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arxiv: 2512.16637 · v3 · pith:76STAN35new · submitted 2025-12-18 · ✦ hep-ph

Probing Excited qbar{q} Mesons via QCD Sum Rules

Shuang-Hong Li , Wei-Yang Lai , Hong-Ying Jin This is my paper

Pith reviewed 2026-05-21 17:32 UTC · model grok-4.3

classification ✦ hep-ph
keywords excitedrulesseveralcovariantderivativesexperimentsgaussianmasses
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The pith

QCD sum rules at NLO with derivative-inserted currents yield masses for several J^P=2± nonets and J=0,1 states that match experimental values.

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

Quantum chromodynamics describes how quarks bind into mesons, but excited states are hard to calculate directly. This work builds special mathematical operators called interpolating currents that include extra derivative terms to better overlap with higher-energy meson states. They apply the operator product expansion up to dimension-eight terms, add next-to-leading-order perturbative corrections, and use both Gaussian and Laplace sum rules to extract masses. For states with total angular momentum and parity 2+ and 2-, the calculated masses line up with known experimental resonances. The same framework also produces some J=0 and J=1 candidates. The key technical step is showing that one particular current can couple to two different 2++ resonances. This demonstrates that adding derivatives to the currents gives a workable way to study excited hadrons without needing full lattice simulations.

Core claim

Employing Gaussian sum rules, we obtain several J^P=2± nonets with masses that agree well with experiments. Several J=0,1 states compatible with experiments are also obtained using both Gaussian and Laplace sum rules.

Load-bearing premise

The assumption that the chosen interpolating currents with inserted covariant derivatives have sufficient overlap with the desired excited states and that higher-dimensional condensates beyond dimension 8 can be neglected without spoiling the mass extraction for these states.

Figures

Figures reproduced from arXiv: 2512.16637 by Hong-Ying Jin, Shuang-Hong Li, Wei-Yang Lai.

Figure 1
Figure 1. Figure 1: FIG. 1: The diagrams involved in the renormalization ←→ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Diagrams for perturbative contributions. The last two diagrams are related to the counterterms. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Diagrams for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Diagrams for [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: NLO GSR mass predictions for [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The two-resonance GSR fitting results of the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: NLO LSR mass predictions [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: NLO LSR mass predictions for [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Condensates’ contributions after Gaussian [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: GSR mass prediction versus [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
read the original abstract

We present a systematic study of the masses of light excited $q\bar{q}$ mesons using QCD sum rules at next-to-leading order (NLO). To probe excited states, we construct several interpolating currents with covariant derivatives inserted. The calculation is carried out up to dimension-8 condensates, including NLO perturbative and $m\langle\bar{q}q\rangle$ corrections. Employing Gaussian sum rules, we obtain several $J^P=2^\pm$ nonets with masses that agree well with experiments. Several $J=0,1$ states compatible with experiments are also obtained using both Gaussian and Laplace sum rules. In particular, the $J^{PC}=2^{++}$ current couples to two distinct $2^{++}$ resonances. This work demonstrates the efficacy of operators with covariant derivatives for studying excited hadrons.

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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.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The calculation rests on the standard operator product expansion truncated at dimension 8, on numerical values of quark and gluon condensates taken from earlier literature, and on the choice of Borel window and continuum threshold for each channel; these are the main external inputs not derived inside the paper.

free parameters (3)
  • continuum threshold s0
    Standard sum-rule parameter adjusted per channel to isolate the ground or excited state contribution.
  • Borel mass M^2
    Stability parameter whose window is chosen to balance OPE convergence and ground-state dominance.
  • quark and gluon condensate values
    Numerical inputs taken from prior phenomenological determinations rather than computed ab initio in this work.
axioms (2)
  • domain assumption Validity of the operator product expansion for the chosen currents at the relevant Euclidean momenta.
    Invoked when the sum rules are written and truncated at dimension 8.
  • domain assumption Quark-hadron duality with a simple continuum model above the threshold.
    Used to subtract the continuum contribution in both Gaussian and Laplace sum rules.

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    Relation between the paper passage and the cited Recognition theorem.

    Employing Gaussian sum rules, we obtain several J^P=2± nonets with masses that agree well with experiments... The calculation is carried out up to dimension-8 condensates, including NLO perturbative and m⟨q̄q⟩ corrections.

