Constraining η/s through high-p⊥ theory and data

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/ Constraining η/s through high-p theory and data © 2023 the Authors Published version Karmakar, Bithika; Zigic, Dusan; Salom, Igor; Djordjevic, Magdalena; Auvinen, Jussi; Huovinen, Pasi; Djordjevic, Marko Karmakar, B., Zigic, D., Salom, I., Djordjevic, M., Auvinen, J., Huovinen, P., & Djordjevic, M. (2023). Constraining η/s through high-p theory and data. Physical Review C, 108, Article 044907. https://doi.org/10.1103/PhysRevC.108.044907 2023 PHYSICAL REVIEW C 108, 044907 (2023) Constraining η/sthrough high-p⊥theory and data Bithika Karmakar , Dusan Zigic , Igor Salom , and Magdalena Djordjevic * Institute of Physics Belgrade, University of Belgrade, Belgrade 11080, Serbia Jussi Auvinen Institute of Physics Belgrade, University of Belgrade, Belgrade 11080, Serbia and University of Jyväskylä, Jyväskylä P.O. Box 35, FI-40014, Finland Pasi Huovinen Incubator of Scientific Excellence—Centre for Simulations of Superdense Fluids, University of Wrocław, Wrocław 50-204, Poland Marko Djordjevic Faculty of Biology, University of Belgrade, Belgrade 11000, Serbia (Received 19 June 2023; accepted 26 September 2023; published 24 October 2023) We study whether it is possible to use high-p⊥data/theory to constrain the temperature dependence of the shear viscosity over entropy density ratio η/sof the matter formed in ultrarelativistic heavy-ion collisions at the BNL Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC). We use two approaches: (i) We calculate high-p⊥RAA and flow coefficients v2,v3,andv4assuming different (η/s)(T)of the fluid-dynamically evolving medium. (ii) We calculate the quenching strength ( ˆq/T3) from our dynamical energy loss model and convert it to η/sas a function of temperature. It turned out that the first approach cannot distinguish between different (η/s)(T) assumptions when the evolution is constrained to reproduce the low-p⊥ data. In distinction, (η/s)(T) calculated using the second approach agrees surprisingly well with the (η/s)(T) inferred through state-of-the-art Bayesian analyses of the low-p⊥data even in the vicinity of Tc, while providing much smaller uncertainties at high temperatures. DOI: 10.1103/PhysRevC.108.044907 I. INTRODUCTION Quantum chromodynamics (QCD) predicts that at extremely high densities matter undergoes a transition to a state consisting of deconfined and interacting quarks, antiquarks, and gluons [1,2]. According to the current cosmology, this new state of matter, called quark-gluon plasma (QGP) [3], existed immediately after the big bang [4]. Today, QGP is created in “little bangs,” when heavy ions collide at ultrarelativistic energies [5] in experiments at the BNL Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC). Such collisions lead to an expanding fireball of quarks and gluons, which thermalizes to form QGP. The QGP cools down, and quarks and gluons hadronize when the temperature Tdrops to the critical temperature Tc. *Email: [email protected] 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. Extracting useful information from “little bangs” requires comparing theoretical predictions with experimental data. By such comparisons, it is established that QGP is formed in the RHIC and LHC experiments [6] through two main lines of evidence [5–7]: (i) by comparison of low transverse momentum (p⊥) measurements with relativistic hydrodynamical predictions, which imply that created QGP is consistent with the description of a nearly perfect fluid [8–10], and (ii) by comparison of perturbative QCD (pQCD) predictions with high-p⊥data [11–14], which showed that high-p⊥partons (jets) significantly interact with the opaque medium [5]. Beyond the discovery phase, the current challenge is to investigate the properties of this extreme form of matter [15–25]. The QGP was expected to behave as a weakly interacting gas based on ideas of asymptotic freedom and color screening [26]. Thus, the agreement of the fluid-dynamical predictions, which assumed the QGP to behave as a nearly inviscid fluid, with the data came as a surprise [10]. Furthermore, subsequent calculations revealed [10] that reproduction of the data required the shear viscosity to entropy density ratio (η/s)of QGP to be near the lower bound predicted by