Beam Energy Dependence of the Linear and Mode-Coupled Flow Harmonics Using the a Multi-Phase Transport Model

Abstract

In the framework of the A Multi-Phase Transport (AMPT) model, the multi-particle azimuthal cumulant method is used to calculate the linear and mode-coupled contributions to the quadrangular flow harmonic ($v_4$) and the mode-coupled response coefficient as functions of centrality in Au+Au collisions at $\sqrt{s_{NN}}$ = 200, 39, 27 and 19.6 GeV. This study indicates that the linear and mode-coupled contributions to $v_4$ are sensitive to beam energy change. Nevertheless, the correlations between different-order flow symmetry planes and the mode-coupled response coefficients show weak beam energy dependence. In addition, the presented results suggest that the experimental measurements that span a broad range of beam energies can be an additional constraint for the theoretical model calculations.

1. Introduction

Experimental investigations of heavy-ion collisions demonstrate the formation of Quantum Chromodynamic (QCD) matter called Quark–Gluon Plasma (QGP) at the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) [1,2,3]. One of the primary purposes of previous and present experimental studies of heavy-ion collisions is to understand the QGP transport properties, such as shear viscosity divided by entropy density, $\eta /s$ [4,5,6,7,8,9,10].

In recent years, experimental measurements of anisotropic flow resumed being a beneficial route to the extraction of $\eta /s$ [5,11,12,13,14,15,16,17,18,19,20,21,22,23]. In addition, anisotropic flow gives the viscous hydrodynamic response to the initial-state energy density anisotropy described by complex eccentricity vectors ${\mathcal{E}}_n$ [24,25,26,27,28].

$$\begin{array}{ccc}\hfill {\mathcal{E}}_n& \equiv & {\epsilon }_n{e}^{in{\Phi }_n}\hfill \\ \hfill & \equiv & -\frac{\int dxdy{r}^n{e}^{in\phi }E(r,\phi )}{\int dxdy{r}^{n}E(r,\phi )},(n>1),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill x& =& rcos\phi ,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill y& =& rsin\phi ,\hfill \end{array}$$

(3)

where ${\epsilon }_n$ and ${\Phi }_n$ are the magnitude and the angle of the n^th-order eccentricity vector, $\phi$ is the spatial azimuthal angle and $E(r,\phi )$ is the initial anisotropic energy density profile [27,29,30].

The anisotropic flow can be given by the Fourier expansion of the azimuthal anisotropy of particles emitted relative to the collision symmetry planes [31].

$$E\frac{d^{3}N}{d^{3}p}={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{d^{2}N}{p_{T}d{p}_{T}dy}}\left(1+\sum _{i=1}^{\infty }2v_{n}cos\left(n\left(\varphi -{\psi }_m\right)\right)\right),$$

(4)

where $v_n$ stands for the $n^{th}$ flow coefficient, y is the rapidity, $\varphi$ represents the particle azimuthal angle, $p_T$ gives the transverse momentum and ${\psi }_m$ is the $m^{th}$-order symmetry plane. Flow harmonics $v_n$ (n = 1, 2 and 3) are called directed, elliptic and triangular flow, respectively.

Prior and current studies of $v_2$ and $v_3$ suggest that to a reasonable degree, they are linearly related to the medium response [28,32,33,34,35,36,37,38,39,40,41].

$$\begin{array}{c}\hfill v_n={\kappa }_n{\epsilon }_n,\end{array}$$

(5)

where ${\kappa }_n$ (for n = 2 and 3) encodes the importance of QGP $\eta /s$ [42,43]. Higher-order flow harmonic $v_4$ [19,34,38,42,44,45,46,47] exhibits a linear response to same-order eccentricity as well as a mode-coupled response to lower-order eccentricity ${\epsilon }_2$ [21,29,30,48].

$$\begin{array}{ccc}\hfill V_4& =& v_4{e}^{i4{\psi }_4}={\kappa }_4{\epsilon }_4{e}^{4i{\Phi }_4}+{\kappa }_4^{{}^{\prime }}{\epsilon }_2^2{e}^{4i{\Phi }_2}\hfill \\ & =& V_4^{\mathrm{Linear}}+{\chi }_{4,22}{V}_4^{\mathrm{MC}},\hfill \end{array}$$

(6)

where ${\kappa }_4^{{}^{\prime }}$ represents the mixed effect of the medium properties and the coupling between lower- and higher-order eccentricity harmonics. $V_4^{\mathrm{Linear}}$, $V_4^{\mathrm{MC}}$ and ${\chi }_{4,22}$ are the linear and the mode-coupled contributions to $V_4$ and the mode-coupled response coefficients, respectively.

The mode-coupled response to $V_4$ represents additional constraints for initial-stage dynamics and $\eta /s$ extraction [29,32,33,37,49,50,51,52,53,54]. Therefore, ongoing work suggests that leveraging comprehensive measurements of $v_4^{\mathrm{Linear}}$ and $v_4^{\mathrm{MC}}$ could provide additional constraints to differentiate between various initial-state models [9,10,32,55]. In addition, these measurements could pin down the $\eta /s$ dependence on temperature (T) and baryon chemical potential (${\mu }_B$).

