⚛️ phenomenology

A limit on top quark pair production at future electron-proton colliders

This paper analyzes the ratio of structure functions and establishes bounds on top quark pair production at future electron-proton colliders (LHeC and FCC-eh) using collinear factorization and modified Bjorken scaling, while also examining dipole cross-sections and Higgs boson production probabilities in photon-gluon interactions.

Original authors: G. R. Boroun

Published 2026-02-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: G. R. Boroun

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

Imagine you are trying to understand the internal structure of a giant, invisible fortress (the proton) by firing tiny, high-speed spears (electrons) at it. This is what physicists do at particle colliders like the proposed LHeC and FCC-eh.

This paper is a theoretical "blueprint" for what happens when those spears hit the fortress hard enough to create the heaviest known particle: the Top Quark. Because the Top Quark is so heavy (about as heavy as a gold atom), creating it requires a massive amount of energy, like trying to smash two cars together to create a new, heavy engine part.

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

1. The Goal: Finding the "Top" of the Mountain

The main goal is to predict how often these heavy Top Quarks will be created when electrons smash into protons. The author, G.R. Boroun, is trying to set a speed limit (a bound) on this production.

Think of it like a traffic cop at a busy intersection. The cop wants to know: "What is the absolute maximum number of heavy trucks (Top Quarks) that can pass through this intersection per hour?" If we know the limit, and we see more trucks than that, we know our traffic laws (our physics theories) are wrong.

2. The "Shadow" and the "Wall" (Structure Functions)

In particle physics, we can't see the inside of the proton directly. Instead, we look at "shadows" cast by the particles inside, called Structure Functions.

The Analogy: Imagine shining a flashlight through a foggy window. You can't see the individual dust motes, but you can see how the light dims (the shadow).
The paper looks at two specific shadows: one for light hitting straight on (Longitudinal) and one for light hitting from the side (Transverse).
The author calculates a ratio between these two shadows. He finds that at the highest energies, this ratio hits a "ceiling" or a bound. It can't go higher than a certain number (around 0.26 to 0.29). This is a crucial rule for future experiments.

3. The "Dipole" and the "Rubber Band"

To understand how the Top Quark is made, the paper uses a model called the Color Dipole.

The Analogy: Imagine the proton is a giant trampoline. When a photon (a particle of light) hits it, it doesn't just bounce; it stretches the trampoline fabric, creating a pair of connected weights (the Top Quark and its anti-particle). This pair is the "dipole."
The paper studies how big this "rubber band" (the dipole size, denoted as r) can get before it snaps or behaves differently.
They found that for very small rubber bands, the physics is predictable. But as the bands get larger, the behavior changes, and the "saturation" (the point where the trampoline is fully stretched) becomes important.

4. The "Heavy Suit" Problem (The Mass Correction)

This is one of the most interesting parts of the paper. Usually, physicists use a standard rule (Bjorken scaling) to predict how particles behave. However, the Top Quark is so heavy that it wears a "heavy suit" that slows it down and changes the rules.

The Analogy: Imagine running a race. If you are running in sneakers, you follow one set of rules. If you are running in 50-pound lead boots, you have to change your stride.
The author applies a correction for this "heavy suit" (the mass of the Top Quark). He shows that if you ignore the heavy suit, your predictions are wrong, especially at lower energies. When you add the correction, the predictions become much smoother and more accurate, especially when the energy isn't quite high enough to easily create the heavy pair.

5. The "Higgs" Connection

The paper also briefly mentions the Higgs Boson (the particle that gives mass to others).

The Analogy: The Top Quark is the Higgs Boson's "best friend" because it has the strongest connection to it.
The author notes that in these new colliders, we might see the Top Quark pair production acting as a factory for Higgs Bosons. It's like seeing a car crash (Top pair production) that accidentally creates a rare, shiny coin (Higgs) in the debris.

The Bottom Line

This paper is a theoretical safety net for future experiments.

