⚛️ phenomenology

Determining the Threshold Excess in Top-Quark Pair Production through a Global PDF Analysis

This paper presents a global PDF analysis of top-quark pair production data from ATLAS and CMS, revealing no significant threshold excess at 8 TeV but a potential excess exceeding two standard deviations at 13 TeV, while highlighting a critical correlation between the threshold excess and the top-quark mass.

Original authors: Richard D. Ball, Jaco ter Hoeve

Published 2026-10-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Richard D. Ball, Jaco ter Hoeve

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, invisible landscape of the subatomic world, the top quark holds a unique position as the heaviest known elementary particle. It is so massive that its existence is fleeting, decaying almost instantly after it is created. Because of this immense weight, the top quark acts as a critical anchor for our understanding of the universe's stability, influencing how the fundamental forces that bind matter together evolve over time. When scientists collide protons at the Large Hadron Collider, they occasionally create pairs of these heavy particles. Just as two magnets might snap together before flying apart, theory suggests that a top quark and its antimatter partner, the anti-top quark, could briefly form a tight, bound state before they decay. This fleeting moment, occurring right at the edge of the energy required to create them, is known as the threshold. Physicists have long wondered if this specific moment leaves a detectable signature—a small, localized bump in the data that would signal the formation of this exotic, short-lived state.

A team of researchers at the University of Edinburgh has taken a fresh, data-driven approach to hunt for this signature. Instead of relying on a single theoretical model to predict exactly what this signal should look like, they treated the potential excess of events as a free variable to be measured directly from the data. They combined a massive collection of measurements from the ATLAS and CMS experiments, which have been recording top-quark pair production at two different energy levels: 8 trillion and 13 trillion electron volts. By analyzing the distribution of the mass of these particle pairs, the researchers looked for any deviation from the smooth background expected by standard calculations. Crucially, they did not treat the mass of the top quark or the strength of the strong nuclear force as fixed, known numbers. Instead, they allowed all these quantities to float and adjust simultaneously, ensuring that the uncertainty in one did not hide or distort the signal of the other.

The results of this global analysis reveal a clear difference between the two energy levels. At the lower energy of 8 trillion electron volts, the data shows no significant evidence of this threshold excess; the measurements align well with the standard predictions, showing no sign of the exotic bound state. However, at the higher energy of 13 trillion electron volts, the picture changes. When combining all available measurements from that energy, the researchers found a distinct signal that sits slightly above the standard background prediction. This excess is statistically significant, standing at just over two standard deviations, which suggests a real effect rather than a random fluctuation, though it does not yet reach the level of a definitive discovery. The size of this excess is estimated to be around 14 picobarns, a unit of probability for particle interactions, with a range of uncertainty that keeps the true value likely between roughly 8 and 20 picobarns.

A key discovery in this work is the deep connection between the size of this excess and the mass of the top quark itself. The researchers found that these two quantities are strongly linked: if the top quark is slightly heavier, the data can be explained with a smaller excess, and if the quark is lighter, a larger excess is needed to fit the observations. This relationship means that any attempt to measure the top quark's mass without accounting for this potential excess could lead to a biased result, and conversely, measuring the excess without knowing the precise mass is equally unreliable. The study demonstrates that these parameters must be determined together to get an accurate picture. Furthermore, the analysis highlights that the strength of this effect depends heavily on the collision energy. The excess observed at 13 trillion electron volts is significantly larger than what is seen at 8 trillion, a behavior that aligns with the expectation that the particles responsible for forming this state behave differently at higher energies.

While the combined data at 13 trillion electron volts points toward this excess, the researchers note that no single experiment or specific measurement channel on its own shows a tension with the standard model strong enough to claim a discovery. The signal emerges only when all the different datasets are brought together, suggesting that the effect is subtle and requires the full weight of the available evidence to be seen. The team also compared their findings to previous claims of a similar excess reported by the experimental collaborations, noting that their own uncertainties are larger, likely because they are accounting for the complex interplay with the internal structure of the proton. Ultimately, this work provides a rigorous, empirical determination of the threshold behavior in top-quark production. It confirms that while there is a hint of new physics or a more complex interaction at the highest energies, the full story requires a joint understanding of the particle's mass, the strength of the forces involved, and the energy of the collision, all of which must be weighed together to avoid misinterpreting the subtle signals of the quantum world.

Technical Summary: Determining the Threshold Excess in Top-Quark Pair Production

Problem Statement
Top-quark pair ( $t\bar{t}$ ) production near the kinematic threshold ( $m_{t\bar{t}} \approx 2m_t$ ) is a unique regime in the Standard Model where the possible formation of a color-singlet quasi-bound state, known as toponium, could manifest as a localized excess in the invariant mass distribution. While theoretical calculations (NLO NRQCD and NNLO QCD) predict varying magnitudes for this excess, recent experimental measurements by ATLAS and CMS at $\sqrt{s} = 13$ TeV have reported cross-sections significantly higher than theoretical predictions, suggesting a potential "threshold excess" ( $\Delta$ ).

