pith. sign in

arxiv: 2605.18350 · v1 · pith:ARJIEIRCnew · submitted 2026-05-18 · ✦ hep-ph · hep-th· nucl-th

Gravitational form factors of light mesons from Basis Light-Front Quantization

Amrita Sain , Sreeraj Nair , Chandan Mondal , Xingbo Zhao , James P. Vary This is my paper

Pith reviewed 2026-05-20 09:43 UTC · model grok-4.3

classification ✦ hep-ph hep-thnucl-th
keywords gravitational form factorspionkaonbasis light-front quantizationD-termmechanical radiienergy-momentum tensorlight mesons
0
0 comments X

The pith

Light-front wave functions produce gravitational form factors for the pion and kaon that match lattice results for A(Q^2) but show an enhanced D-term at low Q^2.

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

The paper computes the gravitational form factors A and D of the pion and kaon from light-front wave functions obtained in the Basis Light-Front Quantization framework. These wave functions solve an effective Hamiltonian that adds three-dimensional confinement to a color-singlet Nambu-Jona-Lasinio interaction between constituent quarks. The resulting A(Q^2) lies in overall agreement with recent lattice QCD and dispersive determinations, while D(Q^2) is larger in magnitude at low Q^2. The enhancement traces to the use of transverse components of the QCD energy-momentum tensor, which are more sensitive to the small-x region and to light-front zero-mode effects within the truncated basis. The same GFFs are then used to extract the mass and mechanical radii of each meson and to map their internal pressure and shear-force distributions.

Core claim

Within the Basis Light-Front Quantization framework the gravitational form factor A(Q^2) of the pion and kaon is found to be in overall agreement with recent lattice QCD and dispersive results. In contrast, D(Q^2) is enhanced in magnitude at low Q^2 relative to both lattice QCD and dispersive determinations. This behavior arises from extracting the D-term using transverse components of the QCD energy-momentum tensor, which are more sensitive to the small-x region and to light-front zero-mode effects in the present truncated framework. Using the resulting GFFs, the mass and mechanical radii of the pion and kaon are determined and their mechanical structure is analyzed through the pressure and

What carries the argument

Basis Light-Front Quantization light-front wave functions solved from an effective Hamiltonian with three-dimensional confinement plus color-singlet Nambu-Jona-Lasinio quark-antiquark interaction.

If this is right

  • Mass and mechanical radii of the pion and kaon follow directly from the computed GFFs.
  • Pressure and shear-force distributions inside each meson can be reconstructed from the GFFs.
  • The D-term enhancement isolates the contribution of small-x physics to the mechanical structure of light mesons.
  • The overall agreement in A(Q^2) provides a consistency check on the light-front wave functions for quantities less sensitive to zero modes.

Where Pith is reading between the lines

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

  • The discrepancy in D(Q^2) suggests that future work could test whether enlarging the basis or restoring zero modes reduces the enhancement toward lattice values.
  • The same framework could be applied to vector mesons or to heavier quarkonia to see whether the transverse-component sensitivity persists across the light-hadron spectrum.
  • If the enhancement survives higher-resolution calculations, it may point to a genuine difference between light-front and Euclidean extractions of the D-term that requires theoretical reconciliation.

Load-bearing premise

The truncated BLFQ basis with the chosen effective Hamiltonian sufficiently captures or correctly represents the small-x region and light-front zero-mode contributions when the D-term is extracted from transverse components of the energy-momentum tensor.

What would settle it

A direct comparison of D(Q^2) obtained from longitudinal versus transverse components of the energy-momentum tensor in a larger-basis BLFQ calculation, or an explicit inclusion of zero-mode contributions, against the same lattice QCD data points.

Figures

Figures reproduced from arXiv: 2605.18350 by Amrita Sain, Chandan Mondal, James P. Vary, Sreeraj Nair, Xingbo Zhao.

Figure 1
Figure 1. Figure 1: Pion GFFs in comparison with lattice QCD results from Ref. [ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Kaon gravitational form factors in comparison with the dispersive analysis of Ref. [ [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pion and kaon gravitational form factors together with their quark flavor decompositions. The left panel displays [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Weighted transverse pressure and shear distributions of the pion and kaon. The four panels display [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Pion and kaon weighted transverse pressure and shear distributions and their quark flavor decompositions. The left [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

We compute the gravitational form factors (GFFs) of the pion and kaon using their light-front wave functions within the Basis Light-Front Quantization framework. The wave functions are obtained by solving a light-front effective Hamiltonian that incorporates three-dimensional confinement along with a color-singlet Nambu--Jona-Lasinio interaction between the constituent quark and antiquark. The form factor $A(Q^2)$ is found to be in overall agreement with recent lattice QCD and dispersive results. In contrast, $D(Q^2)$ is enhanced in magnitude at low $Q^2$ relative to both lattice QCD and dispersive determinations. This behavior arises from extracting the $D$-term using transverse components of the QCD energy--momentum tensor, which are more sensitive to the small-$x$ region and to light-front zero-mode effects in the present truncated framework. Using the resulting GFFs, we determine the mass (matter) and mechanical radii of the pion and kaon and analyze their mechanical structure through the corresponding pressure and shear-force distributions.

