Beauty Hadron Spectrum in a Screened Potential Model

Bhaghyesh; Raghavendra Kaushal

Beauty Hadron Spectrum in a Screened Potential Model

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Pith's one-line read A six-parameter screened-potential model fitted to eight bottomonium states predicts excited bottomonium, diquark, and triply-bottom-baryon masses, with Upsilon(10753) assigned as a D-wave state.

arxiv 2505.13987 v1 pith:3KZNLKSW submitted 2025-05-20 hep-ph

Raghavendra Kaushal , Bhaghyesh This is my paper · ORCID

classification hep-ph

keywords potentialmassmodelspectrumupsilonbaryonsbeautybottomonium

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

The paper models heavy particles made of bottom quarks as non-relativistic two-body systems. The force between the quarks is a Coulomb attraction at short range and a 'screened' string-like force at long range, which flattens out at large separation. The authors add corrections proportional to 1/mass taken from lattice QCD and pNRQCD studies, and a smeared spin-spin interaction is included directly in the Schrodinger equation solver, while spin-orbit and tensor forces are added later as small perturbations.

Six parameters (bottom quark mass, string strength, screening length, two O(1/m) constants, and the width of the spin-spin smearing) are fit to the masses of eight well-measured bottomonium states. The resulting model reproduces those states and gives predictions for higher S, P, and D states. The paper assigns Upsilon(10753) to the 3D1 bottomonium level, and Upsilon(10860) and Upsilon(11020) to the 5S1 and 6S1 levels, though the predicted masses sit 50-70 MeV away from the measured values, which the authors attribute to S-D mixing and coupled-channel effects they do not include.

For triply bottom baryons, the paper uses the diquark-quark model: the two bottom quarks form a diquark, treated as a point particle, which then binds to a third bottom quark. It predicts a ground state mass of 14.243 GeV and a ladder of excited states. No such baryon has been observed yet, so these numbers are untested predictions.

Extended reading notes

Core claim

The mass spectrum of beauty hadrons (bb and bbb baryons) and bb-diquarks are computed, and the paper states: 'We interpret Upsilon(10753) as D-wave bottomonium state and Upsilon(10860) and Upsilon(11020) as S-wave bottomonium states.' Additionally, the ground state mass of the triply bottom baryon is given as 14.243 GeV. If the model is correct, these are the masses of the corresponding physical states.

Load-bearing premise

The diquark-quark model treats the bb diquark as a point-like color source whose interaction with the third quark is identical to the quark-antiquark interaction (color factor halved) and uses the same O(1/m) correction terms and the same fitted parameters as bottomonium (Section II.3, Eqs 17-20). The validity of this mapping is load-bearing for every bbb baryon mass, and it is adopted from Refs [70,76] rather than derived.

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Editorial analysis

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Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a fitted six-parameter Hamiltonian (Table I), the screened-Coulomb-plus-O(1/m) potential form (Eq 9), and the diquark-quark mapping (Section II.3). No new particles or forces are introduced; the diquark is an existing modeling device. The main uncharged inputs are the quenched O(1/m) corrections and the assumption that the baryon interaction equals the meson interaction with halved color factor.

free parameters (8)

mb (bottom quark mass) = 4.680 GeV
Fitted to reproduce eight bottomonium masses (Table I).
lambda (string strength) = 0.241 GeV^2
Fitted to bottomonium spectrum; sets the linear part of the confining potential at short distance.
nu (screening parameter) = 0.078 GeV
Fitted; controls the flattening of the confinement potential at large r.
C (O(1/m) logarithmic coefficient) = 0.100 GeV
Fitted; strength of the ln(ar) correction term in Eq 9.
a (scale in logarithmic correction) = 0.430 GeV
Fitted; scale inside the logarithm of the 1/m correction.
sigma (spin-spin smearing width) = 3.920 GeV
Fitted; width of the Gaussian smearing of the spin-spin contact term.
Lambda_QCD = 0.130 GeV
Chosen within PDG range; determines alpha_s via Eq 10 with n_f=4.
Delta M_exp (constant experimental uncertainty in fit) = 5 MeV
Assumed constant for all fitted states following Ref [73]; affects chi-square weights and fitted parameters.

assumptions (7)

domain assumption The non-relativistic Schrodinger equation (Eq 1) with a reduced-mass two-body Hamiltonian is adequate for bottomonium and bbb systems.
Kinetic energy of heavy quarks is assumed negligible compared to rest mass; used throughout Section II.
domain assumption The potential (Eq 9) is the sum of a one-gluon Coulomb term, a screened confinement term, and O(1/m) corrections taken from quenched LQCD/pNRQCD (Refs [28,60]).
This specific form is postulated, not derived; the quenched nature of the O(1/m) corrections is acknowledged in Section I.
domain assumption Spin-spin interaction is included nonperturbatively as a smeared Gaussian delta, while spin-orbit and tensor are treated as first-order perturbations (Eqs 11-13).
Follows Refs [70,71]; the choice affects the hyperfine and fine splittings.
ad hoc to paper The scalar confinement potential entering the spin-orbit operator is V_S = lambda(1-e^{-nu r})/r (Eq 15), even though the static confinement in Eq 9 and 11 is lambda(1-e^{-nu r})/nu.
This appears to be an internal inconsistency or a typo; the derivative dV_S/dr used in Eq 12 is not the derivative of the confinement potential used in the Hamiltonian.
domain assumption For bb diquarks, the full static potential is half the quark-antiquark potential (color factor kappa=-2/3) (Eq 17).
Adopted from Refs [70,76] based on color wavefunction arguments; central to all diquark and baryon masses.
ad hoc to paper The diquark-quark potential for bbb baryons is the same as the quark-antiquark potential with masses m_d and m_b and the same parameters (Eqs 20-22).
This mapping is assumed, not derived; the diquark is treated as a point-like particle with the mass computed in the same model.
standard math Pauli principle restricts the bb diquark S-wave state to J_d=1 (since the color antitriplet diquark is antisymmetric).
Used to select diquark spin states in Section II.3 and Table II.

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Pith. "Pith review of Beauty Hadron Spectrum in a Screened Potential Model." pith.science (2026). https://pith.science/paper/3KZNLKSW

@misc{pith2026250513987,
  author       = {Pith},
  title        = {Pith review of: Beauty Hadron Spectrum in a Screened Potential Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3KZNLKSW}},
  note         = {Machine review of arXiv:2505.13987}
}

abstract

The mass spectrum of beauty hadrons ($b\overline{b}$ and $bbb$ baryons) and $bb$-diquarks are computed in a non-relativistic phenomenological potential model. The potential comprises of a short-range Coulomb potential, a screened confinement potential, and $O(1/m)$ corrections predicted from lattice and pNRQCD studies. Among the spin-dependent interactions, spin-spin interaction is considered non-perturbatively, whereas spin-orbit and tensor interactions are considered perturbatively. The Matrix-Numerov method is used to numerically solve the non-relativistic Schrodinger equation to evaluate the mass spectra. We interpret $\Upsilon(10753)$ as $D$-wave bottomonium state and $\Upsilon(10860)$ and $\Upsilon(11020)$ as $S$-wave bottomonium states. The mass spectrum of $bbb$ baryons are evaluated under the diquark-quark model. The excited masses are computed by considering various radial and orbital excitations of the diquark as well as the diquark-quark system.

Figures

Figures reproduced from arXiv: 2505.13987 by the authors.

**Figure 2.** FIG. 2: Comparison of mass spectra of bottomonium obtained from [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3: Comparison of mass spectra of [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: Mass spectra of triply bottom baryons corresponding to d [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5: Mass spectra of triply bottom baryons corresponding to d [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6: Comparison of masses of lowest state triply bottom baryon [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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680 0 . 241 0 . 078 0 . 100 0 . 430 3 . 920 II.1. Bottomonium: The color factor in equation (9) for bottomonium, which is a bound st ate of bottom and anti-bottom quarks will be κ = − 4 3 . Hence, the potential (9) for bottomonium takes the form, V (r) = − 4 3 α s(m2 b ) r + λ(1 −e− νr ) ν + ( 1 mb + 1 mb ) (−9α 2 8r2 +C ln(ar) ) + 32πα s(m2 b ) 9m2 b ( σ...

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4: Mass spectra of triply bottom baryons corresponding to d iﬀerent diquark states with nd = 1

543 12 1S1s 1S2s 1S1d 1S2d 1S1p 1S2p 1P1p 1P2p 1P1s 1P2s 1P1d 1P2d 1D1s 1D2s 1D1d 1D2d 1D1p 1D2p 1S d 1 d 1D d 1 1 1 15.5 M ( G ) FIG. 4: Mass spectra of triply bottom baryons corresponding to d iﬀerent diquark states with nd = 1. 2S1s 2S2s 2S1d 2S2d 2S1p 2S2p 2P1p 2P2p 2P1s 2P2s 2P1d 2P2d 2D1s 2D2s 2D1d 2D2d 2D1p 2D2p 2S 2 2 15.5 ( ! " # ) FIG. 5: Mass s...

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243 14 . 468 14 . 396 14 . 496 14 . 834 14 . 432 5 2 +

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915 14 . 895 14 . 894 15 . 293 15 . 101 14 . 981 7 2 +

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936 14 . 909 14 . 894 15 . 286 15 . 101 14 . 988 1 2 −