Dynamical collisional energy loss and transport properties of on- and off-shell heavy quarks in vacuum and in the Quark Gluon Plasma

Dynamical collisional energy loss and transport properties of on- and off-shell heavy quarks in vacuum and in the Quark Gluon Plasma

H. Berrehrah berrehrah@fias.uni-frankfurt.de Frankfurt Institute for Advanced Studies and Institute for Theoretical Physics, Johann Wolfgang Goethe Universität, Ruth-Moufang-Strasse 1,
60438 Frankfurt am Main, Germany
   P.B. Gossiaux gossiaux@subatech.in2p3.fr Subatech, UMR 6457, IN2P3/CNRS, Université de Nantes, École des Mines de Nantes, 4 rue Alfred Kastler, 44307 Nantes cedex 3, France
   J. Aichelin aichelin@subatech.in2p3.fr Subatech, UMR 6457, IN2P3/CNRS, Université de Nantes, École des Mines de Nantes, 4 rue Alfred Kastler, 44307 Nantes cedex 3, France
   W. Cassing wolfgang.cassing@theo.physik.uni-giessen.de Institut für Theoretische Physik, Universität Giessen, 35392 Giessen, Germany
   E. Bratkovskaya brat@th.physik.uni-frankfurt.de Frankfurt Institute for Advanced Studies and Institute for Theoretical Physics, Johann Wolfgang Goethe Universität, Ruth-Moufang-Strasse 1,
60438 Frankfurt am Main, Germany

In this study we evaluate the dynamical collisional energy loss of heavy quarks, their interaction rate as well as the different transport coefficients (drag and diffusion coefficients, , etc). We calculate these different quantities for i) perturbative partons (on-shell particles in the vacuum with fixed and running coupling) and ii) for dynamical quasi-particles (off-shell particles in the QGP medium at finite temperature with a running coupling in temperature as described by the dynamical quasi-particles model). We use the perturbative elastic cross section for the first case, and the Infrared Enhanced Hard Thermal Loop cross sections for the second. The results obtained in this work demonstrate the effects of a finite parton mass and width on the heavy quark transport properties and provide the basic ingredients for an explicit study of the microscopic dynamics of heavy flavors in the QGP - as formed in relativistic heavy-ion collisions - within transport approaches developed previously by the authors.

Quarks Gluons Plasma, Heavy quark, Collisional, Energy loss, Transport coefficients, pQCD, DQPM, PHSD, On-shell, Off-shell.
24.10.Jv, 02.70.Ns, 12.38.Mh, 24.85.+p

I Introduction

In ultrarelativstic heavy-ion collisions there is strong circumstantial evidence that gluons and light quarks (,,) come to an equilibrium and form a plasma of quarks and gluons (QGP) Adams et al. (2005); Adcox et al. (2005); Back et al. (2005); Arsene et al. (2005); Zajc (2008); Cassing and Bratkovskaya (2008). As a consequence, mesons and baryons containing only light quarks are formed with a multiplicity which corresponds to that expected for a system in statistical equilibrium. This observation limits strongly the use of such mesons and baryons to study the properties of the QGP during its expansion because after thermal freeze out the information about the previous phase is lost. Heavy quarks ( and ) do not come to an equilibrium with the plasma degrees of freedom and may therefore serve as a probe to study the properties of the deconfined system in the early phase of the reaction.

