Hydrodynamic description of D meson production in high-energy heavy-ion collisions
SSubmitted to Chinese Physics C
Hydrodynamic description of D meson production in high-energyheavy-ion collisions Chi Ding , Wei-Yao Ke , , Long-Gang Pang a and Xin-Nian Wang , , bc Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan 430079, China Physics Department, University of California, Berkeley, CA 94720, USA and Nuclear Science Division, Lawrence BerkeleyNational Laboratory, Berkeley, CA 94720, USA
Abstract
The large values and the constituent-quark-number (NCQ) scaling of the elliptic flow of low- p T D mesons imply that charm quarks, initially produced through hard processes, might be partiallythermalized through the strong interactions with the quark-gluon plasma (QGP) in high-energyheavy-ion collisions. To quantify the degree of thermalization of low- p T charm quarks, we comparethe D meson spectra and elliptic flow from a hydrodynamic model to the experimental data aswell as transport model simulations. We use an effective charm chemical potential at the freeze-outtemperature to account for the initial charm quark production from hard processes and assumethat they are thermalized in local comoving frame of the medium before freeze-out. D mesons aresampled statistically from the freeze-out hyper-surface of the expanding QGP as described by theevent-by-event (3+1)D viscous hydrodynamic model CLVisc. Both hydrodynamic and transportmodel can describe the elliptic flow of D mesons at p T < GeV/ c as measured in Au+Au collisionsat √ s NN = 200 GeV. Though the experimental data on D spectra are consistent with the hydrody-namic result at small p T ∼ GeV/ c , they deviate from the hydrodynamic model at high transversemomentum p T > GeV/ c . The diffusion and parton energy loss mechanisms in the transportmodel can describe the measured spectra reasonably well within the theoretical uncertainty. Ourcomparative study indicates that charm quarks only approach to local thermal equilibrium at small p T even though they acquire sizable elliptic flow that is comparable to light-quark hadrons at bothsmall and intermediate p T . a email: [email protected] b email: [email protected] c Current address: Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,USA a r X i v : . [ nu c l - t h ] J a n . INTRODUCTION The large collective flow and momentum anisotropy as manifested in the final hadronspectra in high-energy heavy-ion collisions indicate a strong collectivity of the produceddense and hot nuclear matter during its dynamic evolution [1, 2]. The observed approximateconstituent quark number (NCQ) scaling of the elliptic flow for light quark hadrons [3–6]and strong jet quenching [7–10] in high-energy heavy-ion collisions at both the RelativisticHeavy-ion Collider (RHIC) and the Large Hadron Collider (LHC) suggest the formationof a hot, deconfined and opaque quark-gluon plasma (QGP). Similar phenomena are alsoobserved for heavy quark mesons [11–17]. Experimental data from RHIC and LHC show alarge elliptic flow of D mesons at low- p T , which also approximately obeys the NCQ scalingas light quark hadrons [13, 18]. Heavy quarks are predominantly produced through initialhard processes and have mass scales much larger than the typical temperature of the QGPmedium. Therefore, it still remains an interesting question as to whether and to what degreecharms quarks become thermalized [2] and flow with the QGP due to their strong interactionwith the hot medium.Past studies usually describe the thermalization processes of heavy quarks with transportequations. Their low-momentum ( p < ∼ M ) dynamics in the QGP medium is usually treatedas a Brownian motion [19–21], considering masses of heavy quarks being much larger thanthe typical temperatures in the medium M (cid:29) T . For the transport of heavy quarks withintermediate momentum p > ∼ M , both the collisional and radiative energy loss have to beconsidered. Most of these transport calculations consider the interaction between heavyquarks and the medium as a perturbative process or use transport coefficients with weakly-coupled assumptions as inputs.It was found that transport models need a large heavy-quark diffusion parameter to de-scribe the observed medium modification of the heavy-quark meson spectra and elliptic flowcoefficients [22, 23]. There have been several works trying to evaluate the heavy quarkdiffusion coefficient at zero momentum non-perturbatively from lattice QCD [24–26]. Theestimated interaction strength is comparable to the values from phenomenological determi-nation [27, 28], which is much larger than the natural expectation of the weakly-coupledtheory at the leading order. This poses questions on the weak-coupling assumption on thenature of the interaction between heavy quarks