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Holographic QCD model for heavy and exotic mesons at finite density: A self-consistent dynamical approach

Bruno Toniato1,2,*, David Dudal2,†, Subhash Mahapatra3,‡, Roldao da Rocha4,§, and Siddhi Swarupa Jena3,∥

  • *Contact author: bruno.toniato@ufabc.edu.br
  • †Contact author: david.dudal@kuleuven.be
  • ‡Contact author: mahapatrasub@nitrkl.ac.in
  • §Contact author: roldao.rocha@ufabc.edu.br
  • ∥Contact author: 519ph2015@nitrkl.ac.in
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Phys. Rev. D 111, 126021 – Published 27 June, 2025

DOI: https://doi.org/10.1103/pn9x-ycnh

Abstract

We present a self-consistent dynamical holographic quantum chromodynamics (QCD) model to investigate the mass spectra and melting behavior of heavy and exotic mesons at finite temperature and finite density. Our approach is based on the Einstein-Maxwell-dilaton (EMD) framework and incorporates an elsewhere already introduced, albeit by hand, phenomenological nonquadratic dilaton profile. This allows one to capture the nonlinear Regge trajectories of heavy-flavor mesons and model certain exotic states. We show how to construct such models by actually solving the coupled Einstein, Maxwell, and dilaton field equations, ensuring mathematical self-consistency to replace any ad hoc input. At finite temperature, we analyze the confinement-deconfinement transition via a Hawking-Page phase transition. We compute the spectral functions, revealing the sequential melting of quarkonia as the temperature is increased. Extending to finite density, we explore the impact of baryon chemical potential on meson stability, showing significant modifications in spectral peaks and effective potentials that indicate a more rapid melting of mesonic states as the chemical potential increases in the deconfined phase. The dual of the small/large black hole transition now indicates toward a first order phase transition line ending at a second order critical point. Interestingly, the spectral functions smoothly cross this phase transition line.

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References (113)

  1. A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Linear confinement and AdS/QCD, Phys. Rev. D 74, 015005 (2006).
  2. T. Branz, T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Vega, Light and heavy mesons in a soft-wall holographic approach, Phys. Rev. D 82, 074022 (2010).
  3. P. Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeau, and S. Nicotri, Light scalar mesons in the soft-wall model of AdS/QCD, Phys. Rev. D 78, 055009 (2008).
  4. S. J. Brodsky, G. F. de Teramond, H. G. Dosch, and J. Erlich, Light-front holographic QCD and emerging confinement, Phys. Rep. 584, 1 (2015).
  5. J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, QCD and a holographic model of hadrons, Phys. Rev. Lett. 95, 261602 (2005).
  6. T. Sakai and S. Sugimoto, Low energy hadron physics in holographic QCD, Prog. Theor. Phys. 113, 843 (2005).
  7. T. Sakai and S. Sugimoto, More on a holographic dual of QCD, Prog. Theor. Phys. 114, 1083 (2005).
  8. R. Rougemont, J. Grefa, M. Hippert, J. Noronha, J. Noronha-Hostler, I. Portillo, and C. Ratti, Hot QCD phase diagram from holographic Einstein–Maxwell–dilaton models, Prog. Part. Nucl. Phys. 135, 104093 (2024).
  9. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  10. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  11. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998).
  12. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Large N field theories, string theory and gravity, Phys. Rep. 323, 183 (2000).
  13. M. Ammon and J. Erdmenger, Gauge/Gravity Duality: Foundations and Applications (Cambridge University Press, Cambridge, England, 2015), 10.1017/CBO9780511846373.
  14. S. S. Afonin, Bottom-up holographic approach to meson spectroscopy, Phys. Part. Nucl. 53, 96 (2022).
  15. W. de Paula and T. Frederico, Scalar mesons within a dynamical holographic QCD model, Phys. Lett. B 693, 287 (2010).
  16. A. Ballon-Bayona, H. Boschi-Filho, L. A. H. Mamani, A. S. Miranda, and V. T. Zanchin, Effective holographic models for QCD: Glueball spectrum and trace anomaly, Phys. Rev. D 97, 046001 (2018).
  17. A. Ballon-Bayona, L. A. H. Mamani, and D. M. Rodrigues, Spontaneous chiral symmetry breaking in holographic soft wall models, Phys. Rev. D 104, 126029 (2021).
  18. G. Karapetyan, W. de Paula, and R. da Rocha, Configurational information measure of mesonic states in 4-flavor AdS/QCD, Phys. Lett. B 846, 138174 (2023).