It has been shown in experiments at the Relativistic Heavy Ion Collider (RHIC) and at the CERN Large Hadron Collider (LHC) accelerators that high momentum charm and bottom quarks experience a remarkable quenching in the QGP despite of their large mass. This quenching is reflected in the nuclear modification factor , i.e. the ratio of the transverse momentum spectra of heavy quarks in heavy-ion collisions to that in proton-proton collisions, properly scaled by the number of binary proton-proton collisions in a heavy-ion reaction. Usually, such a quenching is attributed to the energy loss of heavy quarks in the QGP, although alternative explanations are possible. Besides the , the elliptic flow of heavy quarks has turned out to be an important observable. Initially neither the heavy quarks, produced in hard collisions, nor the constituents of the plasma have sizeable nonzero elliptic flow. In the hydrodynamically expanding plasma the spatial eccentricity of the overlap region of projectile and target is converted into an elliptic flow in momentum space. In turn, heavy quarks may acquire an elliptic flow by interactions with the plasma constituents. It is remarkable that experimentally the elliptic flow of heavy mesons (including c-quark degrees of freedom) equals almost that of light mesons.

These observations have triggered many studies Dutt-Mazumder et al. (2005); Mustafa (2005); Svetitsky (1988); van Hees and Rapp (2005); Moore and Teaney (2005); van Hees et al. (2006); Gossiaux et al. (2005, 2006); Armesto et al. (2006); van Hees et al. (2008); Molnar (2005); Zhang et al. (2004); Gossiaux and Aichelin (2008a, 2009); Linnyk et al. (2008), first by exploiting the single-electron data Wicks et al. (2007); Gossiaux and Aichelin (2008b); Bielcik (2006); Adler et al. (2006); Rapp and van Hees (), later by investigating identified -mesons. Despite of the many efforts a consensus on how to treat the interactions of heavy quarks with the plasma has not been reached yet. Early predictions adopted the natural starting point to calculate the elastic interaction of the heavy quarks with the constituents of the plasma, the light quarks and gluons, on the basis of perturbative QCD (pQCD) scattering cross sections, making a rather arbitrary choice of the coupling constant and the infrared regulator of the gluon propagator. These calculations underestimate significantly the energy loss and the elliptic flow of heavy quarks, i.e. the coupling of heavy quarks with the QGP. Later on radiative energy loss has been considered. It increases the energy loss but it is difficult to calculate due to the interplay between Landau Pomeranschuck Migdal (LPM) effects and the rapid medium expansion. In addition, it has also been suggested that heavy hadronic resonances could be formed in the plasma which might yield an additional energy loss Rapp and van Hees ().

To be comparable with the experimental data these elementary interactions between heavy quarks and the plasma constituents have to be embedded in a model which describes the expansion of the QGP itself. The description of this expansion is not unique and different expansion scenarios may easily change the and values by a factor of two Gossiaux et al. (2011). This observation, as well as the many parameters which enter the calculations of the elementary interaction between heavy quarks and plasma constituents, which include quark and gluon masses, coupling constants, spectral functions and infrared cutoffs, make it useful to study transport coefficients in which the complicated reaction dynamics is reduced to a couple of coefficients which can be directly compared between different models (or with correlators from lattice QCD). It is the purpose of this article to present these transport coefficients for different models in continuation of Ref. Berrehrah et al. (2014) in which the different elementary cross sections have been computed.

In line with Ref. Berrehrah et al. (2014) we concentrate on the following models:

  • HTL-GA (Hard Thermal Loop-Gossiaux Aichelin) approach. The details of this model can be found in Refs. Berrehrah et al. (2014); Gossiaux and Aichelin (2008a); Gossiaux et al. (2009). As compared to former pQCD approaches this model differs in the description of the interaction between the heavy quarks and the plasma particles in two respects:

    • an effective running coupling constant extended in the non-perturbative domain that remains finite at vanishing four-momentum transfer in line with Dokshitzer et al. (1996).

    • an infrared regulator in the -channel which is determined from hard thermal loop calculations of Braaten and Thoma Braaten and Thoma (1991) including a running coupling . In practice, we use an effective gluon propagator with a ’global’ , where , for and scattering. The parameter is determined by requiring that the energy loss , obtained with this effective propagator, reproduces the running coupling result in the extended HTL calculation in the t-channel from Refs. Gossiaux and Aichelin (2008b, 2009); Gossiaux et al. (2009). The resulting value is . The details of this model are described in the appendix of Ref. Gossiaux and Aichelin (2008a).

