Micron-sized Extra Dimensions and Primordial Black Holes: Charges, Rotating, and Memory Burdened

This paper proposes that six-dimensional primordial black holes with masses ranging from sub-gram to $10^8$ grams, stabilized by the memory burden effect or near-extremality in a TeV-scale extra dimension framework, can serve as dark matter candidates while offering distinctive signatures for detection at future colliders and atmospheric neutrino experiments.

The Gist

Imagine the universe as a giant, multi-layered cake. For a long time, physicists have been puzzled by why gravity is so incredibly weak compared to other forces (like magnetism). It's like trying to lift a car with a feather while a tiny magnet can easily pick up a paperclip.

This paper proposes a delicious new way to slice that cake. It suggests that our universe actually has two extra hidden dimensions that are about the size of a micron (one-millionth of a meter)—roughly the width of a bacterium. These dimensions are so small we can't see them, but they act like a "leak" that dilutes gravity, making it seem weak to us.

Here is the breakdown of their discovery, explained simply:

1. The "Tiny Black Hole" Dark Matter

The authors suggest that the mysterious "Dark Matter" holding galaxies together isn't made of invisible particles, but of tiny black holes born in the very first moments of the universe.

• The Problem: In our normal 4D world, tiny black holes are like popcorn kernels in a hot pan—they evaporate (disappear) almost instantly due to Hawking radiation.
• The Twist: In this 6D universe (4 normal + 2 hidden), things change.
  • Charged & Spinning: If these black holes have an electric charge or are spinning, they become "near-extremal." Think of this like a spinning top that is perfectly balanced; it slows down its evaporation significantly.
  • The "Memory Burden" Effect: This is the paper's biggest surprise. They propose a new rule where, as a black hole gets older, it gets "burdened" by the information (memory) it has absorbed. This acts like a heavy backpack that slows the black hole down.
  • The Result: Because of this "memory burden," even black holes smaller than a grain of sand (sub-gram) can survive for billions of years. They don't vanish; they just sit there, invisible, making up the Dark Matter.

2. The "Neutrino Coincidence"

The paper points out a funny coincidence. The size of these hidden dimensions predicts a specific "gap" in energy levels for particles (called Kaluza-Klein modes).

• The Analogy: Imagine a guitar string. The size of the guitar determines the notes it can play. The size of these hidden dimensions predicts a "note" (energy gap) that happens to match the weight of atmospheric neutrinos (ghostly particles that pass through us constantly).
• Why it matters: This suggests the same hidden dimensions that explain Dark Energy and Dark Matter might also be the reason neutrinos have the tiny mass they do. It connects three big mysteries with one simple geometric shape.

3. Catching Them at the "Future Circular Collider" (FCC)

The paper claims we might be able to catch these tiny black holes in the act.

• The Setup: If we build a super-powerful particle accelerator (like the proposed Future Circular Collider) that smashes particles together at 100 TeV, we might create these micro-black holes.
• The Explosion: These black holes wouldn't last long. They would instantly "pop" (evaporate) into a shower of particles.
• The Signature: Unlike normal particle collisions that produce a few particles, a black hole explosion would be a firework display.
  • The paper predicts that a 100 TeV black hole would explode into about 21 particles at once.
  • These particles would have a specific "thermal" (heat-like) energy pattern.
• The Goal: If we see a burst of ~21 particles with this specific pattern, we can measure exactly how big the hidden dimensions are and confirm the new energy scale of the universe.

Summary of the "Menu"

The paper categorizes these black holes by how long they last:

1. Standard Evaporation: Only very heavy black holes (larger than a mountain) survive to today.
2. Spinning/Charged: Lighter black holes can survive if they spin or have charge.
3. Memory Burden: Even tiny black holes (smaller than a speck of dust) can survive to today if the "memory burden" effect is real. This opens up a whole new world of "light" dark matter candidates.

In a nutshell: The authors propose that our universe has two tiny, hidden dimensions. These dimensions allow tiny, ancient black holes to survive as Dark Matter, explain why neutrinos are light, and could be detected in the future by smashing particles together to create a "firework" of 21 particles.

Technical Summary

1. Problem Statement

The paper addresses two major challenges in theoretical physics:

• The Hierarchy Problem: The vast discrepancy between the electroweak scale ($M_{EW} \sim 100$ GeV) and the four-dimensional Planck scale ($M_{Pl} \sim 10^{19}$ GeV).
• The Nature of Dark Matter: Identifying a viable candidate that explains the observed dark matter density without conflicting with current observational constraints.

The authors explore a specific extension of the Swampland program, specifically the Dark Dimension scenario. This scenario posits the existence of extra spatial dimensions with a size determined by the cosmological constant ($\Lambda$). While the original scenario suggested one extra dimension, this paper investigates a model with two micron-sized extra dimensions ($n=2$). In this framework, the fundamental quantum gravity scale ($M_*$) is lowered to the TeV range ($\sim 10$ TeV), making it accessible to future colliders. The central question is whether Primordial Black Holes (PBHs) in this 6D spacetime can constitute all of the universe's dark matter, considering charged, rotating, and "memory-burdened" configurations.

2. Methodology

The authors employ a combination of theoretical frameworks and phenomenological analysis:

• Theoretical Framework:

  • Swampland Conjectures: Utilization of the AdS Distance Conjecture (linking the size of extra dimensions to $\Lambda$) and the Weak Gravity Conjecture (WGC) (constraining the spectrum of light states and charge-to-mass ratios).
  • Higher-Dimensional Gravity: Analysis of the Reissner-Nordström (charged) and Kerr-like (rotating) metrics in $D=6$ spacetime dimensions.
  • Black Hole Thermodynamics: Calculation of Hawking temperature, entropy, and decay rates for higher-dimensional black holes.
  • Memory Burden Effect: Incorporation of a quantum mechanical suppression mechanism where the evaporation rate is slowed down by a factor related to the black hole's entropy ($S$) after the Page time.

