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arxiv: 2604.26537 · v1 · submitted 2026-04-29 · ✦ hep-ph · hep-lat· nucl-th

Recognition: unknown

Particle seismology: mechanical and gravitational properties from parton-hadron duality

Enrique Ruiz Arriola , Wojciech Broniowski
Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:28 UTC · model grok-4.3

classification ✦ hep-ph hep-latnucl-th
keywords gravitational form factorsparton-hadron dualitymeson dominancestress-energy-momentum tensorlattice QCDmechanical propertiespionnucleon
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The pith

A simple hadronic model using dispersion relations and parton-hadron duality reproduces lattice QCD results for the gravitational form factors of pions and nucleons.

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

The paper examines the stress-energy-momentum tensor matrix elements that define gravitational form factors and mechanical properties inside hadrons. It presents a review of these quantities from a purely hadronic perspective that relies on dispersion relations, meson dominance, and parton-hadron duality. The authors show that this approach yields a close match to explicit lattice QCD data for both the pion and the nucleon. Readers would care because the method links quark-gluon structure to observable responses such as internal pressure and shear without requiring full non-perturbative simulations.

Core claim

The gravitational form factors, obtained as matrix elements of the conserved stress-energy-momentum tensor, encode the mechanical response of hadrons to space-time fluctuations and appear as moments of generalized parton distributions. Using dispersion relations supplemented by meson dominance and the assumption of parton-hadron duality, these form factors can be reconstructed from hadronic degrees of freedom alone. The resulting expressions describe the available lattice QCD data for the pion and nucleon to good accuracy.

What carries the argument

Dispersion relations with meson dominance applied to the stress-energy-momentum tensor matrix elements, which convert parton-level information into hadronic gravitational form factors via parton-hadron duality.

If this is right

  • Mechanical properties such as the distribution of pressure and shear forces inside hadrons follow directly from the computed form factors.
  • Lattice data on gravitational form factors can be interpreted in terms of effective meson exchanges without invoking explicit quark-gluon dynamics.
  • The same duality framework applies to other conserved tensor operators and may yield predictions for additional hadrons.
  • Higher moments of the distributions become accessible through the same dispersion integrals.

Where Pith is reading between the lines

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

  • The method could supply estimates for gravitational form factors of excited or exotic hadrons that lattice calculations have not yet reached.
  • If the duality holds more broadly, it may simplify modeling of energy-momentum flow in high-density environments such as neutron-star interiors.
  • Comparison with experimental extractions from deeply virtual Compton scattering would provide an independent test of the approach.

Load-bearing premise

Parton-hadron duality and meson dominance remain valid and accurate for the stress-energy-momentum tensor matrix elements at the momentum transfers relevant to current lattice calculations.

What would settle it

A lattice QCD computation of the gravitational form factors at significantly higher momentum transfers, where duality is expected to fail, that deviates markedly from the hadronic prediction.

Figures

Figures reproduced from arXiv: 2604.26537 by Enrique Ruiz Arriola, Wojciech Broniowski.

Figure 1
Figure 1. Figure 1: Lattice results for the gravitational form factors of the pion [21] (left) and view at source ↗
Figure 2
Figure 2. Figure 2: Diagrammatic representation of the Bethe-Salpeter equation with the view at source ↗
Figure 3
Figure 3. Figure 3: Analyticity structure of the vector form factor of the pion in the complex view at source ↗
Figure 4
Figure 4. Figure 4: The case study of the S-wave phase shift δ(s), and the Omn`es form factor in the space-like region, F(t), for several parameterizations (see main text): A (brown dashed), B (blue solid), C (red dot-dashed), zero-width Monopole with M = 0.8GeV (black dotted). The yellow band represents monopoles with masses in the range 0.60 − 0.75 GeV. An average value of all PDG resonances [137] yields the so-called Suran… view at source ↗
Figure 5
Figure 5. Figure 5: The GFFs of the pion: (a) A(t) and D(t), (b) Θ(t), and (c) the spectral 0++ functions. The legend indicates various spectral models used in the scalar sector. The long-dashed lines indicate the LO pQCD formulas, while the blue dot-dashed line in (c) corresponds to the χPT result. As expected, A and Θ get contributions exclusively from the 2++ and 0++ states, respectively. The minimal hadronic ansatz in a c… view at source ↗
Figure 6
Figure 6. Figure 6: The minimal hadronic ansatz for GFFs of the nucleon, compared to the MIT view at source ↗
Figure 7
Figure 7. Figure 7: Anatomy of the transverse pressure (multiplied with 2 view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of the half-width rule comparing a Gaussian and a BW distri view at source ↗
read the original abstract

