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