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arxiv: 2604.04846 · v1 · submitted 2026-04-06 · ✦ hep-ph · hep-ex

Recognition: 2 theorem links

· Lean Theorem

boldsymbol{B_c} Meson Spectroscopy from Bayesian MCMC: Probing Confinement and State Mixing

Christas Mony A. , Rohit Dhir
Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:16 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords B_c mesonmeson spectroscopyCornell potentialBayesian MCMCRegge trajectoriesconfinementheavy quarkonium
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The pith

Bayesian MCMC sampling of Cornell and log-modified potentials reproduces known B_c states and forecasts excited ones with uncertainties.

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

The paper uses Markov Chain Monte Carlo sampling to fit parameters of the non-relativistic Cornell potential and a version with an added logarithmic term at intermediate distances. This setup tests how the precise form of confinement affects the spectrum of the B_c meson while consistently handling unequal heavy-quark masses and perturbative spin-dependent corrections. Both models match the observed low-lying states within the computed uncertainties, though the spread widens for higher radial and orbital excitations. The logarithmic correction produces only modest systematic shifts in the higher states. The work supplies updated mass predictions with error bars intended as benchmarks for experiments that will search for these excitations.

Core claim

Markov Chain Monte Carlo sampling of the parameters in both the standard Cornell potential and its logarithmically modified form shows that each reproduces the known B_c states within uncertainties, with errors remaining small for low-lying levels and increasing for higher radial and orbital excitations. The modified potential induces modest, systematic shifts in the higher states. Radial and orbital Regge trajectories display pronounced nonlinearity at low S-waves that trends toward linearity at higher excitations. Updated theoretical predictions for excited B_c states, complete with uncertainties, are provided as benchmarks for ongoing and future experiments.

What carries the argument

Markov Chain Monte Carlo sampling of potential parameters in the Cornell and logarithmically modified Cornell models, which propagates correlated uncertainties through perturbative spin-dependent interactions to all predicted masses.

If this is right

  • Both the standard and logarithmically modified potentials reproduce known B_c states within uncertainties, with errors growing for higher excitations.
  • The logarithmic term produces only modest systematic shifts in higher excited states.
  • Radial and orbital Regge trajectories are nonlinear at low S-waves and become more linear at higher excitations.
  • The resulting mass predictions with uncertainties serve as direct benchmarks for experiments searching for excited B_c states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same MCMC framework could be applied to other heavy-meson systems to test whether the logarithmic correction remains minimal across different quark-mass regimes.
  • Comparison of these predictions against future lattice QCD results at intermediate distances would directly test whether the added logarithmic flexibility captures the correct confining physics.
  • If high-precision data on a few higher states become available, the posterior distributions could be used to constrain possible state-mixing angles beyond the perturbative treatment.

Load-bearing premise

The non-relativistic approximation together with perturbative treatment of spin-dependent interactions remains adequate even for higher radial and orbital excitations.

What would settle it

A precise experimental mass for a high radial or orbital B_c excitation that lies outside the uncertainty bands predicted by both potentials.

Figures

Figures reproduced from arXiv: 2604.04846 by Christas Mony A., Rohit Dhir.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of the Cornell (Potential I, Eq. (1)) and Modified Cornell (Potential II, Eq. (2)) [PITH_FULL_IMAGE:figures/full_fig_p032_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Corner plot for Cornell Potential (Potential I). [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Corner plot for Modified Cornell Potential (Potential II). [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The [PITH_FULL_IMAGE:figures/full_fig_p038_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Radial wave functions of the [PITH_FULL_IMAGE:figures/full_fig_p043_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Radial dependence of the modified Cornell Potential [PITH_FULL_IMAGE:figures/full_fig_p044_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Radial Regge trajectories of [PITH_FULL_IMAGE:figures/full_fig_p045_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Parent and daughter orbital Regge trajectories of [PITH_FULL_IMAGE:figures/full_fig_p046_8.png] view at source ↗
read the original abstract

