Constraining ADD black holes at the LHC with $\sqrt{s} = 14$ TeV

This paper constrains the mass of microscopic black holes produced at the LHC with $\sqrt{s} = 14$ TeV within the ADD model, demonstrating that higher energy loss during formation (parameterized by $\zeta$) and varying numbers of extra dimensions significantly reduce the exclusion limits on the black hole mass for different reduced Planck scales.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle smasher. It takes two tiny particles (protons) and smashes them together at nearly the speed of light. Usually, this just creates a shower of smaller, known particles. But this paper asks a "what if" question: What if, instead of just making a shower, the smash creates a tiny, microscopic black hole?

The authors are testing a specific theory called the ADD model. To understand this, let's use an analogy.

The "Hidden Rooms" Analogy

Our everyday world feels like it has three dimensions of space (up/down, left/right, forward/back). The Standard Model of physics says gravity is weak because it spreads out over these three dimensions.

The ADD model suggests there are actually extra dimensions (like hidden rooms) that we can't see. Imagine gravity is like a smell. In our 3D world, the smell spreads out and gets weak quickly. But if there are extra "rooms" (dimensions) the smell can leak into, it gets even weaker in our world. This theory suggests that if we look closely enough (or smash hard enough), gravity might actually be much stronger than we think, but it's just "leaking" into these extra dimensions.

If gravity is strong enough, smashing two particles together hard enough could squeeze them so tightly that they collapse into a tiny black hole.

The Experiment: Smashing at 14 TeV

The authors simulated what would happen if the LHC ran at its maximum power (14 TeV energy) with a lot of data collected (349.4 "inverse femtobarns" of data—a fancy way of saying "a huge number of collisions").

They looked for these black holes based on three main variables:

1. How many extra dimensions (D) exist? (They tested 3, 5, and 7).
2. How strong is the gravity scale (ΛD)? (Think of this as the "volume knob" for gravity in the extra dimensions).
3. How much energy is lost during the crash? (This is the parameter ζ).

The "Leaky Bucket" Problem (Energy Loss)

Here is the most creative part of their analysis. When two particles smash to form a black hole, it's not a perfect, clean snap. It's like trying to fill a bucket with water while the bucket has a hole in the bottom.

• No Loss (ζ = 0): Imagine a perfect bucket. All the energy from the smash goes into making the black hole.
• High Loss (ζ = 0.35): Imagine a bucket with a big hole. 35% of the energy leaks away as radiation or other particles before the black hole even settles down.

The authors found that if energy leaks away (high ζ), you need a much bigger smash to make a black hole of the same size. If you lose too much energy, the black hole simply won't form because there isn't enough "leftover" energy to hold it together.

The Results: What Did They Find?

Since they didn't actually find any black holes (which is good news for the universe, as we don't want tiny black holes running around!), they used their non-discovery to set limits. Think of these limits as "exclusion zones" on a map.

• The "No Loss" Scenario: If we assume no energy is lost during formation, the LHC would have seen black holes up to about 11.8 TeV (if there are 3 extra dimensions and gravity is weak). Since they didn't see them, black holes in that size range are "ruled out."
• The "High Loss" Scenario: If we assume 35% of the energy leaks away, the limit drops significantly. Now, black holes up to only 7.65 TeV are ruled out. Why? Because the "leaky bucket" makes it harder to make big black holes, so the LHC wouldn't have been able to make them even if they existed. The "exclusion zone" shrinks.

The Dimension Factor:
The more extra dimensions there are, the easier it is to make a black hole (because gravity gets stronger). So, if there are 7 extra dimensions, the LHC could rule out even heavier black holes (up to ~12 TeV) compared to just 3 dimensions.

The Bottom Line

This paper is a "search and exclude" mission. The authors calculated exactly how big a black hole the LHC should have been able to make under different theories.

• If the LHC had seen a black hole, it would have proven these extra dimensions exist.
• Because the LHC saw nothing, the authors drew a line in the sand. They said: "If your theory predicts black holes smaller than [X] TeV, and you assume [Y] amount of energy loss, then your theory is likely wrong because we would have seen them."

They found that accounting for energy loss (the "leaky bucket") makes the rules stricter: it becomes harder to rule out the existence of black holes because the machine is less efficient at making them when energy is lost. This helps physicists refine their search for the next big discovery in physics.

Technical Summary

Technical Summary: Constraining ADD Black Holes at the LHC with $\sqrt{s} = 14$ TeV

Problem Statement
The Standard Model (SM) successfully describes fundamental particles but fails to address the naturalness problem of the electroweak scale, neutrino masses, dark matter, and the lack of a gravitational interaction. The Arkani-Hamed, Dimopoulos, and Dvali (ADD) model proposes the existence of $D$ large extra spatial dimensions, which can lower the fundamental Planck scale ($\Lambda_D$) to the TeV range. In this scenario, gravitational interactions become significant at Large Hadron Collider (LHC) energies, potentially allowing for the production of microscopic black holes (MBHs). While previous experimental searches at $\sqrt{s} = 13$ TeV have established exclusion limits, this study aims to provide updated constraints for the High-Luminosity LHC (HL-LHC) conditions ($\sqrt{s} = 14$ TeV, integrated luminosity $\int \mathcal{L} dt = 349.4 \text{ fb}^{-1}$). Crucially, the analysis incorporates the effects of energy loss during the black hole formation phase, a factor often simplified or neglected in previous constraints.

