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

Transport simulations with a constrained momentum-dependent chiral mean-field equation of state at various isospin fractions

Jan Steinheimer1,2,* and Marcus Bleicher3,4

  • *Contact author: j.steinheimer-froschauer@gsi.de

Phys. Rev. C 114, 044904 – Published 13 October, 2026

DOI: https://doi.org/10.1103/6j14-cm4m

Abstract

We present a comparison of the ultrarelativistic quantum molecular dynamics model using a chiral mean field (CMF) equation of state with flow and pion production data in heavy ion collisions at low beam energies Elab=0.2–2.0A GeV and varying isospin fraction. The CMF model parameters are constrained by known properties of the high density equation of state of quantum chromodynamics at varying isospin fractions. This allows us to calculate the equation of state as well as nuclear interactions for different physical systems like neutron stars and heavy ion collisions in a consistent way. It is found that heavy ion reactions at the upcoming FAIR facility will only have marginal sensitivity on the isospin dependence of the high-density equation of state. At lower beam energies, comparing to FOPI, HADES, and SπRIT data, the sensitivity is higher but the observed deviations and systematic uncertainties of the existing data are larger than the sensitivity to the symmetry energy.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (60)

  1. M. Oertel, M. Hempel, T. Klähn, and S. Typel, Equations of state for supernovae and compact stars, Rev. Mod. Phys. 89, 015007 (2017).
  2. A. Sorensen, K. Agarwal, K. W. Brown, Z. Chajęcki, P. Danielewicz, C. Drischler, S. Gandolfi, J. W. Holt, M. Kaminski, C.-M. Ko, R. Kumar, B.-A. Li, W. G. Lynch, A. B. McIntosh, W. G. Newton, S. Pratt, O. Savchuk, M. Stefaniak, I. Tews, M. B. Tsang, et al., Dense nuclear matter equation of state from heavy-ion collisions, Prog. Part. Nucl. Phys. 134, 104080 (2024).
  3. A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, Equation of state of nucleon matter and neutron star structure, Phys. Rev. C 58, 1804 (1998).
  4. J. Lattimer and M. Prakash, Neutron star observations: Prognosis for equation of state constraints, Phys. Rep. 442, 109 (2007).
  5. J. Aichelin, A. Rosenhauer, G. Peilert, H. Stoecker, and W. Greiner, Importance of momentum-dependent interactions for the extraction of the nuclear equation of state from high-energy heavy-ion collisions, Phys. Rev. Lett. 58, 1926 (1987).
  6. W. Cassing, G. Batko, B. Blättel, V. Koch, A. Lang, U. Mosel, K. Niita, K. Weber, and G. Wolf, Dynamics of heavy-ion reactions and the nuclear equation of state, Nucl. Phys. A 519, 357 (1990).
  7. P. Danielewicz, Determination of the mean-field momentum-dependence using elliptic flow, Nucl. Phys. A 673, 375 (2000).
  8. C. Hartnack, H. Oeschler, and J. Aichelin, Hadronic matter is soft, Phys. Rev. Lett. 96, 012302 (2006).
  9. O. Soloveva, J. Aichelin, and E. Bratkovskaya, Transport properties and equation-of-state of hot and dense QGP matter near the critical endpoint in the phenomenological dynamical quasiparticle model, Phys. Rev. D 105, 054011 (2022).
  10. M. Omana Kuttan, A. Motornenko, J. Steinheimer, H. Stoecker, Y. Nara, and M. Bleicher, A chiral mean-field equation-of-state in UrQMD: Effects on the heavy ion compression stage, Eur. Phys. J. C 82, 427 (2022).
  11. J. Steinheimer, A. Motornenko, A. Sorensen, Y. Nara, V. Koch, and M. Bleicher, The high-density equation of state in heavy-ion collisions: Constraints from proton flow, Eur. Phys. J. C 82, 911 (2022).
  12. J. Zhao, J. Aichelin, and E. Bratkovskaya, Formulation of fully covariant Quantum-Molecular Dynamics for an N-body system with scalar and vector potentials, arXiv:2511.13975.
