Topped Mesons: Heavy-Light Bound States

• Topped mesons are hypothetical heavy–light states where a top quark binds with a light antiquark, forming transient resonances near the top mass threshold.
• Studies use Bethe–Salpeter methods, QCD sum rules, HQET, and potential models to predict masses nearly equal to the free-top mass and narrow widths around 1.4 GeV.
• Experimental strategies focus on detecting threshold enhancements and resonant signatures in final states with W bosons, b-jets, and light jets at the LHC.

Topped mesons are hypothetical heavy–light hadronic states in which a single Standard-Model top quark is bound by QCD to an antiquark, typically written as $t\bar q$ with $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$. In the recent literature they are treated not as established asymptotic hadrons but as transient quasi-bound or near-threshold resonant configurations whose masses lie close to the free-top mass and whose widths are expected to be narrower than those of toponium because only one constituent undergoes weak decay (Zhang et al., 10 Feb 2026, Zhang et al., 5 Aug 2025). Their theoretical study has been driven in part by reported near-threshold enhancements in top-pair production and has proceeded through relativistic Bethe–Salpeter methods, two-point QCD sum rules, HQET, and relativistic potential models (Zhang et al., 10 Feb 2026, Najjar et al., 1 May 2026, Luo et al., 25 Aug 2025).

1. Conceptual status and physical interpretation

The central conceptual issue is whether a top quark, with Standard-Model lifetime $\tau_t\sim5\times10^{-25}\,$s, can participate in hadronic binding at all. The recent topped-meson literature answers this in a qualified way: despite the very short lifetime, transient quasi-bound $t\bar q$ resonances may form under special kinematic configurations at the LHC, especially near threshold, where the top quark may capture a nearby antiquark before decaying (Zhang et al., 10 Feb 2026).

This framework differs from toponium. A $t\bar t$ system contains two unstable constituents and is therefore assigned an effective width of order $2\Gamma_t$, whereas a topped meson contains only one weakly decaying top quark and is expected to have total width of order $\Gamma_t$ (Zhang et al., 10 Feb 2026, Zhang et al., 5 Aug 2025). That distinction underlies the repeated claim that topped mesons should be longer-lived and narrower than toponium.

A second interpretive issue concerns whether the predicted states are genuinely bound. The two-point QCD sum-rule analysis of hypothetical single-top hadrons states that several extracted central masses lie slightly below the corresponding sums of constituent quark masses, which may indicate nontrivial binding dynamics or near-threshold multiquark configurations within the uncertainties of the method, and further states that the possibility of loosely bound configurations cannot be excluded for most of the considered mesonic channels (Najjar et al., 1 May 2026). The same paper also remarks that the small binding energies suggest that these mesons behave almost as threshold enhancements, so amplitude-analysis methods will be essential. This establishes a persistent ambiguity between a discrete-meson interpretation and a threshold-structure interpretation.

The threshold context is reinforced by the relativistic potential-model study that uses the same parameter set to predict a ground pseudoscalar topponium mass of $343.290~\mathrm{GeV}$, almost identical to the threshold enhancement reported by CMS at $343.3~\mathrm{GeV}$ and by ATLAS, and then extends the same framework to single-top mesons (Luo et al., 25 Aug 2025).

2. Bound-state frameworks

The most explicitly relativistic treatment uses the Bethe–Salpeter formalism under the instantaneous approximation. In that approach the momentum-space BS wave function

$$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$

satisfies

$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$0

and the projected positive-energy Salpeter component obeys

$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$1

The interaction kernel is taken to be a Cornell-type potential,

$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$2

with a linear confining term and a one-gluon-exchange term, using $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$3GeV$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$4, $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$5GeV, and $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$6 (Zhang et al., 10 Feb 2026).

A second line of work uses two-point QCD sum rules with pseudoscalar and vector interpolating currents

$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$7

for $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$8. The OPE includes perturbative contributions and nonperturbative condensates up to dimension eight, with spectral densities reconstructed after Dirac and color traces and then Borel transformed to extract masses (Najjar et al., 1 May 2026). In this setup the numerical analysis uses channel-dependent Borel windows and continuum thresholds fixed by the criteria that the pole contribution is at least $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$9, the highest dimension-8 term is at most $\tau_t\sim5\times10^{-25}\,$0 of the OPE, and the extracted mass depends weakly on $\tau_t\sim5\times10^{-25}\,$1 and $\tau_t\sim5\times10^{-25}\,$2 within the quoted intervals (Najjar et al., 1 May 2026).

