Toponium Formation Effects in Colliders

• Toponium is the QCD bound state of a top-antitop quark pair, exhibiting unique threshold effects with modified cross sections and kinematics.
• The NRQCD effective Hamiltonian and Monte Carlo reweighting techniques enable precise simulation of binding energies and event signatures in high-energy collisions.
• Distinct observables such as recoil momentum peaks near 20 GeV and low lepton-jet separations provide robust markers for separating toponium signals from t-tbar backgrounds.

Toponium, the QCD bound state of a top quark and anti-top quark ($t\bar t$), represents the smallest and shortest-lived hadronic system currently accessible to experiment. Its effects, referred to as “toponium formation effects,” arise near the $t\bar t$ threshold in high-energy collisions and are characterized by distinctive modifications of the cross section, event kinematics, and angular correlations, all tied to the nonperturbative interplay between QCD binding and the large weak decay width of the top quark.

1. Non-Relativistic QCD Framework and Bound-State Dynamics

The theoretical description of toponium formation at colliders is governed by the non-relativistic QCD (NRQCD) effective Hamiltonian for the $t\bar t$ relative motion: $H = - \frac{\nabla^2}{m_t} + V(r)$ where $m_t$ is the top-quark pole mass. The static potential $V(r)$ includes Coulomb ($\propto\!1/r$) and higher-loop corrections: $$V(r) = -\frac{C_F\,\alpha_s(\mu)}{r} \left[ 1 + \frac{\alpha_s(\mu)}{4\pi} a_1 + \left(\frac{\alpha_s(\mu)}{4\pi}\right)^2 a_2 + \cdots \right]$$ with $C_F=4/3$ and coefficients $a_1, a_2$ encoding running and two-loop effects (Fuks et al., 3 Sep 2025, Llanes-Estrada, 28 Nov 2024, Fu et al., 15 Dec 2024). The binding energies of the S-wave levels follow (to leading order)

$$E_n \simeq -\frac{m_t\,C_F^2\,\alpha_s^2}{4n^2},\quad n=1,2,\ldots$$

For $n=1$, typical binding energies are $E_1 \sim -2.5$ to $-3.5$ GeV, depending on the scale $\mu$ and the precise definition of $\alpha_s$ (Jiang et al., 24 Dec 2024, Fu et al., 15 Dec 2024, Llanes-Estrada, 28 Nov 2024).

The dynamics include the finite top-quark width $\Gamma_t \sim 1.3$–$1.4$ GeV, which acts as an infrared regulator in the Green’s function formalism: $$G(E)\;=\;\langle \mathbf r=0|(H - E - i\,\Gamma_t)^{-1}|\mathbf r=0\rangle$$ This Green’s function encodes both the would-be bound-state poles ($E_n<0$) and their smearing into the continuum due to the top width (Fuks, 6 May 2025, Fuks et al., 3 Nov 2025).

2. Monte Carlo Implementation and Event Simulation Schemes

Realistic simulations of toponium effects in $pp$ collisions require embedding the NRQCD dynamics into event generators. This is achieved by re-weighting hard-scattering matrix elements using the ratio of interacting to free Green’s functions at given kinematics: $$|\mathcal M|^2 \rightarrow |\mathcal M|^2 \times w(E, p^*)\,,\quad \text{where}\quad w(E, p^*) = \left|\frac{\widetilde G(E; p^*)}{\widetilde G_0(E; p^*)}\right|^2$$ with $p^*$ the recoil momentum in the $t\bar t$ rest frame. This reweighting is implemented at the parton level (e.g., in MadGraph5_aMC@NLO) and is matched to parton showers (e.g., Pythia8), with QCD radiation below $\sim$20 GeV off the $t\bar t$ singlet system being explicitly vetoed by color reassignment (Fuks et al., 3 Sep 2025, Fuks et al., 28 Nov 2024, Fuks, 6 May 2025).

In the full-resonant approach, direct $\eta_t\,t\bar t$ vertices are avoided to prevent double-counting with continuum production. Instead, resonance and continuum diagrams are combined in a way that maintains perturbative consistency and suppresses unphysical interference (Fu et al., 17 Apr 2025).

Alternative prescriptions for the modeling of below-threshold events have been cross-validated. The default "four-mass hybrid" in Pythia 8 samples Breit–Wigner-distributed top masses and incorporates both the above- and below-threshold Green’s-function weights for a smooth transition across threshold (Sjöstrand, 6 Oct 2025).

3. Experimental Signatures in the Single-Lepton Channel

In $pp$ collisions at $\sqrt{s}=13$ TeV (Run 2), the NRQCD-based simulation yields a toponium signal cross section $\sigma_{\rm top}=1.51$ pb in the single-lepton + jets final state, compared to $\sigma_{t\bar t}=243$ pb for the inclusive NNLO+NNLL $t\bar t$ background (Fuks et al., 3 Sep 2025).

The recommended selection employs:

• Exactly one $e/\mu$ ($p_T>10$ GeV, $|\eta|<2.5$),
• Two $b$-jets + two light jets ($p_T>25$ GeV, $|\eta|<2.5$),
• Missing $E_T>30$ GeV, and $\Delta R(\ell, j) > 0.4$,
• $m_{t\bar t}<350$ GeV (to isolate threshold region),
• Angular separation $\min R^2_{\ell j} \leq 0.5$ to enhance the bound-state fraction,
• Optionally, $p^*<40$ GeV for further signal-to-background discrimination.

With these cuts, in 140 fb$^{-1}$, the predicted event yields are $N_{\text{top}}=3,060$ (signal) and $N_{t\bar t}=81,600$ (background), corresponding to a statistical significance $s \simeq 10.5$ and $S/B\simeq 3.8$\%. The toponium contribution is concentrated below $m_{t\bar t}\sim 350$ GeV, with characteristic peaks in $p^*\approx20$ GeV (Bohr radius scale) and small $\min R^2_{\ell j}$, while $p_T^\ell$ and $E_T^{\rm miss}$ are modestly softer for toponium (Fuks et al., 3 Sep 2025).

Systematic uncertainties are dominated by PDF/scale ($\sim4\%$), $b$-tag efficiency ($\sim3$–$5\%$), and jet energy scale. The background normalization is robustly controlled using high-mass sidebands $m_{t\bar t}>400$ GeV.

4. Discriminating Observables and Phenomenological Implications

The most powerful distinguishing observables in the single-lepton channel are:

• Angular separation $\min R^2_{\ell j}$: signal peaks at low values, background is flatter.
• Top recoil momentum in the $t\bar t$ rest frame $p^*$: signal sharply peaks at $\sim$20 GeV, linked to the inverse Bohr radius, while the background extends far higher.
• $m_{t\bar t}$: signal confined just below threshold, background populates higher masses.

Selection windows $m_{t\bar t} \in [300,350]$ GeV and $p^* < 40$ GeV maximize $S/B$. Combining these observables in multivariate fits can further enhance the sensitivity. With future datasets, the expected significances scale as $s \sim 10.5 \sqrt{L/140\,\mathrm{fb}^{-1}}$, e.g., $s \sim 15$ at 300 fb$^{-1}$ (Run 3), $s \gtrsim 45$ at HL-LHC (3 ab$^{-1}$), with further gains at higher-energy hadron colliders (Fuks et al., 3 Sep 2025).

A combined fit to $m_{t\bar t}$, $\min R^2_{\ell j}$, and $p^*$ allows precise (sub-10\%) determination of the threshold line shape and over-constrains the NRQCD potential parameters, particularly $\alpha_s$ at the Bohr scale. This provides a stringent test of QCD and indirect sensitivity to new physics entering the potential (Llanes-Estrada, 28 Nov 2024).

The single-lepton channel is complementary to the dilepton mode (better kinematic reconstruction, higher branching ratio $\sim14.6\%$ vs.\ $\sim7.4\%$) but less sensitive for spin/color analysis due to lower spin analyzing power and increased combinatorics (Aguilar-Saavedra, 29 Jul 2024).

5. QCD Versus Exotic Binding Mechanisms

Standard QCD ("glue") binding yields a series of Coulombic $t\bar t$ bound states with modest binding energies ($2$–$3$ GeV) and moderate resonance peaks. The line shape near threshold consists of a "shoulder"-like enhancement, which becomes more pronounced if the ground-state binding energy is much larger than $\Gamma_t$, and is further filled in by excited toponium states ($nS$), each contributing with $|\psi_n(0)|\sim1/n^{3/2}$ scaling. This results in a net $\Delta\sigma\sim1$ pb "fill-in" below threshold (Llanes-Estrada, 28 Nov 2024).

In contrast, exotic short-range ("nail") interactions, e.g., contact $\delta$-potentials, give a much more sharply peaked resonance (by a factor of $\sim40$ in $|\psi(0)|^2$ for the same binding energy), in strong tension with LHC data unless the interaction strength is significantly smaller. Consequently, current observed excesses ($\sim7$ pb in the threshold region) are fully compatible with glue-driven toponium, and improvements in cross-section precision below $1$ pb would be required to probe new short-range physics (Llanes-Estrada, 28 Nov 2024).

6. Outlook and Future Sensitivities

The single-lepton search strategy achieves robust evidence for toponium with existing data ($>10\sigma$), and future data will over-constrain the QCD potential, offering a laboratory for nonperturbative QCD studies at the electroweak scale. In parallel, precise measurements in the single-lepton and dilepton channels facilitate novel determinations of fundamental parameters such as $m_t$ and $\alpha_s$ at short distances, with constraint potential for light new-physics mediators in the QCD potential (Fuks et al., 3 Sep 2025, Llanes-Estrada, 28 Nov 2024).

At higher luminosities and at future colliders (HE-LHC, FCC-hh), the increase in production rates ($\sigma_{t\bar t}$ and $\sigma_{\rm top}$ both rising by factors $\sim$2–3) will refine these measurements and allow searches for subleading rare decays (e.g., $\eta_t\to ZH$) and more exotic signatures.

The combination of advanced simulation tools, robust event selection in distinctive final states, and systematic theoretical frameworks rooted in NRQCD provides a comprehensive methodology for exploiting toponium as a precision probe of strong-interaction physics and a unique window on novel short-range dynamics.

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Table: Key Simulation and Analysis Parameters for Single-Lepton Toponium Searches

Variable/Selection	Signal Feature / Implementation	Note
$p^*$(top momentum in $t\bar t$ rest frame)	Signal peak at $\sim$20 GeV	Corresponds to Bohr radius scale
$\min R^2_{\ell j}$ (lepton-jet separation)	Signal peaks at $\lesssim0.3$	Powerful for S/B enhancement
$m_{t\bar t}$ (reco $t\bar t$ mass)	Signal below 350 GeV	Background flatter, distribution extends higher
Event selection cuts	$p_T(\ell)>10$ GeV, 2 $b$-jets, 2 light jets, $E_T^{\rm miss}>30$ GeV, $m_{t\bar t}<350$ GeV, $\min R^2_{\ell j}\le0.5$, optionally $p^*<40$ GeV	Standardized for optimal signal extraction
Typical yields/140 fb$^{-1}$	$N_{\rm top}=3,060$, $N_{t\bar t}=81,600$	$s\sim10.5$, $S/B\sim3.8\%$
Dominant systematics	PDF/scale ($\sim4\%$), $b$-tag ($\sim3$–$5\%$), jet energy scale	Controlled by sidebands and high-mass extrapolation

This framework defines the current state-of-the-art in identifying and characterizing toponium formation, ensuring robust interpretation of threshold-region data and enabling systematic searches for deviations arising from QCD or exotic new-physics binding mechanisms (Fuks et al., 3 Sep 2025, Llanes-Estrada, 28 Nov 2024, Fu et al., 17 Apr 2025).