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Constraints on effective field theories via quadruple-differential angular decay rates from $t$ -channel single-top-quark production at $\sqrt{s}=13$ TeV with the ATLAS detector

Using 140 fb $^{-1}$ of 13 TeV proton-proton collision data collected by the ATLAS detector, this study employs Fourier techniques on quadruple-differential angular decay rates from $t$ -channel single-top-quark events to tightly constrain Wilson coefficients of effective field theory operators, finding results consistent with Standard Model predictions.

Original authors: ATLAS Collaboration

Published 2026-08-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: ATLAS Collaboration

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 the universe as a giant, cosmic Lego set. For decades, scientists have been building models of how these blocks interact using a set of blueprints called the Standard Model. It's a brilliant set of instructions that explains almost everything we see, from the spark of a firefly to the heart of a star. But there's a nagging feeling among physicists that the blueprints are incomplete. There are gaps—mysterious forces and particles that the current instructions don't mention. To find them, scientists don't just look for new blocks; they look for tiny, subtle glitches in how the existing blocks snap together. They treat the universe like a high-stakes game of billiards, where if a ball bounces off a cushion at a slightly weird angle, it might mean there's a hidden bump on the table we can't see yet. This paper is all about watching the most energetic "billiard balls" in the universe—the top quarks—to see if they are behaving exactly as the Standard Model predicts, or if they are hinting at a secret new rulebook.

The paper comes from the ATLAS Collaboration, a massive team of scientists working at the Large Hadron Collider (LHC) in Europe. They are studying a specific type of particle collision where a single top quark is created. Think of the top quark as the heavyweight champion of the particle world; it's so heavy and short-lived that it doesn't even have time to form a "family" (like other particles do) before it explodes into other particles. Because it dies so fast, its final moments are a perfect snapshot of the forces acting on it. The scientists are looking for "Effective Field Theory" (EFT) operators. You can think of these as invisible, subtle nudges in the laws of physics that might come from a much heavier, undiscovered particle. If these nudges exist, they would slightly change the way the top quark spins and decays, just like a tiny, invisible wind might change the path of a falling leaf.

The researchers used a massive dataset of 140 fb⁻¹ of proton-proton collisions recorded at a center-of-mass energy of 13 TeV. That's a lot of data—imagine watching trillions of particle collisions to find a few thousand that look exactly like the top quark events they needed. They focused on a specific process called "t-channel" single-top production, which is the most common way these particles are made in the lab. To get the most information possible, they didn't just count how many top quarks they found; they looked at the angles at which the decay products flew out. They measured four specific angles for every event, creating a complex, four-dimensional map of the decay.

To make sense of this four-dimensional maze, the team used a clever mathematical trick involving Fourier analysis. Imagine trying to describe the shape of a complex, wobbly jelly. Instead of trying to draw the whole thing at once, you break it down into a set of simple, standard waves (like the notes on a piano). By measuring how much of each "wave" is present in the data, they could reconstruct the entire shape of the decay distribution. They compared these "wave coefficients" from their real data against what the Standard Model predicts. If the Standard Model is the only rulebook, the data should match the prediction perfectly. If there are those invisible nudges (the EFT operators), the waves would look different.

The results are a triumph of precision. After analyzing the data, the team found that the top quarks behaved exactly as the Standard Model predicted. The "invisible nudges" they were looking for were not there, or at least, they are so tiny that the current experiment couldn't detect them. They placed very tight constraints on the "Wilson coefficients," which are the numbers that measure the strength of these potential new forces. In plain English, they measured the strength of these possible new interactions and found them to be consistent with zero.

Specifically, they looked at seven different parameters (the Wilson coefficients) that could describe how the top quark interacts with the W boson and other particles. They found that the data is compatible with the Standard Model for all of them. For some parameters, like the one describing the top quark's interaction with the W boson, they improved the precision of previous measurements by a factor of two or three. They also looked at a parameter related to the production of the top quark itself, finding a slight deviation that suggests the current computer simulations might need a tiny upgrade (perhaps including more complex physics calculations), but the overall conclusion remains: no new physics was found.

