Top quark FCNC in Randall-Sundrum models: post-LHC allowed rates and searches at $e^+e^-$ and $μ^+ μ^-$ colliders

This paper evaluates the sensitivity of future $e^+e^-$ and $\mu^+\mu^-$ colliders to top quark Flavor Changing Neutral Currents within Randall-Sundrum models, incorporating current and projected HL-LHC limits to determine that while the HL-LHC may reach branching ratios of $10^{-6}$, high-energy lepton colliders offer the potential to probe even smaller coupling strengths.

The Gist

Imagine the universe is a giant, high-stakes game of billiards. Usually, the balls (particles) bounce off each other in very predictable ways. But sometimes, a ball might suddenly change color or switch places with another ball without anyone touching it. In the world of particle physics, this is called a "Flavor Changing Neutral Current" (FCNC). It's a rare, forbidden dance that the Standard Model of physics says shouldn't happen easily, but if it does, it's a huge clue that there are new, hidden rules of the game.

This paper is about hunting for a specific, very rare dance move involving the Top Quark (the heaviest particle in the known universe) and the Charm Quark. Specifically, the authors are looking for a moment where a Top Quark turns into a Charm Quark while interacting with a Z boson (a force-carrier particle).

Here is the breakdown of their quest, using simple analogies:

1. The Problem: The "Ghost" in the Machine

The Large Hadron Collider (LHC) at CERN is like a massive, high-speed crash test facility. Scientists smash protons together to see what breaks. They have been looking for this Top-to-Charm switch.

• The Paper's Finding: The LHC hasn't found the switch yet, but it has tightened the net. It's like saying, "We know the thief isn't hiding in the basement anymore; if they are here, they must be very small and very quiet."
• The Model: The authors use a specific theory called the Randall-Sundrum model. Think of this model as a map that predicts where the "thief" (the new physics) might be hiding. It suggests that the "thief" is actually a heavy, invisible particle (a Kaluza-Klein excitation) that is too heavy for the LHC to catch directly, but its "shadow" (the FCNC effect) might be visible.

2. The Strategy: Changing the Game Plan

Since the LHC is getting better at finding heavy particles, the authors ask: If we can't catch the heavy particle directly, can we catch its shadow in a different way?

They propose using two new types of "microscopes" (colliders) that haven't been built yet:

• The Higgs Factory (e+e−): A circular machine that smashes electrons and positrons together at a "sweet spot" energy (around 240 GeV).
• The Muon Collider (µ+µ−): A much more powerful machine that smashes muons together at incredibly high energies (10 TeV).

3. The Analogy: The Fishing Trip

Imagine you are trying to catch a very shy fish (the Top-Charm interaction).

• The LHC Approach: The LHC is like a giant trawler dragging a massive net through the ocean. It's great for catching big, heavy fish (new heavy particles), but the water is so muddy (lots of background noise) that it's hard to see the tiny, shy fish.
• The Electron Machine (Higgs Factory): This is like a quiet, clear pond. The water is crystal clear. Even though the pond isn't as deep as the ocean, the clarity allows you to spot the shy fish if you look closely. The authors found that by lowering the speed of the "boat" (energy) slightly, they could actually catch more fish because the pond is calmer and they can spend more time there (higher luminosity).
• The Muon Collider: This is like a high-powered laser beam shooting through the ocean. It's so powerful that it can spot the shy fish even if it's hiding deep down or moving very fast.

4. The Results: What They Found

The authors did a lot of computer simulations (like running a video game of the collision) to see what these new machines could achieve.

• The "Cut" Method: They tried simple rules to filter out the noise (like "only look at fish bigger than X"). This worked okay.
• The "BDT" Method: They used an Artificial Intelligence (a "Brain") to learn the difference between the signal and the noise. This was like hiring a master fisherman who can tell the difference between a real fish and a piece of seaweed just by looking at the ripples. This method was much better.