and the QGP medium. It is possible that,2ith a strong coupling, the dynamics of the low- p T charm quarks in the QGP is betterdescribed by a hydrodynamic approach.In this paper, we will compute the production of low- p T D mesons in an extreme limit ofhydrodynamic evolution with the following assumptions. First, the number of (anti)charmquarks is conserved during the lifetime of the fireball, i.e, thermal production of heavy quarkpairs is negligible. Second, the coupling is so strong that the initially produced charm quarksquickly diffuse into the QGP and reach local kinetic equilibrium. Finally, the differentialyield of D mesons is computed using the Cooper-Frye formula [29] on the hydrodynamicfreeze-out hypersurface, with an effective charm chemical potential to guarantee the charmyield is the same as measured in the experiment. We compare the resultant p T -dependentspectra and elliptic flow of low- p T D mesons, which are considered as the manifestation ofthe extreme limit of complete thermalization of heavy quarks, to the experimental data aswell as transport calculations to quantify the degree of heavy-quark thermalization in heavy-ion collisions. We will use the (3+1)D hydrodynamic model (CLVisc)[30] and a linearizedBoltzmann-Langevin model [28] for heavy quark transport.The paper is organized as follows. In Section II, we introduce the relativistic hydrody-namic model and the Boltzmann-Langevin transport model for the calculations of p T spectraand elliptic flow of charmed mesons. In Section III, we first discuss the calibration of thehydro model to the experimental data on light quark hadron spectra and elliptic flow inAu+Au collisions at the RHIC energy. We then compare the hydro results on the p T spec-tra and elliptic flow of charmed mesons at low p T to the experimental data and transportcalculations and discuss the implication on the degree of thermalization for heavy quarks inthe QGP formed in high-energy heavy-ion collisions. II. MODEL DESCRIPTIONSA. Charm meson production in the limit of local kinetic equilibrium
In the usual paradigm for charmed meson production in high-energy heavy-ion collisions,charm quarks are produced through the initial hard scatterings [31]. These heavy quarkswill experience energy loss and momentum diffusion in the QGP through both elastic andinelastic collisions with the medium, which can be modeled as the drag and diffusion coeffi-3ients in the Boltzmann-Langevin equations. The final D mesons are formed through charmquark fragmentation at high p T [32–34] or charm-light quark recombination at low and inter-mediate p T [35]. Such transport models for initial heavy quark and final meson productioncan describe the charm meson spectra and elliptic flow well in high-energy heavy-ion colli-sions [22, 23]. However, it is still interesting to investigate whether and to what degree theheavy quarks achieve kinematic equilibrium. If these heavy quarks indeed reach local ther-mal equilibrium, one should expect that the hydrodynamic model can also describe heavymeson spectra and elliptic flow. Furthermore, a hydrodynamic picture also overcomes thedifficulty of the transport equation when the coupling becomes large between charm quarksand mesons and the medium.In this study, we will consider the extreme scenario and investigate the charmed mesonspectra and elliptic flow in the limit of the fully thermalized low- p T charm quarks. Theselow- p T charm quarks are still produced initially through hard processes, but the interactionsare assumed to be strong enough that they quickly diffuse into the medium and lose thememory of their initial distribution in phase space, both the spatial location and momentumof the initial production through hard processes. In the momentum space, they are assumedto reach full kinetic equilibrium and comove with the medium, flowing with the stronglycoupled quark-gluon plasma as light quarks and gluons. Finally, the charm quarks transit tocharm hadrons with a phase transition of the bulk medium where we assume the interactionin the hadronic phase is still strong enough to maintain the kinetic equilibrium and theequilibrium ratios of different species of charm mesons. In this situation, the low- p T Dmesons are produced in the same way as other light hadrons on the freeze-out hypersurfacein a relativistic hydrodynamic model. The temperature and the fluid velocity profiles onthe freeze-out hypersurface are crucial for a reasonable estimate of the D -meson spectra andelliptic flow in the limit of a complete heavy-quark thermalization. We will use the rapiditydistribution, p T spectra, and elliptic flow of charged pions to calibrate the hydrodynamicmodel.For completeness, we have to consider D mesons from the feed-down of D ∗ . Using the4eed-down tables from [35, 36], D ∗ (2007) −−−→ D (1865) + π + γ, (1) D ∗ (2010) + 68% −−→ D (1865) + π + , (2) D ∗ (2010) − −−→ D (1865) + π − , (3)Notice that the D ∗ ’s are vector mesons with spin 1. So the spin degeneracy ratio for D ∗ /D is . B. Relativistic hydrodynamics
