Ultraheavy Atomic Dark Matter Freeze-Out through Rearrangement

This paper proposes that symmetric ultraheavy atomic dark matter (ranging from $10^6$ to $10^{10} \,\mathrm{GeV}$) can be thermally produced in the early Universe via a freeze-out mechanism dominated by the geometrically enhanced atomic rearrangement annihilation channel.

The Gist

The Big Mystery: What is Dark Matter?

Imagine the universe is a giant party. We can see the people dancing (stars, planets, us), but we know there are invisible guests in the room because the furniture is moving on its own. These invisible guests are Dark Matter. They make up 80% of the "stuff" in the universe, but we don't know what they are made of.

For a long time, scientists thought these invisible guests were like tiny, heavy marbles (called WIMPs) that bump into each other and disappear. But experiments haven't found them yet.

The New Idea: Dark Atoms

This paper proposes a different idea. Instead of being tiny marbles, maybe dark matter is made of Dark Atoms.

Think of our normal atoms: a heavy nucleus (proton) with a light electron orbiting it. This paper suggests there is a "Dark Sector" with its own version of this:

• A Dark Proton (very heavy).
• A Dark Electron (very light).
• They stick together to form a Dark Atom.

Usually, scientists thought these dark atoms were created in an "unbalanced" way (like having only 100 dark atoms and 0 anti-atoms). But this paper asks: What if they were created in perfect balance? What if there were 100 dark atoms and 100 dark anti-atoms?

The Problem: The "Self-Destruct" Button

If you have equal numbers of matter and anti-matter, they usually annihilate (destroy each other) and vanish, leaving nothing behind. If dark matter worked this way, there would be no dark matter left today to hold galaxies together.

However, the authors found a clever loophole involving a process called "Atomic Rearrangement."

The Analogy: The Dance Floor Swap

Imagine a dance floor where couples (Dark Atoms) are dancing.

1. The Old Way (Elementary Particles): If the dancers were just single people (elementary particles), they would have to get extremely close to bump into each other and disappear. It's like trying to high-five someone in a huge stadium; it's hard to do.
2. The New Way (Dark Atoms): Now, imagine the dancers are holding hands in pairs (atoms). Because they are holding hands, they are much bigger targets.
3. The Rearrangement: When a Dark Atom meets a Dark Anti-Atom, they don't just vanish immediately. Instead, they do a "dance swap." The Dark Proton from one atom grabs the Dark Electron from the other, and vice versa.
   • Result: They form two new, temporary "super-couples" (like a proton-proton pair and an electron-electron pair) that are unstable. These new pairs immediately explode (annihilate) into pure energy (dark light), leaving the original atoms gone.

Why is this important?
Because the "dance swap" (rearrangement) has a much larger target area than the single dancers. It's like trying to hit a beach ball (the atom) instead of a marble (the single particle). The chance of them meeting and destroying each other is massively higher.

The "Ultraheavy" Surprise

Here is the twist. Because this "dance swap" is so efficient at destroying dark matter, almost all of it gets wiped out.

To have any dark matter left over today to form galaxies, the universe had to start with a HUGE amount of it.

• Think of it like a leaky bucket. If the hole is tiny, you need a little water to fill it. If the hole is massive (like our efficient atomic rearrangement), you need a tsunami of water just to have a cup left over at the end.

Because the "hole" is so big, the remaining dark matter particles must be Ultraheavy.

• Normal dark matter theories suggest particles weighing about 100 to 1,000 times a proton.
• This paper suggests these dark atoms weigh 1,000,000 to 10,000,000,000 times the mass of a proton. They are essentially microscopic black holes in terms of weight, but they are still "atoms."

How It All Happened (The Timeline)

1. The Hot Start: The universe was hot. Dark protons and electrons were floating around, but too hot to stick together.
2. Cooling Down: As the universe cooled, the heavy dark protons slowed down and "froze out" (stopped disappearing).
3. The Marriage: The temperature dropped enough for the light dark electrons to pair up with the heavy protons, forming Dark Atoms.
4. The Great Cull: Once the atoms formed, the "dance swap" (rearrangement) kicked in. Dark atoms and anti-atoms met, swapped partners, and exploded into energy. This happened so fast and so efficiently that it wiped out 99.999...% of the dark matter.
5. The Survivors: A tiny, tiny fraction survived because the universe expanded too fast for them to find each other. These survivors are the Ultraheavy Dark Atoms we see today.

