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

The study, carried out by Lin-Qing Song and Hai-Qing Zhou of Southeast University in Nanjing and published in The European Physical Journal C, presents a systematic comparison of three levels of approximation applied to exactly the same underlying interaction. The authors solved the full four-dimensional Bethe–Salpeter equation, the gold-standard relativistic bound-state equation of quantum field theory, alongside two of its commonly used three-dimensional reductions: the Salpeter equation, which freezes the interaction in time, and the nonrelativistic Schrödinger equation, which most phenomenological analyses of hadronic molecules employ. Crucially, the couplings and masses fed into each equation were kept identical, with no refitting of parameters to make the results agree. The differences that emerged therefore isolate a single question: how much does the dimensional reduction itself distort the answer?

The formalism begins with a one-boson-exchange model, in which two particles of mass m interact by swapping a boson of mass m_ex, the hadronic analogue of two electrons exchanging a photon. In the full Bethe–Salpeter treatment, the relative momentum between the particles carries both a spatial and a temporal component, and the interaction kernel depends on all four components. Solving this equation is notoriously difficult, so the authors employed a Wick rotation, continuously deforming the integration contour from real time into Euclidean space. They proved that for a massive exchanged boson, none of the poles of the constituent or exchange propagators crosses the rotating contour, so the continuation is clean and generates no extra residue terms. The resulting Euclidean equation was then discretized on an 800 by 800 grid, producing 640,000 coupled linear equations that were solved with the PETSc computational package, with bound-state energies located by a bisection search.

The benchmark calculations used a scalar Wick–Cutkosky-type model with constituent masses of 1 GeV and exchanged-boson masses ranging from 0.01 GeV, a long-range theoretical control, up to 0.5 GeV, representing genuinely short-ranged forces. The results were striking. When the Schrödinger equation predicted a binding energy of roughly 20 MeV, the full Bethe–Salpeter equation gave only about 60 percent of that value for the lightest exchanged boson, but a mere 5 percent for the heaviest one. In other words, for short-ranged, strongly coupled interactions, the nonrelativistic reduction can overstate the binding by more than an order of magnitude, even when the nominal binding energy is small compared with the constituent masses.

This finding cuts against a widespread intuition. It is often assumed that if a bound state is shallow, meaning its binding energy is tiny compared with the masses involved, then relativistic corrections must be negligible and the Schrödinger equation is safe. The new results show that shallow binding alone guarantees nothing of the sort. The reason lies in the structure of the reduction itself. When the Bethe–Salpeter kernel is reduced to a static potential, an exact diagrammatic series relates the original kernel to the reduced one, and the Schrödinger equation keeps only the leading term of that series. In quantum electrodynamics, the tiny fine-structure constant suppresses the neglected higher-order terms, which is why the Schrödinger equation describes hydrogen so beautifully. Hadronic molecules, by contrast, require couplings of order one or larger to bind at all, and under those conditions the discarded terms are anything but small.

The numerical analysis reinforced this picture in an unexpected way. The authors varied the momentum cutoff of their integration domain from 1 GeV up to 1000 GeV and watched how the binding energies responded. Raising the cutoff from 1 to 10 GeV changed the Schrödinger result dramatically, tripling its magnitude, while the Bethe–Salpeter answer was the least sensitive of the three. This demonstrates that momenta well outside the strict nonrelativistic regime contribute substantially even within the reduced equations, another sign that the nonrelativistic approximation is straining against its own assumptions. The authors were careful to note that this cutoff is purely a numerical device for testing convergence, not a physical regulator, and that the results stabilize completely beyond 10 GeV.

To connect the abstract scalar model to real hadrons, the authors repeated the comparison for the isoscalar D-Dbar system, a pair of charmed and anticharmed mesons interacting through the exchange of rho, omega, and sigma mesons, with hadronic form factors softening the short-distance behavior. Using the same parameter set that a previous phenomenological analysis had employed in a Schrödinger framework, they found that across the region where the Schrödinger equation predicts binding energies between roughly 1 and 50 MeV, the full Bethe–Salpeter equation yields only 10 to 30 percent of those values at the same couplings. The discrepancy grows steadily as the vector coupling strength increases, with all exchanged-meson masses and the cutoff held fixed. Their Schrödinger results reproduced the earlier work exactly, confirming that the gap is not a matter of input choices but of the equation itself.

Importantly, the authors emphasize that their conclusion does not overturn the well-established universality of shallow bound states. Effective field theory teaches that when a bound state is much larger than the range of its interaction, its low-energy properties depend only on a few parameters, such as the scattering length, once the short-distance dynamics have been properly matched. The deuteron, with its inverse binding momentum of about 4.3 femtometers, is described superbly by chiral nuclear effective theory, and a D-Dbar state bound by 1 MeV has a similar size of about 4.6 femtometers. But universality applies after parameters have been fitted or matched to the relevant dynamics. The new results address a different and more uncomfortable question: whether a fixed set of one-boson-exchange parameters can be carried unchanged from a Schrödinger calculation into a relativistic Bethe–Salpeter calculation. The answer, emphatically, is no.

The practical implications ripple across the entire exotic-hadron community. Most molecular-state analyses in the literature rely on Schrödinger or reduced Bethe–Salpeter equations with couplings and cutoffs fitted to data or borrowed from related calculations. The new study does not claim that such effective approaches are worthless; a good fitted potential can still reproduce the observables it was tuned to. Rather, the warning is about blind parameter transfer. If a coupling set calibrated in a nonrelativistic framework is plugged into a four-dimensional ladder equation, or vice versa, the resulting binding energies can differ by factors of three to ten or more. Any claim that a candidate molecule is bound by, say, a few hundred kiloelectronvolts rather than a few MeV, or that it is unbound altogether, may hinge entirely on which equation was solved.

The authors also set honest boundaries on their own work. The ladder Bethe–Salpeter equation serves here as a four-dimensional reference for testing the reductions, not as a complete physical prediction. Corrections beyond the ladder approximation, such as crossed-ladder diagrams, can themselves be large; in the scalar model with a heavy exchanged boson, earlier calculations show that including the crossed-ladder contribution can more than double the magnitude of the binding energy at strong coupling. Coupled-channel effects, where a molecular state mixes with nearby scattering channels, are likewise outside the comparison and may shift the numbers further. What remains is a clear and quantified message: the three-dimensional reduction of a relativistic bound-state equation is not a harmless technical step, and for the strongly coupled, short-ranged forces that govern hadronic molecules, it can be the dominant source of uncertainty in predicting how deeply nature binds its most puzzling particles.

Subject of Research: Relativistic corrections to the binding energies of two-body hadronic molecular states calculated via Bethe–Salpeter, Salpeter, and Schrödinger equations

Article Title: Relativistic correction to the binding energies of two-body hadronic molecular states

Article References: Song, L.-Q., & Zhou, H.-Q. (2026). Relativistic correction to the binding energies of two-body hadronic molecular states. The European Physical Journal C, 86(9), Article 1120. https://doi.org/10.1140/epjc/s10052-026-16420-6

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16420-6

Keywords: hadronic molecules, Bethe–Salpeter equation, Schrödinger equation, relativistic corrections, binding energy, exotic hadrons, one-boson exchange, D-Dbar system, quantum field theory, Wick rotation, effective field theory, XYZ states