Charm quarks are more hydrodynamic than light quarks in final-state elliptic flow

We study the charm quark elliptic flow ($v_{2}$) in heavy ion as well as small system collisions by tracking the evolution history of quarks of different flavors within a multi-phase transport model. The charm quark $v_{2}$ is studied as a function of the number of collisions the charm quark suffers with other quarks and then compared to the $v_{2}$ of lighter quarks. We find that the common escape mechanism is at work for both the charm and light quark $v_{2}$. However, contrary to the naive expectation, the hydrodynamics-type flow is found to contribute more to the final state charm $v_{2}$ than light quark $v_{2}$. This could be explained by the smaller average deflection angle the heavier charm quark undergoes in each collision, so that heavy quarks need more scatterings to accumulate a significant $v_{2}$, while lighter quarks can more easily change directions with scatterings with their $v_{2}$ coming more from the escape mechanism. Our finding thus suggests that the charm $v_{2}$ is a better probe for studying the hydrodynamic properties of the quark-gluon plasma.


INTRODUCTION
One of the primary goals of relativistic heavy ion collisions is to create the quark-gluon plasma (QGP) where quantum chromodynamics (QCD) can be studied under the extreme conditions of high temperature and energy density [1][2][3][4][5]. In the hydrodynamics picture, the high pressure buildup in the center of the collision zone drives the system to expand at relativistic speed, generating a large collective flow. The collectivity is sensitive to the pressure gradient and therefore regarded as a valuable probe of the pressure and energy density of the transient QGP. Of particular interest are non-central collisions where the overlap collision zone is anisotropic in the transverse plane perpendicular to beam. The pressure gradient would generate an anisotropic expansion and final-state anisotropic flow in momentum space, whose leading term is elliptical [6]. This large elliptic anisotropy (v 2 ) has been observed in experiments and is regarded as a strong evidence for the formation of the QGP [1][2][3][4][5].
Both hydrodynamics and transport models can describe the measured large v 2 [7]. Hydrodynamics is a macroscopic approach to describe the dynamical evolution of a heavy-ion collision [7]. The fact that hydrodynamics with a small viscosity to entropy density ratio (η/s) can well describe the experimental data suggests that the collision system is strongly interacting and the created QGP is a nearly perfect fluid (sQGP) [8]. On the other hand, A Multi-Phase Transport (AMPT) [9,10] description of data using perturbative QCD motivated σ of 3 mb requires the string melting mechanism. Recently observed large v 2 in small systems of high multiplicity p + p and p+Pb collisions at the Large Hadron Collider (LHC) and d + Au collisions at the Relativistic Heavy Ion Collider (RHIC) can also be described well by AMPT with string melting [11].
A recent study within the AMPT framework indicates that at RHIC energies the large v 2 in the model comes mainly from the escape mechanism [12,13], where the partons have a larger probability to escape along the shorter axis of the overlap volume and the hydrodynamic flow plays only a minor role in AMPT. The escape mechanism would naturally explain the similar azimuthal anisotropies in heavy ion and small systems [12][13][14][15]. It is further shown by AMPT that the main origin of the mass splitting of identified hadron elliptic anisotropies is hadronic scatterings [16][17][18], thus the mass ordering is not a unique signature of hydrodynamics as naively perceived [7]. In this paper, we extend those previous studies to charm quarks at both RHIC and LHC energies. We address the particular question of whether the escape mechanism is also responsible for the development of charm v 2 .

