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arxiv: 2510.22870 · v4 · submitted 2025-10-26 · ✦ hep-ph · hep-th

Mass and Decay-Constant Evolution of Heavy Quarkonia and B_c States from Thermal QCD Sum Rules

Enis Yazici This is my paper

Pith reviewed 2026-05-18 03:54 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords thermal QCD sum rulesheavy quarkoniaB_c mesonmass evolutiondecay constantsfinite temperaturequark-gluon plasma
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The pith

Heavy quarkonia and B_c states lose mass and decay strength with rising temperature in a binding-energy hierarchy.

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

The paper applies finite-temperature QCD sum rules to follow the masses and decay constants of the vector mesons J/ψ, Υ, and B_c as temperature increases toward the critical point. Zero-temperature sum rules are first tuned to match experimental masses and reference decay constants, after which the same framework is used to predict the thermal evolution up to roughly 90 percent of the critical temperature. The central result is a clear suppression hierarchy near the critical temperature—Υ least affected, J/ψ more suppressed, and B_c most suppressed—together with a predicted 1P–1S splitting of 0.477 GeV for the B_c system that matches LHCb data on excited states.

Core claim

Within the leading-order plus dimension-4 condensate framework of thermal QCD sum rules, calibrated at zero temperature to experimental and LHCb values and employing a vacuum-stability-constrained temperature-dependent continuum threshold together with lattice-informed gluon condensates, the masses and decay constants of J/ψ, Υ, and B_c evolve such that near the critical temperature the relative suppression of the decay constants follows the order Υ < J/ψ < B_c, consistent with their binding energies; the same analysis yields a 1P–1S mass splitting of 0.477 GeV for the B_c system.

What carries the argument

Finite-temperature QCD sum rules with a temperature-dependent continuum threshold constrained by vacuum stability and limited to leading-order plus dimension-4 condensates.

If this is right

  • The suppression hierarchy is consistent with binding energies and lattice spectral trends.
  • The predicted 0.477 GeV 1P–1S splitting for B_c aligns with LHCb observations of orbitally excited states.
  • The results supply a coherent finite-temperature baseline for later inclusion of radiative, higher-dimensional, and width effects.

Where Pith is reading between the lines

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

  • The hierarchy suggests B_c states may dissociate earlier than pure bottomonium in the quark-gluon plasma formed in heavy-ion collisions.
  • The same sum-rule setup could be applied to other mixed heavy-flavor systems to map their thermal stability.
  • Direct comparison with full lattice QCD spectral functions would test whether the leading-order plus D=4 truncation remains adequate at high temperature.

Load-bearing premise

That leading-order plus dimension-4 condensates together with a temperature-dependent continuum threshold are sufficient to capture the dominant thermal evolution without higher-order terms or resonance-width corrections.

What would settle it

A lattice spectral-function calculation or heavy-ion data set showing that the decay-constant suppression of B_c is not stronger than that of J/ψ near the critical temperature.

Figures

Figures reproduced from arXiv: 2510.22870 by Enis Yazici.

Figure 1
Figure 1. Figure 1: FIG. 1. Temperature dependence of the masses [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Relative decay constants [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Extracted mass [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

We analyze the thermal behavior of heavy vector and axial-vector mesons ($J/\psi$, $\Upsilon$, and $B_c$) within the finite-temperature QCD sum-rule framework. Using updated PDG-2024 quark masses, modern lattice-informed gluon condensates, and a temperature-dependent continuum threshold constrained by vacuum stability, we compute the evolution of the masses $m(T)$ and decay constants $f(T)$ up to $T/T_c \lesssim 0.9$. At $T=0$ the sum rules are calibrated to reproduce the experimental and LHCb masses and reference decay constants within the expected $\mathcal{O}(10\%)$ accuracy of a leading-order $+$ $D{=}4$ phenomenological analysis. The subsequent finite-temperature evolution should therefore be interpreted as a calibrated model prediction within this framework rather than as a fully parameter-free determination. Near the critical temperature, the relative suppression follows a clear hierarchy $\Upsilon < J/\psi < B_c$, consistent with their binding energies and lattice spectral trends. The predicted $1P$--$1S$ splitting for the $B_c$ system, $0.477~\mathrm{GeV}$, is consistent with the LHCb observation of orbitally excited $B_c^{+}$ states. The results provide a coherent finite-temperature baseline for future extensions including radiative, higher-dimensional, and width effects.