Why Should We Care?

1. Solving the "Small Scale" Problem: Because these dark atoms are so heavy and interact with each other (via the rearrangement), they might behave differently than standard dark matter. This could explain why galaxies look the way they do on small scales, which current theories struggle to predict.
2. New Physics: It suggests that dark matter isn't just a boring, invisible particle, but a complex, heavy, atomic system.
3. Detectability: Because these atoms are so heavy and interact so strongly, they might leave different fingerprints in the cosmic background radiation or in how galaxies cluster, giving us new ways to hunt for them.

In a Nutshell

The authors propose that dark matter isn't made of tiny, lonely particles, but of giant, heavy atoms. These atoms were created in perfect balance with their "anti-matter" twins. They tried to destroy each other by swapping partners (rearrangement), which was so effective that it wiped out almost everything. The only reason we have dark matter left today is that the survivors are so incredibly heavy that a tiny number of them is enough to hold the universe together.

Technical Summary

1. Problem Statement

The nature of Dark Matter (DM) remains one of the most significant mysteries in physics. While the Weakly Interacting Massive Particle (WIMP) paradigm is well-established, it faces challenges such as the "small-scale problems" in structure formation and stringent constraints from direct detection experiments. Furthermore, standard thermal freeze-out mechanisms impose a theoretical upper limit on DM mass (the unitarity bound, $\sim 100$ TeV) to achieve the observed relic density.

Existing models of Atomic Dark Matter (ADM) typically assume an asymmetric production mechanism (analogous to baryogenesis), where dark atoms and anti-atoms are not produced in equal numbers. This paper addresses a gap in the literature by asking: Can symmetric atomic dark matter be thermally produced via freeze-out? Specifically, the authors investigate whether the annihilation of symmetric dark atoms can deplete the DM density sufficiently to allow for ultraheavy masses (far exceeding the unitarity bound) while matching the observed relic density ($\Omega_{DM}h^2 \approx 0.12$).

2. Methodology and Model Setup

Theoretical Framework

The authors propose a symmetric dark sector containing:

• A heavy dark fermion (dark proton, $\chi_p$) with mass $m_{\chi_p}$.
• A light dark fermion (dark electron, $\chi_e$) with mass $m_{\chi_e}$.
• Their respective antiparticles ($\bar{\chi}_p, \bar{\chi}_e$).
• A dark $U(1)_X$ gauge boson ($A'$) with a small mass, mediating a long-range force.
• A real scalar field ($\phi$) introduced to solve a specific yield mismatch problem (detailed below).

The Lagrangian includes kinetic mixing between the dark photon and the Standard Model photon ($\epsilon F^{\mu\nu}F'_{\mu\nu}$) and Yukawa couplings for the scalar $\phi$. The dark sector temperature $T_\chi$ is decoupled from the SM temperature $T_{SM}$ by a ratio $\xi \approx 0.2$.

Key Physical Mechanism: Atomic Rearrangement

The core innovation of the paper is the identification of Atomic Rearrangement (AR) as the dominant annihilation channel for symmetric dark matter.

1. Formation: As the universe cools below the binding energy ($E_b$), $\chi_p$ and $\chi_e$ form bound states (dark atoms, $\chi_A$).
2. Annihilation: Unlike elementary fermion annihilation (which has a cross-section $\sigma \sim \alpha_D^2/m^2$), the annihilation of a dark atom and an anti-atom proceeds via rearrangement:
   $$(\chi_p \chi_e) + (\bar{\chi}_p \bar{\chi}_e) \to (\bar{\chi}_p \chi_p) + \bar{\chi}_e + \chi_e$$
   The intermediate states ($\bar{\chi}_p \chi_p$ and $\bar{\chi}_e \chi_e$) are unstable and decay into dark photons.
3. Cross-Section Enhancement: The rearrangement cross-section is geometrical, proportional to the square of the atomic Bohr radius ($r_b^2$). Since $r_b \propto 1/(\alpha_D m_{\chi_e})$, the cross-section is enhanced by a factor of roughly $(m_{\chi_p}/m_{\chi_e})^2 \alpha_D^{-4}$ compared to point-particle annihilation. This massive enhancement allows for efficient depletion of DM even at ultra-high masses.