MODEL AND ANALYSIS METHOD
The string melting version of the AMPT model [9,10] consists of four components: fluctuating initial conditions, parton elastic scatterings, quark coalescence for hadronization, and hadronic interactions. We employ the same string melting version of AMPT, v2.26t5, as in our earlier studies [12,13,[16][17][18]. The initial partons information is provided by HIJING, and then string melting liberates strings into a larger number of quarks and antiquarks.
The two-body parton-parton elastic scatterings are modeled by Zhang's Parton Cascade (ZPC) [19]. We use a Debye-screened differential cross-section dσ/dt ∝ α 2 s /(t − µ 2 D ) 2 [10], with strong coupling constant α s = 0.33 and Debye screening mass µ D = 2.265/fm (corresponding to σ = 3 mb) for all AMPT simulations in this work. Once partons stop interacting, a simple quark coalescence model is applied to combine two (three) nearest partons into a meson (baryon or antibaryon). The subsequent hadronic interactions are described by an extended ART model [10]. In this study, however, we only study parton-level observables.
We simulate three collision systems: Au+Au collisions with b = 6.6-8.1 fm at the nucleon- 2 ) , where φ is the azimuthal angle of a parton in momentum space [21] and ψ (r) 2 is the harmonic plane of each event from its initial spatial configuration of all partons [22]. All results shown in this study are for partons within the pseudo-rapidity window |η| < 1. with the expectation that the heavier charm quarks are produced by hard scatterings at earlier times from perturbative QCD processes. The flavor dependence is also related to the initial production; for example, Fig. 1(b) shows that charm quarks are produced in the more inner region of the overlap volume than light quarks. This is partly responsible for the larger charm quark N coll than light quark's as shown in Fig. 1(a). We also find that the features shown in Fig. 1 are qualitatively the same for the p+Pb and Pb+Pb collision systems [23].
Our previous studies [12,13] have shown that the overall quark v 2 at RHIC energies comes mainly from the anisotropic escape mechanism. The question is whether or not this is also true for charm v 2 . To address this question, we compare in Fig. 2 the v 2 of freezeout partons of different flavors in both the normal AMPT calculation and the azimuthrandomized AMPT calculation. Since v 2 values for quarks and antiquarks of the same flavor are practically identical at these high energies, we combine the quarks and antiquarks of the same flavor in the v 2 plots. In the normal calculations, there is a mass ordering in v 2 at the same N coll : the light quark v 2 is larger than the charm quark v 2 at small N coll , while the ordering is reversed at large N coll in collisions of large systems. It suggests that charm quarks need more scatterings to accumulate large v 2 while the light quarks need only a few collisions. In the randomized case, the parton azimuthal angles are randomized after each collision and hence their v 2 only comes from the anisotropic escape mechanism [12,13]. The freezeout partons still have positive v 2 but the values are reduced from those in the normal case; this is because of no contribution from the anisotropic collective flow. The results shown in Fig. 2 indicate that the escape mechanism is qualitatively at work for both charm v 2 and light quark (u,d,s) v 2 . Quantitatively, however, we see that the hydrodynamics-type flow has a larger contribution to charm v 2 than the light quark v 2 at large N coll . Note that the significant contribution of hydrodynamic flow to the charm v 2 has been observed in a linearized Boltzmann transport model of heavy quarks coupled with hydrodynamical background [24].   Fig. 2, the escape mechanism contribution as shown by the azimuth-randomized AMPT results is finite [12] for each quark flavor. The mass splitting of quark v 2 is present at low p ⊥ for both φ-randomized and normal AMPT. At high p ⊥ , however, their v 2 's from normal AMPT simulations approach each other as expected because the mass difference becomes less important. We also see that the escape mechanism contributes less to the charm v 2 than to the light quark v 2 , therefore the hydrodynamic flow contributes more to the charm v 2 than to the light quark v 2 ; this is especially clear for Au+Au and Pb+Pb collisions. Since the escape mechanism dominates more the v 2 of small systems [12,13], the percentage contribution from the hydrodynamic flow to the charm v 2 is larger in Pb+Pb collisions than p+Pb collisions.  Table I lists the values of N coll , v 2 , and the ratios of v 2 from azimuth-randomized AMPT to that from normal AMPT in Au+Au, p+Pb and Pb+Pb collisions. The N coll value of freezeout partons of a given flavor increases with the collision system size and beam energy as expected. Note that we also find that the N coll values are larger in the randomized case compared to normal AMPT; this is because the randomization tends to destroy the preferred outward direction of partons. We see in Table I  However, the v 2 values of freezeout partons in the azimuth-randomized AMPT strongly depend on the quark flavor: the light quark v 2 is higher than charm quark v 2 for the same collision system.
The ratio of v 2 from φ-randomized AMPT to that from normal AMPT represents the fraction of v 2 that comes from the escape mechanism.  hydrodynamic contribution to the final quark v 2 is more important for charm quarks. This result suggests that the charm v 2 better reflects the hydrodynamic properties of the quarkgluon plasma, especially for large systems at high energies. This is consistent with several recent findings [25,26].

