Measurement of differential $t$-channel single top (anti)quark production cross-sections at 13 TeV with the ATLAS detector

Using the full Run 2 dataset of 140 fb$^{-1}$ collected by the ATLAS detector at 13 TeV, this study presents differential measurements of $t$-channel single top and top antiquark production cross-sections, finding good agreement with theoretical predictions and setting constraints on the Wilson coefficient $C_{Qq}^{3,1}$ within an effective field theory framework.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful, high-speed particle racetrack. Inside this ring, scientists smash protons together at nearly the speed of light to see what happens when the universe's building blocks collide.

This paper is a detailed report card from the ATLAS experiment, one of the giant detectors watching these collisions. The team is studying a very specific, rare event: the creation of a single top quark.

The Big Picture: Finding a Needle in a Haystack

Top quarks are the heaviest known elementary particles. Usually, they are created in pairs (like twins) when protons collide. But sometimes, through a specific "exchange" process involving a virtual particle called a W boson, a single top quark (or its antimatter twin, the top antiquark) pops out all by itself.

Think of it like a game of billiards. Usually, when you hit a ball, you might get two balls rolling away. But in this specific "t-channel" game, one ball hits another, and they swap a cue stick (the W boson), causing only one new ball to fly off the table. The scientists wanted to measure exactly how often this happens and how fast these "lonely" top quarks are moving.

The Data: A Massive Library of Collisions

The researchers didn't just look at a few collisions; they analyzed data from 2015 to 2018. This corresponds to 140 inverse femtobarns of data. To put that in perspective, if a femtobarn is a single grain of sand, this dataset is like a mountain of sand. They sifted through billions of collisions to find the few thousand that contained the specific "signature" of a single top quark event:

• One isolated electron or muon (a heavy cousin of the electron).
• A lot of "missing" energy (carried away by invisible neutrinos).
• Exactly two jets of particles, with one of them coming from a bottom quark (a "b-tagged" jet).

The Challenge: Cleaning Up the Mess

The problem is that the "signal" (the top quark) is buried under a mountain of "background noise" (other common particle collisions that look similar).

To solve this, the team used a Neural Network (NN). Think of this as a highly trained digital detective. It was taught to look at the shapes, speeds, and angles of the particles in a collision and assign a "suspicion score." If the score was high enough, the event was kept; if low, it was discarded. This allowed them to separate the rare top quark events from the common background noise with high precision.

The Measurement: Mapping the Terrain

Once they isolated the events, the scientists didn't just count them. They wanted to know where and how fast these top quarks were going. They measured the "cross-section" (a fancy word for the probability of the event happening) in two ways:

1. Absolute: How many events happened in total.
2. Normalized: What percentage of the total events fell into specific speed or position ranges.

They mapped these events based on:

• Transverse Momentum ($p_T$): How hard the top quark is moving sideways.
• Rapidity ($|y|$): How far forward or backward the top quark is traveling relative to the beam.

They did this separately for top quarks and top antiquarks. Why? Because protons are made of different ingredients (more "up" quarks than "down" quarks). Theoretically, creating a top quark should be slightly easier than creating a top antiquark. The data confirmed this, showing a higher rate for tops than anti-tops.

The Results: Theory vs. Reality

The team compared their measurements against the best theoretical predictions available, which are like complex mathematical recipes for how the universe should behave.

• The Verdict: The measurements matched the predictions very well. The "recipes" (specifically those using Next-to-Next-to-Leading Order calculations) were accurate.
• The Limitation: While the match was good, the scientists couldn't yet tell the difference between different versions of the recipes because their own measurement "fuzziness" (systematic uncertainties) was still a bit too large. It's like trying to hear a whisper in a noisy room; you know someone is speaking, but you can't quite make out the specific words yet.

The Twist: Searching for New Physics

Finally, the team used their data to test for "New Physics" using a framework called Effective Field Theory (EFT).

• The Analogy: Imagine the Standard Model (our current best theory) is a perfect map of a city. EFT asks, "What if there are secret tunnels or hidden shortcuts we don't know about yet?"
• The Test: They looked for a specific type of "shortcut" involving a four-quark interaction. If this shortcut existed, it would change the speed distribution of the top quarks, especially the very fast ones.
• The Result: They found no evidence of these secret tunnels. They set a strict limit on how big these "shortcuts" could possibly be, improving upon previous limits. They also had to account for the fact that if these shortcuts did exist, they would change how easy it is to spot the events in the first place (the "selection efficiency"), and they corrected for that in their math.

Summary

In simple terms, this paper is a high-precision audit of how single top quarks are created at the LHC. The ATLAS team successfully mapped out the speed and direction of these particles, confirmed that our current theories of physics are working correctly, and tightened the rules on where "new physics" might be hiding. They didn't find any new particles, but they proved that the universe is behaving exactly as our best maps predict, even in these extreme conditions.

Technical Summary

Technical Summary: Measurement of Differential $t$-Channel Single Top (Anti)quark Production Cross-Sections at 13 TeV with the ATLAS Detector

Problem and Motivation
The production of single top quarks via the $t$-channel exchange of a virtual $W$ boson is a fundamental process in the Standard Model (SM), providing a direct probe of the top quark's properties and the $Wtb$ coupling. While total cross-sections have been measured at various LHC energies, differential measurements offer deeper insights into parton distribution functions (PDFs), specifically the $u/d$-quark ratio and the $b$-quark density. Furthermore, precise differential distributions are essential for testing the SM against higher-order Quantum Chromodynamics (QCD) predictions and for constraining Beyond the Standard Model (BSM) physics through Effective Field Theory (EFT). Previous measurements at $\sqrt{s}=13$ TeV were limited by partial data samples or combined $tq$ and $\bar{t}q$ analyses. This paper addresses the need for a high-precision, separate measurement of differential $tq$ and $\bar{t}q$ cross-sections using the full Run 2 dataset, extending the kinematic reach and improving constraints on EFT operators.

