The S-wave topped meson

Inspired by the recent observation of the near-threshold enhancement in top-quark pair production by CMS and ATLAS, we investigate the mass spectrum of S-wave topped meson which containing a single top quark, i.e., $t\bar{q}$, $t\bar{c}$, and $t\bar{b}$, in the framework of Bethe-Salpeter formalism. These states are expected to exhibit significantly enhanced lifetimes and correspondingly narrower decay widths compared to toponium, primarily because only a single top quark participates in the weak decay process. The numerical results indicate that the masses of topped mesons are close to the top-quark mass. For the $t\bar{b}$ states, the $1S$ state is approximately 5.08GeV heavier than the top quark, while the $2S$, $3S$, and $4S$ states are about 5.37GeV, 5.57GeV, and 5.74GeV heavier, respectively. For the $t\bar{c}$ states, the corresponding mass differences are 1.90GeV, 2.23GeV, 2.45GeV, and 2.60GeV. The possible production and decay properties are also analyzed, which could be measured in LHC experiments.

💡 Research Summary

In this work the authors explore a novel class of heavy‑light bound states that contain a single top quark, denoted “topped mesons” (t ¯q, t ¯c, t ¯b). The motivation stems from recent CMS and ATLAS observations of a near‑threshold excess in top‑pair production, which revived interest in bound‑state dynamics involving the top quark. Because the top quark’s lifetime (τ ≈ 5 × 10⁻²⁵ s) is much shorter than the typical QCD hadronisation time (≈10⁻²³ s), conventional top‑onium (t ¯t) is expected to be only a quasi‑bound resonance. However, a system with only one top quark can decay solely via the weak process t → W⁺b, potentially leading to a narrower width and a longer effective lifetime for the bound state.

The theoretical framework employed is the Bethe–Salpeter (BS) equation in the instantaneous approximation. The interaction kernel combines a long‑range linear confining scalar potential V_S(r)=λ r (1−e^{−αr})/(αr) and a short‑range one‑gluon‑exchange vector potential V_V(r)=−(4/3)α_s(r) e^{−αr}/r. The exponential factor e^{−αr} regularises the infrared behaviour and mimics colour‑screening effects. Model parameters (λ=0.18 GeV², α=0.06 GeV, α_s(m_t)=0.11) are fixed by fitting the spectra of well‑known light and heavy mesons, ensuring consistency with the Particle Data Group values.

After projecting the BS wave function onto positive‑energy components (φ_++ and φ_−−) and discarding mixed components under the instantaneous approximation, the mass eigenvalue equations reduce to two coupled algebraic relations involving the quark energies ω_i=√(m_i²+q_⊥²) and projection operators Λ^±. Solving these equations numerically yields the masses of the S‑wave topped mesons up to the fourth radial excitation (n=4). The results are summarised in Table I (re‑presented in the paper):

• For the t ¯b system the 1S state lies ≈5.08 GeV above the top‑quark mass, with the 2S, 3S and 4S states at +5.37 GeV, +5.57 GeV and +5.74 GeV respectively.
• For the t ¯c system the corresponding splittings are +1.90 GeV, +2.23 GeV, +2.45 GeV and +2.60 GeV.
• For t ¯s, t ¯d and t ¯u the mass differences are essentially the same (≈1.73–1.78 GeV) because the light‑quark masses are negligible compared with the top mass.

The spin‑singlet (0⁻, ¹S₀) and spin‑triplet (1⁻, ³S₁) states are found to be degenerate within numerical accuracy, confirming that spin‑dependent splittings are suppressed by the large top mass. The BS wave functions f₁(q) and f₂(q) exhibit the expected node structure: the ground state has no node, the 2S state one node, etc., and all are peaked at low relative momentum, reflecting the dominance of the confining potential.

Production mechanisms at the LHC are discussed qualitatively. The dominant sources of top quarks are gluon‑gluon fusion (gg → t ¯t) and quark‑antiquark annihilation (q ¯q → t ¯t). If a produced top quark happens to be nearly at rest relative to a nearby antiquark, it may capture the antiquark before decaying, forming a topped meson. Partonic channels such as g b → t W⁻ followed by t + ¯b → (t ¯b) are highlighted as especially promising for the t ¯b system because the heavier b‑quark reduces the relative velocity and enhances the wave‑function overlap ψ_T(0). Nevertheless, the formation probability is heavily suppressed by the top’s ultra‑short lifetime, so the overall cross‑section is expected to be tiny.

Decay properties are dominated by the weak decay of the top quark inside the bound state: (t ¯q) → W⁺ + b + ¯q. Subsequent W⁺ decays give the familiar leptonic (ℓ⁺ν_ℓ) or hadronic (q ¯q′) final states. For the t ¯b meson an additional channel (t ¯b) → W⁺ + b + ¯b is possible, leading to b ¯b hadronisation into bottomonium or B‑mesons. Excited states could in principle undergo electromagnetic (γ) or gluonic (g) cascade transitions before the weak decay, but the short top lifetime makes such cascades extremely rare.

Experimentally, a topped‑meson signal would appear as a narrow resonance near the top‑mass region, with a high‑p_T W boson, one or two b‑jets, and possibly an extra light‑flavour jet from the spectator antiquark. Distinguishing this from the overwhelming tt̄ background would require precise reconstruction of the invariant mass m(W b ¯q) (or m(W b ¯b) for t ¯b) and analysis of angular correlations that differ from standard top decay kinematics. The authors suggest that high‑luminosity LHC runs, as well as future colliders such as the FCC‑hh, could set meaningful limits or perhaps observe hints of these exotic states with dedicated triggers and sophisticated multivariate analyses.

In summary, the paper provides a self‑consistent BS‑based calculation of the S‑wave topped‑meson spectrum, demonstrates that the masses lie only a few GeV above the top quark, and outlines plausible production and decay channels. While the practical observation is challenging due to the top’s rapid weak decay, the work establishes a solid theoretical baseline for future experimental searches for heavy‑light bound states containing a top quark.