1 Introduction

In high-energy nuclear collisions, heavy-flavor quarks are produced predominantly in the initial phase because of their large mass (mc/b >> Λ_QCD). The modification of their production in transverse momentum (p_T) due to energy loss and radial flow and in azimuth due to anisotropic flows is sensitive to the heavy quark dynamics in the hot and dense strongly interacting quark-gluon plasma (QGP) [1-3]. Recent experimental measurements of high-p_T D-meson production at RHIC and LHC energies show strong suppression of central heavy-ion collisions [4-6], suggesting significant energy loss by charm quarks inside the QGP medium.

There is a great deal of interest in the use of heavy-flavor quarks to constrain the transport properties of the QGP [7, 8]. During their propagation through the QGP, heavy-flavor quarks interact with the constituents of the medium and lose some of their momentum, and thus can reveal some of the QGP properties. However, the mechanism of the in-medium modification of heavy quarks is not fully understood [9]. A theoretical study of charm production has found that the cross section of gluons makes a large contribution, where heavy flavor appears from gluon splitting only in the late stages of the parton shower evolution [10]. We have studied the collisional energy loss term by adding an additional $c\overline{c}$ pair production in a multiphase transport model (AMPT), which significantly improves the description of the charm hadron p_T spectrum and elliptic flow in Au+Au collisions at $\sqrt{s_{\text{NN}}}=200\text{GeV}$ [11]. Here, we extend the previous study and focus on the collision centrality dependence of D-meson production, the nuclear modification factor, and the azimuthal angular correlation.

Analysis of multiparticle angular correlations is a powerful tool for exploring the properties of the QGP and the underlying mechanism of particle production in hot quantum chromodynamic (QCD) matter [12-20]. Unlike light quarks and gluons, heavy quarks suppress small-angle gluon radiation, which results in a small radiated energy loss [21, 22]. In hadron-hadron collisions, which are dominated by the initial production effect, a distinct feature appears in the azimuthal angular correlation of $\text{D}\overline{D}$, with clear suppression on the near side and enhancement on the away side [23]. In nuclear-nuclear collisions, the angular correlation may be affected by the interactions of charm quarks in the QGP, which is argued to be a signature of a strongly coupled QGP owing to the collective partonic wind effect [23]. The evolution of the azimuthal angular correlation of charm hadron pairs in the QGP has not been carefully studied in a realistic transport calculation, and that lack motivates the current work.

2 Charm quark production in AMPT model

The AMPT model is a hybrid model [24]. It has been extensively used for studying the bulk medium using the microscopic dynamical processes of evolving systems, as reported in recent papers [25-27]. In this model, the initial conditions are taken from the HIJING event generator [28, 29]. In the default AMPT model, the partonic matter consists only of minijet partons from HIJING. It is different in the string melting scenario, in which the hadrons generated by HIJING are dissociated according to their valence quark structures, and the resulting partonic matter is thus much denser. The evolution of the partonic matter is simulated using the parton cascade model ZPC [30], and the partons are converted to hadrons after they stop scattering. In the default AMPT model, the partons are first combined with their parent strings, which then fragment using the Lund string fragmentation model [31]. In the string melting AMPT model, the nearest partons are converted into hadrons via the coalescence model [24]. In both versions of the AMPT model, the scattering of hadrons is described by a relativistic transport model [32].

The initial production of charm quarks in the AMPT model is handled by the HIJING model. It includes pair production ($q+\overline{q}\to Q+\overline{Q}$, $g+g\to Q+\overline{Q}$) and gluon splitting ($g\to Q+\overline{Q}$). The gluon splitting process is implemented using a parton shower method similar to that in a general Monte Carlo event generator [31]. The pair production cross section in perturbative QCD at the leading order can be expressed as

Here, y₁ and y₂ are the rapidities of the produced charm quarks, respectively. Further, x₁ and x₂ are the fractions of momentum carried by the initial partons; the relationship between them is x₁=x_T(e^y1+e^y2)/2, x₂=x_T(e^-y1+e^-y2), where $x_{\text{T}}=2p_{\text{T}}/\sqrt{s}$. K is a factor that accounts for higher-order corrections of the charm quark production and typically has a value of K 2 [29]. In addition, $f(x\mathrm{,}{p}_{\text{T}}^2)$ are the default parton distribution functions in HIJING, which are taken to be the Duke–Owens structure function set 1 [33].

