SMEFT everywhere: a NLO study of $\boldsymbol{pp \to t\bar{t}H}$ with decaying tops

This paper presents a comprehensive next-to-leading order QCD study of the $pp \to t\bar{t}H$ process in the di-lepton channel at 13.6 TeV within the SMEFT framework, demonstrating that consistently including higher-order corrections and dimension-6 operators in both top-quark production and decay is crucial for accurately predicting kinematic distributions and interpreting LHC Run III data.

The Gist

Imagine the Standard Model of particle physics as a perfectly tuned, high-performance race car. For decades, it has won every race we've thrown at it. But scientists suspect there might be a hidden engine upgrade or a secret turbocharger installed by a mysterious mechanic (New Physics) that we haven't seen yet. We can't see the mechanic directly, but we can look for subtle changes in how the car behaves at very high speeds.

This paper is like a team of expert mechanics and data analysts who decided to study a very specific, complex race: the collision of protons that creates a "top quark" pair and a "Higgs boson" (the car's most valuable cargo). They wanted to see if the car's behavior changes when we account for those potential hidden upgrades.

Here is a breakdown of their work using everyday analogies:

1. The "Rulebook" vs. The "Real Race" (SMEFT)

The scientists used a theoretical tool called SMEFT (Standard Model Effective Field Theory). Think of the Standard Model as the official rulebook for how the race car should drive. SMEFT is like adding a set of "what-if" clauses to that rulebook. These clauses describe how the car might behave if there were a hidden turbo (new physics) that only kicks in at extremely high speeds.

The paper focuses on four specific "what-if" clauses (operators) that could tweak how the top quark and Higgs boson interact with the engine (gluons) and the steering wheel (Higgs field).

2. The "Stable" vs. "Exploding" Car Analogy

In previous studies, scientists often treated the top quark like a stable, solid object that just sits there after being created. They calculated the race results based on the car arriving at the finish line intact.

However, in reality, the top quark is like a highly unstable firework that explodes almost instantly into other particles (leptons and bottom quarks).

• The Old Way: Calculating the race based on the firework before it explodes.
• This Paper's Way: Calculating the race based on the debris from the explosion, while also accounting for how the explosion itself might be slightly different if the hidden turbo is active.

The authors argue that you can't just look at the car before it explodes; you have to look at the debris, because the "explosion" (decay) is where the hidden turbo might leave its fingerprints.

3. The "Double-Check" (NLO Corrections)

Calculating particle collisions is like trying to predict the weather. A simple forecast (Leading Order) might say "it will rain," but it misses the wind, humidity, and temperature shifts.

• Leading Order (LO): A rough sketch of the race.
• Next-to-Leading Order (NLO): A high-definition, 3D simulation that includes wind resistance, friction, and tiny bumps in the road.

This paper performs the "high-definition" calculation. They found that when you add these fine details (NLO corrections), the results change significantly. Sometimes the "hidden turbo" effects look 50% or even 100% stronger or weaker than the rough sketch suggested. Ignoring these details is like trying to drive a race car using only a hand-drawn map; you might miss the sharp turns.

4. The "Filter" Problem (Kinematic Cuts)

In a real experiment, scientists can't see every single particle. They put up "filters" (kinematic cuts) to only look at the cleanest, most interesting crashes.

• The Analogy: Imagine a security camera that only records cars moving faster than 100 mph.
• The Discovery: The authors found that these filters change the story. If you look at the "stable" car (before explosion), the hidden turbo seems to do one thing. But if you look at the "debris" (after explosion) through the security camera filters, the turbo seems to do something completely different.

The paper shows that filters, explosions, and high-speed details all interact. If you ignore any one of them, your conclusion about whether the hidden turbo exists will be wrong.