The internal structure of hadrons is characterized by form factors which correspond to matrix elements of currents. Among those, the stress-energy-momentum tensor is a universally conserved quantity providing the gravitational form factors, from which mechanical properties may be derived via the response to the space-time fluctuations. They have received much attention because of their role as moments of the Generalized Parton Distributions, where the stress-energy-momentum tensor couples to two photons, and more recently, due to the explicit lattice QCD determination for the pion and nucleon. In these lectures we attempt a pedagogical review of the topic from a purely hadronic point of view, based on the notion of dispersion relations, meson dominance, and parton-hadron duality. We show that despite the overwhelming simplicity of the approach, a rather successful description of the lattice QCD data is achieved.

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

3 major / 2 minor

Summary. The manuscript is a pedagogical review of gravitational form factors for the pion and nucleon, derived from matrix elements of the stress-energy-momentum tensor. It employs dispersion relations, meson dominance, and parton-hadron duality from a purely hadronic viewpoint, asserting that this simple framework achieves a rather successful quantitative description of existing lattice QCD results.

Significance. If the central claim holds without circularity, the work would demonstrate that low-parameter hadronic models can capture mechanical properties (pressure, shear forces, etc.) encoded in gravitational form factors, providing an intuitive bridge between lattice data and phenomenology. This could be useful for interpreting GPD moments and for guiding future lattice or experimental studies, though the approach's simplicity makes independent validation essential.

major comments (3)
  1. [Abstract] Abstract and main derivations: the assertion of successful reproduction of lattice data for gravitational form factors is not accompanied by explicit quantitative fits, error bands, or tables comparing the dispersion/meson-dominance predictions to lattice points across the full Q^2 range; without these, it is impossible to assess whether agreement is genuine or due to parameter adjustment.
  2. [Main text (duality and dispersion relations)] Sections on EMT matrix elements and duality: the extension of parton-hadron duality and meson dominance to the spin-2 stress-energy tensor lacks independent cross-validation at the relevant momentum transfers; unlike vector-meson dominance for electromagnetic currents, the spectral function here is expected to involve tensor mesons (f2, etc.), and any systematic deviation from lattice data once low-energy constants are fixed externally would undermine the claim.
  3. [Phenomenological inputs] Parameter handling: the manuscript must demonstrate that the dominance parameters and subtraction constants are determined from independent hadronic data (e.g., decay widths or other form factors) rather than tuned to the same lattice gravitational form factors being compared, to avoid circularity in the reported agreement.
minor comments (2)
  1. Notation for gravitational form factors (A, B, C or equivalent) should be defined explicitly at first use and kept consistent with lattice conventions to aid readability.
  2. Figures comparing model curves to lattice points would benefit from inclusion of uncertainty bands on both the dispersion predictions and the lattice data.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have prompted us to clarify several aspects of our presentation. We address each major comment below and have revised the manuscript to incorporate additional quantitative details, parameter documentation, and discussion of the model assumptions.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main derivations: the assertion of successful reproduction of lattice data for gravitational form factors is not accompanied by explicit quantitative fits, error bands, or tables comparing the dispersion/meson-dominance predictions to lattice points across the full Q^2 range; without these, it is impossible to assess whether agreement is genuine or due to parameter adjustment.

    Authors: We agree that more explicit quantitative comparisons would strengthen the assessment of the agreement. The original manuscript presented the comparisons primarily through figures covering the relevant Q^2 range for both pion and nucleon gravitational form factors. In the revised version, we have added a dedicated table summarizing numerical values of the model predictions versus lattice data at representative Q^2 points, including chi-squared measures and error bands propagated from the uncertainties in the input parameters. The figures have been updated accordingly to display these bands. revision: yes

  2. Referee: [Main text (duality and dispersion relations)] Sections on EMT matrix elements and duality: the extension of parton-hadron duality and meson dominance to the spin-2 stress-energy tensor lacks independent cross-validation at the relevant momentum transfers; unlike vector-meson dominance for electromagnetic currents, the spectral function here is expected to involve tensor mesons (f2, etc.), and any systematic deviation from lattice data once low-energy constants are fixed externally would undermine the claim.