A critical challenge in quantifying this excess is its strong correlation with the top-quark pole mass ( $m_t$ ) and the strong coupling constant ( $\alpha_s$ ). Previous analyses often treated these parameters independently or fixed them, potentially biasing the extraction of the excess. Furthermore, the interplay between the excess and the background continuum requires a rigorous treatment of correlated uncertainties, including Parton Distribution Functions (PDFs), theoretical scale variations, and electroweak (EW) corrections.

Methodology
The authors employ a purely data-driven approach using the Theory Covariance Method (TCM) within a global PDF analysis framework. Unlike previous works that fixed the excess or determined it sequentially, this study treats the threshold excess $\Delta$ as a free nuisance parameter to be extracted simultaneously with $m_t$ and $\alpha_s(m_Z)$ , alongside the PDFs.

Key methodological components include:

Formalism: The TCM is extended to determine the posterior distribution of $\Delta$ . Nuisance parameters induce shifts in theoretical predictions ( $T \to T + \lambda_a \beta_a$ ). By generating an ensemble of data replicas and theoretical predictions, the authors compute the posterior mean and covariance for the parameters of interest.
Data Set: The analysis utilizes differential $t\bar{t}$ cross-section measurements in the invariant mass $m_{t\bar{t}}$ from ATLAS and CMS at $\sqrt{s} = 8$ TeV and $13$ TeV. These datasets cover various decay channels (all-hadronic, lepton+jets, dilepton).
Theoretical Predictions: Theoretical baselines are computed at NNLO QCD accuracy using the MATRIX code, interfaced with PineAPPL. These are complemented by NLO EW corrections at orders $O(\alpha_s \alpha)$ and $O(\alpha_s^2 \alpha)$ obtained via MadGraph5_aMC@NLO. Missing Higher Order Uncertainties (MHOUs) are included following established prescriptions.
Priors: A prior central value for $\Delta$ is set based on previous estimates (6.43 pb at 13 TeV, 2.19 pb at 8 TeV) with a deliberately large relative uncertainty (200%) to avoid biasing the result toward the prior. The prior for $m_t$ is 172.5 GeV and $\alpha_s$ is 0.120.
Correlation Handling: The method explicitly constructs cross-covariance matrices between the theoretical predictions and the parameters ( $m_t, \alpha_s, \Delta$ ) to account for their interdependencies.

Key Results
The analysis yields the following findings:

13 TeV Combination: When combining all available 13 TeV measurements, the study finds evidence of a non-zero threshold excess. The result is:
$\Delta = 14.14^{+6.44}_{-6.31} \text{ pb}$
This value represents a tension with the pure NNLO QCD + EW background hypothesis at approximately 2.2 $\sigma$ . The corresponding top-quark mass is determined to be $m_t = 172.85^{+0.59}_{-0.58}$ GeV.
8 TeV Combination: At $\sqrt{s} = 8$ TeV, the evidence for an excess is negligible. The combined result is:
$\Delta = 3.28^{+2.66}_{-2.32} \text{ pb} \quad (\text{with } \Delta > 0 \text{ imposed})$
Without the positivity constraint, the result is consistent with zero ( $\Delta = 0.57 \pm 3.83$ pb). No significant tension with the background is observed at this energy.
Correlations: A strong positive correlation is identified between the threshold excess $\Delta$ and the top-quark mass $m_t$ (correlation coefficient $\rho \approx 0.84$ at 13 TeV). This arises because a higher $m_t$ shifts the threshold to higher invariant masses, which can be compensated by an increased excess to fit the same cross-section data. Conversely, the correlation between $\Delta$ and $\alpha_s$ is negligible.
Dataset Breakdown: Individual datasets (ATLAS and CMS, various channels) generally show excesses consistent with zero within their uncertainties. The significant 2.2 $\sigma$ tension emerges only when the datasets are combined, driven by the specific behavior of the CMS dilepton channel where the PDF fit tends to undershoot the data in the threshold region, requiring an excess or lower mass to compensate.
Energy Dependence: The study observes a significant dependence of the extracted excess on the center-of-mass energy, with the excess being much larger at 13 TeV than at 8 TeV. This is attributed to the steep fall-off of the gluon PDF at higher Bjorken- $x$ , which is probed more deeply at lower energies for a fixed threshold mass.

Significance and Claims
The paper claims that a joint extraction of the threshold excess, top-quark mass, and strong coupling is essential for precision physics. The authors argue that:

Ignoring the threshold excess in mass determinations leads to a biased $m_t$ value, as the mass fit will absorb part of the excess.
Conversely, experimental determinations of the excess must account for the sensitivity of the threshold region to $m_t$ .
The observed 2.2 $\sigma$ tension at 13 TeV suggests that current theoretical descriptions (NNLO QCD + EW) may be incomplete or that the interplay of parameters requires a more sophisticated treatment than fixed-order calculations alone.
The strong correlation between $\Delta$ and $m_t$ implies that future precision measurements must treat these quantities simultaneously to avoid systematic biases.

The authors remain modest, noting that the tension is only visible in the combined dataset and not in individual measurements, and that the uncertainties remain relatively large compared to some experimental claims. They suggest that future theoretical predictions for toponium cross-sections should cover a range of energies to test the observed energy dependence, and that spin correlation observables could be added to future fits to break the $m_t$ - $\Delta$ degeneracy.

Technical Summary: Determining the Threshold Excess in Top-Quark Pair Production

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