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 / 1 minor

Summary. The manuscript computes the gravitational form factors A(Q²) and D(Q²) of the pion and kaon from light-front wave functions obtained by diagonalizing an effective light-front Hamiltonian in the BLFQ framework. The Hamiltonian combines three-dimensional confinement with a color-singlet NJL interaction. A(Q²) is reported to agree overall with lattice QCD and dispersive results, while D(Q²) shows an enhancement in magnitude at low Q². This discrepancy is attributed to the greater sensitivity of transverse components of the QCD energy-momentum tensor to the small-x region and light-front zero-mode effects within the truncated Fock-space basis. The resulting GFFs are used to extract mass and mechanical radii and to analyze pressure and shear-force distributions inside the mesons.

Significance. If the central results hold, the work provides a direct, non-perturbative computation of meson GFFs from solved wave functions without fitting to GFF data itself, which is a methodological strength. The agreement found for A(Q²) lends support to the BLFQ effective Hamiltonian for longitudinal observables, while the D(Q²) behavior illustrates the framework's current limitations for transverse EMT matrix elements. The extraction of radii and mechanical distributions from the GFFs adds phenomenological value, though the interpretation of the D-term enhancement requires further control over truncation effects to be fully convincing.

major comments (1)
  1. [Results and discussion of D(Q²)] The attribution of the low-Q² enhancement in |D(Q²)| to transverse EMT components and their sensitivity to small-x and zero modes (as stated in the abstract and the results discussion) is load-bearing for the paper's interpretation. However, no explicit basis-size convergence study or comparison against a zero-mode-regularized variant is presented for the D-term extraction, even though such checks would be needed to distinguish physical sensitivity from numerical artifact in the truncated model. This is in contrast to the reported agreement for A(Q²), which suggests the model is adequate for other components.
minor comments (1)
  1. [Formalism section] Notation for the transverse components of the EMT and the precise definition of the zero-mode contributions could be clarified with an additional equation or appendix reference to aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the positive assessment of the methodological approach and the identification of areas where additional checks would strengthen the interpretation of the D-term results. We address the major comment below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

  1. Referee: [Results and discussion of D(Q²)] The attribution of the low-Q² enhancement in |D(Q²)| to transverse EMT components and their sensitivity to small-x and zero modes (as stated in the abstract and the results discussion) is load-bearing for the paper's interpretation. However, no explicit basis-size convergence study or comparison against a zero-mode-regularized variant is presented for the D-term extraction, even though such checks would be needed to distinguish physical sensitivity from numerical artifact in the truncated model. This is in contrast to the reported agreement for A(Q²), which suggests the model is adequate for other components.

    Authors: We agree that the interpretation of the observed enhancement in |D(Q²)| at low Q² depends on demonstrating its origin in the truncated Fock-space treatment, and that explicit convergence studies would help separate physical sensitivity from possible numerical effects. The manuscript already notes the greater sensitivity of transverse EMT components to small-x and zero-mode contributions compared with the longitudinal ones that enter A(Q²). To address the referee's concern directly, the revised manuscript will include an additional subsection and accompanying figure that explicitly shows the dependence of both A(Q²) and D(Q²) on the basis truncation parameters K_max and N_max. These checks confirm that A(Q²) converges rapidly while D(Q²) exhibits slower stabilization at low Q², consistent with the small-x sensitivity argument. For zero-mode regularization, our effective Hamiltonian incorporates a three-dimensional confining potential that is designed to suppress unphysical zero-mode contributions; we will expand the discussion to include a brief comparison with results obtained by varying the strength of this term and will reference related light-front studies on zero-mode handling. These additions will be presented without altering the main conclusions or the reported values of the GFFs. revision: yes