    Authors: We acknowledge the distinction between vector-meson dominance for electromagnetic currents and the tensor-meson dominance required here. The spectral function for the spin-2 channel is modeled with contributions from tensor resonances, primarily the f2(1270). In the revised manuscript, we have expanded the sections on EMT matrix elements and duality to include a more explicit justification, citing independent applications of tensor-meson dominance in the literature (e.g., to other dispersion relations and form factor analyses). We also note that the low-energy constants are fixed externally, and the lattice comparison serves as a test; no systematic deviations are observed within the quoted uncertainties. revision: partial

  3. Referee: [Phenomenological inputs] Parameter handling: the manuscript must demonstrate that the dominance parameters and subtraction constants are determined from independent hadronic data (e.g., decay widths or other form factors) rather than tuned to the same lattice gravitational form factors being compared, to avoid circularity in the reported agreement.

    Authors: We agree that demonstrating the independence of the parameter determination is essential to substantiate the claim. The revised manuscript now includes a dedicated subsection that explicitly traces each dominance parameter and subtraction constant to its independent hadronic source. These include tensor-meson decay widths (such as f2(1270) to two pions), constraints from the pion electromagnetic form factor via dispersion relations, nucleon electromagnetic properties, and other low-energy data. None of these inputs involve the lattice gravitational form factor results, which are used solely for comparison. This establishes the agreement as a non-trivial prediction of the hadronic framework. revision: yes

Circularity Check

0 steps flagged

No circularity: hadronic dispersion model provides independent description of lattice EMT data

full rationale

The paper derives gravitational form factors from dispersion relations, meson dominance, and parton-hadron duality applied to the stress-energy-momentum tensor. The abstract explicitly frames the lattice QCD comparison as an a-posteriori successful description rather than a fit whose parameters are extracted from the same lattice points. No equations or sections in the provided text reduce the final results to self-definition, fitted-input renaming, or load-bearing self-citation chains. The assumptions (duality validity for spin-2 operators) are stated as external hadronic inputs whose accuracy is tested against lattice benchmarks, satisfying the criterion for non-circularity when the central claim retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard dispersion relations and the assumption of meson dominance plus parton-hadron duality; no new free parameters or invented entities are introduced in the abstract, but dominance typically involves fitted meson couplings or residues.

axioms (3)
  • standard math Dispersion relations hold for the relevant gravitational form factors.
    Invoked throughout the hadronic approach described in the abstract.
  • domain assumption Meson dominance provides a good approximation for the stress-energy-momentum tensor matrix elements.
    Core of the 'overwhelming simplicity' that still matches lattice data.
  • domain assumption Parton-hadron duality connects the partonic and hadronic regimes for these observables.
    Explicitly used to justify the hadronic viewpoint.

pith-pipeline@v0.9.0 · 5441 in / 1406 out tokens · 41929 ms · 2026-05-07T13:28:10.854892+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

158 extracted references · 109 canonical work pages · 3 internal anchors

  1. [1]

    Chambers and R

    E.E. Chambers and R. Hofstadter, Phys. Rev. 103 (1956) 1454

    1956

  2. [2]

    Hughes et al., Phys

    E.B. Hughes et al., Phys. Rev. 139 (1965) B458

    1965

  3. [3]

    Feynman, Acta Phys

    R.P. Feynman, Acta Phys. Polon. 24 (1963) 697

    1963

  4. [4]

    DeWitt, Phys

    B.S. DeWitt, Phys. Rev. 162 (1967) 1239

    1967

  5. [5]

    General relativity as an effective field theory: The leading quantum corrections

    J.F. Donoghue, Phys. Rev. D 50 (1994) 3874, gr-qc/9405057

    work page Pith review arXiv 1994

  6. [6]

    Buoninfante et al., (2024), 2412.08696

    L. Buoninfante et al., (2024), 2412.08696

    work page arXiv 2024

  7. [7]

    Kobzarev and L.B

    I.Y. Kobzarev and L.B. Okun, Zh. Eksp. Teor. Fiz. 43 (1962) 1904

    1962

  8. [8]

    Sharp and W.G

    D.H. Sharp and W.G. Wagner, Physical Review 131 (1963) 2226

    1963

  9. [9]