The Big Takeaways:

1. Lower Energy Can Be Better: For certain types of interactions, running the electron machine at a slightly lower energy (around 200–240 GeV) actually gives better results than running it at the maximum energy, because you get more "collisions" (luminosity) to study.
2. High Energy is a Powerhouse: The 10 TeV Muon Collider is a beast. It can probe interactions so rare that the LHC would never see them. It could detect a Top-to-Charm switch happening only one time in a million (or even less), whereas the LHC is currently limited to seeing it happen about one time in 100,000.
3. Different Tools for Different Jobs:
   • Some "shy" interactions (involving the Higgs) are best found in the quiet, clear pond (lower energy electron machine).
   • Other "fast" interactions (involving direct contact between particles) are best found with the high-powered laser (high energy muon collider).

5. The Conclusion

The paper concludes that while the LHC has done a great job of ruling out the "easy" places where new physics might hide, the future of finding the Top-Charm switch lies in these new, specialized machines.

• If we build the Electron Machine, we can look for these rare events with incredible precision, potentially finding clues that the LHC missed.
• If we build the Muon Collider, we can look so deep into the "forbidden" territory that we might finally catch a glimpse of the heavy particles that the Randall-Sundrum model predicts.

In short: The LHC has swept the floor, but to find the tiny, hidden dust bunnies (the rare Top-Charm interactions), we need either a very clean, quiet room (the electron machine) or a super-powered vacuum (the muon collider).

Technical Summary

Technical Summary: Top quark FCNC in Randall-Sundrum models: post-LHC allowed rates and searches at $e^+e^-$ and $\mu^+\mu^-$ colliders

Problem and Motivation
The study of top quark interactions is a primary objective for future lepton colliders. While significant attention has been devoted to top quark pair production ($e^+e^- \to t\bar{t}$) near the threshold (350–500 GeV) to precisely determine the top mass and study the $Zt\bar{t}$ coupling, this work investigates the potential of lower center-of-mass energies ($E_{\text{cm}} \in [200, 240]$ GeV) and higher-energy muon colliders ($E_{\text{cm}} = 10$ TeV) to probe Flavor Changing Neutral Currents (FCNC). Specifically, the authors focus on the reaction $\ell^+\ell^- \to t + j$ (with $j=c$), mediated by the $Ztc$ vertex or four-fermion contact interactions.

The motivation stems from the observation that luminosity in circular $e^+e^-$ machines scales roughly as $L \sim E_{\text{cm}}^{-4}$. Consequently, operating at lower energies (near the $t+c$ threshold of $\sim 175$ GeV) could yield significantly higher integrated luminosity, potentially compensating for the phase-space suppression of the signal cross-section. Furthermore, while the Large Hadron Collider (LHC) sets stringent bounds on the microscopic resonances generating these FCNCs, direct searches for the FCNC effects themselves in top decays ($t \to cZ$) are limited. The authors aim to determine the maximal strength of effective FCNC couplings allowed by current LHC data and assess the sensitivity of future colliders to probe couplings beyond these limits.

Methodology
The study employs a multi-step approach combining Effective Field Theory (SMEFT), specific Beyond the Standard Model (BSM) scenarios, and collider phenomenology:

1. Theoretical Framework:

   • SMEFT: The analysis utilizes dimension-6 Warsaw basis operators contributing to the $Ztc$ coupling (vector and dipole operators) and four-fermion contact interactions.
   • Randall-Sundrum (RS) Model: To incorporate realistic limits from the LHC, the authors map the SMEFT operators to the RS model. In this framework, FCNCs arise from the mixing of the SM $Z$ boson with Kaluza-Klein (KK) excitations ($Z_{KK}$).
   • Recasting LHC Limits: The authors recast existing LHC searches for Heavy Vector Triplet (HVT) resonances (specifically $W_{KK}$ and $Z_{KK}$) into bounds on the RS model. They account for differences in branching ratios and production cross-sections between the HVT benchmark and the RS model, utilizing full Run 2 data ($138 \text{ fb}^{-1}$) and projecting to High-Luminosity LHC (HL-LHC) sensitivities ($3 \text{ ab}^{-1}$).