Circularity Check

0 steps flagged

No significant circularity: GFFs computed as matrix elements from Hamiltonian-solved wave functions

full rationale

The derivation proceeds by constructing an effective light-front Hamiltonian (3D confinement + color-singlet NJL), tuning its parameters to reproduce meson masses and decay constants, solving for the light-front wave functions in a truncated Fock-space basis, and then evaluating the gravitational form factors directly as matrix elements of the QCD energy-momentum tensor between those wave functions. This yields A(Q^2) and D(Q^2) as model outputs that are compared to external lattice QCD and dispersive results. The paper's attribution of the low-Q^2 D-term enhancement to transverse EMT components' sensitivity to small-x and zero-modes is an interpretive statement about the numerical outcome within the truncated framework, not a definitional reduction or a fitted input renamed as a prediction. No load-bearing self-citation chain, self-definitional loop, or ansatz smuggling is exhibited in the abstract or described procedure; the central results remain independent of the target GFF observables.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central results rest on parameters that define the effective Hamiltonian and on standard assumptions of light-front quantization and effective quark models.

free parameters (2)
  • confinement scale
    Strength parameter of the three-dimensional confinement term in the light-front Hamiltonian.
  • NJL coupling constant
    Coupling strength of the color-singlet Nambu-Jona-Lasinio interaction between constituent quark and antiquark.
axioms (2)
  • domain assumption Light-front quantization provides a valid framework for bound-state wave functions of light mesons.
    Invoked by the choice of Basis Light-Front Quantization.
  • domain assumption The effective Hamiltonian with confinement plus NJL interaction approximates the relevant QCD dynamics for pion and kaon.
    Stated in the abstract as the model used to obtain the wave functions.

pith-pipeline@v0.9.0 · 5733 in / 1384 out tokens · 47395 ms · 2026-05-20T09:43:34.771902+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages · 14 internal anchors

  1. [1]

    Electron Ion Collider: The Next QCD Frontier - Understanding the glue that binds us all

    A. Accardi et al., Eur. Phys. J. A52, 268 (2016), arXiv:1212.1701 [nucl-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  2. [2]

    Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report

    R. Abdul Khalek et al., Nucl. Phys. A1026, 122447 (2022), arXiv:2103.05419 [physics.ins-det]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  3. [3]

    D. P. Anderle et al., Front. Phys. (Beijing)16, 64701 (2021), arXiv:2102.09222 [nucl-ex]

    work page arXiv 2021

  4. [4]

    Accardiet al., Eur

    A. Accardi et al., Eur. Phys. J. A60, 173 (2024), arXiv:2306.09360 [nucl-ex]

    work page arXiv 2024

  5. [5]

    Gross, E

    F. Gross et al., Eur. Phys. J. C83, 1125 (2023), arXiv:2212.11107 [hep-ph]

    work page arXiv 2023

  6. [6]

    M. V. Polyakov and P. Schweitzer, Int. J. Mod. Phys. A 33, 1830025 (2018), arXiv:1805.06596 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  7. [7]

    V. D. Burkert, L. Elouadrhiri, F. X. Girod, C. Lorc´ e, P. Schweitzer, and P. E. Shanahan, Rev. Mod. Phys. 95, 041002 (2023), arXiv:2303.08347 [hep-ph]

    work page arXiv 2023

  8. [8]

    Ji, Phys

    X.-D. Ji, Phys. Rev. D55, 7114 (1997), arXiv:hep- ph/9609381

    work page arXiv 1997

  9. [9]

    A. V. Radyushkin, Phys. Rev. D56, 5524 (1997), arXiv:hep-ph/9704207

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  10. [10]

    Diehl, Generalized parton distributions, Phys

    M. Diehl, Phys. Rept.388, 41 (2003), arXiv:hep- ph/0307382

    work page arXiv 2003

  11. [11]

    Ji, Phys

    X.-D. Ji, Phys. Rev. Lett.78, 610 (1997), arXiv:hep- ph/9603249

    work page arXiv 1997

  12. [12]

    d’Hose, S

    N. d’Hose, S. Niccolai, and A. Rostomyan, Eur. Phys. J. A52, 151 (2016)

    work page 2016

  13. [13]

    GPD phenomenology and DVCS fitting - Entering the high-precision era

    K. Kumericki, S. Liuti, and H. Moutarde, Eur. Phys. J. A52, 157 (2016), arXiv:1602.02763 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  14. [14]

    A. V. Radyushkin, Phys. Lett. B385, 333 (1996), arXiv:hep-ph/9605431

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  15. [15]

    J. C. Collins, L. Frankfurt, and M. Strikman, Phys. Rev. D56, 2982 (1997), arXiv:hep-ph/9611433

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  16. [16]

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

    work page 2018

  17. [17]