From Eq. (1), one can obtain the total inclusive cross section ${\sigma }^{c\overline{c}}$ by integration at a low p_T cutoff p₀:

For a nucleon-nucleon collision, the average number of semihard parton collisions at impact parameter b is ${\sigma }^{c\overline{c}}$ T_N(b), where T_N(b) is the partonic overlap function between the two nucleons at impact parameter b [28]. The probability of multiple charm pair production is given by

with j≥1. The HIJING model has two components with two key parameters, the minijet transverse momentum cutoff p₀ and the soft interaction cross section _soft. _soft controls the elastic, inelastic, and total cross sections. The production probability for soft interactions without hard processes is

The elastic, inelastic, and total cross sections of binary collisions are thus calculated in the model [28, 29].

Because the differential cross section of charm quark production is smaller than that of light quarks, we trigger charm production using a specific p_T threshold in the normal AMPT event to increase the simulation efficiency. This is done by adding the trigger parameters to the AMPT code to call the related subroutine, where IHPR2(3) = 3 is used to trigger heavy quark hard scattering, IHPR2(18) = 0 is used for charm production, and HIPR1(10) is used for the trigger p_T threshold [29]. The trigger will then change the probability of charm quark production, and thus the entire event structure [28]. In particular, if we select a large $p_{\text{T}}^{\text{t}rig}$ value (i.e., 20 GeV/c), such rare processes with large p_T scattering occur most frequently when the impact parameter of the collisions is small, and thus the partonic overlap is large. At a small impact parameter, the production of the charm quark pair is enhanced. The probability of charm quark pair production in the triggered event is

From the above equation, when $p_{\text{T}}^{\text{t}rig}=p_0$, $g_j^{\text{t}rig}\left(b\right)$ is consistent with gj(b) in Eq. (3). Summing over j≥1 leads to the expected total probability of having at least one charm quark pair with $p_{\text{T}}\ge p_{\text{T}}^{\text{t}rig}$:

Because $g_j^{\text{t}rig}(b)$ differs from gj(b), the triggering of charm quark pair production changes the production rate of other particles in the same event [28]. This trigger algorithm was initially implemented in HIJING to study rare processes such as high-momentum jets [28, 29], and it is extended to study charm quark production and transport dynamics in the hot QCD medium [11].

In this study, the default and string melting AMPT models (version v2.26t5) are used, where additional high-p_T (20 GeV/c) charm quark pair triggers are implemented to improve the simulation efficiency. In both AMPT models, the event statistics are 2 × 10⁵, 2 × 10⁶, and 1 × 10⁶ for inclusive Au+Au and pp collisions at $\sqrt{s_{\text{NN}}}=200\text{GeV}$ and p-Pb collisions at $\sqrt{s_{\text{NN}}}=5.02\text{TeV}$, respectively. A few key input parameters of the program, including the parton cascade cross section, p=3 mb, and the Lund string fragmentation function, a= 0.55, b= 0.15 GeV^-2, are applied [25]. For details on the other parameters, readers are referred to the program manual [24].

3 Results and Discussion

3.1 D⁰ meson p_T distribution and R_AA in AMPT model

Figure 1 presents the D⁰ meson p_T spectrum in 0–10% central Au+Au collisions at $\sqrt{s_{\text{NN}}}=200\text{GeV}$ and compares them with the AMPT model calculations. One sees that the AMPT model predicts only ∼1/4 of the D⁰ yield for the centrality analyzed. Ref. [34] reports that when an extended version of the AMPT model is obtained by removing the p₀ cutoff for the heavy quark production cross section, their inclusion in the total minijet cross section will improve the description of the charm production significantly. We learned that the procedure implemented in Ref. [34] is similar to the approach applied in our earlier study [11]. Because the extended version developed in Ref. [34] is not publicly accessible yet, we further explore the study using our approach. The green open triangles in Fig. 1 show the D⁰ p_T spectrum in the AMPT model with an additional charm-anticharm quark trigger. It overpredicts the data. Because the slopes of the two AMPT calculations in Fig. 1 are the same, one can build a new event sample to describe the data with a fraction determined by the data. The red crosses in Fig. 1 represent the results in such a new event sample, with 80% of the normal AMPT distribution (pink open stars) plus 20% of the AMPT with an additional trigger on charm quark pair production (green triangles). We refer to the new sample as the AMPT mixed sample and use it to study the charm production and evolution in the hot and dense QCD matter in the following.

Fig. 1.Full-screen View

(Color online) D⁰ meson p_T spectra in central Au+Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV. Open symbols represent results of AMPT model calculation for different configurations; solid symbols are experimental data [4].

