- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Scaling functions in the soft-wall AdS/QCD models

Zhongzheng Zhang1, Danning Li1,*, Lang Yu2,†, Zhibin Li3, and Xinyang Wang4

  • *Contact author: lidanning@jnu.edu.cn
  • †Contact author: yulang@jlu.edu.cn
PDF

Phys. Rev. D 113, 086010 – Published 28 April, 2026

DOI: https://doi.org/10.1103/7njb-6v24

Abstract

We investigate the static scaling behavior of the chiral condensate near the two-flavor critical point within the framework of the soft-wall AdS/QCD. The scaling functions are extracted from the chiral order parameters and are found to precisely match those obtained through mean-field calculations. Additionally, it is also checked that the scaling functions are independent of the specific construction of the holographic model. Furthermore, we develop the formalism for calculating the chiral susceptibility and demonstrate that the pseudocritical temperatures obey the scaling law for moderate quark masses. It is shown that the temperature scaling could be comparable with those obtained from Dyson-Schwinger equations and lattice simulations. While the soft-wall AdS/QCD framework predicts mean-field critical exponents that are universal, the critical coefficients and the crossover dynamics are model dependent and provide quantitative constraints for phenomenological model building. These findings could help improve the effectiveness of the soft-wall AdS/QCD.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (71)

