Elliptic flow of hadrons in Au + Au collisions at $\sqrt{{s}_{\mathrm{NN}}}=7.7,19.6$ and 39 GeV from a multi-phase transport model based on a dynamical quark coalescence mechanism

The purpose of relativistic heavy-ion collision experiments is to study the properties of nuclear matter under extremely high temperature and pressure. One of the most widely studied observables is the azimuthal anisotropy of produced particles. In noncentral heavy-ion collisions, the initial spatial anisotropy of the overlapped region transfers to momentum anisotropy due to rescattering among produced particles. The Fourier expansion of the particle azimuthal distribution relative to the reaction plane can be written as [1–3]

$$\begin{eqnarray}&&E\displaystyle \frac{{{\rm{d}}}^{3}N}{{{\rm{d}}}^{3}p}=\displaystyle \frac{1}{2\pi }\displaystyle \frac{{{\rm{d}}}^{2}N}{{p}_{{\rm{T}}}{\rm{d}}{p}_{{\rm{T}}}{\rm{d}}y}\left(1+\displaystyle \sum _{n=1}^{\infty }2{v}_{n}\cos [n(\phi -{{\rm{\Psi }}}_{n})]\right),\end{eqnarray} \tag{ 1 }$$

where ϕ denotes the azimuthal angle of produced particles and Ψ_n is the nth order event plane angle reconstructed by the produced particles.

The second order coefficient (v₂) of the Fourier decomposition of the particle azimuthal distribution is called the elliptic flow. It is sensitive to the properties of the system at the early stage [4–7]. At a given rapidity window, the second coefficient v₂ is estimated by

$$\begin{eqnarray}&&{v}_{2}=\langle \cos [2(\phi -{{\rm{\Psi }}}_{2})]\rangle ,\end{eqnarray} \tag{ 2 }$$

where $\langle \cdots \rangle$ denotes particle averaging over all events.

In order to explain features observed at RHIC, the quark coalescence model is developed by the assumption that recombination proceeds via corresponding constituent quarks [8–13]. In particular, Molnar and Voloshin have studied recombination of quarks with similar momenta, and found an enhancement of elliptic flow compared to that of partons when transverse momentum is above 2 GeV/c [11]. The large antiproton to pion ratio can be explained by the coalescence of minijet partons with QGP partons [14]. A dynamical quark coalescence model (DQCM) includes quark Wigner phase-space functions inside hadrons was used to study the transverse momentum spectra and anisotropic flow of ϕ mesons and Ω baryons [13]. Recently, the dynamical coalescence model is used to study the directed flow and elliptic flow of open charm mesons [15, 16], and the production of Ω and ϕ is also studied with the AMPT model [17].

The phase I of the beam-energy-scan (BES) program involving Au + Au collisions at lower energies of $\sqrt{{s}_{\mathrm{NN}}}$ = 7.7–39 GeV has been carried out at RHIC, to search for the critical point which lies in the baryon chemical potential and temperature plane. Although there is no definitive conclusions on the location of the critical point, many interesting phenomenons have been observed by RHIC-BES I program. Among them is the increasing elliptic flow splitting between particles and their antiparticles with decreasing collision energy [18, 19]. This result indicates that, the number of constituent quark (NCQ) scaling of v₂ which was observed at $\sqrt{{s}_{\mathrm{NN}}}=200\,\mathrm{GeV}$ has been broken down at lower collision energies [20, 21]. Various theoretical works have been proposed to explain this experimental result [22–27]. It should be noted that the NCQ-scaling also breaks down in Pb + Pb collisions at $\sqrt{{s}_{\mathrm{NN}}}=2.76\,\mathrm{TeV}$ [28], and it can be partially explained by the AMPT model [29].

In this study, we determine the elliptic flow and the transverse momentum spectra of π^±, K^±, p and $\bar{{p}}$ through quark coalescence in phase space instead of the simple coalescence formalism in the original AMPT, to see the effect of quark coalescence mechanism to the transverse momentum spectra of charged hadrons and elliptic flow splitting between p and $\bar{{p}}$.

