Next-to-Minimal Freeze-in Dark Matter

This paper extends the concept of Minimal Freeze-in Dark Matter to next-to-minimal scenarios by investigating the impacts of non-instantaneous reheating and exploring higher SU(2)$_L$ representation fermion dark matter models, which can revive excluded WIMP candidates and enhance the discoverability of freeze-in models in the Boltzmann-suppressed regime.

The Gist

Imagine the universe as a giant, bustling party. For decades, physicists have been trying to figure out what the "Dark Matter" guests are. The leading theory for a long time was that these guests were WIMPs (Weakly Interacting Massive Particles). The idea was that they were like partygoers who arrived early, mingled with everyone, and then slowly left as the party cooled down.

But, the party search has been tough. We've looked everywhere, and we haven't found these WIMPs yet. This paper proposes a new, slightly weirder, but very elegant way to think about these guests. It's called "Next-to-Minimal Freeze-in Dark Matter."

Here is the breakdown of the paper's ideas using simple analogies:

1. The "Too Hot to Handle" Problem

In the old WIMP story, the Dark Matter guests needed to be light enough to mingle with the hot, energetic Standard Model particles (the other guests) when the universe was young.

This paper suggests a different scenario: What if the Dark Matter guests are so incredibly heavy that the universe was never hot enough for them to even show up?

• The Analogy: Imagine the universe is a swimming pool. The water temperature represents the energy of the universe.
  • Old Theory: The guests (Dark Matter) are swimmers who can easily jump in and swim around when the water is warm.
  • New Theory: The guests are giant, heavy boulders. The water (universe) gets hot, but never hot enough to melt the boulders or make them float. They are "Boltzmann suppressed"—a fancy way of saying the heat isn't strong enough to create them easily.

2. The "Freeze-In" Mechanism

If the universe isn't hot enough to make them, how do they get there?

• The Analogy: Think of the universe as a factory.
  • Freeze-Out (Old Way): The factory is running at full capacity, churning out products (Dark Matter) until it's saturated, then it stops.
  • Freeze-In (New Way): The factory is barely running. It's a slow, drip-drip-drip production line. The Dark Matter particles are so rare and heavy that they are only created in tiny, tiny amounts by the "heat" of the universe, but they never build up enough to interact with each other. They just slowly "freeze in" to existence over time.

3. The "Next-to-Minimal" Twist

The authors previously proposed a "Minimal" version of this, which used the simplest possible building block: a Doublet (a pair of particles). It was like building a house with just two bricks.

This paper explores the "Next-to-Minimal" versions. They ask: What if we use slightly bigger building blocks?

• They look at Triplets (3 particles), Quintuplets (5 particles), and Septuplets (7 particles).
• Why do this? It's like testing if a house built with 5 bricks is more stable or easier to find than one built with 2. It turns out, these heavier "multi-brick" structures have some cool properties:
  • Stability: Some of these shapes are naturally stable. They don't need a "security guard" (a made-up symmetry) to keep them from falling apart; their shape just keeps them safe.
  • Detectability: Because they are heavier and come in these specific shapes, they might leave a different "fingerprint" that our detectors can catch.

4. The "Reheating" Drama

The paper also changes the story of how the universe got hot in the first place (after the Big Bang).

• The Old Story: The universe got hot instantly, like a light switch flipping on.
• The New Story: The universe might have heated up slowly, like a pot of water on a stove.
• The Result: If the heating was slow, it changes the recipe for how many Dark Matter boulders get made. The authors show that even with this slow heating, the math still works out to give us the right amount of Dark Matter we see today.

5. Why Should We Care? (The "Detectability" Hook)

This is the most exciting part.

• The Problem: Usually, "Freeze-in" Dark Matter is so weakly interacting that it's impossible to detect. It's like trying to hear a whisper in a hurricane.
• The Solution: Because these particles are so heavy (heavier than the universe ever got hot), they are actually easier to find than the old light WIMPs.
  • Direct Detection: If these heavy boulders hit a detector on Earth, they would hit with massive force, like a bowling ball hitting a pin. Experiments like DARWIN (a future giant water tank detector) might be able to hear this "thud."
  • Indirect Detection: Some of these heavy particles might eventually decay (fall apart) and release high-energy signals (like gamma rays or neutrinos) that telescopes like KM3NeT or CTAO could spot.