Solving the Yield Mismatch

A critical issue in symmetric models is that the light fermion ($\chi_e$) usually freezes out with a much lower yield than the heavy fermion ($\chi_p$) due to its faster annihilation rate. This would leave excess $\chi_p$ that cannot form atoms.

• Solution: The authors introduce a scalar $\phi$ (mass $2m_{\chi_e} < m_\phi < 2m_{\chi_p}$) that is produced via freeze-in. $\phi$ continuously decays into $\chi_e \bar{\chi}_e$ pairs during the atomic formation epoch, ensuring an abundant supply of $\chi_e$ to pair with the remaining $\chi_p$ and form atoms.

3. Key Contributions

1. Proposal of Symmetric Thermal Freeze-Out for ADM: The paper demonstrates that symmetric atomic dark matter can be a viable thermal relic, challenging the prevailing assumption that ADM must be asymmetric.
2. Identification of Atomic Rearrangement as the Freeze-Out Driver: The authors show that the freeze-out is not determined by the annihilation of free fermions, but by the geometric rearrangement of bound states. This mechanism is significantly more efficient than standard WIMP annihilation.
3. Resolution of the Unitarity Bound: By leveraging the geometric cross-section of atomic rearrangement, the model naturally produces Dark Matter with masses in the range of $10^6$ to $10^{10}$ GeV, comfortably exceeding the standard unitarity bound ($\sim 100$ TeV).
4. Introduction of the $\phi$-Mediated Yield Mechanism: The paper provides a concrete mechanism (scalar decay) to ensure the necessary chemical balance between heavy and light dark fermions to facilitate complete atomic formation.

4. Results

• Relic Density Calculation: The final relic abundance is determined by the freeze-out of the atomic rearrangement process. The final yield $Y_{\chi_A}^f$ is derived as:
  $$Y_{\chi_A}^f \simeq \frac{3\sqrt{5}}{\pi^{3/2}\sqrt{g_*}} \frac{m_{\chi_e} x_f \xi}{M_P}$$
  where $x_f = E_b/T_f$ is the freeze-out parameter.
• Mass Scaling: To match the observed relic density ($\Omega h^2 \approx 0.12$), the dark atom mass scales as:
  $$m_{\chi_A} \propto \frac{1}{m_{\chi_e}}$$
  For typical parameters ($m_{\chi_e} \sim 1$ GeV, $\alpha_D \sim 0.2$), the required DM mass is approximately $10^9$ GeV.
• Parameter Space: The viable parameter space allows for:
  • Dark Proton Mass ($m_{\chi_p}$): $10^6 - 10^{10}$ GeV.
  • Dark Electron Mass ($m_{\chi_e}$): $0.1 - 10^3$ GeV.
  • Dark Coupling ($\alpha_D$): $0.05 - 0.5$.
• Constraints: The model is constrained by Big Bang Nucleosynthesis (BBN) and the overproduction of millicharged dark electrons. The "bottom-right" region of the parameter space (low binding energy) is excluded because atomic formation would occur during BBN, altering light element abundances.

5. Significance and Implications

• New DM Paradigm: This work establishes a robust pathway for ultraheavy thermal dark matter, a regime previously thought inaccessible to thermal production mechanisms without fine-tuning or non-thermal processes.
• Self-Interacting Dark Matter (SIDM): The large geometric cross-section implies that these ultraheavy atoms are strongly self-interacting. This could resolve small-scale structure formation issues (e.g., the "cusp-core" problem) in galaxy halos.
• Observational Signatures:
  • Indirect Detection: Unlike standard heavy WIMPs (which are hard to detect due to low number density), the large rearrangement cross-section means symmetric atomic DM could produce detectable signals in cosmic rays or gamma rays, despite the high mass.
  • CMB Distortions: The annihilation of symmetric atoms could leave imprints on the Cosmic Microwave Background (CMB) power spectrum.
  • Halo Profiles: The strong self-interaction would result in distinct dark matter halo density profiles compared to collisionless CDM models.

In conclusion, Qiu et al. provide a compelling theoretical framework where the unique physics of atomic bound states (specifically rearrangement) allows for the thermal production of ultraheavy dark matter, offering a fresh perspective on the nature of the dark sector.