Toward a simultaneous description of $R_{AA}$ and $v_2$ for heavy quarks

The two key observables related to heavy quarks that have been measured in RHIC and LHC energies are the nuclear suppression factor $R_{AA}$ and the elliptic flow $v_2$. Simultaneous description of these two observables is a top challenge for all the existing models. We have highlighted how a consistent combination of four ingredients i.e the temperature dependence of the energy loss, full solution of the Boltzmann collision integral for the momentum evolution of heavy quark, hadronization by coalescence, then the hadronic rescattering, responsible to address a large part of such a puzzle. We have considered four different models to evaluate the temperature dependence of drag coefficients of the heavy quark in QGP. All these four different models are set to reproduce the same $R_{AA}$ as of the experiments. We have shown that for the same $R_{AA}$, the $v_2$ could be quite different depending on the interaction dynamics as well as other ingredients.


Introduction
Theoretical calculations predict that hadronic matter at high temperatures and densities dissolves into a deconfined state of quarks and gluons -called Quark Gluon Plasma (QGP).The experimental efforts at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies is aimed at to create and characterized the properties of QGP. The heavy hadrons (hadrons which contain at least one heavy quark, mainly c and b ) constitute a unique probe of the QGP properties, because they are produced in the early stage of the collisions and they are the witness to the entire space-time evolution of the system.
The two key observables related to heavy quarks that have been measured in RHIC and LHC are the nuclear suppression factor R AA [1,2,3], which is the ratio between the the p T spectra of heavy flavored hadrons (D and B) produced in Au+Au collisions with respect to those produced in p+p collisions, and the elliptic flow v 2 [2,4], which is a measure of the anisotropy in the angular distribution. Several theoretical efforts have been made to study the R AA and the v 2 measured in experiments within the Fokker Planck (FP) approach [5,6,7,8,9,10] and the relativistic Boltzmann approach [11,12,13,14,17,16]. However all the approaches shown some difficulties to describe both the R AA and v 2 simultaneously.

Results
To address the simultaneous description of R AA and v 2 , we start with the time evolution of R AA and v 2 to know how they develop during the expansion of the QGP. To study the time evolution of R AA and v 2 , we have solved the Fokker Planck (FP) equation stochastically in terms of the Langevin equation, for detail we refer to our earlier work [10]. The elastic collisions of heavy quarks with the bulk has been considered within the framework of pQCD. The divergence associated with the t-channel diagrams due to massless intermediate particle exchange has been regularized introducing the Debye screening mass m D = 4πα s T with the running coupling α s . For the bulk evolution we are using a transport bulk which can reproduce some of the gross features of the bulk [19,18]. At RHIC energy, Au + Au at √ s = 200, we simulate the fireball with initial temperature in the center is T i = 340 MeV and the initial time is τ i = 0.6 fm/c. for the detail of the initialization, we refer to our early work. Initially the charm quark are distributed according to the charm production in pp collisions [20]. As shown Fig 1, the R AA is such which is develop at the very early stage of the evolution and get saturated within 3-4 fm where as the v 2 (in 2) is develop at the latter stage of the evolution. Initially the bulk have zero v2. First the bulk will develop its own v 2 and then it will transfer it to the heave flavor. Hence the R AA is sensitive to the magnitude of the drag coefficient at the early stage of the evolution i.e at T i where as the v 2 is very sensitive to the magnitude of the drag coefficient at the latter stage of the evolution i.e at T c . This highlights the T-dependence of the drag coefficient may play a significance role for a simultaneous description of R AA and v 2 as they are sensitive to different range of T. To study the impact of T-dependence of drag coefficients on heavy quark observable, we have consider four different model to calculate the drag and diffusion coefficients of heavy quark in QGP.
Model-I (pQCD): The elastic collisions of heavy quarks with the bulk has been considered within the framework of pQCD. Model-II (AdS/CFT): In this case we have considered the drag force from the gauge/string duality i.e. AdS/CFT [21] which reads Γ conf = C T 2 QCD M HQ , where C = 2.1 ± 0.5.(also see [22]) Model-III: In the third case, we have employed a quasi particle model (QPM) model [23,24,25] with T-dependent quasi-particle masses, m q = 1/3g 2 T 2 , m g = 3/4g 2 T 2 , plus a T-dependence background field known as bag constant, tuned to the thermodynamics of the lattice QCD. Such a fit lead to the coupling, g 2 (T ) = This fourth case has been considered to have a drag which decreasing with T as obtained in the T-matrix approach [6,26].The T-dependence of the drag coefficients have been shown in    Figure 5. v 2 as a function of p T .
In Fig 4 we have shown the R AA as a function of p T for the four different models obtained within the Langevin dynamics at RHIC energy. The v 2 for the same R AA has been plotted in Fig 5 for all models as a function of p T . We found that for the same R AA , the v 2 build up can be quite different depending on the T-dependence of the interaction. Larger the interaction at T c larger is the v 2 [10]. Similar effect has also been found in light quark sector [27,28]. This suggests, the correct temperature dependence of drag coefficient has a significance role for a simultaneous reproduction of R AA and v 2 .  Recently it has been shown [13] that in case of the full solution of the Boltzmann (BM) integral i.e. without the assumption of small collisional exchanged momenta, leads in general to a large v 2 than that of the Fokker Plank case which is a approximation of the Boltzmann equation under the assumption of small exchanged momenta. Also heavy quark hadronization by coalescence further enhancement the v 2 [6,29].
To study the role of the hadronic phase (HP) on the heavy quark observables, we study the propagation of D mesons in the hadronic medium consist of pions, kaons and eta. The elastic interaction between the D meson with the hadronic matter has been treated within the framework of chiral perturbation theory [30]. In Fig 6 we have shown the impact of the hadronic medium on R AA which is almost unnoticeable. This is because the R AA develop at the early stage of the evolution and get saturated due to radial flow within 3-4 fm. The impact of the hadronic medium on v 2 has been shown in Fig 7. The hadronic medium further enhance the v 2 around 20%. In Fig 8 we have shown how the v 2 build up depending on the T-dependence of the drag coefficients for the same R AA . Then the v 2 get a boost from the Boltzmann equation (BM) in terms of evolution over the Fokker Planck equation. The v 2 further enhanced due to hadronization by coalescence. Finally the v 2 get the hadronic phase boost where the R AA remain the same.

Summary and outlook
In summary, we have shown how the v 2 build up for the same R AA depending on the Tdependence of the drag coefficients which is key for a simultaneous description of R AA and v 2 . We have also highlighted how the v 2 get a boost from the Boltzmann dynamics to study the momentum evolution of heavy quark and from hadronization by coalescence. Then we have shown the effect of the hadronic medium on R AA and v 2 . The impact of radiative processes on heavy quark observables will also be investigated in an upcoming study along with all these four ingredients within a single framework . It will also be interesting to study the role of pre-equilibrium phase [31] on R AA and v 2 relation.