Summary

This paper is essentially saying:

"We've been looking for light, chatty Dark Matter guests and haven't found them. Let's try looking for giant, silent, heavy guests who were born because the universe was too cold to make them easily. Even though they are rare, their sheer weight makes them easier to spot if they ever bump into us. We've checked the math for different 'shapes' of these guests, and they all look promising for future experiments."

It's a shift from looking for a needle in a haystack to looking for a boulder in a sandpit. The boulder is harder to miss!

Technical Summary

1. Problem Statement

The paper addresses the tension between experimental constraints on Weakly Interacting Massive Particles (WIMPs) and the theoretical appeal of minimal dark matter (DM) models.

• The WIMP Crisis: Standard WIMP models, particularly those involving electroweak doublets (like Higgsinos or Winos), are increasingly excluded by direct detection experiments (e.g., LZ, XENONnT) and indirect detection limits, especially in the TeV mass range.
• The Freeze-in Opportunity: If the DM mass ($m$) exceeds the maximum temperature of the thermal bath ($T_{\text{max}}$), DM production is Boltzmann suppressed ($e^{-m/T}$). In this regime, DM never thermalizes. This "Boltzmann suppressed freeze-in" allows for:
  • Reviving experimentally excluded WIMP-like models (by pushing masses to $m \gtrsim 10^{10}$ GeV).
  • Making "invisible" freeze-in models potentially discoverable, as the required couplings are fixed to electroweak strength rather than being arbitrarily weak.
• The Gap: Previous work (Minimal Freeze-in or MFI) focused on a single electroweak doublet fermion with instantaneous reheating. This paper seeks to extend the framework to Next-to-Minimal scenarios, exploring higher $SU(2)_L$ representations and non-standard cosmological histories.

2. Methodology

The authors employ a combination of quantum field theory calculations and cosmological Boltzmann equation analysis.

• Particle Physics Framework:

  • Model Content: They consider vector-like pairs of Weyl fermions transforming under $SU(2)_L$ with dimension $n$ (representations: doublet $n=2$, triplet $n=3$, quintuplet $n=5$, septuplet $n=7$) and hypercharge $Y$.
  • Couplings: They derive the vector and axial-vector couplings to $Z$ and $W$ bosons for arbitrary representations. A key finding is that for $Y=0$ and odd $n$, the tree-level $Z$ coupling to the neutral DM state vanishes, suppressing direct detection cross-sections to loop levels.
  • Production Mechanisms: They calculate production cross-sections for $q\bar{q} \to \Psi\bar{\Psi}$ via $\gamma, Z$ (neutral current) and $q\bar{q}' \to \Psi\bar{\Psi}'$ via $W$ (charged current/co-production).
  • Stability Analysis: They analyze the cosmological stability of higher representations. While $n=5$ and $n=7$ are often stable in TeV-scale Minimal Dark Matter (MDM) due to accidental symmetries, the authors examine decay operators induced by higher-dimensional Planck-suppressed operators ($\Lambda \sim M_{\text{Pl}}$) in the superheavy regime.

• Cosmological Framework:

  • Reheating Scenarios: They move beyond the assumption of instantaneous reheating. They model the pre-radiation era with an equation of state parameter $\omega$ and a temperature evolution scaling $\alpha$.
  • Boltzmann Equations: They solve for the freeze-in yield ($Y_{\text{FI}}$) integrating reaction densities over the thermal history, accounting for the exponential suppression factor $e^{-2m/T}$.
  • Gravitational Production: They include the sub-dominant contribution from DM production via graviton exchange ($s$-channel) to ensure completeness.

3. Key Contributions

1. Extension to Higher Representations: The paper systematically analyzes DM candidates in $SU(2)_L$ representations up to the septuplet ($n=7$).
2. Non-Instantaneous Reheating: They derive analytical expressions for the freeze-in yield in scenarios where the Universe undergoes a period of matter domination ($\omega=0$) or kination ($\omega=1$) prior to radiation domination, showing how the equation of state significantly alters the required reheating temperature ($T_{\text{rh}}$).
3. Stability Constraints in the Superheavy Regime: They demonstrate that while $n=5$ and $n=7$ are "accidentally stable" in the TeV regime, the superheavy masses required for Boltzmann-suppressed freeze-in make them susceptible to decay via dimension-6 and dimension-8 operators. This imposes strict upper bounds on the DM mass unless the cutoff scale $\Lambda$ is sufficiently high.
4. Direct Detection Re-evaluation: They clarify that for $Y=0$ odd representations, the direct detection cross-section is loop-suppressed ($\sim 10^{-47} - 10^{-46}$ cm$^2$), rendering these models invisible to current and near-future direct detection experiments (LZ, DARWIN), unlike the $Y \neq 0$ cases which are strongly constrained.