A multi-phase transport model (AMPT) model [30] is used in this paper to analyze the beam energy dependence and the transverse momentum dependence of the elliptic flow. The AMPT model is a hybrid model with the initial condition generated by the Heavy Ion Jet Interaction Generator (HIJING) model [31]. There are two versions of it: the string melting AMPT model and the default AMPT model. In our study, we use the string melting version to reproduce the elliptic flow of particles and antiparticles. Hadrons produced in the HIJING model are converted to quarks and anti-quarks, and Zhang's parton cascade model [32] is used to describe the time and space evolution of these (anti)quarks. At hadronization stage, quarks and anti-quarks are recombined to form hadrons through a simple quark coalescence model, and the scatterings between hadrons are modeled by a relativistic transport model [33].

The current quark coalescence process in AMPT searches for a meson partner first, all mesons are formed by searching combining q and $\bar{{q}}$ close in coordinate space, then (anti)baryons are formed combining three nearby (anti) quarks. Recently, He and Lin found that the AMPT with an improved quark coalescence component can better describe low p_T baryon spectra and antibaryon-to-baryon ratios for multi-strange baryons [34]. Our previous work indicates that the elliptic flow splitting between proton and antiproton will be affected by the quark coalescence process [35].

In this paper, we deal with the quark coalescence process in the framework of dynamical quark coalescence mechanism [13]. In this model, the probability of producing a meson is given by

$$\begin{eqnarray}\begin{array}{rcl}{\rho }^{W}\left({\bf{r}},{\bf{k}}\right) & = & \displaystyle \int \psi \left({\bf{r}}+\displaystyle \frac{{\bf{R}}}{2}\right){\psi }^{* }\left({\bf{r}}-\displaystyle \frac{{\bf{R}}}{2}\right)\exp (-{\rm{i}}{\bf{k}}\cdot {\bf{R}}){{\rm{d}}}^{3}{\bf{R}}\\ & = & 8\exp \left(-\displaystyle \frac{{r}^{2}}{{\sigma }_{{\rm{m}}}^{2}}-{\sigma }_{{\rm{m}}}^{2}{k}^{2}\right),\end{array}\end{eqnarray} \tag{ 3 }$$

where ${\bf{r}}={{\bf{r}}}_{1}-{{\bf{r}}}_{2}$ and ${\bf{k}}=({{\bf{k}}}_{1}-{{\bf{k}}}_{2})/2$ are relative distance in coordinate space and momentum space, respectively, calculated in the rest frame of the two-quark system. ψ represents the quark wave function

$$\begin{eqnarray}&&\psi ({{\bf{r}}}_{1},{{\bf{r}}}_{2})=1/{\left(\pi {\sigma }_{{\rm{m}}}^{2}\right)}^{3/4}\exp [-{r}^{2}/(2{\sigma }_{{\rm{m}}}^{2})],\end{eqnarray} \tag{ 4 }$$

here σ_m is the size parameter of meson. The normalized wave function leads to a root-mean-square (RMS) radius of ${R}_{{\rm{m}}}={\left(3/8\right)}^{1/2}{\sigma }_{{\rm{m}}}$.

Similarly, for baryons, the probability to form a baryon from three quarks or anti-quarks can be expressed as

$$\begin{eqnarray}&&{\rho }^{W}(\rho ,\lambda ,{{\bf{k}}}_{\rho },{{\bf{k}}}_{\lambda })={8}^{2}\exp \left(-\displaystyle \frac{{\rho }^{2}+{\lambda }^{2}}{{\sigma }_{{\rm{b}}}^{2}}\right)\exp [-({{\bf{k}}}_{\rho }^{2}+{{\bf{k}}}_{\lambda }^{2}){\sigma }_{{\rm{b}}}^{2}],\end{eqnarray} \tag{ 5 }$$

where σ_b is the size parameter of baryon, (ρ, λ) and $({{\bf{k}}}_{\rho },{{\bf{k}}}_{\lambda })$ are relative coordinates and relative momenta calculated in the rest frame of three-quark system, see [13] for detail.