The paper is organized as follows: Section 2 describes the AMPT model and the analysis technique employed in this work. Section 3 conveys the results of this work. The summary is presented in Section 4.

2. Method

AMPT [56] model (version ampt-v2.26t9b)-simulated events were used in the present investigation of Au+Au collisions at $\sqrt{s_{NN}}$ = 200, 39, 27 and 19.6 GeV. The used AMPT model has both string-melting mechanism and hadronic cascade turned on. The AMPT model has been widely employed to investigate relativistic heavy-ion collision physics [56,57,58,59,60,61,62,63,64,65]. In the AMPT model with string melting on, the HIJING model is used for hadron creation. These hadrons are then transformed into their valence quarks and anti-quarks. In addition, their time and space evolution is evaluated with the ZPC parton cascade model [66].

The AMPT has four essential components: (i) the HIJING model [67,68] in the initial parton production stage, (ii) the parton scattering stage and (iii) hadronization through coalescence, followed by (iv) a hadronic interaction stage [69]. In the stage of parton scattering, the parton scattering cross-section is given as

$$\begin{array}{ccc}\hfill {\sigma }_{pp}& =& {\displaystyle \frac{9\pi {\alpha }_s^2}{2{\mu }^2}},\hfill \end{array}$$

(7)

where $\mu$ = $4.6$ gives the partonic matter screening mass and ${\alpha }_s$ = $0.47$ represents the QCD coupling constant. Parameters $\mu$ and ${\alpha }_s$ generally give the expansion dynamics of A–A collision systems [66,70,71,72]. In the current work, ${\sigma }_{pp}$ was fixed to $1.5$ mb.

In the current work, the centrality intervals were obtained by cutting on the charged particle multiplicity in midrapidity. Then, the AMPT-simulated events were analyzed using the multi-particle cumulant method [49,73,74,75] using particles with pseudorapidities $\left|\eta \right|<1$ and with transverse momentum $0.2<p_T<2.0$ GeV/$c$.

The multi-particle cumulant technique was here used for correlation analysis. The framework of the multi-particle cumulant using one and many sub-events is described in Refs. [49,73,74,75]. Here, I used two-, three- and four-particle correlations in this work by applying the two-sub-event cumulant technique [74]. The two sub-events $A$ and $B$ with $\left|\Delta \eta \right|>0.7$ (i.e., ${\eta }_A>0.35$ and ${\eta }_B<-0.35$) were used. Using the two-sub-event method helps reduce non-flow correlations [76]. Two-, three- and four-particle correlations are given using the two-sub-event cumulant method [74] as

$$\begin{array}{ccc}\hfill v_n^{\mathrm{Inclusive}}& =& v_n={\langle \langle cos\left(n({\phi }_1^A-{\phi }_2^B)\right)\rangle \rangle }^{1/2},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill C_{n+m,nm}& =& \langle \langle cos((n+m){\phi }_1^A-n{\phi }_2^B-m{\phi }_3^B)\rangle \rangle ,\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \langle v_n^2{v}_m^2\rangle & =& \langle \langle cos(n{\phi }_1^A+m{\phi }_2^A-n{\phi }_3^B-m{\phi }_4^B)\rangle \rangle ,\hfill \end{array}$$

(10)

where $\langle \langle \rangle \rangle$ represents the average over all particles and all events, and ${\phi }_i$ is the $i^{th}$ particle azimuthal angle.

Using Equations (8)–(10), the mode-coupled response to $v_{n+m}$ is [30,77]

$$\begin{array}{ccc}\hfill v_{n+m}^{\mathrm{MC}}& =& \frac{C_{n+m,nm}}{\sqrt{\langle v_n^2{v}_m^2\rangle }},\hfill \\ & \sim & \langle v_{n+m}cos((n+m){\Psi }_{n+m}-n{\Psi }_n-m{\Psi }_m)\rangle .\hfill \end{array}$$

(11)

Moreover, the linear response to $v_{n+m}$ is

$$\begin{array}{ccc}\hfill v_{n+m}^{\mathrm{Linear}}& =& \sqrt{{\left(v_{n+m}^{\mathrm{Inclusive}}\right)}^2-{\left(v_{n+m}^{\mathrm{MC}}\right)}^2}.\hfill \end{array}$$

(12)

The ratio of the mode-coupled response to inclusive $v_{n+m}$ gives the correlations between different-order flow symmetry planes.

$$\begin{array}{ccc}\hfill {\rho }_{n+m,nm}& =& \frac{v_{n+m}^{\mathrm{MC}}}{v_{n+m}^{\mathrm{Inclusive}}},\hfill \\ \hfill & \sim & \langle cos((n+m){\Psi }_{n+m}-n{\Psi }_n-m{\Psi }_m)\rangle .\hfill \end{array}$$

