- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

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
  • Access by Xinjiang University

Asymptotics of superfluid Bjorken flow

Alexander Soloviev1,* and Michał Spaliński2,3,†

  • *Contact author: alexander.soloviev@fmf.uni-lj.si
  • †Contact author: michal.spalinski@ncbj.gov.pl
PDF

Phys. Rev. D 114, 034008 – Published 6 August, 2026

DOI: https://doi.org/10.1103/ghpg-qmn9

Abstract

We consider the dynamics of an expanding superfluid modeled by Mueller-Israel-Stewart theory coupled to a complex scalar field with a U(1) symmetry that is spontaneously broken. This is a manageable theoretical setting for explorations of the chiral phase transition of expanding quark-gluon plasma. We study the late proper-time behavior of Bjorken flow in this physical system and find that asymptotic solutions can be expressed as a transseries of a novel form, which contains factors like τ−lnτ. This transseries describes how the information encoded in the initial data is diluted in the course of dissipative evolution. These solutions retain memory of the symmetry-breaking transition and describe two qualitatively different late-time behaviors of the dynamical variables, depending on condensate relaxation rate: either a purely damped falloff or damped oscillations. The possibility that such oscillations could be imprinted in the observed outcomes of heavy ion collision experiments is the main physical insight that follows from our analysis.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (50)

  1. P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45, 473001 (2012).
  2. P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium and Applications to Relativistic Nuclear Collisions (Cambridge University Press, 2019).
  3. D. T. Son, Hydrodynamics of nuclear matter in the chiral limit, Phys. Rev. Lett. 84, 3771 (2000).
  4. D. T. Son, Low-energy quantum effective action for relativistic superfluids, arXiv:hep-ph/0204199.
  5. D. T. Son and M. A. Stephanov, Real time pion propagation in finite temperature QCD, Phys. Rev. D 66, 076011 (2002).
  6. W. Busza, K. Rajagopal, and W. van der Schee, Heavy ion collisions: The big picture, and the big questions, Annu. Rev. Nucl. Part. Sci. 68, 339 (2018).
  7. E. Grossi, A. Soloviev, D. Teaney, and F. Yan, Transport and hydrodynamics in the chiral limit, Phys. Rev. D 102, 014042 (2020).
  8. E. Grossi, A. Soloviev, D. Teaney, and F. Yan, Soft pions and transport near the chiral critical point, Phys. Rev. D 104, 034025 (2021).
  9. A. Florio, E. Grossi, A. Soloviev, and D. Teaney, Dynamics of the O(4) critical point in QCD, Phys. Rev. D 105, 054512 (2022).
  10. A. Florio, E. Grossi, and D. Teaney, Dynamics of the O(4) critical point in QCD: Critical pions and diffusion in model G, Phys. Rev. D 109, 054037 (2024).
  11. A. Florio, E. Grossi, A. Mazeliauskas, A. Soloviev, and D. Teaney, Quenching through the QCD chiral phase transition, Phys. Rev. D 112, 114019 (2025).
  12. A. Florio, E. Grossi, A. Mazeliauskas, A. Soloviev, and D. Teaney, Supercooled Goldstone bosons at the QCD chiral phase transition, Phys. Rev. Lett. 135, 242303 (2025).
  13. I. Müller, Zum Paradoxon der Wärmeleitungstheorie, Z. Phys. 198, 329 (1967).
  14. W. Israel and J. Stewart, Transient relativistic thermodynamics and kinetic theory, Ann. Phys. (N.Y.) 118, 341 (1979).
  15. T. Mitra, A. Mukhopadhyay, and A. Soloviev, Hydrodynamic attractor and novel fixed points in superfluid Bjorken flow, Phys. Rev. D 103, 076014 (2021).
  16. G. K. Buza, T. Mitra, and A. Soloviev, Superfluids in expanding backgrounds and attractor times, Phys. Rev. D 112, 076022 (2025).
  17. L. Gavassino and A. Soloviev, Relativistic superfluid profiles near critical surfaces, arXiv:2512.16797.
  18. M. P. Heller and M. Spalinski, Hydrodynamics beyond the gradient expansion: Resurgence and resummation, Phys. Rev. Lett. 115, 072501 (2015).
  19. J. D. Bjorken, Highly relativistic nucleus-nucleus collisions: The central rapidity region, Phys. Rev. D 27, 140 (1983).
  20. M. P. Heller, R. Jefferson, M. Spaliński, and V. Svensson, Hydrodynamic attractors in phase space, Phys. Rev. Lett. 125, 132301 (2020).
  21. M. Spaliński, Far from equilibrium attractors in phase space, arXiv:2508.21039.
  22. W. Florkowski, M. P. Heller, and M. Spaliński, New theories of relativistic hydrodynamics in the LHC era, Rep. Prog. Phys. 81, 046001 (2018).
  23. A. Soloviev, Hydrodynamic attractors in heavy ion collisions: A review, Eur. Phys. J. C 82, 319 (2022).
  24. J. Jankowski and M. Spaliński, Hydrodynamic attractors in ultrarelativistic nuclear collisions, Prog. Part. Nucl. Phys. 132, 104048 (2023).
  25. M. Spaliński, Small systems and regulator dependence in relativistic hydrodynamics, Phys. Rev. D 94, 085002 (2016).
  26. M. P. Heller, A. Serantes, M. Spaliński, and B. Withers, Rigorous bounds on transport from causality, Phys. Rev. Lett. 130, 261601 (2023).
  27. I. Aniceto and M. Spaliński, Resurgence in extended hydrodynamics, Phys. Rev. D 93, 085008 (2016).
  28. I. Aniceto, B. Meiring, J. Jankowski, and M. Spaliński, The large proper-time expansion of Yang-Mills plasma as a resurgent transseries, J. High Energy Phys. 02 (2019) 073.
  29. I. Aniceto, D. Hasenbichler, and A. O. Daalhuis, The late to early time behaviour of an expanding plasma: Hydrodynamisation from exponential asymptotics, J. Phys. A 56, 195201 (2023).
  30. Y. Akamatsu, A. Mazeliauskas, and D. Teaney, Bulk viscosity from hydrodynamic fluctuations with relativistic hydrokinetic theory, Phys. Rev. C 97, 024902 (2018).
  31. R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, and M. A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance, and holography, J. High Energy Phys. 04 (2008) 100.
  32. S. Bhattacharyya, V. E. Hubeny, S. Minwalla, and M. Rangamani, Nonlinear fluid dynamics from gravity, J. High Energy Phys. 02 (2008) 045.
  33. I. Aniceto, R. Schiappa, and M. Vonk, The resurgence of instantons in string theory, Commun. Num. Theor. Phys. 6, 339 (2012).
  34. I. Aniceto, G. Basar, and R. Schiappa, A primer on resurgent transseries and their asymptotics, Phys. Rep. 809, 1 (2019).
  35. D. Teaney, The effects of viscosity on spectra, elliptic flow, and HBT radii, Phys. Rev. C 68, 034913 (2003).
  36. M. P. Heller, A. Kurkela, M. Spaliński, and V. Svensson, Hydrodynamization in kinetic theory: Transient modes and the gradient expansion, Phys. Rev. D 97, 091503 (2018).
  37. M. P. Heller and V. Svensson, How does relativistic kinetic theory remember about initial conditions?, Phys. Rev. D 98, 054016 (2018).
  38. X. An and M. Spaliński, QGP physics from attractor perturbations, Phys. Rev. D 110, 114043 (2024).
  39. P. K. Kovtun and A. O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72, 086009 (2005).
  40. M. Spaliński, Universal behaviour, transients and attractors in supersymmetric Yang–Mills plasma, Phys. Lett. B 784, 21 (2018).
  41. S. S. Gubser and A. Yarom, Conformal hydrodynamics in Minkowski and de Sitter spacetimes, Nucl. Phys. B846, 469 (2011).
  42. S. Grozdanov, A family of three maximally symmetric boost-invariant flows in relativistic hydrodynamics, arXiv:2510.10769.
  43. A. Soloviev, Close encounters with attractors of the third kind, Phys. Rev. D 113, 074006 (2026).
  44. M. Martinez and C. Plumberg, Maximally symmetric Boost-Invariant solutions of the Boltzmann equation in foliated geometries, Phys. Rev. D 113, 105001 (2026).
  45. R. Singh and A. Soloviev, Spin hydrodynamics on a hyperbolic expanding background, arXiv:2603.02296.
  46. T. van Ritbergen, J. A. M. Vermaseren, and S. A. Larin, The four loop beta function in quantum chromodynamics, Phys. Lett. B 400, 379 (1997).
  47. E. Gardi, G. Grunberg, and M. Karliner, Can the QCD running coupling have a causal analyticity structure?, J. High Energy Phys. 07 (1998) 007.
  48. D. J. Gross, Nobel lecture: The discovery of asymptotic freedom and the emergence of QCD, Rev. Mod. Phys. 77, 837 (2005).
  49. A. Deur, The QCD running coupling, arXiv:2502.06535.
  50. M. P. Heller, A. Mazeliauskas, and T. Preis, Prescaling relaxation to nonthermal attractors, Phys. Rev. Lett. 132, 071602 (2024).

Outline

Information

Phys. Rev. D 114, 034008– Published 6 August, 2026
  • Received 21 April 2026
  • Accepted 13 July 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