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  • Open Access

Non-Markovian dynamics of bottomonia in the QGP

Vyshakh B. R.* and Rishi Sharma†

  • Department of theoretical physics TIFR, Mumbai, India, 400005

  • *Contact author: vyshakh.br@theory.tifr.res.in
  • †Contact author: rishi@theory.tifr.res.in
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Phys. Rev. D 112, 114038 – Published 23 December, 2025

DOI: https://doi.org/10.1103/1tjb-hkwj

Abstract

The evolution of quarkonia in the QGP medium can be described through the formalism of open quantum systems (OQS). In previous works with OQS, the quarkonium evolution was studied by working in either quantum Brownian or optical regime. In this paper, we set up a general non-Markovian master equation to describe the evolution of the quarkonia in the medium. Because of the non-Markovian nature, it cannot be cast in Lindblad form, which makes it challenging to solve. We numerically solve the master equation for ϒ(1S) without considering stochastic jumps for Bjorken and viscous hydrodynamic background at energies relevant to the large hadron collider (LHC) and relativistic heavy ion collider (RHIC). We quantify the effect of the hierarchy between the system timescale, τS∼1/Eb, and the environment timescale, τE, on quarkonium evolution and show that it significantly affects the nuclear modification factor. A comparison of our results with the existing experimental data from the LHC and RHIC is presented.

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References (124)

  1. M. E. Peskin, Short-distance analysis for heavy-quark systems: (I). Diagrammatics, Nucl. Phys. B156, 365 (1979).
  2. G. Bhanot and M. E. Peskin, Short distance analysis for heavy quark systems. 2. Applications, Nucl. Phys. B156, 391 (1979).
  3. T. Matsui and H. Satz, J/ψ suppression by quark-gluon plasma formation, Phys. Lett. B 178, 416 (1986).
  4. L. Grandchamp and R. Rapp, Thermal versus direct J/Psi production in ultrarelativistic heavy ion collisions, Phys. Lett. B 523, 60 (2001).
  5. M. Laine, O. Philipsen, M. Tassler, and P. Romatschke, Real-time static potential in hot QCD, J. High Energy Phys. 03 (2007) 054.

  6. Additionally, regeneration [4, 7, 8, 9] via coalescence of Q and Q¯ created at different hard vertices is important for charmonia. For bottomonia, this contribution is expected to be small [10].

