Freeze-In Dark Matter and Leptogenesis: a $\psi'$SM route

This paper proposes a $\psi'$SM model based on an $E_6$ extension of the Standard Model that simultaneously explains the observed dark matter relic abundance through freeze-in production of a singlet fermion and the baryon asymmetry via leptogenesis, while also generating light neutrino masses through the type-I seesaw mechanism.

The Gist

Imagine the Universe as a giant, bustling party. For decades, physicists have been trying to figure out two massive mysteries about this party:

1. The Invisible Guest: We know there's a ton of "stuff" holding the party together (gravity), but we can't see it. We call this Dark Matter.
2. The Missing Guests: We know there should be equal numbers of "matter" guests and "antimatter" guests, but somehow, the antimatter guests vanished, leaving only matter behind. This is Baryogenesis.

For a long time, scientists thought the "Invisible Guest" was a strong, heavy character (a WIMP) that bumped into regular matter occasionally. But after years of searching, no one has found this heavy guest. So, scientists are now looking for a different kind of guest: a FIMP (Feebly Interacting Massive Particle). Think of a FIMP as a ghost that is so shy it barely touches anything, making it incredibly hard to catch.

This paper proposes a new story about how this shy ghost (Dark Matter) and the missing antimatter guests came to be, all within a specific, elegant framework called the $\psi'$SM (pronounced "psi-prime SM").

Here is the story, broken down with some everyday analogies:

1. The Grand Blueprint (The $E_6$ Extension)

Imagine the Standard Model (our current best understanding of physics) as a basic house. The authors of this paper say, "Let's build a mansion on top of that house." They use a blueprint based on a complex mathematical shape called $E_6$.

When you break this mansion down, it reveals a hidden room with a special door labeled $U(1)_{\psi'}$. This door acts like a security system.

• The Rule: Inside this room, there are two types of people:
  • Right-Handed Neutrinos (RHNs): These are heavy, invisible particles that help explain why regular neutrinos are so light (like a heavy anchor pulling a tiny boat).
  • Singlet Fermions ($N_1$): These are the Dark Matter candidates. They are "singlets," meaning they don't interact with the usual forces, but they do have a special charge under the new security system.

2. The "Freeze-In" Mechanism (The Ghost Entering the Party)

In the old theory (Freeze-Out), Dark Matter was like a popular guest who arrived early, hung out with everyone, and then left when the party got too cold.

In this new theory (Freeze-In), the Dark Matter is like a ghost that never really enters the party.

• The Analogy: Imagine a very quiet, shy ghost ($N_{DM}$) standing outside a warm, crowded room (the early Universe).
• The Process: Inside the room, there is a special scalar particle (let's call it $N$, a heavy "mother" particle). Occasionally, this mother particle decays (breaks apart) and accidentally spits out a tiny piece of the ghost.
• The Catch: Because the ghost is so shy (its interactions are incredibly weak), it never gets enough "handshakes" to join the party properly. It just slowly accumulates in the room over time.
• The Result: By the time the party cools down, there is just the perfect amount of ghostly matter to explain the Dark Matter we see today. The paper shows this works for ghosts ranging from very light (a few MeV) to moderately heavy (a few hundred GeV).

3. The "Leptogenesis" (The Great Imbalance)

While the ghost is slowly filling the room, something else is happening with the heavy Right-Handed Neutrinos (RHNs).

• The Analogy: Imagine the heavy RHNs are like a pair of twins who are almost identical but have a tiny, almost invisible difference in their DNA.
• The Resonance: Because they are so similar (nearly degenerate), when they decay, they create a "resonance" effect—like pushing a swing at just the right moment to make it go higher.
• The Outcome: This resonance amplifies a tiny difference between matter and antimatter. The heavy neutrinos decay, creating a slight surplus of "matter" particles over "antimatter" particles.
• The Sphaleron: Later, a cosmic process (called a sphaleron) converts this lepton imbalance into a baryon imbalance. This is why we have stars, planets, and people, and no antimatter galaxies.

4. The Cosmic Strings (The Fingerprint)

Here is the coolest part of the paper. When the "security door" ($U(1)_{\psi'}$) was locked shut in the early Universe, it didn't just close; it left behind some cosmic scars called Cosmic Strings.

• The Analogy: Think of these strings like cracks in a cooling sidewalk or wrinkles in a bedsheet. They are long, thin, and carry energy.
• The Detection: Because the Dark Matter is so shy, we can't catch it in a lab. But these cosmic strings vibrate and wiggle, sending out Gravitational Waves (ripples in space-time).
• The Future: The authors suggest that if we can detect these specific ripples with future telescopes, we won't just see the strings; we can figure out exactly how heavy the Dark Matter ghost is and how the symmetry was broken.

