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REVIEW 4 major objections 7 minor 57 references

Mass and Decay Properties of Toponium Using SUSY QM Factorization Method

T0 review · 4 major / 7 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read SUSY QM applied to the Cornell potential yields toponium ground-state masses near 344.1 GeV, a 2.57 MeV di-gluon width, and radii of 0.015–0.025 fm.

desk verdict Clean SUSY-QM Cornell application that reproduces its own numbers, but the MeV-precision 1S masses describe a broad threshold bump, not an isolated bound state. read the letter →

arxiv 2607.23643 v1 pith:BC2Y4YN3 submitted 2026-07-26 hep-ph

Bhavishya Tepan , Manan Shah , P C Vinodkumar This is my paper
classification hep-ph
keywords toponiumSUSYQMCornellpotentialhyperfinesplittingdecaywidthsquarkoniumwavefunctionattheoriginbindingenergy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper treats toponium—the fleeting top–antitop bound state—with the same Cornell potential used for lighter quarkonia, but solves it by supersymmetric quantum-mechanical factorization. Expanding the confining piece as a perturbation around the Coulomb problem gives an analytic wave function at the origin through second order in the string tension. From that single object the authors extract a binding energy of about −0.9 GeV, pseudoscalar and vector masses of 344.114 GeV and 344.141 GeV, a 27 MeV hyperfine splitting, a suite of annihilation widths (led by η_t → gg at 2.57 MeV), and spatial sizes of only 0.015–0.025 fm. A sympathetic reader cares because the top quark is so heavy that its bound state sits deep in the perturbative regime, offers a clean handle on the top–Higgs coupling, and has now been glimpsed near threshold at the LHC; an analytic, controlled calculation of its spectrum and decays therefore supplies concrete numbers that future precision colliders can test.

What carries the argument

The SUSY-QM factorization of the radial Schrödinger equation for the Cornell potential, which expands the superpotential in the dimensionless string-tension parameter λ and yields the compact analytic formula |Ψ(0)|² = (1/π a₀³)(1 + 3/2 λ − 31/16 λ²) that controls every mass, radius and decay width.

What would settle it

A precision threshold scan at a future e⁺e⁻ or muon collider that measures the 1S peak position and the η_t → gg (or γγ) partial width; disagreement with 344.11 GeV and 2.57 MeV at the few-MeV level would falsify the O(λ²) SUSY-QM prediction.

Watch

Extended reading notes

Core claim

Within the SUSY-QM factorization of the Cornell potential, the ground-state toponium system has spin-averaged mass 344.134 GeV, binding energy −0.9056 GeV, hyperfine splitting 27.2 MeV (giving η_t = 344.114 GeV and Θ_t = 344.141 GeV), dominant di-gluonic width 2.57 MeV, and mean/RMS/most-probable radii all lying between 0.015 and 0.025 fm—all quantities fixed by the O(λ²) expansion of |Ψ(0)|².

Load-bearing premise

That a stable non-relativistic Schrödinger bound state still makes sense when each top quark already decays with a width larger than the computed binding energy, so the system is at best a broad quasi-bound resonance whose wave function is still given by the same expansion.

Editorial extensions

If this is right

  • Pseudoscalar and vector toponium masses sit only ~27 MeV apart near 344.1 GeV, fixing the location of the near-threshold excess already seen by ATLAS and CMS.
  • The di-gluon channel dominates annihilation (2.57 MeV), while electroweak modes such as Z⁰H (~595 keV) and W⁺W⁻ supply the cleanest experimental tags.
  • The system is extremely compact (radii ≲ 0.025 fm), placing it firmly inside the perturbative QCD regime and making the 1S mass a sharp definition of the top-quark mass.
  • Because the wave function at the origin is analytic, every partial width scales directly with |Ψ(0)|² and can be updated instantly when α_s or the string tension is refined.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the single-top width really exceeds the binding energy, the extracted |Ψ(0)|² should be re-interpreted as the residue of a complex pole rather than a normalizable bound-state density; that reinterpretation would systematically shift all quoted widths.
  • The same O(λ²) SUSY expansion can be applied without change to the first radial excitations, giving a parameter-free prediction for the 2S–1S splitting that threshold scans could test next.
  • Because the calculation is fully analytic in λ, it supplies a transparent uncertainty budget for the top Yukawa extraction once the Z⁰H channel is observed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The manuscript applies the SUSY-QM factorization/Supersymmetric Expansion Algorithm to the Cornell potential for the toponium ground state. Treating the linear term perturbatively through O(λ²), it obtains |Ψ(0)|², a binding energy E_B=−0.9056 GeV, pseudoscalar and vector masses of 344.114 and 344.141 GeV, a 27.2 MeV hyperfine splitting, several characteristic radii, and a broad set of annihilation widths. The reported numbers are compared with earlier potential-model calculations, and the independently decaying top-quark width is also estimated.

