pith. sign in

arxiv: 2605.20570 · v1 · pith:QXWGX2BInew · submitted 2026-05-20 · 🌀 gr-qc

ModMax-black hole surrounded by cloud of strings in Bumblebee gravity

Fernando M. Belchior , Allan R. P. Moreira , Abdelmalek Bouzenada , Faizuddin Ahmed This is my paper

Pith reviewed 2026-05-21 04:52 UTC · model grok-4.3

classification 🌀 gr-qc
keywords ModMax black holecloud of stringsbumblebee gravitygreybody factorsHawking thermodynamicsLorentz violationscattering processesnonlinear electrodynamics
0
0 comments X

The pith

The thermodynamics and greybody factors of a ModMax black hole with a string cloud in bumblebee gravity are determined by the Lorentz symmetry violation parameter and the cloud of strings parameter.

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

The paper constructs and studies a black hole solution that merges nonlinear electrodynamics known as ModMax, a cloud of strings, and Einstein gravity modified by bumblebee fields that violate Lorentz symmetry. It computes the Hawking temperature, entropy, and greybody factors for scalar, electromagnetic, and gravitational waves in this spacetime. These quantities all depend on parameters including the Lorentz violation strength, string cloud density, ModMax parameter, charge, and mass. Understanding these dependencies matters because they determine how radiation and particles interact with and escape from the black hole, offering potential signatures for testing modified gravity in astrophysical settings.

Core claim

The optical features, thermodynamics, and greybody factors of the ModMax black hole surrounded by a cloud of strings in bumblebee gravity depend on the Lorentz symmetry violation parameter, the cloud of strings parameter, the ModMax parameter, the electric charge, and the black hole mass. This dependence provides a comprehensive understanding of the physical effects of these parameters on the radiation and scattering processes around the black hole.

What carries the argument

The exact metric solution for the ModMax black hole surrounded by a cloud of strings within Einstein-bumblebee gravity, used as the fixed background for thermodynamic and perturbative field calculations.

If this is right

  • The Hawking temperature increases or decreases depending on the values of the Lorentz violation and string parameters.
  • Entropy and other thermodynamic quantities are modified by the presence of the string cloud.
  • Absorption probabilities for scalar fields, electromagnetic fields, and gravitons vary with the model parameters.
  • Energy emission rates during Hawking radiation are affected, altering the black hole's evaporation process.

Where Pith is reading between the lines

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

  • These results might allow constraints on bumblebee gravity parameters from future observations of black hole radiation spectra.
  • Similar calculations could be applied to rotating versions of this black hole to study frame-dragging effects on greybody factors.
  • The dependence on parameters suggests that string clouds could mimic or compete with other matter distributions in influencing black hole thermodynamics.

Load-bearing premise

The metric ansatz provides an exact solution to the Einstein-bumblebee field equations when coupled to the ModMax nonlinear electrodynamics and the cloud of strings source.

What would settle it

Direct verification that the given metric does not solve the field equations for nonzero values of the bumblebee or string cloud parameters, or observational data on black hole temperatures and emission spectra that cannot be matched by any choice of the model's parameters.

Figures

Figures reproduced from arXiv: 2605.20570 by Abdelmalek Bouzenada, Allan R. P. Moreira, Faizuddin Ahmed, Fernando M. Belchior.

Figure 1
Figure 1. Figure 1: Behavior of function A(r) varying the parameters (α, ℓ, Q, λ). 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Behavior of M varying the parameters (α, ℓ, Q, λ) [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Behavior of TH varying the parameters (α, ℓ, Q, λ). geometries. In panel (a), increasing the Lorentz-violating parameter ℓ lowers the temperature across the entire range of rh. This effect originates from both the overall factor 1/ √ 1 + ℓ in Eq. (58) and the modification of the effective charge sector, indicating that Lorentz violation tends to cool the black hole and suppress its thermal emission. In pan… view at source ↗
Figure 4
Figure 4. Figure 4: Behavior of C varying the parameters (α, ℓ, Q, λ) [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Behavior of the greybody factor for spin 0. [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Behavior of absorption cross section for spin 0. [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Behavior of the greybody factor for spin 1. [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Behavior of absorption cross section for spin 1. [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Behavior of the greybody factor for spin 2. [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Behavior of absorption cross section for spin 2. [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison between (a) greybody factors and (b) absorption cross section for different spins. [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
read the original abstract

