Toponia at the HL-LHC and FCC-ee

This paper computes the properties of toponium states and evaluates their discovery prospects at the HL-LHC and FCC-ee, highlighting the superior sensitivity of lepton colliders for detecting the vector $\psi_t$ state while noting the challenges in observing $P$-wave resonances and the potential for these measurements to probe top-quark properties and new physics.

The Gist

Imagine the subatomic world as a bustling, chaotic dance floor. Usually, particles like quarks are like wild dancers who spin around each other for a split second and then immediately fly apart. But sometimes, under very specific conditions, two dancers get so close and hold on so tightly that they form a temporary, wobbling couple before letting go.

This paper is about a very special, very short-lived couple made of the heaviest dancers in the universe: the Top Quark and its anti-mate, the Anti-Top Quark. When they pair up, they form a "dance couple" called Toponium.

Here is a breakdown of the paper's findings using simple analogies:

1. What is Toponium? (The "Flash-in-the-Pan" Couple)

In the world of atoms, electrons orbit a nucleus. In the world of heavy particles, quarks can orbit each other too.

• The Analogy: Think of Charmonium (charm quarks) and Bottomonium (bottom quarks) as a stable, long-term marriage. They hold hands, spin around each other many times, and then eventually break up.
• Toponium is different. Because the Top quark is so heavy and unstable, it's like a couple that gets engaged, spins around once, and immediately breaks up because one of them has to leave for an emergency. They are a "quasi-bound state." They exist just long enough to be a "thing," but they vanish almost instantly.

2. The Big Discovery: A Hint of a Ghost

The authors mention that experiments at the Large Hadron Collider (LHC) have recently seen a "ghost" in the data.

• The Analogy: Imagine you are listening to a noisy party. Suddenly, you hear a specific musical note that is slightly louder than the background chatter. The scientists at the LHC (CMS and ATLAS) heard this "note" in the energy of the Top quarks. It suggests that a Toponium state (specifically a "pseudoscalar" one, let's call it $\eta_t$) is actually forming, even if just for a fleeting moment.

3. The Challenge: Finding the Right "Camera"

The paper asks: How do we take a clear photo of this fleeting couple before they vanish?

• At the LHC (The Hadron Collider):

  • The Problem: The LHC smashes protons together. It's like a mosh pit. It's very hard to spot a specific dance couple because the crowd is too loud and chaotic.
  • The "Landau-Yang" Rule: There's a rule in physics that says you can't create a specific type of Toponium (the "vector" one, $\psi_t$) just by smashing protons together. It's like trying to start a specific dance move by throwing two people into a room; the physics simply forbids it.
  • The Solution: The paper suggests looking for a different type of Toponium ($\eta_t$) by looking at the debris left behind when the Top quarks decay. They found a promising way to spot this by looking for a specific pattern of particles (Top + Anti-Top + Bottom quarks) at the future High-Luminosity LHC (HL-LHC).

• At the FCC-ee (The Future Lepton Collider):

  • The Advantage: This future machine will smash electrons and positrons together. It's like a quiet, controlled ballroom dance.
  • The Solution: In this clean environment, we can easily create the $\psi_t$ couple. The paper calculates that if we tune the energy of the collider to the exact "heartbeat" of the Top quark mass, we can see this state clearly. It's like tuning a radio to the exact frequency of a song so you can hear it perfectly.

4. Why Does This Matter? (The "Microscope" for New Physics)

Why should we care about a couple that lasts for a fraction of a nanosecond?

• Testing the Rules of the Universe: Toponium is a perfect laboratory to test Quantum Chromodynamics (QCD), the theory of how the strong force works. It's like using a high-speed camera to see if the rules of gravity work exactly as Einstein predicted, but for the strong force.
• Hunting for New Physics: The paper suggests that if we measure the mass and behavior of Toponium precisely, we might find cracks in our current theories.
  • The Analogy: Imagine you are measuring the weight of a known object. If your scale says it's 1 gram heavier than it should be, you know there's a hidden piece of tape or a bug stuck to it.
  • The authors show that if there are new, invisible particles (like a new type of scalar field) interacting with the Top quarks, they would change the "weight" (mass) and "dance steps" (decay rate) of Toponium. By measuring Toponium, we can indirectly detect these invisible intruders.

5. The Verdict

• $\eta_t$ (The Pseudoscalar): We have a good chance of seeing this at the upgraded LHC soon.
• $\psi_t$ (The Vector): This is the "star" for the future electron collider (FCC-ee). It will be hard to see at the LHC, but easy at the FCC-ee.
• $\chi_t$ (The P-wave states): These are the "ghosts of ghosts." They are so hard to produce and detect that they will likely remain invisible for a long time.

Summary

This paper is a roadmap for the next generation of particle physics. It tells us that the "Toponium" state is real and waiting to be studied. It argues that while the current LHC can give us a glimpse, the future FCC-ee will be the perfect place to hold a mirror up to the Top quark, allowing us to see the fundamental forces of nature with unprecedented clarity and potentially discover new physics hiding in the shadows.

Technical Summary

1. Problem Statement

The paper addresses the phenomenology of toponium ($t\bar{t}$), a unique quasi-bound state in the Standard Model (SM) formed by a top quark and an anti-top quark. Unlike charmonium or bottomonium, toponium is a "quasi-bound" state because the top quark's large decay width ($\Gamma_t \approx 1.42$ GeV) is comparable to the binding energy and the inverse of the formation time. Consequently, the system decays before completing a full orbital motion, appearing as a broad resonance rather than a sharp peak.

Recent experimental hints from CMS and ATLAS at the Large Hadron Collider (LHC) suggest a cross-section enhancement near the $t\bar{t}$ production threshold in the pseudoscalar channel, potentially indicating the existence of the pseudoscalar toponium state $\eta_t$. However, theoretical uncertainties regarding the mass spectrum, production mechanisms, and the distinction between bound-state effects and continuum scattering states remain. The paper aims to:

• Compute precise masses and decay widths for the lowest S-wave ($\eta_t, \psi_t$) and P-wave ($\chi_{t0}, \chi_{t1}$) states.
• Assess the discovery potential of these states at the High-Luminosity LHC (HL-LHC) and the Future Circular Collider (FCC-ee).
• Investigate how toponium measurements can constrain Beyond-Standard-Model (BSM) physics, specifically scalar extensions.

2. Methodology

The authors employ a multi-faceted theoretical approach combining perturbative QCD (pQCD) with non-relativistic quantum mechanics:

• Mass Spectrum Calculation:

  • They solve the non-relativistic Schrödinger equation using a static potential $V_{static}(r)$.
  • Short-distance ($r \leq 0.1$ fm): Uses the pQCD potential up to two-loop order, incorporating running coupling $\alpha_s(\mu_R)$.
  • Long-distance ($r > 0.1$ fm): Matches to the Cornell potential ($V_{Cornell} = -A/r + br$).
  • Spin-dependent terms: Included to calculate mass splittings between states with different quantum numbers ($J^{PC}$).
  • Approximation: For wavefunction calculations, they demonstrate that a pure Coulomb potential with a fitted coupling $\alpha_{Coul}^s \approx 0.189$ yields results comparable to the full static potential, simplifying subsequent calculations.

• Decay and Production Formalisms:

  • Projection Method: Used to calculate annihilation decay widths and production cross-sections for specific bound states (e.g., $gg \to \eta_t$). This isolates the partial-wave state from the $t\bar{t}$ continuum.
  • Green Function Method: Used to incorporate the interplay between the continuous $t\bar{t}$ spectrum and the bound-state resonance. This method utilizes the optical theorem and the imaginary part of the Green function to calculate the Sommerfeld enhancement near the threshold.
  • Validation: The authors compare the projection method (using Breit-Wigner formulas) and the Green function method, finding they yield consistent cross-sections for $\eta_t$ production around the threshold.

• Phenomenological Analysis:

  • Monte Carlo simulations (MadGraph5_aMC@NLO, Pythia 8, Delphes 3) are used to estimate signal efficiencies and backgrounds for specific decay channels at the LHC and FCC-ee.
  • BSM Probe: The paper applies the framework to the $Z_2$-symmetric Real Singlet Extension (SSM) of the SM, analyzing how a light scalar $\phi$ modifies the static potential and affects $\eta_t$ production.