(13)

The mode-coupled response coefficient gives the coupling to the higher-order anisotropic flow harmonics and is given as

$$\begin{array}{ccc}\hfill {\chi }_{n+m,nm}& =& \frac{v_{n+m}^{\mathrm{MC}}}{\sqrt{\langle v_n^2{v}_m^2\rangle }}.\hfill \end{array}$$

(14)

3. Results and Discussion

Extracting the linear and the mode-coupled (i.e., non-linear) contributions to $v_4$ depends on two- and four-particle correlations. Therefore, it is instructive to investigate the model’s potential to simulate the experimental measurements of two- and four-particle flow harmonics [78,79]. Figure 1 and Figure 2 show a comparison of the centrality dependence of $v_n\left\{2\right\}$ and $v_2\left\{4\right\}$ in Au–Au collisions at 200 (a), 39 (b), 27 (c) and 19.6 (d) GeV according to the AMPT model. The AMPT calculations exhibited sensitivity to beam energy change and harmonic order n. They also indicated similar patterns to the data reported by the STAR experiment [78,79] (solid points). The data model comparisons suggest that the AMPT model contains the proper ingredient to describe the experimental data.

The centrality dependence of the three-particle correlators, $C_{4,22}$ (panel (a)) and $C_{5,23}$ (panel (b)), are shown in Figure 3 for Au+Au collisions at 200, 39, 27 and 19.6 GeV according to the AMPT model. My results demonstrate that $C_{4,22}$ and $C_{5,23}$ depend on beam energy. These dependencies indicate that $C_{4,22}$ and $C_{5,23}$ are susceptible to the change in viscous attenuation according to the AMPT model (i.e., $\langle p_T\rangle$ and charged particle multiplicity) and initial-state eccentricity. My results reflect the capability of the three-particle correlators to constrain the interplay between the final- and initial-state effects in the AMPT model. The AMPT calculations qualitatively reproduced the trend observed in the experimental data [10]. However, the AMPT model overestimated $C_{4,22}$ for centrality larger than 30% and $C_{5,23}$ for mid-central (10–50%) region values.

Figure 4 shows the centrality and beam energy dependence of inclusive (a), mode-coupled and linear $v_4$ in Au+Au collisions according to the AMPT model. My results indicate that the linear contribution is the dominant contribution to inclusive $v_4$ in central collisions at all presented energies. In addition, $v_4^{Linear}$ showed weak centrality dependence. In addition, the difference between linear and mode-coupled $v_4$ in central collisions is derived from the difference in ${\epsilon }_4$ and ${\epsilon }^2$, respectively. The presented results show that inclusive, linear and mode-coupled $v_4$ are sensitive to beam energy variation. The AMPT results are in qualitative agreement with the experimental measurements from the STAR experiment in Au+Au collisions at $\sqrt{s_{N}N}$ = 200 GeV [10,80].

Mode-coupling response coefficient ${\chi }_{4,22}$, which gives the coupling strength between lower and higher flow harmonics, is presented in Figure 5a as a function of centrality for Au+Au at 200, 39, 27 and 19.6 GeV. The ${\chi }_{4,22}$ calculations indicated weak centrality and beam energy dependence, which implies that (i) ${\chi }_{4,22}$ is dominated by initial-state eccentricity couplings and (ii) mode-coupled $v_4$ centrality and energy dependence arise from lower-order flow harmonics. Figure 5b illustrates the centrality and energy dependence of the correlation between flow symmetry planes, ${\rho }_{4,22}$, in Au+Au collisions at $\sqrt{s_{NN}}$ = 200, 39, 27 and 19.6 GeV according to the AMPT model. The AMPT calculations of ${\rho }_{4,22}$ indicated stronger event plane correlations in peripheral collisions at all presented energies. Nevertheless, ${\rho }_{4,22}$ magnitudes were shown to be independent of beam energies. Such observation implies that initial-state eccentricity direction correlations dominate the correlation between flow symmetry planes. In addition, these calculations are in agreement with the STAR experiment measurements in Au+Au collisions at $\sqrt{s_{N}N}$ = 200 GeV [10,80].

4. Conclusions

I have presented comprehensive AMPT model calculations to evaluate the beam energy dependence of the linear and mode-coupling contributions to $v_4$, ${\chi }_{4,22}$ and ${\rho }_{4,22}$. The AMPT calculations indicate similar patterns and values to the experimental measurements of $v_n\left\{2\right\}$ and $v_2\left\{4\right\}$. The AMPT calculations of mode-coupled $v_4$ indicate strong centrality dependence; however, they show weak centrality dependence for linear $v_4$. In addition, three-particle correlations and $v_4$ show strong beam energy dependence. In contrast, ${\chi }_{4,22}$ and ${\rho }_{4,22}$ show magnitudes and trends that are weakly dependent on beam energy. The AMPT model calculations suggest that initial-state effects might be the dominant factors behind the correlations of event plane angles and the non-linear response coefficients.