches what we see in the universe, without violating current safety limits from particle colliders.

In short, this paper builds a unified kitchen where the reason neutrinos are light, the reason Dark Matter exists, and the reason particles have their specific "flavors" are all tied together by a single, elegant set of geometric rules.

Technical Summary

Technical Summary: Two-loop neutrino mass model with modular $S_4$ symmetry

Problem Statement
The Standard Model (SM) fails to explain the origin of neutrino masses, the nature of dark matter (DM), and the hierarchical structure of lepton masses and mixings. While radiative neutrino mass models (scotogenic models) offer a framework to link small neutrino masses to loop-suppressed mechanisms and DM candidates, minimal realizations often rely on arbitrarily small parameters, such as scalar mass splittings, to generate the correct mass scale. Furthermore, the SM does not address the flavor structure of fermions. The authors propose a model that addresses these issues simultaneously by elevating the neutrino mass generation to a two-loop level to naturally suppress the mass scale and by employing modular flavor symmetries to reduce the arbitrariness of flavor parameters.

Methodology and Model Construction
The authors construct a model extending the SM with an inert scalar doublet ($\eta$), two scalar singlets ($\sigma, \phi$), and four right-handed neutral fermions (three $N_R$ forming an $S_4$ triplet and one singlet $\Omega_R$). The symmetry group is enlarged by a finite modular flavor symmetry $\Gamma_4 \simeq S_4$ and a discrete $Z_3$ symmetry.

• Symmetry Breaking: The $\tau$-modulus acquires a vacuum expectation value (VEV), spontaneously breaking the modular $S_4$ symmetry. The singlet $\sigma$ also acquires a VEV, breaking the $Z_3$ symmetry. This pattern leaves a remnant $Z_2$ parity.
• Particle Assignments: Fields with non-trivial modular weights inherit odd $Z_2$ charges. Consequently, the inert scalars ($\eta, \phi$) and the right-handed neutrinos ($N_R$) are $Z_2$-odd, while the SM fields are even.
• Neutrino Mass Mechanism: The $Z_3$ symmetry forbids the quartic scalar interaction $(\phi^\dagger \eta)^2$ that would generate neutrino masses at the one-loop level. Additionally, the modular weight assignments forbid the tree-level mass splitting between CP-even and CP-odd inert states. Instead, the required mass splitting is radiatively generated via a box diagram involving heavy fermions and the $\sigma$ field. This forces the active neutrino masses to arise from an effective two-loop scotogenic mechanism.
• Dark Matter: The remnant $Z_2$ symmetry stabilizes the lightest $Z_2$-odd state, providing a DM candidate. The model allows for both scalar (lightest inert complex scalar $\chi_1$) and fermionic (lightest $N_1$ or metastable $\Omega_R$) DM scenarios.

Key Results

1. Lepton Sector:

   • The model successfully reproduces charged lepton masses and neutrino oscillation data for normal ordering (NO). The numerical analysis shows that inverted ordering (IO) is not allowed within the model's constraints, even with complex Yukawa couplings.
   • The modular structure links the neutrino mass matrix textures directly to the vacuum value of the modulus $\tau$, reducing parameter arbitrariness.
   • The model predicts observable rates for charged lepton flavor violation (LFV). The process $\mu \to e\gamma$ provides the dominant constraint, with predicted branching ratios potentially within the reach of the MEG II experiment. Conversely, $\tau \to e\gamma$ and $\tau \to \mu\gamma$ rates are predicted to be well below current and near-future sensitivities in the viable parameter space.

2. Scalar Sector:

   • The inert scalar sector features a complex scalar DM candidate ($\chi_1$) formed by the mixing of CP-even and CP-odd states. At tree level, these states are degenerate; the splitting is radiatively induced and remains small compared to the freeze-out temperature, validating the complex scalar description.
   • The phenomenology depends on the mixing angle $\beta$ (singlet-doublet composition). Pure singlet limits yield larger relic abundances but are constrained by direct detection limits, while pure doublet limits lead to underabundance due to efficient annihilation into gauge bosons.
   • Benchmark Points: The authors identify viable benchmark points (BP1, BP2, BP3) that satisfy relic density and direct detection constraints. BP1 represents a testable scenario saturating the relic abundance and lying within the projected reach of XENONnT. BP2 and BP3 represent mixed and doublet-like scenarios, respectively.

3. Fermionic Dark Matter:

   • $N_1$ Scenario: If the lightest $Z_2$-odd fermion is $N_1$, the relic density is controlled by leptophilic annihilation channels ($N_1 N_1 \to \ell^+ \ell^-$) mediated by the charged inert scalar. This creates a strong correlation between DM abundance and LFV. The viable region is shaped more efficiently by $\mu \to e\gamma$ constraints than by direct detection, as the latter is suppressed by the alignment limit.
   • $\Omega_R$ Scenario: A non-standard scenario exists where the singlet $\Omega_R$ is the lightest state. Although not stabilized by $Z_2$, it becomes metastable due to kinematic suppression. Its relic density is controlled by annihilation into Higgs bosons, restricting the viable mass to a narrow region near the Higgs resonance.

Significance and Claims
The paper claims to provide a coherent framework where neutrino masses, lepton flavor violation, and dark matter are intertwined through a specific symmetry structure. The primary significance lies in the dynamical generation of the small mass splitting required for radiative neutrino masses. By forbidding the one-loop contribution and tree-level splitting via $Z_3$ and modular symmetries, the model naturally explains the smallness of the effective parameter without arbitrary tuning.

The model is presented as a predictive framework that:

• Unifies the explanation of neutrino mass generation and DM stability via a remnant $Z_2$ symmetry.
• Offers testable predictions for future LFV experiments (specifically $\mu \to e\gamma$) and direct detection experiments (XENONnT, DARWIN), particularly for the mixed scalar DM regime.
• Demonstrates that a two-loop mechanism can naturally emerge from symmetry principles, addressing the "arbitrariness" often found in minimal scotogenic models.

The authors conclude that the most promising probes for this model are the continued improvement in $\mu \to e\gamma$ searches and future direct-detection experiments targeting the mixed scalar region, which can test a significant fraction of the viable parameter space.