  7. L. Grandchamp and R. Rapp, Charmonium suppression and regeneration from SPS to RHIC, Nucl. Phys. A709, 415 (2002).
  8. L. Grandchamp, R. Rapp, and G. E. Brown, In medium effects on charmonium production in heavy ion collisions, Phys. Rev. Lett. 92, 212301 (2004).
  9. V. Greco, C. M. Ko, and R. Rapp, Quark coalescence for charmed mesons in ultrarelativistic heavy ion collisions, Phys. Lett. B 595, 202 (2004).
  10. X. Du, M. He, and R. Rapp, Color screening and regeneration of bottomonia in high-energy heavy-ion collisions, Phys. Rev. C 96, 054901 (2017).
  11. A. Andronic et al., Heavy-flavour and quarkonium production in the LHC era: From proton–proton to heavy-ion collisions, Eur. Phys. J. C 76, 107 (2016).
  12. A. Andronic et al., Comparative study of quarkonium transport in hot QCD matter, Eur. Phys. J. A 60, 88 (2024).
  13. N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Potential NRQCD: An effective theory for heavy quarkonium, Nucl. Phys. B566, 275 (2000).
  14. N. Brambilla, J. Ghiglieri, A. Vairo, and P. Petreczky, Static quark-antiquark pairs at finite temperature, Phys. Rev. D 78, 014017 (2008).
  15. N. Brambilla, M. Á. Escobedo, J. Ghiglieri, J. Soto, and A. Vairo, Heavy quarkonium in a weakly-coupled quark-gluon plasma below the melting temperature, J. High Energy Phys. 09 (2010) 038.
  16. N. Brambilla, M. Á. Escobedo, J. Ghiglieri, and A. Vairo, Thermal width and gluo-dissociation of quarkonium in pNRQCD, J. High Energy Phys. 12 (2011) 116.
  17. N. Brambilla, M. A. Escobedo, J. Ghiglieri, and A. Vairo, Thermal width and quarkonium dissociation by inelastic parton scattering, J. High Energy Phys. 05 (2013) 130.
  18. R. Sharma and B. Singh, Energy hierarchies governing quarkonium dynamics in heavy ion collisions, Phys. Rev. C 109, 054905 (2024).
  19. P. Petreczky, Lattice QCD at non-zero temperature, J. Phys. G 39, 093002 (2012).
  20. A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Fermionic contributions to the three-loop static potential, Phys. Lett. B 668, 293 (2008).
  21. C. Anzai, Y. Kiyo, and Y. Sumino, Static QCD potential at three-loop order, Phys. Rev. Lett. 104, 112003 (2010).
  22. A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Three-loop static potential, Phys. Rev. Lett. 104, 112002 (2010).
  23. A. Pineda and J. Segovia, Improved determination of heavy quarkonium magnetic dipole transitions in potential nonrelativistic QCD, Phys. Rev. D 87, 074024 (2013).
  24. Y. Burnier and A. Rothkopf, Complex heavy-quark potential and Debye mass in a gluonic medium from lattice QCD, Phys. Rev. D 95, 054511 (2017).
  25. A. Rothkopf, T. Hatsuda, and S. Sasaki, Complex heavy-quark potential at finite temperature from lattice QCD, Phys. Rev. Lett. 108, 162001 (2012).
  26. D. Bala and S. Datta, Nonperturbative potential for the study of quarkonia in QGP, Phys. Rev. D 101, 034507 (2020).
  27. D. Bala and S. Datta, Interaction potential between heavy QQ¯ in a color octet configuration in the QGP from a study of hybrid Wilson loops, Phys. Rev. D 103, 014512 (2021).
  28. D. Bala, O. Kaczmarek, R. Larsen, S. Mukherjee, G. Parkar, P. Petreczky, A. Rothkopf, and J. H. Weber (HotQCD Collaboration), Static quark-antiquark interactions at nonzero temperature from lattice QCD, Phys. Rev. D 105, 054513 (2022).
  29. N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Effective field theories for heavy quarkonium, Rev. Mod. Phys. 77, 1423 (2005).
  30. A. M. Eller, J. Ghiglieri, and G. D. Moore, Thermal heavy quark mass shift from euclidean correlators, Phys. Rev. D 99, 094042 (2019); 102, 039901(E) (2020).
  31. T. Binder, K. Mukaida, B. Scheihing-Hitschfeld, and X. Yao, Non-Abelian electric field correlator at NLO for dark matter relic abundance and quarkonium transport, J. High Energy Phys. 01 (2022) 137.
  32. B. Scheihing-Hitschfeld and X. Yao, Gauge invariance of non-Abelian field strength correlators: The axial gauge puzzle, Phys. Rev. Lett. 130, 052302 (2023).