The hydrodynamic model we use, CLVisc, is a (3+1)D viscous hydrodynamic modelparallelized on GPU using OpenCL [30]. The program is well tested against several analyticalsolutions and can describe the bulk hadron spectra and anisotropic flow in high-energy heavy-ion collisions at top RHIC and LHC energy. It simulates the fluid dynamic evolution of thestrongly interacting QCD matter created in high-energy heavy-ion collisions by solving thefluid dynamic equations, ∇ µ T µν = 0 , with T µν = εu µ u ν − P ∆ µν + π µν , (4)where ε is the energy density, P is the pressure as a function of energy density given bythe equation of state (EoS), ∆ µν = g µν − u µ u ν is a projection operator, u µ is the fluidfour-velocity obeying u µ u µ = 1 and π µν is the shear stress tensor.The initial condition for entropy density distribution in the transverse plane is providedby Trento Monte Carlo model [37, 38]. An envelope function is used to approximate thelongitudinal distribution along the space-time rapidity, H ( η s ) = exp (cid:34) − ( | η s | − η w ) σ η θ ( | η s | − η w ) (cid:35) , (5)where σ η = 1 . and σ w = 1 . are used for Au+Au collisions at √ s NN = 200 GeV.We have assumed an initial time for the hydrodynamics τ = 0 . fm. In the present study,we use the partial chemical equilibrium EoS with chemical freeze-out temperature 165 MeVand a smooth crossover between a QGP at high temperature and the hadron resonance gas(HRG) EoS at low temperature [39] as inspired by the lattice QCD study.5aryons and mesons passing through the freeze-out hyper-surface are assumed to obeyFermi-Dirac and Bose-Einstein distributions, respectively. Their momentum distributionsare given by the Cooper-Frye formula [29], dN i dY p T dp T dφ = g i (2 π ) ˆ p µ d Σ µ f ( p · u )(1 + δf ) , (6)where g i = 2spin + 1 is the spin degeneracy, p µ is the four-momenta of produced particlesin lab frame, Σ µ is the freeze-out hyper-surface, f ( p · u ) is the Fermi-Dirac/Bose-Einsteindistribution function, f ( p · u ) = 1exp [( p · u − µ i ) /T frz ] ± , (7)and δf is the non-equilibrium correction, δf = (1 ∓ f eq ) p µ p ν π µν T ( ε + P ) . (8)We have chosen the freeze-out temperature T fz = 137 M eV for light flavor hadrons. Thefreeze-out temperature for D mesons will be different and we will consider several values toprovide an estimate of the uncertainties.The elliptic flow of D mesons is defined as the second coefficient of the Fourier decompo-sition of their azimuthal angle distributions with respect to the event plane of light hadrons, d Np T dp T dydφ = d N πp T dp T dy (cid:34) ∞ (cid:88) n =1 v n cos ( n ( φ − Ψ EP )) (cid:35) . (9) C. Transport approach of heavy flavor evolution
We will compare the charm meson spectra and flow from the hydrodynamic calculationto that of a transport calculation [28]. The transport approach assumes that heavy quarks,including those with low momentum in the comoving frame of the medium, remain goodquasi-particles in the QGP. Therefore, the dynamics of low- p T heavy quarks can be describedby a Boltzmann-type transport equation, (cid:18) ∂∂t + v · ∇ (cid:19) f Q ( t, x , p ) = ˆ (cid:20) dRd q ( p + q, q ) f Q ( p + q ) − dRd q ( p, q ) f Q ( p ) (cid:21) d q . (10)Here, f Q ( t, x , p ) is the phase-space density of heavy quarks (or anti-heavy quarks), dR ( p, q ) /dq is the differential rate for a heavy quark with momentum p to transfer three-momentum q to the local medium with flow velocity u and temperature T . The rate includes both the6ontribution from momentum diffusion and energy loss caused by soft-momentum trans-fer and those induced by large-momentum transfer scatterings and medium-induced gluonbremsstrahlung. The initial spectrum that initializes the transport equation is obtainedfrom the perturbative QCD based FONLL [40, 41] calculation with EPPS16 nuclear partondistribution function [42].The hydrodynamic and the transport model have both an overlapping and distinctregimes of application. Both models contain the equilibrium situation. Compared to the hy-drodynamic approach, a transport model also governs the far-from-equilibrium dynamics ofheavy flavor particles. The traces of off-equilibrium effects can be important in a finite andexpanding plasma with moderate coupling between heavy quarks and the medium. In themeantime, hydrodynamics can describe the evolution with large couplings and in the non-quasiparticle regime, which is beyond the applicability of the transport approach. Therefore,both models provide complimentary pictures to understand the experimental data. III. RESULTSA. Model calibration with experimental data