We present a comprehensive Bayesian study of the $B_c$ meson spectrum using non-relativistic Cornell and logarithmically modified Cornell potentials, introducing the logarithmic term as the minimal deformation that preserves short-range Coulombic and long-range linear confinement while adding controlled flexibility at intermediate distances to probe the sensitivity of higher excited states to the confining form. Model parameters are sampled via Markov chain Monte Carlo (MCMC), enabling rigorous propagation of correlated uncertainties to all predictions. Spin-dependent interactions are treated perturbatively, with unequal heavy-quark masses accounted for consistently. Both potentials reproduce the known states within uncertainties, with small errors for low-lying states that grow for higher radial and orbital excitations. Analyzing radial and orbital Regge trajectories using linear and nonlinear parametrizations, we observe pronounced nonlinearity for low $S$-waves trending toward linearity at higher excitations. The modified potential yields modest, systematic shifts in higher excited states, reflecting the logarithmic correction's impact. We provide updated theoretical predictions for excited $B_c$ states with uncertainties, serving as benchmarks for ongoing and future experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript performs a Bayesian MCMC analysis of the B_c meson spectrum using non-relativistic Cornell and logarithmically modified Cornell potentials. Parameters are sampled to reproduce known states, spin-dependent interactions are included perturbatively with unequal quark masses, and predictions with uncertainties are provided for excited states. The work also examines radial and orbital Regge trajectories, finding nonlinearity at low S-waves that trends toward linearity at higher excitations, and notes that the logarithmic term induces modest shifts in higher states.

Significance. If the non-relativistic framework and perturbative treatment remain adequate, the MCMC-based uncertainty propagation and controlled introduction of the logarithmic term offer a transparent way to assess sensitivity of higher B_c excitations to the confining potential. The resulting predictions with correlated errors would serve as useful benchmarks for ongoing LHCb and Belle II searches.

major comments (1)
  1. [higher excitations] Discussion of higher excitations (abstract and results): The paper states that errors grow for higher radial and orbital excitations and that both potentials reproduce known states within uncertainties, yet provides no independent estimate (e.g., via v^2 scaling or comparison to relativistic quark models) of the size of neglected O(v^2) or coupled-channel effects in precisely the regime where the logarithmic term is introduced for flexibility. MCMC propagates only parametric uncertainty; this systematic is load-bearing for the reliability of the excited-state predictions.
minor comments (2)
  1. [title and abstract] The title references 'State Mixing' but the abstract and provided description focus exclusively on potential forms and Regge trajectories without explicit discussion of mixing; clarify how (or whether) state mixing is treated in the full text.
  2. [methods] The precise functional form of the logarithmic modification (coefficient, argument, etc.) and its impact on the intermediate-distance regime should be stated explicitly with an equation number for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the treatment of higher excitations. We address the concern directly below and outline the planned revision.

read point-by-point responses

  1. Referee: Discussion of higher excitations (abstract and results): The paper states that errors grow for higher radial and orbital excitations and that both potentials reproduce known states within uncertainties, yet provides no independent estimate (e.g., via v^2 scaling or comparison to relativistic quark models) of the size of neglected O(v^2) or coupled-channel effects in precisely the regime where the logarithmic term is introduced for flexibility. MCMC propagates only parametric uncertainty; this systematic is load-bearing for the reliability of the excited-state predictions.