  13. H. Stöcker and W. Greiner, High energy heavy ion collisions—Probing the equation of state of highly excited hardronic matter, Phys. Rep. 137, 277 (1986).
  14. P. Danielewicz, R. Lacey, and W. G. Lynch, Determination of the equation of state of dense matter, Science 298, 1592 (2002).
  15. M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano, Z. Li, W. G. Lynch, and A. W. Steiner, Constraints on the density dependence of the symmetry energy, Phys. Rev. Lett. 102, 122701 (2009).
  16. A. Le Fèvre, Y. Leifels, W. Reisdorf, J. Aichelin, and C. Hartnack, Constraining the nuclear matter equation of state around twice saturation density, Nucl. Phys. A 945, 112 (2016).
  17. D. Oliinychenko, A. Sorensen, V. Koch, and L. McLerran, Sensitivity of Au+Au collisions to the symmetric nuclear matter equation of state at 2–5 nuclear saturation densities, Phys. Rev. C 108, 034908 (2023).
  18. M. Omana Kuttan, J. Steinheimer, K. Zhou, and H. Stoecker, QCD equation of state of dense nuclear matter from a Bayesian analysis of heavy-ion collision data, Phys. Rev. Lett. 131, 202303 (2023).
  19. J. Hong and P. Danielewicz, Subthreshold pion production within a transport description of central Au + Au collisions, Phys. Rev. C 90, 024605 (2014).
  20. C. Fuchs, A. Faessler, E. Zabrodin, and Y.-M. Zheng, Probing the nuclear equation of state by K+ production in heavy-ion collisions, Phys. Rev. Lett. 86, 1974 (2001).
  21. S. Huth, P. T. H. Pang, I. Tews, T. Dietrich, A. Le Fèvre, A. Schwenk, W. Trautmann, K. Agarwal, M. Bulla, M. W. Coughlin, and C. Van Den Broeck, Constraining neutron-star matter with microscopic and macroscopic collisions, Nature (London) 606, 276 (2022).
  22. B. Li, L. Chen, and C. Ko, Recent progress and new challenges in isospin physics with heavy-ion reactions, Phys. Rep. 464, 113 (2008).
  23. M. B. Tsang, J. R. Stone, F. Camera, P. Danielewicz, S. Gandolfi, K. Hebeler, C. J. Horowitz, J. Lee, W. G. Lynch, Z. Kohley, R. Lemmon, P. Möller, T. Murakami, S. Riordan, X. Roca-Maza, F. Sammarruca, A. W. Steiner, I. Vidaña, and S. J. Yennello, Constraints on the symmetry energy and neutron skins from experiments and theory, Phys. Rev. C 86, 015803 (2012).
  24. B.-A. Li, Isospin dependence of the π−/π+ ratio and density dependence of the nuclear symmetry energy, Phys. Rev. C 67, 017601 (2003).
  25. B.-A. Li, G.-C. Yong, and W. Zuo, Near-threshold pion production with radioactive beams, Phys. Rev. C 71, 014608 (2005).
  26. J. Estee, W. G. Lynch, C. Y. Tsang, J. Barney, G. Jhang, M. B. Tsang, R. Wang, M. Kaneko, J. W. Lee, T. Isobe, M. Kurata-Nishimura, T. Murakami, D. S. Ahn, L. Atar, T. Aumann, H. Baba, K. Boretzky, J. Brzychczyk, G. Cerizza, N. Chiga, et al., Probing the symmetry energy with the spectral pion ratio, Phys. Rev. Lett. 126, 162701 (2021).
  27. J. Steinheimer, T. Reichert, Y. Nara, and M. Bleicher, Momentum dependent potentials from a parity doubling CMF model in UrQMD: Results on flow and particle production, J. Phys. G: Nucl. Part. Phys. 52, 035103 (2025).
  28. J. Steinheimer, S. Schramm, and H. Stöcker, Hadronic SU(3) parity doublet model for dense matter and its extension to quarks and the strange equation of state, Phys. Rev. C 84, 045208 (2011).
  29. A. Motornenko, S. Pal, A. Bhattacharyya, J. Steinheimer, and H. Stoecker, Repulsive properties of hadrons in lattice QCD data and neutron stars, Phys. Rev. C 103, 054908 (2021).
  30. J. Steinheimer, M. O. Kuttan, T. Reichert, Y. Nara, and M. Bleicher, Simultaneous description of high density QCD matter in heavy ion collisions and neutron star observations, Phys. Lett. B 867, 139605 (2025).