To calibrate the relativistic hydrodynamic model, we have computed the pseudorapiditydensity of charged particles dN ch /dη as shown in Fig. 1, the p T spectra and the elliptic flow v of π + as shown in Fig. 2 for central Au+Au collisions with centrality range − at √ s NN = 200 GeV. The calibrated parameters are the scale factor of 53 multiplied to theinitial entropy density from the Trento model [37, 38], the starting time for hydro τ = 0 . fm,the shear viscosity over entropy density ratio η/s = 0 . , and the freeze-out temperaturefor light hadrons T fz = 137 MeV. These values have been tuned such that the predicted dN ch /dη , p T spectra and v ( p T ) of charged pions from the relativistic hydrodynamics agreewith the experimental data. Relativistic hydrodynamics with the same set of parameters alsodescribes data well at other centralities and we refer readers to Ref. [30] for a more detaileddescription of CLVisc and the parameters. These parameters are fixed in the followingcalculations of the heavy quark meson spectra, except the freeze-out temperatures for D mesons. We will vary the value of the D meson freeze-out temperature and study its effecton the final D meson spectra and the elliptic flow.7 − − − η d N / d η √ s NN = 200GeV , charged hadrons CLViscPHOBOS
Figure 1. The pseudorapidity density of charged hadrons from integrating the Cooper-Frye formulaover the freeze-out hyperspace obtained in the CLVisc calculation. It agrees with data of thePHOBOS experiment for Au + Au collisions in 0-6% centrality at √ s NN = 200 GeV [43]. . . . . . . . . p T [GeV] − − − ( / π ) d N / d Y p T d p T [ G e V ] − √ s NN = 200GeV , π + PHENIXCLVisc . . . . . . p T [GeV] . . . . . . . . . v √ s NN = 200GeV , π + PHENIXCLVisc
Figure 2. Left: the transverse momentum spectra of π + in 0-5% central Au + Au collisions.Right: the anisotropic flow coefficient v in 20-40% central Au + Au collisions using the event-plane method. In both panels, CLVisc calculations (lines) are compared to data from the PHENIXexperiment [44]. . Effects of the freeze-out temperature and resonance decay on D spectra p T [GeV]2 . . . . . . D ( w i t hd ec a y ) / D ( d i r ec t) T frz = 155 MeVT frz = 137 MeVT frz = 100 MeV20-30% Au+Au at √ s NN = 200GeV , D p T [GeV]0 . . . . . . . v √ s NN = 200GeV , D T frz = 155 MeVT frz = 137 MeVT frz = 100 MeVno decaywith decay Figure 3. Left: the freeze-out temperature dependence of the effect of resonance decay on the D p T spectra. Right: the freeze-out temperature dependence of the effect of resonance decay onthe D flow coefficient v . Calculations in both panels use one-shot hydrodynamic simulation of20-30% central Au+Au collisions at √ s NN = 200 GeV.
The effects of freeze-out temperature and resonance decays on D meson production arestudied in this section. In the left panel of Fig. 3, we show the ratio between D p T spectrawith and without contributions from the resonance decays. In the right panel of Fig. 3,we compare the D meson elliptic flow as a function of p T with (dot-dashed) and without(solid) resonance decays for three different values of the D meson freeze-out temperatures, T fz = p T spectra of D (left panel) than that on the elliptic flow v of D (right panel). For a given D meson freeze-out temperature, resonance decays from D ∗ contribute more D mesons atlow p T than that at high p T . The ratio between D with and without decay decreases as the p T increase for all three freeze-out temperatures considered here. As one increases freeze-outtemperature from MeV to
MeV and
MeV, not only the D yield increases, butalso the ratio between D with and without decay increases, from about . to . and . at p T < GeV. The elliptic flow without decay (black solid), on the other hand, almostoverlap that with decays (red dash-dotted) for all three different freeze-out temperatures,therefore, it is insensitive to the contributions from the resonance decays. The freeze-outtemperature, however, has a large effect on the shape and magnitude of the elliptic flow9s a function of transverse momentum. At small transverse momentum ( p T < 2 GeV), theelliptic flow increases with the freeze-out temperature. At large transverse momentum ( p T > 2 GeV), the trend goes the opposite way – the elliptic flow decreases as the freeze-outtemperature increases. C. Model and data comparisons of the p T spectra and v of D mesons p T [GeV]10 − − − − ( / π ) d N / d Y p T d p T [ G e V ] − Au+Au at √ s NN = 200GeV , D - % ( *10 ) - % - % ( *100 ) CLVisc T frz = 150MeV µ c = 700MeVCLVisc T frz = 137MeV µ c = 800MeVSTARtransport modelpp . . . . . v Au+Au at √ s NN = 200GeV10-40% CLVisc D T frz = 150MeVCLVisc π + T frz = 150MeVtransport model D p T [GeV] . . . . .