    Determining the gluonic gravitational form factors of the proton,

    B. Duran et al., Nature615, 813 (2023), arXiv:2207.05212 [nucl-ex]

    work page arXiv 2023

  18. [18]

    Adhikariet al.[The GlueX Collaboration], Phys

    S. Adhikari et al. (GlueX), Phys. Rev. C108, 025201 (2023), arXiv:2304.03845 [nucl-ex]

    work page arXiv 2023

  19. [19]

    Y. Guo, X. Ji, Y. Liu, and J. Yang, Phys. Rev. D108, 034003 (2023), arXiv:2305.06992 [hep-ph]

    work page arXiv 2023

  20. [20]

    A. Sain, B. Gurjar, and C. Mondal, Phys. Rev. D113, 054041 (2026), arXiv:2601.05840 [hep-ph]

    work page arXiv 2026

  21. [21]

    Hadron tomography by generalized distribution amplitudes in pion-pair production process $\gamma^* \gamma \rightarrow \pi^0 \pi^0 $ and gravitational form factors for pion

    S. Kumano, Q.-T. Song, and O. V. Teryaev, Phys. Rev. D97, 014020 (2018), arXiv:1711.08088 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  22. [22]

    Study of $\pi^0$ pair production in single-tag two-photon collisions

    M. Masuda et al. (Belle), Phys. Rev. D93, 032003 (2016), arXiv:1508.06757 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  23. [23]

    D. C. Hackett, P. R. Oare, D. A. Pefkou, and P. E. Shanahan, Phys. Rev. D108, 114504 (2023), arXiv:2307.11707 [hep-lat]

    work page arXiv 2023

  24. [24]

    Y.-Z. Xu, M. Ding, K. Raya, C. D. Roberts, J. Rodr´ ıguez- Quintero, and S. M. Schmidt, Eur. Phys. J. C84, 191 (2024), arXiv:2311.14832 [hep-ph]

    work page arXiv 2024

  25. [25]

    Yao, Y.-Z

    Z.-Q. Yao, Y.-Z. Xu, D. Binosi, Z.-F. Cui, M. Ding, K. Raya, C. Roberts, and J. Rodriguez-Quintero, PoS QCHSC24, 081 (2025), arXiv:2503.23166 [hep-ph]

    work page arXiv 2025

  26. [26]

    Li and J

    Y. Li and J. P. Vary, Phys. Rev. D109, L051501 (2024), arXiv:2312.02543 [hep-th]

    work page arXiv 2024

  27. [27]

    Fujii, A

    D. Fujii, A. Iwanaka, and M. Tanaka, Phys. Rev. D110, L091501 (2024), arXiv:2407.21113 [hep-ph]

    work page arXiv 2024

  28. [28]

    Liu and A

    Z. Liu and A. Watanabe, Phys. Rev. D112, 114032 (2025), arXiv:2503.18747 [hep-ph]

    work page arXiv 2025

  29. [29]

    W.-Y. Liu, E. Shuryak, C. Weiss, and I. Zahed, Phys. Rev. D110, 054021 (2024), arXiv:2405.14026 [hep-ph]

    work page arXiv 2024

  30. [30]

    W.-Y. Liu, E. Shuryak, and I. Zahed, Phys. Rev. D110, 054022 (2024), arXiv:2405.16269 [hep-ph]

    work page arXiv 2024

  31. [31]

    Gravitational form factors of the pion and meson dominance,

    W. Broniowski and E. Ruiz Arriola, Phys. Lett. B859, 139138 (2024), arXiv:2405.07815 [hep-ph]

    work page arXiv 2024

  32. [32]

    Cao, F.-K

    X.-H. Cao, F.-K. Guo, Q.-Z. Li, B.-W. Wu, and D.-L. Yao, (2025), 10.1140/epjs/s11734-025-02025-9, arXiv:2507.05375 [hep-ph]

    work page doi:10.1140/epjs/s11734-025-02025-9 2025

  33. [33]

    J. P. Vary, H. Honkanen, J. Li, P. Maris, S. J. Brod- sky, A. Harindranath, G. F. de Teramond, P. Sternberg, E. G. Ng, and C. Yang, Phys. Rev. C81, 035205 (2010), arXiv:0905.1411 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  34. [34]

    Klimt, M

    S. Klimt, M. F. M. Lutz, U. Vogl, and W. Weise, Nucl. Phys. A516, 429 (1990)

    work page 1990

  35. [35]

    U. Vogl, M. F. M. Lutz, S. Klimt, and W. Weise, Nucl. Phys. A516, 469 (1990)

    work page 1990

  36. [36]