    Authors: We agree that the manuscript currently lacks an explicit, independent estimate of the size of neglected O(v^2) and coupled-channel effects for the higher states. The MCMC procedure propagates only the parametric uncertainties within the chosen non-relativistic framework, and the growth of those uncertainties with excitation number is already visible in our results. To strengthen the discussion, we will add a concise paragraph (and a short table) in the results section that (i) extracts a rough v^2 estimate from the known low-lying states and extrapolates it to higher excitations, and (ii) compares a subset of our predictions with published results from relativistic quark models. This addition will clarify the regime in which the logarithmic modification is being tested and will make the limitations of the quoted uncertainties explicit. The core numerical results and the Regge-trajectory analysis remain unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper fits potential parameters via MCMC to reproduce known low-lying B_c states and then extrapolates to predict excited states with propagated uncertainties. This is standard phenomenological calibration followed by genuine extrapolation to states outside the fit data set, not a reduction by construction. The reported growth in discrepancies for higher excitations is presented as an observation rather than forced agreement. No self-definitional loops, fitted inputs renamed as predictions, load-bearing self-citations, or smuggled ansatzes appear in the provided text. The Regge trajectory analysis and logarithmic modification are direct consequences of the model outputs without circular renaming or uniqueness claims.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central results rest on phenomenological potential parameters that are fitted to data and on standard domain assumptions of non-relativistic quark models; no new particles or forces are introduced.

free parameters (1)
  • Cornell and log-modified potential parameters (Coulomb coefficient, string tension, logarithmic strength, etc.)
    Determined by MCMC sampling to reproduce known B_c states
axioms (2)
  • domain assumption Non-relativistic approximation for heavy-quark bound states
    Invoked for B_c as bottom and charm quarks are heavy
  • domain assumption Perturbative treatment of spin-dependent interactions
    Stated explicitly for fine-structure corrections

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Reference graph

Works this paper leans on

72 extracted references · 34 canonical work pages · 2 internal anchors

  1. [1]

    The pseudoscalar statesB c(1S) andB c(2S) are reproduced in excellent agreement 4 We have also applied the modified Cornell potential to bottomonium, where the larger number of observed states enables a more robust assessment; a systematic improvement in higher excited states is found and will be detailed in a forthcoming publication. 13 with the PDG [7] ...

  2. [2]

    Among the reference models the spread is 39−67 MeV, with MBK’s value of 39 MeV [20] indicating a more pronounced suppression of the contact interaction

    The 1Shyperfine splitting, ∆M I 1S = 55.71+3.63 −3.97 MeV, ∆M II 1S = 55.65+3.90 −3.91 MeV, agrees with the LQCD results of 55(3) MeV [41] and 54(3) MeV [56], validating the spin-contact interaction parametrization. Among the reference models the spread is 39−67 MeV, with MBK’s value of 39 MeV [20] indicating a more pronounced suppression of the contact i...

  3. [3]

    The mass gapδM n ≡M I(n1S0)−M II(n1S0) accumulates at∼14 MeV per unitn, reaching∼70 MeV by 6S

    TheS-wave predictions from both potentials agree closely through 2S, beyond which a systematic inter-potential separation develops and grows nearly linearly withn. The mass gapδM n ≡M I(n1S0)−M II(n1S0) accumulates at∼14 MeV per unitn, reaching∼70 MeV by 6S. This divergence originates in the long-range confining sector where the two potentials differ, as ...

  4. [4]

    Forn≥5, LLWL [19], LZ [11], and LTFWP [14] fall below our median predictions, while the remaining models follow similar trends

    For low-lying states, both potentials agree with all reference models except AAMS [12], which consistently underestimates excited-state masses. Forn≥5, LLWL [19], LZ [11], and LTFWP [14] fall below our median predictions, while the remaining models follow similar trends

  5. [5]

    ForP-wave states (Table V), the 1 3P0 and axial-vector 1P 1 medians from both po- tentials agree excellently with LQCD [41, 56, 57], and the 1P ′ 1 and 1 3P2 states are reproduced within LQCD uncertainties [57]. The 1Pfine-structure spread, ∆M I fine(1P) =M(1 3P2)−M(1 3P0) = 49.95 MeV, ∆M II fine(1P) = 52.56 MeV, is governed by the spin-orbit and tensor i...