20 0-80% CLVisc D T frz = 137MeVCLVisc π + T frz = 137MeVSTAR D Figure 4. Left: hydrodynamic calculations (lines) of D p T spectra in Au + Au collisions at √ s NN = 200 GeV for three different centrality bins, using two different sets of freeze-out tempera-tures and effective charm chemical potentials. They are compared to transport calculations in thesame collision system (blue bands), and data from the STAR experiment [12]. We also compare theD-meson production in p + p collisions from FONLL calcualtion (green band) [40, 41] to the STARmeasurement [45]. Right: hydrodynamic calculations of D meson v are compared to transportcalculations and STAR data [13]. In Fig. 4, we show the hydrodynamic calculations of the transverse momentum ( p T )spectra (left) and the p T -dependent elliptic flow (right) of D mesons in Au + Au collisionsat √ s NN = 200 GeV for different centralities as compared with the results of the transportapproach and experimental data from the STAR experiment [12]. Since the thermal produc-10ion of charm quark pairs is negligible in the QGP at the temperatures reached in heavy-ioncollisions at the RHIC and LHC colliding energies, we assume the charm quark number fromthe initial hard processes is conserved via an introduction of charm chemical potential at thefreeze-out. The value of the chemical potential µ c at the freeze-out temperature T frz is ad-justed to fit the magnitude of the experimental data on the p T spectra at low p T in the mostcentral collisions. For T frz = 150 MeV, the effective charm chemical potential µ c = 700 MeVis found to fit the experimental data on D spectra (red) at low p T for 0-10% (solid), 10-40%(dot-dashed) and 0-80% (dashed) centrality. With the same freeze-out temperature and theeffective charm chemical potential in the most central collisions, the hydrodynamic modelcan describe the low p T spectra well at other centralities. However, the hydro calculationsover predict the p T spectra at p T > GeV/ c .Different hadrons can freeze-out at different temperatures in the hydrodynamic model ofhadron production. For example, it was shown in the UrQMD model studies that protonsfreeze-out earlier than pions and kaons [46]. In the hydro calculation of D spectra (redin the left panel), we have assumed that D mesons freeze out ( T frz = 150 MeV) earlierthan the light quark hadrons ( T frz = 137 MeV). If we set the freeze-out temperature of D as the same as the light quark hadrons at 137 MeV with a re-adjusted charm chemicalpotential µ c = 800 MeV, the slope of the p T spectra becomes slightly larger (blue) due toincreased radial expansion, further away from experimental data. Increasing the freeze-outtemperature will slightly decrease the slope of the spectra at high p T . However, the changeis too small to describe the data at high p T .In the left panel of Fig. 4, we also compare to the transport calculations (blue bands)of the D spectra in Au+Au collisions. We can see that the transport approach, whichinclude effects of non-equilibrium evolution, describes the experimental data on p T spectraat large p T > GeV/ c . At low p T , the calculations are still consistent with experimentswithin error bands but the central points of the band under-predict the experimental data.The uncertainty of the transport calculation is dominated by the uncertainty in the initialcharm spectra that initializes the transport equation. Initial charm spectra are computedusing the FONLL program [40, 41], whose major uncertainty stems from the variation ofthe renomalization scale m T / < µ R < m T , where m T = (cid:112) M + p T is the transversemass of the heavy quark. The central FONLL prediction in proton-proton collisions (thegreen band) is systemically below the STAR data on D production in p+p collisions at11ow- p T , while the upper band of the prediction with µ R = m T / is found to be closer tothe measurements. Moreover, in nuclear collisions, the production of low- p T D mesons issensitive to the small- x region of the nuclear parton distribution function (nPDF), whichis less well-constrained. Note that the EPPS16 also published error estimating nPDF sets;however, we have only shown the calculation using the central fit of EPPS16. Therefore, thelow- p T part of the spectra of the transport calculation has quite large uncertainty and hasto be taken into account seriously when compared to data.Comparison of the high- p T part of the spectra between the hydrodynamic results andexperimental