⚛️ phenomenology

Dijets with a large rapidity separation in the next-to-leading order BFKL formalism for searches of large extra dimensions at colliders

This paper estimates the signal for large extra dimensions and the corresponding next-to-leading order BFKL QCD background for high-mass dijet production with large rapidity separation to assess the discovery potential at the HL-LHC and future colliders.

Original authors: Anatolii Iu. Egorov, Victor T. Kim, Viktor A. Murzin, Vadim A. Oreshkin

Published 2026-03-03

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CC BY 4.0

Original authors: Anatolii Iu. Egorov, Victor T. Kim, Viktor A. Murzin, Vadim A. Oreshkin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Hunting for Hidden Dimensions

Imagine the universe is like a giant, multi-story building. We humans live on the ground floor (our 4 dimensions: 3 of space and 1 of time). But what if there are hidden attics or basements (extra dimensions) that we can't see?

Physicists have a theory called the ADD model (named after Arkani-Hamed, Dimopoulos, and Dvali). It suggests that gravity is weak not because it's inherently weak, but because it "leaks" into these hidden extra dimensions, while other forces (like electricity) are stuck on the ground floor.

The goal of this paper is to figure out how to find evidence of these hidden dimensions using giant particle smashers (colliders) like the Large Hadron Collider (LHC) and future super-colliders.

The Problem: The "Noise" is Too Loud

To find these extra dimensions, scientists smash particles together at incredibly high speeds. They are looking for a specific "signal": a pair of jets (sprays of particles) that fly apart with a huge gap between them in speed and direction (called "rapidity separation").

However, there is a massive problem: Background Noise.

When you smash particles, they naturally create jets all the time due to the strong nuclear force (QCD). This is like trying to hear a whisper (the extra dimension signal) in a stadium full of screaming fans (the QCD background).

For years, scientists used a standard rulebook (called DGLAP) to predict how loud the "screaming fans" would be. But recent experiments showed that this rulebook was wrong. It predicted the stadium would be much louder than it actually is. If you use the old rulebook, you think the whisper is just noise, and you miss the discovery.

The Solution: A Better Rulebook (BFKL)

The authors of this paper say, "Stop using the old rulebook! We need a new one called BFKL."

The Old Rulebook (DGLAP): Good for small gaps between jets, but it overestimates the noise when the jets are far apart. It's like using a map of a city to navigate a dense forest; it gets you lost.
The New Rulebook (BFKL): This is a more advanced mathematical tool designed specifically for high-energy collisions where particles fly far apart. It accounts for the "chaos" of the quantum world much better.

The Analogy:
Imagine you are trying to spot a rare, glowing firefly in a field.

The Old Way: You assume the field is filled with thousands of fake, bright lanterns. You decide, "There are too many lanterns; I can't see the firefly," and you give up.
The New Way: You realize the lanterns aren't actually that bright. You look again, and suddenly, you see the firefly glowing clearly against the dimmer background.

By using the new BFKL math, the "noise" (background) drops significantly. This makes the "signal" (extra dimensions) stand out much more clearly.

The "Trans-Planckian" Regime: The High-Speed Chase

The paper focuses on a very specific, extreme scenario called the "Trans-Planckian Eikonal Regime."

Trans-Planckian: This means the energy of the collision is so high that it exceeds the "Planck scale" (the theoretical limit where gravity usually becomes strong).
Eikonal: This is a fancy way of saying the particles are grazing each other. They don't crash head-on; they pass by, exchange a "graviton" (a particle of gravity), and fly off in opposite directions.

The Metaphor:
Imagine two race cars speeding past each other on a track.

In a normal crash, they hit head-on and explode (this is what we usually study).
In this study, they are driving so fast and so close that they don't even touch, but the air pressure between them (gravity) is so intense it creates a massive wake.
If extra dimensions exist, this "wake" would be much stronger than expected, creating a pair of jets that fly off with a massive gap between them.

What Did They Find?

The authors ran the numbers for current colliders (LHC) and future ones (like the FCC, which would be 7 times more powerful).

