Radiative Neutrino Mass in a Nonholomorphic $T'$ Modular Invariant Model

This paper proposes a non-holomorphic modular-invariant model based on the $T'$ group that successfully realizes the radiative $\texttt{T4-2-i}$ neutrino mass topology while naturally suppressing tree-level seesaw contributions and stabilizing dark matter via a residual $\mathbb{Z}_2$ symmetry, thereby satisfying all current neutrino, flavor, and cosmological constraints.

The Gist

Imagine the universe as a giant, complex machine. For a long time, physicists have had two major mysteries about how this machine works:

1. The Ghost Particles: Neutrinos are tiny, invisible particles that were thought to have no weight (mass), but experiments proved they do have a tiny bit of weight. We don't know how they got it.
2. The Invisible Stuff: There is a huge amount of "Dark Matter" holding galaxies together, but we can't see it or touch it. We don't know what it is.

Usually, scientists try to solve these two puzzles separately. This paper proposes a clever way to solve both at the same time using a specific set of rules called "Modular Symmetry."

Here is a simple breakdown of what the authors did:

1. The "Forbidden" Recipe

The authors are trying to build a model where neutrinos get their mass not directly, but through a "loop" process. Think of it like baking a cake.

• The Old Way (Tree-Level): Usually, you might just mix flour and eggs (a direct process) to make a cake. In physics, this would be a direct way for neutrinos to get mass.
• The Problem: In this specific recipe (called the T4-2-i topology), if you just mix the ingredients, you accidentally create a "bad cake" (unwanted physics that contradicts what we see in the real world).
• The New Solution: The authors use a special set of rules (based on a group called T prime) to act like a strict head chef. This chef says, "No direct mixing allowed! You must go through a complex, one-step loop process to get the mass." This ensures the "bad cake" never gets made.

2. The Magic Ingredient: "Modular Forms"

How does the chef know which ingredients to mix? They use a mathematical tool called Modular Forms.

• Imagine these forms as a magic cookbook. In older versions of this theory, the cookbook only had recipes for "even numbers" (like 2, 4, 6).
• This paper introduces a new edition of the cookbook that includes "odd numbers" (1, 3, 5) as well.
• By using both even and odd numbers, the authors can create a much more flexible menu. This flexibility allows them to:
  • Block the "bad cake" (forbidden tree-level mass).
  • Create the "good cake" (the correct neutrino mass).
  • Crucially: It naturally creates a "security guard" (a symmetry) that keeps the Dark Matter candidate safe from decaying. You don't have to invent a security guard by hand; the math creates one automatically.

3. The Cast of Characters

To make this work, the model introduces new particles:

• Inert Scalars: These are like "ghostly twins" of the Higgs boson. They don't interact with normal matter directly, but they run around in the loop helping to generate neutrino mass.
• Heavy Neutrinos: Big, heavy cousins of the neutrinos we know.
• The Dark Matter Candidate: The authors focus on the lightest of the "odd" particles (a heavy Majorana fermion named N1). Because of the "security guard" mentioned above, this particle cannot decay into normal stuff, so it survives from the Big Bang until today as Dark Matter.

4. The "Loop" Connection

The paper explains that the neutrino mass is generated in a loop involving these new particles.

• Analogy: Imagine a relay race. The neutrino passes a baton (mass) to a heavy particle, which passes it to a ghostly scalar, which passes it back. By the time the baton gets back to the neutrino, it has gained a tiny bit of weight.
• Because this process is so complex (it happens in a loop), the resulting mass is naturally very small, which explains why neutrinos are so light compared to other particles.

