^{1,2,3}

^{*}

^{1}

^{4}

^{2}

^{†}

lish@ctgu.edu.cn

liaoji@indiana.edu

^{3}.

The heavy quark propagation behavior inside the quark-gluon plasma (QGP), is usually described in terms of the Boltzmann dynamics, which can be reduced to the Langevin approach by assuming a small momentum transfer for the scattering processes between heavy quarks and the QGP constituents. In this work, the temperature and energy dependence of the transport coefficients are calculated in the framework of both Boltzmann and Langevin dynamics, by considering only the elastic scattering processes to have a better comparison and understanding of these two models. The extracted transport coefficients are found to be larger in the Boltzmann approach as compared with the Langevin, in particular in the high-temperature and high-energy region. Within each of the two theoretical frameworks, we simulate the charm quark production and the subsequent evolution processes in relativistic heavy-ion collisions. We find that the energy loss due to elastic scattering is larger from the Boltzmann dynamics, resulting in a smaller

In ultrarelativistic collisions of heavy nuclei such as Au or Pb, an extreme high temperature and energy density environment can be produced around the collision point, which allows formation of a new state of nuclear matter consisting of the deconfined quarks and gluons, namely the quark-gluon plasma, QGP

Heavy quark (HQ), including charm and bottom, are of particular interest

During the propagating through the QGP medium, the HQ dynamics is usually described by the Boltzmann or Langevin model

Many models were developed from the Boltzmann

As mentioned, the Langevin approach is a very convenient and widely used model, and it allows us to establish, directly, a link between the observables and transport coefficients, which can be extracted from the lattice QCD calculations. However, the condition

The paper is organized as follows. In Sec.

The Boltzmann transport equation (BTE) reads

In the local rest frame (LRF) of the cell, the heavy quark transport is performed within a given time step

It was argued

While traversing the quark-gluon plasma (QGP), HQ suffers frequent but soft momentum kicks from the medium partons, therefore, HQ behaves the Brownian motion, which can be described by the Langevin transport equation (LTE)

The deterministic drag force reads

The stochastic force that acts on the HQ is expressed as

In this section, we mainly focus on the comparison of the transport coefficients obtained via the Boltzmann and Langevin approaches with considering only the elastic scattering (

Scattering rate obtained in

The spatial diffusion coefficient

Spatial diffusion coefficient

With Eq.

In Fig.

The drag coefficient

Concerning the drag coefficient

In the previous analysis

The initialization of the heavy quark pairs is performed in the spatial and momentum space, respectively. In the transverse direction, the initial spatial distribution is sampled according to the initial binary collision density that is modeled by a Glauber-based approach

The above initial-state configuration allows providing the relevant entropy density distribution, which will be taken as the input of the subsequent hydrodynamical evolution. All the parameters in this procedure are tuned by the model-to-data comparison

The underlying medium evolution is modeled by a (3+1)D relativistic viscous hydrodynamics, vHLLE

The QGP medium expands and cools down, and the (local) temperature drops below the critical one

We refer to Ref.

sample a given number of HQ pairs at the position and momentum (

move all the HQs from

search the fluid cell at the same position as HQ,

make a discrete time step

Boltzmann dynamics: for the current HQ with

Langevin dynamics: fix the drag and momentum diffusion coefficient with the fluid cell temperature

and then, modify the HQ momentum

update the HQ position after the time step

repeat the above steps (iii)–(v) when the local temperature

The heavy quark will suffer the instantaneous hadronization procedure via a dual approach, including fragmentation and heavy-light coalescence mechanisms, when the local temperature drops below the critical one

Concerning the universal fragmentation function, the Braaten approach

The momentum distributions of heavy-flavor mesons (

In Fig.

Comparison of the coalescence probability, for

As displayed in Fig.

In order to study the difference between the Boltzmann and Langevin dynamics, in this section, we focus on the time evolution of the charm quark momentum distribution, which is obtained inside a static medium with temperature fixed at

In Fig.

(a) Charm quark momentum distribution based on the Boltzmann dynamics at different times during the hydrodynamical evolution of the medium with a constant temperature

Figure

Elastic energy loss of charm quarks obtained via Boltzmann approach (thick curve) and Langevin approach with

Figure

(a) Comparison of the charm quark

Concerning the relative azimuthal angle distribution, the yields of the initially back-to-back generated

Comparison of the relative azimuthal angle between

Figure

(a) Comparison of the nuclear modification factor

Based on the Bayesian model-to-data analysis, the original Lido hybrid model

Concerning the scattering inelastically with the light (anti)quarks and gluons of QGP in Boltzmann, both the

The gluon radiation incorporated Langevin transport model is expressed as ^{1}

According to its definition,

We can see that the implementations of radiative energy loss are different in the Boltzmann and Langevin approaches, which will apparently introduce the source of uncertainty when comparing these two models. However, it is still necessary to check further the modifications for each dynamics.

Figure

Same as Fig.

Figure

Same as Fig.

Figure

Same as Fig.

In this work, we investigated the charm quark evolution via the Boltzmann and Langevin dynamics in relativistic heavy-ion collisions. By including only the elastic scattering contributions, the extracted drag coefficient (

It is found that

It is necessary to mention that Ref.

Finally, it is interesting to note that the resolution of the above model-to-data challenge may require the inclusion of nonperturbative dynamics in the medium. It may be noted that a similar challenge was previously investigated for light-flavor jet energy loss and a viable solution was previously proposed by introducing a nontrivial medium color structure that includes both chromoelectric and chromomagnetic degrees of freedom

The authors are grateful to Weiyao Ke for providing the Boltzmann module in Duke model, as well as the data shown in Fig.