  19. S. P. Bartz, A. Dhumuntarao, and J. I. Kapusta, Dynamical AdS/Yang-Mills model, Phys. Rev. D 98, 026019 (2018).
  20. A. Ballon-Bayona, T. Frederico, L. A. H. Mamani, and W. de Paula, Dynamical holographic QCD model for spontaneous chiral symmetry breaking and confinement, Phys. Rev. D 108, 106016 (2023).
  21. I. Aref’eva and K. Rannu, Holographic anisotropic background with confinement-deconfinement phase transition, J. High Energy Phys. 05 (2018) 206.
  22. I. Y. Aref’eva, A. Hajilou, P. Slepov, and M. Usova, Running coupling for holographic QCD with heavy and light quarks: Isotropic case, Phys. Rev. D 110, 126009 (2024).
  23. J. Erdmenger, N. Evans, W. Porod, and K. S. Rigatos, Gauge/gravity dual dynamics for the strongly coupled sector of composite Higgs models, J. High Energy Phys. 02 (2021) 058.
  24. U. Gursoy and E. Kiritsis, Exploring improved holographic theories for QCD: Part I, J. High Energy Phys. 02 (2008) 032.
  25. U. Gursoy, E. Kiritsis, and F. Nitti, Exploring improved holographic theories for QCD: Part II, J. High Energy Phys. 02 (2008) 019.
  26. U. Gursoy, E. Kiritsis, L. Mazzanti, G. Michalogiorgakis, and F. Nitti, Improved holographic QCD, From Gravity to Thermal Gauge Theories: The AdS/CFT Correspondence, Lecture Notes in Physics (Springer, Berlin, Heidelberg, 2011), Vol. 828, p. 79.
  27. S. He, S.-Y. Wu, Y. Yang, and P.-H. Yuan, Phase structure in a dynamical soft-wall holographic QCD model, J. High Energy Phys. 04 (2013) 093.
  28. T. Alho, M. Järvinen, K. Kajantie, E. Kiritsis, C. Rosen, and K. Tuominen, A holographic model for QCD in the Veneziano limit at finite temperature and density, J. High Energy Phys. 04 (2014) 124.
  29. J. Alanen, K. Kajantie, and V. Suur-Uski, A gauge/gravity duality model for gauge theory thermodynamics, Phys. Rev. D 80, 126008 (2009).
  30. C. Csaki and M. Reece, Toward a systematic holographic QCD: A braneless approach, J. High Energy Phys. 05 (2007) 062.
  31. T. Gherghetta, J. I. Kapusta, and T. M. Kelley, Chiral symmetry breaking in the soft-wall AdS/QCD model, Phys. Rev. D 79, 076003 (2009).
  32. S. P. Bartz, R. C. Meadows, and G. Brock, Chiral phase transition in soft-wall AdS/QCD with scalar-dilaton coupling, Phys. Rev. D 110, 026027 (2024).
  33. N. R. F. Braga and L. F. Ferreira, Bottomonium dissociation in a finite density plasma, Phys. Lett. B 773, 313 (2017).
  34. L. F. Ferreira and R. da Rocha, Pion family in AdS/QCD: The next generation from configurational entropy, Phys. Rev. D 99, 086001 (2019).
  35. R. da Rocha, Deploying heavier η meson states: Configurational entropy hybridizing AdS/QCD, Phys. Lett. B 814, 136112 (2021).
  36. L. F. Ferreira and R. da Rocha, Tensor mesons, AdS/QCD and information, Eur. Phys. J. C 80, 375 (2020).
  37. N. R. F. Braga, L. F. Ferreira, and R. da Rocha, Thermal dissociation of heavy mesons and configurational entropy, Phys. Lett. B 787, 16 (2018).
  38. N. R. F. Braga, L. F. Ferreira, and O. C. Junqueira, Configuration entropy of a rotating quark-gluon plasma from holography, Phys. Lett. B 847, 138265 (2023).
  39. P. Colangelo and F. Loparco, Configurational entropy can disentangle conventional hadrons from exotica, Phys. Lett. B 788, 500 (2019).
  40. L. F. Ferreira and R. da Rocha, Nucleons and higher spin baryon resonances: An AdS/QCD configurational entropic incursion, Phys. Rev. D 101, 106002 (2020).
  41. A. E. Bernardini and R. da Rocha, Informational entropic Regge trajectories of meson families in AdS/QCD, Phys. Rev. D 98, 126011 (2018).