It tells the scientists at the LHeC and FCC-eh: "If you see Top Quarks being produced at a rate higher than our calculated 'ceiling,' something is wrong with our understanding of the universe."
It provides a better way to calculate these rates by accounting for the Top Quark's massive weight, ensuring that when the new colliders turn on, the data can be interpreted correctly.

In short, it's a guidebook for how to look for the heaviest particle in the universe without getting lost in the math, ensuring that when the new "microscopes" (colliders) are built, we know exactly what to look for.

1. Problem Statement

The paper addresses the theoretical challenges and kinematic limits associated with top quark pair ( $t\bar{t}$ ) production in future high-energy electron-proton ($ep$) colliders, specifically the Large Hadron Electron Collider (LHeC) and the Future Circular Collider electron-hadron (FCC-eh).

Key issues include:

Kinematic Limits: Determining the behavior of top quark production at the extreme kinematic limit where the Bjorken scaling variable $x$ reaches its minimum value, defined as $x_{min} = Q^2/s$ (where $Q^2$ is the momentum transfer and $s$ is the center-of-mass energy squared).
Structure Function Bounds: Establishing rigorous upper bounds on the ratio of longitudinal to transverse structure functions ( $F_L/F_2$ ) for top quarks in Deep Inelastic Scattering (DIS).
Scale Dependence: Investigating the impact of renormalization ( $\mu_R$ ) and factorization ( $\mu_F$ ) scales, particularly distinguishing between the standard scale $\mu^2 = Q^2$ and the heavy-quark specific scale $\mu^2 = Q^2 + 4m_t^2$ .
Saturation Physics: Understanding the behavior of the dipole cross-section and gluon saturation in the small- $x$ regime for heavy quark production.

2. Methodology

The author employs a multi-faceted theoretical approach combining collinear factorization and the color dipole picture (CDP):

Collinear Factorization & DAS Approach:
- The study utilizes the Double Asymptotic Scaling (DAS) approach to analyze top quark structure functions at small $x$ .
- The ratio of structure functions $F_L/F_2$ is derived using coefficient functions ( $B_{k,g}^{(n)}$ ) at Leading Order (LO) and Next-to-Leading Order (NLO).
- A compact formula for the ratio $F_{L2}(Q^2/s, Q^2, m_t)$ is developed to extract bounds on the top structure function.
Unintegrated Gluon Distribution (UGD):
- The integrated gluon distribution $xg(x, Q^2)$ is derived from the UGD using the correspondence between the color dipole picture and $k_t$ -factorization.
- The UGD is modeled using an analytical expression involving a saturation scale $Q_s^2(x) = Q_0^2(x/x_0)^{-\lambda}$ .
Dipole Cross-Section (BGK Model):
- The Bartels-Golec-Biernat-Kowalski (BGK) model is used to calculate the dipole cross-section $\sigma_{dip}$ .
- The virtual photon dissociates into a $t\bar{t}$ dipole pair, interacting with the proton's color field. The cross-section is calculated via an integral over the dipole size $r$ and the light-cone wave function $\Psi$ .
Scale Modifications:
- The paper applies a modified Bjorken scaling variable: $x = Q^2 / (s + 4m_t^2)$ , specifically for the renormalization scale $\mu^2 = Q^2 + 4m_t^2$ . This modification is crucial for improving scaling quantities when $Q^2 < 4m_t^2$ .
Higgs Production Probability:
- The probability of Higgs boson production via $\gamma^* g$ interactions is compared to the $gg$ process, scaled by the order $O(\alpha_{em}/\alpha_s)$ .