  33. N. Brambilla, V. Leino, P. Petreczky, and A. Vairo, Lattice QCD constraints on the heavy quark diffusion coefficient, Phys. Rev. D 102, 074503 (2020).
  34. N. Brambilla, M. A. Escobedo, A. Islam, M. Strickland, A. Tiwari, A. Vairo, and P. Vander Griend, Regeneration of bottomonia in an open quantum systems approach, Phys. Rev. D 108, L011502 (2023).
  35. N. Brambilla, T. Magorsch, M. Strickland, A. Vairo, and P. Vander Griend, Bottomonium suppression from the three-loop QCD potential, Phys. Rev. D 109, 114016 (2024).
  36. G. Nijs, B. Scheihing-Hitschfeld, and X. Yao, Chromoelectric field correlator for quarkonium transport in the strongly coupled N=4 Yang-Mills plasma from AdS/CFT, J. High Energy Phys. 06 (2023) 007.
  37. G. Nijs, B. Scheihing-Hitschfeld, and X. Yao, Generalized gluon distribution for quarkonium dynamics in strongly coupled N=4 Yang-Mills theory, Phys. Rev. D 109, 094043 (2024).
  38. J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal, and U. Achim Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions (Cambridge University Press, Cambridge, England, 2014).
  39. V. Leino, Adjoint chromoelectric and -magnetic correlators with gradient flow, Proc. Sci., LATTICE2023 (2024) 385 [arXiv:2401.06733].
  40. J. Mayer-Steudte, Adjoint chromoelectric correlators for heavy quarkonium diffusion, Proc. Sci., LATTICE2024 (2025) 205.
  41. P. Romatschke, New developments in relativistic viscous hydrodynamics, Int. J. Mod. Phys. E 19, 1 (2010).
  42. D. A. Teaney, Viscous hydrodynamics and the quark gluon plasma, in Quark-Gluon Plasma 4, edited by R. C. Hwa and X.-N. Wang (World Scientific, Singapore, 2010), pp. 207–266, 10.1142/9789814293297_0004.
  43. T. Hirano, Dynamics of relativistic heavy ion collisions and the quark gluon plasma, Prog. Theor. Phys. Suppl. 195, 1 (2012).
  44. H. Song, Hydrodynamic modelling for relativistic heavy-ion collisions at RHIC and LHC, Pramana 84, 703 (2015).
  45. C. Gale, S. Jeon, and B. Schenke, Hydrodynamic modeling of heavy-ion collisions, Int. J. Mod. Phys. A 28, 1340011 (2013).
  46. U. Heinz and R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, Annu. Rev. Nucl. Part. Sci. 63, 123 (2013).
  47. A. Jaiswal and V. Roy, Relativistic hydrodynamics in heavy-ion collisions: General aspects and recent developments, Adv. High Energy Phys. 2016, 9623034 (2016).
  48. C. Chattopadhyay, R. S. Bhalerao, J.-Y. Ollitrault, and S. Pal, Effects of initial-state dynamics on collective flow within a coupled transport and viscous hydrodynamic approach, Phys. Rev. C 97, 034915 (2018).
  49. R. S. Bhalerao, A. Jaiswal, and S. Pal, Collective flow in event-by-event partonic transport plus hydrodynamics hybrid approach, Phys. Rev. C 92, 014903 (2015).
  50. C. Young and K. Dusling, Quarkonium above deconfinement as an open quantum system, Phys. Rev. C 87, 065206 (2013).
  51. Y. Akamatsu and A. Rothkopf, Stochastic potential and quantum decoherence of heavy quarkonium in the quark-gluon plasma, Phys. Rev. D 85, 105011 (2012).
  52. Y. Akamatsu, Real-time quantum dynamics of heavy-quark systems at high temperature, Phys. Rev. D 87, 045016 (2013).
  53. H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007).
  54. Y. Akamatsu, Quarkonium in quark–gluon plasma: Open quantum system approaches re-examined, Prog. Part. Nucl. Phys. 123, 103932 (2022).
  55. X. Yao, Open quantum systems for quarkonia, Int. J. Mod. Phys. A 36, 2130010 (2021).
  56. R. Sharma, Quarkonium propagation in the quark–gluon plasma, Eur. Phys. J. Spec. Top. 230, 697 (2021).
  57. N. Borghini and C. Gombeaud, Dynamical evolution of heavy quarkonia in a deconfined medium, arXiv:1103.2945.
  58. N. Borghini and C. Gombeaud, Heavy quarkonia in a medium as a quantum dissipative system: Master-equation approach, Eur. Phys. J. C 72, 2000 (2012).
  59. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  60. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N level systems, J. Math. Phys. (N.Y.) 17, 821 (1976).
  61. X. Yao and B. Müller, Approach to equilibrium of quarkonium in quark-gluon plasma, Phys. Rev. C 97, 014908 (2018); 97, 049903(E) (2018).