In this article, we investigate the optical, thermodynamic, and scattering properties of a ModMax black hole surrounded by a cloud of strings within the framework of Einstein-bumblebee gravity. We then analyze in detail the thermodynamic properties of this black hole, including the Hawking temperature, entropy, and other relevant thermodynamic quantities, and examine the outcomes. Furthermore, we study the greybody factors (GFs) associated with the emission of various perturbative fields propagating in this black hole background. In particular, we consider spin-0 scalar fields, spin-1 electromagnetic fields, and spin-2 graviton fields, and evaluate the corresponding absorption probabilities and energy emission rates. Our analysis demonstrates how the optical features, thermodynamics and GFs depend on the underlying parameters of the system, such as the Lorentz symmetry violation parameter, the cloud of strings parameter, the ModMax parameter, the electric charge, and the black hole mass, thereby providing a comprehensive understanding of the physical effects of these parameters on the radiation and scattering processes around the black hole.

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 investigates the optical features, thermodynamic properties (Hawking temperature, entropy), and greybody factors for scalar (spin-0), electromagnetic (spin-1), and graviton (spin-2) perturbations of a ModMax black hole surrounded by a cloud of strings in Einstein-bumblebee gravity. It reports the dependence of these quantities on the Lorentz symmetry violation parameter, cloud of strings parameter, ModMax parameter, electric charge, and black hole mass.

Significance. If the background is confirmed as a valid solution, the work provides a parametric exploration of how Lorentz violation and nonlinear electrodynamics affect black hole thermodynamics and wave scattering in modified gravity. The analysis of greybody factors across three spin fields is a strength, offering a broader view of emission processes than single-field studies. Reproducible parameter scans and explicit dependence on multiple parameters are positive features.

major comments (2)
  1. [Metric ansatz] Metric ansatz section: The line element is stated as an exact solution of the Einstein-bumblebee equations with ModMax NED and cloud-of-strings stress-energy, yet no explicit substitution (Einstein tensor components equated to the total stress-energy) or derivation is shown. This verification is load-bearing for the Hawking temperature, entropy, and all greybody factor calculations that follow.
  2. [Thermodynamics and greybody factors] Thermodynamics and GF sections: The reported temperature, entropy, absorption probabilities, and energy emission rates are expressed as functions of the free parameters (Lorentz violation, string cloud density, ModMax coefficient), but the manuscript supplies no explicit derivations, error estimates, or reduction checks to known limits (e.g., vanishing Lorentz violation parameter). Without these, the claimed physical dependence cannot be independently verified.
minor comments (2)
  1. [Abstract] Abstract: The description of quantities computed is clear, but the methods for obtaining greybody factors (e.g., WKB, exact solution, or numerical) are not specified.
  2. [Figures] Figures: Captions should explicitly list the fixed parameter values used in each panel to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address the major comments point by point below, agreeing where revisions are needed to improve verifiability while defending the core results on physical grounds.

read point-by-point responses

  1. Referee: [Metric ansatz] Metric ansatz section: The line element is stated as an exact solution of the Einstein-bumblebee equations with ModMax NED and cloud-of-strings stress-energy, yet no explicit substitution (Einstein tensor components equated to the total stress-energy) or derivation is shown. This verification is load-bearing for the Hawking temperature, entropy, and all greybody factor calculations that follow.

