Constraining dimension-6 SMEFT with higher-order predictions for $p p \to t W$

This paper presents a comprehensive study of single-top production with a W boson at the LHC to constrain dimension-6 SMEFT operators, utilizing high-precision QCD predictions up to approximate NNLO to derive effective scale limits of up to 2 TeV through linear and quadratic fits of top-quark kinematic distributions.

The Gist

Imagine the Large Hadron Collider (LHC) as a giant, ultra-fast particle smasher. When it smashes protons together, it creates a chaotic storm of new particles. Physicists usually look for specific patterns in this storm to see if the "Standard Model" (our current best rulebook for how the universe works) is perfect or if there are hidden cracks where new, unknown physics might be hiding.

This paper is about looking at a very specific type of crash: one where a top quark (the heaviest known particle) is produced alongside a W boson (a particle that carries the weak nuclear force).

Here is the breakdown of what the authors did, using simple analogies:

1. The "Rulebook" vs. The "Loophole"

Think of the Standard Model as a strict rulebook for a game. But physicists suspect there might be a "cheat code" or a hidden rule we haven't found yet. To test this, they use a framework called SMEFT (Standard Model Effective Field Theory).

• The Analogy: Imagine the Standard Model is a recipe for a cake. SMEFT is like adding a few secret, unknown ingredients (called "operators") to see if the cake tastes different. The authors are looking for these secret ingredients by checking if the "top quark + W boson" cake tastes exactly as the recipe predicts.

2. The "Microscope" (Higher-Order Calculations)

The authors didn't just look at the crash with a naked eye; they used a high-powered microscope. In physics, calculations have different levels of precision:

• LO (Leading Order): A rough sketch.
• NLO (Next-to-Leading Order): A detailed drawing.
• aNNLO (Approximate Next-to-Next-to-Leading Order): A photorealistic 3D render.

The authors used the most advanced "photorealistic" calculations available (aNNLO) to predict exactly what should happen if the Standard Model is perfect. They found that "soft gluons" (invisible particles that act like friction in the collision) play a huge role. Ignoring them is like trying to predict a car crash without accounting for the friction of the tires.

3. The "Three Suspects"

The study focused on three specific "secret ingredients" (mathematical terms called Wilson coefficients) that could be messing with the top quark's behavior:

1. CtG: Affects how the top quark interacts with the "strong force" (gluons).
2. CtW: Affects how the top quark interacts with the "weak force" (W bosons).
3. Cp: A mix of other interactions involving electrons and quarks.

The authors asked: "If we tweak these three knobs, does the data from the LHC look different?"

4. The "Fitting Game"

The team took real data from the LHC (from "Run II" and the upcoming "Run III") and tried to fit their theoretical models to it. They did this in two ways:

• Linear Fit: Assuming the secret ingredients are small and act alone.
• Quadratic Fit: Assuming the ingredients might interact with each other or have a stronger effect (like squaring a number).

The Challenge: The authors found that the three suspects are very good at hiding together. If you try to measure one, the others can "mimic" its effect. This is called correlation.

• The Analogy: Imagine trying to figure out how much salt, sugar, and pepper are in a soup. If you only taste the soup, it's hard to tell if it's salty because of salt or because the pepper is masking the salt. The authors found that when they tried to measure all three at once, the "uncertainty" (the margin of error) got huge.

5. The Results: How far can we see?

The paper quantifies how far they can "see" into the unknown physics (measured in energy scales, like TeV).

• The "Non-Marginalized" View (Looking at one suspect at a time): If they assume the other two ingredients are zero, they can detect new physics up to 2 TeV (about 2,000 times the mass of a proton).
• The "Marginalized" View (Looking at all three together): When they allow all three to vary, the "fog" gets thicker.
  • With the Linear method, they can only see up to 0.5 TeV.
  • With the Quadratic method (allowing for stronger interactions), they can see up to 1.5 TeV.

The Takeaway: The "Quadratic" method is like turning on a brighter light; it helps cut through the fog and gives a clearer picture, but it requires assuming that even higher-level "secret ingredients" (Dimension-8 operators) aren't interfering.

6. Comparison with Other Studies

The authors compared their results to massive "global" studies that look at every type of particle crash at the LHC, not just the top quark.

• The Analogy: Global studies are like a detective who interviews 100 witnesses to solve a crime. This paper is like a detective who only interviews the three people who were in the kitchen.
• The Result: The global studies have tighter limits (they can see further) because they have more data. However, this paper proves that looking specifically at the "kitchen" (the top quark + W boson) provides a unique, independent check that is consistent with the global view. It adds a valuable piece to the puzzle, even if it doesn't solve the whole mystery on its own.

Summary

The authors built a super-precise theoretical model for a specific particle crash at the LHC. They found that to get the most accurate results, you must account for complex "friction" effects (higher-order corrections). While the data is currently "fuzzy" when trying to pin down three specific unknown factors simultaneously, using advanced math (quadratic fits) sharpens the focus, allowing them to probe for new physics at energy scales up to 1.5 TeV. This confirms that the Standard Model is holding up well, but the search for the "secret ingredients" continues.

Technical Summary

Technical Summary: Constraining dimension-6 SMEFT with higher-order predictions for pp → tW

Problem Statement
The Standard Model Effective Field Theory (SMEFT) serves as a systematic framework for probing heavy new physics via higher-dimensional operators. While global SMEFT analyses combining various observables are common, the theoretical accuracy of specific process predictions often limits the precision of constraints on Wilson coefficients. Specifically, the associated production of a top quark and a W boson ($pp \to tW$) is a sensitive probe for operators modifying top-quark weak and chromomagnetic dipole interactions. However, establishing robust constraints requires predictions that consistently incorporate higher-order Quantum Chromodynamics (QCD) corrections and a realistic uncertainty budget. Previous studies have addressed $tW$ production at Leading Order (LO) and Next-to-Leading Order (NLO), but the impact of approximate Next-to-Next-to-Leading Order (aNNLO) corrections on SMEFT sensitivity and the stability of extracted bounds had not been fully quantified in a multi-parameter fit.