The Old Math is Dangerous: If you use the old DGLAP math, you might think the background noise is so high that you'll never see the extra dimensions. You might conclude, "No extra dimensions here!" and miss the discovery.
The New Math is Promising: Using the new BFKL math, the background noise is much lower. This means:
- At the current LHC (13 TeV), we could potentially see extra dimensions if the gravity scale is around 3 TeV.
- At future colliders (100 TeV), we could potentially see them if the scale is as high as 20 TeV.

Why This Matters

This paper is a "check-engine light" for physicists. It warns them: "Don't trust the old maps! If you use the new, more accurate math, the path to discovering new physics (like extra dimensions) is much clearer."

It suggests that by looking at high-mass particle pairs that are far apart in the detector, and by using the correct mathematical tools to filter out the noise, the next generation of colliders might finally prove that our universe has hidden extra dimensions.

Summary in One Sentence

This paper argues that by using a more accurate mathematical model (BFKL) to filter out the background noise of particle collisions, we can significantly improve our chances of spotting the "whisper" of extra dimensions in the "scream" of the universe's highest-energy collisions.

1. Problem Statement

The paper addresses the challenge of searching for Large Extra Dimensions (ADD gravity) at high-energy colliders (HL-LHC, FCC-pp, CEPC-SppC) within the trans-Planckian eikonal regime.

The Regime: The search focuses on kinematic conditions where the parton-parton center-of-mass energy is much larger than the fundamental Planck scale ( $\sqrt{\hat{s}} \gg M_D$ ), but the momentum transfer is small ( $\sqrt{-\hat{t}} \ll M_D$ ). This regime is characterized by the production of high-mass dijets with a large rapidity separation ( $\Delta y$ ).
The Background Problem: Accurately estimating the Standard Model (SM) QCD background in this semi-hard regime ( $\sqrt{\hat{s}} \gg Q \gg \Lambda_{QCD}$ $\overset{s}{^} ≫ Q ≫ Λ_{QC D}$ ) is critical.
- Traditional DGLAP evolution (used in standard Monte Carlo generators like Pythia) tends to overestimate the dijet cross-section at large rapidity separations by up to a factor of 6 compared to experimental data.
- Leading Logarithmic (LL) BFKL approximations often overestimate the data due to missing higher-order corrections.
- Using an inaccurate background model (like DGLAP) could lead to false exclusions of new physics or the misinterpretation of data, potentially masking the ADD gravity signal.

2. Methodology

The authors perform a comparative analysis of the ADD gravity signal against QCD backgrounds calculated using different theoretical frameworks.

Signal Calculation (ADD Gravity):
- Modeled using the eikonal approximation for multiple Kaluza-Klein (KK) graviton exchanges in the trans-Planckian regime ( $\sqrt{\hat{s}} \gg M_D \gg \sqrt{-\hat{t}}$ ).
- The parton-parton cross-section is derived from an infinite series of graviton exchanges, convoluted with Parton Distribution Functions (PDFs).
- Parameters varied: Number of extra dimensions ( $n_D = 2, 6$ ) and the fundamental Planck scale ( $M_D$ ).
Background Calculation (QCD):
- DGLAP Approach: Calculated via Leading Order (LO) matrix elements convoluted with PDFs evolved via Leading Logarithmic (LL) DGLAP equations, and via the Pythia8 Monte Carlo generator (CP5 tune, which includes NLO coupling and NNLO PDFs with rapidity ordering).
- BFKL Approach: Calculated at both Leading Logarithmic (LL) and Next-to-Leading Logarithmic (NLL) accuracy.
  - The NLL BFKL calculation utilizes the BFKLP (Brodsky-Fadin-Kim-Lipatov-Pivovarov) method to fix renormalization and factorization scales, resolving scale ambiguities.
  - It includes process-independent Green's functions and process-dependent impact factors.
- Systematic Uncertainties: Estimated by varying the high-energy asymptotic scale parameter ( $s_0$ ) and using MC replicas from the PDF4LHC15 set to account for PDF uncertainties.
Observable Definition:
- Mueller-Navelet (MN) Dijets: Defined as the pair of jets in an event with the largest rapidity separation ( $\Delta y$ ) and transverse momentum $p_\perp > 20$ GeV.
- Selection Cuts: Jets restricted to $|y| < 4.7$ ; high-mass cuts ( $M_{jj}$ ) applied to enhance sensitivity to the trans-Planckian regime.
- Collision Energies: Simulations performed for $\sqrt{s} = 13$ TeV (HL-LHC), 40 TeV, and 100 TeV (FCC-pp/CEPC-SppC).

