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Relaxation for massive particles: Transport and causality

Matej Bajec* and Alexander Soloviev†

  • *Contact author: matej.bajec@fmf.uni-lj.si
  • †Contact author: alexander.soloviev@fmf.uni-lj.si
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Phys. Rev. D 112, 065002 – Published 9 September, 2025

DOI: https://doi.org/10.1103/v4tj-fj5k

Abstract

Correlators at finite density and temperature encode important information about a physical system, such as transport and causality. For a massive gas of particles in the relaxation time approximation of kinetic theory, we provide complete analytic results for all first order transport coefficients, including thermoelectric coefficients. We demonstrate the first complete picture of the complex structure of the correlators as a function of mass, providing an interpretation in terms of the causal structure of light cones. We provide a simple criterion to extract the effective light cone velocity from the cut of the retarded correlator.

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References (55)

  1. L. Pitaevskii and E. Lifshitz, Physical Kinetics: Volume 10, let. 10 (Butterworth-Heinemann, Washington, DC, 2012).
  2. S. de Groot, W. van Leeuwen, and C. van Weert, Relativistic Kinetic Theory (North-Holland, Amsterdam, 1980).
  3. J. Anderson and H. Witting, Physica (Utrecht) 74, 466 (1974).
  4. R. Liboff, Kinetic Theory (Springer-Verlag, New York, 2003).
  5. S. Grozdanov, N. Kaplis, and A. O. Starinets, J. High Energy Phys. 07 (2016) 151.
  6. P. Romatschke, Eur. Phys. J. C 76, 352 (2016).
  7. W. Florkowski, R. Ryblewski, and M. Strickland, Nucl. Phys. A916, 249 (2013).
  8. W. Florkowski, R. Ryblewski, and M. Strickland, Phys. Rev. C 88, 024903 (2013).
  9. W. Florkowski, E. Maksymiuk, R. Ryblewski, and M. Strickland, Phys. Rev. C 89, 054908 (2014).
  10. A. Kurkela and U. A. Wiedemann, Eur. Phys. J. C 79, 776 (2019).
  11. A. Kurkela, U. A. Wiedemann, and B. Wu, Eur. Phys. J. C 79, 965 (2019).
  12. D. Dash, S. Bhadury, S. Jaiswal, and A. Jaiswal, Phys. Lett. B 831, 137202 (2022).
  13. V. E. Ambrus, S. Schlichting, and C. Werthmann, Phys. Rev. D 107, 094013 (2023).
  14. R. Brants, Phys. Rev. D 110, 116027 (2024).
  15. J. Hu, arXiv:2409.05131.
  16. S. K. Singh, S. Bhadury, M. Kurian, and V. Chandra, Phys. Rev. D 111, 114007 (2025).
  17. A. Mukherjee, S. Bhadury, and P. Singha, arXiv:2504.11572.
  18. J.-P. Blaizot and L. Yan, Phys. Rev. C 104, 055201 (2021).
  19. I. Aniceto, J. Noronha, and M. Spalínski, Phys. Rev. D 111, 076025 (2025).
  20. V. E. Ambrus, E. Molnár, and D. H. Rischke, Phys. Rev. D 106, 076005 (2022).
  21. S. Jaiswal, J.-P. Blaizot, R. S. Bhalerao, Z. Chen, A. Jaiswal, and L. Yan, Phys. Rev. C 106, 044912 (2022).
  22. V. E. Ambruş, E. Molnár, and D. H. Rischke, Phys. Rev. D 109, 076001 (2024).
  23. S. Ochsenfeld and S. Schlichting, J. High Energy Phys. 09 (2023) 186.
  24. M. Bajec, S. Grozdanov, and A. Soloviev, J. High Energy Phys. 08 (2024) 065.
  25. N. Abbasi and D. H. Rischke, J. High Energy Phys. 05 (2025) 241.
  26. G. S. Rocha and G. S. Denicol, arXiv:2505.14823.
  27. K. Dusling and T. Schäfer, Phys. Rev. C 85, 044909 (2012).
  28. A. Das, H. Mishra, and R. K. Mohapatra, Phys. Rev. D 102, 014030 (2020).
  29. M. Zebarjadi, S. E. Rezaei, M. S. Akhanda, and K. Esfarjani, Phys. Rev. B 103, 144404 (2021).
  30. P. Romatschke, Phys. Rev. D 85, 065012 (2012).
  31. A. Hataei, R. Heydari, and F. Taghinavaz, arXiv:2504.14591.
  32. X. Lin, Q.-Z. Sun, X.-H. Wu, and J. Hu, arXiv:2505.04444.
  33. S. Grozdanov, A. Lucas, and N. Poovuttikul, Phys. Rev. D 99, 086012 (2019).
  34. M. P. Heller, A. Serantes, M. Spaliński, and B. Withers, Phys. Rev. Lett. 130, 261601 (2023).
  35. M. P. Heller, A. Serantes, M. Spaliński, and B. Withers, Nat. Phys. 20, 1948 (2024).
  36. L. Gavassino, M. M. Disconzi, and J. Noronha, Phys. Rev. Lett. 132, 162301 (2024).
  37. S. A. Hartnoll and S. P. Kumar, J. High Energy Phys. 12 (2005) 036.
  38. G. S. Rocha, I. Danhoni, K. Ingles, G. S. Denicol, and J. Noronha, Phys. Rev. D 110, 076003 (2024).
  39. J. Casalderrey-Solana, S. Grozdanov, and A. O. Starinets, Phys. Rev. Lett. 121, 191603 (2018).
  40. S. Grozdanov and A. O. Starinets, J. High Energy Phys. 04 (2019) 080.
  41. U. Gursoy and E. Kiritsis, J. High Energy Phys. 02 (2008) 032.
  42. U. Gursoy, E. Kiritsis, and F. Nitti, J. High Energy Phys. 02 (2008) 019.
  43. P. Betzios, U. Gürsoy, M. Järvinen, and G. Policastro, Phys. Rev. D 97, 081901 (2018).
  44. I. S. Gradshteyn, I. M. Ryzhik, D. Zwillinger, and V. Moll, Table of Integrals, Series, and Products, 8th ed. (Academic Press, Amsterdam, 2015).
  45. P. Kovtun, J. Phys. A 45, 473001 (2012).
  46. K. Hattori, M. Hongo, and X.-G. Huang, Symmetry 14, 1851 (2022).
  47. C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications (Birkhäuser, Basel, 2002).
  48. S. A. Hartnoll, P. K. Kovtun, M. Muller, and S. Sachdev, Phys. Rev. B 76, 144502 (2007).
  49. S. A. Hartnoll, A. Lucas, and S. Sachdev, arXiv:1612.07324.
  50. A. Kurkela, U. A. Wiedemann, and B. Wu, Eur. Phys. J. C 79, 759 (2019).
  51. S. Grozdanov and A. Soloviev, arXiv:2501.00099.
  52. D. Bazow, G. S. Denicol, U. Heinz, M. Martinez, and J. Noronha, Phys. Rev. Lett. 116, 022301 (2016).
  53. A. Kurkela, A. Mukhopadhyay, F. Preis, A. Rebhan, and A. Soloviev, J. High Energy Phys. 08 (2018) 054.
  54. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1964).
  55. L. V. Ahlfors, Complex Analysis, 2nd ed. (McGraw-Hill Book Company, New York, 2018).

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Phys. Rev. D 112, 065002– Published 9 September, 2025
  • Received 3 July 2025
  • Accepted 13 August 2025
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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.

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