Toponium at the LHC: A Ghostly Quark Pair Reshapes the Search for New Physics

For nearly five decades, physicists have catalogued the bound states of heavy quarks with remarkable success. Charmonium and bottomonium, the hydrogen-like atoms of the strong nuclear force, have served as precision laboratories for testing quantum chromodynamics, the theory that describes how quarks and gluons interact. Yet one member of the heavy-quark family has always been conspicuously absent from the experimental record: toponium, the bound state of a top quark and its antiparticle. The reason is brutal in its simplicity. The top quark, with a mass of roughly 173 GeV, decays through the weak interaction in about 10 to the minus 25 seconds, far faster than the roughly 10 to the minus 24 seconds needed for hadronization to glue quarks into measurable particles. Any would-be toponium meson is doomed to dissolve before it can exist as an object in the conventional sense.

That long-standing picture changed dramatically when the CMS Collaboration, analyzing the full Run-2 proton-proton dataset at 13 TeV corresponding to 138 inverse femtobarns of integrated luminosity, reported a statistically significant excess of top-antitop events localized just above the kinematic production threshold. The excess, measured in dileptonic final states with multiple jets, amounts to a cross section of 8.8 picobarns with uncertainties of plus 1.2 and minus 1.4 picobarns above the fixed-order perturbative QCD background. CMS interpreted the signal with a simplified pseudoscalar toponium hypothesis, describing a color-singlet quasi-bound state in the spin-singlet configuration labeled one-S-zero. Soon afterward, ATLAS independently confirmed a compatible enhancement in its own full Run-2 sample, rejecting the pure continuum hypothesis at a significance of 7.7 standard deviations and measuring an enhancement of 9.0 plus or minus 1.3 picobarns. Together, the two measurements have converted toponium from a theoretical curiosity into an empirical fact demanding explanation.

A new theoretical study published in The European Physical Journal C by E. J. Thompson takes up that challenge from an unusual direction. Rather than treating the excess purely within standard nonrelativistic QCD, the work reinterprets the threshold enhancement in the framework of nonlocal quantum field theory, in which the ultraviolet behavior of the theory is tamed by entire-function regulators. The central idea is elegant: instead of modifying QCD at the energies probed by most experiments, the theory smooths the short-distance structure of the interaction kernel that binds the top-antitop pair, while leaving the infrared spectrum and all established low-energy tests untouched. The result is a data-driven framework in which a single new parameter, the kernel scale Lambda-kernel, can be constrained directly by the observed near-threshold excess.

The mathematical machinery rests on a venerable tool of bound-state physics, the Bethe-Salpeter equation, which describes relativistic two-body bound states in quantum field theory. Thompson extends this equation to the nonlocal setting by inserting exponential regulator factors of the form exp of minus p-squared over Lambda-UV-squared into the propagators and the single-gluon-exchange kernel. Crucially, the regulator is built as an entire function of the covariant d’Alembertian operator, which means the deformation intertwines the same symmetry action as the original operator. Lorentz covariance, gauge covariance, and the associated Ward and Slavnov-Taylor identities are therefore preserved, and the theory introduces no additional physical poles that could signal pathological ghost states. In the nonrelativistic potential region relevant to threshold dynamics, the regulated Coulomb potential acquires a smooth error-function form, replacing the singular one-over-r attraction at short distances with a softened profile controlled by Lambda-kernel.

The consequences for toponium spectroscopy are calculable and physically intuitive. Expanding the regulated potential in powers of the momentum transfer generates a tower of contact interactions, the leading one being a repulsive delta-function term proportional to four-pi times C-F times alpha-s divided by Lambda-kernel-squared. Acting on the ground-state wavefunction, this contact term shifts the one-S energy level upward, reducing the binding energy and pushing the would-be resonance closer to the production threshold. The same perturbation suppresses the wavefunction at the origin, the quantity that controls the production rate, since near-threshold cross sections scale as the square of the wavefunction at zero separation, itself proportional to alpha-s cubed. In effect, the nonlocal regulator reshapes both the position and the strength of the threshold enhancement in a way that can be confronted directly with the LHC line shape.

