Probing $t$-channel single top-quark and antiquark production via differential cross-section measurements at $\sqrt{s}=$\SI{13}{\TeV} with the ATLAS detector

Using the full ATLAS Run 2 dataset at $\sqrt{s}=13$ TeV, this paper presents the first measurement of the differential ratio between $t$-channel single top-quark and antiquark production cross-sections, comparing these results to theoretical predictions and deriving constraints on the Wilson coefficient $C^{3,1}_{Qq}$ within an effective field theory framework.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful smash-up derby. Scientists shoot tiny particles (protons) at each other at nearly the speed of light to see what happens when they collide. Usually, these collisions create a chaotic mess of debris, but sometimes, they create something very special: a top quark.

The top quark is the "heavyweight champion" of the particle world. It's so heavy and short-lived that it's incredibly hard to catch a glimpse of it.

This paper is about a specific way these champions are made, called the "t-channel", and how the ATLAS experiment at the LHC took a very detailed look at them. Here is the breakdown in simple terms:

1. The Main Event: The "t-Channel" Dance

In the world of particle physics, particles interact by swapping other particles, kind of like two people throwing a ball back and forth.

• The Process: In this specific "t-channel" event, a proton (which is like a bag of smaller particles called quarks) smashes into another proton. Inside, a "u-quark" from one proton and a "d-quark" from the other swap a virtual W boson (a force-carrying particle).
• The Result: This swap creates a single top quark (the heavy champion) and a bottom quark.
• The Twist: Sometimes, the collision creates a top antiquark (the "anti-champion"). Because protons have more "u-quarks" than "d-quarks," it's much easier to make a top quark than a top antiquark. It's like trying to catch a red ball vs. a blue ball when the bag has 100 red ones and only 10 blue ones.

2. The Goal: Not Just "How Many," But "How Fast and Where"

Previous studies were like a census: "We counted 1,000 top quarks."
This study is much more detailed. It's like a traffic camera that doesn't just count cars, but measures exactly how fast they are going and which lane they are in.

• The Measurements: The scientists measured the transverse momentum (how hard the top quark is flying sideways) and the rapidity (how far forward or backward it's moving).
• The Big First: For the very first time, they didn't just count the top quarks and antiquarks separately; they calculated the ratio of top quarks to top antiquarks across different speeds. This is like comparing the speed of red cars to blue cars to see if the difference in their numbers changes when they drive fast vs. slow.

3. The Detective Work: Cleaning Up the Mess

The collision data is messy. There is a lot of "background noise" (other particles flying around that aren't top quarks).

• The Filter: The team used a Neural Network (a type of AI) acting like a super-smart bouncer. It looks at every event and gives it a score. If the score is high enough, the bouncer lets it in as a "Top Quark Candidate."
• The Unfolding: The detectors aren't perfect; they blur the picture a bit. The scientists used a mathematical technique called "unfolding" to reverse the blur. Imagine taking a blurry photo of a race car and using software to sharpen it until you can see the exact speedometer reading. This allowed them to see what the particles were doing before they hit the detector.

4. The Comparison: Checking the Recipe Books

The scientists compared their real-world data against theoretical predictions (recipe books written by mathematicians).

• They checked if the data matched different computer simulations (like different chefs trying to cook the same dish).
• They also checked different "PDF sets." Think of a PDF (Parton Distribution Function) as a map of the inside of a proton. It tells you where the quarks are hiding. The data helped them see which map was the most accurate.
• The Verdict: The real data matched the recipe books very well! The "chefs" (theoretical models) got the flavor right.

5. The "New Physics" Hunt: The EFT Interpretation

Finally, the scientists asked: "Is there anything weird here that our current rules of physics can't explain?"

• They used a framework called Effective Field Theory (EFT). Think of this as a "sensitivity test." They asked, "If there were a hidden force or a new particle we don't know about, how would it change the speed of our top quarks?"
• They looked for a specific number (called a Wilson Coefficient) that would indicate new physics.
• The Result: They found nothing suspicious. The number was zero (within a very small margin of error). This is actually a good result because it means our current understanding of the universe is still holding up strong, and they have set a very tight "fence" around where new physics could be hiding.

Summary

In short, this paper is a high-definition traffic report on the heaviest particles in the universe.

1. They caught a massive dataset of top quarks and antiquarks.
2. They measured exactly how fast and where they were going.
3. They compared this to our best theories and maps of the proton.
4. Everything matched perfectly, confirming our current understanding of how the universe works, while also setting strict limits on where "new, weird physics" might be hiding.

It's a victory for the Standard Model, but a victory that gives us even sharper tools to look for the unknown in the future.

Technical Summary

Based on the provided paper, here is a detailed technical summary of the ATLAS collaboration's work on probing t-channel single top-quark production.

1. Problem and Motivation

The paper addresses the need for precise measurements of single top-quark production via the t-channel process at the Large Hadron Collider (LHC). This process is the dominant electroweak top-quark production mechanism and offers unique sensitivity to:

• Proton Structure: The cross-section ratio between top quarks ($tq$) and top antiquarks ($\bar{t}q$) is sensitive to Parton Distribution Functions (PDFs), specifically the $u$-quark vs. $d$-quark content in the proton.
• Theoretical Modelling: Differential distributions test the accuracy of parton showers, matrix-element generators, and fixed-order calculations.
• New Physics: Deviations from Standard Model (SM) predictions can be interpreted within the framework of Effective Field Theory (EFT) to constrain Wilson coefficients associated with four-fermion operators.

