An EFT study of the $pp \to \bar{t} t Z(ll) h(bb)$ process at the FCC-$\boldsymbol{hh}$

This paper presents an Effective Field Theory study of the $pp \to \bar{t} t Zh$ process at the Future Circular Collider (FCC-$hh$), demonstrating its unique capability to probe anomalous $\bar{t} t Zh$ and $\bar{t} t Z$ couplings through a comprehensive collider analysis of the $4b + 3 \ell + \ge 2j + \slashed{E}_T$ final state.

The Gist

Imagine the universe is a giant, complex machine built from tiny Lego bricks. For decades, scientists have been trying to figure out exactly how these bricks snap together. The "Standard Model" is their instruction manual, but they suspect there are hidden pieces or secret rules they haven't found yet.

This paper is a proposal for a massive, future experiment to find those missing pieces, specifically focusing on the Top Quark.

Here is the story of the paper, broken down into simple concepts:

1. The Star of the Show: The Top Quark

Think of the Top Quark as the "heavyweight champion" of the particle world. It's the heaviest known particle, and because it's so heavy, it interacts very strongly with the Higgs boson (the particle that gives everything mass) and the Z boson (a messenger of the weak force).

Scientists suspect that the Top Quark might be holding a secret handshake with these other particles that doesn't follow the standard instruction manual. If we can find this secret handshake, it could prove the existence of "New Physics"—a whole new layer of reality we haven't seen yet.

2. The Detective Work: Effective Field Theory (EFT)

Since we can't build a machine big enough to see these secret handshakes directly right now, the scientists use a clever trick called Effective Field Theory (EFT).

Imagine you are trying to guess the rules of a game by watching players from far away. You can't see their hands, but you can see if they are moving strangely.

• The Standard Model is the "normal" way the game is played.
• EFT is like adding a "magnifying glass" to your view. It allows scientists to look for tiny, subtle deviations in how the particles behave. If the Top Quark moves even a little bit differently than the manual says it should, EFT helps us measure exactly how weird it is.

3. The Arena: The Future Circular Collider (FCC-hh)

To catch these rare events, we need a racetrack that is impossibly huge. The current Large Hadron Collider (LHC) is like a local high school track. The proposed FCC-hh (Future Circular Collider) would be a super-highway circling the entire Earth (or at least a massive chunk of it).

• The Goal: Smash protons together at energies 7 times higher than the LHC.
• The Target: Create a very specific, rare crash: Top Quark + Anti-Top Quark + Z Boson + Higgs Boson.
• Why this crash? It's like looking for a specific needle in a haystack. It happens very rarely, but when it does, it tells us everything about how the Top Quark interacts with the Higgs and Z bosons.

4. The Challenge: Finding a Needle in a Haystack

The paper describes a massive simulation of what this crash would look like.

• The Signal: The crash produces a specific set of debris: 4 "b-jets" (particles from bottom quarks), 3 charged particles (electrons or muons), and some missing energy.
• The Noise: The problem is that the universe is messy. There are billions of other crashes that look almost the same. It's like trying to hear a whisper at a rock concert.
• The Solution: The authors wrote a computer program to act as a super-detective. They simulated millions of crashes, applied strict rules (like "only count the ones where the particles are moving this fast"), and filtered out the noise.

5. The "Energy Growth" Trick

Here is the most exciting part of the physics.
In normal physics, if you double the energy of a crash, the result usually just doubles. But in this "New Physics" scenario, the weird effects grow quadratically.

• Analogy: Imagine pushing a swing. If you push it normally, it goes a little higher. But if there's a hidden spring (the New Physics), pushing it harder makes it fly way higher, much faster than expected.
• The scientists realized that if they look at the crashes with the highest energy, the "New Physics" signal will pop out like a flare, while the normal background noise stays quiet.

6. The Results: What Did They Find?

The team ran the numbers for the FCC-hh with a massive amount of data (30 "inverse attobarns"—a unit that represents a huge number of collisions).

• The Verdict: They found that the FCC-hh would be able to measure these Top Quark interactions with incredible precision.
• The Sensitivity: They could detect deviations as small as 1%.
• The Comparison: This is so precise that it rivals what we hope to get from future electron-positron colliders, but it does it in a completely different way. It proves that the FCC-hh is a unique tool for this job.

