The challenge of seeing toponium is fundamentally one of disentanglement. When a top quark pair is produced and both W bosons from the top decays decay leptonically, the final state contains two b-quark jets, two oppositely charged leptons, and two neutrinos that escape the detector unseen. The neutrinos carry away energy and momentum that can only be inferred indirectly, through an imbalance in the measured transverse momentum of the event. Worse still, because protons are composite objects, the initial energy of the partons that collided is unknown, so the neutrino momenta along the beam axis are irretrievably lost. On top of these missing-information problems sits a combinatorial puzzle: which b-jet belongs with which lepton, and how should the invisible momentum be divided between the two neutrinos?

Previous experimental strategies have attacked this problem with analytical and geometrical techniques. ATLAS employed the Ellipse Method in its threshold analysis, while CMS turned to the Sonnenschein Method, both of which impose kinematic constraints to solve for the neutrino momenta. The Adelaide team instead adopted Recursive Jigsaw Reconstruction, a framework built on a different philosophy. Rather than solving a system of equations, the method defines a decay tree that includes every intermediate particle, measured or invisible, and then applies a sequence of jigsaw rules, each of which specifies how to divide unknown quantities between the two branches at every step of the chain. The rules are applied recursively until the entire decay tree has been assigned a consistent set of reference frames.

The power of this approach lies in what it leaves behind. Once the reconstruction is complete, the analyst can access kinematic variables in the rest frames of intermediate particles, quantities that are washed out in more global reconstructions. For toponium searches, where the signal differs from ordinary top-quark pair production in subtle angular and mass correlations, such frame-specific observables are exactly the kind of discriminating tools needed to separate a quasi-bound state from the smooth continuum background.

To test the idea, the researchers generated Monte Carlo samples simulating both LHC Run 2 and Run 3 conditions, at centre-of-mass energies of 13 and 13.6 teraelectronvolts respectively, with integrated luminosities of 140 and 300 inverse femtobarns. The toponium signal was modelled using a non-relativistic QCD prescription implemented in MadGraph5_aMC@NLO, which incorporates matrix element re-weighting through NRQCD Green’s functions and allows the bound state to be simulated near the threshold region without introducing an explicit intermediate particle. The background sample of conventional top-quark pairs was generated at next-to-leading order in QCD, and both samples were passed through Pythia8 for parton showering and hadronisation, with jets reconstructed using the anti-kT algorithm. In total, one million signal events were generated at 13 teraelectronvolts and two million at 13.6 teraelectronvolts.

The team compared four different jigsaw rules for dividing the invisible dineutrino momentum between the two decay branches. One rule forces the two reconstructed top-quark masses to be equal, another imposes equality of the two W masses, and two others minimise functions of the reconstructed top masses. All four methods showed reasonable correlation between reconstructed and truth-level values of the top-pair invariant mass and the top-quark transverse momenta, but the equal-top-mass rule, Method A, gave the most consistent reconstruction of the top-quark mass and was adopted for the rest of the analysis. Crucially, the reconstructed top-pair invariant mass faithfully captured the narrow peak structure of the toponium signal, which exists only in the mass window between roughly 330 and 350 gigaelectronvolts.

The current experimental strategy, used by both ATLAS and CMS, bins events in two angular variables called c-hel and c-han, which are built from the scalar products of lepton momenta boosted into the top-quark rest frames. These variables exploit the fact that toponium, being a pseudoscalar, produces a characteristically steep slope in the c-hel distribution, while the background distribution is nearly flat in c-han. Dividing the phase space into nine regions formed by three bins in each variable, the Adelaide team found that the optimal region yields a significance of about 12.4 standard deviations under Run 3 conditions, and 8.4 standard deviations for Run 2, using the standard significance metric of signal divided by the square root of signal plus background.

The new proposal introduces two variables that the Recursive Jigsaw framework makes natural to construct. The first, Nc-hel, is a modified version of c-hel in which the top quark is not first boosted into the top-pair centre-of-mass frame before the lepton boosts are applied. The second, the difference in azimuthal angle between the reconstructed top quark and antiquark, probes the geometry of the pair production. Combining three bins in each of these variables into nine phase-space regions, the team found that four regions now exceed five standard deviations of significance, compared with only two under the conventional strategy, and the optimal region reaches 15.5 standard deviations before stricter event selection is applied, or 10.2 standard deviations in the Run 2 configuration.

When the researchers applied a realistic set of event selection criteria, designed to mimic the experimental vetoing of non-top-pair backgrounds, requiring the dilepton mass to exceed 15 gigaelectronvolts, to lie more than 10 gigaelectronvolts from the Z boson mass, the missing transverse energy to exceed 40 gigaelectronvolts, and the reconstructed top-pair mass to stay below 550 gigaelectronvolts, the significances dropped as expected. The conventional strategy gave 12.8 standard deviations in the optimal Run 3 region, while the new variable combination gave 13.1. The real gain emerged after numerical optimisation: restricting the azimuthal angle region and scanning over a cut on Nc-hel revealed that requiring Nc-hel to be greater than zero pushes the significance to 14.3 standard deviations, a 12 percent improvement over the current strategy in the Run 3 configuration.

Beyond the immediate numbers, the study highlights a broader lesson about how the LHC can probe the top-quark threshold region. The Recursive Jigsaw Reconstruction framework provides access to a whole family of scale-sensitive and angular observables in the rest frames of intermediate states, and the authors argue that these variables promise to improve the discrimination between toponium and conventional top-pair production beyond the current level, while also enhancing sensitivity to any additional resonant states that might lurk near the threshold. As the collider accumulates more data in Run 3 and beyond, techniques like this one may transform toponium from a tantalising hint into a precision object of study, offering a unique laboratory for the physics of quark binding at the heaviest mass scale nature provides.

Subject of Research: Reconstruction of the toponium quasi-bound state in dileptonic top-quark pair decays at the LHC using Recursive Jigsaw Reconstruction

Article Title: Analysing toponium at the LHC using Recursive Jigsaw Reconstruction

Article References: Desai, A., Lovison, A., & Jackson, P. (2026). Analysing toponium at the LHC using Recursive Jigsaw Reconstruction. The European Physical Journal C, 86(8), Article 1023. https://doi.org/10.1140/epjc/s10052-026-16276-w

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16276-w

Keywords: toponium, Large Hadron Collider, top quark, Recursive Jigsaw Reconstruction, ATLAS, CMS, non-relativistic QCD, dilepton final state, MadGraph5_aMC@NLO, particle physics, threshold region, event reconstruction

The story begins with the CMS and ATLAS collaborations, which have reported evidence for an unexpectedly large production rate of top quark pairs in the threshold region, where the combined energy of the colliding particles barely suffices to create the pair. Both experiments observe an excess of roughly 5 to 10 picobarns relative to what standard continuum perturbative calculations predict. The CMS collaboration, analysing the dilepton channel, measured a cross section of 8.8 picobarns for a pseudoscalar toponium-like state, while ATLAS found an excess of 9.3 picobarns in the same channel. A second CMS analysis, using the single-lepton channel, reported a smaller but still significant excess of 5.1 picobarns. All three measurements were performed in proton-proton collisions at 13 teraelectronvolts and corrected for branching ratios, detector efficiencies and backgrounds.

The natural interpretation is that these excess events arise from so-called pseudo-bound states — short-lived configurations in which the top and antitop quark orbit each other briefly before decaying. Unlike the charmonium and bottomonium systems studied since the 1970s, toponium is a peculiar beast. The top quark is so heavy, and decays so quickly through the weak interaction, that a genuine bound state in the traditional sense cannot form. For a top mass above roughly 125 gigaelectronvolts, the top quark decays faster than the pair can complete even a single orbit. The large decay width, about 1.34 gigaelectronvolts, smears out the distinction between below-threshold bound states and the above-threshold Coulomb enhancement caused by the mutual attraction of the slowly moving quarks.

This is precisely where the Green’s function formalism of non-relativistic quantum chromodynamics earns its keep. In the language of Feynman diagrams, the formalism sums an infinite series of so-called Coulomb ladders — chains of soft gluon exchanges between the quark and antiquark that become important when their relative velocity is small. Rather than treating below-threshold and above-threshold physics separately, the Green’s function provides a unified description of the entire threshold region. Fadin and Khoze first applied it to top production at electron-positron colliders, and it was later extended to hadron colliders in work with Torbjörn Sjöstrand of Lund University.