In this paper, the simple quark coalescence model in the original AMPT model is replaced by a DQCM. We take the values of RMS radii, R_m = 0.61 fm for meson and R_b = 0.877 fm for baryon, as used in [36] for π⁺ and p, respectively. We take the Lund string fragmentation parameters as a = 0.55 and b = 0.15 GeV⁻², and the strong coupling constant α_s = 0.33. These values are the same as those used in [37].

We first show in figure 1 the midrapidity ($| y| \lt 0.1$) transverse momentum spectrum for Au + Au collisions at $\sqrt{{s}_{\mathrm{NN}}}=19.6\,\mathrm{GeV}$. It is seen that the calculated transverse momentum spectra from the DQCM AMPT is a little stiffer than that of original AMPT. And both the original AMPT and the DQCM AMPT describe the midrapidity π⁺ and proton data well, consistent with [38] using the original AMPT model.

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We note that the difference between original AMPT and DQCM AMPT is largest for $\bar{{p}}$, so the hadron size parameter dependence of $\bar{{p}}$ spectra needs to be studied. Figure 2 shows the transverse momentum spectra of $\bar{{p}}$ from the original AMPT and the DQCM AMPT with different meson size parameters scaled by the result of STAR at $\sqrt{{s}_{\mathrm{NN}}}=19.6\,\mathrm{GeV}$. All of the spectra used in the calculation of $\bar{{p}}(\mathrm{AMPT},\mathrm{norm})/\bar{{p}}(\mathrm{STAR},\mathrm{norm})$ in figure 2 are normalized to one unity. It can be seen from figure 2 that the transverse momentum spectra of $\bar{{p}}$ depends strongly on the size parameter of mesons, the spectra of $\bar{{p}}$ will become more stiffer if larger meson size parameter is applied in the DQCM AMPT. We also found that the transverse momentum spectra of $\bar{{p}}$ shows slightly dependence on the size parameter of baryons at $\sqrt{{s}_{\mathrm{NN}}}=19.6\,\mathrm{GeV}$ (not shown here).

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In figure 3, we show the transverse momentum dependence of elliptic flow of midrapidity π⁺, p and $\bar{{p}}$ for Au + Au collisions at $\sqrt{{s}_{\mathrm{NN}}}=$ 19.6 GeV and b = 8 fm with different values of RMS radius R_m for meson and fixed R_b = 0.877 fm for baryon. It can be seen that the v₂ of π⁺ and $\bar{{p}}$ depends strongly on the size parameter of mesons. For π⁺, the saturation value of v₂ becomes larger with increasing R_m, and the saturation point moves to higher transverse momentum at larger R_m. The v₂ of $\bar{{p}}$ decreases with increasing size parameter of meson, this may be due to the competition of anti-quarks between meson and antibaryon in the hadronization process. Compared to $\bar{{p}}$, the v₂ of p show different scenario due to the fact that the quarks of meson and baryon are non-interchangeable in the quark coalescence process of the AMPT model.

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In figure 4, we show the v₂ of p and $\bar{{p}}$ in Au + Au collisions at $\sqrt{{s}_{\mathrm{NN}}}=19.6\,\mathrm{GeV}$ and b = 8 fm with different values of RMS radius R_b for baryon and fixed R_m = 0.61 fm for meson. It is seen that the v₂ of p and $\bar{{p}}$ are sensitive to the size parameter of themselves, with the larger ${p}(\mathrm{or}\,\bar{{p}})$ size parameter giving a smaller v₂.

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The elliptic flow splitting of p and $\bar{{p}}$ are also studied. Figure 5 shows the elliptic flow of p and $\bar{{p}}$ of Au + Au collisions at $\sqrt{{s}_{\mathrm{NN}}}=19.6\,\mathrm{GeV}$ and b = 8 fm. The comparison is also often performed in terms of ${\rm{\Delta }}{v}_{2}=[{v}_{2}({p})-{v}_{2}(\bar{{p}})]$. In the upper panel of figure 5, we show the results of Δv₂ without hadron cascade from (a) original AMPT and (b) DQCM AMPT. The v₂ of $\bar{{p}}$ decreases in the DQCM AMPT compared to that of original AMPT when the transverse momentum is above 1 GeV/c, and Δv₂ (negative value here) increases, as expected. The Δv₂ of p and $\bar{{p}}$ with hadron cascade (figure 5(c) for original AMPT and figure 5(d) for DQCM AMPT) seen to increase again, and a positive Δv₂ has been observed in DQCM AMPT at transverse momentum range of p_T > 1 GeV/c. From figure 5, it is seen that quark coalescence in phase space has a clear effect on the elliptic flow splitting of p and $\bar{{p}}$, and it makes Δv₂ more close to the experimental result [18] at higher transverse momentum(p_T > 1 GeV/c) in the energy of 19.6 GeV.