data show that charm quarks have not reached complete equilibrium for p T > GeV/ c , since the spectra from experiments are still below the hydrodynamic results whichassumes a fully equilibrated system of charm quarks that flow with the QGP fluid.Both the relativistic hydro and transport model can describe the experimental data onthe elliptic flow of D mesons within the error bars as shown in the right panel of Fig. 4.One can see the mass ordering of the elliptic flow by comparing it to the elliptic flow ofcharged pions (blue) from the same hydrodynamic calculations. The hydrodynamic resultson v ( p T ) at low p T are insensitive to the value of the freeze-out temperature.We stress that even though the hydrodynamics fails to describe the p T spectra of D beyond p T > GeV/ c , its results on the elliptic flow agree with the experimental data andthe transport model calculation (solid blue) for p T up to 4 GeV/ c . This implies that theexperimental data on the elliptic flow cannot provide a stringent constraint on the kineticequilibration of heavy quarks. The strong interaction between heavy quarks and the medium,however, can "drag" the heavy quarks along collective flow developed for the bulk mediumeven though the interaction might not drive the heavy quarks to full kinetic equilibration. IV. SUMMARY
We have calculated the D meson spectra and elliptic flow in Au+Au collisions at theRHIC energy within a relativistic viscous hydrodynamic model assuming that charm quarksinitially produced through hard processes become fully kinetically equilibrated in the QGP.We neglect the thermal charm quark pair production in the QGP and use an effectivecharm chemical potential at the freeze-out which is tuned to describe the number of charmquarks produced in the initial hard processes. With the charm freeze-out temperature12 frz = 137 − MeV, the hydrodynamic model can describe the experimental data on D elliptic flow for p T < GeV/ c as well as the elliptic flow of light quark hadrons. Themass ordering of the elliptic flow for pions and D mesons in the experimental data is alsoobserved for a freeze-out temperature between 137 and 150 MeV. We have also comparedthe hydrodynamic results with that from a transport model with both energy loss andmomentum diffusion, which can describe the observed elliptic flow of D mesons at both lowand high p T .The hydrodynamics, however, fails to describe the p T spectra of D for p T > GeV/ c ,significantly over-predicting the D p T spectra at large p T . The transport model, on theother hand, can describe well the p T spectra at large p T > c due to parton energyloss. But its central values under-predict the spectra at low p T due to the baseline spectraof the initial charm production through hard processes in p+p collisions.Our comparative study indicates that only the low- p T D mesons in the experiments mighthave reached kinetic equilibrium while charm quarks in the intermediate region of p T arepartially thermalized. Parton energy loss and momentum diffusion in the transport modelcan describe well the non-equilibrium behavior of D mesons spectra and elliptic flow atlarge p T . The interaction of charm quarks in partial equilibrium with the medium in theintermediate p T , however, can develop elliptic flow as large as the fully equilibrated charmquarks. ACKNOWLEDGMENTS
This work is supported in part by the National Science Foundation of China under GrantNos. 11935007, 11221504, 11861131009, 12075098 and 11890714, and by the Director, Officeof Energy Research, Office of High Energy and Nuclear Physics, Division of Nuclear Physics,of the U.S. Department of Energy (DOE) under grant No. DE- AC02-05CH11231, bythe U.S. National Science Foundation under grant No. ACI-1550228 within JETSCAPECollaboration, No. OAC- 2004571 within the X-SCAPE Collaboration by the UCB-CCNUCollaboration Grant. Computations are performed at Nuclear Science Computer Center atCCNU (NSC3) and the National Energy Research Scientific Computing Center (NERSC),a U.S. Department of Energy Office of Science User Facility operated under Contract No.13E-AC02- 05CH11231. [1] Boris A. Gelman, Edward V. Shuryak, and Ismail Zahed. Classical strongly coupled QGP. I.The Model and molecular dynamics simulations.
Phys. Rev. C , 74:044908, 2006.[2] Ulrich W. Heinz. The Strongly coupled quark-gluon plasma created at RHIC.
J. Phys. A ,42:214003, 2009.[3] S.S. Adler et al. Elliptic flow of identified hadrons in Au+Au collisions at √ s NN = 200 GeV.
Phys. Rev. Lett. , 91:182301, 2003.[4] John Adams et al. Particle type dependence of azimuthal anisotropy and nuclear modificationof particle production in Au + Au collisions at √ s NN = 200 GeV . Phys. Rev. Lett. , 92:052302,2004.[5] Betty Bezverkhny Abelev et al. Elliptic flow of identified hadrons in Pb-Pb collisions at √ s NN = 2 . TeV.
JHEP , 06:190, 2015.[6] C. Adler et al. Disappearance of back-to-back high p T hadron correlations in central Au+Aucollisions at √ s NN = 200 GeV.
Phys. Rev. Lett. , 90:082302, 2003.[7] K. Adcox et al. Suppression of hadrons with large transverse momentum in central Au+Aucollisions at √ s NN = 130-GeV. Phys. Rev. Lett. , 88:022301, 2002.[8] S.S. Adler et al. Suppressed π production at large transverse momentum in central Au+ Aucollisions at √ s NN = 200 GeV. Phys. Rev. Lett. , 91:072301, 2003.[9] C. Adler et al. Centrality dependence of high p T hadron suppression in Au+Au collisions at √ s NN = 130-GeV. Phys. Rev. Lett. , 89:202301, 2002.[10] Serguei Chatrchyan et al. Study of high-pT charged particle suppression in PbPb comparedto pp collisions at √ s NN = 2 . TeV.
Eur. Phys. J. C , 72:1945, 2012.[11] L. Adamczyk et al. Observation of D Meson Nuclear Modifications in Au+Au Collisions at √ s NN = 200 GeV.
Phys. Rev. Lett. , 113(14):142301, 2014. [Erratum: Phys.Rev.Lett. 121,229901 (2018)].[12] Jaroslav Adam et al. Centrality and transverse momentum dependence of D -meson productionat mid-rapidity in Au+Au collisions at √ s NN = 200 GeV . Phys. Rev. C , 99(3):034908, 2019.[13] L. Adamczyk et al. Measurement of D Azimuthal Anisotropy at Midrapidity in Au+AuCollisions at √ s NN =200 GeV. Phys. Rev. Lett. , 118(21):212301, 2017.
14] Betty Abelev et al. Suppression of high transverse momentum D mesons in central Pb-Pbcollisions at √ s NN = 2 . TeV.
JHEP , 09:112, 2012.[15] B. Abelev et al. D meson elliptic flow in non-central Pb-Pb collisions at √ s NN = 2.76TeV. Phys. Rev. Lett. , 111:102301, 2013.[16] Albert M Sirunyan et al. Nuclear modification factor of D mesons in PbPb collisions at √ s NN = 5 . TeV.
Phys. Lett. B , 782:474–496, 2018.[17] Albert M Sirunyan et al. Measurement of prompt D meson azimuthal anisotropy in Pb-Pbcollisions at √ s NN = 5.02 TeV. Phys. Rev. Lett. , 120(20):202301, 2018.[18] A. M. Sirunyan et al. Elliptic flow of charm and strange hadrons in high-multiplicity pPbcollisions at √ s NN = Phys. Rev. Lett. , 121(8):082301, 2018.[19] Ming Chen Wang and G. E. Uhlenbeck. On the theory of the brownian motion ii.
Rev. Mod.Phys. , 17:323–342, Apr 1945.[20] Ralf Rapp and Hendrik van Hees. Heavy Quarks in the Quark-Gluon Plasma. pages 111–206,2010.[21] Guy D. Moore and Derek Teaney. How much do heavy quarks thermalize in a heavy ioncollision?
Phys. Rev. C , 71:064904, 2005.[22] A. Beraudo et al. Extraction of Heavy-Flavor Transport Coefficients in QCD Matter.
Nucl.Phys. A , 979:21–86, 2018.[23] Shanshan Cao et al. Toward the determination of heavy-quark transport coefficients in quark-gluon plasma.
Phys. Rev. C , 99(5):054907, 2019.[24] Debasish Banerjee, Saumen Datta, Rajiv Gavai, and Pushan Majumdar. Heavy Quark Mo-mentum Diffusion Coefficient from Lattice QCD.
Phys. Rev. D , 85:014510, 2012.[25] H.T. Ding, A. Francis, O. Kaczmarek, F. Karsch, H. Satz, and W. Soeldner. Charmoniumproperties in hot quenched lattice QCD.