UNDERSTANDING WITH A TOY MODEL
For parton scatterings, a main difference between light and heavy quarks is their masses.
In the AMPT model used for the current study, the mass difference leads to a difference in the average deflection angle in a 2→2 scattering. Figure 4(a) shows the root-mean-square (rms) change of the azimuth angle (σ ∆φ ) as a function of N coll for different quark flavors in Au+Au collisions. The results are similar for p+Pb and Pb+Pb collisions [23]. The rms change of the azimuthal angle is approximately σ ∆φ =1.0, 0.65, 0.25 for light, strange, and charm quarks, respectively; so it is much smaller for heavier quarks [27].
For better understanding, we now consider a toy model where partons start from the center (x, y) = (0, 0) and propagate out to the boundary of an ellipse. We take the eccentricity of the elliptical area to be 2 = 0.17, corresponding to semi-central Au+Au collisions. The number of collisions a parton suffers, assuming straightline propagation, can be written as where φ i is the initial azimuthal angle. Note that N coll here is a measure of the system size. would correspond to a mid-central collision between two hypothetical large nuclei. The cumulative deflection in the azimuth angle after the parton leaves the elliptical area, assuming straightline propagation, would be Gaussian distributed with the width of σ ∆φ N coll (φ i ).
Then the parton average elliptic flow v 2 can be calculated as The results are shown in Fig. 4(b). The v 2 of light quarks is larger than charm quarks at small N coll but is the opposite at large N coll . Partons along the longer y-axis suffer more collisions than those along the x-axis, and each collision deflects the parton from its original direction to a range of directions. As a result, more y-going partons will be deflected towards an isotropic distribution. This results in a positive v 2 , and this is a generic mechanism for the v 2 generation in any cascade model. Since σ ∆φ is large for light quarks, a small number of collisions is already strong enough to reshuffle the φ directions to produce a large v 2 .
However, also because σ ∆φ is large, light quarks easily forget their original direction, so the light quark v 2 quickly drops to zero at modest N coll in this toy model. For charm quarks, on the other hand, it takes many collisions to build up a sizable v 2 because each collision only deflects the charm quark direction a little. In addition, since v 2 only depends on the variable σ ∆φ N coll , the average elliptic flow v 2 has the same peak value for all flavors, while the peak occurs at a larger N coll value for heavier quarks. We note that this toy model does not have much dynamics and thus does not capture certain important features of dynamical models, such as the finite asymptotic v 2 after infinite number of collisions. Instead, the v 2 from the toy model basically represents the v 2 generated by the escape mechanism. In the toy model partons of all flavors will be randomized and reach zero v 2 after a large number of collisions, where the hydrodynamic collective flow would be the dominant source of anisotropic flows [12,13]. Because our toy model assumes straightline propagation, it cannot be trusted at large N coll when σ ∆φ N coll is of order 1. However, our toy model calculation helps to illustrate the importance of the average scattering deflection angle, thus also the importance of quark mass, for the generation of v 2 by parton scatterings.
It is commonly expected that transport with large number of scatterings would approach hydrodynamic limit. If hydrodynamic evolution proceeds with a quasiparticle picture, then it is conceivable that light quarks do not accumulate v 2 efficiently because they can easily change directions, and heavy flavors attain better the accumulated v 2 to the final state.
In other words, heavy flavor final-state flow would appear more hydrodynamic than the light flavor flow, even though light flavor approaches hydrodynamic equilibrium earlier in the collision. This would imply that light quarks can partially lose their evolution memory because of thermalization, and this includes also their v 2 which has been generally perceived to accumulate to the final state.

SUMMARY
Recent transport model studies have shown that the overall azimuthal anisotropies v n in heavy ion collisions come mainly from the anisotropic escape probability in heavy ion collisions at RHIC energies and in small collision systems. Here we have extended those previous studies to quarks of different flavors, particularly the charm quarks. We find that the common escape mechanism is at work not only for light quark but also for strange and charm quark v 2 , where it dominates the generation of v 2 for small-enough systems but would become insignificant for large-enough systems. However, we find that charm v 2 has a larger fraction coming from the hydrodynamic collective flow (and thus less coming from the escape mechanism) than the light quark v 2 , in contrast to naive expectations. We further find that this is closely related to the mass dependence of the average scattering angles.
Light quarks have large deflection angles and thus cannot efficiently accumulate their v 2 (their final state v 2 comes largely from the escape mechanism), while heavy quarks deflect at smaller angles and thus need more scatterings with the light-quark-dominated medium in order to accumulate a significant v 2 . We conclude that the charm v 2 is more hydrodynamic than the light quark v 2 , therefore it is a better probe of the hydrodynamic properties of QGP.