5. Did It Work? (The Results)

The authors ran a massive computer simulation to see if this model fits with real-world data. They checked:

• Neutrino Data: Does it match the known mass differences and mixing angles? Yes.
• Dark Matter: Does it produce the right amount of Dark Matter in the universe? Yes.
  • How? The Dark Matter particles don't just vanish on their own; they "co-annihilate" with their ghostly scalar partners. It's like a group of friends leaving a party together; they clear out the room efficiently, leaving just the right amount of people (Dark Matter) behind.
• Safety Checks: Does it break any known laws of physics (like creating too much energy or messing up the Higgs boson)? No. The model passes all current tests.
• Detection: If we try to catch this Dark Matter in a detector, will we see it?
  • The paper says probably not easily. Because the Dark Matter only interacts with normal matter through a very complex, "loop-generated" path (like a secret tunnel), the signal is extremely weak. It's like trying to hear a whisper in a hurricane. This is actually a good thing, as it explains why we haven't found it yet.

Summary

This paper builds a theoretical machine that:

1. Explains why neutrinos have mass (using a complex loop recipe).
2. Explains what Dark Matter is (a stable particle protected by math).
3. Solves both problems using a single, elegant mathematical framework (T' Modular Symmetry) without needing to invent extra "fixes" by hand.

The authors conclude that this model is a viable, consistent way to describe our universe, and it works for both possible arrangements of neutrino masses (Normal and Inverted). Future experiments looking for Dark Matter or rare particle decays will be the ultimate test to see if this "recipe" is the one nature actually uses.

Technical Summary

Problem Statement

The origin of neutrino masses and the identity of dark matter (DM) remain open questions in the Standard Model (SM). Radiative neutrino mass models offer a unified framework where the new fields generating neutrino masses at the loop level also provide a DM candidate. Specifically, the $T_{4-2-i}$ topology represents a one-loop realization of the Majorana neutrino mass, viewed as a radiative extension of the type-II seesaw mechanism. This topology involves a scalar triplet ($\Delta$), two inert scalar doublets ($\rho, \phi$), and singlet fermions ($N_i$) propagating in the loop.

However, realizing this topology presents two significant challenges:

1. Tree-level Contamination: The presence of both a scalar triplet and singlet fermions generically allows for tree-level type-I and type-II seesaw contributions. These must be forbidden to ensure neutrino masses arise purely radiatively.
2. Dark Matter Stability: The topology itself does not guarantee the stability of the lightest beyond-SM state. Previous realizations often relied on ad hoc discrete symmetries (e.g., $D_4 \times Z_3 \times Z_5 \times Z_2$) to forbid unwanted operators and stabilize DM, leading to complex symmetry structures and enlarged scalar sectors.

Methodology

The authors propose a non-holomorphic modular-invariant framework based on the double-cover group $T'$ (the binary tetrahedral group, isomorphic to $\Gamma'_3$) to address these challenges.

• Modular Symmetry and Forms: Unlike traditional holomorphic modular models (often supersymmetric) restricted to even weights, the non-holomorphic framework utilizes polyharmonic Maaß forms. These forms depend on both $\tau$ and $\bar{\tau}$ and satisfy a Laplace-type constraint rather than holomorphicity. Crucially, the homogeneous group $T'$ allows for both even and odd modular weights.
• Field Assignments:
  • Standard Model (SM) fields are assigned even modular weights.
  • Dark sector fields (inert doublets $\rho, \phi$ and the singlet fermion $N_1$) are assigned odd modular weights.
• Mechanism for Suppression and Stability:
  • Forbidding Tree-Level Terms: Modular invariance requires the total modular weight of any operator to vanish. The specific assignments of even and odd weights, combined with the representation structure of $T'$, forbid the tree-level type-I and type-II seesaw operators. For instance, the type-I contraction $3 \otimes 1 \otimes 1$ does not contain a $T'$ singlet, and the type-II operator $LL\Delta$ fails modular weight conservation.
  • DM Stability: The residual $Z_2$ symmetry associated with the vicinity of the fixed point $\tau = i$ naturally stabilizes the lightest odd state. This eliminates the need for an ad hoc discrete symmetry; stability is a consequence of modular invariance.
• Phenomenological Analysis: The model is implemented in FeynRules and CalcHEP, with relic density and direct detection calculations performed using micrOMEGAs 6.2.3. A comprehensive scan is performed over the parameter space, fitting:
  • Charged lepton masses.
  • Neutrino oscillation observables ($\Delta m^2_{21}, \Delta m^2_{3\ell}, \theta_{12}, \theta_{23}, \theta_{13}, \delta_{CP}$).
  • Electroweak (EW) precision observables ($S, T$ parameters).
  • Higgs diphoton signal strength ($R_{\gamma\gamma}$).
  • Dark matter relic abundance ($\Omega h^2$).
  • Constraints from charged-lepton flavor violation (cLFV), cosmological bounds on $\sum m_i$, and direct detection limits.

