Mass and Decay Properties of Toponium Using SUSY QM Factorization Method

Technical Summary: Mass and Decay Properties of Toponium Using SUSY QM Factorization Method

Problem Statement
The paper addresses the theoretical characterization of toponium ($t\bar{t}$), the quasi-bound state of a top quark and its antiquark. Despite the top quark's extremely short lifetime ($\sim 3.29 \times 10^{-25}$ s), which traditionally suggests it decays before forming a bound state, recent observations by ATLAS and CMS (cited as occurring in mid-2025) have confirmed a surplus of top-quark pairs near the production threshold, indicating the existence of a near-threshold $t\bar{t}$ quasi-bound state. The challenge lies in accurately modeling the mass spectrum, binding energy, hyperfine splitting, and decay widths of this system. Unlike lighter quarkonia (charmonium, bottomonium), toponium dynamics are dominated by weak interactions and the large top quark mass, necessitating precise theoretical tools to distinguish its signals from background noise and to probe Standard Model (SM) consistency, particularly regarding the top-Higgs Yukawa coupling and electroweak symmetry breaking.

Methodology
The authors employ the Supersymmetric Quantum Mechanics (SUSY QM) factorization method applied to the Cornell potential. The Cornell potential, $V(r) = -\frac{4}{3}\frac{\alpha_s}{r} + \sigma r$, models the interaction between quarks by combining a short-distance color Coulombic term with a long-distance linear confinement term.

The methodology proceeds as follows:

1. Hamiltonian Factorization: The system's Hamiltonian is factorized using ladder operators (supercharges) to relate the target potential to a partner potential. This allows for the solution of the radial Schrödinger equation via a Riccati equation approach.
2. Perturbative Expansion: The linear confining term is treated as a perturbation around the exactly solvable Coulomb problem. The superpotential is expanded as a power series in the normalized string tension parameter ($\lambda$).
3. Wavefunction Derivation: This approach yields an analytical expression for the square of the wavefunction at the origin, $|\Psi(0)|^2$, up to second order in $\lambda$. This quantity is critical as it governs the decay rates.
4. Parameter Inputs: The analysis uses a fixed strong coupling constant $\alpha_s \approx 0.11$ evaluated at the scale $\mu = m_t = 172.52$ GeV. The top quark mass is taken as $172.52$ GeV, and the system is treated non-relativistically, with relativistic corrections deemed negligible due to the heavy mass.

Key Contributions and Results

• Mass Spectrum and Binding Energy:
  The study calculates a binding energy of $E_B = -0.9056$ GeV. Consequently, the spin-averaged mass of the toponium system is determined to be $344.134$ GeV. The method distinguishes between the pseudoscalar ($\eta_t$, $0^{-+}$) and vector ($\Theta_t$, $1^{--}$) states, yielding masses of 344.114 GeV and 344.141 GeV, respectively. The hyperfine splitting between these states is calculated as 27.2 MeV. These results show good agreement with previous predictions from non-relativistic quark potential models and Salpeter equation calculations.

• Spatial Characteristics:
  The spatial extent of the toponium bound state is found to be extremely compact. The mean radius is calculated as 0.02304 fm, the root-mean-square (RMS) radius as 0.02657 fm, and the most probable radius as 0.01544 fm. These values confirm that the system lies deep within the perturbative regime of Quantum Chromodynamics (QCD).

• Decay Properties:
  The paper computes decay widths for various channels, highlighting a distinct hierarchy compared to lighter quarkonia:

  • Gluonic Decays: The di-gluonic decay ($\eta_t \to gg$) is the dominant hadronic mode with a width of 2.57 MeV. The tri-gluonic decay ($\Theta_t \to ggg$) is significantly suppressed at 57.72 eV.
  • Electroweak Bosonic Decays: These channels are prominent due to the top quark's mass. For the pseudoscalar state, the $\eta_t \to Z^0 H$ channel is the largest electroweak mode (594.68 keV), followed by $\eta_t \to W^+ W^-$ (135.39 keV). For the vector state, $\Theta_t \to W^+ W^-$ is dominant (570.45 keV).
  • Fermionic Decays: Pseudoscalar fermionic decays ($\eta_t \to f\bar{f}$) are strongly suppressed (e.g., $\eta_t \to b\bar{b}$ is 1.49 keV). In contrast, vector fermionic decays are enhanced due to contributions from both virtual photon and $Z^0$ exchange, with $\Theta_t \to b\bar{b}$ reaching 454.98 keV.
  • Total Width: Assuming independent decay of constituent quarks, the total toponium width is estimated at 2.9566 GeV, corresponding to a lifetime of $2.226 \times 10^{-25}$ s.

Significance and Claims
The paper claims that the SUSY QM factorization method provides a consistent and analytically tractable framework for studying heavy quarkonium systems like toponium. By successfully applying this method to the Cornell potential, the authors demonstrate that it can reproduce mass spectra and binding energies in agreement with established models while offering a structured way to calculate decay properties.

The authors assert that their results reinforce the view of toponium as a unique probe for the Standard Model. The calculated compactness of the state supports the dominance of short-range interactions, while the specific hierarchy of decay widths—particularly the dominance of bosonic channels over fermionic ones for the pseudoscalar state—provides clear theoretical signatures for experimental identification. The paper concludes that while fermionic decay channels are too suppressed to be accessible under projected High-Luminosity LHC (HL-LHC) conditions, the bosonic decay channels (specifically $Z^0 H$ and $W^+ W^-$) constitute the primary experimental handles for discovering and characterizing the pseudoscalar toponium meson. The study positions toponium as a valuable system for testing the top-Higgs Yukawa coupling and searching for physics beyond the Standard Model.