The paper explicitly rules out the idea that there are large, obvious deviations from the Standard Model in these specific top quark interactions. They argue against the existence of strong "imaginary" components in these interactions that would signal a violation of time-reversal symmetry (CP violation) in this specific context. While they didn't find the "smoking gun" of new physics, they did something equally important: they tightened the net. By showing that the top quark behaves exactly as expected, they have forced any future theories of new physics to be much more subtle and harder to find. The paper concludes that while the search continues, the Standard Model remains the undisputed champion of our current understanding of the top quark's behavior.

Technical Summary: Constraints on Effective Field Theories via Quadruple-Differential Angular Decay Rates from $t$ -Channel Single-Top-Quark Production

Problem and Motivation
This paper addresses the search for physics beyond the Standard Model (BSM) by constraining non-zero Wilson coefficients (WCs) within the Standard Model Effective Field Theory (SMEFT). The analysis focuses on the top quark due to its unique properties: its heavy mass ( $m_t \approx 172.5$ GeV) and short lifetime ( $1/\Gamma_t \sim O(10^{-25})$ s) allow it to decay before hadronization, preserving spin information. This makes the kinematics of top-quark production and decay a sensitive probe for BSM interactions.

Specifically, the study targets dimension-six operators affecting the $t$ -channel single-top-quark production process and the $t \to Wb$ decay vertex. While previous measurements have constrained these operators, this work aims to extract the maximum amount of information by simultaneously analyzing the joint fourfold probability distribution of four kinematic angles, rather than relying on lower-dimensional projections or total cross-sections alone.

Methodology
The analysis utilizes proton-proton collision data at $\sqrt{s} = 13$ TeV collected by the ATLAS detector at the Large Hadron Collider (LHC), corresponding to an integrated luminosity of $140 \text{ fb}^{-1}$ from 2015–2018.

Event Selection: The signal channel is $t$ -channel single-top-quark production, characterized by a final state containing one lepton, missing transverse momentum (from a neutrino), one $b$ -jet, and one light-flavour spectator jet. Events are selected in a Signal Region (SR) and two Control Regions (CRs) for $W$ +jets and $t\bar{t}$ backgrounds. The dataset is partitioned into four channels based on the lepton charge and the reconstructed top-quark transverse momentum ( $p_T$ ), distinguishing between low- $p_T$ ( $<80$ GeV) and high- $p_T$ ( $\ge 80$ GeV) regions to enhance sensitivity to specific operators.
Effective Field Theory Operators: The analysis constrains seven free parameters corresponding to five dimension-six operators:
1. $\mathcal{O}_{\varphi Q}^{(3)}$ : Modifies the left-handed vector coupling (real WC, $C_{\varphi Q}$ ).
2. $\mathcal{O}_{\varphi tb}$ : Induces a right-handed vector coupling (real WC, $C_{\varphi tb}$ ).
3. $\mathcal{O}_{tW}$ : Contributes as a left-handed tensor coupling (complex WC, $C_{tW} = C_{tW} + iC^I_{tW}$ ).
4. $\mathcal{O}_{bW}$ : Contributes as a right-handed tensor coupling (complex WC, $C_{bW} = C_{bW} + iC^I_{bW}$ ).
5. $\mathcal{O}_{Qq}^{(3,1)}$ : A four-quark operator modifying $t$ -channel production (real WC, $C_{Qq}$ ).
  The imaginary parts of $C_{tW}$ and $C_{bW}$ are sensitive to CP violation.
Fourier Analysis and $M$ -Functions: To handle the challenges arising from the quadruple-differential angular decay rate ( $d\Gamma/d\Omega d\Omega^*$ $d Γ/ d Ω d Ω^{*}$ ), the author employs a Fourier technique using a set of orthogonal basis functions known as $M$ $M$ -functions. These functions are constructed from Wigner $D$ $D$ -functions and depend on four angles: $\theta, \phi$ $θ, ϕ$ (defining the $W$ $W$ -boson direction in the top rest frame) and $\theta^*, \phi^*$ $θ^{*}, ϕ^{*}$ (defining the charged lepton direction in the $W$ $W$ -boson rest frame).
- The differential distribution is projected onto 23 $M$ -function coefficients ( $c_{j_1 j_2}^{m' m}$ with $j_1 \le 1, j_2 \le 2$ ).
- These coefficients are calculated event-by-event and averaged over the dataset. This method captures the full shape and normalization information without binning the four-dimensional phase space.
Morphing and Fitting: A morphing technique is used to model the dependence of the $M$ -function coefficients on the Wilson coefficients. The coefficients are parameterized as polynomials (up to fourth-order for operators affecting both production and decay, and second-order for the production-only operator). A $\chi^2$ fit is performed using the 92 $M$ -function coefficients (23 coefficients $\times$ 4 channels) and 12 event counts (SR and CRs) to extract the best-fit values and confidence intervals for the WCs. Systematic uncertainties (experimental and theoretical) are incorporated via a covariance matrix.