In a new paper published in The European Physical Journal C, Sjöstrand together with Valery Khoze of Durham University and Christian Preuss of RWTH Aachen University have revived and dissected this old formalism, putting it back together in a form suited for modern Monte Carlo event generation. The effort was far from trivial. The original Fadin–Khoze expressions integrate out the Breit–Wigner smearing of the two top quark masses, meaning the formulas depend only on the invariant mass of the top pair. For rapid cross-section studies this is convenient, but for generating complete simulated events, the individual top and antitop masses must be selected event by event — three variables rather than one. An early attempt to combine the old formulas with event generation led to a subtle double-counting of the top width, which the new work identifies and corrects.

The authors traced the mathematical origins of the Green’s functions with some care, since the original derivations were never fully published and the relevant notes have been lost. They show that in the limit where the top width vanishes, the colour-singlet Green’s function decomposes into the familiar Coulomb enhancement factor above threshold plus an infinite series of delta functions below threshold, corresponding to discrete bound-state levels. A remarkable mathematical identity allows the Coulomb factor itself to be rewritten in a form sharing the same pole structure as the bound states, explaining why the full expressions — after convolution with Breit–Wigner distributions — so seamlessly blend both regimes. The colour-octet channel, which lacks bound states, shows perfect agreement with the Coulomb result in the same limit.

With the formalism understood and corrected, the team implemented it in the Pythia event generator, one of the workhorse simulation tools used by both experimental collaborations and theorists worldwide. The implementation offers several modes, including a preferred configuration in which the width assigned to the Green’s function is kept small and the remainder of the top width is handled by explicit Breit–Wigner mass selection, avoiding double-counting while preserving numerical stability. The code also introduces angular correlations in the decay of near-threshold pseudoscalar top pairs, calculated for the full six-particle final state and cross-checked against independent matrix-element computations to machine precision. These correlations, particularly the striking alignment of the two charged leptons from the W decays, are among the most powerful experimental handles for isolating a toponium signal.

The headline numbers are striking. In the preferred configuration, the contribution from below-threshold pseudo-bound states comes out at approximately 4.4 picobarns, while Breit–Wigner smearing raises the observable cross section below the nominal threshold of 345 gigaelectronvolts to about 6.5 picobarns — squarely in the range reported by CMS and ATLAS. The results are also compatible with recent independent NRQCD calculations by Garzelli and colleagues, which found an excess of about 4.2 picobarns in the 340-to-350 gigaelectronvolt mass window after subtracting non-resonant backgrounds, compared with 3.5 picobarns in the new Pythia-based study. Uncertainties from the choice of parton distribution functions, the strong coupling and the assumed colour-singlet fraction of gluon fusion leave some leeway in these figures, but the broad picture is consistent.

The formalism also offers a glimpse back to the future. Applied to a hypothetical 350-gigaelectronvolt electron-positron collider — the machine for which these calculations were originally conceived — the model predicts a characteristic enhancement just below the top pair threshold. Intriguingly, the authors find that QED bremsstrahlung, which effectively smears the collision energy, almost completely washes out the narrow below-threshold resonance peak in the total cross section, though the steeper rise of the cross section in the Green’s function scenario remains as a distinguishing feature. This has direct relevance for precision top-mass measurements at any future lepton collider, where a threshold scan is usually advocated as the cleanest method.

For now, the new code is publicly available within Pythia, allowing experimentalists to perform detailed comparisons that were previously impractical. The authors are appropriately modest about the limitations: the original calculation defined the state of the art thirty-five years ago, but modern higher-order perturbative calculations, more sophisticated QCD potentials and numerical Green’s function approaches have advanced the field since. Still, what they now offer is a simple, better-understood and highly flexible model that captures the essential physics of the threshold region. As the LHC experiments continue to scrutinise their data for signs of ephemeral toponium, a piece of theoretical machinery conceived before anyone had ever seen a top quark has found new life at the frontier of particle physics.

Subject of Research: Non-relativistic QCD Green's function formalism for top quark pair production near threshold and the toponium excess at the LHC

Article Title: Top pair threshold revisited

Article References: Sjöstrand, T., Khoze, V. A., & Preuss, C. T. (2026). Top pair threshold revisited. The European Physical Journal C, 86(10), Article 1145. https://doi.org/10.1140/epjc/s10052-026-16444-y

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16444-y

Keywords: top quark, toponium, threshold production, NRQCD, Green's function, CMS, ATLAS, LHC, Pythia, Monte Carlo event generation, Coulomb enhancement, quantum chromodynamics

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