Significance. The study is timely given the ATLAS and CMS near-threshold tt̄ measurements and provides explicit, readily checkable potential-model predictions for the spectrum, wave function at the origin, radii, and annihilation channels. A particular strength is the analytic O(λ²) treatment: for the parameters implied by the text, λ is small and the reported numerical chain is internally reproducible. The side-by-side decay tables are also useful. If the quasi-bound-state interpretation and uncertainty treatment are made rigorous, the paper can serve as a compact benchmark for conventional NR potential-model expectations, although the method itself is an application of previously published formalism rather than a new theoretical development.

major comments (4)
  1. [§II, Eqs. (8)–(12) and §II.A] The numerical potential is not fully specified. Equation (10) defines λ=2σ/(μ²α³), but the manuscript never states σ, λ, or explicitly defines the reduced mass μ used in the calculation. Inverting the quoted E_B with μ=m_t/2 gives approximately λ=0.0157 and σ=0.184 GeV², but readers should not have to reverse-engineer a central input. Please state these values, show the O(λ²) energy formula used for E_B, justify the chosen σ, and give the associated truncation/parameter uncertainty.
  2. [§II.C and §III] There is an unresolved tension between the bound-state language and the paper's own width estimate. The manuscript obtains Γ_tt̄=2Γ_t=2.9566 GeV, which is more than three times |E_B|=0.9056 GeV and also exceeds the roughly 0.76 GeV 1S–2S spacing implied by the same Coulomb-plus-linear approximation. Thus the object is a broad quasi-bound threshold enhancement, not an isolated narrow 1S level. The tables can still be meaningful as conventional partial annihilation widths evaluated with a frozen real-potential wave function, but the authors must state that approximation, qualify the phrase “tightly bound state,” and discuss the effect of width smearing on the quoted masses and observables.
  3. [Abstract, Table I, and §III] The precision attached to the central results is not supported by an uncertainty analysis. The masses are quoted to 1 MeV even though the input m_t=172.52±0.33 GeV alone gives an uncertainty of about ±0.66 GeV in 2m_t; α_s(μ), σ, omitted higher-order potential terms, and relativistic corrections add further model dependence. Similar uncertainties propagate into |Ψ(0)|² and every width in Tables II–III. An uncertainty budget is needed, or the results should be rounded and presented explicitly as central values within the stated model.
  4. [Abstract and §III] The statement that η_t→gg is the dominant decay mode is accurate only within the restricted set of annihilation channels. The manuscript itself finds the independent weak decays of the two constituents to give Γ_tt̄≈2.96 GeV, roughly three orders of magnitude above Γ(η_t→gg)=2.57 MeV. The abstract, discussion, and table captions should consistently say “dominant annihilation channel” and, where branching fractions or experimental relevance are discussed, normalize to the total quasi-bound-state width rather than only to annihilation widths.
minor comments (7)
  1. [§II.A, Eq. (14)] The hyperfine notation is ambiguous. As written, Eq. (14) multiplies the coefficient by ⟨S₁·S₂⟩, which gives spin-dependent level shifts; the quoted 27.2 MeV is instead the coefficient and hence the vector–pseudoscalar mass difference. Please distinguish δE_S=C|Ψ(0)|²⟨S₁·S₂⟩ from ΔM(Θ_t−η_t)=C|Ψ(0)|².
  2. [Abstract and §II.B] The abstract states that all three radii lie in 0.015–0.025 fm, but the quoted RMS radius is 0.02657 fm. Either adjust the range or correct the reported value.
  3. [§II, Eq. (11)] The one-loop coupling formula uses log(e+μ²/Λ²), rather than the more standard log(μ²/Λ²). Although the Euler-constant offset is numerically negligible at μ=m_t, its origin and the scale uncertainty should be explained. It would also help to state explicitly that μ=m_t/2 in the two-body Schrödinger equation.
  4. [§II.B] The “hyperfine splittings” assigned to r_mp and r_rms (3.72 and 0.88 MeV) are introduced without a defining formula. Since the Fermi contact interaction is controlled by |Ψ(0)|², the meaning and derivation of these radius-based estimates should be given or the statements removed.
  5. [Tables II–III] Please clarify whether entries such as e⁺e⁻, μ⁺μ⁻, τ⁺τ⁻ and the three neutrino channels in Table III are per flavor or summed over flavors. Adding branching fractions relative to Γ_tt̄ would also make the phenomenological hierarchy clearer.
  6. [§I] The experimental statements should be more cautious: the collaborations observe a near-threshold cross-section enhancement, while its specific quasi-bound-state interpretation is model dependent. “Effectively proving that the top and anti-top quarks pair up briefly” overstates the inference.
  7. [General] There are several typographical and wording issues, including “succesfully,” “dimesionless,” “Ricatti,” “Riccati equation” formatting, “event estimated,” and inconsistent spacing around units and particle symbols. The denominator in Eq. (14) should also be typeset unambiguously as 9m_Q².