An HQET formulation specializes to the heavy-quark limit $\tau_t\sim5\times10^{-25}\,$3, replacing the top field by $\tau_t\sim5\times10^{-25}\,$4. The ground-state heavy-light doublet is interpolated by

$\tau_t\sim5\times10^{-25}\,$5

and the hadronic pole is parameterized as

$\tau_t\sim5\times10^{-25}\,$6

Including $\tau_t\sim5\times10^{-25}\,$7 corrections yields

$\tau_t\sim5\times10^{-25}\,$8

The perturbative spectral density and condensate terms are then Borel transformed and matched to the hadronic side (Zhang et al., 5 Aug 2025).

A relativistic potential-model approach based on the Godfrey–Isgur–Capstick Hamiltonian instead solves a spinless-Salpeter equation with relativistic kinetic terms plus Coulomb, linear-confining, contact, spin–orbit, and tensor interactions:

$\tau_t\sim5\times10^{-25}\,$9

using the meson-sector parameters of Godfrey–Isgur with $t\bar q$0 GeV (Luo et al., 25 Aug 2025).

3. Spectroscopy and mass predictions

Across these approaches, the predicted masses remain close to the free-top mass. In the BS treatment, S-wave states are labeled by principal quantum number $t\bar q$1 and by $t\bar q$2, but in the heavy-quark limit the hyperfine splitting is negligible,

$t\bar q$3

so the states are commonly denoted simply as $t\bar q$4 (Zhang et al., 10 Feb 2026). The QCD sum-rule and potential-model calculations likewise find near-degenerate pseudoscalar and vector ground states (Najjar et al., 1 May 2026, Luo et al., 25 Aug 2025).

Framework	Channels	Representative masses
Bethe–Salpeter S-wave (Zhang et al., 10 Feb 2026)	$t\bar q$5	$t\bar q$6, $t\bar q$7, $t\bar q$8, $t\bar q$9 GeV
Bethe–Salpeter S-wave (Zhang et al., 10 Feb 2026)	$t\bar t$0	$t\bar t$1, $t\bar t$2, $t\bar t$3, $t\bar t$4 GeV
Two-point QCD sum rules (Najjar et al., 1 May 2026)	ground-state PS/V	$t\bar t$5, $t\bar t$6, $t\bar t$7, $t\bar t$8 GeV
Relativistic potential model (Luo et al., 25 Aug 2025)	$t\bar t$9	$2\Gamma_t$0, $2\Gamma_t$1, $2\Gamma_t$2, $2\Gamma_t$3 GeV
HQET sum rules (Zhang et al., 5 Aug 2025)	ground-state vector	$2\Gamma_t$4 GeV; non-strange doublet $2\Gamma_t$5 GeV

In the BS calculation the mass differences above the top mass are explicit: for $2\Gamma_t$6 they are $2\Gamma_t$7, $2\Gamma_t$8, $2\Gamma_t$9, and $\Gamma_t$0 GeV for the $\Gamma_t$1 through $\Gamma_t$2 states, while for $\Gamma_t$3 they are $\Gamma_t$4, $\Gamma_t$5, $\Gamma_t$6, and $\Gamma_t$7 GeV; analogous values for $\Gamma_t$8 with $\Gamma_t$9 lie between the $343.290~\mathrm{GeV}$0 and $343.290~\mathrm{GeV}$1 cases (Zhang et al., 10 Feb 2026). The QCD sum-rule study instead quotes the difference $343.290~\mathrm{GeV}$2 and finds positive central values for $343.290~\mathrm{GeV}$3, $343.290~\mathrm{GeV}$4, and $343.290~\mathrm{GeV}$5, but a slightly negative central value, $343.290~\mathrm{GeV}$6 GeV, for $343.290~\mathrm{GeV}$7, interpreting negative $343.290~\mathrm{GeV}$8 as weak binding (Najjar et al., 1 May 2026).

This spread suggests strong model dependence in how binding is encoded. Some frameworks place the light-flavor channels only a few hundred MeV above the top pole mass, while others place the $343.290~\mathrm{GeV}$9 channel roughly $343.3~\mathrm{GeV}$0–$343.3~\mathrm{GeV}$1 GeV above it. The common element is not a deeply bound spectrum but a family of states very close to heavy-quark thresholds.

4. Widths, lifetimes, and decay patterns

The dominant dynamical assumption is that the top quark decays weakly inside the bound state while the antiquark acts as a spectator. In the BS treatment the total decay width of a topped meson is therefore taken to satisfy

$343.3~\mathrm{GeV}$2

roughly half that of toponium, which contains two unstable constituents (Zhang et al., 10 Feb 2026). The HQET analysis gives the corresponding free-top value as $343.3~\mathrm{GeV}$3 and reaches the same qualitative conclusion (Zhang et al., 5 Aug 2025). The two-point QCD sum-rule study similarly states that widths are expected to be of order $343.3~\mathrm{GeV}$4 or larger, dominantly from top decay (Najjar et al., 1 May 2026).