Methodology
The analysis utilizes proton-proton collision data recorded by the ATLAS detector between 2015 and 2018 at a center-of-mass energy of $\sqrt{s}=13$ TeV, corresponding to an integrated luminosity of 140 fb$^{-1}$.

• Event Selection: Signal events are characterized by the leptonic decay of the top quark, requiring exactly one isolated electron or muon ($p_T > 28$ GeV), large missing transverse momentum ($E_T^{miss} > 30$ GeV), and exactly two jets. One jet must be $b$-tagged (identified via the DL1r algorithm with 60% efficiency working point) and have $|\eta| < 2.5$. To suppress multijet backgrounds, kinematic requirements are applied, including $m_T(W) > 50$ GeV and a specific cut on the azimuthal angle between the lepton and the leading jet.
• Signal Extraction: A Neural Network (NN), trained on kinematic variables to distinguish signal from background, is employed. A requirement of $D_{nn} > 0.93$ is imposed to achieve high signal purity, separating the analysis into two signal regions based on lepton charge ($\ell^+$ for $tq$ and $\ell^-$ for $\bar{t}q$).
• Background Estimation: Backgrounds from $t\bar{t}$, single-top ($s$-channel, $tW$), dibosons, and $W/Z$+jets are estimated using Monte Carlo (MC) simulations normalized via a simultaneous profile maximum-likelihood fit. Multijet backgrounds are estimated using a combination of data-driven techniques and simulated dijet samples.
• Unfolding: Measured distributions are corrected for detector effects (resolution, acceptance, efficiency) using D'Agostini's iterative unfolding method. The distributions are unfolded to the parton level for stable top quarks and extrapolated to the full kinematic phase space.
• Theoretical Comparison: Results are compared with SM predictions at Next-to-Leading Order (NLO) and Next-to-Next-to-Leading Order (NNLO) using various matrix-element generators (Powheg Box, MG_aMC@NLO), parton-shower programs (Pythia 8, Herwig 7), and PDF sets (NNPDF, MSHT, CT, ABMP, ATLASPDF).
• EFT Interpretation: The differential cross-sections as a function of top quark transverse momentum ($p_T(t)$) are interpreted within the Standard Model Effective Field Theory (SMEFT). The analysis constrains the Wilson coefficient $C_{Qq}^{3,1}$ associated with a four-quark operator ($O_{Qq}^{3,1}$), explicitly accounting for the impact of non-zero Wilson coefficients on selection efficiency and kinematic acceptance.

Key Contributions

1. First Separate Differential Measurement: This work presents the first separate measurement of differential $tq$ and $\bar{t}q$ production cross-sections at $\sqrt{s}=13$ TeV, as well as the first measurement of their ratio as a function of $p_T$ and rapidity $|y|$.
2. Extended Kinematic Range: The measurement extends the $p_T(t)$ or $p_T(\bar{t})$ range up to 500 GeV, a significant improvement over previous measurements limited to 300 GeV.
3. Precision Improvement: By utilizing the full Run 2 dataset (140 fb$^{-1}$), the analysis achieves a precision of 3% to 6% for normalized differential cross-sections as a function of $|y|$, improving upon the 7% to 15% precision of previous 8 TeV measurements.
4. EFT Constraints with Acceptance Correction: The analysis provides a novel interpretation in the SMEFT framework that incorporates the effect of the four-quark operator on the selection efficiency, a factor often neglected in inclusive measurements.

Results

• Cross-Section Measurements: The absolute and normalized differential cross-sections for $tq$ and $\bar{t}q$ production are measured as functions of $p_T(t)$, $p_T(\bar{t})$, $|y(t)|$, and $|y(\bar{t})|$. The ratio of the cross-sections is also measured.
• Comparison with Theory: Overall, good agreement is observed between the measurements and theoretical predictions.
  • Fixed-order calculations at LO (MCFM) fail to describe the $p_T$ spectra, as expected due to missing gluon radiation.
  • NLO and NNLO predictions show very good agreement with the data, with NNLO generally providing a better fit.
  • Predictions using different PDF sets agree within measurement uncertainties, though the ABMP16 set shows a lower probability for the $\bar{t}q$ cross-section in the $|y(\bar{t})|$ distribution and overestimates the cross-section ratio.
• Uncertainties: The measurements are dominated by systematic uncertainties, primarily from signal modeling and experimental sources (jet energy scale, $b$-tagging). Statistical uncertainties become significant only in the high-$p_T$ region and for the cross-section ratios.
• EFT Constraints: The interpretation of the $p_T$ distributions constrains the Wilson coefficient $C_{Qq}^{3,1}/\Lambda^2$. The 95% confidence level interval is determined to be $-0.12 \text{ TeV}^{-2} < C_{Qq}^{3,1}/\Lambda^2 < 0.12 \text{ TeV}^{-2}$.

Significance
The paper claims that this analysis significantly advances the understanding of single top quark production by providing the most precise differential measurements to date at 13 TeV. The results validate the SM predictions at NNLO and constrain the $u/d$-quark ratio in the proton. Crucially, the EFT interpretation demonstrates that accounting for acceptance changes due to BSM operators is essential for deriving accurate constraints. The obtained limits on the four-quark operator $O_{Qq}^{3,1}$ improve upon previous constraints derived from the inclusive total cross-section measurement, highlighting the sensitivity of differential distributions to new physics effects. The paper concludes that the measurement is limited by systematic uncertainties, suggesting that future improvements will depend on better modeling of signal processes and detector performance.