Summary

This paper tells a unified story:

1. The Setup: A new symmetry ($U(1)_{\psi'}$) exists in a larger version of the Standard Model.
2. The Dark Matter: A shy ghost ($N_1$) is slowly "leaked" into existence by a decaying particle, never reaching thermal equilibrium (Freeze-In).
3. The Matter/Antimatter: Heavy twins (RHNs) use a resonance trick to tip the scales, creating the matter we see today (Leptogenesis).
4. The Proof: The process leaves behind cosmic strings that might be detectable as gravitational waves, offering a way to test this theory even if we can't catch the Dark Matter directly.

It's a clever way to solve two of the universe's biggest puzzles with one elegant, non-supersymmetric mechanism, replacing the "heavy, hard-to-find" WIMP with a "shy, hard-to-catch" FIMP.

Technical Summary

Here is a detailed technical summary of the paper "Freeze-In Dark Matter and Leptogenesis: a $\psi'$SM route" by Adeela Afzal and Rishav Roshan.

1. Problem Statement

The Standard Model (SM) fails to explain two fundamental cosmological puzzles:

1. Dark Matter (DM): The nature of the non-baryonic matter comprising ~27% of the universe's energy density.
2. Baryon Asymmetry: The observed matter-antimatter asymmetry in the universe.

While the Weakly Interacting Massive Particle (WIMP) paradigm has been the leading DM candidate, null results from collider and direct detection experiments have motivated the exploration of Feebly Interacting Massive Particles (FIMPs). These particles are produced via the freeze-in mechanism, where they never reach thermal equilibrium with the early universe plasma due to extremely weak couplings.

Furthermore, traditional thermal leptogenesis (explaining baryogenesis via heavy right-handed neutrino decay) typically requires high reheating temperatures ($T_{RH} > 10^9$ GeV), which can conflict with low-scale DM scenarios or supersymmetry constraints.

The Goal: To propose a unified, non-supersymmetric framework based on an $E_6$ extension of the Standard Model that simultaneously accommodates:

• Freeze-in production of a stable singlet fermion as Dark Matter.
• Neutrino mass generation via the Type-I seesaw mechanism.
• Baryogenesis via resonant leptogenesis at potentially lower energy scales.

2. Methodology and Model Framework

The $\psi'$SM Model

The authors construct a non-supersymmetric extension of the SM called $\psi'$SM, derived from the $E_6$ Grand Unified Theory (GUT).

• Symmetry Breaking Chain:
  $$E_6 \to SO(10) \times U(1)_\psi \to SU(5) \times U(1)_\chi \times U(1)_\psi \to G_{SM} \times U(1)_\chi \times U(1)_\psi \to G_{SM} \times U(1)_{\psi'} \to G_{SM}$$
• Residual Symmetry: The final stage involves a residual gauge symmetry $U(1)_{\psi'}$, defined as a linear combination of $U(1)_\chi$ and $U(1)_\psi$:
  $$Q_{\psi'} = \frac{1}{4}(Q_\chi + \sqrt{15}Q_\psi)$$
• Field Content:
  • Right-Handed Neutrinos (RHNs): The standard RHNs ($N^c_i$) residing in the $16$-plet of$SO(10)$are **singlets** under$U(1)_{\psi'}$. This allows them to generate neutrino masses via the seesaw mechanism without being charged under the new gauge symmetry.
  • Singlet Fermions ($N_i$): The $SO(10)$ singlet fermions ($N_i$) carry a non-zero charge under $U(1)_{\psi'}$.
  • Scalar Field ($N$): A scalar field arising from the $27$-plet of$E_6$(sharing quantum numbers with$N_i$) acquires a vacuum expectation value (VEV),$v_{\psi'}$, breaking$U(1)_{\psi'}$.

Mechanisms Implemented

1. Dark Matter Stability: A discrete $Z_2$ symmetry is imposed under which the lightest singlet fermion ($N_1$, denoted as $N_{DM}$) is odd. This forbids its decay, ensuring cosmological stability.
2. Mass Generation: Since $N_{DM}$ is charged under $U(1)_{\psi'}$, it cannot acquire a mass via renormalizable interactions. Its mass arises from a dimension-five operator:
   $$\mathcal{L} \supset \frac{c}{\Lambda} N_{DM} N_{DM} N^\dagger N$$
   Upon symmetry breaking ($N \to v_{\psi'}$), the DM mass is generated: $M_{DM} = \frac{c v_{\psi'}^2}{2\Lambda}$.
3. Leptogenesis: The model utilizes Resonant Leptogenesis. Two heavy RHNs ($N_1, N_2$) are assumed to be nearly degenerate in mass ($\Delta M \ll M$). This degeneracy enhances the CP asymmetry parameter ($\epsilon$) via self-energy corrections, allowing successful baryogenesis even at lower mass scales ($M \sim 10^7$ GeV) and lower reheating temperatures.