  42. N. Barbosa-Cendejas, R. Cartas-Fuentevilla, A. Herrera-Aguilar, R. R. Mora-Luna, and R. da Rocha, Dynamical tachyonic AdS/QCD and information entropy, Phys. Lett. B 782, 607 (2018).
  43. R. da Rocha, Information entropy in AdS/QCD: Mass spectroscopy of isovector mesons, Phys. Rev. D 103, 106027 (2021).
  44. R. da Rocha and P. H. O. Silva, Deformed AdS/QCD, mesonic mass spectra, and the differential configurational entropy: Still a margin for heavier resonances, Phys. Rev. D 110, 126019 (2024).
  45. M. A. Martin Contreras and A. Vega, Nonlinear Regge trajectories with AdS/QCD, Phys. Rev. D 102, 046007 (2020).
  46. G. Karapetyan and R. da Rocha, Configurational entropy of heavy-quark QCD exotica, Eur. Phys. J. Plus 136, 993 (2021).
  47. M. A. Martin Contreras and A. Vega, Holographic stability for non-qq¯ candidates, Phys. Rev. D 108, 126024 (2023).
  48. O. DeWolfe, S. S. Gubser, and C. Rosen, A holographic critical point, Phys. Rev. D 83, 086005 (2011).
  49. O. DeWolfe, S. S. Gubser, and C. Rosen, Dynamic critical phenomena at a holographic critical point, Phys. Rev. D 84, 126014 (2011).
  50. R. Rougemont, J. Noronha, and J. Noronha-Hostler, Suppression of baryon diffusion and transport in a baryon rich strongly coupled quark-gluon plasma, Phys. Rev. Lett. 115, 202301 (2015).
  51. J. Noronha, Connecting Polyakov loops to the thermodynamics of SU(Nc) gauge theories using the gauge-string duality, Phys. Rev. D 81, 045011 (2010).
  52. J. Noronha, The heavy quark free energy in QCD and in gauge theories with gravity duals, Phys. Rev. D 82, 065016 (2010).
  53. S. He, S.-Y. Wu, Y. Yang, and P.-H. Yuan, Phase structure in a dynamical soft-wall holographic QCD Model, J. High Energy Phys. 04 (2013) 093.
  54. A. Ballon-Bayona, H. Boschi-Filho, E. F. Capossoli, and D. M. Rodrigues, Criticality from Einstein-Maxwell-dilaton holography at finite temperature and density, Phys. Rev. D 102, 126003 (2020).
  55. S. S. Gubser, A. Nellore, S. S. Pufu, and F. D. Rocha, Thermodynamics and bulk viscosity of approximate black hole duals to finite temperature quantum chromodynamics, Phys. Rev. Lett. 101, 131601 (2008).
  56. H. Bohra, D. Dudal, A. Hajilou, and S. Mahapatra, Anisotropic string tensions and inversely magnetic catalyzed deconfinement from a dynamical AdS/QCD model, Phys. Lett. B 801, 135184 (2020).
  57. H. Bohra, D. Dudal, A. Hajilou, and S. Mahapatra, Chiral transition in the probe approximation from an Einstein-Maxwell-dilaton gravity model, Phys. Rev. D 103, 086021 (2021).
  58. D. Dudal, A. Hajilou, and S. Mahapatra, A quenched 2-flavour Einstein–Maxwell–dilaton gauge-gravity model, Eur. Phys. J. A 57, 142 (2021).
  59. R.-G. Cai, S. He, and D. Li, A hQCD model and its phase diagram in Einstein-Maxwell-dilaton system, J. High Energy Phys. 03 (2012) 033.
  60. Y. Yang and P.-H. Yuan, Confinement-deconfinement phase transition for heavy quarks in a soft wall holographic QCD model, J. High Energy Phys. 12 (2015) 161.
  61. A. J. Ferreira-Martins, P. Meert, and R. da Rocha, Deformed AdS4–Reissner–Nordström black branes and shear viscosity-to-entropy density ratio, Eur. Phys. J. C 79, 646 (2019).
  62. S. S. Jena, B. Shukla, D. Dudal, and S. Mahapatra, Entropic force and real-time dynamics of holographic quarkonium in a magnetic field, Phys. Rev. D 105, 086011 (2022).
  63. B. Shukla, D. Dudal, and S. Mahapatra, Anisotropic and frame dependent chaos of suspended strings from a dynamical holographic QCD model with magnetic field, J. High Energy Phys. 06 (2023) 178.