3. Key Contributions and Results

A. Theoretical Properties

• Mass Spectrum: The ground state masses are predicted to be $M_{\eta_t} \approx M_{\psi_t} \approx 342.1$ GeV (with an uncertainty of $\sim 2$ GeV due to $m_t$ and renormalization scale). The P-wave states ($\chi_{t0}, \chi_{t1}$) are slightly heavier.
• Decay Modes:
  • Dominant: Intrinsic decays of the constituent top quarks ($t\bar{t} \to W^+b W^-\bar{b}$) dominate with a width $\Gamma \approx 2.84$ GeV.
  • Annihilation: Two-body annihilation decays (e.g., $\eta_t \to b\bar{b}, \gamma Z$) have branching ratios $\lesssim 10^{-3}$ but are crucial for indirect searches.
  • Selection Rules: The Landau-Yang theorem forbids the direct production of the vector state $\psi_t$ via gluon fusion ($gg \to \psi_t$), making it invisible at hadron colliders in the dominant channel.

B. Discovery Prospects at Colliders

• $\eta_t$ (Pseudoscalar, $0^{-+}$):
  • HL-LHC: The direct $pp \to \eta_t \to bW^+ \bar{b}W^-$ channel is challenging due to large backgrounds. However, the associated production $pp \to \eta_t b\bar{b}$ is identified as a promising channel. With $3 \text{ ab}^{-1}$, this channel could reach a significance of $6.1\sigma$, providing complementary evidence that the threshold excess is indeed a $t\bar{t}$ bound state rather than a new BSM pseudoscalar.
  • FCC-ee: Direct production is suppressed due to $J^{PC}$ constraints, but associated production $e^+e^- \to \eta_t \gamma$ is negligible.
• $\psi_t$ (Vector, $1^{--}$):
  • HL-LHC: Production is heavily suppressed by the Landau-Yang theorem and overwhelmed by continuum backgrounds. Discovery is unlikely.
  • FCC-ee: This is the optimal environment. The state can be produced via $e^+e^- \to \psi_t$ near the threshold.
    • Intrinsic Decay: The $W^+W^-b\bar{b}$ channel offers high sensitivity.
    • Annihilation Decay: The authors analyze the interference between the $\psi_t$ pole and SM $\gamma/Z$ processes in the $b\bar{b}$ channel. They correct a sign error in previous literature (Ref. [40, 52]) regarding the vector coupling. With $420 \text{ fb}^{-1}$ at $\sqrt{s} \approx 340.5$ GeV, the interference effect in the $b\bar{b}$ channel could yield a significance of $7.6\sigma$.
• $\chi_{t0}, \chi_{t1}$ (P-wave, $0^{++}, 1^{++}$):
  • Production cross-sections are suppressed by $|R'_P(0)|^2 \propto \alpha_s^5$.
  • Both hadron and lepton colliders face overwhelming continuum backgrounds. The paper concludes these states are unlikely to be observed in the near future.

C. BSM Constraints (Real Singlet Extension)

• The authors demonstrate that the presence of a light scalar $\phi$ in the SSM modifies the static potential via Yukawa interactions.
• This modification shifts the binding energy and wavefunction at the origin, altering the predicted $\eta_t$ production cross-section.
• Current measurements of $\eta_t$ production (CMS/ATLAS) already constrain the parameter space ($\sin \alpha, \tan \beta$) of the SSM, excluding regions where the scalar coupling constructively enhances the potential. Future precision measurements could tighten these constraints significantly.

4. Significance and Implications

• Validation of QCD: Confirming the existence and properties of toponium would provide a rigorous test of perturbative QCD in the heavy-quark limit and the nature of the QCD potential at short distances.
• Top Quark Precision: A precise measurement of the $\eta_t$ mass at the HL-LHC would help determine the optimal center-of-mass energy for the FCC-ee to scan the $t\bar{t}$ threshold, thereby improving the precision of top quark mass and width measurements.
• New Physics Probe: Toponium serves as a sensitive probe for new scalar fields coupling to the top quark. The interference patterns in annihilation decays at lepton colliders offer a unique method to detect new physics that might be invisible in standard resonance searches.
• Methodological Correction: The paper corrects a sign error in the effective couplings used in previous BSEFT (Bound State Effective Field Theory) calculations, highlighting the importance of careful matching between effective theories and full calculations, particularly for off-shell effects.

In summary, the paper establishes that while P-wave toponia are likely unobservable, the pseudoscalar $\eta_t$ and vector $\psi_t$ offer distinct and complementary discovery channels at the HL-LHC and FCC-ee, respectively, serving as powerful tools for precision SM tests and BSM searches.