- 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

Pion phenomenology from the thermal soft-wall model of holographic QCD

Narmin Nasibova1,2,3,* and Xerxes D. Arsiwalla4,†

  • *Contact author: n.nesibli88@gmail.com
  • †Contact author: x.d.arsiwalla@gmail.com
PDF

Phys. Rev. D 112, 116010 – Published 12 December, 2025

DOI: https://doi.org/10.1103/m327-xs85

Abstract

Within the framework of the thermal soft-wall model of AdS/QCD, we investigate phenomenological properties of pions at finite temperature. This includes the electromagnetic form factor Fπ(Q2,T), the thermal mass Mπ(T), charge radius rπ(T), the generalized parton distribution (GPD) Hπ(x,Q2,T), the charge density ρπ(b,T) of the pion, the pion-nucleon coupling constant gπNN(T), and pion-Δ baryon coupling constant gπΔΔ(T) coupling constant at finite temperature. The thermal pion form factor is extrapolated from the zero-temperature case. Subsequently, the GPD is obtained at finite temperature from this form factor. The above-mentioned quantities were analyzed using a thermal dilaton field in the five-dimensional anti–de Sitter action. Moreover, we determine the theoretical expression for the temperature-dependent pion-nucleon coupling constant, and the pion-Δ baryon coupling constant using thermal profile functions of the nucleon, Δ baryon and the pion. Our results show that the values of these quantities decrease with increasing temperature and vanish near the critical temperature Tc, except for the pion radius, which diverges at Tc. Our results reproduce expected features of low-energy hadron dynamics, thus validating the phenomenological utility of the thermal soft-wall model.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (93)