  62. X. Yao and B. Müller, Quarkonium inside the quark-gluon plasma: Diffusion, dissociation, recombination, and energy loss, Phys. Rev. D 100, 014008 (2019).
  63. X. Yao and T. Mehen, Quarkonium in-medium transport equation derived from first principles, Phys. Rev. D 99, 096028 (2019).
  64. Y. Akamatsu, Heavy quark master equations in the lindblad form at high temperatures, Phys. Rev. D 91, 056002 (2015).
  65. S. Kajimoto, Y. Akamatsu, M. Asakawa, and A. Rothkopf, Dynamical dissociation of quarkonia by wave function decoherence, Phys. Rev. D 97, 014003 (2018).
  66. T. Miura, Y. Akamatsu, M. Asakawa, and A. Rothkopf, Quantum Brownian motion of a heavy quark pair in the quark-gluon plasma, Phys. Rev. D 101, 034011 (2020).
  67. R. Sharma and A. Tiwari, Quantum evolution of quarkonia with correlated and uncorrelated noise, Phys. Rev. D 101, 074004 (2020).
  68. J.-P. Blaizot, D. De Boni, P. Faccioli, and G. Garberoglio, Heavy quark bound states in a quark–gluon plasma: Dissociation and recombination, Nucl. Phys. A946, 49 (2016).
  69. J. P. Blaizot and M. A. Escobedo, Quantum and classical dynamics of heavy quarks in a quark-gluon plasma, J. High Energy Phys. 06 (2018) 034.
  70. D. De Boni, Fate of in-medium heavy quarks via a Lindblad equation, J. High Energy Phys. 08 (2017) 064.
  71. J. P. Blaizot and M. A. Escobedo, Approach to equilibrium of a quarkonium in a quark-gluon plasma, Phys. Rev. D 98, 074007 (2018).
  72. R. Katz and P. B. Gossiaux, Semi-classical approach to J/ψ suppression in high energy heavy-ion collisions, J. Phys. Conf. Ser. 509, 012095 (2014).
  73. P. B. Gossiaux and R. Katz, Upsilon suppression in the Schrödinger–Langevin approach, Nucl. Phys. A956, 737 (2016).
  74. S. Delorme, R. Katz, T. Gousset, P. B. Gossiaux, and J.-P. Blaizot, Quarkonium dynamics in the quantum Brownian regime with non-Abelian quantum master equations, J. High Energy Phys. 06 (2024) 060.
  75. N. Brambilla, M. A. Escobedo, J. Soto, and A. Vairo, Quarkonium suppression in heavy-ion collisions: An open quantum system approach, Phys. Rev. D 96, 034021 (2017).
  76. N. Brambilla, M. A. Escobedo, J. Soto, and A. Vairo, Quarkonium suppression in heavy-ion collisions: An open quantum system approach, Phys. Rev. D 96, 034021 (2017).
  77. N. Brambilla, M. A. Escobedo, M. Strickland, A. Vairo, P. Vander Griend, and J. H. Weber, Bottomonium suppression in an open quantum system using the quantum trajectories method, J. High Energy Phys. 05 (2021) 136.
  78. N. Brambilla, M. A. Escobedo, M. Strickland, A. Vairo, P. Vander Griend, and J. H. Weber, Bottomonium production in heavy-ion collisions using quantum trajectories: Differential observables and momentum anisotropy, Phys. Rev. D 104, 094049 (2021).
  79. N. Brambilla, M. A. Escobedo, A. Islam, M. Strickland, A. Tiwari, A. Vairo, and P. Vander Griend, Heavy quarkonium dynamics at next-to-leading order in the binding energy over temperature, J. High Energy Phys. 08 (2022) 303.
  80. A. G. Redfield, On the theory of relaxation processes, IBM J. Res. Dev. 1, 19 (1957).
  81. K. Blum, Density Matrix Theory and Applications (Plenum Press, New York, 1981).
  82. H.-P. Breuer, B. Kappler, and F. Petruccione, Stochastic wave-function method for non-Markovian quantum master equations, Phys. Rev. A 59, 1633 (1999).
  83. J.-P. Blaizot and M. A. Escobedo, Phenomenological study of quarkonium suppression and the impact of the energy gap between singlets and octets, Phys. Rev. D 104, 054034 (2021).
  84. S. Chatrchyan et al. (CMS Collaboration), Indications of suppression of excited ϒ states in Pb-Pb collisions at sNN=2.76  TeV, Phys. Rev. Lett. 107, 052302 (2011).
  85. S. Chatrchyan et al. (CMS Collaboration), Observation of sequential upsilon suppression in PbPb collisions, Phys. Rev. Lett. 109, 222301 (2012); 120, 199903(E) (2018).
  86. V. Khachatryan et al. (CMS Collaboration), Suppression of ϒ(1S),ϒ(2S) and ϒ(3S) production in PbPb collisions at sNN=2.76  TeV, Phys. Lett. B 770, 357 (2017).

  87. Production of ϒ(nS) mesons in Pb+Pb and pp collisions at 5.02 TeV with ATLAS (2019).

  88. G. Aad et al. (ATLAS Collaboration), Production of ϒ(  nS) mesons in Pb+Pb and pp collisions at 5.02 TeV, Phys. Rev. C 107, 054912 (2023).

  89. Observation of the ϒ(3S) meson and sequential suppression of ϒ states in PbPb collisions at sNN=5.02  TeV (2022).

  90. A. Tumasyan et al. (CMS Collaboration), Observation of the ϒ(3S) meson and suppression of ϒ states in Pb-Pb collisions at sNN=5.02  TeV, Phys. Rev. Lett. 133, 022302 (2024).
  91. M. Strickland and S. Thapa, Bottomonium suppression at RHIC and LHC in an open quantum system approach, Phys. Rev. D 108, 014031 (2023).
  92. B. Aboona et al. (STAR Collaboration), Measurement of sequential ϒ suppression in Au+Au collisions at sNN=200  GeV with the STAR experiment, Phys. Rev. Lett. 130, 112301 (2023).
  93. R. Aaij et al. (LHCb Collaboration), Study of ϒ production in pPb collisions at sNN=8.16  TeV, J. High Energy Phys. 11 (2018) 194; 02 (2020) 93.
  94. M. Aaboud et al. (ATLAS Collaboration), Measurement of quarkonium production in proton–lead and proton–proton collisions at 5.02 TeV with the ATLAS detector, Eur. Phys. J. C 78, 171 (2018).
  95. S. Acharya et al. (ALICE Collaboration), ϒ production in p–Pb collisions at sNN=8.16  TeV, Phys. Lett. B 806, 135486 (2020).
  96. A. Tumasyan et al. (CMS Collaboration), Nuclear modification of ϒ states in pPb collisions at sNN=5.02  TeV, Phys. Lett. B 835, 137397 (2022).
  97. M. Strickland, S. Thapa, and R. Vogt, Bottomonium suppression in 5.02 and 8.16 TeV p-Pb collisions, Phys. Rev. D 109, 096016 (2024).

  98. The master equations are derived for a medium in global equilibrium. For phenomenology in an expanding medium, local thermodynamic variables are used to describe E in the neighbourhood of S.