    Using the latter ingredients the cross section is increased as compared to the former pQCD studies, especially for small momentum transfer Berrehrah et al. (2014); Gossiaux and Aichelin (2008a); Gossiaux et al. (2009). In the HTL-GA approach we will also consider the case of a fixed coupling constant () and of a Debye mass with as infrared regulator, even if such naive pQCD calculations are unable to reproduce the data, neither the energy loss nor the elliptic flow. The aim is to make contact with the previous calculations where the gluon propagator in the -channel Born matrix element has to be IR regulated by a fixed screening mass Combridge (1979); Svetitsky (1988). We will also report the results from standard pQCD calculations if available (cf. Moore and Teaney (Moore and Teaney, 2005)).

    In the HTL-GA approach, the heavy quarks are considered massive while the gluons and light quarks are massless. Nevertheless, we will study the effects of finite masses of gluons and light quarks on the transport properties of the heavy quarks, too.

  • IEHTL (Infrared Enhanced Hard Thermal Loop) approach: This approach takes into account nonperturbative spectral functions and self-energies of the quarks, antiquarks and gluons in the QGP at finite temperature. For this purpose, we use parametrizations of the quark and gluon masses and coupling constants provided by the dynamical quasi-particle model (DQPM) which are determined to reproduce lattice quantum chromodynamics (lQCD) results Peigne and Peshier (2008); Peshier (2004, 2005); Cassing et al. (2013); Cassing (2009); Cassing and Bratkovskaya (2009a); Cassing (2007); Cassing and Bratkovskaya (2009b) at finite temperature and vanishing quark chemical potential . In this approach, the gluons and light and heavy quarks have finite masses and widths.

  • DpQCD (Dressed perturbative QCD) approach: In this approach the spectral functions of the IEHTL approach are replaced by the DQPM pole masses for the incoming, outgoing and exchanged quarks and gluons. In this model the gluons and light and heavy quarks are massive but have zero widths.

The different models presented here aim to describe the heavy meson spectra and are related to the fundamental parameters of the microscopic interactions of heavy quarks with the QGP partons. The mesoscopic quantities, characterizing the transport properties of a heavy quark propagating in the QGP, are the transport coefficients formally expressed by the variable . Their time evolution is given for on-shell partons by


Here the indices stand for the discrete quantum numbers (spin , flavor and color ) of the initial heavy quark (with the momentum , mass , energy and velocity ) and the initial light quark or gluon (with the momentum , mass , energy and velocity ) while denote those of the heavy quark (with the momentum ) and of the light quark/gluon (with the momentum ) in the final state. is the degeneracy factor of the heavy quark () and is the degeneracy factor of the parton ( for light quarks, for massless gluons and for massive gluons). Furthermore, denotes the sum over the light quarks and gluons of the medium while is the transition matrix-element squared, related to the cross section by:




denoting the flux of the incoming particles and .

Depending on the choice of one can address different transport coefficients: For = 1, = , = , = , the formula (I.1) corresponds, respectively, to the interaction rate , the energy loss , the drag coefficient and finally the diffusion coefficient , all evaluated per unit proper time in the rest system of the heavy quark.

In our study we use (I.1) for the case of on-shell heavy quarks and partons with the transition amplitudes . Replacing the on-shell phase-space measure in (I.1) by the off-shell equivalent


where is the spectral function of the off-shell particle and denotes the Heaviside function, one can extend (I.1) to the case of off-shell partons to


with as explained above.

In this study we will address the different transport coefficients of a heavy quark in a static medium at finite temperature. They give the response of the medium to the propagation of a heavy quark under fixed thermodynamical conditions. For out-of-equilibrium microscopic dynamics of the heavy quarks in the QGP, a description of the medium evolution is required. This study is beyond the scope of this paper and can be realized within the Parton-Hadron-String-Dynamics (PHSD) or Monte-Carlo Heavy Quarks (MC@sHQ) transport approaches.

The paper is organized as follows: In Sec. II we first briefly recall the Dynamical Quasi-Particle Model in order to fix the main ingredients used in the DpQCD/IEHTL approaches. In Section III we evaluate the elastic interaction rate and the relaxation time for the approaches introduced above. The calculation of the transport coefficients is performed on the basis of these cross sections. In Sections IV and V we calculate the drag coefficient as well as the energy loss per unit length and the spatial diffusion coefficient. Both the drag and diffusion coefficients are the transport coefficients which are necessary for Fokker-Planck transport approaches describing the time evolution of the heavy-quark momentum distribution. In Section VI we discuss the transport coefficient , the average change of the square of the transverse momentum per unit length which is strongly related to the transverse diffusion. Finally, in Section VII we evaluate the transport cross section and perform a critical analysis of the applicability of the independent collision approach. We conclude in Section VIII with a summary of our findings.

Ii Reminder of the dynamical quasi-particle model

The dynamical quasi-particle model Peshier (2004, 2005); Cassing (2009) describes QCD properties in terms of “resummed” single-particle Green’s functions (in the sense of a two-particle irreducible (2PI) approach) and leads to a quasi-particle equation of state, which reproduces the QCD equation of state extracted from lattice QCD calculations in Ref. Borsanyi et al. (2010a, b). According to the DQPM, the constituents of the strongly coupled Quark Gluon Plasma (sQGP) are strongly interacting massive partonic quasi-particles. Details of the approach can be found in Berrehrah et al. (2014). Due to the finite imaginary parts of the selfenergies the partons of type are described by spectral functions for which we can assume one of the following forms:

  • Lorentzian- form: the spectral function , with , is given by Cassing et al. (2013); Cassing (2009); Cassing and Bratkovskaya (2009a); Cassing (2007); Cassing and Bratkovskaya (2009b)


    This spectral function (II) is antisymmetric in and normalized as


    , are the particle pole mass and width, respectively.

  • Breit-Wigner- form: the spectral function is given by the Breit-Wigner distribution:


    where is the dynamical quasi-particle (DQPM) mass (i.e pole mass). Here is the independent variable; it’s related to by while is related to by the relation .

Figure 1: (Color online) The DQPM spectral functions of light quarks, heavy quarks and gluons of (II)-(II.3) for ( GeV). and can be found in Berrehrah et al. (2014). Left (a) (Center (b)): The Lorentzian spectral function as a function of for T ( T). Right (c): The Breit-Wigner spectral function as a function of for .

The Breit-Wigner- form of the spectral function is a non-relativistic approximation of the Lorentzian- form (neglecting compared to , ). Figure 1 (l.h.s and center) shows the Lorentzian- form as a function of for different values of the momentum (the Breit-Wigner- form as a function of ) of the spectral function of gluons, light and heavy quarks for (r.h.s). One sees that the peaks and the widths of these distributions are different. The Lorentzian- shape depends on the momentum , whereas the Breit-Wigner- form is independent of . Comparing the figures 1-left and center, we notice that an increasing momentum leads to a shift of poles masses to larger values of .

The pole masses and widths for light quarks and gluons used in the DQPM are displayed in Fig. 2-(a) as a function of . The functional forms for the dynamical pole masses of quasiparticles (gluons and quarks) are chosen in a way that they become identical to the perturbative thermal masses in the asymptotic high-temperature regime Cassing (2009); Cassing and Bratkovskaya (2009a); Cassing (2007); Cassing and Bratkovskaya (2009b). Fig. 2-(a) gives also the light quark and gluon masses in HTL, where and , with .

Figure 2-(b) shows the infrared regulator used in the t-channel as a function of the medium temperature following the different models presented in the introduction. We recall that is identical to the DQPM gluon mass squared in the DpQCD model and is given by for the model HTL-GA with constant and by , where for the HTL-GA model with running . The small value of observed in HTL-GA with running as compared to the DpQCD value will have a strong effect on the transport coefficients (see below).

