Scrutinizing Fermionic Dark Matter in Scotogenic Model with Low Reheating Temperature

This paper investigates fermionic dark matter in the scotogenic model within a low reheating temperature cosmology, demonstrating that entropy injection relaxes relic density constraints and that next-generation direct detection and charged lepton flavor violation experiments can effectively probe the resulting expanded parameter space.

The Gist

Here is an explanation of the paper "Scrutinizing Fermionic Dark Matter in Scotogenic Model with Low Reheating Temperature," translated into simple language with creative analogies.

The Big Picture: Solving Two Mysteries at Once

Imagine the universe is a giant, complex puzzle. Scientists have two massive pieces missing:

1. Dark Matter: We know it's there because it holds galaxies together, but we can't see it or touch it.
2. Neutrino Mass: We know these tiny ghost particles exist, but according to the "Standard Model" (the rulebook of physics), they should have no weight. Yet, they do.

This paper proposes a single, elegant solution that fixes both problems at once. The authors are looking at a specific theory called the Scotogenic Model. Think of this model as a "secret society" of new particles that only interact with each other and very rarely with us.

The Cast of Characters

In this secret society, there are two main types of new particles:

• The Ghosts (Fermionic Dark Matter): These are the invisible particles that make up Dark Matter. They are the "lightest" member of the group, which makes them stable and perfect for being the Dark Matter we are looking for.
• The Invisible Twins (Inert Scalars): These are heavy, invisible partners that help the Ghosts do their job.

The "Scotogenic" name comes from the Latin word for "dark" (scotos). The idea is that these particles live in a "dark sector" and only reveal themselves by giving mass to neutrinos through a tricky, one-time loop process (like a secret handshake that only happens once).

The Twist: A "Lazy" Universe (Low Reheating Temperature)

Usually, physicists assume the universe started hot and cooled down smoothly, like a pot of soup cooling on a stove. In this standard scenario, Dark Matter particles would have annihilated (destroyed each other) until just the right amount was left over to match what we see today.

But this paper asks: "What if the universe cooled down differently?"

Imagine the early universe wasn't just cooling; it was being "reheated" by a dying giant (the Inflaton).

• Standard Scenario: The giant dies instantly. The universe is hot, Dark Matter freezes out, and we get our current amount.
• This Paper's Scenario (Low Reheating): The giant dies slowly. It keeps pumping energy (heat) into the universe after the Dark Matter has already stopped interacting.

The Analogy of the Dilution:
Imagine you have a cup of coffee (the Dark Matter) and you add a huge splash of milk (the energy from the dying giant).

• In the standard model, you drink the coffee as is.
• In this paper's model, the giant adds so much milk after you've poured the coffee that the cup gets diluted. The coffee is still there, but it's much weaker.

Because the universe "diluted" the Dark Matter with extra energy, the Dark Matter didn't need to annihilate as much to reach the correct amount we see today. This opens up a new playground for physics. It allows the Dark Matter to be "weaker" (interact less) than previously thought possible, which was a problem in the old models.

The Hunt: How Do We Find Them?

Since these particles are so shy, how do we catch them? The paper suggests two main hunting strategies:

1. The "Silent" Knock (Direct Detection)

Scientists use giant tanks of liquid xenon (like DARWIN or XLZD) buried deep underground to catch Dark Matter.

• The Challenge: Usually, these "Ghost" particles are so shy they never knock on the door.
• The Hope: The paper shows that if the "Invisible Twins" (the heavy scalars) are heavy enough, or if they have a specific relationship with the Higgs boson, they might knock just hard enough to be heard by these future detectors. It's like hoping the ghost knocks on the door just once, loud enough for a super-sensitive microphone to hear.

2. The "Leaky Faucet" (Charged Lepton Flavor Violation)

This is the paper's strongest lead. In this model, the particles that give neutrinos mass also cause a very rare, forbidden event: a muon (a heavy cousin of the electron) turning into an electron and a photon (light) or three electrons.

• The Analogy: Imagine a faucet that is supposed to only drip water. But because of this new physics, it occasionally drips oil or juice instead.
• The Hunt: Experiments are looking for this "oil drip" (specifically the process $\mu \to 3e$). The paper predicts that if the "Lazy Universe" (Low Reheating) scenario is true, these experiments (like MEG II or Mu3e) will be able to see this drip much sooner than expected.

The Conclusion: Why This Matters

The authors found that by allowing the universe to have this "lazy" cooling phase (Low Reheating), they unlocked a huge new area of possibilities.

• Before: We thought the Dark Matter had to be very specific and strong to exist.
• Now: We realize the Dark Matter could be much "weaker" and still exist, because the universe diluted it.

The Takeaway:
This model is not just a math exercise; it's a roadmap. It tells experimentalists exactly where to look.

1. Look for the "drip": The next generation of experiments looking for muons turning into electrons will likely find this model.
2. Look for the "knock": Future giant detectors might finally hear the Dark Matter knocking.

The paper concludes that the "Fermionic Dark Matter" in the Scotogenic model is not a lost cause. It is hiding in plain sight, waiting for us to check the "leaky faucets" and listen for the "knocks" in the next few years. The "Lazy Universe" theory actually makes it easier to find, not harder.

Technical Summary

Here is a detailed technical summary of the paper "Scrutinizing Fermionic Dark Matter in Scotogenic Model with Low Reheating Temperature" by Abhishek Roy and Rameswar Sah.

1. Problem Statement

The Standard Model (SM) fails to explain neutrino masses, the matter-antimatter asymmetry, and the nature of Dark Matter (DM). The Scotogenic Model offers a unified solution by introducing an inert scalar doublet ($\eta$) and three singlet fermions ($N_i$) odd under a discrete $Z_2$ symmetry. While the scalar DM scenario is well-studied, the fermionic DM scenario ($N_1$) faces significant challenges:

• Suppressed Detection: Direct detection signals are loop-induced and highly suppressed.
• Indirect Detection: Annihilation is $p$-wave suppressed due to the Majorana nature of $N_1$.
• Cosmological Tension: Standard thermal freeze-out often requires annihilation cross-sections that conflict with Charged Lepton Flavour Violation (cLFV) constraints or direct detection limits.

The paper addresses whether a non-standard cosmological history, specifically a low reheating temperature ($T_{RH}$), can resolve these tensions. In such a scenario, the inflaton decays late, injecting entropy that dilutes the DM abundance, potentially allowing for viable parameter spaces that are otherwise excluded in standard cosmology.

2. Methodology

The authors perform a comprehensive phenomenological analysis of the fermionic Scotogenic model under the following framework:

• Model Setup:

  • Particle Content: SM fields + Inert Doublet $\eta$ + Singlet Fermions $N_{1,2,3}$.
  • Neutrino Mass: Generated radiatively at the one-loop level via the exchange of $\eta$ and $N_k$. The mass matrix is calculated using the Casas-Ibarra parametrization with the matrix $R$ fixed to the Identity (minimal setup) and Normal Ordering (NO) for neutrino masses.
  • DM Candidate: The lightest singlet fermion $N_1$.
  • Mass Hierarchy: To isolate the fermionic DM effects, the authors enforce $m_{\eta} > 1.5 M_{N_1}$ to suppress scalar-DM co-annihilation, though fermion-fermion co-annihilation is included.

• Cosmological Evolution (Low Reheating):

  • The Boltzmann equation is solved in a background dominated by a decaying inflaton field ($\phi$) before the radiation-dominated era.
  • Key Variable: The inflaton decay width $\Gamma_\phi$ determines the reheating temperature ($T_{RH}$).
  • Mechanism: If $T_{dec} > T_{RH}$ (DM decouples before reheating ends), the subsequent entropy injection from inflaton decay dilutes the DM number density. This allows the model to reproduce the observed relic density ($\Omega_{DM}h^2 \approx 0.12$) with a smaller annihilation cross-section than required in standard freeze-out.

• Constraints & Scanning:

  • Theoretical: Vacuum stability, perturbativity, perturbative unitarity, and global minimum conditions.
  • Experimental:
    • Electroweak Precision Tests (S and T parameters).
    • Higgs diphoton decay rate ($h \to \gamma\gamma$).
    • cLFV limits: $\mu \to e\gamma$, $\mu \to 3e$, and $\mu \to e$ conversion in nuclei.
    • Direct Detection (DD): Current limits from LUX-ZEPLIN (LZ).
  • Numerical Tools: micrOMEGAs for relic density and cross-sections; Scan over parameters ($m_{N_1}, m_\eta, \lambda_{3,4,5}, \Gamma_\phi$).

3. Key Contributions

1. Entropy Dilution Mechanism: The paper demonstrates quantitatively how a low reheating temperature ($T_{RH} < T_{dec}$) relaxes the constraints on the annihilation cross-section. The entropy injection dilutes the DM density, meaning a larger coupling $\lambda_5$ (which suppresses the cross-section) can still yield the correct relic abundance.
2. Parameter Space Expansion: It identifies a new region of viable parameter space where $\lambda_5$ is larger ($> 4 \times 10^{-10}$) and the inflaton decay width is smaller ($\Gamma_\phi < 10^{-15}$ GeV). In standard cosmology, these points would be excluded because the DM would be under-produced.
3. Correlation between Cosmology and cLFV: The study establishes a critical link between the cosmological history ($\Gamma_\phi$) and cLFV observables. Lower $\Gamma_\phi$ (lower $T_{RH}$) allows for larger $\lambda_5$, which in turn suppresses the Yukawa couplings required for neutrino masses, thereby relaxing cLFV constraints.
4. Complementarity of Probes: The work highlights that while direct detection is challenging for fermionic DM, the combination of next-generation cLFV searches and direct detection can cover the viable parameter space, even in non-standard cosmological scenarios.

4. Key Results

• Relic Density & $\Gamma_\phi$:

  • For $\Gamma_\phi > 10^{-15}$ GeV, the scenario mimics standard freeze-out ($T_{dec} < T_{RH}$).
  • For $\Gamma_\phi < 10^{-15}$ GeV, $T_{dec} \gg T_{RH}$. The DM decouples during the inflaton-dominated era. The subsequent entropy injection dilutes the yield, requiring a larger $\lambda_5$ to compensate.
  • The variable $\Delta N_1$ (deviation from standard freeze-out) reaches unity for low $\Gamma_\phi$, indicating maximal dilution.

• cLFV Sensitivity:

  • The model predicts that $N_1$ behaves as a $\mu\tau$-philic particle due to cLFV constraints suppressing couplings to electrons ($Y^\nu_{1e} \ll Y^\nu_{1\mu}, Y^\nu_{1\tau}$).
  • Future Probes:
    • $\mu \to e\gamma$ and $\mu \to 3e$: Next-generation experiments (MEG II, Mu3e) will probe a significant portion of the parameter space, particularly for $\Gamma_\phi \gtrsim 10^{-16}$ GeV.
    • $\mu \to e$ Conversion: Experiments like COMET and Mu2e (sensitivity to $\mu \to e$ conversion in Titanium) will probe even lower $\Gamma_\phi$ values ($\gtrsim 10^{-17}$ GeV), covering the low-reheating regime.

• Direct Detection (DD):

  • The spin-independent cross-section ($\sigma_{SI}$) is loop-suppressed but can be enhanced by the quartic coupling $\lambda_3$.
  • For $\lambda_3 > 10^{-3}$, the predicted $\sigma_{SI}$ falls within the sensitivity reach of future experiments like DARWIN and XLZD (1000 tonne-year exposure).
  • There is no strong correlation between $\sigma_{SI}$ and $\Gamma_\phi$, meaning DD sensitivity is robust regardless of the reheating temperature, provided $\lambda_3$ is sufficiently large.

5. Significance

This paper provides a crucial "rehabilitation" of the fermionic Scotogenic Dark Matter scenario.

• Viability: It proves that fermionic DM in this model is not ruled out by the tension between relic density requirements and cLFV limits, provided the universe underwent a low-reheating phase.
• Testability: It shifts the narrative from "untestable" to "highly testable." The synergy between cLFV searches (which are highly sensitive to the Yukawa structure modified by the cosmological history) and future direct detection (sensitive to $\lambda_3$) offers a comprehensive strategy to test the model.
• Cosmological Implications: It underscores the importance of non-standard cosmological histories in particle physics model building. Ignoring low reheating temperatures could lead to the premature exclusion of viable BSM scenarios.

In conclusion, the authors demonstrate that the fermionic Scotogenic model remains a compelling candidate for Dark Matter, with its parameter space significantly enlarged and made testable by upcoming experiments in the next decade, specifically through the interplay of cLFV and direct detection in a low-reheating cosmological context.