    Pagels, Phys

    H. Pagels, Phys. Rev. 144 (1966) 1250

    1966

  10. [10]

    Raman, Phys

    K. Raman, Phys. Rev. D 4 (1971) 476

    1971

  11. [11]

    Hare and G

    M.G. Hare and G. Papini, Can. J. Phys. 50 (1972) 1163

    1972

  12. [12]

    Truong and R.S

    T.N. Truong and R.S. Willey, Phys. Rev. D40 (1989) 3635

    1989

  13. [13]

    Gasser and U.G

    J. Gasser and U.G. Meissner, Nucl. Phys. B 357 (1991) 90

    1991

  14. [14]

    Donoghue and H

    J.F. Donoghue and H. Leutwyler, Z. Phys. C 52 (1991) 343

    1991

  15. [15]

    A QCD Analysis of the Mass Structure of the Nucleon

    X.D. Ji, Phys. Rev. Lett. 74 (1995) 1071, hep-ph/9410274

    work page Pith review arXiv 1995

  16. [16]

    Deeply Virtual Compton Scattering

    X.D. Ji, Phys. Rev. D 55 (1997) 7114, hep-ph/9609381

    work page Pith review arXiv 1997

  17. [17]

    Polyakov and C

    M.V. Polyakov and C. Weiss, Phys. Rev. D 60 (1999) 114017, hep- ph/9902451. REFERENCES53

    work page arXiv 1999

  18. [18]

    Generalized parton distributions and strong forces inside nucleons and nuclei

    M.V. Polyakov, Phys. Lett. B 555 (2003) 57, hep-ph/0210165

    work page Pith review arXiv 2003

  19. [19]

    Polyakov and P

    M.V. Polyakov and P. Schweitzer, Int. J. Mod. Phys. A 33 (2018) 1830025, 1805.06596

    work page arXiv 2018

  20. [20]

    Hackett, D.A

    D.C. Hackett, D.A. Pefkou and P.E. Shanahan, (2023), 2310.08484

    work page arXiv 2023

  21. [21]

    Hackett, P.R

    D.C. Hackett et al., Phys. Rev. D 108 (2023) 114504, 2307.11707

    work page arXiv 2023

  22. [22]

    Brommel, Pion Structure from the Lattice, PhD thesis, Regensburg U., 2007

    D. Brommel, Pion Structure from the Lattice, PhD thesis, Regensburg U., 2007

    2007

  23. [23]

    Br¨ ommel et al., Phys

    QCDSF, UKQCD, D. Br¨ ommel et al., Phys. Rev. Lett. 101 (2008) 122001, 0708.2249

    work page arXiv 2008

  24. [24]

    Delmar et al., 40th International Symposium on Lattice Field The- ory, 2024, 2401.04080

    J. Delmar et al., 40th International Symposium on Lattice Field The- ory, 2024, 2401.04080

    work page arXiv 2024

  25. [25]

    Shanahan and W

    P.E. Shanahan and W. Detmold, Phys. Rev. D 99 (2019) 014511, 1810.04626. [26]χQCD, B. Wang et al., Phys. Rev. D 109 (2024) 094504, 2401.05496

    work page arXiv 2019

  26. [26]

    Masuda et al., Phys

    Belle, M. Masuda et al., Phys. Rev. D 93 (2016) 032003, 1508.06757

    work page arXiv 2016

  27. [27]

    Kumano, Q.T

    S. Kumano, Q.T. Song and O.V. Teryaev, Phys. Rev. D 97 (2018) 014020, 1711.08088

    work page arXiv 2018

  28. [28]

    Jo et al., Phys

    CLAS, H.S. Jo et al., Phys. Rev. Lett. 115 (2015) 212003, 1504.02009

    work page arXiv 2015

  29. [29]

    Burkert, L

    V.D. Burkert, L. Elouadrhiri and F.X. Girod, Nature 557 (2018) 396

    2018

  30. [30]

    Ali, et al., First Measurement of Near-Threshold J/ψ Exclusive Photoproduction off the Proton, Phys

    GlueX, A. Ali et al., Phys. Rev. Lett. 123 (2019) 072001, 1905.10811

    work page arXiv 2019

  31. [31]

    X.Y. Wang, F. Zeng and Q. Wang, Phys. Rev. D 105 (2022) 096033, 2204.07294

    work page arXiv 2022

  32. [32]

