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On Recent measurements of Toponium Threshold Enhancement in Entire-Function-Regulated Nonlocal Quantum Field Theory

This paper proposes that the recently observed threshold enhancement in top-antitop production at the LHC can be consistently explained within a finite, gauge-covariant, entire-function-regulated nonlocal quantum field theory framework, where the top quark's large decay width uniquely shapes toponium phenomenology while preserving global QCD constraints.

Original authors: E. J. Thompson

Published 2026-05-08

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CC BY 4.0

Original authors: E. J. Thompson

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

Imagine the universe is built out of tiny, invisible Lego bricks called quarks. Usually, these bricks snap together to form larger structures called "mesons" (like a proton or a neutron's cousins). Scientists have spent decades studying two specific types of these structures: Charmonium (made of charm quarks) and Bottomonium (made of bottom quarks). They are like sturdy, long-lasting Lego castles that stay together for a long time, allowing physicists to study how the "glue" (the strong force) holds them together.

Then, there is the Top Quark. It is the heaviest, most energetic brick of them all. Because it is so heavy, it is also incredibly unstable. It's like a Lego castle built out of wet sand; it falls apart almost instantly (in a fraction of a second) before it can even fully form a shape.

The Mystery: A "Ghost" Castle

Recently, two giant experiments at the Large Hadron Collider (LHC)—called CMS and ATLAS—started smashing protons together at record speeds. They were looking for pairs of top quarks being created.

According to standard physics rules, when these top quarks are created, they should just fly apart immediately. However, the data showed something strange: right at the exact energy needed to create a pair, there was a bump or an excess of events. It was as if the top quarks were briefly holding hands and forming a "quasi-castle" (a toponium state) before falling apart. The experiments saw about 8 to 9 extra events (measured in picobarns) that standard physics couldn't explain.

The New Theory: Smoothing the Rough Edges

The author of this paper, E. J. Thompson, proposes a new way to look at this mystery using a framework called "Entire-Function-Regulated Nonlocal Quantum Field Theory."

Here is the simple analogy:

Standard Physics treats the interaction between quarks like a sharp, jagged spike. If you get too close to the center, the math gets infinitely crazy (infinite energy), which is a problem.
The New Theory suggests that at the tiniest scales, the universe isn't actually jagged. Instead, it's smoothed out, like a sharp mountain peak that has been rounded off by a gentle breeze. This "smoothing" is done by a mathematical tool called an entire-function regulator.

Think of it like this: If you try to measure the temperature of a flame with a thermometer that is too sensitive, it might break. This new theory uses a "thermometer" that automatically smooths out the extreme heat so the math stays stable and finite, without breaking the rules of symmetry (like how the laws of physics look the same from every angle).

How It Solves the Puzzle

The paper uses this "smoothed" math to re-calculate what happens when top quarks meet.

The Smoothing Effect: By applying this smoothing rule, the theory predicts that the top quarks can form a slightly more stable, "ghost-like" connection right at the threshold. This connection creates a small bump in the data.
The Result: When the author plugs in the numbers, this new theory predicts an excess of about 8.3 picobarns. This matches almost perfectly with what the CMS experiment actually saw (8.8 ± 1.3 picobarns).
The "Magic" Scale: The theory introduces a specific "smoothing scale" (called $\Lambda_{ker}$ ) that acts like a tuning knob. By setting this knob to a specific value related to the top quark's mass, the math lines up with reality.

A Second Ingredient: The "Slow-Motion" Glue

The paper also suggests a second tweak. It proposes that the "strength" of the glue (the strong force) changes slightly depending on how fast you are looking at it. The author uses a special mathematical shape (involving something called the Lambert W-function) to describe how this glue behaves near the top quark's mass. This "holomorphic" adjustment adds a little extra boost to the prediction, helping it fit the data even better, though the main "smoothing" effect is the heavy lifter.

Why Toponium is Special

The paper compares the three heavy quark systems like different types of vehicles:

Charmonium & Bottomonium: These are like trains on a track. They are heavy, stable, and run for a long time. You can see them clearly and study their tracks.
Toponium: This is like a firework. It explodes the moment it is lit. You can't see the firework itself as a solid object; you only see the flash and the smoke (the "threshold enhancement") right at the moment of ignition. Because the top quark decays so fast, it never becomes a true "meson" like the others; it's just a fleeting moment of connection.

The Bottom Line

The paper claims that by using this "smoothed" version of physics (Nonlocal QFT), we can explain the mysterious bump seen at the LHC without breaking any other known laws of physics. It suggests that the top quark's fleeting "ghost castle" is real, and that the universe might be slightly "smoother" at the tiniest scales than we previously thought.

The author concludes that this framework allows scientists to use the top quark as a unique laboratory to test the very edges of our understanding of the universe, bridging the gap between the very small (quantum mechanics) and the very heavy (high-energy physics).

Technical Summary: Recent Measurements of Toponium Threshold Enhancement in Entire-Function-Regulated Nonlocal Quantum Field Theory

Problem Statement
The paper addresses a recent experimental anomaly reported by the CMS and ATLAS collaborations at the LHC: a statistically significant excess in top-antitop ( $t\bar{t}$ ) production near the kinematic threshold ( $\sqrt{s} \approx 2m_t$ ). While standard perturbative QCD (pQCD) predicts a smooth continuum, the data reveals an enhancement of approximately $8.8 \pm 1.3$ pb (CMS) and $9.0 \pm 1.3$ pb (ATLAS). This excess is interpreted as a quasi-bound state or "toponium" resonance. However, the top quark's ultrashort lifetime ( $\tau_t \sim 10^{-25}$ s) and large decay width ( $\Gamma_t \approx 1.41$ GeV) prevent the formation of conventional, long-lived mesonic bound states. The paper seeks to theoretically reassess these threshold dynamics using a framework that extends beyond purely local formulations, aiming to explain the excess while preserving global QCD constraints.