Previous ATLAS measurements were largely inclusive or reported ratios of $tq$ to the total single top production ($tq + \bar{t}q$) due to statistical limitations. This work aims to provide differential cross-sections at the parton level for both $tq$ and $\bar{t}q$ separately and, for the first time, the differential ratio $\sigma(tq)/\sigma(\bar{t}q)$.

2. Methodology

Dataset and Event Selection

• Dataset: Full Run 2 ATLAS dataset corresponding to an integrated luminosity of 140 fb$^{-1}$ at a center-of-mass energy of $\sqrt{s} = 13$ TeV.
• Selection Criteria: Events are selected to match the t-channel signature:
  • Exactly one isolated charged lepton ($e$ or $\mu$) with $p_T > 28$ GeV.
  • Exactly two jets with $p_T > 30$ GeV and $|\eta| < 4.5$.
  • Exactly one b-tagged jet.
  • Missing transverse energy ($E_T^{miss}$) $> 30$ GeV.
  • Transverse mass of the W boson ($m_T(W)$) $> 50$ GeV.

Signal Extraction and Background Suppression

• A feed-forward Neural Network (NN) is employed to distinguish signal from background.
• Events are required to have an NN score $D_{nn} > 0.93$.
• Signal Regions (SR): Defined by lepton charge sign:
  • $\ell^+$ SR (for $tq$): Signal-to-background ($S/B$) ratio of 6.1.
  • $\ell^-$ SR (for $\bar{t}q$): $S/B$ ratio of 3.8.

Unfolding Procedure

• To obtain parton-level results, detector effects (resolution, efficiency, acceptance) are corrected using Iterative Bayesian Unfolding (IBU).
• Migration matrices are constructed using nominal Monte Carlo (MC) signal predictions.
• Four iterations are used to minimize statistical fluctuations while reducing prior bias.
• The differential cross-section is calculated as:
  $$\frac{d\sigma_k}{dX_k} = \frac{1}{\epsilon_k L_{int} \Delta X_k} \sum_j M^{-1}_{jk} (N^{data}_j - \hat{B}_j)$$

Systematic Uncertainties

Uncertainties are categorized into:

1. Signal Modelling: Dominant source (parton shower, scale variations, PDF uncertainties).
2. Experimental: Jet energy scale, b-tagging efficiencies.
3. Background: Subdominant due to high signal purity in the selected regions.

3. Key Contributions

• First Differential Ratio: This is the first measurement of the differential ratio $\sigma(tq)/\sigma(\bar{t}q)$ as a function of kinematic variables. This ratio benefits from significant cancellations of systematic uncertainties and provides enhanced sensitivity to PDF modelling.
• Parton-Level Differential Measurements: Provides unfolded cross-sections as functions of:
  • Transverse momentum ($p_T$) of $tq$ and $\bar{t}q$.
  • Absolute rapidity ($|y|$) of $tq$ and $\bar{t}q$.
• EFT Interpretation: Performs a model-independent interpretation to constrain the Wilson coefficient $C_{Qq}^{3,1}$ of the four-fermion operator $O_{Qq}^{3,1}$.

4. Results

Comparison with Theory

• Generators: The measured differential cross-sections show good agreement with predictions from various MC generators, including Powheg+Pythia8, MG5_aMC@NLO, and fixed-order calculations from MCFM (at NLO and NNLO).
• PDF Sets: The data is consistent with various modern PDF sets (CT18, MSHT20, NNPDF3.0/4.0, PDF4LHC21, ATLASpdf21, ABMP16).
• Kinematic Distributions:
  • Figure 1 (in the paper) shows the normalized differential $tq$ cross-section vs. $p_T(t)$ and the ratio $\sigma(tq)/\sigma(\bar{t}q)$ vs. $p_T(t)$.
  • Figure 2 shows the differential cross-section vs. $|y(t)|$, confirming consistency across rapidity bins.

EFT Constraints

• The analysis parameterizes the dependence of the differential cross-section on the Wilson coefficient $C_{Qq}^{3,1}$ (scaled by the new physics scale $\Lambda$).
• Using the EFTfitter tool, a 95% confidence level constraint is derived:
  $$-0.12 \, \text{TeV}^{-2} < \frac{C_{Qq}^{3,1}}{\Lambda^2} < 0.12 \, \text{TeV}^{-2}$$
• This represents a significant improvement in precision compared to previous inclusive measurements.

5. Significance

• Precision Physics: The results provide the most stringent differential tests of t-channel single top production to date, validating the SM predictions at the parton level.
• PDF Constraints: The differential ratio measurement offers a powerful new observable to constrain the $u/d$ quark ratio in the proton, reducing PDF uncertainties for future LHC analyses.
• New Physics Sensitivity: The improved EFT constraints narrow the parameter space for potential four-fermion contact interactions, pushing the sensitivity to higher energy scales.
• Methodological Advancement: The successful application of unfolding techniques to separate $tq$ and $\bar{t}q$ differential ratios demonstrates a robust methodology for future high-precision measurements in complex final states.