5. The "Recipe" (The Operators)

The team focused on four specific ingredients (operators) in their recipe:

• Otϕ: Tweaks the connection between the top quark and the Higgs boson.
• OtG: Changes how the top quark interacts with the "glue" (gluons) holding the nucleus together.
• OϕG: Changes how the Higgs boson interacts with the glue.
• OtW: Changes how the top quark decays (explodes) into other particles.

They found that while some ingredients barely changed the taste of the dish, others (like the glue interactions) completely altered the flavor, especially when looking at the high-speed "tails" of the data.

The Bottom Line

The main takeaway is simple: Don't look at the car in isolation.

To find new physics at the Large Hadron Collider (LHC), scientists must stop treating the top quark as a simple, stable block. They must treat it as a complex, exploding firework, and they must calculate the race with extreme precision (NLO), all while respecting the specific filters the experimenters use to collect data. If you mix these three things up, you might think you found a new engine when you were just looking at a shadow, or vice versa.

This paper provides the new, more accurate "blueprint" for how to calculate these collisions, ensuring that future searches for new physics are looking in the right place with the right tools.

Technical Summary

Technical Summary: SMEFT everywhere: a NLO study of $pp \to t\bar{t}H$ with decaying tops

Problem Statement
The Standard Model (SM) is incomplete, and the search for Beyond the Standard Model (BSM) physics at the Large Hadron Collider (LHC) has so far yielded no direct resonances. Consequently, indirect searches via precision measurements of SM processes, particularly those involving the top quark and Higgs boson, are crucial. The Standard Model Effective Field Theory (SMEFT) provides a model-independent framework to parameterize these indirect effects via higher-dimensional operators.

While Next-to-Leading Order (NLO) QCD calculations for SMEFT processes involving top quarks (e.g., $pp \to t\bar{t}$, $pp \to t\bar{t}H$) have been developed, a significant gap remains in the literature. Previous studies typically treated top quarks as stable particles or considered SMEFT effects only in the production matrix elements, neglecting their impact on top-quark decays. Furthermore, the interplay between kinematic fiducial cuts, higher-order QCD corrections, and SMEFT operators acting simultaneously in production and decay has not been fully investigated. This omission is critical because kinematic cuts and decay dynamics can significantly alter the shape of observables and the interpretation of Wilson coefficients.

Methodology
To address these limitations, the authors developed Helac-Smeft, an updated version of the Helac-NLO Monte Carlo framework. The study focuses on the $pp \to t\bar{t}H$ process at the LHC Run III center-of-mass energy of $\sqrt{s} = 13.6$ TeV, specifically in the di-lepton decay channel ($t\bar{t}H \to W^+W^-b\bar{b}H \to e^+\nu_e \mu^-\bar{\nu}_\mu b\bar{b}H$).

The methodology involves:

1. Operator Selection: The study incorporates a minimal, realistic set of dimension-6 SMEFT operators: $O_{t\phi}$ (modifying the top-Higgs Yukawa coupling), $O_{tG}$ (chromomagnetic dipole), $O_{\phi G}$ (gluon-Higgs interaction), and $O_{tW}$ (affecting $Wtb$ vertices in decays).
2. Framework Implementation: Significant modifications were made to the Helac-NLO code to handle the increased complexity of SMEFT, including:
   • Automated generation of vertex functions for 1568 distinct vertices (LO, UV, and R2 types) derived from the Smeft@NLO UFO model.
   • Implementation of color-flow decomposition and flavor-dressing algorithms optimized for SMEFT.
   • Handling of one-loop topologies where the rank of tensor integrals exceeds the number of propagators (a feature of non-renormalizable operators like $O_{\phi G}$).
   • Validation of UV vertices, including a specific fix for a Ward identity violation found in the public $O_{\phi G}$ vertex in the Smeft@NLO model.
3. Calculation Strategy: The authors perform calculations in two modes:
   • Stable Top Quarks: $pp \to t\bar{t}H + X$ (inclusive).
   • Narrow-Width Approximation (NWA): $pp \to t\bar{t}H + X \to \dots$ where QCD corrections and SMEFT effects are consistently included in both production and decay matrix elements.
4. Scale and Uncertainty: Theoretical uncertainties are estimated via 7-point scale variations for renormalization ($\mu_R$) and factorization ($\mu_F$) scales, and a 3-point variation for the effective scale ($\mu_{EFT}$) to account for renormalization group (RG) running and operator mixing.