Using the above p_T distributions, we investigate the nuclear modification factor R_AA of the D⁰ meson in central Au+Au collisions at $\sqrt{s_{\text{NN}}}=200\text{GeV}$, because R_AA is well established as a sensitive observable for the study of the interaction of hard partons with the medium. It is usually defined as

$R_{\text{AA}}\left(p_{\text{T}}\right)=\frac{1}{N_{\text{coll}}}\frac{d^2{N}_{\text{AA}}/\left(\text{d}{p}_{\text{T}}\text{d}y\right)}{{\text{d}}^2{N}_{\text{pp}}/\left(\text{d}{p}_{\text{T}}\text{d}y\right)}\mathrm{,}$ (7)

where N_coll represents the number of binary collisions in the reaction.

Figure 2 compares the results of our calculations with experimental data. The typical calculations for pp collisions using the default or string melting AMPT model both exhibit enhancement at a p_T value of approximately 1.5 GeV/c and strong suppression at intermediate p_T, although their magnitudes are much larger than those of the experimental data. This result, in combination with the normal AMPT distribution presented in Fig. 1, suggests that charm production in pp collisions at $\sqrt{s}=200$ GeV is greatly underpredicted by the AMPT model compared to that in Au+Au collisions. This behavior is also observed by studying the collision energy dependence of the total cross sections of charm-anticharm quark pairs [34]. We thus build the new event sample of pp collisions following the procedure used for Au+Au collisions with the same percentages of normal and triggered AMPT events. The new R_AA describes the experimental data well.

Fig. 2.Full-screen View

(Color online) D⁰ meson R_AA in central Au+Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV. Dashed lines represent results of AMPT model calculation for different configurations. Data points are experimental measurements [4].

In the AMPT model, the interaction between charm quarks and the medium is simulated by parton elastic scattering. It has been noted that the elastic collisional energy loss is dominant for heavy flavors below a moderately high p_T, for example, for the charm hadron at pT≤5-6 GeV/c in Au+Au collisions at $\sqrt{s_{\text{NN}}}$=200 GeV/c [35]. Therefore, it is understood that the AMPT model with only collisional energy loss and no radiative energy loss can describe the R_AA data at low and intermediate p_T. In the parton cascade of the AMPT model, the parton cross section and its angular distribution determine the strength of the interaction between heavy quarks and the expanding medium [11, 34]. The follow-up hadronic interactions are mainly for light-flavor hadrons. For heavy-flavor hadrons, we consider only the decays with large mass resonances [24].

3.2 D⁰ meson azimuthal angular correlation

We use the two-particle azimuthal angular correlation function to study the charm quark dynamics evolution in the hot and dense medium using the AMPT model, in which multi-parton scattering may have a significant effect on the correlation function C(Δϕ). In our study, C(Δϕ) is built as follows:

$C\left(\Delta \eta \mathrm{,}\Delta \varphi \right)=\frac{S\left(\Delta \eta \mathrm{,}\Delta \varphi \right)/N_{\text{pairs}}^{\text{signal}}}{B\left(\Delta \eta \mathrm{,}\Delta \varphi \right)/N_{\text{pairs}}^{\text{mixed}}}\mathrm{,}$ (8)

where Δ≤ta is the relative pseudorapidity, and Δϕ is the relative azimuthal angle between charm hadron pairs. The signal pair distribution, S(Δη,Δϕ), represents the yield of charm hadron pairs that come from the same event:

$S\left(\Delta \eta \mathrm{,}\Delta \varphi \right)=\frac{{\text{d}}^2{N}_{\text{pairs}}^{\text{signal}}}{\text{d}\Delta \eta \text{d}\Delta \varphi }\mathrm{.}$ (9)

In addition, B(Δη,Δϕ) is built from the mixed event charm hadron pairs distribution:

$B\left(\Delta \eta \mathrm{,}\Delta \varphi \right)=\frac{{\text{d}}^2{N}_{\text{pairs}}^{\text{mixed}}}{\text{d}\Delta \eta \text{d}\Delta \varphi }\mathrm{,}$ (10)

which is flat in our calculation.