Figure 2: (Color online) Masses and widths of light quarks and gluons in the DQPM Ozvenchuk et al. (2013) and masses in the HTL as a function of ( GeV) (left). The infrared regulator as a function of the temperature given in HTL-GA as well as in the DpQCD model (right). The parameters , and in the HTL-GA model are described in the introduction.

Iii Elastic Interaction rate and the associated relaxation time

We start out with the elastic interaction rate, i.e. the rate of the interaction of a heavy quark with momentum propagating through a QGP in thermal equilibrium at a given temperature . The quarks of the plasma are described by a Fermi-Dirac distribution whereas the gluons follow a Bose-Einstein distribution . Therefore, the interaction rates are temperature dependent. In our calculation we use the elastic scattering cross sections of Ref. Berrehrah et al. (2014) for and collisions, for on-shell as well as for off-shell partons.

iii.1 Elastic interaction rates for on-shell partons

For on-shell particles (and in the reference system in which the heavy quark has the velocity ) the interaction rate for collisions is given by (see eq.(I.1)). We introduce the invariant quantity




is the momentum of the scattering partners in the c.m. frame. The interaction rate in the plasma rest system for a heavy quark with momentum is given, following (I.1), by


where denotes the sum over the light quarks and gluons of the medium. The integral in the middle of (III.3) does not depend on the reference frame and therefore it is convenient to perform the calculation in the rest frame of the heavy quark,


where is the invariant distribution of the plasma constituents in the rest frame of the heavy quark:


with being the fluid 4-velocity measured in the heavy quark rest frame, while is the angle between and .

iii.2 Elastic interaction rates for off-shell partons

For off-shell partons the elastic interaction rate (I.5) is obtained by replacing by with being the spectral function which can be specific for each particle species (I). For the two forms of spectral functions, the Lorentzian- form and the Breit-Wigner- form, the explicit results are given in appendix A for completeness.

The total interaction rate of the off-shell approach (IEHTL) in the plasma rest system is compared to that of the on-shell calculations (HTL-GA and DpQCD) (III.3) in figure 3-(a) and (b) as a function of the momentum of the heavy quark, and the temperature, respectively. We assume here a Breit-Wigner spectral function and a Boltzmann-Jütner distribution for both, the light quarks and the gluons. Our results are rather independent of the choice of the spectral function and of whether the Boltzmann-Jütner distribution is replaced by a Fermi/Bose distribution. In figures 3-(a) and (b), the thick black line refers to the case where a constant and a Debye mass with as infrared regulator are used, whereas the thin black line gives the result for a constant running coupling and , with and in HTL-GA model. If we give a finite mass to the light quark and gluon () as shown in figure 2-a) in the HTL-GA model the total rate changes only moderately as can be seen from the black dotted dashed line. The yellow dashed and red solid lines in figure 3-(a) and (b) display the results for the DpQCD and IEHTL models, respectively. As expected, the rates decrease when increases because the number of plasma collision particles becomes smaller. The reduction factor due to a finite mass is almost independent of and therefore the ratio Rate(massive )/Rate(massless ) in the HTL-GA model shows only a week temperature dependence.

A running coupling with an effective Debye mass of yields a larger cross section than a fixed coupling constant . Naturally this increases the interaction rate if the mass of the particles does not change. Even in regions where the masses of gluons and quarks in the HTL-GA model are not very different from those in the DpQCD approach (cf. Fig.2-a) the rates differ significantly. Therefore, the finite mass of light quark and gluons explains only part of the much lower interaction rate observed for the former model. The remaining difference is due to the cross sections which are lower in the DpQCD approach as compared to HTL-GA, Ref.Berrehrah et al. (2014).