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We show in figures 6–8 the elliptic flow of π^±, K^±, p and $\bar{{p}}$ based on the η-sub EP method [18] for 0%–80% centrality Au + Au collisions from the original AMPT and DQCM AMPT at $\sqrt{{s}_{\mathrm{NN}}}$ = 7.7, 19.6 and 39 GeV in comparison with the experimental data from STAR [18]. It is known that the v₂ of charged hadrons is sensitive to the parton cross section, with a larger parton cross section giving larger v₂. To compare with the experimental data and see the effect of quark coalescence in phase space to v₂, we chose several parton cross sections for different energies (see figures 6–8). More detailed research on parton cross section dependence of v₂ can be seen in [40].

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Figure 6 shows v₂ from the DQCM AMPT with parton cross section of 3 mb and original AMPT with parton cross section of 1 mb and 3 mb. It is observed that the v₂ decreases in the DQCM AMPT (3 mb) compared to that of original AMPT (3 mb), as expected. And the v₂ from the DQCM AMPT (3 mb) and original AMPT (1 mb) are generally consistent with the experimental data.

The v₂ for Au + Au collisions at 19.6 GeV is shown in figure 7. It seems that v₂ from DQCM AMPT (6 mb) and original AMPT (1.5 mb) describe the data well at lower p_T, and DQCM AMPT (6 mb) together with original AMPT (6 mb) give a better v₂ at higher p_T range (p_T > 1.5 GeV). Similar results for $\sqrt{{s}_{\mathrm{NN}}}=39\,\mathrm{GeV}$ can be seen in figure 8.

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We note that the current quark coalescence process in the AMPT model has some limitations. First, the current AMPT model neglect the finite thickness of colliding nuclei, which has significant effect at lower energies [41]. Besides, in the formation of hadrons, mesons have higher priority than (anti)baryons [34] in the simple quark coalescence model of AMPT. And the present study only improves the quark coalescence process from coordinate space to phase space. The phase space coalescence combined with more reasonable hadron formation order will be considered in the future.

In summary, we have studied the elliptic flow and transverse momentum spectra of π^±, K^±, p and $\bar{{p}}$ of Au + Au collisions at $\sqrt{{s}_{\mathrm{NN}}}=7.7,19.6$ and 39 GeV using the AMPT model based on the framework of a dynamical quark coalescence mechanism. We find that the v₂ of $\bar{{p}}$ decreases with larger size parameter of meson, but the v₂ of p is insensitive to meson's size parameter. This is an interesting observation, which indicates that the DQCM can make a positive contribution to the elliptic flow splitting of p and $\bar{{p}}$ at lower energies. The AMPT results are compared with STAR data. The comparison shows that, with this coalescence mechanism, the yields of hadrons is enhanced at higher transverse momentum (p_T > 1 GeV/c), which makes the transverse momentum spectra from the DQCM AMPT a little more stiffer than that of original AMPT, and they are consistent with the experimental data. The v₂ decreases in the DQCM AMPT compared to that of original AMPT, and they are seen to agree with available experimental data, although the parton cross section needs to be larger in the DQCM AMPT.

Authors would like to thank Dr Zi-Wei Lin for helpful discussion on the quark coalescence process in the AMPT model. We also thank Dr Lie-Wen Chen for giving a kindly reminder about the unit of relative momentum in the dynamical quark coalescence model. We thank the HPC Studio at the Physics Department of the Harbin Institute of Technology for computing resources allocated through INSPUR-HPC@PHY.HIT.