  64. B. Shukla, J. Nongmaithem, D. Dudal, and S. Mahapatra, Interplay of magnetic field and chemical potential induced anisotropy and frame dependent chaos of a QQ¯ pair in holographic QCD, Phys. Rev. D 111, 106002 (2025).
  65. D. Dudal and S. Mahapatra, Thermal entropy of a quark-antiquark pair above and below deconfinement from a dynamical holographic QCD model, Phys. Rev. D 96, 126010 (2017).
  66. S. He, M. Huang, Q.-S. Yan, and Y. Yang, Confront holographic QCD with Regge trajectories, Eur. Phys. J. C 66, 187 (2010).
  67. M. Rinaldi, Double parton correlations in mesons within AdS/QCD soft-wall models: A first comparison with lattice data, Eur. Phys. J. C 80, 678 (2020).
  68. S. S. Afonin and I. V. Pusenkov, Universal description of radially excited heavy and light vector mesons, Phys. Rev. D 90, 094020 (2014).
  69. J.-K. Chen, Regge trajectories for the mesons consisting of different quarks, Eur. Phys. J. C 78, 648 (2018).
  70. F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Hadronic molecules, Rev. Mod. Phys. 90, 015004 (2018).
  71. R. F. Lebed, R. E. Mitchell, and E. S. Swanson, Heavy-quark QCD exotica, Prog. Part. Nucl. Phys. 93, 143 (2017).
  72. R. Oncala and J. Soto, Heavy quarkonium hybrids: Spectrum, decay and mixing, Phys. Rev. D 96, 014004 (2017).
  73. R. L. Jaffe, Exotica, Phys. Rep. 409, 1 (2005).
  74. Particle Data Group, Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  75. C. A. Meyer and E. S. Swanson, Hybrid mesons, Prog. Part. Nucl. Phys. 82, 21 (2015).
  76. L. Bellantuono, P. Colangelo, and F. Giannuzzi, Exotic JPC=1−+ mesons in a holographic model of QCD, Eur. Phys. J. C 74, 2830 (2014).
  77. BESIII Collaboration, Observation of Zc(3900)0 in e+e−→π0π0J/ψ, Phys. Rev. Lett. 115, 112003 (2015).
  78. Y. Liu, M. A. Nowak, and I. Zahed, Heavy holographic exotics: Tetraquarks as Efimov states, Phys. Rev. D 100, 126023 (2019).
  79. S. Godfrey and S. L. Olsen, The exotic XYZ charmonium-like mesons, Annu. Rev. Nucl. Part. Sci. 58, 51 (2008).
  80. Q.-F. Lü, D.-Y. Chen, and Y.-B. Dong, Open charm and bottom tetraquarks in an extended relativized quark model, Phys. Rev. D 102, 074021 (2020).
  81. LHCb Collaboration, Recent LHCb results on heavy hadron spectroscopy, Nucl. Part. Phys. Proc. 318–323, 66 (2022).
  82. LHCb Collaboration, Observation of new resonances decaying to J/ψK++ and J/ψϕ, Phys. Rev. Lett. 127, 082001 (2021).
  83. S. Campanella, P. Colangelo, and F. De Fazio, Excited heavy meson decays to light vector mesons: Implications for spectroscopy, Phys. Rev. D 98, 114028 (2018).
  84. S. Mahapatra, S. Priyadarshinee, G. N. Reddy, and B. Shukla, Exact topological charged hairy black holes in AdS space in D-dimensions, Phys. Rev. D 102, 024042 (2020).
  85. S. Mahapatra and P. Roy, On the time dependence of holographic complexity in a dynamical Einstein-dilaton model, J. High Energy Phys. 11 (2018) 138.
  86. S. Priyadarshinee and S. Mahapatra, Analytic three-dimensional primary hair charged black holes and thermodynamics, Phys. Rev. D 108, 044017 (2023).
  87. S. Priyadarshinee, S. Mahapatra, and I. Banerjee, Analytic topological hairy dyonic black holes and thermodynamics, Phys. Rev. D 104, 084023 (2021).
  88. A. Daripa and S. Mahapatra, Analytic three-dimensional primary hair charged black holes with Coulomb-like electrodynamics and their thermodynamics, Phys. Rev. D 109, 124039 (2024).
  89. M. A. Martin Contreras, S. Diles, and A. Vega, Heavy quarkonia spectroscopy at zero and finite temperature in bottom-up AdS/QCD, Phys. Rev. D 103, 086008 (2021).
  90. L. A. H. Mamani, D. Hou, and N. R. F. Braga, Melting of heavy vector mesons and quasinormal modes in a finite density plasma from holography, Phys. Rev. D 105, 126020 (2022).