anti–de Sitter and conformal field theory (AdS/CFT) correspondence [27]. However, the temperature of the QGP changes significantly [28] during the evolution of the collision system. For example, in the LHC experiments, the temperature is estimated 2469-9985/2023/108(4)/044907(14) 044907-1 Published by the American Physical Society BITHIKA KARMAKAR et al. PHYSICAL REVIEW C 108, 044907 (2023) to span the range from 4Tcto Tc. Even if the QGP behaves as a perfect fluid close to Tc(the “soft,” strongly coupled regime), its η/smay significantly increase with increasing T if the QGP becomes weakly coupled at higher temperatures (the “hard,” weakly coupled regime). We call this possibility the “soft-to-hard” medium hypothesis. Testing this hypothesis has turned out to be surprisingly difficult. Reproduction of the observed anisotropies of low-p⊥ particles necessitates low η/sin the vicinity of Tc,butthe value of shear viscosity in higher temperatures has only a weak effect on anisotropies in collisions at LHC energies, and basically no effect at all at RHIC [29–31]. In the recent Bayesian analyses of the data, this is manifested in a well constrained η/sin the Tc< ∼T< ∼1.5Tctemperature range and weak constraints at larger temperatures [17–20]. Some of the most recent Bayesian analyses [32,33] even suggest that η/s may decrease in the region Tc–2Tc, where the reason for such a decrease still remains to be understood. Thus, it is evident that a complementary theory and observables are needed to investigate the “soft-to-hard” medium hypothesis. Since most of the jet energy loss takes place when the system is hottest, it is reasonable to expect the high-p⊥ observables to be sensitive to the properties of the system at that stage. To use jet energy loss and high-p⊥data to provide constraints to the bulk properties of the collision system, we developed the state-of-the-art DREENA tomography tool [34,35] based on the dynamical energy loss formalism [36–38]. So far, we have used this tool to, e.g., provide constraints to the early evolution of the collision system [22] and map how the shape of the collision system is manifested in the high-p⊥data [23]. In this study, we explore whether high-p⊥data can provide constraints on the η/sratio of QGP at high temperatures. As is known, shear viscosity generates entropy, which means that the system with larger viscosity cools slower or, alternatively, to reach the same final entropy, the system with larger viscosity must have a lower initial temperature. Thus, different assumed η/sduring the early evolution of the system may lead to different jet energy loss and therefore different nuclear suppression factor RAA. As well, azimuthal anisotropy in path lengths and temperature along the paths leads to azimuthal dependence of jet suppression [5], which is measured as vn of high-p⊥particles. High-p⊥vnare known to be sensitive to the details of the medium evolution [34,35,39], and, since viscosity changes the evolution of the anisotropy of the system, the changes in η/scan lead to changes in high-p⊥vn.We choose three different parametrizations of (η/s)(T), adjust the parameters to reproduce the low-p⊥data measured in √sNN = 200 GeV Au +Au collisions (RHIC) and √sNN =5.02 TeV Pb +Pb collisions (LHC), calculate the temperature evolution of the system, and energy loss of jets traversing this system in each case, and evaluate the RAA and high-p⊥v2,v3, and v4to see if different assumptions of η/slead to differences in these observables. Complementary to this phenomenological approach to infer the η/sratio from the experimental data, we also provide a fully theoretical estimate of η/sbased on jet energy loss: The jet quenching strength is quantified through the jet quenching parameter ˆq. It has been argued that in a weakly coupled regime T3/ˆqis directly proportional to η/s[40], and thus evaluating one allows one to know the other. We estimate the quenching parameter ˆqas function of temperature using our dynamical energy loss formalism, convert it to η/s, and compare the resulting (η/s)(T) to constraints obtained from state-of-the-art Bayesian analyses [18,19]. II. METHODS A. Modeling the bulk evolution To calculate the temperature evolution and the low-p⊥ observables we use the version of VISHNEW [41,42]usedin Refs. [17,18,43].1It is a code to