4. Key Results

• Relic Abundance:

  • In the instantaneous reheating limit, the correct relic density ($\Omega_{\text{DM}} h^2 \approx 0.12$) is achieved for a specific ratio $m/T_{\text{rh}} \approx 23 \pm 1$ across all representations.
  • For a fixed $m$, the required $T_{\text{rh}}$ is uniquely determined, making these models highly predictive.
  • Figure 1 & 4: Show that for $m \sim 10^{10} - 10^{18}$ GeV, the required $T_{\text{rh}}$ ranges from $10^4$ to $10^{15}$ GeV. Non-instantaneous reheating (e.g., early matter domination) expands the viable parameter space, allowing for lower $T_{\text{rh}}$ for a given $m$.

• Direct Detection Limits:

  • $Y \neq 0$ (e.g., Doublet, Triplet $Y=1$): These have tree-level $Z$ couplings. The LZ experiment excludes masses up to $\sim 10^{10}$ GeV for doublets. Future experiments (DARWIN) will probe even higher masses.
  • $Y = 0$ (Odd $n$): The tree-level $Z$ coupling vanishes. The scattering cross-section is loop-suppressed, falling well below the "neutrino fog" and the reach of DARWIN. These models are effectively safe from direct detection.

• Indirect Detection & Decay:

  • Annihilation: For $Y=0$ models, indirect detection limits (from Fermi-LAT/HESS) on annihilation set lower mass bounds ($m \gtrsim 3.5$ TeV for triplet, $\gtrsim 20$ TeV for quintuplet/septuplet).
  • Decay: For $n=5$ with $Y=0, \pm 1, +2$, the DM lifetime is constrained by Auger observations of galactic $\gamma$-rays ($\tau \gtrsim 10^{30}$ s). This imposes an upper mass bound of $m \lesssim 1.2 \times 10^{10}$ GeV (assuming $\Lambda \sim M_{\text{Pl}}$).
  • Exception: The quintuplet with $Y=-2$ and the septuplet with $Y=0$ are stable against Planck-suppressed operators up to extremely high scales, making them viable candidates even at $m \sim M_{\text{Pl}}$.

• Gravitational Production:

  • Gravitational production is always sub-dominant to electroweak production in these models due to the $M_{\text{Pl}}^{-4}$ suppression, unless the electroweak channel is completely absent (e.g., singlets).

5. Significance and Implications

• Revival of Excluded Models: The paper demonstrates that models previously ruled out as WIMPs (due to direct detection) can be revived as superheavy Boltzmann-suppressed freeze-in candidates.
• Predictivity: Unlike generic freeze-in models with arbitrary couplings, these "Next-to-Minimal" models are determined by the representation and mass alone. The mass fixes the reheating temperature required to match the observed relic density.
• Discovery Potential:
  • Direct Detection: Models with $Y \neq 0$ are within reach of future experiments like DARWIN.
  • Indirect Detection: The metastable nature of the quintuplet ($n=5$) and septuplet ($n=7$) suggests potential decay signals (e.g., $\Psi_0 \to h\nu$) detectable by KM3NeT or CTAO, particularly if the mass is in the PeV range.
  • Collider Physics: While current colliders cannot reach these masses, future 100 TeV or muon colliders could probe the TeV-scale versions of these multiplets if the reheating temperature is lower than the Boltzmann-suppressed regime.
• Cosmological Flexibility: The analysis highlights that the "standard" assumption of instantaneous reheating is not necessary; non-standard cosmological histories (early matter domination) can significantly alter the relationship between DM mass and the reheating temperature, broadening the viable parameter space.

In conclusion, this work establishes a robust framework for Next-to-Minimal Freeze-in Dark Matter, showing that electroweak multiplets with masses far above the thermal bath temperature constitute a viable, predictive, and potentially discoverable class of dark matter candidates that evade current constraints while offering unique signatures for next-generation experiments.