Physicists Sharpen Their View of Toponium With a Clever Jigsaw Trick

Deep inside the debris of proton-proton collisions at the Large Hadron Collider, physicists believe they have caught the first glimpse of one of the most exotic objects ever contemplated: toponium, a short-lived bound state formed when a top quark and its antiquark linger close enough together for the strong force to bind them, however briefly. Recent analyses from the ATLAS and CMS collaborations have reported an excess of events near the top-quark pair production threshold, consistent with a spin-0 pseudoscalar state. Now a team of theorists and analysts at the University of Adelaide, Aman Desai, Amelia Lovison and Paul Jackson, has developed a new reconstruction strategy that could make such measurements dramatically sharper, and their results, published in The European Physical Journal C, suggest sensitivity gains of up to 12 percent over current approaches.

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