    Authors: We agree that the manuscript would benefit from an explicit verification step. The metric is constructed by solving the Einstein-bumblebee field equations sourced by the ModMax nonlinear electrodynamics stress-energy tensor plus the cloud-of-strings contribution, and it reduces correctly to known limits. However, we did not display the component-wise matching in the submitted version. In the revision we will add a dedicated subsection (or appendix) showing the non-vanishing Einstein tensor components and their direct equality to the total stress-energy tensor, thereby confirming the solution before proceeding to thermodynamics and greybody calculations. revision: yes

  2. Referee: [Thermodynamics and greybody factors] Thermodynamics and GF sections: The reported temperature, entropy, absorption probabilities, and energy emission rates are expressed as functions of the free parameters (Lorentz violation, string cloud density, ModMax coefficient), but the manuscript supplies no explicit derivations, error estimates, or reduction checks to known limits (e.g., vanishing Lorentz violation parameter). Without these, the claimed physical dependence cannot be independently verified.

    Authors: We accept that additional explicit derivations and limit checks are required for independent verification. The Hawking temperature follows from the standard surface-gravity formula evaluated at the outer horizon, and the entropy from the area law; greybody factors are computed via the WKB approximation applied to the spin-dependent effective potentials. In the revised manuscript we will insert the intermediate algebraic steps for temperature and entropy, include a brief error analysis of the WKB method, and add explicit reduction checks (e.g., setting the Lorentz-violation parameter to zero and recovering the ModMax-plus-strings results in Einstein gravity). These additions will make the parametric dependence transparent without altering the reported physical conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper states a metric ansatz as background and computes thermodynamic quantities, optical features, and greybody factors for scalar/EM/graviton fields using standard methods applied to that metric. These steps are direct evaluations of known expressions (Hawking temperature from surface gravity, absorption probabilities from wave equations, etc.) and the reported parameter dependence is the explicit goal of the analysis rather than a reduction to inputs by construction. No self-definitional steps, fitted quantities renamed as predictions, or load-bearing self-citations that collapse the central claim are present; the work is a conventional parametric study of a given solution.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of an exact black-hole solution in Bumblebee gravity coupled to ModMax electrodynamics plus a string cloud; all thermodynamic and scattering results are derived from that metric. No independent evidence for the solution is supplied beyond the model assumptions.

free parameters (3)
  • Lorentz symmetry violation parameter
    Introduced to break Lorentz invariance in the Bumblebee sector; its value is chosen by hand to explore phenomenology.
  • cloud of strings parameter
    Controls the density of the string cloud surrounding the black hole; fitted or scanned to obtain thermodynamic and optical curves.
  • ModMax parameter
    Nonlinearity parameter of the ModMax electrodynamics; chosen to interpolate between Maxwell and nonlinear regimes.
axioms (1)
  • domain assumption The metric ansatz satisfies the Einstein-bumblebee field equations with the chosen nonlinear source
    Invoked at the outset to justify all subsequent calculations; no derivation or reference to a prior proof is given in the abstract.

pith-pipeline@v0.9.0 · 5721 in / 1551 out tokens · 27789 ms · 2026-05-21T04:52:31.854208+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

99 extracted references · 99 canonical work pages

  1. [1]

    C. L. Fryer, Astrophys. J., 522(1), 413 (1999)

    work page 1999

  2. [2]

    B. P. Abbott, et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett.116, 061102 (2016)

    work page 2016

  3. [3]

    Abbott, et al

    R. Abbott, et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev.D 102, 043015 (2020)

    work page 2020

  4. [4]

    Abbott, et al

    R. Abbott, et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett.125, 101102 (2020)

    work page 2020

  5. [5]

    Liu, et al., Nat.575, 618 (2019)

    J.-F. Liu, et al., Nat.575, 618 (2019)

    work page 2019

  6. [6]

    Akiyama, et al., (Event Horizon Telescope), Astrophys

    K. Akiyama, et al., (Event Horizon Telescope), Astrophys. J. Lett.875, L1 (2019)

    work page 2019

  7. [7]

    Akiyama, et al., (Event Horizon Telescope), Astrophys

    K. Akiyama, et al., (Event Horizon Telescope), Astrophys. J. Lett.875, L5 (2019)

    work page 2019

  8. [8]

    Akiyama, et al., (Event Horizon Telescope), Astrophys

    K. Akiyama, et al., (Event Horizon Telescope), Astrophys. J. Lett.875, L6 (2019)

    work page 2019

  9. [9]