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Forward citations

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Works this paper leans on

44 extracted references · 44 canonical work pages · cited by 1 Pith paper · 5 internal anchors

  1. [1]

    +θ(t−s 0)ρ(t),(19) the normalized GSR in Eq. (16d) then becomes eG(s, τ) = 1√ 4πτ h r e− (s−m2 1 )2 4τ + (1−r)e − (s−m2 2 )2 4τ i ,(20) whereris the relative coupling strength r= f2 1 f2 1 +f 2 2 .(21) χ2 3.0 3.5 4.0 4.5 5.0 10-9 10-8 s0(GeV2) m (GeV) 3.0 3.5 4.0 4.5 5.0 1.2 1.4 1.6 1.8 2.0 2.2 m1 m2 s0(GeV2) r 3.0 3.5 4.0 4.5 5.0 0.0 0.1 0.2 0.3 0.4 0.5 ...

    work page 1980

  2. [2]

    + 4m2Q6 −4m 3Q5 −2 γαµνgβρ +γ αβν gµρ 2Q8 1 +Q 8 3 + 2Q8 5 + 4m(Q7 1 −2Q 7 2 −2Q 7

    work page

  3. [3]

    + 4m2Q6 −4m 3Q5 + 2 γβµν gαρ +γ αβµgνρ −2Q8 1 −4Q 8 2 +Q 8 3 + 2Q8 5 + 4Q8 6 −4m(Q 7 1 −2Q 7 2)−4m 2Q6 + 4m3Q5 + 2γανρgβµ −2Q8 1 + 2Q8 2 +Q 8 3 −2Q 8 4 + 2Q8 6 −2m(2Q 7 1 −7Q 7 2 + 2Q7 4)−4m 2Q6 + 4m3Q5 + 2 γβνρ gαµ +γ αµρgβν −2Q8 1 −2Q8 2 + 3Q8 3 −2Q8 4 + 4Q8 5 + 2Q8 6 −2m(2Q7 1 −Q7 2 + 4Q7 3 + 2Q7 4)−4m2Q6 + 4m3Q5 + 2γβµρgαν −2Q8 1 + 2Q8 2 −5Q 8 3 −2Q 8...

    work page

  4. [4]

    + 28m2Q6 + 12m3Q5 −12m 5Q3 +γ ρgανgβµ 8Q8 1 + 2Q8 2 + 7Q8 3 + 10Q8 4 + 20Q8 5 −20Q 8 6 −4m(7Q 7 1 −Q 7 2 −3Q 7 3 +Q 7

    work page

  5. [5]

    + 44m2Q6+12m3Q5 −12m5Q3 +γ ρgαµgβν 16Q8 1 + 2Q8 2 −5Q 8 3 + 10Q8 4 + 12Q8 5 −20Q 8 6 + 4m(Q7 1 +Q 7 2 + 3Q7 3 −Q 7

    work page

  6. [6]

    + 44m2Q6+12m3Q5 −12m5Q3 − γµgαρgβν+γβgαµgνρ 8Q8 1+2Q8 2+Q8 3+2Q8 4+8Q8 5 −8Q8 6+4m(11Q7 1 −2Q7 2 −3Q7 3+3Q7 4)−12m2Q6−12m3Q5+12m5Q3 − γαgβρgµν +γ νgαβgµρ 8Q8 1 −2Q8 2+5Q8 3 −2Q8 4 −4Q8 5+12Q8 6 −4m(Q7 1+Q7 2+3Q7 3+3Q7 4)+4m2Q6−12m3Q5+12m5Q3 −γ ρgαβgµν 24Q8 1 −18Q8 2 + 9Q8 3 + 6Q8 4 + 12Q8 5 + 4Q8 6 + 4m(3Q7 1 + 3Q7 2 −3Q7 3 +Q 7 4)+20m2Q6 −12m3Q5+12m5Q3 +...

    work page

  7. [7]

    [27] are adopted

    + 4(7m2Q6 + 3m3Q5 −3m5Q3) −g αν γµgβρ +γ βgµρ 2Q8 2+Q8 3 + 2Q8 4 + 8Q8 5 −8Q8 6+4m(3Q7 1 −2Q7 2 −3Q7 3+3Q7 4)−12(m 2Q6+m3Q5 −m5Q3) i ji − iA 4(d−4)(d−3)(d−2)(d−1)d γ[µγνγργαγβ] ji.(B2) Here, the condensate notations in Ref. [27] are adopted. In the last line of Eq. (B2), both the term γ[µγνγργαγβ] and the denominator vanish at dimension- 4; therefore, we ...