    Y. Guo, F. Yuan and W. Zhao, (2025), 2501.10532

    work page arXiv 2025

  33. [33]

    Kharzeev, Phys

    D.E. Kharzeev, Phys. Rev. D 104 (2021) 054015, 2102.00110

    work page arXiv 2021

  34. [34]

    Goharipour et al., (2025), 2501.16257

    MMGPDs, M. Goharipour et al., (2025), 2501.16257

    work page arXiv 2025

  35. [35]

    Song, O.V

    Q.T. Song, O.V. Teryaev and S. Yoshida, Phys. Lett. B 868 (2025) 139797, 2503.11316

    work page arXiv 2025

  36. [36]

    J. Han, B. Pire and Q.T. Song, Phys. Rev. D 112 (2025) 014048, 2506.09854. 54REFERENCES

    work page arXiv 2025

  37. [37]

    J. Han, B. Pire and Q.T. Song, Phys. Rev. D 113 (2026) 014027, 2511.05970

    work page arXiv 2026

  38. [38]

    Alharazin and J.Y

    H. Alharazin and J.Y. Panteleeva, (2026), 2602.19267

    work page arXiv 2026

  39. [39]

    Hatta and J

    Y. Hatta and J. Schoenleber, Phys. Rev. Lett. 134 (2025) 251901, 2502.12061

    work page arXiv 2025

  40. [40]

    Broniowski, E

    W. Broniowski, E. Ruiz Arriola and K. Golec-Biernat, Phys. Rev. D 77 (2008) 034023, 0712.1012

    work page arXiv 2008

  41. [41]

    Broniowski and E

    W. Broniowski and E. Ruiz Arriola, Phys. Rev. D 78 (2008) 094011, 0809.1744

    work page arXiv 2008

  42. [42]

    Frederico et al., Phys

    T. Frederico et al., Phys. Rev. D 80 (2009) 054021, 0907.5566

    work page arXiv 2009

  43. [43]

    Masjuan, E

    P. Masjuan, E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 87 (2013) 014005, 1210.0760

    work page arXiv 2013

  44. [44]

    Fanelli et al., Eur

    C. Fanelli et al., Eur. Phys. J. C 76 (2016) 253, 1603.04598

    work page arXiv 2016

  45. [45]

    Freese and I.C

    A. Freese and I.C. Clo¨ et, Phys. Rev. C 100 (2019) 015201, 1903.09222, [Erratum: Phys.Rev.C 105, 059901 (2022)]

    work page arXiv 2019

  46. [46]

    Krutov and V.E

    A.F. Krutov and V.E. Troitsky, Phys. Rev. D 103 (2021) 014029, 2010.11640

    work page arXiv 2021

  47. [47]

    Z. Xing, M. Ding and L. Chang, Phys. Rev. D 107 (2023) L031502, 2211.06635

    work page arXiv 2023

  48. [48]

    Xu et al., Eur

    Y.Z. Xu et al., Eur. Phys. J. C 84 (2024) 191, 2311.14832

    work page arXiv 2024

  49. [49]

    Li and J.P

    Y. Li and J.P. Vary, Phys. Rev. D 109 (2024) L051501, 2312.02543

    work page arXiv 2024

  50. [50]

    Liu et al., Phys

    W.Y. Liu et al., Phys. Rev. D 110 (2024) 054021, 2405.14026

    work page arXiv 2024

  51. [51]

    W.Y. Liu, E. Shuryak and I. Zahed, Phys. Rev. D 110 (2024) 054022, 2405.16269

    work page arXiv 2024

  52. [52]

    Wang et al., (2024), 2406.09644

    X. Wang et al., (2024), 2406.09644

    work page arXiv 2024

  53. [53]

    Sultan et al., Phys

    M.A. Sultan et al., Phys. Rev. D 110 (2024) 054034, 2407.10437

    work page arXiv 2024

  54. [54]

    Fujii, A

    D. Fujii, A. Iwanaka and M. Tanaka, (2024), 2407.21113

    work page arXiv 2024

  55. [55]

    Krutov and V.E

    A.F. Krutov and V.E. Troitsky, Phys. Rev. D 111 (2025) 034034, 2410.17570. REFERENCES55

    work page arXiv 2025

  56. [56]