solve the dissipative fluiddynamical equations in 2+1 dimensions, i.e., assuming boost invariance. Shear stress and bulk pressure are taken as dynamical variables and evolved according to the Israel-Stewart type equations [45]. We use an equation of state (EoS) [43] that combines the lattice QCD-based EoS of the HotQCD Collaboration [46] at large temperatures and a hadron resonance gas EoS at low temperatures. At a constant temperature Tsw =151 MeV hypersurface, we convert the fluid to particle ensembles according to the Cooper-Frye prescription [47]. These ensembles are fed to the UrQMD hadron cascade [48,49], which describes the evolution of the hadronic stage of the system until freeze-out. We generate the event-by-event fluctuating initial states using the TRENTo model [50]. In this model nucleus-nucleus collisions are considered as a superposition of nucleonnucleon collisions. The nucleons are represented by Gaussian distributions, which in this study have the width w=0.5 fm while the minimum nucleon-nucleon distance within the nucleus is also set to d=0.5 fm. The inelastic nucleonnucleon cross section is 70 mb at √sNN =5.02 TeV (energy of Pb +Pb collisions) and 42 mb at √sNN =200 GeV (energy of Au +Au collisions). For the other parameters we use the maximum a posteriori (MAP) values found in Ref. [18]. We do not allow any preequilibrium evolution (free-streaming or otherwise), and use τ0=1fm/cas the initial time for fluiddynamical evolution, since the reproduction of the high-p⊥ observables does not allow strong transverse expansion earlier [22]. We include both bulk and shear viscosity in our fluiddynamical calculation. The temperature dependence of the bulk viscosity coefficient ζis parametrized as a Cauchy distribution [18]: (ζ/s)(T)=(ζ/s)max 1+T−T0 (ζ/s)width 2.(1) We consider a small bulk viscosity with a maximum value (ζ/s)max =0.03, the width parameter (ζ/s)width =0.022, and T0=0.183 GeV. As in the case of the TRENTo parameters described above, the width and T0correspond to MAP parameter values from Ref. [18]. However, the maximum of the bulk viscosity [(ζ/s)max] is decreased compared to the MAP 1Code available at [44]. 044907-2 CONSTRAINING η/sTHROUGH … PHYSICAL REVIEW C 108, 044907 (2023) value in [18] to compensate for the lack of preequilibrium free streaming and still reach agreement with the p⊥spectra. As mentioned, our main objective is to find out whether high-p⊥data can provide constraints to η/sat high temperatures. Naively one can expect the jet energy loss to be proportional to the third power of temperature (T3), but a detailed calculation has shown it to be proportional to only T1.2[51,52]. Since the sensitivity to temperature is weaker than expected, we want to maximize the difference in temperature due to differences in η/s. Therefore we do not take as our (η/s)(T) the upper and lower limits suggested by the Bayesian analyses [18,19] but something more extreme. We parametrize the temperature dependence of η/sas [17,18] (η/s)(T)=(η/s)min,T<Tc, (η/s)min +(η/s)slope(T−Tc)T Tc(η/s)crv ,T>Tc,(2) where (η/s)min is the minimum value of the specific shear viscosity, (η/s)slope is the slope above Tc, and (η/s)crv controls the curvature above Tc.Tcis fixed to the pseudocritical temperature Tc=154 MeV evaluated by the HotQCD Collaboration [46]. We study three different scenarios, each capable of describing a subset of low-p⊥data at RHIC and LHC with reasonable accuracy: (1) constant η/s(0.15 for Pb +Pb collision at LHC and 0.12 for Au +Au collision at RHIC), (2) (η/s)min =0.1, (η/s)slope =1.11, (η/s)crv =−0.48, (3) (η/s)min =0.04, (η/s)slope =3.30, (η/s)crv =0. The parameters in our second scenario are within the 90% credible intervals of the analysis of Ref. [18]. Therefore, we label it as “Nature.” Nevertheless, our (η/s)min is larger than in Ref. [18] since we require the reproduction of the RHIC data, not only the LHC data. As is known, including the RHIC data tends to increase the favored minimum value of η/s[20]. Our third scenario with its very rapidly rising η/s (see Fig. 8) is inspired by the “LHHQ” parametrization in Ref. [30]. Consequently, we label it as such. To calculate the low-p⊥and high-p⊥predictions, we generated 104minimum-bias events and sorted the events in centrality classes according to the number of participants. While using the final particle multiplicity would be closer to the centrality selection done in experiments, participant number sorting allows us to reduce the number of hydrodynamic simulations by focusing on the narrower (10–50)% centrality range, thus saving computational resources (we numerically tested that this approximation would have a negligible effect on theoretical predictions). Finally, we evaluated the eventaveraged observables in each centrality bin. We reproduced the pion, kaon, and proton multiplicities and charged hadron four-particle cumulant elliptic flow v2{4}in Au +Au collisions at √sNN =200 GeV (RHIC) and Pb +Pb collisions at √sNN =5.02 TeV (LHC) in 10–20%, 20–30%, 30–40%, and 40–50% centrality classes by varying only the nucleon-nucleon cross section according to the collision energy and the overall normalization factor according to the collision energy and choice of (η/s)(T). All the other TRENTo parameters were kept the same in all cases. For the LHHQ parametrization, the minimum value of η/sis chosen to get an acceptable agreement of v2{4}with both Pb +Pb and Au +Au collision data. The centrality dependence of charged particle multiplicities (p⊥-integrated yields) for Pb +Pb and Au +Au collisions with three different (η/s)(T) parametrizations found from the hydrodynamical simulation are shown in the left panels of Fig. 1. The four-particle elliptic flow coefficient v2{4}at different centrality classes for Pb +Pb and Au +Au collisions are shown in the right panels of Fig. 1. B. Overview of DREENA framework After evaluating the temperature evolution, we use the “generalized DREENA-A” framework to calculate the high-p⊥ observables: Nuclear suppression factor RAA and high-p⊥flow harmonics v2,v3, and v4.DREENA (Dynamical Radiative and Elastic ENergy loss Approach) is a computationally efficient tool for QGP tomography [34,35], based on generalized hard thermal loop (HTL) perturbation theory [57] with naturally regulated infrared divergences [36,58]. In this formalism both the radiative [37,38] and collisional energy loss [36]ofhigh energy particles have been computed in an evolving QCD medium of finite size at finite temperature. Furthermore, the framework is extended to account for running coupling [59], finite magnetic mass [60], and beyond soft-gluon approximation [58]. We also recently extended the formalism towards finite orders in opacity [61], but showed that higher-order effects can be neglected for high-p⊥predictions. Thus, a computationally more efficient version with one scattering center is used in this study. Additionally, in this framework, all parameters are fixed to standard literature values stated below (i.e., no fitting parameters are used) [22,23]. This allows systematic comparison of data and the predictions from the simulation obtained using the same formalism and parameter set. We use the generic pQCD convolution formula [59,62]to generate the final quenched (q) and unquenched (u) spectra of hadrons as Efd3σq(HQ) dp 3 f=Eid3σ(Q) dp 3 i⊗P(Ei→Ef)⊗D(Q→HQ), (3) Efd3σu(HQ) dp 3 f=Eid3σ(Q) dp 3 i⊗D(Q→HQ),(4) where iand fdenote the initial parton (Q) and the final hadron (HQ) respectively. Eid3σ(Q) dp 3 i represents the initial parton spectrum calculated at the next-to-leading order for light 044907-3 BITHIKA KARMAKAR et al. PHYSICAL REVIEW C 108, 044907 (2023) FIG. 1. Left panels: Centrality dependence of the p⊥-integrated yields of pions, kaons, and protons are shown in different centrality classes. The pion multiplicity is scaled by 0.5. The upper panel corresponds to 5.02 TeV Pb +Pb collisions, where the ALICE experimental data are takenfromRef.[53]. The lower panel corresponds to 200 GeV Au +Au collisions, where the PHENIX experimental data are taken from Ref. [54]. Right panels: v2{4}is shown at different centrality classes. The upper panel corresponds to 5.02 TeV Pb +Pb collisions, where the ALICE experimental data are taken from Ref. [55]. The lower panel corresponds to 200 GeV Au +Au collisions, where the STAR experimental data are taken from Ref. [56]. and heavy partons [63–65]. P(Ei→Ef) is the energy loss probability computed within finite temperature field theory. D(Q→HQ) represents the fragmentation function. DSS [66], BCFY [67,68], and KLP [69] fragmentation functions were used for charged hadrons, Dmesons, and Bmesons, respectively. DREENA-A[35], where “A” stands for adaptive (i.e., arbitrary) temperature profiles, has been optimized to incorporate any event-by-event fluctuating temperature profile [34]. For parameters, we use QCD =0.2GeV[70–72] and the effective numbers of light quark flavors, nf=3 and 2.5 for Pb +Pb and Au +Au collision systems, respectively. We also consider the gluon mass mg=μE/√2[73], where μEis the temperature-dependent Debye mass computed following the procedure in Ref. [70] (outlined in the next subsection). We assume the mass of the light quarks to be μE/6, and the masses of the charm and bottom quarks to be 1.2 and 4.75 GeV, respectively. The magnetic-to-electric mass ratio is μM/μE=0.6[74]. C. Derivation of transport coefficient ˆqfrom dynamical energy loss formalism To derive the transport coefficient ˆq, which is the squared average transverse momentum exchange between the medium and the fast parton per unit path length [75], we start from dynamical perturbative QCD medium, where the interaction between high-p⊥partons and QGP constituents can be characterized by the HTL resummed elastic collision rate [76]: del d2q=4CA1+nf 6T3α2 s q2q2+μ2 E.(5) While αsin Eq. (5) is presumed to be constant, for RHIC and LHC, it is necessary to include running coupling constant in the kernel due to the wide kinematic range covered in these experiments. To include running coupling in dynamical energy loss formalism, we adopt the procedure from Ref. [77] where α2 s→αs(ET)αsμ2 E,(6) and μEis obtained [70] as a self-consistent solution to μ2 E=1+nf 64παμ2 ET2,(7) where α(t)=4π 11 −2 3nf 1 ln t 2,(8) 044907-4 CONSTRAINING η/sTHROUGH … PHYSICAL REVIEW C 108, 044907 (2023) leading to μE=2ξ(T) W(ξ(T)),(9) where ξ(T)=1+nf 6 11 −2 3nf4πT 2 ,(10) and Wis Lambert’s Wfunction. Note that μEobtained through this procedure agrees with lattice QCD results [70]. By using Eqs. (9) and (6), Eq. (5) reduces to del d2q=CA πTα(ET)μ2 E q2q2+μ2 E,(11) which reproduces Eq. (16) from Ref. [76] in the case of constant coupling α(ET)=g2/4π. Following Ref. [60], finite magnetic mass can be introduced into Eq. (11), reducing the collision rate to del d2q=CA πTα(ET)μ2 E−μ2 M q2+μ2 Eq2+μ2 M,(12) where μMis the magnetic mass defined in the previous subsection, and CA=4/3. This expression [i.e., Eq. (12)] can be further reduced to del d2q=CA πTα(ET)1 q2+μ2 M−1 q2+μ2 E.(13) In the fluid rest frame, the transport coefficient ˆqcan then be computed as [76,78] ˆq=√6ET 0 d2qq 2del d2q =CATα(ET)6ET 0 dq2q21 q2+μ2 M−1 q2+μ2 E =CAT4π 11 −2 3nfμ2 Eln 6ET+μ2 E μE2−μ2 Mln 6ET+μ2 M μ2 M ln ET 2. (14) In the limit ET →∞,Eq.(14) reduces to an expression independent of jet E: ˆq=CAT4π 11 −2 3nF⎛ ⎝μ2 E ln ET μ2 E/6 ln ET 2−μ2 M ln ET μ2 M/6 ln ET 2⎞ ⎠ ≈CAT4π 11 −2 3nFμ2 E−μ2 M =CAT4π 11 −2 3nF 1+nF 6 11 −2 3nF (4π)2T2 W(ξ(T))1−x2 ME =CA4π 11 −2 3nF24π1+nF 6 W(ξ(T))1−x2 MET3,(15) where xME =μM/μEis the magnetic-to-electric mass ratio. It is worth noticing that this is expected behavior: As a property of the medium, ˆqshould be independent (or weakly dependent) on jet energy [76]. Nevertheless, many models/approaches fail to describe this behavior [76]. III. RESULTS A. Constraining η/sthrough high-p⊥data To examine the sensitivity of the high-p⊥observables on the specific shear viscosity of the medium, we compare in Fig. 2the experimental charged hadron RAA and high-p⊥flow harmonics v2,v3, and v4in Pb +Pb collisions at √sNN = 5.02 TeV to the theoretical predictions calculated using three different (η/s)(T) parametrizations (see Sec. II A). The highp⊥flow harmonics are computed using the scalar product method [34]. As seen in all three cases, the calculated charged hadron RAA and flow anisotropies are almost indistinguishable from each other. Furthermore, the predicted charged hadron v4significantly underestimates the experimental data even when the current large experimental uncertainties are taken into account. We previously reported a similar observation in Ref. [34], where high-p⊥v4was calculated using several different initializations of the fluid dynamical evolution. Unfortunately, the heavy flavor high-p⊥observables shown in Fig. 3are hardly more sensitive to the (η/s)(T) parametrizations. The calculated Dand Bmeson RAA,v2, and v3in Pb +Pb collisions at √sNN =5.02 TeV do not depend on our assumptions about η/s, whereas v4in the 10–30% centrality class shows some sensitivity. Nevertheless, given the large experimental uncertainties of v2and v3,itis doubtful whether the small difference in v4is experimentally detectable, especially when our v4predictions are very close to 0. Since the collisions at LHC reach larger initial temperatures than collisions at RHIC, we may expect them to be more sensitive to η/sat large temperatures, and thus to our (η/s)(T) parametrizations. Nevertheless, for the sake of completeness and to allow for surprises in the evolution, we checked whether the high-p⊥observables measured in collisions at the full RHIC energy (√sNN =200 GeV) allow us to distinguish between different (η/s)(T) parametrizations. The theoretical predictions for charged hadron and D and Bmeson high-p⊥observables in Au +Au collisions at √sNN =200 GeV collision energy are shown in Figs. 4 and 5, respectively. Again, we calculated our predictions using the generalized DREENA-Aframework with three different (η/s)(T) parametrizations. As can be seen, the high-p⊥ observables are not sensitive to the η/sratio at high temperatures, and thus we cannot further constrain (η/s)(T)using high-p⊥observables. As was argued in the Introduction, different η/srequire different initial temperatures. However, as shown in Fig. 6, temperature difference during the evolution is small and, as demonstrated above, insufficient to lead to observable differences in high-p⊥observables. In Fig. 6, we characterize the system temperature using the so-called average jet-perceived temperature: At each time τwe average the system temperature in the transverse plane using the number of jets at each point as weight; e.g., while the average initial temperature in (10–20)% centrality class for Pb +Pb collisions at 044907-5 BITHIKA KARMAKAR et al. PHYSICAL REVIEW C 108, 044907 (2023) FIG. 2. Charged hadron RAA (first row) and high-p⊥flow harmonics v2(second row), v3(third row), and v4(fourth row) as a function of transverse momentum in Pb +Pb collisions at √sNN =5.02 TeV for different (η/s)(T) parametrizations indicated in the legend. CMS (blue squares) [79,80], ALICE (red circles) [81,82], and ATLAS (green triangles) [83,84] experimental data are also shown for comparison. Columns 1–4 represent the centrality classes 10–20%, 20–30%, 30–40%, and 40–50%, respectively. √s=5.02 TeV is 370 MeV, the maximal temperature experienced by the jet can reach up to 600 MeV. The jet-perceived temperatures calculated using all three (η/s)(T) parametrizations are almost identical, differing less than 4% at the early stages of the evolution and settling for less than 2% for most of the evolution. The differences in the anisotropy of the jet-perceived temperature, jT2, introduced in Ref. [23], are equally small (not shown). The investigated high-p⊥observables turned out to be insensitive to such small differences in temperature. Even if our calculated high-p⊥RAA and v2agree with the data (see Fig. 2), the calculated jT2are slightly below the experimentally favored values. This deviation is possible since our results for both RAA and v2are at the lower end of experimental uncertainty. When taking the ratio v2/(1 −RAA), which constrains jT2, this deviation from the data is magnified, and the calculated jT2is below the experimental constraint. Nevertheless the values of jT2obtained in these calculations are close to the largest values obtained in Ref. [23] for various initialization models. B. Calculating η/sfrom the dynamical energy loss ˆq In our previous publications, we have seen that the DREENA framework is capable of reproducing the observed RAA without fitting parameters [59,93,94] (see also comparison to RAA in the previous subsection). This agreement suggests that the dynamical energy loss formalism can adequately describe interactions between high-p⊥particles and the QCD medium. Thus, it seems reasonable to estimate (η/s)(T) theoretically using the dynamical energy loss model. 