2. Collider Simulations:

   • Signal and Background: The signal process is $\ell^+\ell^- \to \bar{c}t \to \bar{c}b\ell^+\nu$. The dominant background is $\ell^+\ell^- \to W^+W^- \to \bar{c}s\ell^+\nu$, where the $s$-quark jet is misidentified as a $b$-jet.
   • Detector Effects: Simulations are performed at Leading Order (LO) using MG5_aMC@NLO with the dim6top UFO model. Detector effects are modeled via Gaussian energy smearing (1% and 5%) for jets and leptons.
   • Analysis Strategies: Two approaches are used:
     • Cut-based Analysis: Selections based on kinematic variables: recoil top mass ($m_{t,\text{recoil}}$), dijet invariant mass ($m_{jj}$), and missing transverse momentum.
     • BDT Analysis: A Boosted Decision Tree is trained on kinematic variables ($m_{t,\text{recoil}}$, $m_{jj}$, $E_c$) to optimize the signal-to-background ratio ($S/\sqrt{B}$).

3. Energy Scenarios:

   • HTE Factory: Circular $e^+e^-$ colliders operating at $E_{\text{cm}} \in [200, 240]$ GeV.
   • Muon Collider: A high-energy $\mu^+\mu^-$ collider at $E_{\text{cm}} = 10$ TeV.

Key Results

• LHC Constraints on RS Models:

  • Recasting HVT searches yields a lower bound on the mass of EW gauge KK resonances ($M_{KK}$) of approximately 3.7 TeV (from CMS $WH$ resonance searches).
  • Projections for HL-LHC suggest this bound could improve to 4.5 TeV for a single experiment and exceed 5 TeV for a combined limit.
  • These mass limits translate to an upper bound on the branching ratio $\text{BR}(t \to cZ) \simeq 1.2 \times 10^{-6}$ for $M_{KK} > 5$ TeV. This represents an improvement over previous indirect limits (e.g., from LEP).

• Sensitivity at $e^+e^-$ Colliders (HTE Factory):

  • The study finds that running at lower energies ($E_{\text{cm}} \in [200, 240]$ GeV) does not result in a loss of sensitivity to $Ztc$ couplings, provided the luminosity scales as $E_{\text{cm}}^{-4}$.
  • Using BDT analysis, the sensitivity to Wilson coefficients is comparable across the entire $[200, 240]$ GeV range.
  • The projected sensitivity corresponds to probing $\text{BR}(t \to cZ)$ at the level of $10^{-6}$, comparable to the best possible sensitivity of direct exotic decay searches at a top factory.

• Sensitivity at 10 TeV Muon Collider:

  • The 10 TeV muon collider offers significantly enhanced sensitivity for dipole-like and four-fermion operators due to the favorable energy scaling of their cross-sections ($\sigma \sim E_{\text{cm}}^0$ and $\sigma \sim E_{\text{cm}}^2$, respectively).
  • For four-fermion operators, the collider can probe effective mass scales up to tens of TeV (specifically $M_{KK} \sim 30$ TeV for certain operators), far exceeding the reach of the HL-LHC.
  • For Higgs-current-fermion-current operators, the sensitivity is less improved at high energy due to the $E_{\text{cm}}^{-2}$ scaling, but the process $\mu^+\mu^- \to t\bar{c}h$ is noted as a potentially more effective probe at these energies.

Significance and Claims
The paper claims that future lepton colliders can probe FCNC couplings in the top sector at levels ($\text{BR} \sim 10^{-6}$) that are currently inaccessible to direct decay searches at the LHC, even after the HL-LHC upgrade.

• Complementarity: The work highlights a complementarity between low-energy high-luminosity machines (optimal for Higgs-current operators) and high-energy machines (optimal for dipole and four-fermion operators).
• Flavor Physics Frontier: The authors assert that high-energy machines will reach the frontier of flavor physics, with sensitivity to SMEFT Wilson coefficients significantly exceeding that of low-energy flavor experiments.
• Model Discrimination: The study emphasizes that the different energy dependencies of various operator classes allow for the characterization of the underlying New Physics structure (e.g., distinguishing between RS scenarios with different flavor symmetries or chiral structures).
• Methodological Contribution: The paper provides a validated framework for recasting LHC HVT searches into RS constraints and demonstrates the efficacy of BDT-based analyses in optimizing FCNC searches across varying center-of-mass energies.

The authors conclude that while electroweak precision observables generally provide the strongest constraints on RS models, a signal in top quark flavor observables would point to a non-trivial flavor structure within the model, warranting the dedicated exploration proposed in this work.