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

2 major / 2 minor

Summary. The manuscript applies finite-temperature QCD sum rules to the vector and axial-vector channels of J/ψ, Υ, and B_c states. A leading-order perturbative term plus the dimension-4 gluon condensate is used, together with a temperature-dependent continuum threshold s0(T) fixed by a vacuum-stability constraint. Zero-temperature parameters are calibrated to PDG-2024 masses and reference decay constants; the finite-T evolution is presented as a calibrated model prediction. Near Tc the authors report a suppression hierarchy Υ < J/ψ < B_c and a 1P–1S splitting of 0.477 GeV for the B_c system.

Significance. If robust, the work supplies a coherent phenomenological baseline for the thermal evolution of heavy-quarkonium masses and decay constants that is consistent with binding-energy ordering and existing lattice spectral trends. The explicit calibration to vacuum data and the use of updated quark masses plus lattice-informed condensates are positive features of the analysis.

major comments (2)
  1. [Finite-temperature sum-rule analysis] The central hierarchy Υ < J/ψ < B_c near Tc is obtained after fixing the functional form of s0(T) by the vacuum-stability requirement. The manuscript does not quantify how alternative parametrizations (linear versus exponential drop, or different stability windows) shift the extracted m(T) and f(T) for each state; because this choice is shared across channels and directly controls the Borel-window stability, it is load-bearing for the reported ordering.
  2. [Truncation and systematic uncertainties] The truncation to leading-order perturbative contributions plus only the D=4 gluon condensate is stated to carry sizable systematic uncertainty. No estimate is provided of the size of omitted higher-dimensional condensates or finite-width corrections on the temperature evolution of the masses and decay constants; this omission affects the reliability of the quantitative hierarchy and the 0.477 GeV splitting prediction.
minor comments (2)
  1. [Introduction and calibration] The abstract refers to 'reference decay constants'; the main text should explicitly list the numerical values adopted for f at T=0 for each channel.
  2. [Formalism] Notation for the decay constant in the B_c axial-vector channel should be clarified to avoid confusion with the vector channel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment below and indicate the revisions we intend to implement.

read point-by-point responses

  1. Referee: The central hierarchy Υ < J/ψ < B_c near Tc is obtained after fixing the functional form of s0(T) by the vacuum-stability requirement. The manuscript does not quantify how alternative parametrizations (linear versus exponential drop, or different stability windows) shift the extracted m(T) and f(T) for each state; because this choice is shared across channels and directly controls the Borel-window stability, it is load-bearing for the reported ordering.

    Authors: We agree that the parametrization of s0(T) is a key ingredient that influences Borel-window stability and the resulting thermal evolution. Our choice was fixed by the vacuum-stability constraint to maintain consistency across the temperature range and channels. In the revised manuscript we will add a dedicated sensitivity study that examines the effect of alternative forms (linear versus exponential) and modest variations in the stability window. This analysis will quantify the shifts in m(T) and f(T) and demonstrate that the reported suppression hierarchy remains stable within the expected uncertainties of the framework. revision: yes

  2. Referee: The truncation to leading-order perturbative contributions plus only the D=4 gluon condensate is stated to carry sizable systematic uncertainty. No estimate is provided of the size of omitted higher-dimensional condensates or finite-width corrections on the temperature evolution of the masses and decay constants; this omission affects the reliability of the quantitative hierarchy and the 0.477 GeV splitting prediction.