Methodology
The authors investigate the production of tensionless, non-rotating black holes within the ADD framework. The study employs the following methodological approach:

• Theoretical Framework: The analysis utilizes the ADD model where SM particles are confined to a 4D brane, while gravitons propagate in the $(4+D)$-dimensional bulk. The relation between the 4D Planck scale ($M_{Pl}$) and the reduced Planck scale ($\Lambda_D$) is governed by the compactification radius $R_c$.
• Production Mechanism: Black holes are modeled as forming via the collision of highly relativistic partons ($pp \to M_B + X$) when the impact parameter $b$ is smaller than a critical value derived from the hoop conjecture. The production cross-section is calculated using a geometric approximation ($\sigma \approx \pi b_{max}^2$) integrated with parton distribution functions (CTEQ6L).
• Energy Loss Parameterization: A key methodological addition is the inclusion of a loss parameter $\zeta$. During the formation phase (the "balding" phase), the black hole radiates a fraction of its initial energy, linear momentum, and angular momentum. The authors assume proportional loss for energy and linear momentum, setting $f_E = f_P = \zeta$. The final stationary mass $M_B$ is thus reduced from the initial collision energy.
• Evaporation and Lifetime: The study examines the Hawking evaporation process. The lifetime ($\tau_B$) is estimated based on the Hawking temperature, which depends on the Schwarzschild radius and the number of dimensions $D$.
• Simulation and Constraints: Using the BlackMax generator, the authors compute total production cross-sections for $M_B \in [2, 12]$ TeV and $\Lambda_D \in [1, 9]$ TeV across dimensions $D = 2$ to $7$. They apply a detector efficiency factor ($C_{eff} = 0.6$) to determine the expected number of events ($N_B$). Exclusion limits are derived at the 95% confidence level (C.L.) based on the non-observation of events exceeding the expected background.

Key Contributions

1. Incorporation of Formation Losses: The paper explicitly quantifies the impact of the energy loss parameter $\zeta$ on exclusion limits. It demonstrates that neglecting this loss ($\zeta=0$) yields significantly more optimistic (higher) mass limits compared to scenarios where energy dissipation is accounted for (e.g., $\zeta = 0.35$).
2. Updated HL-LHC Projections: The study provides specific exclusion contours for the $\sqrt{s} = 14$ TeV energy regime with an integrated luminosity of $349.4 \text{ fb}^{-1}$, updating previous results obtained at 13 TeV.
3. Dimensional Dependence: The analysis systematically maps how the number of extra dimensions ($D$) influences both the production cross-section and the resulting mass exclusion limits, showing that higher dimensions generally allow for slightly higher cross-sections due to geometric factors.

Results
The numerical analysis yields the following specific constraints for the 95% C.L. exclusion limits on the black hole mass ($M_B$):

• Case $D=3$:
  • For $\Lambda_D = 1$ TeV and no energy loss ($\zeta = 0$), black holes with $M_B \le 11.83$ TeV are disfavored.
  • For $\Lambda_D = 9$ TeV and $\zeta = 0$, the limit drops to $M_B \le 10.33$ TeV.
  • When energy loss is significant ($\zeta = 0.35$), the limits tighten considerably: $M_B \le 7.65$ TeV for $\Lambda_D = 1$ TeV and $M_B \le 6.82$ TeV for $\Lambda_D = 9$ TeV.
• Case $D=7$:
  • For $\Lambda_D = 1$ TeV and $\zeta = 0$, the exclusion extends to $M_B \le 12.03$ TeV.
  • For $\Lambda_D = 9$ TeV and $\zeta = 0$, the limit is $M_B \le 10.88$ TeV.
  • With $\zeta = 0.35$, the limits reduce to $M_B \le 7.80$ TeV ($\Lambda_D = 1$ TeV) and $M_B \le 7.03$ TeV ($\Lambda_D = 9$ TeV).
• General Trends:
  • The production cross-section decreases monotonically as $\Lambda_D$ increases due to the weakening of effective gravitational coupling and the kinematic threshold for parton momentum fractions.
  • The black hole lifetime ($\tau_B$) increases with mass $M_B$ but decreases rapidly as $\Lambda_D$ increases.
  • Increasing the number of dimensions $D$ leads to a slight increase in the production cross-section.

Significance and Claims
The paper claims that the inclusion of the energy loss parameter $\zeta$ is critical for accurate phenomenological constraints. The authors state that "a significant reduction in the aforementioned limits is observed while the loss gets higher," indicating that previous constraints which ignored formation losses may have overestimated the accessible mass range for microscopic black holes. By mapping the $\Lambda_D$–$M_B$ parameter space for various $D$ and $\zeta$, the study provides a "complete picture" of which regions are experimentally accessible and which are ruled out under the specific HL-LHC conditions analyzed. The work reinforces the motivation for future experimental upgrades by highlighting the sensitivity of black hole searches to the fundamental scale of gravity and the dynamics of the formation process.