  1. P. Braun-Munzinger, K. Redlich, and J. Stachel, Quark-Gluon Plasma 3 (World Scientific, Singapore, 2004), p. 491.
  2. E. Shuryak, Rev. Mod. Phys. 89, 035001 (2017).
  3. D. H. Rischke, Prog. Part. Nucl. Phys. 52, 197 (2004).
  4. M. Huang and P. Zhuang, Symmetry 15, 541 (2023).
  5. R. D. Pisarski and F. Wilczek, Phys. Rev. D 29, 338 (1984).
  6. S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo (Wuppertal-Budapest Collaboration), J. High Energy Phys. 09 (2010) 073.
  7. M. Cheng et al., Phys. Rev. D 74, 054507 (2006).
  8. H.-T. Ding, F. Karsch, and S. Mukherjee, Int. J. Mod. Phys. E 24, 1530007 (2015).
  9. Z. Bai, L. Chang, J. Chao, F. Gao, and Y.-X. Liu, Phys. Rev. D 104, 014005 (2021).
  10. J. B. Kogut and D. K. Sinclair, Phys. Rev. D 73, 074512 (2006).
  11. S. Ejiri, F. Karsch, E. Laermann, C. Miao, S. Mukherjee, P. Petreczky, C. Schmidt, W. Soeldner, and W. Unger, Phys. Rev. D 80, 094505 (2009).
  12. A. Bazavov et al., Phys. Rev. D 85, 054503 (2012).
  13. F. Burger, E.-M. Ilgenfritz, M. Kirchner, M. P. Lombardo, M. Müller-Preussker, O. Philipsen, C. Urbach, and L. Zeidlewicz (tmfT Collaboration), Phys. Rev. D 87, 074508 (2013).
  14. A. Y. Kotov, M. P. Lombardo, and A. Trunin, Phys. Lett. B 823, 136749 (2021).
  15. A. Y. Kotov, M. P. Lombardo, and A. Trunin, Symmetry 13, 1833 (2021).
  16. H. T. Ding et al. (HotQCD Collaboration), Phys. Rev. Lett. 123, 062002 (2019).
  17. F. Gao and J. M. Pawlowski, Phys. Rev. D 105, 094020 (2022).
  18. J. Bernhardt and C. S. Fischer, Phys. Rev. D 108, 114018 (2023).
  19. J. Braun, B. Klein, and P. Piasecki, Eur. Phys. J. C 71, 1576 (2011).
  20. J. Braun et al., Phys. Rev. D 111, 094010 (2025).
  21. Z. Fodor and S. D. Katz, Phys. Lett. B 534, 87 (2002).
  22. G. ’t Hooft, Conf. Proc. C 930308, 284 (1993), arXiv:gr-qc/9310026.
  23. L. Susskind, J. Math. Phys. (N.Y.) 36, 6377 (1995).
  24. J. M. Maldacena, AIP Conf. Proc. 484, 51 (1999).
  25. G. Policastro, D. T. Son, and A. O. Starinets, Phys. Rev. Lett. 87, 081601 (2001).
  26. A. Buchel and J. T. Liu, Phys. Rev. Lett. 93, 090602 (2004).
  27. P. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005).
  28. S. J. Brodsky, G. F. de Téramond, H. G. Dosch, and J. Erlich, Phys. Rep. 584, 1 (2015).
  29. U. Gursoy, M. Jarvinen, and G. Nijs, Phys. Rev. Lett. 120, 242002 (2018).
  30. S. S. Gubser, A. Nellore, S. S. Pufu, and F. D. Rocha, Phys. Rev. Lett. 101, 131601 (2008).
  31. U. Gursoy, E. Kiritsis, L. Mazzanti, and F. Nitti, Phys. Rev. Lett. 101, 181601 (2008).
  32. D. Li, S. He, M. Huang, and Q.-S. Yan, J. High Energy Phys. 09 (2011) 041.
  33. S. I. Finazzo, R. Rougemont, H. Marrochio, and J. Noronha, J. High Energy Phys. 02 (2015) 051.
  34. R. Zöllner and B. Kämpfer, Eur. Phys. J. Plus 135, 304 (2020).
  35. Y.-Q. Zhao, S. He, D. Hou, L. Li, and Z. Li, J. High Energy Phys. 04 (2023) 115.
  36. X. Chen and M. Huang, Phys. Rev. D 109, L051902 (2024).
  37. J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, Phys. Rev. Lett. 95, 261602 (2005).
  38. A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Phys. Rev. D 74, 015005 (2006).
  39. T. Gherghetta, J. I. Kapusta, and T. M. Kelley, Phys. Rev. D 79, 076003 (2009).
  40. D. Li and M. Huang, J. High Energy Phys. 11 (2013) 088.
  41. K. Chelabi, Z. Fang, M. Huang, D. Li, and Y.-L. Wu, J. High Energy Phys. 04 (2016) 036.
  42. K. Chelabi, Z. Fang, M. Huang, D. Li, and Y.-L. Wu, Phys. Rev. D 93, 101901 (2016).
  43. D. Li and M. Huang, J. High Energy Phys. 02 (2017) 042.
  44. S. P. Bartz and T. Jacobson, Phys. Rev. C 97, 044908 (2018).
  45. Z. Fang, Y.-L. Wu, and L. Zhang, Phys. Lett. B 762, 86 (2016).
  46. H. A. Ahmed, M. Kawaguchi, and M. Huang, Phys. Rev. D 110, 046002 (2024).
  47. J. Chen, S. He, M. Huang, and D. Li, J. High Energy Phys. 01 (2019) 165.
  48. F. R. Brown, F. P. Butler, H. Chen, N. H. Christ, Z.-h. Dong, W. Schaffer, L. I. Unger, and A. Vaccarino, Phys. Rev. Lett. 65, 2491 (1990).
  49. Y. Chen, M. Ding, D. Li, K. Bitaghsir Fadafan, and M. Huang, Phys. Rev. D 111, 126010 (2025).
  50. M. Lv, D. Li, and S. He, J. High Energy Phys. 11 (2019) 026.
  51. H. Nishihara and M. Harada, Phys. Rev. D 89, 076001 (2014).
  52. X. Chen, D. Li, D. Hou, and M. Huang, J. High Energy Phys. 03 (2020) 073.
  53. D. Li, M. Huang, Y. Yang, and P.-H. Yuan, J. High Energy Phys. 02 (2017) 030.
  54. D. M. Rodrigues, D. Li, E. Folco Capossoli, and H. Boschi-Filho, Phys. Rev. D 98, 106007 (2018).
  55. P. Colangelo, J. J. Sanz-Cillero, and F. Zuo, J. High Energy Phys. 11 (2012) 012.
  56. X. Cao, H. Liu, and D. Li, Phys. Rev. D 102, 126014 (2020).
  57. X. Cao, S. Qiu, H. Liu, and D. Li, J. High Energy Phys. 08 (2021) 005.
  58. X. Cao, M. Baggioli, H. Liu, and D. Li, J. High Energy Phys. 12 (2022) 113.
  59. D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 88, 202302 (2002).
  60. A. Cherman, T. D. Cohen, and E. S. Werbos, Phys. Rev. C 79, 045203 (2009).
  61. J. P. Boyd, Chebyshev and Fourier Spectral Methods (Courier Corporation, New York, 2001).
  62. W. Liang, X. Cao, H. Liu, and D. Li, Phys. Rev. D 108, 096019 (2023).
  63. L. P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V. Palciauskas, M. Rayl, J. Swift, D. Aspnes, and J. Kane, Rev. Mod. Phys. 39, 395 (1967).
  64. J. Engels and F. Karsch, Phys. Rev. D 85, 094506 (2012).
  65. E. Grossi, A. Soloviev, D. Teaney, and F. Yan, Phys. Rev. D 104, 034025 (2021).
  66. J. Engels and F. Karsch, Phys. Rev. D 85, 094506 (2012).
  67. J. Engels and F. Karsch, Phys. Rev. D 90, 014501 (2014).
  68. F. Gao and Y.-x. Liu, Phys. Rev. D 94, 076009 (2016).
  69. G. Aarts et al., Phys. Rev. D 105, 034504 (2022).
  70. J. Braun, W.-j. Fu, J. M. Pawlowski, F. Rennecke, D. Rosenblüh, and S. Yin, Phys. Rev. D 102, 056010 (2020).
  71. A. Holl, P. Maris, and C. D. Roberts, Phys. Rev. C 59, 1751 (1999).

Outline

Information

Phys. Rev. D 113, 086010– Published 28 April, 2026
  • Received 6 August 2025
  • Accepted 27 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation