Cogenesis of visible and dark matter in a scotogenic model

This paper proposes a scotogenic model where the CP-violating decay of a heavy right-handed neutrino simultaneously generates the visible matter asymmetry and a sub-GeV dark matter relic density via freeze-in, successfully reconciling these phenomena with neutrino parameters and flavor observables.

The Gist

The Big Mystery: Two Worlds, One Origin

Imagine the universe as a giant, dark ocean. We know there are two types of "stuff" floating in it:

1. Visible Matter (The Luminous Islands): This is the stuff we see—stars, planets, and us. It makes up about 15% of the ocean.
2. Dark Matter (The Invisible Currents): This is the invisible stuff that holds galaxies together. It makes up about 85% of the ocean.

For a long time, scientists thought these two things were created by completely different rules. But this paper asks a bold question: What if they were born from the same event?

The authors propose a story where the "islands" (visible matter) and the "currents" (dark matter) were created at the same time, from the same "parent," and in a way that explains why there is more dark matter than visible matter (a ratio of roughly 5 to 1).

The Stage: The "Scotogenic" Model

To tell this story, the authors use a specific set of rules called the Scotogenic Model. Think of this model as a special kitchen where the ingredients are hidden.

• The Hidden Ingredient (The $Z_2$-Odd Neutrino): In this kitchen, there is a heavy, invisible particle (let's call him N2) that doesn't play by the usual rules. It's "odd" in a mathematical sense, meaning it's forbidden from mixing with normal matter unless it decays.
• The Secret Recipe (CP Violation): When N2 decays, it doesn't split evenly. It's like a biased coin flip. It prefers to create one type of particle over another. This bias is called CP Violation.

The Story: How Everything Was Born

Here is the step-by-step drama of how the universe got its matter:

1. The Big Split (The Decay)

Imagine the heavy particle N2 is a giant, unstable balloon floating in the early universe. It pops (decays).

• The Visible Side: It releases a burst of "leptons" (particles like electrons). Because of the biased coin flip (CP violation), it releases slightly more matter than antimatter. This tiny imbalance is the seed for all the stars and galaxies we see today.
• The Dark Side: It also releases a "dark particle" (an inert scalar, let's call it $\eta$). This particle is the parent of our Dark Matter.

2. The Great Delay (The "Traffic Jam")

This is the most clever part of the paper. Usually, when a particle is created, it immediately turns into something else. But here, the dark particle $\eta$ gets stuck in traffic.

• The Analogy: Imagine $\eta$ is a runner trying to reach the finish line (turning into stable Dark Matter). But the track is crowded with other particles (the "primordial soup").
• The Interaction: Because $\eta$ interacts with the Higgs field (the "glue" of the universe), it keeps getting bumped around and slowed down by the crowd. It bounces back and forth, staying in a state of "thermal equilibrium" (staying warm and active with the crowd) for a very long time.
• The Result: This delay is crucial. It stops $\eta$ from turning into Dark Matter too early.

3. The Late Release (Freeze-In)

Eventually, the universe expands and the "traffic" thins out. The crowd can no longer keep $\eta$ busy.

• The Release: $\eta$ finally decays into the stable, lightest dark particle (N1).
• The "Freeze-In": Because this happens so late and the connection to normal matter is so weak, these new Dark Matter particles don't interact with anything else. They just "freeze" into existence, filling the universe.

Why This Matters: Solving Three Puzzles at Once

This single story solves three massive problems in physics:

1. Why is there more Dark Matter than Visible Matter?
   The paper shows that the competition between the "traffic" (scattering) and the "decay" naturally creates a ratio where Dark Matter is about 5 times more abundant than visible matter. It's like a factory assembly line that accidentally produces 5 dark widgets for every 1 light widget.

2. Why do neutrinos have mass?
   In this model, the heavy particles and the dark scalar loop around to give tiny masses to neutrinos (ghostly particles that pass through us). The math works out perfectly to match what we observe in experiments.

3. Is the Dark Matter "Cold" enough?
   Dark Matter needs to be heavy enough to hold galaxies together but light enough to not break the laws of physics. The authors show that because the Dark Matter was born from a "delayed decay" (the traffic jam scenario), it ends up with the perfect mass (around 3.8 MeV) to be a "Freeze-In" candidate that fits all cosmological rules.

The "Safety Check"

The authors ran thousands of computer simulations (like testing millions of different recipes) to make sure their story holds up.

• No Leaks: They checked if this model would cause particles to decay in ways we haven't seen (like a muon turning into an electron and a photon). The answer: No, the rates are so low they are undetectable, which is good.
• Structure Formation: They checked if the Dark Matter would be too "hot" (moving too fast) to form galaxies. The "delayed decay" ensures the Dark Matter is slow enough to build the cosmic web we see today.

The Takeaway

This paper proposes a beautiful, unified story: The visible universe and the dark universe are siblings, born from the same heavy parent.

The "secret sauce" is a delayed decay. Just like a parent holding back a child from running out the door until the coast is clear, the dark sector particles waited in the "traffic" of the early universe before finally becoming the Dark Matter we see today. This simple mechanism elegantly explains the mass of neutrinos, the existence of Dark Matter, and the reason we exist at all.

Technical Summary

1. Problem Statement

The paper addresses three fundamental mysteries in cosmology and particle physics simultaneously:

1. The Origin of Matter: Explaining the observed baryon asymmetry of the universe ($\eta_b \sim 10^{-10}$).
2. Dark Matter (DM) Nature: Identifying the particle nature of Dark Matter, which constitutes $\sim 85\%$ of the gravitating matter.
3. Neutrino Mass: Explaining the origin of non-zero neutrino masses and their oscillation parameters.

Standard models often treat these problems separately. The authors propose a unified "cogenesis" framework where the generation of the baryon asymmetry and the production of Dark Matter share a common origin in the early universe, specifically within the context of the Scotogenic model (a radiative neutrino mass model).

2. Theoretical Framework and Model

The authors utilize a minimal extension of the Standard Model (SM) known as the Scotogenic Model, characterized by:

• $Z_2$ Symmetry: A discrete symmetry under which new particles are odd, ensuring the stability of the lightest dark sector particle.
• Field Content:
  • Three heavy right-handed neutrinos (RHNs), denoted $N_1, N_2, N_3$, with hierarchical masses ($M_{N_3} > M_{N_2} \gg M_{N_1}$).
  • One inert scalar doublet $\eta$ (which does not acquire a vacuum expectation value).
• Mass Generation: Neutrino masses are generated radiatively at the one-loop level via the exchange of $N_i$ and $\eta$.
• Key Assumption: The lightest RHN ($N_1$) is the Dark Matter candidate, while the next-to-lightest particle (NLSP) in the dark sector is the inert scalar $\eta$.

3. Methodology: The Cogenesis Mechanism

The core mechanism relies on the CP-violating decay of the second-heaviest sterile neutrino ($N_2$).

• Common Origin: The decay $N_2 \to L + \eta$ (where $L$ is a lepton and $\eta$ is the inert scalar) occurs out of thermal equilibrium.
  • The Lepton ($L$) sector generates a lepton asymmetry, which is later converted into a Baryon Asymmetry via standard sphaleron processes.
  • The Inert Scalar ($\eta$) populates the dark sector.
• Dark Matter Production (Freeze-in):
  • Unlike standard thermal freeze-out, the Dark Matter ($N_1$) is produced via freeze-in.
  • The $\eta$ particles, initially in thermal equilibrium with the primordial plasma (via Higgs portal couplings $\lambda_{3,4,5}$), eventually decay into the stable $N_1$ ($\eta \to N_1 + \dots$).
  • Crucial Dynamic: The decay of $\eta$ is delayed. There is a competition between the scattering of $\eta$ with the thermal bath (which keeps it in equilibrium) and its decay into $N_1$.
  • This delay determines the timing of DM production, which is critical for satisfying cosmological constraints on structure formation.
• Numerical Simulation: The authors solve a coupled system of Boltzmann equations to track the evolution of comoving number densities ($Y$) for $N_2$, $\eta$, the asymmetry $\delta(B-L)$, and $N_1$.

4. Key Contributions

1. Unified Cogenesis in Scotogenic Model: This is the first study to demonstrate that a minimal scotogenic model can simultaneously explain neutrino masses, baryon asymmetry, and sub-GeV Dark Matter through a single CP-violating decay channel.
2. Dynamical Regulation of Relic Density: The paper highlights how the interplay between thermal scattering (Higgs portal) and decay of the NLSP ($\eta$) dynamically regulates the ratio of Dark Matter to Baryon density ($\Omega_{DM}/\Omega_b \sim 5$).
3. Sub-GeV Freeze-in DM: The model successfully produces a sub-GeV mass Dark Matter ($M_{N_1} \sim \text{MeV}$ scale) via a delayed decay mechanism, distinguishing it from standard freeze-in scenarios where DM is produced at higher temperatures.
4. Constraint Satisfaction: The authors perform a rigorous scan of the parameter space to ensure consistency with:
   • Neutrino oscillation data (masses and mixing angles).
   • Lepton Flavor Violation (LFV) limits (e.g., $\mu \to e\gamma$, $\mu \to 3e$).
   • Electric Dipole Moment (EDM) constraints.
   • Cosmological bounds (CMB, BBN, and crucially, structure formation bounds on free-streaming length).

5. Results

• Benchmark Point: A specific benchmark point (Table 3) was identified with:
  • $M_{N_3} = 5 \times 10^{10}$ GeV, $M_{N_2} = 5 \times 10^9$ GeV, $M_{\eta} = 5 \times 10^8$ GeV.
  • Dark Matter Mass: $M_{N_1} \approx 3.8$ MeV.
  • CP Asymmetry: $\epsilon_2 \approx 7.6 \times 10^{-5}$.
  • Baryon Asymmetry: Matches the observed value ($\sim 8.8 \times 10^{-11}$).
  • Relic Density: Satisfies $\Omega_{DM}h^2 = 0.12$.
• Structure Formation: The delayed decay of $\eta$ results in a momentum distribution for $N_1$ that satisfies the lower mass bound for freeze-in DM ($M_{N_1} \gtrsim 3.5$ MeV), avoiding the "warm dark matter" problem that would erase small-scale structures.
• LFV and EDM: Due to the high mass scale of the inert doublet and the specific Yukawa couplings required, the predicted rates for LFV processes (e.g., $Br(\mu \to e\gamma) \sim 10^{-41}$) and EDMs are orders of magnitude below current experimental limits, leaving the parameter space unconstrained by these observables.
• Parameter Space Scan: Extensive Monte Carlo scans (Figures 6 and 8) show that a viable region exists where all constraints (neutrino data, BAU, DM relic density, structure formation) are met simultaneously.

6. Significance

• Theoretical Economy: The model offers a "minimal" solution, linking the origin of neutrino mass, the matter-antimatter asymmetry, and Dark Matter without introducing excessive new fields beyond the standard Scotogenic setup.
• New Production Mechanism: It introduces a novel mechanism where the competition between scattering and decay in the dark sector is the primary driver for setting the correct Dark Matter abundance and mass scale.
• Testability: While the heavy states ($N_2, N_3, \eta$) are too heavy for direct collider detection, the model predicts specific correlations between neutrino parameters and Dark Matter properties. Future improvements in neutrinoless double beta decay experiments or precision measurements of LFV could further probe the parameter space, although current bounds are already satisfied.
• Sub-GeV DM Viability: It provides a robust theoretical justification for sub-GeV Dark Matter, a regime often difficult to reconcile with structure formation constraints in standard freeze-in scenarios.

In conclusion, the paper demonstrates that the minimal scotogenic model is a viable candidate for a unified explanation of the visible and dark sectors of the universe, driven by the CP-violating decay of a heavy sterile neutrino and regulated by the thermal history of the inert scalar.