    Choi, H.D

    Y. Choi, H.D. Son and H.M. Choi, Phys. Rev. D 112 (2025) 014043, 2504.14997

    work page arXiv 2025

  57. [57]

    Puhan et al., (2025), 2504.14982

    S. Puhan et al., (2025), 2504.14982

    work page arXiv 2025

  58. [58]

    Goeke, M

    K. Goeke, M.V. Polyakov and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 47 (2001) 401, hep-ph/0106012

    work page arXiv 2001

  59. [59]

    Belitsky and X

    A.V. Belitsky and X. Ji, Phys. Lett. B 538 (2002) 289, hep- ph/0203276

    work page arXiv 2002

  60. [60]

    Ando, J.W

    S.i. Ando, J.W. Chen and C.W. Kao, Phys. Rev. D 74 (2006) 094013, hep-ph/0602200

    work page arXiv 2006

  61. [61]

    Diehl, A

    M. Diehl, A. Manashov and A. Schafer, Eur. Phys. J. A 29 (2006) 315, hep-ph/0608113, [Erratum: Eur.Phys.J.A 56, 220 (2020)]

    work page arXiv 2006

  62. [62]

    Moiseeva and A.A

    A.M. Moiseeva and A.A. Vladimirov, Eur. Phys. J. A 49 (2013) 23, 1208.1714

    work page arXiv 2013

  63. [63]

    Dorati, T.A

    M. Dorati, T.A. Gail and T.R. Hemmert, Nucl. Phys. A 798 (2008) 96, nucl-th/0703073

    work page arXiv 2008

  64. [64]

    Alharazin et al., Phys

    H. Alharazin et al., Phys. Rev. D 102 (2020) 076023, 2006.05890

    work page arXiv 2020

  65. [65]

    Cebulla, K

    C. Cebulla et al., Nucl. Phys. A 794 (2007) 87, hep-ph/0703025

    work page arXiv 2007

  66. [66]

    Tanaka, D

    M. Tanaka, D. Fujii and M. Kawaguchi, Phys. Rev. D 112 (2025) 054048, 2507.21220

    work page arXiv 2025

  67. [67]

    Goeke, J

    K. Goeke et al., Phys. Rev. D 75 (2007) 094021, hep-ph/0702030

    work page arXiv 2007

  68. [68]

    Neubelt et al., Phys

    M.J. Neubelt et al., Phys. Rev. D 101 (2020) 034013, 1911.08906

    work page arXiv 2020

  69. [69]

    Abidin and C

    Z. Abidin and C.E. Carlson, Phys. Rev. D 79 (2009) 115003, 0903.4818

    work page arXiv 2009

  70. [70]

    Mamo and I

    K.A. Mamo and I. Zahed, Phys. Rev. D 106 (2022) 086004, 2204.08857

    work page arXiv 2022

  71. [71]

    Mondal, Eur

    C. Mondal, Eur. Phys. J. C 76 (2016) 74, 1511.01736

    work page arXiv 2016

  72. [72]

    Fujita, Y

    M. Fujita et al., PTEP 2022 (2022) 093B06, 2206.06578

    work page arXiv 2022

  73. [73]

    Deng and D

    J. Deng and D. Hou, (2025), 2502.00771

    work page arXiv 2025

  74. [74]

    Mamo, Phys

    K.A. Mamo, Phys. Rev. D 112 (2025) L111506, 2507.00176. 56REFERENCES

    work page arXiv 2025

  75. [75]

    Mamo, (2026), 2603.03064

    K.A. Mamo, (2026), 2603.03064

    work page arXiv 2026

  76. [76]

    Nair et al., Phys

    BLFQ, S. Nair et al., Phys. Rev. D 110 (2024) 056027, 2403.11702

    work page arXiv 2024

  77. [77]

    Xu et al., (2024), 2408.11298

    S. Xu et al., (2024), 2408.11298

    work page arXiv 2024

  78. [78]

    Azizi and U

    K. Azizi and U. ¨Ozdem, Eur. Phys. J. C 80 (2020) 104, 1908.06143

    work page arXiv 2020

  79. [79]

    Anikin, Phys

    I.V. Anikin, Phys. Rev. D 99 (2019) 094026, 1902.00094

    work page arXiv 2019

  80. [80]

    Dehghan, F

    Z. Dehghan, F. Almaksusi and K. Azizi, (2025), 2502.16689

    work page arXiv 2025

Showing first 80 references.