Multi-component Dark Matter and leptogenesis with double seesaw in an extended left-right symmetric theory

This paper extends the Left-Right Symmetric model with a sterile neutrino dark matter candidate and $A_4$ modular symmetry to investigate how varying modular weights influence the relic abundance of dark matter and the mechanism of leptogenesis.

The Gist

Imagine the universe as a giant, complex puzzle. For decades, scientists have been trying to fit the pieces together using a rulebook called the Standard Model. But there are two massive pieces missing from this puzzle:

1. Dark Matter: The invisible "glue" that holds galaxies together, making up about 27% of the universe, but which we can't see or touch.
2. The Great Imbalance: Why is the universe made of matter (us, stars, planets) instead of equal parts matter and antimatter (which would have annihilated each other instantly)?

This paper proposes a new, more elegant way to solve both mysteries at once using a specific theoretical framework. Here is the breakdown in simple terms.

1. The New Rulebook: Left-Right Symmetry

Think of the Standard Model as a house with only a left wing. Scientists have long suspected there should be a "Right Wing" to make the house symmetrical. This is the Left-Right Symmetric Model (LRSM).

In this paper, the author (Ankita Kakoti) adds a new room to this house: a Sterile Neutrino.

• Normal Neutrinos are like shy ghosts that interact very weakly with the rest of the world.
• Sterile Neutrinos are even shyer; they are "sterile" because they don't interact with the weak force at all.
• The Twist: The author suggests that these sterile neutrinos aren't just random particles; they are the Dark Matter we've been looking for.

2. The "Double Seesaw" Mechanism

How do we get these particles to have the right mass? The paper uses a mechanism called the Double Seesaw.

• The Analogy: Imagine a playground seesaw. If a heavy kid sits on one end, the other end goes up. In physics, heavy particles can make light particles (like the neutrinos we know) very light.
• The Double Seesaw: This paper suggests a two-stage seesaw. It's like having a seesaw on top of another seesaw. This double-layered effect explains why neutrinos are incredibly light (almost massless) while the new sterile neutrinos (the Dark Matter) have a specific, tiny mass (in the keV range).

3. The Magic Ingredient: Modular Symmetry (The "Flavor" Symmetry)

This is the most creative part of the paper. Usually, to make these models work, physicists have to invent new, invisible particles called "flavons" to force the math to work out.

The author uses Modular Symmetry (specifically the A4 group) instead.

• The Analogy: Imagine you are arranging a dinner party. Instead of inviting random guests, you have a strict seating chart based on a mathematical pattern (the modular symmetry).
• The Weights: The author assigns different "modular weights" (think of these as different colors or sizes of place settings) to the particles.
  • Weight 4: A specific color arrangement.
  • Weight 6: A different color arrangement.
  • Weight 8 & 10: Even more complex arrangements.

The paper tests these different "color arrangements" to see which one creates a universe that matches what we actually observe.

4. The Results: Which "Color" Works?

The author ran the numbers for these different weights to see if they could explain two things simultaneously:

1. Dark Matter: Is there enough of it? Is it stable enough? Does it decay too fast (which would give off X-rays we would have seen)?
2. Leptogenesis: Did the decay of these heavy particles create the matter/antimatter imbalance we see today?

Here is what the "taste test" revealed:

• Weight 4: The math worked for Dark Matter, but it couldn't explain the matter/antimatter imbalance perfectly.
• Weight 6: The math didn't work out at all; the results were messy.
• Weight 8: It explained Dark Matter well, but failed to explain the matter/antimatter imbalance.
• Weight 10: This was the winner.
  • It successfully predicted a Dark Matter mass range (between 10 and 30 keV) that fits all current telescope constraints (Lyman-alpha forest and X-ray limits).
  • It successfully explained how the universe ended up with more matter than antimatter.

5. The Big Picture Conclusion

The paper argues that the universe might be a "Multi-Component" system. Instead of having just one type of Dark Matter particle, we might have three sterile neutrinos acting together as a team to make up the Dark Matter we observe.

By using the "Modular Symmetry" (the mathematical seating chart) with Weight 10, the author found a single, elegant solution that:

1. Explains why neutrinos are so light.
2. Identifies sterile neutrinos as the Dark Matter.
3. Explains why we exist (the matter/antimatter imbalance).

In a nutshell: The author took a complex theory, added a "double seesaw" to fix the masses, used a mathematical pattern (Modular Symmetry) to organize the particles, and found that one specific pattern (Weight 10) solves the biggest mysteries of the universe without needing to invent any extra, unnecessary particles. It's a "two birds, one stone" solution for particle physics.

Technical Summary

1. Problem Statement

The Standard Model (SM) of particle physics fails to explain two major cosmological phenomena:

1. Dark Matter (DM): The SM lacks a suitable candidate for the non-baryonic matter comprising ~26.8% of the universe's energy density.
2. Baryon Asymmetry of the Universe (BAU): The observed matter-antimatter asymmetry cannot be explained by the minimal CP violation present in the SM quark sector.

While the Left-Right Symmetric Model (LRSM) successfully addresses neutrino mass generation and offers a framework for leptogenesis, the specific mechanism for generating light neutrino masses and the nature of the Dark Matter candidate require further refinement. Specifically, the paper aims to investigate how modular symmetry (specifically $A_4$) and varying modular weights affect the phenomenology of a Double Seesaw mechanism extended with sterile neutrinos acting as multi-component Dark Matter.

2. Methodology

Model Framework

• Extended LRSM: The authors extend the standard LRSM by introducing a sterile neutrino ($S_L$) per generation.
• Scalar Sector: Unlike the triplet scalar model, this work utilizes a Doublet LRSM containing:
  • One Higgs bidoublet $\phi(1, 2, 2, 0)$.
  • Two scalar doublets $H_L(1, 2, 1, -1)$ and $H_R(1, 1, 2, -1)$.
  • Note: $H_L$ does not acquire a Vacuum Expectation Value (VEV), preserving left-right invariance without breaking $SU(2)_L$.
• Mass Generation Mechanism: The light neutrino mass is generated via the Double Seesaw Mechanism. This involves a two-step suppression:
  1. Type-I seesaw involving heavy right-handed neutrinos ($N_R$).
  2. A second suppression involving the sterile neutrino Majorana mass ($\mu$).
     The resulting light neutrino mass matrix is: $m_\nu \approx M_D (M_{RS}^T)^{-1} \mu M_{RS}^{-1} M_D^T$.

Modular Symmetry Implementation

• Symmetry Group: The model is realized using $A_4$ modular symmetry (isomorphic to the $\Gamma_3$ modular group).
• Yukawa Couplings: Instead of introducing flavon fields, Yukawa couplings are expressed as modular forms ($Y$) dependent on the complex modulus $\tau$.
• Variable Modular Weights ($k_Y$): The core methodology involves assigning different modular weights ($k_Y = 4, 6, 8, 10$) to the particle content. The condition that the sum of modular weights in every superpotential term must be zero is strictly enforced.
• Analysis: The authors derive the mass matrices ($M_D$, $M_{RS}$, $\mu$, $M_R$, and $m_\nu$) for each weight case and numerically analyze the resulting phenomenology.

Phenomenological Calculations

• Dark Matter: The sterile neutrinos are treated as multi-component Dark Matter (all three generations are degenerate in mass). The relic abundance ($\Omega_{DM}h^2$) is calculated using the Dodelson-Widrow (DW) mechanism, and decay rates are computed to check against X-ray constraints.
• Leptogenesis: The CP asymmetry ($\epsilon_1$) generated by the decay of the lightest right-handed neutrino ($N_1$) is calculated to determine the baryon asymmetry parameter ($\eta_B$).