Summary

This paper is a blueprint for a future treasure hunt. The authors are saying:

"If we build this giant collider (FCC-hh) and look for this specific, rare crash (Top + Top + Z + Higgs), we can use a mathematical magnifying glass (EFT) to see if the Top Quark is breaking the rules. We believe we can find these rule-breakers with 95% confidence, opening a door to understanding the universe at a level we've never seen before."

It's a mix of high-stakes detective work, theoretical math, and the promise of a machine that will push the boundaries of human engineering.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the need to probe anomalous interactions between the top quark ($t$), the $Z$ boson, and the Higgs boson ($h$). While the Standard Model (SM) describes these interactions, many Beyond the Standard Model (BSM) scenarios predict deviations in the $\bar{t}tZh$ and $\bar{t}tZ$ couplings.

• Current Limitations: Existing constraints on top-quark couplings are weak compared to lighter quarks. Previous studies at the LHC and proposed Electron-Ion Colliders have not sufficiently constrained the $\bar{t}tZh$ contact interactions.
• The Gap: The $\bar{t}tZh$ interaction is uniquely sensitive to higher-dimensional Effective Field Theory (EFT) operators. While future $e^+e^-$ colliders (like FCC-ee) will constrain $\bar{t}tZ$ couplings, they cannot directly probe the $\bar{t}tZh$ contact terms.
• Objective: To evaluate the discovery potential of the Future Circular Collider in proton-proton mode (FCC-hh, $\sqrt{s} = 100$ TeV) for detecting anomalous $\bar{t}tZh$ couplings using the specific final state $pp \to \bar{t}tZ(\ell^+\ell^-)h(b\bar{b})$.

2. Methodology

Theoretical Framework (EFT)

The authors utilize the Standard Model Effective Field Theory (SMEFT) and Higgs EFT (HEFT) frameworks.

• Lagrangian: They focus on dimension-6 operators that generate anomalous $\bar{t}tZh$ couplings. The relevant Lagrangian terms involve vector and tensor couplings:
  $$\Delta \mathcal{L} \supset g^h_{ZtL} \frac{h}{v} Z_\mu \bar{t}_L \gamma^\mu t_L + g^h_{ZtR} \frac{h}{v} Z_\mu \bar{t}_R \gamma^\mu t_R + g^h_{ZtLR} \frac{h}{v} \bar{t}_L \sigma_{\mu\nu} t_R Z^{\mu\nu} + \text{h.c.}$$
• SMEFT Correlations: In SMEFT, these couplings are correlated with deviations in the $\bar{t}tZ$ vertex ($\delta g^Z_{tL,R}$). The authors derive the relationship $g^h_{Zf} = 2\delta g^Z_f$, allowing bounds on one to translate to the other.
• Energy Growth: A key feature of the EFT contribution is that the amplitude grows quadratically with the invariant mass of the $Zh$ system ($m_{Zh}$) relative to the SM, due to the absence of the $Z$ propagator in the contact interaction. This enhances sensitivity in the high-energy tails.

Simulation and Event Generation

• Collider: FCC-hh with $\sqrt{s} = 100$ TeV and an integrated luminosity of $30 \text{ ab}^{-1}$.
• Tools:
  • FeynRules: Used to encode SMEFT vertices and generate Universal FeynRules Output (UFO) models.
  • Sherpa 2.2.11: Used for event generation at Leading Order (LO) with the Comix matrix element generator.
  • PDFs: CT14nlo set.
• Signal Process: $pp \to \bar{t}tZ(\ell^+\ell^-)h(b\bar{b})$ with $Z \to e^+e^-, \mu^+\mu^-, \tau^+\tau^-$ and $h \to b\bar{b}$.
• Backgrounds: A comprehensive set of backgrounds was generated, including $\bar{t}th$, $\bar{t}tZ$, $\bar{t}tZZ$, $W^+W^-Z$, $ZZZ$, and processes with mistagged jets.
• Phase Space Splitting: To ensure sufficient statistics in the high-energy tails (where EFT effects dominate), signal samples were generated in two regions: $m_{Zh} > 750$ GeV and $m_{Zh} > 1700$ GeV.