The one-dimensional correlation function along Δϕ can be constructed from the C(Δη,Δϕ) distribution by integrating over Δη:

$C\left(\Delta \varphi \right)=A\times \frac{{\displaystyle \int }S\left(\Delta \eta \mathrm{,}\Delta \varphi \right)\text{d}\Delta \eta }{{\displaystyle \int }B\left(\Delta \eta \mathrm{,}\Delta \varphi \right)\text{d}\Delta \eta }\mathrm{,}$ (11)

where the normalization constant A is estimated from $N_{\text{pairs}}^{\text{mixed}}/N_{\text{pairs}}^{\text{signal}}$, which depends on the details of the mixed event process. For example, the current event is mixed with eight other events in the same collision centrality interval to improve the statistical power of the background estimation. The two-dimensional (2D) charm hadron correlation function as a function of Δ≤ta and Δϕ is analyzed in different centrality classes of Au+Au collisions at $\sqrt{{\text{s}}_{\text{NN}}}=200$ GeV and high-multiplicity p-Pb collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV with 0.3 < p_T < 5 GeV/c for trigger particles and associated particles of interest.

Figure 3 presents the centrality dependence of the D⁰ meson azimuthal angular correlations in the default and string melting AMPT models, where the former represents a combination of initial production plus the multi-parton scattering effect, and the latter is dominated by the initial production effect. The open charm hadron azimuthal angular correlations in the default AMPT model show clear suppression at |Δϕ| 0 and enhancement at |Δϕ| π. This is expected from the initial back-to-back charm-anticharm quark pair production due to their large mass and momentum. However, the distribution in the string melting AMPT model differs from that in the default AMPT model. In peripheral Au+Au collisions, finite parton scatterings smear the angular correlation with a similar concave distribution versus |Δϕ|. In central Au+Au collisions, strong parton scatterings affect the correlation with a flat or near convex distribution. If the dynamics of parton cascade is traced by counting the number of collisions of charm quark with other partons, the value in central collisions could be 5–6 times higher than that in peripheral collisions [11]. The multi-parton scatterings reshape the charm hadron angular correlations. The evolution of the shape of the correlation function associated with the number of parton collisions suggests that the charm hadron azimuthal angular correlations may be an experimental observable that can be used to quantitatively measure the charm hadron interaction with the medium. Comparisons of future measurements using upgrades of existing experiments or new experiments with current calculations will elucidate the charm quark dynamics in hot and dense media.

Fig. 3.Full-screen View

D⁰ meson azimuthal angular correlations vs. centrality in Au+Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV for default or string melting AMPT models and various ratios of the two.

To eliminate the trigger effect in the initial stage and focus on the multi-parton scattering effect, we define the ratio of the correlation functions of the default and string melting AMPT models:

$\text{ratio}=\frac{C{\left(\Delta \varphi \right)}_{\text{melting}}}{C{\left(\Delta \varphi \right)}_{\text{default}}}\mathrm{.}$ (12)

The results are shown as open triangles in Fig. 3. They show a convex distribution versus Δϕ owing to the multi-parton scattering interaction in the string melting AMPT model. The result for peripheral collisions is far from the flat distribution among the three centralities, probably because charm quarks usually lose the most energy in the first collision [11].

As noted in [23], the strong collective flow of pp or Pb–Pb collisions at LHC energies will significantly modify the charm hadron azimuthal correlation. We performed the transport calculation in this study. Figure 4 shows the open charm hadron azimuthal angular correlation function C(Δϕ) in high-multiplicity p-Pb collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV, where the multiplicity is calculated using the charged particle pseudorapidity distributions within |η| < 0.5 [25]. The result in the default AMPT model is similar to the distribution of peripheral Au+Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV, whereas the string melting AMPT model shows a distribution closer to that of central Au+Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV. The ratio of the string melting and default AMPT models shows a clear convex distribution. These results, together with the elliptic flow study of p-Pb collisions from the AMPT model in comparison with data [17], confirm that strong partonic collectivity affects the charm hadron pair azimuthal angular correlation.

Fig. 4.Full-screen View

D⁰ azimuthal angular correlation function C(Δϕ) in p-Pb collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV within the multiplicity range N_track>110 of AMPT model calculations.

4 Summary and outlook

In summary, we studied the dynamic evolution of the charm hadron in the AMPT model with additional charm quark pair production. The model describes the D⁰ meson nuclear modification factor in central Au+Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV up to p_T=6 GeV/c. The results suggest that collisional energy loss is dominant for the D⁰ meson in the low and intermediate pT regions. The D⁰ meson pair azimuthal angular correlations are studied for different centralities and from Au+Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV to high-multiplicity p-Pb collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV. The AMPT model shows clear smearing of the away-side charm hadron pair azimuthal angular correlation induced by parton scattering. Our study provides a method of quantitative study of the heavy-flavor dynamics in the hot and dense QGP created in high-energy nuclear collisions. It is also useful for future experiments with upgraded detectors at LHC-ALICE or for new experiments such as the RHIC-sPHENIX.