  6. [6]

    TheP-wave inter-potential agreement holds through 2P, after which masses diverge progressively, reaching∼80 MeV between the potentials at 6 3P2. The splitting be- tween mixed axial-vector states,M nP ′ 1 −MnP1, remains small and nearlyn-independent (∼5 MeV at 1P,∼7 MeV at 6P), a consequence of the weakn-dependence of the underlying spin-dependent matrix e...

  7. [7]

    For higher excitations, EFG [15, 16] predicts slightly larger masses than our medians, while LLWL [19], LZ [11], and LTFWP [14] fall below them

    The low-lyingP-wave predictions agree with all reference models except AAMS [12]. For higher excitations, EFG [15, 16] predicts slightly larger masses than our medians, while LLWL [19], LZ [11], and LTFWP [14] fall below them. Differences in the adopted mixing-angle convention preclude direct comparison ofθ nP across models. 15

  8. [8]

    TheD-wave predictions (Table VI) agree between the two potentials for low-lying states and diverge steadily at highn, mirroring theS- andP-wave behavior. The gap at 63D3 reaches∼90 MeV, larger than forS- orP-waves at the samen, because higher- lstates are more spatially extended and thus more sensitive to long-range potential differences. Unlike theP-wave...

  9. [9]

    For higher excitations, LLWL [19], AAMS [12], and LTFWP [14] fall systematically below our medians

    The 1Dpredictions from both potentials agree with the majority of reference mod- els, with MBK [20] and AAMS [12] as outliers. For higher excitations, LLWL [19], AAMS [12], and LTFWP [14] fall systematically below our medians

  10. [10]

    This non-standard ordering persists through 4Dbefore reverting to the conventional hierarchy at higher excitations

    The mass ordering of the low-lyingD-wave multiplets follows 1D 2 <1 3D3 <1 3D1 < 1D′ 2 and 2D 2 <2 3D1 <2 3D3 <2D ′ 2, which deviates from the naive expectation n3D1 < nD 2 < nD ′ 2 < n 3D3. This non-standard ordering persists through 4Dbefore reverting to the conventional hierarchy at higher excitations. This level-dependent sign reversal is not a numeri...

  11. [11]

    7) exhibit pro- nounced non-linearity under both potentials, which diminishes forP- andD-wave trajectories

    The radial Regge trajectories ofS-wave singlet and triplet states (Fig. 7) exhibit pro- nounced non-linearity under both potentials, which diminishes forP- andD-wave trajectories. This behavior reflects the potential structure: (a) forS-waves, the Coulomb and linear contributions are comparable over the relevant radial range, lead- ing to non-linear traje...

  12. [12]

    Trajectories from Potential II lie systematically below those of Potential I for interme- diate and high excitations, reflecting the softened long-range confining slope discussed in Sec. III C. The shift grows withn, tracking the inter-potential mass gaps (∼14 MeV per radial level forS-waves, larger for higher-ltrajectories), and is most pronounced in the...

  13. [13]

    8) show a characteristic pattern that parent tra- jectories are distinctly non-linear, while daughter trajectories are comparatively more linear and approximately parallel

    The orbital Regge trajectories (Fig. 8) show a characteristic pattern that parent tra- jectories are distinctly non-linear, while daughter trajectories are comparatively more linear and approximately parallel. Potential II predicts systematically lower masses across all orbital trajectories, consistent with the radial-trajectory behavior. The inter-potent...

  14. [14]

    For the highest excitations, where MCMC uncertainties are larger, linear and non-linear fits yield comparable quality for some daughter trajectories, indicating that any re- maining curvature is unresolved at the current level of precision. The Regge behavior ofB c is intermediate between charmonium and bottomonium [38], with trajectories more closely res...