  31. R. Negreiros, L. Brodie, J. Steinheimer, V. Dexheimer, and R. D. Pisarski, Enhanced neutrino cooling from parity-doubled nucleons in neutron star cooling simulations, arXiv:2603.06789.
  32. R. L. Workman, V. D. Burkert, V. Crede, E. Klempt, U. Thoma, L. Tiator, K. Agashe, G. Aielli, B. C. Allanach, C. Amsler, M. Antonelli, E. C. Aschenauer, D. M. Asner, H. Baer, S. Banerjee, R. M. Barnett, L. Baudis, C. W. Bauer, J. J. Beatty, V. I. Belousov, et al., Review of particle physics, Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  33. D. Bollweg, D. A. Clarke, J. Goswami, O. Kaczmarek, F. Karsch, S. Mukherjee, P. Petreczky, C. Schmidt, and S. Sharma et al. (HotQCD Collaboration), Equation of state and speed of sound of (2+1)-flavor QCD in strangeness-neutral matter at nonvanishing net baryon-number density, Phys. Rev. D 108, 014510 (2023).
  34. S. Borsányi, G. Endrődi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabó, QCD equation of state at nonzero chemical potential: Continuum results with physical quark masses at order μ2, J. High Energy Phys. 08 (2012) 053.
  35. J. Keller, K. Hebeler, and A. Schwenk, Nuclear equation of state for arbitrary proton fraction and temperature based on chiral effective field theory and a Gaussian process emulator, Phys. Rev. Lett. 130, 072701 (2023).
  36. S. Hama, B. C. Clark, E. D. Cooper, H. S. Sherif, and R. L. Mercer, Global Dirac optical potentials for elastic proton scattering from heavy nuclei, Phys. Rev. C 41, 2737 (1990).
  37. J. P. Blaizot, Nuclear compressibilities, Phys. Rep. 64, 171 (1980).
  38. D. H. Youngblood, Y. W. Lui, H. L. Clark, B. John, Y. Tokimoto, and X. Chen, Isoscalar E0−E3 strength in Sn116, Sm144, Sm154, and Pb208, Phys. Rev. C 69, 034315 (2004).
  39. U. Garg and G. Colò, The compression-mode giant resonances and nuclear incompressibility, Prog. Part. Nucl. Phys. 101, 55 (2018).
  40. A. Jinno, K. Murase, Y. Nara, and A. Ohnishi, Repulsive Λ potentials in dense neutron star matter and binding energy of Λ in hypernuclei, Phys. Rev. C 108, 065803 (2023).
  41. T. E. Riley, A. L. Watts, S. Bogdanov, P. S. Ray, R. M. Ludlam, S. Guillot, Z. Arzoumanian, C. L. Baker, A. V. Bilous, D. Chakrabarty, K. C. Gendreau, A. K. Harding, W. C. G. Ho, J. M. Lattimer, S. M. Morsink, and T. E. Strohmayer, A NICER view of PSR J0030+0451: Millisecond pulsar parameter estimation, Astrophys. J. Lett. 887, L21 (2019).
  42. T. E. Riley, A. L. Watts, P. S. Ray, S. Bogdanov, S. Guillot, S. M. Morsink, A. V. Bilous, Z. Arzoumanian, D. Choudhury, J. S. Deneva, K. C. Gendreau, A. K. Harding, W. C. G. Ho, J. M. Lattimer, M. Loewenstein, R. M. Ludlam, C. B. Markwardt, T. Okajima, C. Prescod-Weinstein, R. A. Remillard, et al., A NICER view of the massive pulsar PSR J0740+6620 informed by radio timing and XMM-Newton spectroscopy, Astrophys. J. Lett. 918, L27 (2021).
  43. D. Choudhury, T. Salmi, S. Vinciguerra, T. E. Riley, Y. Kini, A. L. Watts, B. Dorsman, S. Bogdanov, S. Guillot, P. S. Ray, D. J. Reardon, R. A. Remillard, A. V. Bilous, D. Huppenkothen, J. M. Lattimer, N. Rutherford, Z. Arzoumanian, K. C. Gendreau, S. M. Morsink, and W. C. G. Ho, A NICER view of the nearest and brightest millisecond pulsar: PSR J0437–4715, Astrophys. J. Lett. 971, L20 (2024).
  44. J. R. Oppenheimer and G. M. Volkoff, On massive neutron cores, Phys. Rev. 55, 374 (1939).
  45. P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, A two-solar-mass neutron star measured using Shapiro delay, Nature (London) 467, 1081 (2010).
  46. J. Antoniadis, P. C. C. Freire, N. Wex, T. M. Tauris, R. S. Lynch, M. H. van Kerkwijk, M. Kramer, C. Bassa, V. S. Dhillon, T. Driebe, J. W. T. Hessels, V. M. Kaspi, V. I. Kondratiev, N. Langer, T. R. Marsh, M. A. McLaughlin, T. T. Pennucci, S. M. Ransom, I. H. Stairs, J. van Leeuwen, et al., A massive pulsar in a compact relativistic binary, Science 340, 1233232 (2013).