• Analytical Approach:

  • Derivation of lifetime scaling laws for static, near-extremal charged, and rotating black holes under standard Hawking evaporation.
  • Application of the Schwinger effect (pair production) to analyze the discharge of near-extremal black holes.
  • Calculation of the Memory Burden suppression factor ($\Gamma \propto S^{-p}$) to determine how it extends the black hole lifetime.
  • Collider Phenomenology: Estimation of production cross-sections, decay multiplicities, and temperature-mass relations for micro-black holes at future colliders (e.g., FCC).

3. Key Contributions

The paper makes several distinct theoretical and phenomenological contributions:

1. Extension to 6D with Two Extra Dimensions: It moves beyond the single-dimension Dark Dimension scenario to a model with $n=2$ extra dimensions, resulting in a fundamental scale $M_* \sim 10$ TeV.
2. Analysis of Charged and Rotating PBHs: It provides detailed lifetime calculations for 6D black holes that are not just static but also charged (near-extremal) and rotating, showing how these properties affect stability.
3. The Memory Burden Mechanism: The paper rigorously applies the "memory burden" scenario to 6D PBHs. It demonstrates that this mechanism drastically suppresses evaporation, allowing black holes with masses far below the gram scale (sub-gram) to survive to the present day.
4. Unified Neutrino Mass Connection: The authors identify a striking coincidence where the Kaluza-Klein (KK) mass splitting ($\Delta m$) in the $n=2$ scenario ($\sim 0.04$ eV) aligns with the observed atmospheric neutrino mass scale.
5. Collider Signatures: It proposes specific experimental signatures for the Future Circular Collider (FCC), including high-multiplicity events with thermal spectra.

4. Key Results

A. Dark Matter Viability (Lifetime Analysis)

The viability of PBHs as dark matter depends heavily on the evaporation mechanism:

• Standard Hawking Evaporation ($p=0$):
  • Static 6D PBHs require masses $M \gtrsim 10^8$ g to survive to the present.
  • Near-extremal charged PBHs have extended lifetimes due to temperature suppression ($T \propto \sqrt{\beta/S}$), allowing masses down to $M \gtrsim \sqrt{\beta}$ g.
• Memory Burden Scenario ($p \ge 1$):
  • The evaporation rate is suppressed by $S^{-p}$.
  • For $p=1$ and $p=2$, the lifetime scales as $\tau \propto M^{11/3}$ and $\tau \propto M^5$ respectively.
  • Result: This opens a massive new window for dark matter. Sub-gram black holes (down to $10^{-7.5}$ g for charged and $10^{-9.5}$ g for static cases) can survive the age of the universe, effectively evading current gamma-ray and microlensing constraints that rule out lighter 4D PBHs.

B. Collider Phenomenology

• Production: Micro-black holes can be produced at colliders with $\sqrt{s} \sim 10$ TeV (FCC).
• Multiplicity: The average number of decay particles $\langle N \rangle$ is derived as $\langle N \rangle \approx \frac{2\pi}{3} (M_{BH}/M_*)^{4/3}$.
  • For a $100$ TeV black hole at the FCC, $\langle N \rangle \approx 21$.
  • This high multiplicity with a thermal spectrum is a distinct signature separating it from Standard Model backgrounds.
• Measurement: By measuring the slope of $\log(T_H)$ vs. $\log(M_{BH})$, one can directly extract the number of extra dimensions ($n$) and the fundamental scale ($M_*$).

C. Neutrino Mass Connection

The mass splitting between KK modes is calculated as:
$$\Delta m \sim \frac{M_*^2}{M_{Pl}} \approx 4.2 \times 10^{-2} \text{ eV}$$
This value coincides with the atmospheric neutrino mass difference ($\sqrt{\Delta m^2_{atm}} \sim 0.05$ eV), suggesting a unified origin for dark energy, dark matter (PBHs), and neutrino masses within the Swampland framework.

5. Significance and Implications

• Unified Framework: The paper presents a cohesive model connecting the Swampland program, early-universe cosmology (PBHs), collider physics (micro-black holes), and low-energy phenomenology (neutrino masses).
• New Dark Matter Candidate: It revitalizes the PBH dark matter hypothesis by utilizing the "memory burden" effect to allow for very light black holes that were previously thought to have evaporated instantly.
• Testability: Unlike many quantum gravity scenarios, this model is falsifiable.
  • Direct Detection: Future colliders (FCC) can search for high-multiplicity thermal events.
  • Indirect Detection: The specific mass ranges ($10^8 - 10^{21}$ g for standard, sub-gram for memory-burdened) can be cross-referenced with gamma-ray background limits and gravitational wave observatories.
• Theoretical Consistency: The scenario satisfies the Weak Gravity Conjecture and the AdS Distance Conjecture, providing a consistent UV completion for the effective field theory.

Conclusion

The authors conclude that the two-micron-sized extra dimensions scenario with a fundamental scale of $\sim 10$ TeV is a highly attractive framework. It not only solves the hierarchy problem but also provides a robust mechanism for Primordial Black Holes to serve as the totality of dark matter, particularly when the memory burden effect is considered. The prediction of a specific neutrino mass scale and distinct collider signatures makes this a compelling target for next-generation experimental physics.