  99. X. Yao and T. Mehen, Quarkonium semiclassical transport in quark-gluon plasma: Factorization and quantum correction, J. High Energy Phys. 02 (2021) 062.
  100. D. Banerjee, R. Gavai, S. Datta, and P. Majumdar, Temperature dependence of the static quark diffusion coefficient, Nucl. Phys. A1038, 122721 (2023).
  101. A. Messiah, Quantum Mechanics. Vol. 2 (German Translation) (North-Holland Publishing Company, Amsterdam, 1962).
  102. W. M. Alberico, A. Beraudo, A. De Pace, A. Molinari, M. Monteno, M. Nardi, F. Prino, and M. Sitta, Heavy flavors in AA collisions: Production, transport and final spectra, Eur. Phys. J. C 73, 2481 (2013).
  103. M. L. Miller, K. Reygers, S. J. Sanders, and P. Steinberg, Glauber modeling in high energy nuclear collisions, Annu. Rev. Nucl. Part. Sci. 57, 205 (2007).
  104. Z.-W. Lin, C. M. Ko, B.-A. Li, B. Zhang, and S. Pal, Multiphase transport model for relativistic heavy ion collisions, Phys. Rev. C 72, 064901 (2005).
  105. A. Sirunyan, A. Tumasyan et al. (CMS Collaboration), Measurement of nuclear modification factors of ϒ(1S), ϒ(2S) and ϒ(3S) mesons in PbPb collisions at sNN=5.02  TeV, Phys. Lett. B 790, 270 (2019).
  106. J. Kim, J. Park, B. Hong, J. Hong, E.-J. Kim, Y. Kim, M. Kweon, S. H. Lee, S. Lim, and J. Seo, Investigation of suppression of ϒ(nS) in relativistic heavy-ion collisions at sNN=200  GeV and 5.02 TeV, Phys. Rev. C 111, 014902 (2025).
  107. S. Gupta and R. Sharma, Thermalization of quarkonia at energies available at the CERN Large Hadron Collider, Phys. Rev. C 89, 057901 (2014).
  108. D. Kumar, N. Sarkar, P. P. Bhaduri, and A. Jaiswal, Examination of thermalization of quarkonia at energies available at the CERN Large Hadron Collider, Phys. Rev. C 107, 064906 (2023).
  109. N. Brambilla, M. A. Escobedo, J. Soto, and A. Vairo, Heavy quarkonium suppression in a fireball, Phys. Rev. D 97, 074009 (2018).
  110. N. Brambilla, M. A. Escobedo, A. Vairo, and P. Vander Griend, Transport coefficients from in medium quarkonium dynamics, Phys. Rev. D 100, 054025 (2019).
  111. S. Kim, P. Petreczky, and A. Rothkopf, Quarkonium in-medium properties from realistic lattice NRQCD, J. High Energy Phys. 11 (2018) 088.
  112. R. Larsen, S. Meinel, S. Mukherjee, and P. Petreczky, Thermal broadening of bottomonia: Lattice nonrelativistic QCD with extended operators, Phys. Rev. D 100, 074506 (2019).
  113. D. Banerjee, S. Datta, R. Gavai, and P. Majumdar, Heavy quark momentum diffusion coefficient from lattice QCD, Phys. Rev. D 85, 014510 (2012).
  114. H. T. Ding, A. Francis, O. Kaczmarek, F. Karsch, H. Satz, and W. Soldner, Heavy Quark diffusion from lattice QCD spectral functions, J. Phys. G 38, 124070 (2011).
  115. D. Banerjee, S. Datta, and M. Laine, Lattice study of a magnetic contribution to heavy quark momentum diffusion, J. High Energy Phys. 08 (2022) 128.
  116. N. Brambilla, V. Leino, O. Philipsen, C. Reisinger, A. Vairo, and M. Wagner, Lattice gauge theory computation of the static force, Phys. Rev. D 105, 054514 (2022).
  117. N. Brambilla, H. S. Chung, A. Vairo, and X.-P. Wang, QCD static force in gradient flow, J. High Energy Phys. 01 (2022) 184.
  118. G. D. Moore and D. Teaney, How much do heavy quarks thermalize in a heavy ion collision?, Phys. Rev. C 71, 064904 (2005).
  119. M. G. Mustafa, Energy loss of charm quarks in the quark-gluon plasma: Collisional vs radiative losses, Phys. Rev. C 72, 014905 (2005).
  120. S. Caron-Huot and G. D. Moore, Heavy quark diffusion in perturbative QCD at next-to-leading order, Phys. Rev. Lett. 100, 052301 (2008).
  121. C. H. Simon and G. D. Moore, Heavy quark diffusion in QCD and N=4 SYM at next-to-leading order, J. High Energy Phys. 02 (2008) 081.
  122. J. I. Kapusta and C. Gale, Finite-Temperature Field Theory: Principles and Applications, 2nd ed., Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2006).

  123. The analysis is a little formal since the integral over d3q is UV divergent and needs to be regularized by an ultraviolet completion of the theory [17]. This detail does not change the arguments.

  124. M. L. Bellac, Thermal Field Theory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2011).

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Phys. Rev. D 112, 114038– Published 23 December, 2025
  • Received 16 May 2025
  • Accepted 22 October 2025
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