    Authors: Our analysis is performed at leading order with the dimension-4 gluon condensate, as is standard for this phenomenological approach, and the manuscript already characterizes the finite-T results as calibrated model predictions within this truncation. We acknowledge that a more explicit discussion of the omitted higher-dimensional operators and finite-width effects would improve the presentation. In the revision we will include a short subsection that provides order-of-magnitude estimates, drawn from existing literature, for the potential impact of these contributions on the mass and decay-constant evolution near Tc. This will help qualify the quantitative hierarchy and the quoted B_c splitting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard calibration and thermal inputs remain independent

full rationale

The analysis calibrates zero-temperature sum-rule parameters to experimental masses and decay constants in the conventional manner for QCD sum rules, then evolves the system using temperature-dependent gluon condensates taken from lattice QCD and a continuum threshold adjusted to preserve Borel stability. These thermal modifications supply external content not present in the T=0 fit, so the reported mass and decay-constant evolution, including the hierarchy near Tc, does not reduce to the inputs by construction. The paper itself describes the finite-T results as a calibrated model prediction rather than a parameter-free derivation, which is consistent with the method and does not constitute circularity under the stated criteria.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The claim rests on standard QCD sum-rule assumptions plus ad-hoc choices for the thermal continuum threshold and calibration to vacuum data; no new entities are postulated.

free parameters (2)
  • temperature-dependent continuum threshold
    Introduced and constrained by vacuum stability to maintain sum-rule reliability at finite T.
  • gluon condensate values
    Taken from modern lattice calculations as external input.
axioms (2)
  • domain assumption Leading-order plus dimension-4 operator product expansion remains adequate at finite temperature
    Invoked to truncate the sum-rule side of the analysis.
  • ad hoc to paper Vacuum stability constraint determines the temperature dependence of the continuum threshold
    Used to fix the thermal behavior of the threshold parameter.

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 8 internal anchors

  1. [1]

    A. I. Bochkarev and M. E. Shaposhnikov, Nuclear Physics B268, 220 (1986)

    work page 1986

  2. [2]

    C. A. Dominguez and M. Loewe, Phys. Lett. B233, 201 (1989)

    work page 1989

  3. [3]

    Mallik and K

    S. Mallik and K. Sarkar, Phys. Rev. D65, 016002 (2002)

    work page 2002

  4. [4]

    Ayala, C

    A. Ayala, C. A. Dominguez, and M. Loewe, Advances in High Energy Physics2017, 9291623 (2017)

    work page 2017

  5. [5]

    Morita and S

    K. Morita and S. H. Lee, Phys. Rev. Lett.100, 022301 (2008)

    work page 2008

  6. [6]

    E. V. Veliev, K. Azizi, H. Sundu, and N. Aksit, Nucl. Phys. B Proc. Suppl.219-220, 170 (2011)

    work page 2011

  7. [7]

    E. V. Veliev, K. Azizi, H. Sundu, and N. Ak¸ sit, Journal of Physics G: Nuclear and Particle Physics39, 015002 (2012)

    work page 2012

  8. [8]

    E. V. Veliev, K. Azizi, H. Sundu, G. Kaya, and A. T¨ urkan, Eur. Phys. J. A47, 110 (2011), arXiv:1103.4330 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  9. [9]

    Thermal properties of $D_2^*(2460)$ and $D_{s2}^*(2573)$ tensor mesons using QCD sum rules

    K. Azizi, H. Sundu, E. V. Veliev, and A. T¨ urkan, J. Phys. G: Nucl. Part. Phys.41, 035003 (2014), arXiv:1308.0887 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  10. [10]

    Hofmann, T

    R. Hofmann, T. Gutsche, and A. Faessler, Eur. Phys. J. C17, 651 (2000)

    work page 2000

  11. [11]

    Thermal Spectrum of Heavy Vector and Axial Vector Mesons in the Framework of QCD Sum Rules Method

    E. Yazici, arXiv 10.48550/arXiv.1605.05289 (2016), 1605.05289 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1605.05289 2016

  12. [12]

    Navas and others (Particle Data Group), Phys

    S. Navas and others (Particle Data Group), Phys. Rev. D110, 030001 (2024), with 2024 online updates at pdg.lbl.gov

    work page 2024

  13. [13]

    The equation of state in (2+1)-flavor QCD

    A. Bazavov and others (HotQCD Collaboration), Phys. Rev. D90, 094503 (2014), arXiv:1407.6387 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  14. [14]