3. Key Contributions

1. Multi-component DM in LRSM: The paper proposes a scenario where all three sterile neutrinos are degenerate in mass and contribute to Dark Matter, rather than just the lightest one.
2. Modular Weight Dependence: It systematically investigates how assigning different modular weights ($k_Y = 4, 6, 8, 10$) to the $A_4$ modular forms alters the texture of the mass matrices and the resulting phenomenological constraints.
3. Double Seesaw in Doublet LRSM: It provides a concrete realization of the Double Seesaw mechanism within a doublet scalar sector LRSM, avoiding the need for scalar triplets.
4. Comprehensive Constraints: The study simultaneously checks consistency with:
   • Neutrino oscillation data.
   • Lyman-$\alpha$ forest constraints (structure formation).
   • X-ray constraints (sterile neutrino decay).
   • Planck limits on the sum of neutrino masses ($\sum m_\nu$).
   • Observed Baryon Asymmetry ($\eta_B \approx 6 \times 10^{-10}$).

4. Results

The numerical analysis yielded distinct results for different modular weights:

• Case $k_Y = 4$:

  • Dark Matter: The data points failed to satisfy Lyman-$\alpha$ and X-ray constraints simultaneously. No specific mass range was found where both constraints were met, though the decay rate was negligible ($10^{-37}$ to $10^{-22}$ s$^{-1}$).
  • Leptogenesis: Successfully reproduced the observed baryon asymmetry ($\eta_B$) while satisfying DM and neutrino mass sum constraints.
  • Verdict: Satisfactory for Leptogenesis, but DM constraints were not fully met simultaneously.

• Case $k_Y = 6$:

  • The derived mass matrices did not yield satisfactory results for the phenomenological parameters. Consequently, no plots or detailed results were presented for this weight.

• Case $k_Y = 8$:

  • Dark Matter: A viable mass range was identified: 10 keV – 33.98 keV (Normal Hierarchy) and 10 keV – 26.63 keV (Inverted Hierarchy). These ranges satisfy both Lyman-$\alpha$ and X-ray constraints.
  • Leptogenesis: Failed to reproduce the observed baryon asymmetry. The calculated $\eta_B$ values were significantly higher than the observed limit ($> 10^{-7}$), except for extremely small neutrino mass sums ($\sim 10^{-12}$ eV), which are physically unrealistic.
  • Verdict: Satisfactory for Dark Matter, but failed for Leptogenesis.

• Case $k_Y = 10$:

  • Dark Matter: Allowed mass ranges were found (up to ~30.4 keV for NH and ~28.2 keV for IH). However, the relic abundance data points satisfying the X-ray constraints did not align perfectly with the observed relic density in all regions.
  • Leptogenesis: The baryon asymmetry parameter $\eta_B$ was not satisfied when plotted against the DM mass. However, when plotted against the sum of neutrino masses, the results were satisfactory.
  • Verdict: Mixed results; failed to simultaneously satisfy DM mass constraints and BAU in the same parameter space.

Summary Table of Success:

Modular Weight ($k_Y$)	Dark Matter (Constraints Met)	Baryon Asymmetry (Constraints Met)	Both Simultaneously
4	Partial (No simultaneous mass range)	Yes	No
6	N/A	N/A	N/A
8	Yes	No	No
10	Partial	Partial (Dependent on $\sum m_\nu$)	No

5. Significance and Conclusion

The paper demonstrates that while the Double Seesaw mechanism in an extended Doublet LRSM with $A_4$ modular symmetry is a viable framework for explaining neutrino masses and Dark Matter, the choice of modular weight is critical.

• Key Insight: Different modular weights lead to different textures in the active-sterile mixing matrix ($M_{RS}$) and the heavy neutrino mass matrix ($M_R$). These structural differences drastically alter the phenomenological outcomes.
• Main Conclusion: No single modular weight ($k_Y = 4, 8, 10$) investigated in this work successfully explained both the Dark Matter relic abundance (within all observational constraints) and the Baryon Asymmetry of the Universe simultaneously.
  • $k_Y=4$ works for Leptogenesis but fails DM constraints.
  • $k_Y=8$ works for DM constraints but fails Leptogenesis.
  • $k_Y=10$ shows promise but requires fine-tuning.

The study highlights the tension between explaining Dark Matter and Baryogenesis in this specific theoretical framework and suggests that further model building (e.g., different symmetry groups, additional fields, or different modular weight assignments) is necessary to achieve a unified solution. The work also reinforces the utility of modular symmetry in reducing the number of free parameters (eliminating flavons) in neutrino mass models.