Heavy quark diffusion coefficient during hydrodynamization : non-equilibrium vs. equilibrium

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-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ Heavy quark diffusion coefficient during hydrodynamization : non-equilibrium vs. equilibrium © 2024 the Authors Published version Peuron, Jarkko; Boguslavski, Kirill; Kurkela, Aleksi; Lappi, Tuomas; Lindenbauer, Florian Peuron, J., Boguslavski, K., Kurkela, A., Lappi, T., & Lindenbauer, F. (2024). Heavy quark diffusion coefficient during hydrodynamization : non-equilibrium vs. equilibrium. In HardProbes2023: 11th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions (Article 091). Sissa Medialab. POS Proceedings of Science, 438. https://doi.org/10.22323/1.438.0091 2024 PoS(HardProbes2023)091 Heavy quark diffusion coefficient during hydrodynamization - non-equilibrium vs. equilibrium K. Boguslavski,𝑎A. Kurkela,𝑏T. Lappi,𝑐,𝑑 F. Lindenbauer𝑎and J. Peuron𝑐,𝑑,𝑒,∗ 𝑎Institute for Theoretical Physics, Technische Universität Wien, 1040 Vienna, Austria 𝑏Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway 𝑐Department of Physics, P.O. Box 35, 40014 University of Jyväskylä, Finland 𝑑Helsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland 𝑒Dept. of Physics, Lund University, Sölvegatan 14A, Lund,SE-223 62, Sweden E-mail: [email protected],[email protected], [email protected],[email protected], [email protected] We compute the heavy quark momentum diffusion coefficient using effective kinetic theory for a system going through bottom-up isotropization until approximate hydrodynamization. We find that when comparing the nonthermal diffusion coefficient to the thermal one for the same energy density, the observed deviations throughout the whole evolution are within 30% from the thermal value. For thermal systems matched to other quantities we observe considerably larger deviations. We also observe that the diffusion coefficient in the transverse direction dominates at large occupation number, whereas for an underoccupied system the longitudinal diffusion coefficient dominates. Similarly, we study the jet quenching parameter, where we obtain a smooth evolution connecting the large values of the glasma phase with the smaller values in the hydrodynamical regime. HardProbes2023 26-31 March 2023 Aschaffenburg, Germany ∗Speaker ©Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ PoS(HardProbes2023)091 Heavy quark diffusion coefficient during hydrodynamization - non-equilibrium vs. equilibrium J. Peuron 10−210−1100 hpλfi/hpi 100 101 102 PT/PL λ= 0.5 λ= 1 λ= 2 λ= 5 λ= 10 Figure 1: Trajectory of the system during the bottom-up thermalization on the occupation number anisotropy plane. Solid and dashed curves correspond to different initial conditions. Reproduced from [13]. 1. Introduction Recent studies on transport coefficients out of equilibrium have indicated that the glasma stage can have considerable impact on the coefficients [1–8]. However, there has been a literature gap until very recently: equilibrium transport coefficients are relatively well known but the evolution of transport coefficients during hydrodynamization remained poorly understood. We report here of our recent studies where we aimed to close the gap and investigated the heavy quark momentum diffusion coefficient 𝜅[9] and the jet quenching parameter ˆ𝑞[10] during hydrodynamization using effective kinetic theory. The two main questions these proceedings address involve the magnitude of 𝜅compared to its equilibrium value during hydrodynamization and the relative importance of transverse and longitudinal diffusion coefficients during the hydrodynamization process. 2. Method: effective kinetic theory and bottom-up thermalization We reproduce the bottom-up thermalization [11] scenario using effective kinetic theory [12], as in [13]. The evolution of the system is illustrated in Fig. 1. In order to make a connection to the evolution of the system and other quantities, we have placed a few markers in Fig. 1. The star marker is placed at 𝑓=1 /𝜆=1 /(4𝜋𝑁𝑐𝛼𝑠),where 𝑁𝑐is the number of colors and 𝛼𝑠is the strong coupling constant. For weak couplings this also corresponds to maximum anisotropy. The circle marker is placed at minimum occupancy, which in the bottom-up thermalization picture corresponds to 𝛼𝑠. Finally, the triangle marker is placed at 𝑃𝑇/𝑃𝐿=2, corresponding to approximate isotropy. The crosses at the bottom correspond to the expected values at thermal equilibrium. In effective kinetic theory the dynamical degree of freedom is the gluon phase space density 𝑓(𝒑)=1 𝜈𝑔 d𝑁 d3𝑥d3𝒑,whose time-evolution is given by the Boltzmann equation −𝜕 𝑓 (𝒑) 𝜕𝜏 =C1↔2[𝑓(𝒑)] + C2↔2[𝑓(𝒑)] − 𝑝𝑧 𝜏 𝜕 𝜕𝑝𝑧 𝑓(𝒑).(1) 2 PoS(HardProbes2023)091 Heavy quark diffusion coefficient during hydrodynamization - non-equilibrium vs. equilibrium J. Peuron 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 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 λ= 2 λ= 5 λ= 10 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 Figure 2: Three different ways to compare equilibrium and nonequilibrium. Left: comparing for the same effective temperature of the infrared modes. Center: for the same energy density. Right: for the same screening mass. The dominant contribution to the diffusion coefficient 𝜅arises from scattering with the medium gluons via t-channel gluon exchange [14]. The coefficient is given by 3𝜅=Δ𝑘2 Δ𝑡=1 2𝑀∫𝒌𝒌′𝒑′ (2𝜋)3𝛿3(𝒑+𝒌−𝒑′−𝒌′)2𝜋𝛿 (𝑘′−𝑘)𝒒2|M𝜅|2𝑓(𝒌)(1+𝑓(𝒌′)), (2) where 𝑘, 𝑘′are the ingoing and outgoing gluon momenta, 𝑞=𝑘−𝑘′,is the momentum transfer and 𝑝, 𝑝′are the ingoing and outgoing heavy quark momenta. The integration measure is given by ∫𝒑=∫d𝑝3/2𝑝0(2𝜋)3. The matrix element corresponding to this process is |M|2 𝜅= 𝑁𝑐𝐶𝐻𝑔416𝑀2𝑘2(1+cos2𝜃𝒌𝒌′) (𝑞2+𝑚2 𝐷)2.The effective temperature of the infrared modes is 𝑇∗=2𝜆 𝑚𝐷∫d3𝑝/(2𝜋)3𝑓(𝑝)(1+𝑓(𝑝)),where the Debye screening mass is 𝑚2 𝐷=4∫d3𝑝/(2𝜋)3𝜆 𝑓 (𝑝)/𝑝. When comparing equilibrium and nonequilibrium systems, we need an estimate for the temperature of the corresponding equilibrium system. This temperature is defined through energy density 𝜀 as 𝑇𝜀=(30 𝜀/𝜋2𝜈𝑔)1/4.In equilibrium the quantities above (𝑇∗, 𝑚𝐷, 𝑇𝜀)are computed using the Bose-Einstein distribution. 3. Results Since there is no unambiguous way to compare equilibrium and nonequilibrium systems, we will try to compare the equilibrium and nonequilibrium systems for the same 𝑚𝐷, 𝑇∗and 𝜀. The comparison is done as a function of time. As a consequence, the corresponding thermal system changes during the time-evolution. We rescale the time with the thermalization timescale 𝜏BMSS =𝛼−13/5 𝑠/𝑄𝑠. The results are shown in Fig. 2. We observe that when matching for the same screening mass 𝑚𝐷or infrared temperature 𝑇∗there are large deviations during the equilibration. However when matching for the same energy density 𝜀the deviations are (depending on the coupling) within approximately ∼30% during the evolution. Thus matching for the same energy density (Landau matching) is the best way to compare equilibrium and nonequilibrium systems in this case. We can also break the comparison down into transverse and longitudinal components as we have done in Fig. 3. We observe that the transverse (𝜅𝑇) and longitudinal (𝜅𝐿) diffusion coefficients 3 PoS(HardProbes2023)091 Heavy quark diffusion coefficient during hydrodynamization - non-equilibrium vs. equilibrium J. Peuron 10−510−410−310−210−1100 τ/τBMSS 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 κT/κ eq λ= 0.5 λ= 1 λ= 2 λ= 5 λ= 10 10−510−410−310−210−1100 τ/τBMSS 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 κz/κ eq λ= 0.5 λ= 1 λ= 2 λ= 5 λ= 10 Figure 3: Transverse and longitudinal diffusion coefficients compared to their equilibrium values for the same energy density. 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 Figure 4: Ratio of the transverse and longitudinal diffusion coefficients during the hydrodynamization process. behave qualitatively similarly to the full coefficient (except in the case of the longitudinal diffusion coefficient at very early times). For smaller coupling 𝜆we observe larger deviations. This is most likely due to the fact that for small coupling the bottom-up thermalization is reproduced more accurately. Fig. 4shows the ratio of transverse and longitudinal diffusion coefficients during the evolution. We observe that the transverse diffusion coefficient is initially enhanced compared to the longitudinal coefficient. When the system becomes underoccupied, the hierarchy is inverted, and the longitudinal coefficient is enhanced compared to the transverse coefficient. The anisotropy of the coefficients can become sizable, of the order 10 - 40 %, depending on the coupling strength. The plot also shows the emergence of a limiting attractor, which we will discuss elsewhere in more detail. Then we proceed to the jet quenching factor ˆ𝑞defined as ˆ𝑞𝑖 𝑗 =d⟨𝑞𝑖𝑞𝑗⟩ d𝐿. Here we use the following convention: ˆ𝑥jet direction, ˆ𝑧beam direction. The jet quenching factor is given by ˆ𝑞𝑖 𝑗 =1 4𝑑𝑅 lim |p|→∞ ∫kk′p′ 𝑞⊥<Λ⊥ 𝑞𝑖 ⊥𝑞𝑗 ⊥(2𝜋)4𝛿4(𝑃+𝐾−𝑃′−𝐾′)M𝑎𝑔 𝑎𝑔 2 |p|𝑓k(1+𝑓k′),(3) where M𝑎𝑔 𝑎𝑔 2is the matrix element corresponding to elastic scatterings off in-medium gluons. Here we consider a quark jet. However the value of ˆ𝑞for a gluon jet can be obtained by scaling with a simple Casimir factor. The curves shown in Fig. 5are obtained as follows: We match 𝜀to glasma as in [6] to obtain the value of 𝑄𝑠at the initial condition. Then ˆ𝑞is matched to the result of JETSCAPE [15] at the triangle marker to obtain a value for the transverse momentum transfer 4 PoS(HardProbes2023)091 Heavy quark diffusion coefficient during hydrodynamization - non-equilibrium vs. equilibrium J. Peuron 10−1100101 τ(fm/c) 0 2 4 6 8 10 ˆ q(GeV2/fm) Glasma Kinetic theory Hydrodynamics ˆqfrom glasma λ= 10 Qs= 1.4GeV Ejet = 100 GeV Ejet = 20 GeV Figure 5: The value of the jet quenching factor ˆ𝑞computed according to the procedure described in the text. cutoff Λ⊥at that time. The bands correspond to different cutoff models and initial conditions. We observe that our results match the glasma simulation at early times relatively well and smoothly connect to the hydrodynamic evolution. 4. Conclusions The two main conclusions of these proceedings are, that during the hydrodynamization 𝜅 is within 30 % from its equilibrium value when the equilibrium and nonequilibrium systems are matched for the same energy density. The second conclusion is that there is a clear hierarchy between transverse and longitudinal diffusion coefficients. Initially the transverse diffusion coefficient 𝜅𝑇 dominates. At underoccupation 𝜅𝑧is larger. In both cases the deviation is roughly a factor of two. Our results may be used for phenomenological descriptions of heavy quark diffusion, quarkonium dynamics and jet quenching. Our future plans involve studying limiting attractors using 𝜅and ˆ𝑞as test observables. Acknowledgments The authors would like to thank N. Brambilla, M. Escobedo, D.I. Müller, A. Rothkopf and M. Strickland for valuable discussions. This work is supported by the European Research Council, ERC-2018-ADG-835105 YoctoLHC. This work was also supported under the European Union’s Horizon 2020 research and innovation by the STRONG-2020 project (grant agreement No. 824093). 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. This work was funded in part by the Knut and Alice Wallenberg foundation, contract number 2017.0036. TL and JP have been supported by the Academy of Finland, by the Centre of Excellence in Quark Matter (project 346324) and project 321840. KB and FL would like to thank the Austrian Science Fund (FWF) for support under project P 34455, and FL is additionally supported by the Doctoral Program W1252-N27 Particles and Interactions. The authors wish to acknowledge CSC – IT Center for Science, Finland, for computational resources. 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