Ω and ϕ production in Au+Au collisions at sNN = 11.5 and 7.7 GeV in a dynamical quark coalescence model

Ω and ϕ production in Au+Au collisions at <math id="M1"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><msub><mrow/><mrow><mtext>N</mtext><mi>N</mi></mrow></msub></mrow></msub></mrow></msqrt></mrow></math> = 11.5 and 7.7 GeV in a dynamical quark coalescence model 1 2 first-author 1 corresp chenjinhui@sinap.ac.cn, chenjinhui@sinap.ac.cn, chenjinhui@sinap.ac.cn 1 3 corresp ygma@sinap.ac.cn ygma@sinap.ac.cn ygma@sinap.ac.cn 1 1 2 1 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China ShanghaiTech University, Shanghai 200031, China ShanghaiTech University, Shanghai 200031, China The Ω and ϕ production in relativistic heavy-ion collisions is studied in a dynamical quark coalescence model using the phase space information of strange quarks from a multiphase transport (AMPT) model. Enhanced local parton density fluctuation is implemented in the AMPT to simulate the QCD phase transition dynamics. By studying the transverse momentum p T spectra and the elliptic flow of the multi-strangeness particles, such as Ω and ϕ , and the Ω / ϕ ratio as a function of p T in the AMPT, we find that the new development improves the description of experimental data. The study motivates further experimental investigations of Ω and ϕ production in Phase-II of the Beam Energy Scan program at RHIC. QCD phase transition multi-strangeness particles elliptic flow AMPT 1 Introduction 1 Searching for the QCD critical point and mapping the QCD phase diagram are major scientific goals of the Beam Energy Scan (BES) program in Relativistic Heavy-Ion Collisions (RHIC) [ 1 1 ]. Finite temperature lattice QCD calculations show that at high temperature and low baryon chemical potential ( μ B ), a phase transition from hadron gas to the quark gluon plasma (QGP) will happen, and the phase transition is smooth and continuous [ 2 2 ]. At large μ B region, the QCD-based models predict that the phase transition is of the first order and there should exist a QCD critical point as the endpoint of the first order phase boundary [ 3 3 - 5 5 3-5 ]. Heavy ion collisions at relativistic energy provide a unique opportunity to study the properties of QGP and the QCD phase diagram. Many exciting results in the field show rich structure of QCD matter, such as the non-monotonic behaviour of conserved quantity fluctuation and final state correlations [ 6 6 - 15 15 6-15 ], net-proton direct flow [ 16 16 ], the strangeness enhancement [ 17 17 - 20 20 17-20 ], the local baryon-strangeness correlation measurement from the hypernuclei system [ 21 21 - 23 23 21-23 ], novel quantum phenomena induced by strong magnetic fields in heavy-ion collisions [ 24 24 ], the breaking of the number of constituent quark scaling on identified particles [ 25 25 - 27 27 25-27 ], and the deviation of the Ω / ϕ ratio in BES in comparison with data at top RHIC energy.[ 28 28 ] Multi-strange hadrons, such as Ω and ϕ , are clean probes to explore the QCD phase diagram [ 29 29 , 30 30 ], as they are expected to have a relatively small hadronic interaction cross section and little contribution from resonance decay [ 31 31 , 32 32 ]. Therefore, they can carry the information directly from the chemical freeze-out stage with little or no distortion due to hadronic rescattering. As a result, the production of Ω and ϕ particles offers an advantage in probing the transition from partonic to hadronic dynamics [ 29 29 ]. In a recent measurement of Ω and ϕ production at midrapidity from Au+Au collisions at <math id="M2"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><msub><mrow/><mrow><mtext>N</mtext><mi>N</mi></mrow></msub></mrow></msub></mrow></msqrt></mrow></math> = 7.7, 11.5, 19.6, 27, and 39 GeV from the STAR experiment at RHIC, one finds that the Ω / ϕ ratios scaled by their number of constituent quark show a suppression of strange quark production in central collisions at 11.5 GeV compared to results at <math id="M3"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><msub><mrow/><mrow><mtext>N</mtext><mi>N</mi></mrow></msub></mrow></msub></mrow></msqrt></mrow></math> ≥ 