The basic weak decay chain is

$343.3~\mathrm{GeV}$5

with the partial width

$343.3~\mathrm{GeV}$6

used as the basic input (Zhang et al., 10 Feb 2026).

Beyond the inclusive $343.3~\mathrm{GeV}$7 signature, the HQET study lists possible exclusive hadronic final states after spectator hadronization. For $343.3~\mathrm{GeV}$8 these include $343.3~\mathrm{GeV}$9, $$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$0, $$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$1, and $$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$2; for $$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$3 the listed possibilities include $$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$4, $$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$5, and $$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$6 (Zhang et al., 5 Aug 2025). These channels are presented as experimentally favorable reconstruction modes rather than as alternatives to top decay.

5. Production mechanisms and experimental search strategies

The production picture at the LHC is that a top quark is first produced perturbatively and then captures a nearby antiquark before decaying. The BS study identifies three representative subprocesses: gluon fusion $$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$7 followed by $$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$8; quark annihilation $$\chi_P(q)=\int d^4x_1\,d^4x_2\,e^{-i\,q\cdot(x_1-x_2)-i\,P\cdot X}\, \langle0|T\{\psi(x_1)\bar\psi(x_2)\}|P\rangle$$9 followed by the same capture step; and associated production $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$00 followed by $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$01 (Zhang et al., 10 Feb 2026). In that treatment the production amplitude scales as

$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$02

so more compact states, such as $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$03, are favored by larger wave function at the origin, although the overall cross sections are expected to be very small, $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$04pb (Zhang et al., 10 Feb 2026).

The characteristic inclusive signature is

$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$05

corresponding experimentally to one high-$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$06 $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$07 boson, one or two $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$08-jets, and one additional light-flavor jet. Suggested handles are an anomalous resonance in the invariant mass $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$09, angular correlations among decay products, and deviations of single-top kinematic distributions from Standard-Model expectations. CMS and ATLAS are identified as the natural venues, with searches in single-lepton, dilepton, or fully hadronic final states at high luminosity (Zhang et al., 10 Feb 2026).

The 2026 QCD sum-rule study argues that practical searches should focus on small threshold enhancements or mild bumps in invariant-mass distributions of $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$10 final states around $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$11–$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$12 GeV, and specifically recommends broad-structure searches in $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$13jet invariant-mass spectra with optimized jet-flavor tagging for $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$14, $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$15, and $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$16 jets (Najjar et al., 1 May 2026). It also states that the near-threshold character of these states makes finite-width and threshold-resummation methods important.

The relativistic potential model identifies the $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$17 system as the most promising target. In that framework the $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$18 pair-production threshold is $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$19 GeV, and one proposed strategy is to tag top decay leptons together with $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$20 from the companion bottom sector. The same paper suggests single-meson searches near $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$21 GeV and pair-production searches in the 350–380 GeV region using lepton+jets triggers with high-$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$22 muon pairs from $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$23 decays (Luo et al., 25 Aug 2025).

6. Related systems, nomenclature, and unresolved questions

The term “topped meson” is not completely uniform across the literature. In the Standard-Model-focused papers it denotes a single-top heavy–light state $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$24 or its flavor extensions $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$25 and $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$26 (Zhang et al., 10 Feb 2026, Najjar et al., 1 May 2026). In a different context, the 331–bilepton model uses the same term for mesons built from an exotic quark $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$27 of electric charge $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$28 bound to light antiquarks. Those states are assigned ground-state masses around $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$29TeV for $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$30TeV and are discussed in connection with same-sign dilepton plus jet signatures at a future $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$31 collider (Frampton, 2021). These are conceptually distinct objects and should not be conflated with Standard-Model single-top mesons.

Related but also distinct are toponium and top-hybrid states. The mixed top-quark hybrid meson of QCD sum rules employs a $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$32 current with $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$33 and $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$34 components, with $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$35, a nearly $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$36 admixture, and a mass estimate

$\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$37

That analysis treats production in heavy-ion collisions, quotes a crude Pb–Pb estimate of $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$38 at $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$39TeV, and frames the state as a possible quark–gluon-plasma probe rather than as a single-top meson (Kisslinger et al., 2019).

The principal unresolved question within the Standard-Model topped-meson program is interpretive. The BS formalism predicts a family of narrow S-wave topped mesons with discrete radial excitations up to $\bar q=\bar u,\bar d,\bar s,\bar c,\bar b$40 (Zhang et al., 10 Feb 2026). The QCD sum-rule analyses, by contrast, repeatedly emphasize weak binding, threshold proximity, and the possibility that the observed effect, if any, may be closer to a threshold enhancement than to a deeply bound hadron (Najjar et al., 1 May 2026). This suggests that future progress will depend not only on resonance hunting but also on threshold-sensitive amplitude analyses capable of separating loosely bound configurations, transient quasi-bound states, and nonresonant kinematic enhancements.