⚛️ phenomenology

Tri-Resonant Leptogenesis in a Non-Holomorphic Modular A $_4$ Scotogenic Model

This paper proposes a non-holomorphic modular $A_4$ scotogenic model with tri-resonant leptogenesis that successfully generates baryogenesis at the low scale of $\sim 537$ GeV for both normal and inverted neutrino mass hierarchies, while predicting specific neutrino oscillation parameters and being constrained by cosmological data that currently disfavors the inverted hierarchy scenario.

Original authors: Tapender, Surender Verma

Published 2026-02-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tapender, Surender Verma

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, complex machine. For a long time, physicists thought they had the instruction manual (the Standard Model), but they realized the manual was missing two crucial chapters: why the universe is made of matter instead of antimatter (the "Great Imbalance"), and what the invisible "dark matter" holding galaxies together actually is.

This paper proposes a new, elegant solution that ties these two mysteries together, along with a third: why neutrinos (ghostly particles) have mass.

Here is the story of their discovery, explained without the heavy math.

1. The Setup: A New Rulebook

The authors are building a model called the Scotogenic Model. Think of this as a "secret society" of particles.

The Problem: Usually, to explain why there is more matter than antimatter, you need very heavy, invisible particles. But if they are too heavy, we can't test them. If they are light, they usually get "washed out" by other interactions before they can do their job.
The Solution: They use a special mathematical symmetry called Modular $A_4$ . Imagine this as a strict set of rules for how particles can dance together. In this dance, the rules are so specific that they naturally force three heavy particles (Right-Handed Neutrinos) to have almost identical weights.

2. The "Twin" Trick: Tri-Resonant Leptogenesis

This is the core magic of the paper.

The Analogy: Imagine three tuning forks. If you strike one, it vibrates. If you have three tuning forks that are exactly the same size and weight, and you strike them together, they don't just vibrate; they resonate. The sound becomes incredibly loud and powerful.
The Physics: In the early universe, these three heavy neutrinos are like those tuning forks. Because they are nearly identical in mass (degenerate), they resonate. This resonance acts like a volume knob turned up to maximum for a specific effect called "CP violation" (the difference between matter and antimatter behavior).
The Result: Even though the particles are relatively light (around 500–600 GeV, which is heavy for a particle but light enough to be tested at the Large Hadron Collider), this "resonance" allows them to generate enough matter to fill the universe. It's like getting a massive explosion from a tiny spark because the conditions were perfectly tuned.

3. The "One-Handed" Clock

Usually, physics models have many "knobs" (parameters) that scientists can turn to make the math work. This paper is special because it has only one knob.

The Metaphor: Imagine a clock where the only thing that moves is the second hand, but that second hand controls the time, the date, and the temperature.
The Reality: The model uses a complex number called $\tau$ (tau). This single number dictates everything: the mass of the neutrinos, how they mix, and the amount of matter created. Because there is only one knob, the model is highly predictive. It doesn't just say "it could happen"; it says, "If this is true, then this specific thing must happen."

4. The Two Scenarios: Normal vs. Inverted

The authors tested two different "shapes" for the neutrino masses:

Normal Hierarchy (NH): Like a staircase where each step is higher than the last.
- Prediction: The universe behaves "normally." The mixing of neutrinos is moderate, and the total mass of neutrinos is low enough that future telescopes won't see them.
Inverted Hierarchy (IH): Like a staircase where the bottom two steps are huge, and the top one is tiny.
- Prediction: This is the "tougher" scenario. It predicts that the neutrinos mix in a very specific way (almost perfectly 50/50).
- The Catch: The authors found that if the universe follows this "Inverted" path, the total mass of neutrinos would be so high that it might conflict with new data from the DESI telescope (which maps the universe). This suggests the "Inverted" path might be a dead end, or at least very hard to sustain.

5. The "Dark Matter" Sidekick

The model also includes a "Dark Matter" candidate (a particle called $\eta$ ).

Because the rules of the "dance" (the symmetry) forbid certain interactions, this particle is stable and doesn't decay. It hangs around, providing the invisible mass that holds galaxies together.
The model predicts that this Dark Matter particle should weigh between 530 and 1500 GeV. This is a very specific target for future experiments to hunt for.

6. Why This Matters

Testability: Most theories about the early universe involve particles so heavy we can never build a machine to see them. This model suggests the particles are light enough (around 500 GeV) that we might be able to create them in particle colliders soon.
Precision: Because the model has only one "knob" ( $\tau$ $τ$ ), it makes sharp predictions.
- It predicts the angle at which neutrinos mix ( $\theta_{23}$ ) should be very close to a specific value (maximal mixing) if the "Inverted" scenario is true.
- It predicts the "effective mass" of neutrinos ( $m_{ee}$ ), which can be tested by experiments looking for neutrinoless double beta decay (a rare process where a nucleus decays without emitting neutrinos).

The Bottom Line

The authors have built a "Goldilocks" model. It's not too heavy to be untestable, and not too light to be impossible. By using a clever mathematical symmetry ( $A_4$ ) and a "resonance" trick (making three particles almost identical), they explain why we exist, what dark matter is, and why neutrinos have mass, all while making specific predictions that future experiments can prove or disprove.

It's like solving a jigsaw puzzle where you only have one piece, but that piece is so uniquely shaped that it forces the rest of the picture to fall into place perfectly.

1. Problem Statement

The Standard Model (SM) fails to explain three critical phenomena: the origin of neutrino masses, the nature of Dark Matter (DM), and the observed matter-antimatter asymmetry (baryogenesis).

The Scotogenic Model: Proposed by Ma, this model generates neutrino masses radiatively (at the one-loop level) and provides a DM candidate (an inert scalar doublet). However, generating the observed baryon asymmetry via leptogenesis in this framework is challenging at low energy scales (below 1 TeV).
The Challenge: Successful leptogenesis typically requires large Yukawa couplings to generate sufficient CP asymmetry, but large couplings induce strong "washout" effects that erase the asymmetry. Conversely, reducing couplings to suppress washout diminishes the CP asymmetry.
The Goal: The authors aim to realize successful baryogenesis at low scales ( $< 1$ TeV) without resorting to unnaturally small Yukawa couplings, while simultaneously explaining neutrino masses, DM, and CP violation within a unified symmetry framework.

2. Methodology and Theoretical Framework

A. Model Construction

The authors implement the scotogenic model within a non-holomorphic modular $A_4$ symmetry framework.

Particle Content: Standard Model fields plus three right-handed (RH) neutrinos ( $N$ ) and an inert scalar doublet ( $\eta$ ).
Symmetry Mechanism: Instead of an ad-hoc discrete $Z_2$ symmetry to forbid tree-level neutrino masses, the model uses modular weights. Fields with odd modular weights are effectively $Z_2$ -odd, placing them in the dark sector.
CP Violation: A Generalized CP (gCP) symmetry is imposed. This renders all coupling constants real, leaving the complex modulus $\tau$ as the sole source of CP violation. This makes the model highly predictive.
Yukawa Couplings: These are not arbitrary parameters but are determined by modular forms (functions of $\tau$ ).

B. Mechanism for Tri-Resonant Leptogenesis

To overcome the washout problem, the model employs Tri-Resonant Leptogenesis, which requires three nearly degenerate RH neutrinos.

Mass Spectrum Generation: The RH neutrino mass matrix ( $M_N$ $M_{N}$ ) is constructed from the $3 \otimes 3$ $3 \otimes 3$ decomposition of the $A_4$ $A_{4}$ group.
- The dominant term comes from the singlet representation, creating a $(2,3)$ symmetric structure.
- The symmetric contribution from the triplet representation is treated as a small perturbation.
- This perturbation softly breaks the degeneracy, naturally yielding three nearly degenerate RH neutrinos without fine-tuning.
Resonance Condition: The mass splitting ( $\delta_{ij} = |m_{Ni} - m_{Nj}|$ ) is tuned to be comparable to the decay width ( $\Gamma$ ), leading to a resonant enhancement of the CP asymmetry.

C. Numerical Analysis

Flavor Effects: Since the mass scale is low ( $< 1$ TeV), flavor effects are crucial. The authors solve the flavor-dependent density matrix equations rather than simple Boltzmann equations.
Parameter Scanning: They scan the parameter space (modulus $\tau$ $τ$ , coupling constants, scalar masses) to satisfy:
1. Neutrino oscillation data (mixing angles, mass squared differences).
2. Dark Matter relic density constraints (assuming the lightest inert scalar $\eta_I$ is the DM).
3. Observed baryon asymmetry ( $\eta_B \approx 6.12 \times 10^{-10}$ ).
Hierarchies: Both Normal Hierarchy (NH) and Inverted Hierarchy (IH) of light neutrino masses are investigated.

3. Key Contributions and Results

A. Neutrino Phenomenology

Predictive Power: The complex modulus $\tau$ is constrained to very narrow regions in the complex plane for both NH and IH.
Mixing Angles:
- NH: $\theta_{23}$ is not strongly constrained.
- IH: $\theta_{23}$ is predicted to be close to maximal mixing in the lower octant ( $\theta_{23} \in [44.50^\circ, 45.04^\circ]$ ).
CP Phases:
- NH: Sizable CP violation is predicted.
- IH: CP violation is highly suppressed; the Dirac phase $\delta_{CP}$ and Majorana phase $\alpha_{21}$ are predicted to be close to $0^\circ$ or $360^\circ$ .
Observables:
- Effective Majorana Mass ( $m_{ee}$ ): For NH, $m_{ee}$ is below future $0\nu\beta\beta$ sensitivity. For IH, $m_{ee} \in [48.43, 49.75]$ meV, which is within the reach of future experiments like nEXO.
- Sum of Neutrino Masses ( $\sum m_i$ ):
  - NH: $\sum m_i \approx 60.5 - 62.2$ meV (consistent with DESI+BAO and Planck).
  - IH: $\sum m_i \approx 110.5 - 113.7$ meV. This is close to the Planck limit but disallowed by DESI+BAO data, effectively ruling out the IH scenario in this specific model.

B. Baryogenesis Results

Low Scale Success: The model successfully generates the observed baryon asymmetry with RH neutrino masses as low as 537 GeV.
Washout Regime:
- NH: Successful baryogenesis is achieved even in the deep strong-washout regime (decay parameters $K \sim 10^5$ ). This is possible due to the tri-resonant enhancement of CP asymmetry and flavor effects.
- IH: Requires a stronger mass degeneracy ( $O(10^{-8})$ ) compared to NH ( $O(10^{-7} - 10^{-6})$ ) because the intrinsic CP violation is smaller.
Benchmark Points: Two benchmark points (BP1 and BP2) for each hierarchy were identified, confirming that the correct baryon asymmetry can be reproduced without extreme fine-tuning of the resonance condition.

4. Significance and Implications

Natural Degeneracy: The paper demonstrates a model-building mechanism where the $A_4$ symmetry naturally produces the nearly degenerate RH neutrino spectrum required for resonant leptogenesis, avoiding the need for ad-hoc fine-tuning.
Low-Scale Testability: By lowering the RH neutrino mass scale to $\sim 500$ GeV, the model becomes experimentally testable at current or near-future colliders (e.g., LHC, future circular colliders), unlike high-scale leptogenesis models.
Predictive CP Violation: The identification of $\tau$ as the unique source of CP violation links neutrino oscillation parameters directly to baryogenesis. The model makes sharp predictions for $\theta_{23}$ and CP phases, particularly for the Inverted Hierarchy.
Cosmological Constraints: The study highlights the power of cosmological data (DESI+BAO) in constraining particle physics models. The IH scenario is effectively ruled out by current cosmological bounds on the sum of neutrino masses within this framework.
Flavor Dynamics: The successful generation of baryon asymmetry in the deep strong-washout regime for NH underscores the critical importance of including flavor effects and tri-resonant enhancements in low-scale leptogenesis calculations.

In conclusion, this work presents a highly predictive, unified framework that simultaneously addresses neutrino mass generation, Dark Matter, and baryogenesis at the TeV scale, offering clear pathways for experimental verification through neutrinoless double beta decay, neutrino oscillation measurements, and cosmological surveys.