    work page

  8. [8]

    Chen, Z.-R

    Z.-S. Chen, Z.-R. Huang, H.-Y. Jin, T. Steele, and Z.-F. Zhang, Mixing of X and Y states from QCD sum rules analysis *, Chinese Physics C46, 063102 (2022)

    work page 2022

  9. [9]

    Takizawa, S

    M. Takizawa, S. Takeuchi, and K. Shimizu, The Charmonium-molecule hybrid structure of the X(3872), AIP Conference Proceedings1388, 348 (2011)

    work page 2011

  10. [10]

    XYZ States

    W. Chen, W.-Z. Deng, J. He, N. Li, X. Liu, Z.-G. Luo, Z.-F. Sun, and S.-L. Zhu, XYZ States (2013), arXiv:1311.3763 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  11. [11]

    R. D. Matheus, F. S. Navarra, M. Nielsen, and C. M. Zanetti, QCD sum rules for theX(3872)as a mixed molecule-charmonium state, Phys. Rev. D80, 056002 (2009)

    work page 2009

  12. [12]

    QCD Sum Rules, a Modern Perspective

    P. COLANGELO and A. KHODJAMIRIAN, QCD SUM RULES, A MODERN PERSPECTIVE, inAt The Fron- tier of Particle Physics(WORLD SCIENTIFIC, 2001) pp. 1495–1576, hep-ph/0010175v1

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  13. [13]

    C. A. Dominguez,Quantum Chromodynamics Sum Rules, SpringerBriefs in Physics (Springer Cham, 2018)

    work page 2018

  14. [14]

    A. V. Radyushkin, Introduction to QCD Sum Rule Ap- proach (2001), arXiv:hep-ph/0101227 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  15. [15]

    An Introduction to Sum Rules in QCD

    E. de Rafael, An Introduction to Sum Rules in QCD (1998), arXiv:hep-ph/9802448 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  16. [16]

    Narison and E

    S. Narison and E. de Rafael, On QCD sum rules of the Laplace transform type and light quark masses, Physics Letters B103, 57 (1981)

    work page 1981

  17. [17]

    Reinders, S

    L. Reinders, S. Yazaki, and H. Rubinstein, L = 1 light quark mesons in QCD, Nuclear Physics B196, 125 (1982)

    work page 1982

  18. [18]

    Reinders, H

    L. Reinders, H. Rubinstein, and S. Yazaki, Hadron prop- erties from QCD sum rules, Physics Reports127, 1 (1985)

    work page 1985

  19. [19]

    Kim and S

    H. Kim and S. H. Lee, QCD sum rule forΛ(1405), Zeitschrift für Physik A Hadrons and Nuclei357, 425 (1997)

    work page 1997

  20. [20]

    A. L. Kataev, QCD Sum Rules and Radial Excitations of Light Pseudoscalar and Scalar Mesons, Physics of Atomic Nuclei68, 567 (2005)

    work page 2005

  21. [21]

    S. H. Lee, K. Morita, and K. Ohnishi, Sigma meson in QCD sum rules using a two-quark current with deriva- tives, Phys. Rev. D79, 074025 (2009)

    work page 2009

  22. [22]

    L. Liu, G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, P. Vilaseca, J. J. Dudek, R. G. Edwards, B. Joó, D. G. Richards, and F. t. H. S. collaboration, Excited and ex- otic charmonium spectroscopy from lattice QCD, Journal of High Energy Physics2012, 126 (2012)

    work page 2012

  23. [23]

    and and and, PossibleJP C = 0+− exotic states, Chinese Physics C37, 033104 (2013)

    work page 2013

  24. [24]

    Su and H.-X

    N. Su and H.-X. Chen,S- andP-wave fully strange tetraquark states from QCD sum rules, Phys. Rev. D 106, 014023 (2022)

    work page 2022

  25. [25]

    T. M. Aliev, S. Bilmis, and M. Savci, Analysis of the light J P = 3− mesons in QCD sum rules, Phys. Rev. D108, 034020 (2023)

    work page 2023

  26. [26]

    Tan, W.-Y

    W.-H. Tan, W.-Y. Liu, H.-Z. Xi, and H.-X. Chen, QCD sum rule study of excited light meson operators, Phys. Rev. D112, 054040 (2025)

    work page 2025

  27. [27]