19.6 GeV, and the data of 19.6 GeV or above show little beam energy dependence, which may suggest strange quark equilibration is achieved in higher energy but not in lower energy [ 28 28 ]. We have carried out a detail study on the underlying dynamics of strange quark matter at the RHIC based on AMPT model previously, and found that the AMPT model presents the trend of Ω / ϕ vs p T of the data, though under-predicts the Ω yield by a factor of 5 [ 30 30 ]. New development of the naive quark coalescence model in the AMPT model improves the description of baryon production in heavy-ion collisions, such as protons and Λ , but still significantly under-predicts the Ω and Ξ yield [ 33 33 ]. An alternative solution using a dynamical quark coalescence model to replace the naive quark coalescence model in the AMPT model predicted the Ω and ϕ yield in Au+Au collisions at <math id="M4"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><msub><mrow/><mrow><mtext>N</mtext><mi>N</mi></mrow></msub></mrow></msub></mrow></msqrt></mrow></math> = 200 GeV reasonably well [ 34 34 ]. In this paper, we follow the procedure described in Ref. [ 34 34 ] and introduce the local parton density fluctuation effect in the AMPT model to study the Ω and ϕ production as a possible signal of the QCD phase transition. We noted that similar idea was discussed in Ref. [ 35 35 ]. 2 The ampt model 1 The AMPT model is a hybrid model developed to describe heavy-ion collisions at relativistic energies [ 36 36 ]. It has two versions: default AMPT and string melting AMPT. In our study, we adopt the string melting version to study the QCD fluctuation effect. In the string melting version, all excited strings are converted into partons [ 36 36 ], it has a clear advantage over the default version in describing the flow of harmonic and pion interferometry [ 33 33 , 37 37 - 39 39 ]. The AMPT model with the string melting scenario consists of four components. The initial stage is described by a heavy-ion jet interaction generator (HIJING) [ 40 40 , 41 41 ], which is designed to simulate multiple jets and particle production in heavy ion collisions. Zhang’s parton cascade (ZPC) [ 42 42 ] is used to describe scattering among partons. It only included two-body elastic interaction until now. The hadronization of parton is based on a naive quark coalescence model from coordinate space distribution [ 36 36 ]. The scattering of resulting hadrons is described by a relativistic transport (ART) model [ 43 43 , 44 44 ]. In the transport approach, interactions among partons are described by their scattering cross sections, which in the AMPT model are given by[ 42 42 ] <math display="block" id="M5"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>σ</mi><mi>p</mi></msub></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>≃</mo><mfrac><mrow><mn>9</mn><mi>π</mi><msubsup><mi>α</mi><mi>s</mi><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><msup><mi>μ</mi><mn>2</mn></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M6"><mrow><msub><mi>σ</mi><mi>p</mi></msub><mo>≃</mo><mfrac><mrow><mn>9</mn><mi>π</mi><msubsup><mi>α</mi><mtext>s</mtext><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><msup><mi>μ</mi><mn>2</mn></msup></mrow></mfrac><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>μ</mi><mn>2</mn></msup><mo>/</mo><mi>s</mi></mrow></mfrac><mn>.</mn></mrow></math> where α s is the strong coupling constant with a typical value of 0.47 in AMPT, μ is the screening mass which depends on the medium effect [ 42 42 ], and s and t are the usual Mandelstam variables. By applying a large parton scattering cross sections of 6-10 mb, the AMPT model can reproduce the centrality and transverse momentum dependence of hadron elliptic flow [ 45 45 ]. Recent development on the input parameters of AMPT model seems to obtain a smaller value, and more close to the value used in pQCD calculation [ 38 38 ]. For our current study, we follow the instruction from Ref. [ 38 38 ] and take the parton scattering cross section of 3 mb to simulate parton interaction. In