044907-6 CONSTRAINING η/sTHROUGH … PHYSICAL REVIEW C 108, 044907 (2023) FIG. 3. Predictions for D(left 4 ×2 panel) and Bmeson (right 4 ×2 panel) RAA (first row) and high-p⊥flow harmonics v2(second row), v3(third row), and v4(fourth row) using three different (η/s)(T) parametrizations at various centralities in Pb +Pb collisions at √sNN = 5.02 TeV. The theoretical predictions for Dmesons are compared with the CMS (blue squares) [85] and ALICE (red circles) [86,87] data, whereas Bmeson predictions are compared to preliminary CMS (blue squares) [88] and preliminary ALICE (red circles) [89] data. For this purpose, we need to estimate the jet quenching parameter ˆq, quantifying the transverse momentum broadening of fast parton due to its elastic scatterings with the medium [75]. This parameter is a key quantity in estimating the interaction strength between jet partons and nuclear matter [76,97– 101]. It has been proposed to be a valuable tool for various purposes, including gaining insights into the jet quenching phenomenon, estimating the bulk medium property (η/s)(T) [40,102], and, more recently, exploring the QCD phase diagram [103]. We presented the derivation of the transport coefficient ˆq from our dynamical energy loss model in Sec. II C. We note that ˆqis weakly dependent on Edue to ln(ET) appearing both in the numerator and the denominator of Eq. (14), as desired for a medium property such as transport coefficient. Before discussing our results further, we outline the theoretical expectations for ˆqand its relationship to η/s.To account for the temperature dependence of the coefficients η and ˆq, it is common practice to examine their dimensionless counterparts: η/s, and ˆq/T3[102]. Both quantities are sensitive to the effective coupling strength in QGP. If the coupling is weak, η/sis large, while ˆq/T3is small. Conversely, when the coupling is strong, η/sbecomes small, while ˆq/T3is large. In the case of weak coupling, it has been argued that these two quantities are related by η s ˆq T3≈const, i.e., more specifically [40,102], η s≈1.25 T3 ˆq.(16) Furthermore, to explain the large observed high-p⊥v2, it was proposed in Ref. [104], that the jet-quenching factor ˆq/T3must rise rapidly when approaching Tcfrom above. We schematically depicted such behavior in Fig. 7(a).This behavior is neither straightforward nor trivial to obtain from a model calculation. The expected (qualitative) relation of the T3/ˆqand (η/s)(T)—based on the existing knowledge from previous 044907-7 BITHIKA KARMAKAR et al. PHYSICAL REVIEW C 108, 044907 (2023) FIG. 4. The calculated charged hadron RAA (first row), high-p⊥v2(second row), v3(third row), and v4(fourth row) in √sNN =200 GeV Au +Au collisions. The experimental data from the STAR (orange stars) [90] and PHENIX (purple diamonds) [91,92] Collaborations are also shown. Columns 1–4 represent the centrality classes 10–20%, 20–30%, 30–40%, and 40–50%, respectively. studies—is schematically depicted in Fig. 7(b) [40]. At large temperatures, we expect the system to be weakly coupled. At that limit, our dynamical energy loss model should be applicable, and Eq. (16) should be a good approximation. Thus, we expect the calculated T3/ˆqto agree well with the inferred η/s as shown in the grey area at the right part of Fig. 7(b).Onthe other hand, in the strongly coupled limit [the left gray area in Fig. 7(b)] close to Tc, the calculated T3/ˆqis expected to significantly deviate from the inferred η/s. Interestingly, the T3/ˆqcalculated using weak coupling methods is expected to drop below the inferred η/s[40], which, as known, is very close to the AdS-CFT lower limit of 1/(4π) in the vicinity of Tc. The region between strongly and weakly coupled limits is the so-called “soft-to-hard” boundary [105], i.e., the region where the transition from a strongly to a weakly coupled regime could take place. Therefore, plotting together η/sand T3/ˆqas a function of Tmight allow estimating the “soft-tohard” boundary as the region where these two curves start to deviate, as schematically shown in Fig. 7(b). We calculate ˆq/T3from our dynamical energy loss using Eq. (14) in the initial jet energy range 3 <E<10 GeV, as p⊥has to be low enough to mimic interactions of partons within the medium. The obtained result is shown in Fig. 8(a), and qualitatively similar to the expectation shown in Fig. 7(a). In particular, near Tc, we obtain an enhanced quenching, which is considerably larger than quenching in other energy loss models [76] (with the exception of [101], which got a substantial increase in ˆq/T3near Tc, due to a very large coupling in their model). Some models even 044907-8