3. Key Results

A. Dark Matter Phenomenology (Freeze-In)

• Production Channel: The DM is produced primarily via the decay of the scalar field $N$ into two DM particles: $N \to N_{DM} N_{DM}$.
• Thermalization Avoidance: The coupling is kept sufficiently small ($c v_{\psi'}/\Lambda \ll 1$) to ensure the interaction rate $\Gamma < H$ (Hubble rate), preventing thermalization.
• Relic Abundance: Solving the Boltzmann equation for the comoving number density $Y_{DM}$, the authors find that the observed relic density ($\Omega_{DM}h^2 \approx 0.12$) can be achieved for a wide range of parameters.
• Mass Range: The model supports DM masses ranging from a few MeV to a few hundred GeV, depending on the symmetry breaking scale $v_{\psi'}$ and the scalar mass $M_N$.
• Benchmark Points:
  • BP1: $v_{\psi'} \approx 2.9 \times 10^8$ GeV, $M_N = 10^6$ GeV, $M_{DM} = 1$ GeV.
  • BP2: $v_{\psi'} \approx 9.2 \times 10^9$ GeV, $M_N = 10^4$ GeV, $M_{DM} = 10$ GeV.
• Constraints: The parameter space is constrained by the kinematic condition ($M_N > 2M_{DM}$), the Planck scale limit ($\Lambda < M_{Pl}$), and Lyman-$\alpha$ forest bounds (excluding very light, warm DM).

B. Leptogenesis and Neutrino Masses

• Resonant Enhancement: By tuning the mass splitting $\Delta M$ and the complex angle $z_R$ in the Casas-Ibarra parametrization, the CP asymmetry is maximized.
• Baryon Asymmetry: The model successfully reproduces the observed baryon asymmetry ($Y_B \approx 8.75 \times 10^{-11}$) via the decay of nearly degenerate RHNs.
• Low Reheating Temperature: Unlike standard thermal leptogenesis, this resonant scenario allows for successful baryogenesis at $T_{RH} \sim M_{RHN}$, which can be as low as $10^7$GeV. This flexibility is crucial for compatibility with the freeze-in DM scenario, which often prefers lower$T_{RH}$ to avoid overproduction or thermalization.

C. Cosmic Strings and Gravitational Waves

• The symmetry breaking chain $E_6 \to \dots \to U(1)_{\psi'}$ leads to the formation of metastable current-carrying cosmic strings.
• These strings radiate energy as a stochastic gravitational wave background (SGWB). This provides a unique, indirect detection pathway for the model, potentially constraining the $U(1)_{\psi'}$ breaking scale and, by extension, the DM mass.

4. Significance and Contributions

1. Unified Non-SUSY Framework: The paper presents a minimal, non-supersymmetric $E_6$-derived model that solves the DM, neutrino mass, and baryogenesis problems simultaneously without introducing a large spectrum of superpartners.
2. FIMP Viability: It demonstrates that a singlet fermion, naturally arising from $E_6$ representations, is a viable FIMP candidate produced via scalar decay, covering a broad mass spectrum (MeV to GeV).
3. Resonant Leptogenesis Synergy: The work highlights the synergy between low-scale resonant leptogenesis and freeze-in DM. The flexibility in the reheating temperature allows both mechanisms to coexist without the high-temperature constraints of standard thermal leptogenesis.
4. Observational Prospects: While the DM is "invisible" to direct detection due to feeble couplings, the model predicts gravitational wave signatures from cosmic strings. This offers a concrete target for future observatories (like LISA or Einstein Telescope) to test the $E_6$ symmetry breaking scale.
5. Theoretical Consistency: The model elegantly separates the roles of $SO(10)$ singlets (DM) and $SO(10)$ 16-plet neutrinos (Leptogenesis) via the specific charge assignments of the residual $U(1)_{\psi'}$ symmetry.

Conclusion

The authors successfully construct a $\psi'$SM model where the spontaneous breaking of a residual $U(1)_{\psi'}$ symmetry generates the mass for a stable singlet fermion (Dark Matter) via a dimension-five operator. This DM is produced via freeze-in from scalar decays. Concurrently, the model utilizes nearly degenerate right-handed neutrinos to generate light neutrino masses and the baryon asymmetry via resonant leptogenesis. The framework is testable through future gravitational wave observations of cosmic strings, offering a promising alternative to WIMP-based scenarios.