order to understand the QCD phase diagram on a wide range of collision energies, we add local parton density fluctuation in AMPT. Viewed from the thermodynamics, a critical point is a point at which a single thermodynamic state bifurcates into two macroscopically distinct states. This bifurcation may lead to long-range thermal fluctuations. To model this effect, a large density fluctuation is introduced in the end of parton scattering in ZPC. Specifically, we assume that the partonic matter prior to the QCD phase transition consists of clusters of various sizes. We redistribute partons produced in the AMPT model to a few clusters but keep their momenta unchanged. In our algorithm, the center of a cluster is determined by the maximum number density of parton distributions in that event. Cluster positions are selected based on the freeze out positions of partons, and then far away partons in the transverse plane are moved close to the nearest cluster. All partons then coalescence into hadrons. To demonstrate this effect, we only allow in the present study the formation of four quark clusters before hadronization. An example of the parton spatial distribution is shown in panel (b) of Fig. 1 Fig. 1 . For comparison, the distribution from the AMPT model is shown in panel (a). The subsequent distribution of hadrons formed immediately after hadronization from partons are shown in panel (c) and (d), respetively. From Fig. 1 Fig. 1 , it is seen that the enhanced local parton density fluctuation has a clear effect on the spatial distributions of hadronic matter in AMPT. The fluctuation effect (panel (d)) is stronger than the original AMPT model (panel (c)). In this paper, we perform our calculations with string melting AMPT in version 2.24 for Au+Au collision. 3 The dynamical quark coalescence model of Ω and ϕ 1 A dynamical quark coalescence model has been used to study the production of Ω and ϕ [ 34 34 ]. In this model, the probability for producing a hadron from partons is given by the overlap of parton phase space distributions with the parton Wigner function inside the hadron [ 46 46 ]. The multiplicity of a M -parton hadron in an heavy-ion collision is given by <math display="block" id="M8"><mtable columnalign="left"><mtr><mtd><msub><mi>N</mi><mi>M</mi></msub><mo>=</mo><mi>G</mi><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mtext>d</mtext><msub><mi>r</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mtext>d</mtext><msub><mi>q</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>…</mo><mtext>d</mtext><msub><mi>r</mi><mrow><msub><mi>i</mi><mrow><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mtext>d</mtext><msub><mi>q</mi><mrow><msub><mi>i</mi><mrow><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub></mtd></mtr><mtr><mtd><mo>×</mo><mo>〈</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>i</mi><mn>1</mn></msub><mo>></mo><msub><mi>i</mi><mn>2</mn></msub><mo>></mo><mo>…</mo><mo>></mo><msub><mi>i</mi><mi>M</mi></msub></mrow></munder></mstyle><msubsup><mi>ρ</mi><mi>i</mi><mtext>W</mtext></msubsup><mo stretchy="false">(</mo><msub><mi>r</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mn>,</mn><msub><mi>q</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>…</mo><msub><mi>r</mi><mrow><msub><mi>i</mi><mrow><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mn>,</mn><msub><mi>q</mi><mrow><msub><mi>i</mi><mrow><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">)</mo><mo>〉</mo><mn>.</mn></mtd></mtr></mtable></math> In Eq. (3), ⟨⟩ represents the event averaging; <math id="M9"><mrow><msub><mi>r</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub></mrow></math> ,, <math id="M10"><mrow><msub><mi>r</mi><mrow><msub><mi>i</mi><mrow><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub></mrow></math> and <math id="M11"><mrow><msub><mi>q</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub></mrow></math> ,, <math id="M12"><mrow><msub><mi>q</mi><mrow><msub><mi>i</mi><mrow><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub></mrow></math> are the M -1 relative coordinates and momenta in the M -parton rest frame; <math