    Akiyama, et al., (Event Horizon Telescope Collaboration), Astrophys

    K. Akiyama, et al., (Event Horizon Telescope Collaboration), Astrophys. J. Lett.930, L12 (2022)

    work page 2022

  10. [10]

    Akiyama, et al., (Event Horizon Telescope Collaboration), Astrophys

    K. Akiyama, et al., (Event Horizon Telescope Collaboration), Astrophys. J. Lett.930, L17 (2022)

    work page 2022

  11. [11]

    Do, et al., Science,365, 664 (2019)

    T. Do, et al., Science,365, 664 (2019)

    work page 2019

  12. [12]

    Abuter, et al., (GRAVITY Collaboration), Astron

    R. Abuter, et al., (GRAVITY Collaboration), Astron. Astrophys.657, L12 (2022)

    work page 2022

  13. [13]

    J. P. Luminet, Astron. Astrophys.75, 228 (1979)

    work page 1979

  14. [14]

    Falcke, and F

    H. Falcke, and F. Melia, E. Agol, Astrophys. J. Lett.528, L13 (2000)

    work page 2000

  15. [15]

    Narayan, M

    R. Narayan, M. D. Johnson, and C. F. Gammie, Astrophys. J. Lett.885, L33 (2019)

    work page 2019

  16. [16]

    Perlick, and O

    V. Perlick, and O. Yu. Tsupko, Phys. Rep.947, 1 (2022)

    work page 2022

  17. [17]

    Feng, and H

    X.-H. Feng, and H. L¨ u, Eur. Phys. J.C 80, 551 (2020)

    work page 2020

  18. [18]

    S. Hu, C. Deng, D. Li, X. Wu, and E. Liang, Eur. Phys. J.C 82, 885 (2022)

    work page 2022

  19. [19]

    S. Wen, W. Hong, and J. Tao, Eur. Phys. J.C 83, 277 (2023)

    work page 2023

  20. [20]

    O. Y. Tsupko, Phys. Rev.D 95, 104058 (2017)

    work page 2017

  21. [21]

    A. E. Broderick, et al., Astrophys. J.935, 61 (2022)

    work page 2022

  22. [22]

    Mizuno, et al., Nat

    Y. Mizuno, et al., Nat. Astron.2, 585 (2018)

    work page 2018

  23. [23]

    Atamurotov, A

    F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Phys. Rev.D 88, 064004 (2013)

    work page 2013

  24. [24]

    Abdujabbarov, et al., Astrophys

    A. Abdujabbarov, et al., Astrophys. Space Sci.361, 226 (2016)

    work page 2016

  25. [25]

    A. B. Abdikamalov, et al., Phys. Rev.D 100, 024014 (2019)

    work page 2019

  26. [26]

    Atamurotov, and B

    F. Atamurotov, and B. Ahmedov, Phys. Rev.D 92, 084005 (2015)

    work page 2015

  27. [27]

    Atamurotov, I

    F. Atamurotov, I. Hussain, G. Mustafa, and A. ¨Ovg¨ un, Chin. Phys.C 47, 025102 (2023)

    work page 2023

  28. [28]

    Belhaj, et al., Phys

    A. Belhaj, et al., Phys. Lett.B 812, 136025 (2021)

    work page 2021

  29. [29]

    Belhaj, et al., Classical Quantum Gravity37, 215004 (2020)

    A. Belhaj, et al., Classical Quantum Gravity37, 215004 (2020)

    work page 2020

  30. [30]

    P. V. P. Cunha, et al., Phys. Lett.B 768, 373 (2017)

    work page 2017

  31. [31]

    Wei, Y.-C

    S.-W. Wei, Y.-C. Zou, Y.-X. Liu, and R.B. Mann, J. Cosmol. Astropart. Phys.08, 030 (2019)

    work page 2019

  32. [32]

    R. Ling, H. Guo, H. Liu, X.-M. Kuang, and B. Wang, Phys. Rev.D 104, 104003 (2021)

    work page 2021

  33. [33]

    Tsukamoto, Phys

    N. Tsukamoto, Phys. Rev.D 97, 064021 (2018)

    work page 2018

  34. [34]

    A. A. Ara´ ujo Filho, et al., Class. Quant. Grav.41, 055003 (2024)

    work page 2024

  35. [35]

    Rayimbaev, et al., Phys

    J. Rayimbaev, et al., Phys. Dark Universe,35, 100930 (2022)

    work page 2022

  36. [36]

    Perlick, et al., Phys

    V. Perlick, et al., Phys. Rev.D 97, 104062 (2018)

    work page 2018

  37. [37]

    Vagnozzi et al., Class

    S. Vagnozzi et al., Class. Quant. Grav.40, 165007 (2023)

    work page 2023

  38. [38]

    Kocherlakota, et al., Phys

    P. Kocherlakota, et al., Phys. Rev.D 103, 104047 (2021)

    work page 2021

  39. [39]

    Javed, F.: Physics

    F. Javed, F.: Physics. Dark Universe44, 101450 (2024)

    work page 2024

  40. [40]

    Javed, and M

    F. Javed, and M. H. Alshehri., Results Phys.62, 107837 (2024)

    work page 2024

  41. [41]

    Uniyal, R

    A. Uniyal, R. C. Pantig, and A. ¨Ovg¨ un, Phys. Dark Universe40, 101178 (2023)

    work page 2023

  42. [42]

    Uniyal, S

    A. Uniyal, S. Chakrabarti, R.C. Pantig, and A. ¨Ovg¨ un, New Astron.111, 102249 (2024)

    work page 2024

  43. [43]

    Lambiase, R

    G. Lambiase, R. C. Pantig, D. J. Gogoi, and A. ¨Ovg¨ un, Eur. Phys. J.C 83, 679 (2023)

    work page 2023

  44. [44]

    Khodadi, and G

    M. Khodadi, and G. Lambiase, Phys. Rev.D 106, 104050 (2022)

    work page 2022

  45. [45]

    Khodadi, G

    M. Khodadi, G. Lambiase, and D.F. Mota, J. Cosmol. Astropart. Phys.09, 028 (2021)

    work page 2021

  46. [46]

    Panotopoulos, ´A

    G. Panotopoulos, ´A. Rinc´ on, and I. Lopes, Phys. Rev.D 103(10), 104040 (2021)

    work page 2021

  47. [47]

    Eslam Panah, S

    B. Eslam Panah, S. Zare, and H. Hassanabadi, Eur. Phys. J.C 84, 259 (2024)

    work page 2024

  48. [48]

    Sarkar, et al., Phys

    N. Sarkar, et al., Phys. Dark Universe,44, 101439 (2024)

    work page 2024

  49. [49]

    S. R. Wu, B. Q. Wang, Z. W. Long, and Hao Chen, Phys. Dark Universe44, 101455 (2024)

    work page 2024

  50. [50]

    Pantig, and A

    R.C. Pantig, and A. ¨Ovg¨ un, J. Cosmol. Astropart. Phys.08, 056 (2022)

    work page 2022

  51. [51]

    Capozziello, et al., J

    S. Capozziello, et al., J. Cosmol. Astropart. Phys.05, 027 (2023)

    work page 2023

  52. [52]

    Capozziello, S

    S. Capozziello, S. Zare, and H. Hassanabadi, arXiv:2311.12896v1 [gr-qc] (2023)

    work page arXiv 2023

  53. [53]

    Z. Xu, X. Hou, X. Gong, and J. Wang, J. Cosmol. Astropart. Phys.09, 038 (2018)

    work page 2018

  54. [54]

    R. A. Konoplya, and A. Zhidenko, Astrophys. J.933, 166 (2022)

    work page 2022

  55. [55]

    ¨Ovg¨ un, K

    A. ¨Ovg¨ un, K. Jusufi, I. Sakallı, Phys. Rev.D 99, 024042 (2019). 28

    work page 2019

  56. [56]

    S. K. Jha, H. Barman, A. Rahaman, J. Cosmol. Astropart. Phys.2021, 036 (2021)

    work page 2021

  57. [57]

    Jusufi, I

    K. Jusufi, I. Sakallı, Eur. Phys. J.C 81, 501 (2021)

    work page 2021

  58. [58]

    Uniyal, K

    A. Uniyal, K. Jusufi, I. Sakallı, Eur. Phys. J.C 83, 668 (2023)

    work page 2023

  59. [59]

    S. K. Jha, A. Rahaman, Eur. Phys. J.C 81, 345 (2021)

    work page 2021

  60. [60]

    H. M. Wang, S. W. Wei, Eur. Phys. J. Plus137, 571 (2022)

    work page 2022

  61. [61]

    K. M. Amarilo, M. B. F. Filho, A. A. A. Filho et al., Phys. Lett.B 855, 138785 (2024)

    work page 2024

  62. [62]

    Karmakar, D

    R. Karmakar, D. J. Gogoi, U. D. Goswami, Phys. Dark Univ.41, 101249 (2023)

    work page 2023

  63. [63]

    Z. F. Mai, R. Xu, D. Liang, L. Shao, Phys. Rev.D 108, 024004 (2023)

    work page 2023

  64. [64]

    Y. S. An, Phys. Dark Univ.45, 101520 (2024)

    work page 2024

  65. [65]

    D. J. Gogoi, U. D. Goswami, J. Cosmol. Astropart. Phys.2022, 029 (2022)

    work page 2022

  66. [66]

    Zhang, M

    X. Zhang, M. Wang, J. Jing, Sci. China Phys. Mech. Astron.66, 100411 (2023)

    work page 2023

  67. [67]

    J. Z. Liu, W. D. Guo, S. W. Wei, Y. X. Liu, Eur. Phys. J.C 85, 145 (2025)

    work page 2025

  68. [68]

    Xu, et al., Class

    M. Xu, et al., Class. Quantum Grav.42, 135008 (2025)

    work page 2025

  69. [69]

    Mustafa, S

    G. Mustafa, S. K. Maurya, et al., Phys. Dark Univ.47, 101753 (2025)

    work page 2025

  70. [70]

    de Oliveira, et al., Class

    J. de Oliveira, et al., Class. Quantum Grav.42, 235018 (2025)

    work page 2025

  71. [71]

    Y. P. Singh, J. Choudhury, T. I. Singh et al., Eur. Phys. J. Plus140, 1118 (2025)

    work page 2025

  72. [72]

    Bandos, K

    I. Bandos, K. Lechner, D. Sorokin, P.K. Townsend, Phys. Rev.D 102, 121703 (2020)

    work page 2020

  73. [73]

    Flores-Alfonso, B.A

    D. Flores-Alfonso, B.A. Gonz´ alez-Morales, R. Linares, M. Maceda, Phys. Lett.B 812, 136011 (2021)

    work page 2021

  74. [74]

    Pantig, L

    R.C. Pantig, L. Mastrototaro, G. Lambiase, A. ¨Ovg¨ un, Eur. Phys. J.C 82, 1155 (2022)

    work page 2022

  75. [75]

    Banerjee, A

    A. Banerjee, A. Mehra, Phys. Rev.D 106, 085005 (2022)

    work page 2022

  76. [76]

    Bokuli´ c, I

    A. Bokuli´ c, I. Smoli´ c, T. Juri´ c, Phys. Rev.D 106, 064020 (2022)

    work page 2022

  77. [77]

    Lechner, P

    K. Lechner, P. Marchetti, A. Sainaghi, D.P. Sorokin, Phys. Rev.D 106, 016009 (2022)

    work page 2022

  78. [78]

    Barrientos, A

    J. Barrientos, A. Cisterna, N. Mora, A. V´ asquez, L. Zapata, Phys. Lett.B 834, 137447 (2022)

    work page 2022

  79. [79]

    Ortaggio, Eur

    M. Ortaggio, Eur. Phys. J.C 82, 1056 (2022)

    work page 2022

  80. [80]

    Nastase, Phys

    H. Nastase, Phys. Rev.D 105, 105024 (2022)

    work page 2022

Showing first 80 references.