    Lorcé, Cédric, The gauge-invariant canonical energy- momentum tensor, EPJ Web of Conferences112, 01013 (2016)

    work page 2016

  28. [28]

    A.Freese,Noether’stheoremsandtheenergy-momentum tensor in quantum gauge theories, Phys. Rev. D106, 125012 (2022)

    work page 2022

  29. [29]

    Wikipedia contributors, Gordon decomposition — Wikipedia, The Free Encyclopedia (2025), [Online; ac- cessed 21-August-2025]

    work page 2025

  30. [30]

    Gleisberg, F

    T. Gleisberg, F. Krauss, K. T. Matchev, A. Schälicke, S. Schumann, and G. Soff, Helicity formalism for spin- 2 particles, Journal of High Energy Physics2003, 001 (2003)

    work page 2003

  31. [31]

    Jafarzade, A

    S. Jafarzade, A. Koenigstein, and F. Giacosa, Phe- nomenology ofJ P C = 3 −− tensor mesons, Phys. Rev. D103, 096027 (2021)

    work page 2021

  32. [32]

    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)

    work page 2024

  33. [34]

    GROZIN, METHODS OF CALCULATION OF HIGHER POWER CORRECTIONS IN QCD, Interna- tional Journal of Modern Physics A10, 3497 (1995)

    A. GROZIN, METHODS OF CALCULATION OF HIGHER POWER CORRECTIONS IN QCD, Interna- tional Journal of Modern Physics A10, 3497 (1995)

    work page 1995

  34. [35]

    Man, S.-Q

    Z.-L. Man, S.-Q. Luo, and X. Liu, Is the3S−2Dmixing strong for the charmoniaψ(4040)andψ(4160)?, Phys. Rev. D112, 074025 (2025)

    work page 2025

  35. [36]

    Nesterenko and A

    V. Nesterenko and A. Radyushkin, Local quark-hadron duality and nucleon form factors in QCD, Physics Letters B128, 439 (1983)

    work page 1983

  36. [37]

    Bertlmann, G

    R. Bertlmann, G. Launer, and E. de Rafael, Gaussian sum rules in quantum chromodynamics and local duality, Nuclear Physics B250, 61 (1985)

    work page 1985

  37. [38]

    Orlandini, T

    G. Orlandini, T. Steele, and D. Harnett, Gaussian sum- rules and prediction of resonance properties, Nuclear Physics A686, 261 (2001)

    work page 2001

  38. [39]

    Gubler,A Bayesian Analysis of QCD Sum Rules, Springer Theses (Springer Tokyo, 2013)

    P. Gubler,A Bayesian Analysis of QCD Sum Rules, Springer Theses (Springer Tokyo, 2013)

    work page 2013

  39. [40]

    J. Ho, R. Berg, T. G. Steele, W. Chen, and D. Harnett, Is theY(2175)a strangeonium hybrid meson?, Phys. Rev. D100, 034012 (2019)

    work page 2019

  40. [41]

    J. Ho, R. Berg, T. G. Steele, W. Chen, and D. Harnett, Masscalculationsoflightquarkonium, exoticJ P C = 0+− hybrid mesons from Gaussian sum rules, Phys. Rev. D 98, 096020 (2018)

    work page 2018

  41. [42]

    Ioffe, V

    B. Ioffe, V. Fadin, and L. Lipatov,Quantum Chromody- namics: Perturbative and Nonperturbative Aspects,Cam- bridge Monographs on Particle Physics, Nuclear Physics and Cosmology (Cambridge University Press, 2010)

    work page 2010

  42. [43]

    Narison, Mini-review on QCD spectral sum rules, Nu- clear and Particle Physics Proceedings258-259, 189 (2015), 17th International Conference in Quantum Chro- modynamics (QCD 14)

    S. Narison, Mini-review on QCD spectral sum rules, Nu- clear and Particle Physics Proceedings258-259, 189 (2015), 17th International Conference in Quantum Chro- modynamics (QCD 14)

    work page 2015

  43. [44]

    "GammaMaP" - A Mathematica Package for Clifford Algebras, Gamma Matrices and Spinors

    P. Kuusela, "GammaMaP" - A Mathematica Package for Clifford Algebras, Gamma Matrices and Spinors (2019), arXiv:1905.00429 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  44. [45]

    Ioffe and K

    B. Ioffe and K. Zyablyuk, The V-A sum rules and the operator product expansion in complexq2-plane fromτ- decay data, Nuclear Physics A687, 437 (2001)

    work page 2001