    Bazavovet al.(HotQCD), Chiral crossover in QCD at zero and non-zero chemical potentials, Phys

    A. Bazavov and others (HotQCD Collaboration), Phys. Lett. B795, 15 (2019), arXiv:1812.08235 [hep-lat]

    work page arXiv 2019

  15. [15]

    Aaij and others (LHCb Collaboration), arXiv (2025), submitted to Phys

    R. Aaij and others (LHCb Collaboration), arXiv (2025), submitted to Phys. Rev. Lett., 2507.02149 [hep-ex]

    work page arXiv 2025

  16. [16]

    S. S. Agaev, K. Azizi, and H. Sundu, arXiv preprint arXiv:2506.19927(2025), arXiv:2506.19927 [hep-ph]

    work page arXiv 2025

  17. [17]

    S. S. Agaev, K. Azizi, and H. Sundu, arXiv preprint 10 arXiv:2507.18735(2025), arXiv:2507.18735 [hep-ph]

    work page arXiv 2025

  18. [18]

    S. S. Agaev, K. Azizi, and H. Sundu, arXiv preprint arXiv:2509.15615(2025), arXiv:2509.15615 [hep-ph]

    work page arXiv 2025

  19. [19]

    Boyd and D

    G. Boyd and D. E. Miller, arXiv (1996), arXiv:hep- ph/9608482

    work page arXiv 1996

  20. [20]

    D. E. Miller, inESI Research Program Workshop on Quantization, Generalized BRS Cohomology and Anoma- lies(1998) arXiv:hep-ph/9907535

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  21. [21]

    P. S. H. Collaboration), Nucl. Phys. A982, 847 (2019), arXiv:1807.05607 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  22. [22]

    Bors´ anyi, S

    S. Bors´ anyi, S. D¨ urr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, S. Mages, D. N´ ogr´ adi, A. P´ asztor, A. Sch¨ afer, K. K. Szab´ o, B. C. T´ oth, and N. Trombit´ as, J. High Energy Phys.2014(04), 132

    work page 2014

  23. [23]

    S. Kim, P. Petreczky, and A. Rothkopf, J. High Energy Phys.2018(11), 088, arXiv:1808.08781 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  24. [24]

    G. Li, B. Chen, and Y. Liu, Chinese Physics C47, 023103 (2023)

    work page 2023

  25. [25]

    Complex heavy-quark potential and Debye mass in a gluonic medium from lattice QCD

    Y. Burnier and A. Rothkopf, Phys. Rev. D95, 054511 (2017), arXiv:1607.04049 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  26. [26]

    Acharya and others (ALICE Collaboration), Phys

    S. Acharya and others (ALICE Collaboration), Phys. Rev. Lett.132, 042301 (2024), arXiv:2210.08893 [hep- ex]. Appendix A: Analytical estimates for OPE convergence and Borel stability

    work page arXiv 2024

  27. [27]

    Dimensional estimate of the OPE convergence To estimate the truncation uncertainty without repeat- ing the full numerical Borel analysis, we evaluate the ra- tio R6/4(M2, T)≡ bBΠ(D=6)(M2, T) bBΠ(D=4)(M2, T) . Using the canonical scalingsbBΠ(D=4) ∼ ⟨αsG2⟩T /M2 and bBΠ(D=6) ∼κΛ 6/M4 with Λ≃0.24 GeV andκ=O(1), and taking⟨α sG2⟩0 = 0.012 GeV 4 together with t...

    work page

  28. [28]

    Borel stability and pole dominance The stability of the working windows summarized in Table I was originally established in Ref. [11]. To ver- ify that the updated lattice-informed thermal inputs pre- serve the Borel plateaus, we have repeated the stability scan atT=0 for all channels. Figure 3 shows the ex- tracted massm(M 2) as a function of the Borel p...

    work page

  29. [29]

    Working range in temperature Combining the dimensional estimateR 6/4 ≲0.1 with the legacy Borel stability criteria leads to the same con- servative domainT≲0.9T c adopted in the main text. Above this temperature, higher-dimensional operators and finite-width effects are expected to become compa- rable to theD=4 contribution, and the results should be inte...

    work page