  91. C. P. Herzog, A holographic prediction of the deconfinement temperature, Phys. Rev. Lett. 98, 091601 (2007).
  92. J.-K. Chen, Regge trajectories for the mesons consisting of different quarks, Eur. Phys. J. C 78, 648 (2018).
  93. M. D. Scadron, R. Delbourgo, and G. Rupp, Constituent quark masses and the electroweak standard model, J. Phys. G 32, 735 (2006).
  94. N. R. F. Braga, L. F. Ferreira, and A. Vega, Holographic model for charmonium dissociation, Phys. Lett. B 774, 476 (2017).
  95. B. Lucini, M. Teper, and U. Wenger, The high temperature phase transition in SU(N) gauge theories, J. High Energy Phys. 01 (2004) 061.
  96. W.-S. Hou, C.-S. Luo, and G.-G. Wong, Glueball states in a constituent gluon model, Phys. Rev. D 64, 014028 (2001).
  97. D. Dudal and T. G. Mertens, Radiation gauge in AdS/QCD: Inadmissibility and implications on spectral functions in the deconfined phase, Phys. Lett. B 751, 352 (2015).
  98. S. S. Jena, J. Barman, B. Toniato, D. Dudal, and S. Mahapatra, A dynamical Einstein-Born-Infeld-dilaton model and holographic quarkonium melting in a magnetic field, J. High Energy Phys. 12 (2024) 096.
  99. G. A. Baker and P. Graves-Morris, Padé Approximants, 2nd ed., Encyclopedia of Mathematics and Its Applications (Cambridge University Press, Cambridge, England, 1996).
  100. A. Iserles, On the generalized Padé approximations to the exponential function, SIAM J. Numer. Anal. 16, 631 (1979).
  101. A. S. Miranda, C. A. Ballon Bayona, H. Boschi-Filho, and N. R. F. Braga, Black-hole quasinormal modes and scalar glueballs in a finite-temperature AdS/QCD model, J. High Energy Phys. 11 (2009) 119.
  102. L. A. H. Mamani, A. S. Miranda, and V. T. Zanchin, Melting of scalar mesons and black-hole quasinormal modes in a holographic QCD model, Eur. Phys. J. C 79, 1 (2019).
  103. M. Fujita, K. Fukushima, T. Misumi, and M. Murata, Finite-temperature spectral function of the vector mesons in an AdS/QCD model, Phys. Rev. D 80, 035001 (2009).
  104. M. Fujita, T. Kikuchi, K. Fukushima, T. Misumi, and M. Murata, Melting spectral functions of the scalar and vector mesons in a holographic QCD model, Phys. Rev. D 81, 065024 (2010).
  105. P. de Forcrand and O. Philipsen, The QCD phase diagram for small densities from imaginary chemical potential, Nucl. Phys. B642, 290 (2002).
  106. M. A. Stephanov, QCD phase diagram and the critical point, Prog. Theor. Phys. Suppl. 153, 139 (2004).
  107. S. Borsanyi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto et al., QCD crossover at finite chemical potential from lattice simulations, Phys. Rev. Lett. 125, 052001 (2020).
  108. S. Borsanyi, Z. Fodor, J. N. Guenther, P. Parotto, A. Pasztor, C. Ratti et al., Lattice QCD constraints on the critical point from an improved precision equation of state, arXiv:2502.10267.
  109. M. Hippert, J. Grefa, T. A. Manning, J. Noronha, J. Noronha-Hostler, I. Portillo Vazquez et al., Bayesian location of the QCD critical point from a holographic perspective, Phys. Rev. D 110, 094006 (2024).
  110. O. Andreev, QQ¯qqq quark system, compact pentaquark, and gauge/string duality. II, Phys. Rev. D 108, 106012 (2023).
  111. LHCb Collaboration, Observation of a narrow pentaquark state, Pc(4312)+, and of two-peak structure of the Pc(4450)+, Phys. Rev. Lett. 122, 222001 (2019).
  112. LHCb Collaboration, Observation of J/ψp resonances consistent with pentaquark states in Λb0→J/ψK−p decays, Phys. Rev. Lett. 115, 072001 (2015).
  113. LHCb Collaboration, Observation of a J/ψΛ resonance consistent with a strange pentaquark candidate in B−→J/ψΛp− decays, Phys. Rev. Lett. 131, 031901 (2023).

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Phys. Rev. D 111, 126021– Published 27 June, 2025
  • Received 30 April 2025
  • Accepted 5 June 2025
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