3. Key Contributions

Validation of NLL BFKL: The paper reinforces that NLL BFKL evolution provides the most accurate description of the SM QCD background for dijets with large rapidity separation, aligning with recent CMS measurements that DGLAP fails to reproduce.
Quantification of DGLAP Overestimation: The study demonstrates that using DGLAP dynamics in the semi-hard, large- $\Delta y$ regime can overestimate the QCD background by orders of magnitude (up to 2 orders) compared to NLL BFKL. This highlights a critical risk in current search strategies.
Signal Sensitivity in Trans-Planckian Regime: The authors establish sensitivity limits for ADD gravity in a kinematic regime distinct from previous LHC searches (which focused on $\sqrt{\hat{s}} \sim M_D$ ). Here, the graviton scale is defined by $\hat{t}$ (much lower than $M_D$ ), allowing for the probing of gravity at scales where effective field theories might otherwise break down.
Future Collider Projections: Provides specific sensitivity projections for $M_D$ at future colliders (40 TeV and 100 TeV), showing how sensitivity scales with energy.

4. Results

Background Discrepancy: At large rapidity separations ( $\Delta y \approx 8.7$ ), the NLL BFKL background prediction is significantly lower than DGLAP-based predictions. The systematic uncertainty of the NLL BFKL calculation is estimated at a factor of ~5, which is still smaller than the discrepancy between DGLAP and NLL BFKL.
Sensitivity Limits: Based on the NLL BFKL background, the study establishes the following lower bounds for the ADD Planck scale ( $M_D$ $M_{D}$ ) where a signal could be distinguished:
- $\sqrt{s} = 13$ TeV: Sensitive to $M_D < 3$ TeV.
- $\sqrt{s} = 40$ TeV: Sensitive to $M_D < 10$ TeV.
- $\sqrt{s} = 100$ TeV: Sensitive to $M_D < 20$ TeV.
Parameter Dependence: The signal cross-section is more sensitive to the value of $M_D$ than to the number of extra dimensions ( $n_D$ ).
Black Hole Formation: The study notes that at very large $\Delta y$ and high masses, black hole formation could occur. However, the eikonal approximation used is considered reliable for $\Delta y \gtrsim 3.5$ to $5.5$ depending on the energy, with the most robust results found at $\Delta y \approx 8.7$ .

5. Significance

Methodological Shift: The paper argues that future searches for new physics in the large rapidity gap regime must utilize NLL BFKL formalism rather than DGLAP extrapolations. Relying on DGLAP could lead to the false conclusion that no new physics exists simply because the background was overestimated.
Complementary Search Strategy: This approach probes the trans-Planckian eikonal regime, which is complementary to standard resonance searches (dilepton/diphoton) that operate near the $M_D$ scale. It allows for the exploration of gravity at momentum transfers much lower than the fundamental Planck scale.
Detector Requirements: The results emphasize the need for high-granularity detectors and high integrated luminosity (order of $1 \text{ ab}^{-1}$ ) at future colliders to resolve rare, high-mass dijet events with large rapidity separations and to manage pile-up effects.
Theoretical Consistency: By applying the eikonal approximation, the study avoids the ultraviolet divergences associated with linearized effective gravity theories, offering a more robust framework for calculating cross-sections in the high-energy limit.

In conclusion, the work provides a rigorous framework for searching for large extra dimensions in a previously underutilized kinematic regime, demonstrating that accurate QCD background modeling via NLL BFKL is essential for uncovering potential signals of quantum gravity at future high-energy colliders.