id="M13"><mrow><msubsup><mi>ρ</mi><mi>i</mi><mtext>W</mtext></msubsup></mrow></math> is the Wigner phase space function inside the hadron, and G is the statistical factor for the M partons. To determine the quark Wigner phase space functions inside Ω and ϕ , we need quark wave functions. The quark wave functions can be taken as a spherical harmonic oscillator as done in early work [ 34 34 ]. For the ϕ particle, it can be expressed as <math display="block" id="M14"><mrow><mi>ψ</mi><mtext>(</mtext><msub><mi>r</mi><mtext>1</mtext></msub><mtext>,</mtext><mtext> </mtext><msub><mi>r</mi><mtext>2</mtext></msub><mtext>)=1/(</mtext><mi>π</mi><msubsup><mi>σ</mi><mi>ϕ</mi><mtext>2</mtext></msubsup><msup><mtext>)</mtext><mrow><mtext>3/4</mtext></mrow></msup><mtext> </mtext><mtext>exp</mtext><mrow><mo>[</mo><mrow><mo>−</mo><msup><mi>r</mi><mtext>2</mtext></msup><mtext>/(2</mtext><msubsup><mi>σ</mi><mi>ϕ</mi><mtext>2</mtext></msubsup><mtext>)</mtext></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where r = r 1 - r 2 is the relative coordinate and σ ϕ is a size parameter of ϕ . Its normalized wave function leads to a root-mean-square (RMS) radius of R ϕ = (3/8) 1/2 σ ϕ . The quark Wigner function in the ϕ particle can be expressed as <math display="block" id="M15"><mrow><msubsup><mi>ρ</mi><mi>ϕ</mi><mtext>W</mtext></msubsup><mtext>(r,k)=8exp</mtext><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><msup><mtext>r</mtext><mtext>2</mtext></msup></mrow><mrow><msubsup><mi>σ</mi><mi>ϕ</mi><mtext>2</mtext></msubsup></mrow></mfrac><mo>−</mo><msubsup><mi>σ</mi><mi>ϕ</mi><mtext>2</mtext></msubsup><msup><mtext>k</mtext><mtext>2</mtext></msup></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where k = (k 1 -k 2 )/2 is the relative momentum between s and <math id="M16"><mover accent="true"><mi>s</mi><mo>¯</mo></mover></math> . Similarly, for Ω - and <math id="M17"><mrow><msup><mover accent="true"><mi>Ω</mi><mo>¯</mo></mover><mo>+</mo></msup></mrow></math> particles, their wave function can be described by the following equation <math display="block" id="M18"><mrow><mi>ψ</mi><mtext>(</mtext><msub><mi>r</mi><mtext>1</mtext></msub><mtext>,</mtext><msub><mi>r</mi><mtext>2</mtext></msub><mtext>,</mtext><msub><mi>r</mi><mtext>3</mtext></msub><mtext>)=(3</mtext><msup><mi>π</mi><mtext>2</mtext></msup><msubsup><mi>σ</mi><mi>Ω</mi><mtext>4</mtext></msubsup><msup><mtext>)</mtext><mrow><mo>−</mo><mtext>3/4</mtext></mrow></msup><mtext> </mtext><mtext>exp</mtext><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><msup><mi>ρ</mi><mtext>2</mtext></msup><mo>+</mo><msup><mi>λ</mi><mtext>2</mtext></msup></mrow><mrow><mtext>2</mtext><msubsup><mi>σ</mi><mi>Ω</mi><mtext>2</mtext></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> The quark Wigner phase-space function inside the Ω - and <math id="M19"><mrow><msup><mover accent="true"><mi>Ω</mi><mo>¯</mo></mover><mo>+</mo></msup></mrow></math> baryon can be expressed as <math display="block" id="M20"><mtable columnalign="left"><mtr><mtd><msubsup><mi>ρ</mi><mi>Ω</mi><mtext>W</mtext></msubsup><mtext>(</mtext><mi>ρ</mi><mtext>,</mtext><mi>λ</mi><mtext>,</mtext><msub><mi>k</mi><mi>ρ</mi></msub><mtext>,</mtext><msub><mi>k</mi><mi>λ</mi></msub><mtext>)</mtext><mo>=</mo></mtd></mtr><mtr><mtd><mtext>64</mtext><mtext> </mtext><mtext>exp</mtext><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><msup><mi>ρ</mi><mtext>2</mtext></msup><mo>+</mo><msup><mi>λ</mi><mtext>2</mtext></msup></mrow><mrow><msubsup><mi>σ</mi><mi>Ω</mi><mtext>2</mtext></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow><mtext>exp</mtext><mrow><mo>[</mo><mrow><mo>−</mo><mtext>(</mtext><msubsup><mi>k</mi><mi>ρ</mi><mtext>2</mtext></msubsup><mo>+</mo><msubsup><mi>k</mi><mi>λ</mi><mtext>2</mtext></msubsup><mtext>)</mtext><msubsup><mi>σ</mi><mi>Ω</mi><mtext>2</mtext></msubsup></mrow><mo>]</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> where ρ and λ are relative coordinates of quark, k ρ and k λ are relative momenta, and σ Ω is a size parameter that is related to the RMS radius, R Ω . The two parameters σ ϕ and σ Ω in the quark Wigner phase space