3. Key Contributions

Establishment of Structure Function Bounds: The paper derives a non-trivial upper bound for the ratio $F_L/F_2$ at the limit $y \to 1$ (high inelasticity). The bound is found to be in the range 0.259 – 0.288 for LHeC and FCC-eh energies, consistent with color dipole model predictions.
Reduced Cross-Section Limits: A compact formula for the reduced cross-section $\sigma_r$ is established. The study demonstrates that at low $Q^2$ ( $Q^2 < 100 \text{ GeV}^2$ ), the reduced cross-section is dominated by the transverse structure function ( $\sigma_r \approx F_2$ ), while at high $Q^2$ , the longitudinal component becomes significant.
Impact of Renormalization Scales: The analysis highlights that the choice of scale ( $\mu^2 = Q^2$ vs. $\mu^2 = Q^2 + 4m_t^2$ ) significantly affects the results at low $Q^2$ . The inclusion of the top mass in the scale ( $\mu^2 = Q^2 + 4m_t^2$ ) improves the description of the cross-section in the region $Q^2 < 4m_t^2$ .
Saturation Behavior: The paper identifies a "limited behavior" in observing top saturation. The saturation scale $Q_s^2$ is found to be relatively flat in the region $10 \le Q^2 < 10^5 \text{ GeV}^2$ when using the modified scale.

4. Key Results

Structure Function Ratio ( $F_L/F_2$ ):
- At LO, the ratio $g(Q^2, s, m_t)$ reaches a maximum of $\approx 0.288$ (LHeC) and $\approx 0.259$ (FCC-eh).
- At NLO, the results for LHeC and FCC-eh are very similar and increase with $Q^2$ .
- The ratio $F_L/F_2$ approaches the derived bound in the range $10^5 < Q^2 < 10^7 \text{ GeV}^2$ .
Reduced Cross-Section ( $\sigma_r$ ):
- Low $Q^2$ ( $< 100 \text{ GeV}^2$ ): $\sigma_r \approx F_2$ . The longitudinal contribution is negligible. The values are approximately $0.027$ (LHeC) and $0.049$ (FCC-eh) for $Q^2 \lesssim 4m_t^2$ .
- High $Q^2$ ( $> 4m_t^2$ ): The behavior of $\sigma_r$ becomes similar regardless of whether the Bjorken variable is modified. The cross-section rises with $Q^2$ until a maximum, then falls for $Q^2 \gg 4m_t^2$ .
- Uncertainty: Theoretical uncertainty due to the choice of $\mu$ is negligible at high $Q^2$ but significant at low $Q^2$ .
Dipole Cross-Section:
- The ratio $\sigma_{dip}/\sigma_0$ exhibits a bound in the range of 0.978 – 0.980 for the dipole size range $0.0001 < r < 100$ fm.
- The dipole cross-section shows a violation of standard behavior for $3\times 10^{-2} \lesssim r < 1$ fm at the scale $\mu^2 = Q^2$ , but aligns with the GBW model behavior at $\mu^2 = Q^2 + 4m_t^2$ for $r \gtrsim 1$ fm and $r \lesssim 3\times 10^{-2}$ fm.
Higgs Production:
- The cross-section for Higgs production via $\gamma^* g \to t\bar{t} \to H$ is estimated to be suppressed relative to the $gg$ fusion process by the factor $\alpha_{em}/\alpha_s$ .

5. Significance

Future Collider Physics: The results provide critical theoretical constraints for the LHeC and FCC-eh programs. They define the expected kinematic reach and cross-section limits for top quark pair production, which are essential for experimental design and data analysis.
QCD Validation: The study offers a new test of Quantum Chromodynamics (QCD) in the heavy quark sector, specifically probing the interplay between perturbative QCD (collinear factorization) and non-perturbative saturation effects (color dipole picture).
Precision Measurements: By establishing bounds on $F_L/F_2$ and the reduced cross-section, the paper aids in extracting precise parton distribution functions (PDFs), particularly the gluon density at very small $x$ , which is currently a major uncertainty in high-energy physics.
Higgs-Top Coupling: The analysis supports future investigations into the top-quark Yukawa coupling and the CP nature of the top-Higgs interaction, which are vital for testing the Standard Model and searching for Beyond-the-Standard-Model (BSM) physics.

In conclusion, the paper demonstrates that while top saturation is limited in observability, the modified Bjorken scaling and specific renormalization scales are essential for accurately predicting top quark production rates at future $ep$ colliders, particularly in the low- $Q^2$ regime.