Key Contributions

1. First Non-Supersymmetric Realization: This work presents the first fully non-supersymmetric realization of the $T_{4-2-i}$ topology using non-holomorphic $T'$ modular forms with both even and odd weights.
2. Natural DM Stability: The model demonstrates that DM stability can emerge naturally from the residual modular symmetry at $\tau \approx i$, removing the need for manually imposed discrete symmetries.
3. Predictive Flavor Structure: The use of polyharmonic Maaß forms constrains the Yukawa sector, leading to predictive textures for charged-lepton and neutrino mass matrices while suppressing unwanted operators.
4. Dual DM Candidates: The framework allows for both fermionic ($N_1$) and scalar (inert doublet) DM candidates, though the numerical analysis focuses on the fermionic case.

Results

The numerical scan, focusing on the fermionic Majorana candidate $N_1$ as the lightest odd state, yields the following results for both Normal Ordering (NO) and Inverted Ordering (IO) of neutrino masses:

• Viable Parameter Space: Solutions exist for both NO and IO.
  • NO: The allowed range for the DM mass $M_{N_1}$ is broad ($97 \lesssim M_{N_1} \lesssim 852$ GeV), with a sum of neutrino masses $0.060 \lesssim \sum m_i \lesssim 0.120$ eV.
  • IO: The allowed range is narrower ($127 \lesssim M_{N_1} \lesssim 378$ GeV), with a more compressed spectrum ($0.099 \lesssim \sum m_i \lesssim 0.119$ eV).
• Dark Matter Relic Abundance: The observed relic density is achieved primarily through coannihilation with the inert scalar partners ($\rho, \phi$). The mass splitting between $N_1$ and the lightest inert scalar is constrained to be small ($6-9$ GeV) to facilitate this mechanism.
• Direct Detection: The spin-independent (SI) direct-detection cross-section is naturally suppressed. Since $N_1$ is an electroweak singlet, it lacks tree-level $Z$ or Higgs couplings to quarks. The SI interaction arises only via a loop-generated Higgs portal, resulting in rates well below current XENONnT and LZ limits and the neutrino floor.
• Electroweak and Higgs Constraints:
  • The model satisfies global EW precision fits ($S, T$ parameters) with small deviations.
  • The Higgs diphoton signal strength ($R_{\gamma\gamma}$) remains consistent with ATLAS measurements, with contributions from charged scalars ($H^\pm, \delta^{\pm\pm}, \rho^\pm, \phi^\pm$) staying within experimental bounds.
• Flavor Violation: The model satisfies current bounds on cLFV processes (e.g., $\mu \to e\gamma$, $\mu \to 3e$, $\mu-e$ conversion). While cLFV does not currently exclude the viable parameter space, future experiments (MEG II, Mu3e, Belle II) will probe significant portions of the remaining parameter space.

Significance and Claims

The paper claims that the $T_{4-2-i}$ topology can be consistently embedded in a genuinely non-supersymmetric modular framework based on the $T'$ group. The authors emphasize that this construction:

• Solves the problem of unwanted tree-level seesaw contributions and DM stability simultaneously through the modular structure, without ad hoc symmetries.
• Provides a robust phenomenological realization where the fermionic DM candidate $N_1$ is viable for both neutrino mass orderings.
• Predicts a compressed odd-sector spectrum where coannihilation controls the relic density, and loop-induced effects suppress direct detection rates.

The authors conclude that while the scalar DM candidate was also investigated, the fermionic candidate offers the most robust phenomenological realization under the full set of current constraints. Future progress in neutrinoless double-beta decay, direct detection, and cLFV experiments will provide further tests of the surviving parameter space.