    2022

  15. [15]

    Both are comfortably larger than the respective ESS values

    for Potential II. Both are comfortably larger than the respective ESS values. The posterior corner plots, parameter tables, and propagated spectral uncertainties are presented in Sec. III. 27

  16. [16]

    Abeet al.(CDF), Observation of theB c meson inp¯pcollisions at √s= 1.8 TeV, Phys

    F. Abeet al.(CDF), Observation of theB c meson inp¯pcollisions at √s= 1.8 TeV, Phys. Rev. Lett.81, 2432 (1998), arXiv:hep-ex/9805034

    work page arXiv 1998

  17. [17]

    Abeet al.(CDF), Observation ofB c mesons inp¯pcollisions at √s= 1.8 TeV, Phys

    F. Abeet al.(CDF), Observation ofB c mesons inp¯pcollisions at √s= 1.8 TeV, Phys. Rev. D58, 112004 (1998), arXiv:hep-ex/9804014

    work page arXiv 1998

  18. [18]

    Aaijet al.(LHCb), Precision measurement of theB + c meson mass, JHEP07, 123, arXiv:2004.08163 [hep-ex]

    R. Aaijet al.(LHCb), Precision measurement of theB + c meson mass, JHEP07, 123, arXiv:2004.08163 [hep-ex]

    work page arXiv 2004

  19. [19]

    Aaijet al.(LHCb), Observation of an excitedB + c state, Phys

    R. Aaijet al.(LHCb), Observation of an excitedB + c state, Phys. Rev. Lett.122, 232001 (2019), arXiv:1904.00081 [hep-ex]

    work page arXiv 2019

  20. [20]

    A. M. Sirunyanet al.(CMS), Observation of Two Excited B + c States and Measurement of the B + c (2S) Mass in pp Collisions at √s= 13 TeV, Phys. Rev. Lett.122, 132001 (2019), arXiv:1902.00571 [hep-ex]

    work page arXiv 2019

  21. [21]

    Aadet al.(ATLAS), Observation of an ExcitedB ± c Meson State with the ATLAS Detector, Phys

    G. Aadet al.(ATLAS), Observation of an ExcitedB ± c Meson State with the ATLAS Detector, Phys. Rev. Lett.113, 212004 (2014), arXiv:1407.1032 [hep-ex]

    work page arXiv 2014

  22. [22]

    Navaset al.(Particle Data Group), Review of particle physics, Phys

    S. Navaset al.(Particle Data Group), Review of particle physics, Phys. Rev. D110, 030001 (2024)

    2024

  23. [23]

    Aaij et al

    R. Aaijet al.(LHCb), Observation of Orbitally Excited Bc+ States, Phys. Rev. Lett.135, 231902 (2025), arXiv:2507.02149 [hep-ex]

    work page arXiv 2025

  24. [24]

    Aaij et al

    R. Aaijet al.(LHCb), Study of Bc(1P)+ states in the Bc+γmass spectrum, Phys. Rev. D 112, 112003 (2025), arXiv:2507.02142 [hep-ex]

    work page arXiv 2025

  25. [25]

    E. J. Eichten and C. Quigg, Mesons with Beauty and Charm: New Horizons in Spectroscopy, Phys. Rev. D99, 054025 (2019), arXiv:1902.09735 [hep-ph]

    work page arXiv 2019

  26. [26]

    Li, M.-S

    Q. Li, M.-S. Liu, L.-S. Lu, Q.-F. L¨ u, L.-C. Gui, and X.-H. Zhong, Excited bottom-charmed mesons in a nonrelativistic quark model, Phys. Rev. D99, 096020 (2019), arXiv:1903.11927 [hep-ph]

    work page arXiv 2019

  27. [27]

    Asghar, F

    I. Asghar, F. Akram, B. Masud, and M. A. Sultan, Properties of excited charmed-bottom mesons, Phys. Rev. D100, 096002 (2019), arXiv:1910.02680 [hep-ph]

    work page arXiv 2019

  28. [28]

    Devlani, V

    N. Devlani, V. Kher, and A. K. Rai, Masses and electromagnetic transitions of the B c mesons, Eur. Phys. J. A50, 154 (2014)

    2014

  29. [29]