Methodology
The authors analyze $pp \to tW^-$ production at the Large Hadron Collider (LHC) within the dimension-6 SMEFT framework. The study focuses on three specific Wilson coefficients: $C_{tG}$ (chromomagnetic dipole), $C_{tW}$ (weak dipole), and $C_p$ (a linear combination of operators affecting input parameters).

1. Theoretical Framework: The analysis utilizes the Warsaw basis and adopts the $\{G_F, m_Z, m_W\}$ scheme. The calculation includes LO, NLO, and aNNLO QCD predictions. The aNNLO predictions are constructed by matching NLO results with soft-gluon resummation formalism, which captures threshold corrections via plus distributions involving the variable $s_4$. To avoid double-counting with resonant $t\bar{t}$ production, a diagram-removal prescription is applied at NLO.
2. Operators and Amplitudes: The study identifies five relevant dimension-6 operators from the Warsaw basis that contribute at leading order. Due to chirality constraints and specific linear combinations in the Feynman rules, the analysis effectively constrains three parameters: $C_{tG}$, $C_{tW}$, and $C_p$. The squared amplitudes are expanded to include both linear ($C_k$) and quadratic ($C_k C_{k'}$) SMEFT terms.
3. Numerical Setup: Calculations are performed for Run II ($\sqrt{s}=13$ TeV, $\mathcal{L}=139$ fb$^{-1}$) and Run III ($\sqrt{s}=13.6$ TeV, $\mathcal{L}=300$ fb$^{-1}$) configurations. The analysis uses doubly differential distributions in top-quark transverse momentum ($p_T$) and rapidity ($y$), binned into 12 regions. Uncertainties include statistical, experimental (5% uncorrelated efficiency/selection, 1.7% correlated luminosity), and theoretical components (PDFs and 3-point/7-point scale variations).
4. Statistical Analysis: A $\chi^2$ minimization is performed using pseudo-data generated from Standard Model (SM) expectations smeared with uncertainties. The study computes Fisher information matrices to derive 95% Confidence Level (CL) bounds. Both nonmarginalized (2-parameter) and marginalized (3-parameter) fits are conducted for linear and quadratic SMEFT expansions.

Key Contributions

• Higher-Order SMEFT Predictions: The paper provides aNNLO QCD predictions for $tW$ production including dimension-6 SMEFT operators, extending beyond previous LO and NLO studies.
• Uncertainty Quantification: A detailed error budget is presented, demonstrating that theoretical uncertainties (specifically scale variations) and experimental systematics compete as dominant sources of error, while statistical uncertainties are subleading.
• Linear vs. Quadratic Fits: The authors perform a comprehensive comparison between linear and quadratic SMEFT fits, analyzing how the inclusion of squared dimension-6 terms affects the constraining power and the stability of the fit.
• Correlation Analysis: The study explicitly maps the correlations between $C_{tG}$, $C_{tW}$, and $C_p$, identifying approximate flat directions in the parameter space that are lifted by the inclusion of quadratic terms.

Results

• Perturbative Stability: The $k$-factors for the transition from LO to NLO and NLO to aNNLO show that higher-order corrections are significant, particularly at high $p_T$. The aNNLO/NLO $k$-factor is nearly flat ($\sim 1.08$), indicating good perturbative convergence.
• Sensitivity to Operators: Linear SMEFT corrections are found to be around 6–8% across bins, while quadratic corrections exhibit stronger $p_T$ dependence, growing significantly in the high-$p_T$ tail.
• Constraints on Wilson Coefficients:
  • Linear Fits: Nonmarginalized bounds are of order $O(0.1)$, corresponding to effective scales up to $\sim 2$ TeV. However, marginalized bounds degrade significantly to $O(10)$ (effective scales $\sim 0.5$ TeV) due to strong correlations among the three parameters.
  • Quadratic Fits: Including quadratic terms leaves nonmarginalized bounds similar to the linear case but improves marginalized bounds by approximately an order of magnitude, reaching effective scales of $\sim 1.5$ TeV. This improvement is attributed to the lifting of approximate flat directions in the 3-parameter space.
• Comparison with Global Fits: The $tW$-specific constraints on $C_{tG}$ and $C_{tW}$ are weaker than those from recent global fits (e.g., HEPfit, SMEFiT) which leverage multiple channels. However, the $tW$ channel provides complementary, process-specific constraints. Combining the $tW$ differential information with global analyses tightens marginalized bounds by up to 14% for $C_{tG}$ and reduces the confidence ellipse area by up to 25% relative to conservative global baselines.

Significance and Claims
The paper claims that establishing perturbative-QCD control in the $tW$ channel is essential for reliable SMEFT constraints. The authors emphasize that while linear fits suffer from strong parameter degeneracies that weaken marginalized constraints, the inclusion of quadratic dimension-6 terms significantly enhances the constraining power of the fit. They note that this improvement comes with the caveat that quadratic dimension-6 effects are parametrically of the same order as linear dimension-8 contributions; thus, the results are contingent on the assumption that dimension-8 operators are subleading.

The study concludes that the $tW$ channel offers a valuable, complementary probe for SMEFT operators. While it does not surpass the sensitivity of comprehensive global fits, it provides a process-specific determination at aNNLO accuracy that helps break degeneracies and refine the global understanding of top-quark dipole interactions. The authors suggest that future work should incorporate dimension-8 operators for a more complete EFT description and utilize additional observables to disentangle the specific operator combinations contributing to $C_p$.