Study of $t\bar{t}$ threshold effects in $e\mu$ differential distributions measured in $\sqrt{s}=13$\,TeV $pp$ collisions with the ATLAS detector

The ATLAS collaboration reports evidence exceeding three standard deviations for the formation of quasi-bound states near the $t\bar{t}$ threshold in 13 TeV $pp$ collisions, as the measured normalised $e\mu$ differential distributions are better described by models incorporating these states than by standard perturbative QCD predictions.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle smasher. Scientists at CERN's ATLAS detector are constantly smashing protons together to see what happens. Usually, when they smash protons, they create pairs of "top quarks," which are the heaviest known elementary particles. Think of a top quark as a very heavy, very short-lived bowling ball.

Usually, when two of these heavy bowling balls are created, they fly apart immediately. But this paper asks a specific question: What happens when they are created with just enough energy to barely stick together?

The "Cosmic Velcro" Effect

The scientists were looking at a very specific moment: right at the "threshold" where the energy is just enough to make a top quark and an anti-top quark (its mirror image) form a temporary, quasi-bound state.

In everyday language, imagine two magnets. If you throw them at each other too fast, they bounce off. If you throw them too slow, they don't reach each other. But if you throw them at just the right speed, they might snap together for a split second before flying apart again. The paper suggests that top quarks do exactly this. They briefly form a "quasi-bound state" (a temporary molecule of two top quarks) before decaying.

The Mystery of the "Missing" Data

For a long time, the computer models used to predict these collisions (based on standard physics rules) didn't quite match what the detectors saw.

• The Prediction: The computer models said there should be a certain number of events where the two resulting particles (an electron and a muon) have a specific combined weight (invariant mass).
• The Reality: The actual data from the ATLAS detector showed a "bump" or an excess of events in the low-mass region. It was like the computer predicted 100 cars would pass a checkpoint, but the camera actually saw 120.

Previous studies hinted at this, but this new paper uses a much larger dataset (140 times more data than some earlier studies) and a more sophisticated way of looking at the numbers.

The Detective Work: Testing the Models

The team compared the real data against three different "recipes" for how these collisions should behave:

1. The Standard Recipe: Just the usual physics rules (perturbative QCD).
2. The "Velcro" Recipe: The standard rules plus the idea that the top quarks briefly stick together (quasi-bound states).
3. The "Resonance" Recipe: A simplified version where the sticking happens like a specific, short-lived particle (a pseudo-scalar resonance).

The Result:
The "Standard Recipe" failed to explain the data; it missed the bump. However, the "Velcro" and "Resonance" recipes fit the data beautifully.

• When they added the "sticking together" effect to their models, the predictions matched the ATLAS measurements almost perfectly.
• Specifically, looking at the mass of the electron-muon pair, the data showed a clear signal that the top quarks were indeed forming these temporary bound states.

The Verdict: A "3-Sigma" Discovery

The paper claims that the evidence for this "sticking together" phenomenon is strong. They calculated the statistical significance and found it exceeds three standard deviations (often called "3-sigma").

In the world of particle physics, this is like rolling a die and getting a six three times in a row by pure chance—it's unlikely, but not impossible. It's strong evidence that the "Velcro" effect is real, though scientists usually wait for "5-sigma" (five times in a row) to declare a full, official discovery.

Summary

In short, this paper says:

• We smashed protons together to create heavy top quarks.
• The data showed more low-mass events than standard physics predicted.
• By adding a rule that says "top quarks can briefly stick together like magnets," the predictions finally matched the reality.
• The match is so good that we are very confident (over 99% sure) that this temporary binding is actually happening, confirming a subtle and fascinating behavior of the universe's heaviest particles.

The paper does not discuss medical applications, future technologies, or what this means for the future of the universe; it is strictly a report on observing a specific, rare behavior of particles in a collider.

Technical Summary

1. Problem Statement

The production of top-quark pairs ($t\bar{t}$) at the Large Hadron Collider (LHC) serves as a critical probe for Quantum Chromodynamics (QCD) at high energy scales. Recent measurements by both ATLAS and CMS have identified an excess of events at low invariant masses ($m_{t\bar{t}}$) near the production threshold ($2m_t \approx 345$ GeV) compared to standard perturbative QCD (pQCD) predictions.

This excess suggests the possible formation of quasi-bound states (often referred to as "toponium") where the top quark and antiquark interact via the strong force before decaying. Previous studies (e.g., Ref [7] in the paper) hinted at this phenomenon but relied on HEPData records that lacked full bin-to-bin correlation information and used smaller datasets (36 fb$^{-1}$). This paper aims to:

• Extend the analysis to the full ATLAS Run 2 dataset (140 fb$^{-1}$).
• Rigorously test the hypothesis of quasi-bound state formation using normalized differential cross-sections.
• Incorporate full experimental and theoretical uncertainties, including bin-to-bin correlations, to provide a statistically robust significance for the observation.

2. Methodology

Data and Selection:

• Dataset: $pp$ collisions at $\sqrt{s}=13$ TeV, corresponding to an integrated luminosity of 140 fb$^{-1}$.
• Channel: $t\bar{t} \to e\mu\nu\bar{\nu}b\bar{b}$ (dilepton channel with opposite-charge electron and muon).
• Fiducial Region: Leptons with $p_T > 20$ GeV and $|\eta| < 2.5$. No jet requirements for the primary $t\bar{t} \to e\mu$ analysis; a separate analysis with $b$-tagged jets ($e\mu b\bar{b}$) was performed for comparison.