Phys. Rev. D , 86:014509, 2012.[26] A. Francis, O. Kaczmarek, M. Laine, T. Neuhaus, and H. Ohno. Nonperturbative estimate ofthe heavy quark momentum diffusion coefficient.
Phys. Rev. D , 92(11):116003, 2015.[27] Yingru Xu, Jonah E. Bernhard, Steffen A. Bass, Marlene Nahrgang, and Shanshan Cao. Data-driven analysis for the temperature and momentum dependence of the heavy-quark diffusioncoefficient in relativistic heavy-ion collisions.
Phys. Rev. C , 97(1):014907, 2018.[28] Weiyao Ke, Yingru Xu, and Steffen A. Bass. Linearized boltzmann-langevin model for heavyquark transport in hot and dense qcd matter.
Phys. Rev. C , 98:064901, Dec 2018.
29] Fred Cooper and Graham Frye. Comment on the Single Particle Distribution in the Hydro-dynamic and Statistical Thermodynamic Models of Multiparticle Production.
Phys. Rev. D ,10:186, 1974.[30] Long-Gang Pang, Hannah Petersen, and Xin-Nian Wang. Pseudorapidity distribution anddecorrelation of anisotropic flow within the open-computing-language implementation CLVischydrodynamics.
Phys. Rev. C , 97(6):064918, 2018.[31] John C. Collins, Davison E. Soper, and George F. Sterman. Heavy Particle Production inHigh-Energy Hadron Collisions.
Nucl. Phys. B , 263:37, 1986.[32] Eric Braaten, King-man Cheung, Sean Fleming, and Tzu Chiang Yuan. Perturbative QCDfragmentation functions as a model for heavy quark fragmentation.
Phys. Rev. D , 51:4819–4829, 1995.[33] Matteo Cacciari and Paolo Nason. Charm cross-sections for the Tevatron Run II.
JHEP ,09:006, 2003.[34] Matteo Cacciari, Stefano Frixione, Nicolas Houdeau, Michelangelo L. Mangano, Paolo Nason,and Giovanni Ridolfi. Theoretical predictions for charm and bottom production at the LHC.
JHEP , 10:137, 2012.[35] R. Rapp and E.V. Shuryak. D meson production from recombination in hadronic collisions.
Phys. Rev. D , 67:074036, 2003.[36] C. Patrignani et al. Review of Particle Physics.
Chin. Phys. C , 40(10):100001, 2016.[37] J. Scott Moreland, Jonah E. Bernhard, and Steffen A. Bass. Alternative ansatz to wounded nu-cleon and binary collision scaling in high-energy nuclear collisions.
Phys. Rev. C , 92(1):011901,2015.[38] Jonah E. Bernhard, J. Scott Moreland, Steffen A. Bass, Jia Liu, and Ulrich Heinz. ApplyingBayesian parameter estimation to relativistic heavy-ion collisions: simultaneous characteriza-tion of the initial state and quark-gluon plasma medium.
Phys. Rev. C , 94(2):024907, 2016.[39] Pasi Huovinen and Pter Petreczky. QCD Equation of State and Hadron Resonance Gas.
Nucl.Phys. A , 837:26–53, 2010.[40] Matteo Cacciari, Mario Greco, and Paolo Nason. The P(T) spectrum in heavy flavor hadropro-duction.
JHEP , 05:007, 1998.[41] Matteo Cacciari, Stefano Frixione, and Paolo Nason. The p(T) spectrum in heavy flavorphotoproduction.
JHEP , 03:006, 2001.
42] Kari J. Eskola, Petja Paakkinen, Hannu Paukkunen, and Carlos A. Salgado. EPPS16: Nuclearparton distributions with LHC data.
Eur. Phys. J. C , 77(3):163, 2017.[43] B.B. Back et al. The Significance of the fragmentation region in ultrarelativistic heavy ioncollisions.
Phys. Rev. Lett. , 91:052303, 2003.[44] S.S. Adler et al. Identified charged particle spectra and yields in Au+Au collisions at √ s NN =200 GeV.
Phys. Rev. C , 69:034909, 2004.[45] L. Adamczyk et al. Measurements of D and D ∗ Production in p + p Collisions at √ s = 200 GeV.
Phys. Rev. D , 86:072013, 2012.[46] H. van Hecke, H. Sorge, and N. Xu. Evidence of early multistrange hadron freezeout in high-energy nuclear collisions.
Phys. Rev. Lett. , 81:5764–5767, 1998., 81:5764–5767, 1998.