The description of the interaction between the heavy quark and the partons of the medium differs between the models. Whereas the coupling constant is strong in the HTL-GA (with running) and DpQCD/IEHTL models, the coupling is weak in HTL-GA for constant coupling . On the other hand the interaction has a large range in HTL-GA (with running) due to a small infrared regulator and it has an intermediate range in the DpQCD/IEHTL and HTL-GA for constant coupling .

Figure 3: (Color online) , the total elastic interaction rate of c-quarks in the plasma rest frame for the three different approaches as a function of the heavy quark momentum for ( GeV) (left) and as a function of the temperature for a heavy quark momentum GeV (right). The parameters , and in the HTL-GA model are described in the introduction.

We see from Figs. 3-(a) and (b) that the spectral function decreases the interaction rate of heavy quarks with the medium on the order of 20%. This modification is rather independent of the heavy quark momentum and of the temperature of the plasma.

The difference between the DpQCD and IEHTL rates is related on one side to the propagator, which contains an additional imaginary part proportional to the gluon width in the IEHTL model, and on the other side to the asymmetry of the Breit-Wigner spectral function. For finite width, the contribution of larger masses in the Breit-Wigner spectral function (right side of the pole mass) is larger compared to smaller masses (left side of the pole mass), i.e. with broader gluon and light/heavy quark in the mass distribution, the average mass is shifted to higher values of . This implies that the heavy quark is getting more massive on average and consequently is slowing down in momentum/velocity. Therefore, the number of interactions/per unit length becomes smaller, as shown in figure 3-(a) and (b).

The HTL-GA approach (with running coupling) and massless light quarks and gluons gives by far the largest rate. Besides having the largest cross section (Berrehrah et al., 2014), the massless nature of the QGP constituent increase the number of possible elastic interactions. Even a finite mass lowers the rate not substantially.

The dependence of the rates on the medium temperature for an intermediate heavy quark momentum ( GeV) is illustrated in figure 3-(b). All models show an increase of the interaction rate with temperature. The linear dependence seen for the HTL-GA model with and can be easily explained since , where is the parton density and is the total cross section in this model. As studied in Ref. (Berrehrah et al., 2014) the temperature dependence of the total cross section differs from one model to another due to the different coupling constants and infrared regulators employed. The extra decrease of the rate in the DPQCD/IEHTL models for small temperature is related to the temperature dependence of the DQPM parton masses (cf. Fig.2-a).

The number of heavy quark elastic collisions per Fermi at and for an intermediate heavy quark momentum () is 18.4 for HTL-GA (running coupling constant and ) model, 1.37 for DpQCD and 1.06 for IEHTL. These numbers are larger at with 29.84 elastic collisions per Fermi in the HTL-GA model, 2.07 for DpQCD and 1.72 for IEHTL.

iii.3 Relaxation time for on- and off-shell partons

For the transport coefficients one needs to specify the relaxation time which can be deduced from the elastic interaction rate by averaging over the initial heavy-quark momentum. The relaxation time is given for the on- and off-shell partons by


with and being the Breit-Wigner- spectral functions (Berrehrah et al., 2014) for the incoming and outgoing light and heavy quark. Furthermore, is the heavy quark degeneracy factor while is the Fermi-Dirac distribution and the heavy quark density given for the on-shell case and the off-shell case by


The in-medium elastic cross section for , and scattering processes have been studied in detail in Ref. Berrehrah et al. (2014). The processes involving chemical equilibration (, and ) will not be included in the computation of the relaxation time for heavy quarks since their contribution is negligible due to the small probability that 2 of the few heavy quarks interact.

The results for the relaxation time of charm quarks are shown in figure 4-(a) for the on-shell and off-shell cases at finite temperature and at in a medium composed by light quarks/antiquarks and gluons. We display the results for the DpQCD/IEHTL and HTL-GA models. From Fig. 4-(a) one deduces that the heavy quark relaxation time is about the same for DpQCD/IEHTL and HTL-GA with fixed coupling. A much shorter relaxtion time is seen for HTL-GA with a running coupling.