Methodology
The authors employ a finite, gauge-covariant, nonlocal quantum field theory (QFT) framework characterized by "entire-function regulators." The methodology proceeds through three primary avenues:

Modified Bethe–Salpeter (BS) Formalism: The authors extend a UV-complete nonlocal QFT to the heavy-quark sector. They introduce regulators of the form $e^{-\square/\Lambda_{UV}^2}$ into the quadratic terms of the Lagrangian, modifying the propagators for the top quark and gluon. This deformation preserves Lorentz and gauge covariance while introducing exponential damping in the Euclidean region.
- The modified BS equation is derived for a color-singlet $t\bar{t}$ pair.
- In the nonrelativistic limit, this is matched to an instantaneous Schrödinger equation with a regulated potential $V(r; \Lambda_{ker}) = -C_F \alpha_s \frac{\text{erf}(\Lambda_{ker} r/2)}{r}$ .
- Crucially, the authors distinguish between the hard ultraviolet scale ( $\Lambda_{UV}$ ), constrained by high-energy data, and an effective kernel scale ( $\Lambda_{ker}$ ) specific to the nonrelativistic threshold region.
Holomorphic Renormalization Group (RG) Flow: The authors introduce a deformation to the QCD $\beta$ -function, inspired by holomorphic constructions in unified field theories. The deformed $\beta$ -function is given by $\beta_h(\alpha_s) = -\beta_0 \frac{\alpha_s^2}{4\pi} [1 - \frac{\alpha_s}{\alpha^*}]$ , where $\alpha^*$ represents a nontrivial infrared fixed point.
- This leads to an analytic solution for the running coupling $\alpha_s(\mu)$ involving the Lambert $W$ -function.
- The authors analyze how this deformation affects the coupling strength at the threshold scale $\mu \approx 2m_t$ and subsequently the production cross-section.
Comparative Phenomenology: The study systematically compares the $t\bar{t}$ system with charmonium ( $c\bar{c}$ ) and bottomonium ( $b\bar{b}$ ). This involves quantifying binding energies ( $\Delta E$ ), decay widths, and quantum number assignments to clarify whether toponium constitutes a true meson or a virtual threshold phenomenon.

Key Results

Threshold Enhancement Quantification: By solving the regulated Schrödinger equation and fitting the effective kernel scale $\Lambda_{ker}$ to the observed excess, the authors find that setting $\Lambda_{ker} \approx 2m_t \approx 346$ GeV yields a predicted enhancement of $\approx 8.3$ pb. This is in close agreement with the CMS measurement of $8.8 \pm 1.3$ pb.
RG Deformation Limits: The holomorphic RG deformation alone, with a conservative infrared fixed point ( $\alpha^* \gtrsim 0.5$ ), predicts only a $\sim 10\%$ enhancement in the cross-section (roughly 1–2 pb). The authors note that larger enhancements (e.g., $\alpha^* \sim 0.15$ ) would produce an excess far exceeding experimental data and are thus phenomenologically disfavored. Consequently, the dominant contribution to the observed excess is attributed to the nonlocal kernel deformation rather than the RG flow modification.
Nature of Toponium: The analysis confirms that toponium does not behave as a standard meson. Due to the condition $|\Delta E| \sim \Gamma_t$ , the would-be bound state pole is deeply embedded in the continuum. The paper concludes that toponium is best interpreted as a "threshold resonance" or a quasi-bound state, characterized by a broad width and a distortion of the production cross-section rather than a narrow Breit-Wigner peak.
Consistency with Global Data: The framework ensures that deviations in hard observables (e.g., deep inelastic scattering, jet cross-sections) scale as $Q^2/\Lambda_{UV}^2$ . With $\Lambda_{UV}$ constrained to be in the multi-TeV range, these corrections remain negligible, preserving agreement with global QCD fits and lattice determinations of the heavy-quark potential.

Significance and Claims
The paper claims that the observed $t\bar{t}$ threshold excess can be consistently accommodated within a finite, nonlocal QFT framework without violating established QCD constraints. The primary significance lies in demonstrating that:

Toponium as a Laboratory: The top quark threshold serves as a unique laboratory for probing both infrared bound-state dynamics and ultraviolet completion effects.
Data-Driven Scales: The effective kernel scale $\Lambda_{ker}$ can be directly constrained by the threshold excess data, distinct from the hard UV scale $\Lambda_{UV}$ .
Mechanism for Excess: The "entire-function-regulated" approach provides a mechanism to smooth the short-distance Coulomb kernel, reducing binding energy and shifting the resonance closer to the threshold, thereby explaining the observed cross-section enhancement.
Theoretical Distinction: The work clarifies the distinction between a true meson and a virtual threshold phenomenon, arguing that the top quark's large width fundamentally alters the phenomenology compared to lighter quarkonia.

The authors conclude that their minimal entire-function-regulated framework successfully reproduces the experimental data through a matched threshold kernel, offering a new avenue for precision tests of QCD and potential ultraviolet completions.