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: masses, widths, and radii are computed from external PDG inputs, the standard Cornell form, an independent SUSY expansion (Ref. [34]), and literature decay formulas.

full rationale

The derivation chain is feed-forward and non-circular. Inputs are external and independently specified: PDG top mass and couplings (Eq. 11, §II.C constants), the Cornell potential (Eq. 8), the SUSY QM λ-expansion of |Ψ(0)|² and the radial density taken from Napsuciale et al. [34] (no author overlap with the present paper), and standard annihilation-width formulas cited from the broader literature [23–25, 37–39]. From these the paper evaluates E_B, spin-averaged and hyperfine-split masses (Eqs. 13–14), radii via moments of u²(x) (Eqs. 15–16), and partial widths (Eqs. 20–33). None of the reported outputs is algebraically identical to an input by construction, nor is any parameter fitted to toponium data and then re-presented as a prediction of the same data. Mild model dependence (choice of Λ, fixed-scale α_s, implicit σ/λ) is ordinary potential-model practice, not circularity under the stated criteria. Self-citations ([26] Parmar thesis supervised by one co-author) are background and not load-bearing for the uniqueness or form of the results. Score 0 is therefore appropriate.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central numbers rest on the Cornell potential, a non-relativistic two-body Schrödinger treatment, the SUSY/SEA λ-expansion imported from Ref. [34], a fixed-scale α_s with hand-chosen Λ, an implicit string tension, and standard NR annihilation formulas. No new dynamical entity is postulated; the free choices are conventional potential-model knobs.

free parameters (3)
  • QCD scale Λ in α_s formula = 0.10 GeV
    Set to Λ=0.10 GeV by hand to obtain α_s(m_t)≈0.11; this fixes the Coulomb strength α=4/3 α_s and therefore the leading binding energy and |Ψ(0)|².
  • string tension σ (or normalized λ) = not reported (implicit via E_B)
    Enters the Cornell potential and the O(λ²) SUSY expansion for energies, wave function, and radii; numerical value is never stated, only the output E_B=−0.9056 GeV, so λ is an implicit free choice.
  • renormalization scale choice μ=m_t for α_s = μ=m_t=172.52 GeV
    α_s is frozen at the top mass rather than run or matched at a Bohr-radius scale; this choice directly sets the short-distance coupling used everywhere.
assumptions (6)
  • domain assumption Quarkonium dynamics are described by the non-relativistic Schrödinger equation with the Cornell potential V(r)=−(4/3)α_s/r + σr.
    Stated in §II as the working interaction; relativistic and open-flavor continuum effects are neglected despite Γ_t ∼ |E_B|.
  • domain assumption The linear term may be treated as a perturbation and the superpotential expanded to O(λ²), yielding the quoted |Ψ(0)|² and u²(x) series from the algorithm of Ref. [34].
    Core computational step in §II; validity requires λ≪1, plausible for toponium but not demonstrated numerically in-paper.
  • domain assumption Spin-averaged mass is 2m_t + E_B and hyperfine splitting is the contact formula ΔE_hf=(32πα_s/9m_Q²)|Ψ(0)|²⟨S1·S2⟩.
    Eqs. (13)–(14); standard NRQCD/potential-model contact interaction.
  • domain assumption Annihilation decay widths factorize into known short-distance coefficients times |ψ(0)|² (and listed NLO α_s factors) as written in Eqs. (20)–(33).
    Imported from Refs. [23–25,37–39]; paper does not re-derive the hard coefficients.
  • domain assumption Top and anti-top in the bound state decay independently, so Γ_tt̄≈2Γ_t with the LO three-body weak formula.
    §II.C and Ref. [49]; used to quote the lifetime that actually dominates the phenomenology.
  • standard math SUSY QM factorization, partner Hamiltonians, and Riccati equation for the superpotential are the correct solution method for the radial problem.
    §II setup following Cooper–Khare–Sukhatme and Ref. [34]; standard mathematical framework.

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Cite this review

Pith. "Pith review of Mass and Decay Properties of Toponium Using SUSY QM Factorization Method." pith.science (2026). https://pith.science/paper/BC2Y4YN3

@misc{pith2026260723643,
  author       = {Pith},
  title        = {Pith review of: Mass and Decay Properties of Toponium Using SUSY QM Factorization Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BC2Y4YN3}},
  note         = {Machine review of arXiv:2607.23643}
}
abstract

The Supersymmetric quantum mechanics (SUSY QM) factorization method is applied to study the Cornell potential in the context of toponium $(t\bar{t})$, the heaviest quarkonium system. The method is successfully applied to determine the mass, binding energy, and spin-dependent hyperfine splitting, giving pseudoscalar $(\eta_t)$ and vector $(\Theta_t)$ state masses of 344.114 GeV and 344.141 GeV, respectively. The decay properties of both the pseudoscalar and vector states are also investigated through their respective decay widths, highlighting the crucial role of the bound-state wavefunction. Among the decay channels, the di-gluonic $(\eta_t\rightarrow gg)$ mode is found to be dominant, with an estimated decay width of 2.57 MeV. In addition, the spatial characteristics of toponium are examined by evaluating the mean radius, root-mean-square (RMS) radius, and most probable radius, all of which are found to lie within the range of $0.015\text{--}0.025\,\mathrm{fm}$.

Figures

Figures reproduced from arXiv: 2607.23643 by the authors.

Figure 1
Figure 1. FIG. 1: Comparison of pseudoscalar and vector toponium [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Radial probability distribution of toponium. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Dominant hadronic decay channels of pseudoscalar [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p006_10.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p007_12.png]

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