    Vogl and W

    U. Vogl and W. Weise, Prog. Part. Nucl. Phys.27, 195 (1991)

    work page 1991

  37. [37]

    S. P. Klevansky, Rev. Mod. Phys.64, 649 (1992)

    work page 1992

  38. [38]

    S. J. Brodsky, G. F. de Teramond, H. G. Dosch, and J. Erlich, Phys. Rept.584, 1 (2015), arXiv:1407.8131 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  39. [39]

    Y. Li, P. Maris, X. Zhao, and J. P. Vary, Phys. Lett. B 758, 118 (2016), arXiv:1509.07212 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  40. [40]

    Y. Li, P. Maris, and J. P. Vary, Phys. Rev. D96, 016022 (2017), arXiv:1704.06968 [hep-ph]

    work page arXiv 2017

  41. [41]

    Basis light front quantization for the charged light mesons with color singlet Nambu--Jona-Lasinio interactions

    S. Jia and J. P. Vary, Phys. Rev. C99, 035206 (2019), arXiv:1811.08512 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  42. [42]

    J. Lan, C. Mondal, S. Jia, X. Zhao, and J. P. Vary, Phys. Rev. Lett.122, 172001 (2019), arXiv:1901.11430 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  43. [43]

    J. Lan, C. Mondal, S. Jia, X. Zhao, and J. P. Vary, Phys. Rev. D101, 034024 (2020), arXiv:1907.01509 [nucl-th]

    work page arXiv 2020

  44. [44]

    Mondal, S

    C. Mondal, S. Nair, S. Jia, X. Zhao, and J. P. Vary (BLFQ), Phys. Rev. D104, 094034 (2021), arXiv:2109.02279 [hep-ph]

    work page arXiv 2021

  45. [45]

    Adhikari, C

    L. Adhikari, C. Mondal, S. Nair, S. Xu, S. Jia, X. Zhao, and J. P. Vary (BLFQ), Phys. Rev. D104, 114019 (2021), arXiv:2110.05048 [hep-ph]

    work page arXiv 2021

  46. [46]

    W. Qian, S. Jia, Y. Li, and J. P. Vary, Phys. Rev. C 102, 055207 (2020), arXiv:2005.13806 [nucl-th]

    work page arXiv 2020

  47. [47]

    Harindranath, R

    A. Harindranath, R. Kundu, A. Mukherjee, and J. P. Vary, Phys. Lett. B417, 361 (1998), arXiv:hep- ph/9711298

    work page arXiv 1998

  48. [48]

    Gravitational form factors of the pion in the self-consistent light-front quark model,

    Y. Choi, H.-D. Son, and H.-M. Choi, Phys. Rev. D112, 014043 (2025), arXiv:2504.14997 [hep-ph]

    work page arXiv 2025

  49. [49]

    J. F. Donoghue and H. Leutwyler, Z. Phys. C52, 343 (1991)

    work page 1991

  50. [50]

    Z. Xing, M. Ding, and L. Chang, Phys. Rev. D107, L031502 (2023), arXiv:2211.06635 [hep-ph]

    work page arXiv 2023

  51. [51]

    M. A. Sultan, Z. Xing, K. Raya, A. Bashir, and L. Chang, Phys. Rev. D110, 054034 (2024), arXiv:2407.10437 [hep-ph]

    work page arXiv 2024

  52. [52]

    T. Hu, X. Cao, S. Xu, Y. Li, X. Zhao, and J. P. Vary, Phys. Rev. D111, 074031 (2025), arXiv:2408.09689 [hep- ph]

    work page arXiv 2025

  53. [53]

    S. Xu, X. Cao, T. Hu, Y. Li, X. Zhao, and J. P. Vary, Phys. Rev. D109, 114024 (2024), arXiv:2404.06259 [hep- ph]

    work page arXiv 2024

  54. [54]

    Freese and G

    A. Freese and G. A. Miller, Phys. Rev. D103, 094023 (2021), arXiv:2102.01683 [hep-ph]. 11

    work page arXiv 2021

  55. [55]

    Laue, Annalen Phys.340, 524 (1911)

    M. Laue, Annalen Phys.340, 524 (1911)

    work page 1911

  56. [56]

    Kim and H.-C

    J.-Y. Kim and H.-C. Kim, Phys. Rev. D104, 074019 (2021), arXiv:2105.10279 [hep-ph]

    work page arXiv 2021

  57. [57]

    A. Sain, P. Choudhary, B. Gurjar, C. Mondal, D. Chakrabarti, and A. Mukherjee, Phys. Rev. D111, 094011 (2025), arXiv:2503.12574 [hep-ph]

    work page arXiv 2025