Theoretical Models:
The measured distributions were compared against several theoretical predictions:

1. pQCD Baselines:
   • Powheg + Pythia8: NLO matrix element matched to parton shower.
   • Powheg + Pythia8 NNLO rew: Reweighted to match NNLO QCD + NLO EW predictions for top kinematics.
   • Powheg MiNNLO + Pythia8: NNLO QCD matched to parton shower.
   • Powheg bb4l + Pythia8: Includes $Wt$ contributions and interference effects for the $e\mu b\bar{b}$ channel.
2. Quasi-Bound State Models (added to pQCD):
   • $t\bar{t}$ GFRW: Based on the Green's function of non-relativistic QCD (NRQCD) in the Coulomb gauge. This is a physically motivated model for toponium formation.
   • $\eta_t$ Model: A simplified pseudo-scalar resonance model (mass 343 GeV, width 2.8 GeV), previously used by CMS.

Statistical Analysis:

• Metric: A $\chi^2$ statistic was calculated comparing measured normalized differential cross-sections to predictions.
• Covariance: The analysis explicitly included the full covariance matrix, accounting for experimental uncertainties, theoretical uncertainties (PDF, scale, mass, radiation), and crucially, bin-to-bin correlations.
• Fitting: A signal strength parameter $\mu$ was introduced, where $\mu=1$ corresponds to the theoretical cross-section of 6.43 pb for the quasi-bound state. Fits were performed to determine the best-fit $\mu$ and its uncertainty.

Distributions Analyzed:

• One-dimensional: Dilepton invariant mass ($m_{e\mu}$), azimuthal angle difference ($\Delta\phi_{e\mu}$), lepton $p_T$, etc.
• Two-dimensional: $\Delta\phi_{e\mu}$ vs. $m_{e\mu}$.

3. Key Contributions

• Full Correlation Treatment: Unlike previous studies, this analysis incorporates the full bin-to-bin correlation matrix for both data and theory, significantly improving the statistical rigor of the $\chi^2$ evaluation.
• Larger Dataset: Utilization of the full 140 fb$^{-1}$ Run 2 dataset reduces statistical uncertainties and improves lepton efficiency modeling compared to the 36 fb$^{-1}$ dataset used in earlier ATLAS measurements.
• Dual Model Approach: The study simultaneously tests a rigorous NRQCD Green's function approach ($t\bar{t}$ GFRW) and a simplified resonance model ($\eta_t$), finding consistent results for both.
• Cross-Channel Validation: The analysis validates findings using both the inclusive $t\bar{t} \to e\mu$ selection and the more constrained $e\mu b\bar{b}$ selection, ensuring robustness against modeling of $Wt$ backgrounds.

4. Results

Differential Distributions:

• $m_{e\mu}$ Distribution: The standard pQCD models (e.g., Powheg MiNNLO) show a significant deficit in the low $m_{e\mu}$ region ($< 100$ GeV) compared to data. The addition of the quasi-bound state component ($t\bar{t}$ GFRW or $\eta_t$) significantly improves the description of the data.
  • For the Powheg MiNNLO + Pythia8 model, the $\chi^2$ for the $m_{e\mu}$ distribution improves from 25.9 to 16.5 (14 degrees of freedom) upon adding the quasi-bound state.
  • The quasi-bound state contribution adds approximately 3% to the absolute differential cross-section for $m_{e\mu} < 30$ GeV.
• Other Distributions: Improvements in $\chi^2$ for $\Delta\phi_{e\mu}$ and 2D distributions were less pronounced or negligible, indicating that the $m_{e\mu}$ distribution provides the primary sensitivity to threshold effects due to the kinematic constraints at low mass.

Signal Strength Fits:

• Fits to the $m_{e\mu}$ distribution for the $t\bar{t} \to e\mu$ channel yielded a signal strength of:
  $$\mu = 1.22 \pm 0.38$$
  (using the Powheg MiNNLO + Pythia8 + $t\bar{t}$ GFRW model).
• Significance: The fitted value of $\mu$ differs from zero by more than three standard deviations ($>3\sigma$).
• Cross-Section: This corresponds to a measured quasi-bound state cross-section of $7.8 \pm 2.4$ pb, which is consistent with dedicated studies using fully reconstructed $t\bar{t}$ invariant mass distributions.
• Consistency: The $e\mu b\bar{b}$ channel results were consistent with the inclusive channel, though with larger uncertainties due to $b$-tagging systematics.

5. Significance

This paper provides evidence for the formation of quasi-bound states (toponium) near the $t\bar{t}$ threshold with an observed significance exceeding 3 standard deviations.

• Physics Implication: The results support the existence of a color-singlet quasi-bound state formed by the top quark and antiquark before they decay. This validates the use of non-relativistic QCD (NRQCD) descriptions in the threshold region, a regime where standard perturbative expansions may be insufficient.
• Model Validation: The data are significantly better described by models incorporating these bound-state effects than by pure pQCD predictions, resolving long-standing discrepancies in the low invariant mass region of top-quark pair production.
• Future Impact: The methodology and results set a new standard for precision top-quark physics, demonstrating the necessity of including threshold resummation and bound-state effects in event generators for future high-luminosity LHC analyses. The measured cross-section aligns with theoretical expectations for toponium formation, opening new avenues for studying the strong interaction in the heaviest quark sector.