  47. A. Bauswein, O. Just, H.-T. Janka, and N. Stergioulas, Neutron-star radius constraints from GW170817 and future detections, Astrophys. J. Lett. 850, L34 (2017).
  48. B. Margalit and B. D. Metzger, Constraining the maximum mass of neutron stars from multi-messenger observations of GW170817, Astrophys. J. Lett. 850, L19 (2017).
  49. L. Rezzolla, E. R. Most, and L. R. Weih, Using gravitational-wave observations and quasi-universal relations to constrain the maximum mass of neutron stars, Astrophys. J. Lett. 852, L25 (2018).
  50. A. Motornenko, J. Steinheimer, V. Vovchenko, S. Schramm, and H. Stoecker, Equation of state for hot QCD and compact stars from a mean-field approach, Phys. Rev. C 101, 034904 (2020).
  51. Y. Lu, F. Gao, Y.-x. Liu, and J. M. Pawlowski, Finite density signatures of confining and chiral dynamics in QCD thermodynamics and fluctuations of conserved charges, Phys. Rev. D 113, 054019 (2026).
  52. J. Aichelin and H. Stöcker, Quantum molecular dynamics—A novel approach to N-body correlations in heavy ion collisions, Phys. Lett. B 176, 14 (1986).
  53. J. Aichelin, “Quantum” molecular dynamics—A dynamical microscopic n-body approach to investigate fragment formation and the nuclear equation of state in heavy ion collisions, Phys. Rep. 202, 233 (1991).
  54. S. A. Bass et al., Microscopic models for ultrarelativistic heavy ion collisions, Prog. Part. Nucl. Phys. 41, 255 (1998).
  55. W. Reisdorf, Y. Leifels, A. Andronic, R. Averbeck, V. Barret, Z. Basrak, N. Bastid, M.L. Benabderrahmane, R. Čaplar, P. Crochet, P. Dupieux, M. Dželalija, Z. Fodor, P. Gasik, Y. Grishkin, O.N. Hartmann, N. Herrmann, K.D. Hildenbrand, B. Hong, T.I. Kang, et al., Systematics of azimuthal asymmetries in heavy ion collisions in the regime, Nucl. Phys. A 876, 1 (2012).
  56. J. Adamczewski-Musch, O. Arnold, C. Behnke, A. Belounnas, A. Belyaev, J. C. Berger-Chen, A. Blanco, C. Blume, M. Böhmer, P. Bordalo, S. Chernenko, L. Chlad, I. Ciepal, C. Deveaux, J. Dreyer, E. Epple, L. Fabbietti, O. Fateev, P. Filip, P. Fonte, et al., Directed, elliptic, and higher order flow harmonics of protons, deuterons, and tritons in Au+Au collisions at NN=2.4GeV, Phys. Rev. Lett. 125, 262301 (2020).
  57. J. Adamczewski-Musch, O. Arnold, C. Behnke, A. Belounnas, J. C. Berger-Chen, A. Blanco, C. Blume, M. Böhmer, P. Bordalo, L. Chlad, I. Ciepal, C. Deveaux, J. Dreyer, E. Epple, L. Fabbietti, P. Filip, P. Fonte, C. Franco, J. Friese, I. Fröhlich, et al., Proton, deuteron and triton flow measurements in Au+Au collisions at SNN=2.4GeV, Eur. Phys. J. A 59, 80 (2023).
  58. W. Reisdorf, M. Stockmeier, A. Andronic, M. L. Benabderrahmane, O. N. Hartmann, N. Herrmann, K. D. Hildenbrand, Y. J. Kim, M. Kiš, P. Koczoń, T. Kress, Y. Leifels, X. Lopez, M. Merschmeyer, A. Schüttauf, V. Barret, Z. Basrak, N. Bastid, R. Čaplar, P. Crochet, et al., Systematics of pion emission in heavy ion collisions in the regime, Nucl. Phys. A 781, 459 (2007).
  59. J. Adamczewski-Musch, O. Arnold, C. Behnke, A. Belounnas, A. Belyaev, J. C. Berger-Chen, J. Biernat, A. Blanco, C. Blume, M. Böhmer, P. Bordalo, S. Chernenko, L. Chlad, C. Deveaux, J. Dreyer, A. Dybczak, E. Epple, L. Fabbietti, O. Fateev, P. Filip, et al., Identical pion intensity interferometry at sNN=2.4GeV, Eur. Phys. J. A 56, 140 (2020).
  60. M. D. Cozma, Constraining the density dependence of the symmetry energy using the multiplicity and average pT ratios of charged pions, Phys. Rev. C 95, 014601 (2017).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation