Heavy quark momentum diffusion coefficient during hydrodynamization via effective kinetic theory

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Heavy quark momentum diffusion coefficient during hydrodynamization via effective kinetic theory © 2024 the Authors Published version Boguslavski, Kirill; Kurkela, Aleksi; Lappi, Tuomas; Lindenbauer, Florian; Peuron, Jarkko Boguslavski, K., Kurkela, A., Lappi, T., Lindenbauer, F., & Peuron, J. (2024). Heavy quark momentum diffusion coefficient during hydrodynamization via effective kinetic theory. In R. Bellwied, F. Geurts, R. Rapp, C. Ratti, A. Timmins, & I. Vitev (Eds.), 30th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (Quark Matter 2023) (Article 09001). EDP Sciences. EPJ Web of Conferences, 296. https://doi.org/10.1051/epjconf/202429609001 2024 09001 Heavy quark momentum diffusion coefficient during hydrodynamization via effective kinetic theory Kirill Boguslavski1,Aleksi Kurkela2,Tuomas Lappi3,4,Florian Lindenbauer1, and Jarkko Peuron3,4,∗ 1Institute for Theoretical Physics, Technische Universität Wien, 1040 Vienna, Austria 2Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway 3Department of Physics, P.O. Box 35, 40014 University of Jyväskylä, Finland 4Helsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland Abstract. In these proceedings, we compute the heavy quark momentum diffusion coefficient using QCD effective kinetic theory for a plasma going through the bottom-up thermalization scenario until approximate hydrodynamization. This transport coefficient describes heavy quark momentum diffusion in the quark-gluon plasma and is used in many phenomenological frameworks, e.g. in the open quantum systems approach. Our extracted nonthermal diffusion coefficient matches the thermal one for the same energy density within 30%. At large occupation numbers in the earliest stage, the transverse diffusion coefficient dominates, while the longitudinal diffusion coefficient is larger for the underoccupied system in the later stage of hydrodynamization. 1 Introduction Recent studies on the very early stages of ultrarelativistic heavy-ion collisions indicate that transport coefficients are large during the glasma stage [1–7], where the heavy quark diffusion coefficient is estimated to be roughly κ≈O(10)Gev2/fm [8]. In equilibrium we expect κ≈ O(0.1)Gev2/fm [9], as illustrated in the left panel of Fig. 1. The discrepancy is due to the larger energy density at the early stages. The aim of this work [10] is to study κduring the hydrodynamization process to answer the questions: How large and how anisotropic is κ? 2 Theoretical background In effective kinetic theory [12] we describe the evolution of the gluon phase space density f by numerically solving the Boltzmann equation [11] ∂f(p) ∂τ =C1↔2[f]+C2↔2[f]+Cexp[f],(1) where C1↔2describes the effective one to two splittings, C2↔2two to two processes and Cexp incorporates the longitudinal expansion in the form of an effective scattering term. The right panel of Fig. 1 illustrates the evolution of the distribution function (different linestyles ∗e-mail: [email protected] © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (https://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 296, 09001 (2024) https://doi.org/10.1051/epjconf/202429609001 Quark Matter 2023 α−3/2 s/Qsα−13/5 s/Qs=τBMSS τ O(0.1)Gev2 fm O(10)Gev2 fm Hydrodynamics Kinetic theory Glasma κequil κglasma ? 10−210−1100 pλf/p 100 101 102 PT/PL λ=0.5 λ=1 λ=2 λ=5 λ= 10 Init. cond: fh∼1/αs fh∼αsThermal equil. τf≈τBMSS τi=1/Qs Figure 1. Left: Cartoon of the evolution of κfrom the initial nonequilibrium phase to thermal equilibrium. Right: Trajectory of the system on an occupancy-anisotropy plane [11]. correspond to different initial conditions) on the occupancy-anisotropy plane. In order to make a connection to the bottom-up thermalization picture [13], we use time markers. The star symbol indicates occupancy fh∼1/λ,λ=g2Nc, coinciding with maximum anisotropy for small couplings. The circle marker indicates minimum occupancy. The triangle marker is located at approximate isotropy, quantified by PT/PL=2. 2.1 Heavy quark diffusion coefficient In kinetic theory, the diffusion coefficient can be computed as [14] 3κ=1 2Mkk′p′ (2π)3δ3p+k−p′−k′2πδ k′−kq2|Mκ|2f(k)(1 +f(k′)).(2) Here qis the momentum transfer, kand k′(pand p′) are the inand outgoing gluon (heavy quark) momenta. We use the shorthand notation p=dp3/2p0(2π)3.The dominant contributions in the limit of very large quark mass Marise from t-channel gluon exchange and are described by the matrix element |Mκ|2=NcCHg416M2k21+cos2θkk′ (q2+m2 D)2.In this limit, the momentum transfer is purely spatial and the screening can be implemented by inserting the screening mass mDinto the propagator. The transverse κTand longitudinal κzcoefficients are related to the full coefficient by 3κ=2κT+κz. In order to better understand the nonequilibrium medium, we define three scales associated to it. The effective temperature is given by T∗=4λ /mDppf(p)(1 +f(p)).The Debye screening mass can be computed as m2 D=8pλf(p).The temperature can be defined from the energy density by Tε=(30 ε /π2νg)1/4,where νg=2Nc2−1for pure glue QCD. 3 Results 3.1 Comparing non-equilibrium κresults to thermal equilibrium We compare our nonequilibrium simulations with thermal systems for the same ε(t), mD(t), and T∗(t) as functions of time rescaled by the thermalization timescale τBMSS =α−13/5 s/Qs[13, 15]. The results are shown in Fig. 2. The main result is that for the same ε(Landau matching), the deviation from equilibrium is ∼30% (left panel). When matching for the same mD(center panel) or T∗(right panel) the deviations are considerably larger. 2 EPJ Web of Conferences 296, 09001 (2024) https://doi.org/10.1051/epjconf/202429609001 Quark Matter 2023 10−510−410−310−210−1100 τ/τBMSS 0.0 0.2 0.4 0.6 0.8 1.0 1.2 κ/κε eq λ=0.5 λ=1.0 λ=2.0 λ=5.0 λ= 10.0 30% deviation 10−510−410−310−210−1100101 τ/τBMSS 0 1 2 3 4 5 κ/κmD eq λ= 10 λ=5 λ=2 λ=1 λ=0.5 10−410−310−210−1100 τ/τBMSS 0.0 0.2 0.4 0.6 0.8 1.0 1.2 κ/κT∗ eq λ=0.5 λ=1 λ=2 λ=5 λ= 10 Figure 2. Equilibrium and nonequilibrium κfor the same Tε(left), mD(center) and T∗(right). Figures taken from [10]. 10−510−410−310−210−1100 τ/τBMSS 0.0 0.5 1.0 1.5 2.0 2.5 κT/κz λ=0.5 λ=1 λ=2 λ=5 λ= 10 10−1100101 τ[fm] 0.0 0.2 0.4 0.6 0.8 1.0 κ[GeV2/fm] Hydrodynamics Kinetic theory Glasma λ= 10 Qs=1.4GeV κT κz κGlasma T κGlasma z κ eq Lattice 10−1 100 κ/T3 ε EKT nonequilibrium Lattice equilibrium λ=2 λ=5 λ= 10 Lattice T=1.5Tc Lattice T= 104Tc Figure 3. Left: Comparison of transverse and longitudinal diffusion coefficients as functions of time in the units of the thermalization time. Center: Comparison of our results and [17] in units of GeV. Right: Comparison of our values with lattice simulation results [9]. Figures taken from [10]. 3.2 Transverse vs. longitudinal diffusion coefficient The ratio of the transverse and longitudinal diffusion coefficients is shown in the left panel in Fig. 3. The initial κT/κz>1 arises from the overoccupation and large anisotropy leading to enhanced transverse momentum exchange. After the star marker, i.e., during the second stage of the bottom-up scenario, one finds κT/κz<1. This originates from the large momentum anisotropy of the underoccupied system and is in line with results from squeezed thermal distributions [16]. Between a maximal underoccupation and hydrodynamization, the ratio smoothly evolves towards unity. Throughout the whole evolution, the anisotropy ratio is at most 2. 3.3 Comparison with lattice & glasma The center panel of Fig. 3 shows our and the glasma results [17], which are initially considerably larger. The transverse diffusion coefficients match better than the longitudinal ones. The lattice result [9] at T=1.5Tcis depicted for the same energy density. The right panel of Fig. 3 shows our results for λ=2,5,10 and lattice results [9] at T=1.5Tcand T=104Tcin terms of the ratio κ /T3. At extremely high temperatures, the coupling of the lattice calculation corresponds to λ≈2, while for the lower temperature λ∼10 from the one-loop beta function (breaks down at this scale). The stages of the bottomup evolution are shown by the respective markers. Our result for λ=2 is in rough agreement with the lattice estimate at T=104Tc. However, at 1.5Tcthe lattice result is considerably larger. 3 EPJ Web of Conferences 296, 09001 (2024) https://doi.org/10.1051/epjconf/202429609001 Quark Matter 2023 4 Conclusions & Outlook Our primary aim in this paper is to understand the magnitude and anisotropy of κduring hydrodynamization. We find that the diffusion coefficient is within 30 % from its equilibrium value for the same energy density. For the anisotropy of the diffusion coefficient, we observed that initially κT>κ z. For underoccupied systems, the hierarchy is reversed. The maximal difference between κTand κzthroughout the entire evolution is a factor of ≲2. We expect our results to have applications especially in phenomenological descriptions of heavy quark diffusion and quarkonium dynamics. Acknowledgements This work is supported by the European Research Council, ERC-2018-ADG-835105 YoctoLHC and under the European Union’s Horizon 2020 research and innovation by the STRONG-2020 project (grant agreement No. 824093), Academy of Finland by the Centre of Excellence in Quark Matter (project 346324) and project 321840, the Austrian Science Fund (FWF) under project P 34455, and the Doctoral Program W1252-N27 Particles and Interactions. The authors wish to acknowledge CSC – IT Center for Science, Finland, for computational resources. The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views expressed therein lies entirely with the authors. References [1] M.E. Carrington, A. Czajka, S. Mrowczynski, Phys. Rev. C 105, 064910 (2022), 2202.00357 [2] M.E. Carrington, A. Czajka, S. Mrowczynski, Phys. Lett. B 834, 137464 (2022), 2112.06812 [3] K. Boguslavski, A. Kurkela, T. Lappi, J. Peuron, JHEP 09, 077 (2020), 2005.02418 [4] A. Ipp, D.I. Müller, D. Schuh, Phys. Lett. B 810, 135810 (2020), 2009.14206 [5] P. Khowal, S.K. Das, L. Oliva, M. Ruggieri, Eur. Phys. J. Plus 137, 307 (2022), 2110.14610 [6] Y. Sun, G. Coci, S.K. Das, S. Plumari, M. Ruggieri, V. Greco, Phys. Lett. B 798, 134933 (2019), 1902.06254 [7] K. Boguslavski, A. Kurkela, T. Lappi, F. Lindenbauer, J. Peuron (2023), 2303.12595 [8] D. Avramescu, V. B˘ aran, V. Greco, A. Ipp, D.I. Müller, M. Ruggieri, Phys. Rev. D 107, 114021 (2023), 2303.05599 [9] N. Brambilla, V. Leino, J. Mayer-Steudte, P. Petreczky (TUMQCD), Phys. Rev. D 107, 054508 (2023), 2206.02861 [10] K. Boguslavski, A. Kurkela, T. Lappi, F. Lindenbauer, J. Peuron (2023), 2303.12520 [11] A. Kurkela, Y. Zhu, Phys. Rev. Lett. 115, 182301 (2015), 1506.06647 [12] P.B. Arnold, G.D. Moore, L.G. Yaffe, JHEP 01, 030 (2003), hep-ph/0209353 [13] R. Baier, A.H. Mueller, D. Schiff, D.T. Son, Phys. Lett. B 502, 51 (2001), hep-ph/0009237 [14] G.D. Moore, D. Teaney, Phys. Rev. C 71, 064904 (2005), hep-ph/0412346 [15] K. Boguslavski, A. Kurkela, T. Lappi, F. Lindenbauer, J. Peuron (2023), 2312.11252 [16] P. Romatschke, Phys. Rev. C 75, 014901 (2007), hep-ph/0607327 [17] D. Avramescu, V. B˘ aran, V. Greco, A. Ipp, D.I. Müller, M. Ruggieri, Heavy quark κand jet ˆq transport coefficients in the Glasma early stage of heavy-ion collisions, in Hard Probes 2023 (2023), 2307.07999 4 EPJ Web of Conferences 296, 09001 (2024) https://doi.org/10.1051/epjconf/202429609001 Quark Matter 2023