    T.-y. Li, L. Tang, Z.-y. Fang, C.-h. Wang, C.-q. Pang, and X. Liu, Higher states of the Bc 28 meson family, Phys. Rev. D108, 034019 (2023), arXiv:2204.14258 [hep-ph]

    work page arXiv 2023

  30. [30]

    Ebert, R

    D. Ebert, R. N. Faustov, and V. O. Galkin, Properties of heavy quarkonia andB c mesons in the relativistic quark model, Phys. Rev. D67, 014027 (2003), arXiv:hep-ph/0210381

    work page arXiv 2003

  31. [31]

    Ebert, R

    D. Ebert, R. N. Faustov, and V. O. Galkin, Spectroscopy and Regge trajectories of heavy quarkonia andB c mesons, Eur. Phys. J. C71, 1825 (2011), arXiv:1111.0454 [hep-ph]

    work page arXiv 2011

  32. [32]

    Godfrey,Spectroscopy ofB c mesons in the relativized quark model,Phys

    S. Godfrey, Spectroscopy ofB c mesons in the relativized quark model, Phys. Rev. D70, 054017 (2004), arXiv:hep-ph/0406228

    work page arXiv 2004

  33. [33]

    G.-L. Wang, T. Wang, Q. Li, and C.-H. Chang, The mass spectrum and wave functions of the Bc system, JHEP05, 006, arXiv:2201.02318 [hep-ph]

    work page arXiv

  34. [34]

    Li, Y.-S

    X.-J. Li, Y.-S. Li, F.-L. Wang, and X. Liu, Spectroscopic survey of higher-lying states ofB c meson family, Eur. Phys. J. C83, 1080 (2023), arXiv:2308.07206 [hep-ph]

    work page arXiv 2023

  35. [35]

    A. P. Monteiro, M. Bhat, and K. B. Vijaya Kumar,c ¯bspectrum and decay properties with coupled channel effects, Phys. Rev. D95, 054016 (2017), arXiv:1608.05782 [hep-ph]

    work page arXiv 2017

  36. [36]

    Molina, M

    D. Molina, M. De Sanctis, C. Fern´ andez-Ram´ ırez, and E. Santopinto, Bottomonium spec- trum with a Dirac potential model in the momentum space, Eur. Phys. J. C80, 526 (2020), arXiv:2001.05408 [hep-ph]

    work page arXiv 2020

  37. [37]

    Spin-averaged $B_c$ Spectrum in a Cornell-type Potential Using VMC Baseline and GFMC Evolution

    T. Akan, Spin-averagedB c Spectrum in a Cornell-type Potential Using VMC Baseline and GFMC Evolution, (2025), arXiv:2511.10986 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  38. [38]

    Eichten, K

    E. Eichten, K. Gottfried, T. Kinoshita, J. B. Kogut, K. D. Lane, and T.-M. Yan, The Spectrum of Charmonium, Phys. Rev. Lett.34, 369 (1975), [Erratum: Phys.Rev.Lett. 36, 1276 (1976)]

    1975

  39. [39]

    Eichten, K

    E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T.-M. Yan, Charmonium: Comparison with Experiment, Phys. Rev. D21, 203 (1980)

    1980

  40. [40]

    Brambilla, S

    N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, TheXY Zstates: experimental and theoretical status and perspectives, Phys. Rept.873, 1 (2020), arXiv:1907.07583 [hep-ex]

    work page arXiv 2020

  41. [41]

    Eichten and F

    E. Eichten and F. Feinberg, Spin Dependent Forces in QCD, Phys. Rev. D23, 2724 (1981)

    1981

  42. [42]

    Gromes, Spin Dependent Potentials in QCD and the Correct Long Range Spin Orbit Term, Z

    D. Gromes, Spin Dependent Potentials in QCD and the Correct Long Range Spin Orbit Term, Z. Phys. C26, 401 (1984)

    1984

  43. [43]

    De Rujula, H

    A. De Rujula, H. Georgi, and S. L. Glashow, Hadron Masses in a Gauge Theory, Phys. Rev. D12, 147 (1975)

    1975

  44. [44]

    Godfrey and N

    S. Godfrey and N. Isgur, Mesons in a Relativized Quark Model with Chromodynamics, Phys. 29 Rev. D32, 189 (1985)

    1985

  45. [45]

    Kwong, J

    W. Kwong, J. L. Rosner, and C. Quigg, Heavy Quark Systems, Ann. Rev. Nucl. Part. Sci.37, 325 (1987)

    1987

  46. [46]

    Brambillaet al.(Quarkonium Working Group), Heavy Quarkonium Physics 10.5170/CERN-2005-005 (2004), arXiv:hep-ph/0412158

    N. Brambillaet al.(Quarkonium Working Group), Heavy Quarkonium Physics 10.5170/CERN-2005-005 (2004), arXiv:hep-ph/0412158

    work page doi:10.5170/cern-2005-005 2005

  47. [47]

    Lucha and F

    W. Lucha and F. F. Schoberl, Effective potential models for hadrons, inInternational Summer School for Students on Development in Nuclear Theory and Particle Physics(1995) arXiv:hep- ph/9601263

    work page arXiv 1995

  48. [48]

    Godfrey and R

    S. Godfrey and R. Kokoski, The Properties of p Wave Mesons with One Heavy Quark, Phys. Rev. D43, 1679 (1991)

    1991

  49. [49]

    McClary and N

    R. McClary and N. Byers, Relativistic Effects in Heavy Quarkonium Spectroscopy, Phys. Rev. D28, 1692 (1983)

    1983

  50. [50]

    Eichten and F

    E. Eichten and F. L. Feinberg, Spin Dependent Forces in Heavy Quark Systems, Phys. Rev. Lett.43, 1205 (1979)

    1979

  51. [51]

    Barnes, S

    T. Barnes, S. Godfrey, and E. S. Swanson, Higher charmonia, Phys. Rev. D72, 054026 (2005), arXiv:hep-ph/0505002

    work page arXiv 2005

  52. [52]

    S. M. Ikhdair and R. Sever,B c and heavy meson spectroscopy in the local approximation of the Schrodinger equation with relativistic kinematics, Int. J. Mod. Phys. A20, 4035 (2005), arXiv:hep-ph/0403280

    work page arXiv 2005

  53. [53]

    C. Mony A. and R. Dhir, Spectroscopy and annihilation decay widths of charmonium and bottomonium excitations, J. Phys. G51, 115004 (2024), arXiv:2311.05274 [hep-ph]

    work page arXiv 2024

  54. [54]

    emcee: The MCMC Hammer

    D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, emcee: The MCMC Hammer, Publ. Astron. Soc. Pac.125, 306 (2013), arXiv:1202.3665 [astro-ph.IM]

    work page internal anchor Pith review arXiv 2013

  55. [55]

    Goodman and J

    J. Goodman and J. Weare, Ensemble samplers with affine invariance, Commun. Appl. Math. Comput. Sc.5, 65 (2010)

    2010

  56. [56]

    Mathur, M

    N. Mathur, M. Padmanath, and S. Mondal, Precise predictions of charmed-bottom hadrons from lattice QCD, Phys. Rev. Lett.121, 202002 (2018), arXiv:1806.04151 [hep-lat]

    work page arXiv 2018

  57. [57]

    Foreman-Mackey, corner.py: Scatterplot matrices in python, The Journal of Open Source Software1, 24 (2016)

    D. Foreman-Mackey, corner.py: Scatterplot matrices in python, The Journal of Open Source Software1, 24 (2016)

    2016

  58. [58]

    Regge, Introduction to complex orbital momenta, Nuovo Cim.14, 951 (1959)

    T. Regge, Introduction to complex orbital momenta, Nuovo Cim.14, 951 (1959)

    1959

  59. [59]