Key Contributions

• First Simultaneous NLO SMEFT Study: This work presents the first NLO QCD calculation for $pp \to t\bar{t}H$ that consistently includes SMEFT operators in both the production and the decay of unstable top quarks.
• Helac-Smeft Framework: The development and validation of a new automated tool capable of computing full SMEFT cross-sections (linear, cross, and quadratic terms) with NLO accuracy, including reweighting capabilities for various scale and PDF settings.
• Operator Mixing and Running: The study explicitly includes the effects of RG evolution and operator mixing (e.g., $O_{tG}$ mixing with $O_{t\phi}$ and $O_{\phi G}$) on the Wilson coefficients.
• Decay Sensitivity: The inclusion of the $O_{tW}$ operator, which affects top decays but is absent in stable-top calculations, allowing for the study of spin correlations and $Wtb$ structures.

Results
The authors present integrated and differential cross-section predictions for $\sqrt{s} = 13.6$ TeV:

• Integrated Cross Sections:
  • The K-factors (NLO/LO ratios) for SMEFT contributions vary significantly from the SM case, ranging from $K \approx 1.17$ to $1.99$ depending on the operator and the process (stable vs. NWA).
  • For the operator $O_{\phi G}$, NLO corrections are particularly large (up to $\sim 100\%$ for quadratic terms), driven by new tree-level contributions opening up at NLO that cannot be connected to Born diagrams via QCD splitting.
  • Quadratic and cross terms ($\sigma_{ij}, \sigma_{ii}$) are often comparable to or larger than linear interference terms ($\sigma_i$), particularly for $O_{tG}$ and $O_{\phi G}$, indicating that truncating the EFT expansion at linear order may be insufficient in certain kinematic regimes.
• Differential Distributions:
  • Significant shape distortions are observed in differential distributions (e.g., $p_T(H)$, $p_T(b_1)$, angular separations) when comparing stable top quarks to reconstructed tops in the NWA.
  • For linear terms involving $O_{tG}$ and $O_{\phi G}$, the differences between stable and reconstructed top kinematics can reach 75–100% in specific phase-space regions, exceeding theoretical uncertainties.
  • The $O_{tW}$ operator, active only in decays, introduces distinct shape modifications in lepton kinematics and angular correlations.
• Jet Veto Impact: Applying a jet veto ($p_T^{veto} = 100-150$ GeV) significantly reduces the large theoretical uncertainties associated with $O_{\phi G}$ contributions, though it also suppresses the size of the NLO corrections.

Significance and Conclusions
The paper concludes that kinematic cuts, higher-order QCD effects, and SMEFT operators in top-quark decays must be considered simultaneously and consistently. The study demonstrates that:

1. Kinematic Cuts Matter: Fiducial cuts applied to decay products induce substantial shape differences in observables compared to inclusive (stable top) calculations. These differences are amplified by SMEFT effects and can lead to misinterpretations of Wilson coefficients if not properly modeled.
2. Decay Dynamics are Crucial: Ignoring SMEFT effects in top decays (as done in stable-top approximations) misses significant contributions (e.g., from $O_{tW}$) and fails to capture the correct kinematic correlations.
3. Theoretical Precision: The inclusion of NLO corrections is essential not only for reducing scale uncertainties but also for correctly predicting the shape of distributions, which is vital for global SMEFT fits.

The authors emphasize that their findings highlight the necessity of moving beyond stable-top approximations for precision SMEFT studies at the LHC. Future work will extend these results to include dynamical scale settings, full off-shell effects, and more exclusive fiducial cuts to further constrain operator coefficients.