Key Contributions

Quadruple-Differential Analysis: This analysis simultaneously fits the full four-dimensional angular decay distribution of $t$ -channel single-top-quark events using a Fourier-based approach to handle the associated challenges. This maximizes the information extracted from the data compared to traditional differential or inclusive measurements.
Simultaneous Constraint of Complex WCs: The analysis provides simultaneous constraints on the real and imaginary parts of the tensor coupling WCs ( $C_{tW}$ and $C_{bW}$ ), which are sensitive to CP-violating effects.
Morphing Formalism: The paper generalizes the morphing technique to accommodate a larger set of Wilson coefficients simultaneously, allowing for a robust treatment of interference terms between SM and EFT amplitudes, including quadratic EFT contributions.
Separation of Production and Decay Effects: By utilizing the $p_T$ dependence of the event yields and the angular distributions, the analysis effectively separates the effects of the production operator ( $C_{Qq}$ ) from the decay operators.

Results

Compatibility with SM: All best-fit values for the Wilson coefficients are consistent with the Standard Model prediction (zero for the WCs). The fit yields a $p$ -value of 70.0% for the EFT hypothesis with nine parameters.
Constraints:
- The most restrictive constraints are placed on the real and imaginary parts of $C_{tW}$ .
- The constraint on $C_{tW}$ improves upon previous measurements of $W$ -boson helicity fractions by a factor of two, without assuming other WCs are zero.
- The constraint on the imaginary part of $C_{tW}$ ( $C^I_{tW}$ ) improves upon previous top-quark polarization measurements by a factor of three.
- Constraints on $C_{Qq}$ show a $2.8\sigma$ deviation from the SM, attributed to a slight excess of observed events in the high- $p_T$ signal region compared to the Powheg Box v2 simulation. The author notes that NNLO effects might improve this agreement.
- The constraints on the complex WCs $C_{\varphi tb}$ and $C_{bW}$ are less stringent due to their lack of large interference with the SM amplitude, but the analysis provides constraints along directions where previous helicity fraction measurements were blind.
Uncertainty Breakdown: The total systematic uncertainty is comparable to the statistical uncertainty of the data (except for $C_{\varphi Q}$ ). No single systematic source dominates; the uncertainty is distributed among various experimental and modeling components.

Significance
The paper claims that this analysis represents a significant advancement in probing EFT operators through angular analysis. By utilizing the full quadruple-differential information, the study achieves competitive constraints on all targeted Wilson coefficients, improving the precision on $C_{tW}$ and $C^I_{tW}$ compared to previous ATLAS results. The methodology demonstrates the feasibility of using Fourier techniques to handle high-dimensional distributions in EFT analyses, offering a path toward higher precision with future LHC data (Run 3) and improved theoretical calculations (NNLO). The results are compatible with the Standard Model, placing tight bounds on potential BSM physics in the top-quark sector.

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