    Regge, Bound states, shadow states and Mandelstam representation, Nuovo Cim.18, 947 30 (1960)

    T. Regge, Bound states, shadow states and Mandelstam representation, Nuovo Cim.18, 947 30 (1960)

    1960

  60. [60]

    G. F. Chew and S. C. Frautschi, Principle of Equivalence for All Strongly Interacting Particles Within the S Matrix Framework, Phys. Rev. Lett.7, 394 (1961)

    1961

  61. [61]

    G. F. Chew and S. C. Frautschi, Regge Trajectories and the Principle of Maximum Strength for Strong Interactions, Phys. Rev. Lett.8, 41 (1962)

    1962

  62. [62]

    H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, and S.-L. Zhu, A review of the open charm and open bottom systems, Rept. Prog. Phys.80, 076201 (2017), arXiv:1609.08928 [hep-ph]

    work page arXiv 2017

  63. [63]

    Martin, Regge Trajectories in the Quark Model, Z

    A. Martin, Regge Trajectories in the Quark Model, Z. Phys. C32, 359 (1986)

    1986

  64. [64]

    Lucha, F

    W. Lucha, F. F. Schoberl, and D. Gromes, Bound states of quarks, Phys. Rept.200, 127 (1991)

    1991

  65. [65]

    Ebert, R

    D. Ebert, R. N. Faustov, and V. O. Galkin, Spectroscopy and Regge trajectories of heavy quarkonia in the relativistic quark model, Phys. Atom. Nucl.76, 1554 (2013)

    2013

  66. [66]

    Chen, Regge trajectories for heavy quarkonia from the quadratic form of the spinless Salpeter-type equation, Eur

    J.-K. Chen, Regge trajectories for heavy quarkonia from the quadratic form of the spinless Salpeter-type equation, Eur. Phys. J. C78, 235 (2018)

    2018

  67. [67]

    Feng, J.-K

    X. Feng, J.-K. Chen, and J.-Q. Xie, Regge trajectories for the doubly heavy diquarks, Phys. Rev. D108, 034022 (2023), arXiv:2305.15705 [hep-ph]

    work page arXiv 2023

  68. [68]

    Chen, Regge trajectory relation for the universal description of the heavy-heavy sys- tems: Diquarks, mesons, baryons and tetraquarks, Nucl

    J.-K. Chen, Regge trajectory relation for the universal description of the heavy-heavy sys- tems: Diquarks, mesons, baryons and tetraquarks, Nucl. Phys. A1050, 122927 (2024), arXiv:2302.05926 [hep-ph]

    work page arXiv 2024

  69. [69]

    , title=

    H. Dembinski and P. O. et al., scikit-hep/iminuit 10.5281/zenodo.3949207 (2020)

    work page doi:10.5281/zenodo.3949207 2020

  70. [70]

    James and M

    F. James and M. Roos, Minuit: A System for Function Minimization and Analysis of the Parameter Errors and Correlations, Comput. Phys. Commun.10, 343 (1975)

    1975

  71. [71]

    R. J. Dowdall, C. T. H. Davies, T. C. Hammant, and R. R. Horgan, Precise heavy-light meson masses and hyperfine splittings from lattice QCD including charm quarks in the sea, Phys. Rev. D86, 094510 (2012), arXiv:1207.5149 [hep-lat]

    work page arXiv 2012

  72. [72]

    C. T. H. Davies, K. Hornbostel, G. P. Lepage, A. J. Lidsey, J. Shigemitsu, and J. H. Sloan, B(c) spectroscopy from lattice QCD, Phys. Lett. B382, 131 (1996), arXiv:hep-lat/9602020. 31 TABLE III. Potential parameters inB c system. Parameters Potential I Potential II αs 0.378+0.056 −0.028 0.388+0.051 −0.029 σ(in GeV 2) 0.187 +0.01 −0.02 0.15+0.024 −0.028 ρ(...

    work page arXiv 1996