When Thompson fixes the matched kernel scale to the natural threshold value of two times the top mass, approximately 346 GeV, the framework predicts an enhancement of about 8.3 picobarns, in striking agreement with the CMS measurement of 8.8 plus or minus 1.3 picobarns and the ATLAS value of 9.0 plus or minus 1.3 picobarns. Importantly, this agreement does not come at the cost of wrecking other precision tests. Deviations in hard observables such as deep inelastic scattering structure functions scale as Q-squared over Lambda-UV-squared and fall below current experimental uncertainties once the ultraviolet scale exceeds a few TeV. The extraction of the strong coupling from Z-decay observables remains consistent with the Particle Data Group world average of 0.1181 plus or minus 0.0011, and high-precision lattice QCD determinations of the static quark potential show no deviation from the standard Cornell form, implying that regulator effects in the nonperturbative regime are safely bounded.

The study adds a second, complementary ingredient: a holomorphic deformation of the quantum chromodynamic beta function, the equation governing how the strong coupling runs with energy. Inspired by the exact Novikov-Shifman-Vainshtein-Zakharov beta function of supersymmetric gauge theory, the deformed beta function admits a nontrivial infrared fixed point at a coupling value alpha-star, and its renormalization group equation can be solved in closed form using the Lambert W function. Near the threshold scale, the deformed coupling is mildly enhanced relative to the standard running value, and because the threshold cross section scales as alpha-s cubed, even a modest coupling enhancement translates into a measurable increase in the peak height. Thompson shows that the deformation must be handled with care: aggressive fixed-point values near 0.15 would generate an enhancement of several hundred percent, grossly exceeding the data, so the phenomenologically viable regime restricts alpha-star to values above roughly 0.5, where the holomorphic contribution remains a subleading ten-percent effect on top of the dominant kernel deformation.

Perhaps the most illuminating part of the analysis is the systematic comparison of the three heavy-quark systems. Charmonium, with a ground-state mass near 3.097 GeV and a width of only 93 kiloelectronvolts, and bottomonium, at 9.460 GeV with a width of 54 kiloelectronvolts, are textbook narrow resonances, produced and studied in electron-positron colliders through direct scans of their spectral lines. Toponium is a different beast entirely. Its effective binding energy of roughly 0.2 GeV is dwarfed by the top quark’s decay width of about 1.41 GeV, meaning the would-be bound-state pole lies deeply embedded in the continuum. Rather than a Breit-Wigner meson, toponium is best understood as a threshold resonance, a quasi-bound distortion of the production cross section visible only through its fingerprint on the invariant-mass distribution. Electroweak decay suppresses any well-separated excited states, leaving only the ground-state threshold structure, a pseudoscalar with quantum numbers zero-minus-plus in contrast to the vector one-minus-minus assignments of its lighter cousins.

The broader significance of this work lies in what toponium can now be used for. Because the threshold line shape is sensitive both to infrared bound-state dynamics and to the short-distance structure of the underlying field theory, it offers a rare window onto ultraviolet completion effects at an otherwise inaccessible scale. The distinction between the hard ultraviolet scale Lambda-UV, constrained by global high-energy data, and the matched threshold kernel scale Lambda-kernel, constrained by the LHC excess, gives theorists two independent handles on the same nonlocal framework. As future datasets from the High-Luminosity LHC sharpen the measured invariant-mass spectrum, fits to the threshold region will tighten the bounds on both scales, potentially revealing whether the smoothing of the Coulomb kernel is a genuine signature of nonlocal physics or simply the first precision glimpse of ordinary QCD behaving exactly as predicted. Either outcome would mark a milestone: toponium, the meson that never quite exists, has become one of the most powerful precision tools in the particle physicist’s arsenal.

Subject of Research: Theoretical reinterpretation of the LHC top-antitop threshold enhancement using nonlocal quantum field theory and toponium bound-state dynamics

Article Title: On recent measurements of toponium threshold enhancement in entire-function-regulated nonlocal quantum field theory

Article References: Thompson, E. J. (2026). On recent measurements of toponium threshold enhancement in entire-function-regulated nonlocal quantum field theory. The European Physical Journal C, 86(9), Article 1102. https://doi.org/10.1140/epjc/s10052-026-16177-y

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16177-y

Keywords: toponium, top quark, LHC, quantum chromodynamics, nonlocal quantum field theory, Bethe-Salpeter equation, threshold enhancement, CMS, ATLAS, quarkonium, renormalization group, particle physics