Figure 4: (Color online) The relaxation time (left) and the total transition rate (right) for different models as a function of with GeV at for off-shell (IEHTL, red solid line) and on-shell (DpQCD, yellow dashed line) partons. We consider the DQPM pole masses for the on-shell partons in the DpQCD model and the DQPM spectral functions for the off-shell IEHTL approach. Also shown is a comparison with results in the HTL-GA approaches with constant and running coupling.

Figure 4-(a) shows also that decreases with temperature since the quark and gluon densities are increasing functions of the temperature. We can evaluate -in terms of powers of - the behavior of this relaxation time for the different approaches. The density is proportional to for large temperatures in case of . The transition rate then is proportional to a power law in , i.e. with , for the HTL-GA model and , for the DpQCD/IEHTL models. Hence, the relaxation time and for the DpQCD/IEHTL models. The large exponent in the relaxation times for in the DpQCD/IEHTL models can be traced back to the infrared enhancement of the effective coupling.

With the etablished results for the relaxation time we can now calculate transport coefficients in the quark gluon medium, like the viscosities and conductivities, and study their sensitivity to the off-shellness of the medium partons. Furthermore, from the interaction rate one can also deduce the transition rate for particles in a heat bath of a given temperature . In contradistinction to the elastic interaction rate, here both scattering partners are assumed to have a thermal distribution. The transition rate is derived for the case of on- and off-shell partons by


where is the density of the on-shell parton . The results for the total on- and off-shell transition rates are displayed in Fig. 4-(b). Due to the small DQPM parton widths, the influence of the spectral function is negligible and we obtain for DpQCD and IEHTL about the same transition rate . Parametrizing by we find differences between HTL-GA and DpQCD/IEHTL as discussed above. This different behaviour leads to different transport coefficients as we will show below.

Iv Drag Force and Coefficient

We continue our study with the drag and diffusion coefficient. For heavy masses of and quarks the relaxation times are large as compared to the typical time of an individual and collision. Therefore we might describe the time evolution of heavy quarks in momentum space by a Fokker-Planck (FP) equation Svetitsky (1988); Golam Mustafa et al. (1998):


The drag () and diffusion () coefficients are evaluated according to Svetitsky (1988); Golam Mustafa et al. (1998); Gossiaux et al. (2005) by a Kramers-Moyal power expansion of the collision integral kernel of the Boltzmann equation. Note that the diffusion tensor admits a transverse-longitudinal decomposition (perpendicular and along the direction of the heavy quark in the frame where the fluid is at rest) and contains two independent coefficient and . As realized in Refs. Gossiaux et al. (2005); Walton and Rafelski (2000), the asymptotic distribution coming out of the FP evolution deviates from a (relativistic) Maxwell-Boltzmann distribution, which is a consequence of the truncation of the Kramers-Moyal series.

iv.1 Drag coefficient for on-shell partons

The drag coefficient describes the time evolution of the mean momentum of the heavy quark. For on-shell partons it has been defined in the plasma rest frame by Svetitsky Svetitsky (1988); Golam Mustafa et al. (1998). It is obtained by multiplying (I.1) by , using for the longitudinal component of the momentum transfer, .

In the absence of diffusion, (), the Fokker Planck equation (IV.1) allows to write


In particular, assuming , we obtain


where is the time measured in the heavy-quark rest system. is the spatial part of the covariant




using . We evaluate in the c.m. frame as


where , . is the unit vector in the direction of which we choose to have only a component and is the unit vector in the direction of with a and a component; . Therefore we find


The integration over the transverse component of gives zero and we are left with a four-vector in which only the component differs from zero. Using (with denoting the angle between and in the c.m frame) we can replace the integration by an integration over and obtain


We have to calculate now in the heavy quark rest frame. For this we have to boost to this system. Using and , with being the energy (momentum) of the light quark in the heavy quark rest system we find


By construction