  1. J. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999).
  2. Makoto Natsuume, AdS/CFT Duality User Guide, Lecture Notes in Physics Vol. 903 (Springer, New York, 2015).
  3. Anastasia Golubtsova, Alexander Nikolaev, and Mikhail Podoinitsyn, Holographic RG flows in a 3D gauged supergravity at finite temperature, Phys. Rev. D 110, 066011 (2024).
  4. Anastasia Golubtsova, Alexander Nikolaev, and Mikhail Podoinitsyn, Stability analysis of holographic RG flows in 3D supergravity, Eur. Phys. J. Plus 138, 1 (2023).
  5. H. Boschi-Filho and N. R. F. Braga, Gauge/string duality and scalar glueball mass ratios, J. High Energy Phys. 05 (2003) 009.
  6. H. Boschi-Filho and N. R. F. Braga, QCD/string holographic mapping and glueball mass spectrum, Eur. Phys. J. C 32, 529 (2004).
  7. J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, QCD and a holographic model of hadrons, Phys. Rev. Lett. 95, 261602 (2005).
  8. L. Da Rold and A. Pomarol, Chiral symmetry breaking from five dimensional spaces, Nucl. Phys. B721, 79 (2005).
  9. A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Linear confinement and AdS/QCD, Phys. Rev. D 74, 015005 (2006).
  10. L. Da Rold and A. Pomarol, The scalar and pseudoscalar sectors in a five-dimensional approach to chiral symmetry breaking, J. High Energy Phys. 01 (2006) 157.
  11. S. J. Brodsky, F. de Teramond, H. G. Dosch, and, J. Erlich, Light-front holographic QCD and emerging confinement, Phys. Rep. 584, 1 (2015).
  12. 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).
  13. K. Ghoroku, N. Maru, M. Tachibana, and M. Yahiro, Holographic model for hadrons in deformed AdS5 background, Phys. Lett. B 633, 602 (2006).
  14. W. de Paula, T. Frederico, H. Forkel, and M. Beyer, Dynamical holographic QCD with area-law confinement and linear Regge trajectories, Phys. Rev. D 79, 075019 (2009).
  15. M. Rinaldi and V. Vento, Meson and glueball spectroscopy within the graviton soft wall model, Phys. Rev. D 104, 034016 (2021).
  16. T. M. Aliev, T. Barakat, and M. Savc𝚤, Study of the b→Nℓℓ decay in light cone sum rules, Eur. Phys. J. Plus 136, 635 (2021).
  17. T. M. Aliev, S. Bilmiş, and M. Savc𝚤, Analysis of the light JP=3− mesons in QCD sum rules, Phys. Rev. D 108, 034020 (2023).
  18. T. M. Aliev, A. Özpineci, and M. Savc𝚤, Form factors of Ds+→ϕℓ¯ν decay in QCD light cone sum rule, Eur. Phys. J. C 38, 85 (2004).
  19. M. Shekari Tousi, K. Azizi, and H. R. Moshfegh, Investigation of the semileptonic decay within QCD sum rules, Phys. Rev. D 110, 114001 (2024).
  20. T. Gutsche, V. E. Lyubovitskij, and I. Schmidt, A. Vega, Nucleon structure including high Fock states in AdS/QCD, Phys. Rev. D 86, 036007 (2012).
  21. H. Hashamipour, M. Goharipour, and S. Gousheh, Nucleon axial form factor from generalized parton distributions, Phys. Rev. D 100, 016001 (2019).
  22. H. Hashamipour, M. Goharipour, K. Azizi, and S. V. Goloskokov, Generalized parton distributions at zero skewness, Phys. Rev. D 107, 096005 (2023).
  23. M. Goharipour, H. Hashamipour, F. Irani, and K. Azizi, Impact of JLab data on the determination of GPDs at zero skewness and new insights from transition form factors N→Δ, Phys. Rev. D 109, 074042 (2024).
  24. Satyajit Puhan, Navdeet Kaur, and Harleen Dahiya, Transverse and spatial structure of light to heavy pseudoscalar mesons in light-cone quark model, Phys. Rev. D 111, 014008 (2025).
  25. T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Y. Trifonov, Baryons in a soft-wall AdS-Schwarzschild approach at low temperature, Phys. Rev. D 99, 114023 (2019).
  26. T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Y. Trifonov, Mesons in a soft-wall AdS-Schwarzschild approach at low temperature, Phys. Rev. D 99, 054030 (2019).
  27. Sh. Mamedov and N. Nasibova, Temperature dependence of ρ meson-nucleon coupling constant from the AdS/QCD soft-wall model, Phys. Rev. D 104, 036010 (2021).
  28. N. Nasibova, Meson-Δ and meson-nucleon-Δ transition coupling constants in the soft-wall model of holographic QCD at finite temperature, Lett. High Energy Phys. 326, 31526 (2022).
  29. N. Nasibova, Isospin symmetry of ω meson at finite temperature in the soft-wall model of holographic QCD, Few-Body Syst. 66, 4 (2025).
  30. M. Cheng et al., Transition temperature in QCD, Phys. Rev. D 74, 054507 (2006).
  31. A. Ayala, C. A. Dominguez, M. Loewe, and Y. Zhang, Finite temperature QCD sum rules: A review, Adv. High Energy Phys. 113, 9291623 (2017).
  32. 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).
  33. Sh. Mamedov and N. Nasibova, Axial-vector form factor of nucleons at finite temperature from the AdS/QCD soft-wall model, Int. J. Mod. Phys. A, 38, 2350131 (2023).
  34. T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Y. Trifonov, Electromagnetic properties of the nucleon and the Roper resonance in soft-wall AdS/QCD at finite temperature, Nucl. Phys. B952, 114934 (2020).
  35. Z. Abidin and C. Carlson, Nucleon electromagnetic and gravitational form factors from holography, Phys. Rev. D 79, 115003 (2009).
  36. M. Allahverdiyeva and Sh. Mamedov, Vector meson gravitational form factors and generalized parton distributions at finite temperature within the soft-wall AdS/QCD model, Eur. Phys. J. C 83, 447 (2023).
  37. Sh. Mamedov and Sh. Taghiyeva, a1 meson-nucleon coupling constant at finite temperature from the soft-wall AdS/QCD model, Eur. Phys. J. C 81, 1080 (2021).
  38. W. Broniowski and E. R. Arriola, Impact-parameter dependence of the generalized parton distribution of the pion in chiral quark models, Phys. Lett. B 574, 57 (2003).
  39. A. E. Dorokhov, W. Broniowski, and E. R. Arriola, Generalized quark transversity distribution of the pion in chiral quark models, Phys. Rev. D 84, 074015 (2011).
  40. R. M. Davidson and E. R. Arriola, Parton distributions functions of pion, kaon and eta pseudoscalar mesons in the NJL model, Acta Phys. Pol. B 33, 1791 (2002), https://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=33&page=1791.
  41. S. Dalley, Mesons on a transverse lattice, Phys. Rev. D 64, 036006 (2001).
  42. Xiang Gao, Nikhil Karthik, Swagato Mukherjee, Peter Petreczky, Sergey Syritsyn, and Yong Zhao, Pion form factor and charge radius from lattice QCD at the physical point, Phys. Rev. D 104, 114515 (2021).
  43. J.van der Heide, J. H. Koch, and E. Laermann, The pion form factor on the lattice at zero and finite temperature, Few-Body Syst. 36, 119 (2004).
  44. Navdeep Kaur, Narinder Kumar, Chandan Mondal, and Harleen Dahiya, Generalized parton distributions of pion for non-zero skewness in AdS/QCD, Nucl. Phys. B934, 80 (2018).
  45. A. V. Radyushkin, Generalized parton distributions and their singularities, Phys. Rev. D 83, 076006 (2011).
  46. M. Diehl, Generalized parton distributions, Phys. Rep. 388, 41 (2003).
  47. X. Cao, M. Baggioli, H. Liu, and D. Li, Pion dynamics in a soft-wall AdS-QCD model, J. High Energy Phys. 12 (2022) 113.
  48. John G. Taylor, Michael J. Moravcsik, and Jack L. Uretsky, Determination of the pion-nucleon coupling constant from photoproduction angular distribution, Phys. Rev. 113, 689 (1959).
  49. A. Vega and M. A. Martin Contreras, Melting of scalar hadrons in an AdS/QCD model modified by a thermal dilaton, Nucl. Phys. B942, 410 (2019).
  50. J. Gasser and H. Leutwyler, Light quarks at low temperatures, Phys. Lett. B 184, 83 (1987).
  51. H. C. Ahn, D. K. Hong, C. Park, and S. Siwach, Spin 3/2 baryons and form factors in AdS/QCD, Phys. Rev. D 80, 054001 (2009).
  52. A. Cherman, T. D. Cohen, and E. S. Werbos, Chiral condensate in holographic models of QCD, Phys. Rev. C 79, 045203 (2009).
  53. M. A. Martin Contreras, E. Folco Capossoli, D. Li, A. Vega, and H. Boschi-Filho, Pion form factor from an AdS deformed background, Nucl. Phys. B977, 115726 (2022).
  54. T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Vega, Chiral symmetry breaking and meson wave functions in soft-wall AdS/QCD, Phys. Rev. D 87, 056001 (2013).
  55. M. Gell-Mann, R. J. Oakes, and B. Renner, Behavior of current divergences under SU3SU3, Phys. Rev. 175, 2195 (1968).
  56. J. Jankowski, D. Blaschke, and M. Spaliński, Chiral condensate in hadronic matter, Phys. Rev. D 87, 105018 (2013).
  57. A. Gómez Nicola, J. R. Pelaez, A. Dobado, and F. J. Llanes-Estrada, Thermal meson properties within chiral perturbation theory, AIP Conf. Proc. 660, 156 (2003).
  58. H. Nematollahi and K. Azizi, Unpolarized valence GPDs and form factors of the pion in the modified chiral quark model, Phys. Rev. D 111, 014011 (2025).
  59. M. Shekari Tousi and K. Azizi, Properties of doubly heavy spin- baryons: The ground and excited states, Phys. Rev. D 109, 054005 (2024).
  60. T. Horn et al. (Jefferson Lab Fπ Collaboration), Determination of the pion charge form factor at Q2=1.60 and 2.45(  GeV/c)2, Phys. Rev. Lett. 97, 192001 (2006).
  61. Herry J. Kwee and Richard F. Lebed, Pion form factors in holographic QCD, J. High Energy Phys. 01 (2008) 027.
  62. S. R. Amendolia et al. (NA7 Collaboration), Measurement of the pion form factor in the time-like region for q2 values between 0.1  (GeV/c)2 and 0.18  (GeV/c)2, Phys. Lett. B 138, 454 (1984).
  63. V. Tadevosyan et al. (Jefferson Lab Fπ Collaboration), Determination of the pion charge form factor for Q2=0.60–1.60  GeV2, Phys. Rev. C 75, 055205 (2007).
  64. H. Ackermann, T. Azemoon, W. Gabriel, H. D. Mertiens, H. D. Reich, G. Specht, F. Janata, and D. Schmidt, Determination of the longitudinal and the transverse part in π+ electroproduction, Nucl. Phys. B137, 294 (1978).
  65. C. J. Bebek et al., Electroproduction of single pions at low epsilon and a measurement of the pion form-factor up to q2=10  GeV2, Phys. Rev. D 17, 1693 (1978).
  66. Y. Aoki, T. Blum, G. Colangelo, S. Collins, M. D. Morte, P. Dimopoulos et al. (Flavour Lattice Averaging Group (FLAG), FLAG review 2021, Eur. Phys. J. C 82, 869 (2022).
  67. B. Sheng, Y. Wang, X. Wang, and L. Yu, Pole and screening masses of neutral pions in a hot and magnetized medium: A comprehensive study in the Nambu–Jona-Lasinio model, Phys. Rev. D 103, 094001 (2021).
  68. D. Bećirević and V. Lubicz, Estimate of the chiral condensate in quenched lattice QCD, Phys. Lett. B 600, 83 (2004).
  69. P. A. Zyla et al. (Particle Data Group), Review of particle physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  70. J. A. Oller and E. Oset, Chiral symmetry amplitudes in the S-wave isoscalar and isovector channels and the σ, f0(980), a0(980) scalar mesons, Nucl. Phys. A620, 438 (1997).
  71. D. Fernández-Fraile and A. G. Nicola, Electrical conductivity of a pion gas, Phys. Rev. D 73, 045025 (2006).
  72. J. M. Torres-Rincon, Degeneracy patterns of chiral companions at finite temperature, Symmetry 13, 1400 (2021).
  73. A. Schenk, Pion propagation at finite temperature, Phys. Rev. D 47, 5138 (1993).
  74. J. Gasser and G. R. Zarnauskas, On the pion decay constant, Phys. Lett. B 693, 122 (2010).
  75. C. A. Dominguez, C. Van Gend, and M. Loewe, Pion-nucleon coupling at finite temperature, Phys. Lett. B 429, 64 (1998).
  76. Zhu-Fang Cui, Daniele Binosi, Craig D. Roberts, and Sebastian M. Schmidt, Pion charge radius from pion+electron elastic scattering data, Phys. Lett. B 822, 136631 (2021).
  77. Xiang Gao, Nikhil Karthik, Swagato Mukherjee, Peter Petreczky, Sergey Syritsyn, and Yong Zhao, Pion form factor and charge radius from lattice QCD at the physical point, Phys. Rev. D 104, 114515 (2021).
  78. A. V. Radyushkin and H. R. Grigoryan, Pion form factor in chiral limit of hard-wall AdS/QCD model, Phys. Rev. D 76, 115007 (2007).
  79. S. J. Brodsky and G. F. De Teramond, Light-front dynamics and AdS/QCD correspondence: The pion form factor in the space- and time-like regions, Phys. Rev. D 77, 056007 (2008).
  80. J. de Boer, K. Papadodimas, and E. Verlinde, Holographic neutron stars, J. High Energy Phys. 10 (2010) 020.
  81. X. Arsiwalla, J. de Boer, K. Papadodimas, and E. Verlinde, Degenerate stars and gravitational collapse in AdS/CFT, J. High Energy Phys. 01 (2011) 144.
  82. V. A. Babenko and N. M. Petrov, Relation between the charged and neutral pion-nucleon coupling constants in the Yukawa model, Phys. Part. Nucl. Lett. 14, 58 (2017).
  83. R. L. Altmeyer, M. Gockeler, R. Horsley, E. Laermann, and G. Schierholz, Hadronic coupling constants in lattice QCD, Z. Phys. C 68, 443 (1995).
  84. D. V. Bugg, The pion nucleon coupling constant, Eur. Phys. J. C 33, 505 (2004).
  85. N. Maru and M. Tachibana, Meson–nucleon coupling from AdS/QCD, Eur. Phys. J. C 63, 123 (2009).
  86. K. Chelabi, Z. Fang, M. Huang, D. Li, and Y. L. Wu, Realization of chiral symmetry breaking and restoration in holographic QCD, Phys. Rev. D 93, 101901 (2016).
  87. K. Chelabi, Z. Fang, M. Huang, D. Li, and Y. L. Wu, Chiral phase transition in the soft-wall model of AdS/QCD, J. High Energy Phys. 04 (2016) 036.
  88. C. R. Argüelles and N. E. Grandi, Fermionic halos at finite temperature in AdS/CFT, J. High Energy Phys. 05 (2018) 118.
  89. C. R. Argüelles, T. Canavesi, M. Díaz, and N. E. Grandi, Thermodynamic instabilities in holographic neutron stars at finite temperature, Classical Quantum Gravity 37, 205002 (2020).
  90. X. D. Arsiwalla, Entropy functions with 5D Chern-Simons terms, J. High Energy Phys. 09 (2009) 059.
  91. X. D. Arsiwalla, Supersymmetric black holes as probes of quantum gravity, Ph.D. thesis, University of Amsterdam, 2010, https://pure.uva.nl/ws/files/871677/75420.
  92. X. D. Arsiwalla, More rings to rule them all: Fragmentation, 4D↔5D and split-spectral flows, J. High Energy Phys. 02 (2008) 066.
  93. N. Nasibova and X. D. Arsiwalla, The calculation of GPD, charge density and form factor of pion in the thermal soft wall model, 10.13140/RG.2.2.11587.87847/1 (2025).

Outline

Information

Phys. Rev. D 112, 116010– Published 12 December, 2025
  • Received 13 July 2025
  • Accepted 20 November 2025
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