The analysis is deliberately not a new measurement and not a new model. It is a model-discriminating exercise built entirely from publicly available CMS and LHCb data, with preliminary CMS results in proton-lead and light-ion collisions used only as supporting cross-system evidence. The central question is sharply defined: can the multiplicity-dependent suppression of excited bottomonium and charmonium states in small systems be generated entirely by hadronic final-state interactions, or does the differential structure of the data demand early-time, non-local dynamics? The answer that emerges is that the simplest explanations — local particle density or total multiplicity alone — struggle to survive the combined weight of the evidence.

The starting point is the CMS measurement of event-activity-dependent upsilon production ratios in proton-proton collisions at 7 TeV. CMS observed that the ratios of excited to ground-state yields, R21 and R31, decrease monotonically with the charged-track multiplicity, and that the decrease is stronger for the more weakly bound upsilon(3S) than for the upsilon(2S), exactly as expected for a sequential suppression ordered by binding energy. LHCb independently confirmed the same pattern for bottomonium at 13 TeV and for the charmonium psi(2S)-to-J/psi ratio, using a different detector, a different energy and a different rapidity coverage. The effect is therefore not an experimental artefact of a single apparatus. Crucially, the suppression also weakens as the transverse momentum of the quarkonium increases, consistent with a fast-moving state crossing a finite-sized active region in a shorter time.

The discriminating power of the new analysis comes from how CMS sliced the same dataset along several geometric and topological axes. The first test is cone isolation: events were classified by the number of charged tracks inside a narrow cone of half-opening angle 0.5 radians around the upsilon flight direction. For a local hadronic dissociation mechanism, the expectation is unambiguous — more nearby hadrons should mean more dissociation. The data show no such separation. The multiplicity-dependent suppression trends for empty-cone and dense-cone events are statistically compatible across the measured range, placing direct pressure on any mechanism in which the suppression probability scales with the local density of particles surrounding the quarkonium.

The second test closes a possible escape route. If local density is not the controlling variable, perhaps a diffuse directional mechanism is at work. CMS therefore measured the suppression as a function of the multiplicity in three azimuthal sectors relative to the upsilon direction: forward, transverse and backward. The transverse sector is especially informative, because particles emitted at roughly ninety degrees to the quarkonium can be neither co-moving comovers nor recoil partners. Yet all three sectors yield mutually compatible suppression trends. Direction-based and near-side explanations are disfavoured in their simplest forms; the sector multiplicity behaves as a proxy for global event activity rather than for any local, directional density.

The third and arguably most consequential test is transverse sphericity, an event-shape variable that distinguishes jet-like, two-pronged events from isotropic ones. At fixed track multiplicity, jet-like events with sphericity below 0.55 show a multiplicity dependence of the suppression ratios that is largely absent, while isotropic events above 0.85 show a clear decrease. A mechanism depending only on total multiplicity would produce identical behaviour in both subsamples, which the data do not support. Even more strikingly, inclusively the three upsilon states are accompanied by different mean track multiplicities — 33.9, 33.0 and 32.0 tracks respectively — but within the jet-like class alone this difference vanishes entirely, with all three states accompanied by exactly 22.4 tracks. The multiplicity-state correlation is thus a topology effect, not an intrinsic property of each state. Together, the cone and sphericity results form what the author calls a scissors constraint: local-density pictures are cut by the cone test, pure-multiplicity pictures by the sphericity test, and the two cuts are orthogonal.

Proton-lead data extend the map into denser territory and add a powerful non-locality argument. CMS measurements at 5.02 TeV and, with higher statistics, at 8.16 TeV show that the upsilon suppression correlates with the forward transverse energy measured several units of pseudorapidity away from the quarkonium — a long-range correlation that local comover density at the quarkonium rapidity cannot generate. Meanwhile, LHCb’s comparison of the psi(2S)-to-J/psi ratio in proton-going versus lead-going configurations delivers a qualitative stress test: the ratio decreases with multiplicity in the proton-going direction but is suppressed yet approximately flat in the denser lead-going direction, a pattern LHCb itself concludes cannot be fully explained by comovers alone and suggests an additional mechanism, possibly linked to quark-gluon-plasma-like effects. Preliminary CMS results in oxygen-oxygen and neon-neon collisions further show that the single-ratio suppression trend follows a smooth common curve as a function of event activity across pp, pPb and light-ion systems.

What kind of mechanism survives all six constraints? The timing argument is kinematic and hard to evade. The bottom-antibottom pair requires roughly 0.1 to 0.5 femtoseconds — expressed in natural units, 0.1 to 0.5 fm of formation time — to evolve into a physical bound state, while light hadrons take about 1 fm to form. The suppression must therefore act before either the quarkonium or its hadronic surroundings exist, in a pre-hadronic coloured environment. Order-of-magnitude Bjorken energy-density estimates for high-multiplicity proton-proton collisions at LHC energies yield values of order 1 to 10 GeV per cubic femtometre, comparable to or above the QCD crossover density of roughly 0.5 GeV per cubic femtometre from lattice calculations, making a partonic interpretation physically plausible — though the author is careful to note that pre-hadronic does not mean partonic, and that string-based and colour-glass frameworks also operate in this window and are not excluded by timing alone.

Intriguingly, the multiplicity window in which the suppression sets in overlaps with the window in which an entirely independent analysis by Campanini and Ferri, based purely on soft-sector observables, identified a qualitative change in collective behaviour — an experimental equation-of-state proxy built from the interplay between charged-particle density and mean transverse momentum. The same multiplicity range has since been associated with strangeness enhancement, long-range ridge correlations and partonic flow signatures reported by ALICE and CMS. The author stresses that this convergence is a cross-check, not a proof: no individual observable constitutes evidence for deconfinement in isolation, and the analysis explicitly refrains from claiming that the sphericity dependence is fully independent of transverse-momentum composition, since a definitive separation would require a triple-differential measurement that does not yet exist.

The value of the work lies in the benchmark it establishes. To the author’s knowledge, no published suppression framework — not the comover interaction model in either its hadronic or partonic form, not open-quantum-system approaches based on pNRQCD, not QGP-droplet models, not the hydrodynamic SHINCHON framework — has been tested against the full six-constraint set simultaneously. The paper lays out falsifiable follow-ups: cone and sphericity calculations within the comover framework, a binding-energy-ordered elliptic-flow hierarchy in non-central proton-lead collisions, topology-resolved charmonium measurements at LHCb, and above all the triple-differential CMS measurement that would cleanly separate topology from kinematics. Whatever the ultimate microscopic answer proves to be, the data now make a conventional local-density, multiplicity-only, or purely late-hadronic interpretation very difficult to maintain — and point, with unusual coherence, toward an early, globally correlated, geometry-sensitive coloured medium in the smallest collision systems humanity has ever produced.

Subject of Research: Quarkonium suppression mechanisms in high-multiplicity proton-proton and proton-lead collisions at the LHC

Article Title: A multi-differential constraint map for quarkonium suppression mechanisms in high-multiplicity pp and pPb collisions

Article References: Campanini, R. (2026). A multi-differential constraint map for quarkonium suppression mechanisms in high-multiplicity pp and pPb collisions. The European Physical Journal C, 86(9), Article 1052. https://doi.org/10.1140/epjc/s10052-026-16267-x

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16267-x

Keywords: quarkonium, bottomonium, charmonium, quark-gluon plasma, CMS, LHCb, proton-proton collisions, proton-lead collisions, comover interaction model, sphericity, cone isolation, colour deconfinement

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

That hidden feature is the gluon condensate, a nonperturbative property of the quantum chromodynamics (QCD) vacuum that measures the expectation value of the gluon field strength squared. In plain terms, even empty space seethes with gluonic fluctuations, and the intensity of this background seething shapes how strongly quarks bind into hadrons. In a paper published in The European Physical Journal C, Fei Wang and Zi-qiang Zhang of China University of Geosciences in Wuhan have computed, for the first time within their framework, how varying the strength of the gluon condensate reshapes the thermal spectral functions of charmonium and bottomonium — the bound states of charm and bottom quark pairs. Their conclusion is striking: a stronger gluon condensate systematically narrows the spectral peaks of these states and boosts their spectral weight, meaning the bound states live longer and survive more readily in the plasma. The condensate, far from being a spectator, actively hinders thermal dissociation.

The technical engine behind this result is holography, the remarkable dictionary arising from the AdS/CFT correspondence that translates intractable strongly coupled gauge theories into tractable classical gravity problems in a higher-dimensional spacetime. In this picture, a four-dimensional quantum field theory without a known analytical solution is recast as a five-dimensional gravitational system, with the extra dimension playing the role of an energy scale. Quarkonia — heavy vector mesons such as the J/psi and the Upsilon — correspond to waves of a five-dimensional gauge field propagating in this curved bulk geometry. The authors employed an improved soft-wall AdS/QCD model, in which a background scalar field with three independent energy scales encodes the nonperturbative physics responsible for the masses and decay constants of heavy mesons. Unlike earlier hard-wall and original soft-wall constructions, this refined profile reproduces both the vacuum mass spectrum and the experimentally observed monotonic decrease of decay constants as one climbs the ladder of radial excitations.

Spectral functions are the central objects in this analysis. At zero temperature, the two-point correlator of the heavy quark vector current decomposes into an infinite sum of infinitely sharp delta-function peaks, one for each meson state, with heights set by the decay constants. At finite temperature, these razor-sharp resonances broaden into structures with finite width, and the position, width, and height of each peak encode the quasiparticle mass, lifetime, and dissociation behavior of the corresponding bound state inside the hot medium. A peak that broadens and sinks is a meson dissolving; a peak that stays tall and narrow is a meson holding together. Tracking these peaks as functions of temperature and condensate strength therefore offers a direct, dynamical window into quarkonium melting that bulk thermodynamic measures cannot provide.

To introduce the gluon condensate, the authors extended the setup to finite temperature using a dilaton black hole geometry. In holographic language, temperature appears as a black hole in the fifth dimension, and the gluon condensate enters through a second scalar field — the dilaton — whose back-reaction warps the black hole metric itself. Crucially, the construction keeps two scalar sectors physically distinct: the soft-wall scalar fixes the vacuum masses and decay constants of the mesons, while the dilaton, whose strength parameter is proportional to the expectation value of the gluon field strength squared, deforms only the finite-temperature geometry. This separation has a clean consequence visible in the numerical results: increasing the condensate changes the widths and heights of the spectral resonances but leaves their central positions untouched, because the real part of the quasiparticle energy is governed by the vacuum sector alone.

The numerical machinery relies on the membrane paradigm, a technique that reformulates the computation of retarded Green’s functions as a flow equation for a frequency-dependent conductivity that is integrated from the black hole horizon out to the boundary of spacetime. Regularity at the horizon supplies the boundary condition, and the real part of the resulting boundary conductivity, multiplied by frequency, yields the spectral function. Because no closed-form solution exists, the flow equations must be integrated numerically, using parameter sets fitted in earlier work: for charmonium a soft-wall scale of 1.2 GeV with a string-tension scale of 0.55 GeV, and for bottomonium 2.45 GeV and 1.55 GeV respectively, along with mass-scale parameters tied to non-hadronic decay channels.

The temperature scan with the condensate switched off establishes the baseline physics. At 195 MeV, just above the pseudo-critical deconfinement temperature, the charmonium spectrum still displays a sharp J/psi peak, indicating that the ground state remains largely intact. The bottomonium spectrum at the same temperature is even more robust, showing a dominant Upsilon(1S) resonance accompanied by visible peaks from the 2S and 3S excited states — a reflection of the bottom quark’s larger mass and correspondingly stronger binding. As the temperature climbs to 330, 465, and 600 MeV, the peaks for both systems systematically broaden and lose height, the unmistakable signature of medium-induced dissociation: thermal fluctuations increasingly tear the heavy quark pairs apart, draining spectral weight from the quasiparticle resonances.

The condensate scan reverses this narrative. Holding the temperature fixed at 280 MeV for charmonium and 300 MeV for bottomonium — both comfortably above the deconfinement crossover — the authors varied the condensate parameter through the values 0.2, 0.5, 0.8, and 0.9. Even at these temperatures, the 1S resonances remain clearly distinguishable, while the excited-state peaks, broader and weaker as expected from their larger spatial extent and looser binding, respond most dramatically. As the condensate strengthens, the peaks progressively narrow and grow taller, and the effect builds gradually rather than switching on abruptly. Physically, the interpretation is that a stronger nonperturbative gluonic background reinforces the gluon-mediated interaction between the heavy quark and antiquark, making it harder for thermal fluctuations to pry the pair apart. The result is a longer-lived bound state — a narrower resonance — and a larger occupation probability — a taller peak.

Perhaps the most novel observation is a mass-dependent asymmetry between the two quarkonium families. For the same increment in condensate strength, the narrowing and amplification of the bottomonium 2S peak is more pronounced than for its charmonium counterpart. The authors attribute this to the fact that, at the temperatures considered, the binding energies of bottomonium excited states are larger relative to the thermal energy than those of charmonium excited states, making them more sensitive to the condensate-induced changes in the effective interaction. This differential response, they note, had not been emphasized in previous separate analyses of the two systems, and it offers a potentially testable fingerprint: if the gluon condensate in the plasma is large, excited bottomonium states should show disproportionately enhanced survival compared with excited charmonium.

The findings dovetail with two independent lines of holographic evidence. Earlier calculations of the imaginary part of the interquark potential and of entropic forces in condensate-bearing backgrounds both concluded that gluon condensation impedes quarkonium melting. The new spectral-function results also mesh with the authors’ own prior work using configurational entropy, an information-theoretic stability measure which showed that larger condensates reduce microscopic disorder in the plasma. Together, these converging approaches suggest that condensate-driven stabilization of quarkonia is a robust nonperturbative effect rather than an artifact of any single formalism. The stakes are high for experiment: quarkonium suppression patterns measured at the LHC and at RHIC are among the primary diagnostics of plasma formation, and any condensate-induced stabilization directly alters how those patterns should be interpreted. The authors point to comparisons with lattice QCD and potential models, and to data from LHC Run 4 and the RHIC Beam Energy Scan II, as the next steps in constraining the magnitude of the gluon condensate in hot QCD matter — a quantity that, if these calculations hold, may be quietly deciding which of the plasma’s heaviest messengers survive the fire.

Subject of Research: Holographic study of gluon condensate effects on heavy quarkonium spectral functions and thermal dissociation in the quark–gluon plasma

Article Title: Gluon condensate effects on heavy quarkonium spectral functions and thermal dissociation

Article References: Wang, F., & Zhang, Z.-Q. (2026). Gluon condensate effects on heavy quarkonium spectral functions and thermal dissociation. The European Physical Journal C, 86(9), Article 1129. https://doi.org/10.1140/epjc/s10052-026-16323-6

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16323-6

Keywords: quark–gluon plasma, quarkonium, gluon condensate, spectral functions, holographic QCD, AdS/QCD, charmonium, bottomonium, thermal dissociation, heavy-ion collisions, dilaton black hole, nonperturbative QCD