Analysis Strategy and Reconstruction

• Final State Selection: The analysis targets the signature: 4b-jets + 3 isolated leptons ($\ell$) + $\ge$ 2 light jets + Missing Transverse Energy ($E_T$).
• Detector Modeling:
  • b-tagging: Implemented with $p_T$ and $\eta$-dependent efficiencies specific to FCC-hh projections.
  • Mistag Rates: Included probabilities for charm and light jets to be misidentified as $b$-jets.
  • Kinematic Cuts: Trigger-level cuts on $p_T$ and $\eta$ for leptons and jets.
• Reconstruction Algorithm:
  1. Z Boson: Reconstructed from the pair of opposite-sign, same-flavor (OSSF) leptons with invariant mass closest to $m_Z$.
  2. Higgs Boson: Reconstructed by scanning all pairs of the 4 b-tagged jets to minimize $\chi^2_h = (m_{bb} - m_h)^2 / \Delta_h^2$. A non-standard $m_h = 120$ GeV is used to account for neutrino losses in $b$-hadron decays.
  3. Hadronic Top: Reconstructed using the remaining b-jet and two light jets, minimizing $\chi^2_{th}$.
  4. Leptonic Top: Partially reconstructed using the remaining b-jet and the third lepton.
• Statistical Analysis: A binned $\chi^2$ fit is performed on the $m_{Zh}$ distribution (800 GeV to 1.5 TeV). The fit includes SM contributions, linear interference terms (SM $\times$ EFT), and quadratic EFT terms.

3. Key Contributions

1. First Full Collider Analysis: This is the first comprehensive study of the $pp \to \bar{t}tZh$ process at the FCC-hh, including a realistic treatment of backgrounds, detector efficiencies, and reconstruction algorithms.
2. EFT Validity Cut: The authors explicitly impose a kinematic cut ($m_{Zh} < 1.5$ TeV) to ensure the validity of the EFT expansion, preventing unphysical results from high-energy extrapolation.
3. Correlation with $\bar{t}tZ$: The study demonstrates how bounds on $\bar{t}tZh$ contact terms in SMEFT directly translate to constraints on $\bar{t}tZ$ coupling deviations, providing a complementary probe to $e^+e^-$ colliders.
4. Background Handling: Detailed modeling of complex backgrounds, including those with fewer than 4 b-quarks at the matrix-element level that mimic the signal via mistags.

4. Results

• Sensitivity: The analysis projects 95% Confidence Level (CL) bounds on the anomalous couplings $g^h_{ZtL}$ and $g^h_{ZtR}$.
• Precision: The FCC-hh is expected to probe values of these couplings down to $\mathcal{O}(10^{-2})$ (percent level).
• SMEFT Translation: Using the relation $g^h_{Zf} = 2\delta g^Z_f$, these results imply percent-level constraints on $\bar{t}tZ$ coupling deviations.
• Comparison: These projected bounds are competitive with, and complementary to, the precision expected from the 365 GeV run of the FCC-ee.
• Discrimination Power: The high-energy tail of the $m_{Zh}$ distribution (800 GeV – 1.5 TeV) is identified as the most sensitive region, where the quadratic growth of the EFT contribution significantly enhances the signal-to-background ratio.

5. Significance

• Probing New Physics: The study confirms that the FCC-hh is a unique machine capable of probing the $\bar{t}tZh$ sector, which remains largely unconstrained by current or near-future $e^+e^-$ facilities.
• Validation of EFT Approach: It validates the strategy of using high-mass tails in hadron colliders to constrain dimension-6 operators, even in the presence of large QCD backgrounds.
• Future Roadmap: The paper serves as a foundational study for future global SMEFT fits. It highlights the need for advanced techniques (e.g., machine learning for signal/background separation) and improved detector performance to fully exploit the FCC-hh's potential in top-quark precision physics.

In conclusion, the paper establishes that the $pp \to \bar{t}tZh$ channel at the FCC-hh is a powerful, albeit challenging, probe for anomalous top-Higgs-Z interactions, offering sensitivity that rivals the precision of future lepton colliders for specific coupling deviations.