functions inside the ϕ meson and Ω baryon are related to their RMS radii. We take the same values of RMS, R ϕ = 0.65 fm and R Ω = 1.2 fm, as used in Ref. [ 34 34 ]. One should note that the quark coalescence model violates energy conservation. The effect is much more ambiguous at low p T . In the present study, the coalescence model is considered as a perturbative approach, valid only if the numbers of partons coalesced into hadrons is small compared with the total numbers of partons in the system [ 34 34 ]. This condition is satisfied for Ω and ϕ produced in relativistic heavy-ion collisions because their numbers are indeed significantly smaller than that of kaons. 4 Transverse momentum spectra of Ω and ϕ 1 Using the parton phase space information and the dynamical quark coalescence model, we first study the effect of local parton density fluctuation on the transverse momentum p T spectra of Ω and ϕ . Figure 2 Figure 2 presents the transverse momentum distributions of Ω and ϕ at midrapidity (| y | ≤ 1) in Au+Au collisions at <math id="M21"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><msub><mrow/><mrow><mtext>N</mtext><mi>N</mi></mrow></msub></mrow></msub></mrow></msqrt></mrow></math> = 11.5 and 7.7 GeV. The local parton density fluctuation causes the trend to increase the production rate of Ω and ϕ in comparison with the two AMPT calculations shown in Fig. 2 Fig. 2 . The fluctuation scenario describes the experimental data better than the original AMPT case, both with dynamical quark coalescence. We note that the calculations at p T > 1 GeV/ c are below the data, particularly for the Ω . Our result is similar to that shown in Ref. [ 34 34 ], which may be attributed to the deficiency of the AMPT model in treating baryon production. Development on this direction is on the way and better description on proton and Λ productions have been achieved [ 33 33 ]. Nevertheless, one should note that the AMPT model with naive quark coalescence only predicts 20% of Ω yield [ 30 30 ]. The new calculation certainly improves the description of Ω and ϕ production. We further study the ratio of Ω / ϕ as a function of p T , which is believed to be sensitive to local strange quark fluctuation. The results are presented in Fig. 3 Fig. 3 . The <math id="M23"><mrow><mi>N</mi><mo stretchy="false">(</mo><msup><mi>Ω</mi><mo>−</mo></msup><mo>+</mo><msup><mover accent="true"><mi>Ω</mi><mo>¯</mo></mover><mo>+</mo></msup><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mi>N</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></math> ratio from string melting AMPT increases slowly as p T increases. It is close to the experiment data at p T < 1.0 GeV/ c and underestimates the ratio at high p T significantly. For the result from AMPT with enhanced local parton density fluctuation scenario, the ratio is larger and describes the experimental data well, especially for p T > 1.5 GeV/ c . It seems that the density fluctuation scenario allows a larger possibility for strange quarks or antiquarks to overlap in the coordinate space, as shown in Fig. 1 Fig. 1 . The production rate on Ω and ϕ thus enhances via quark coalescence. The effect is stronger on Ω than ϕ from the Ω / ϕ ratio analysis. 5 Anisotropic flows of Ω and ϕ 1 In this section, we evaluate the effect of variations in local parton density spacial distributions and discuss its consequence on elliptic flow. The collectivity in high-energy heavy-ion collisions can be measured through final particle azimuthal anisotropy [ 47 47 ]. The anisotropy coefficients are generally obtained from Fourier expansion of final particle azimuthal distortion [ 48 48 ]. i.e., <math display="block" id="M25"><mrow><mi>E</mi><mfrac><mrow><msup><mtext>d</mtext><mn>3</mn></msup><mi>N</mi></mrow><mrow><msup><mtext>d</mtext><mn>3</mn></msup><mi>p</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>N</mi></mrow><mrow><msub><mi>p</mi><mtext>T</mtext></msub><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub><mtext>d</mtext><mi>y</mi></mrow></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><mn>2</mn><msub><mi>v</mi><mi>n</mi></msub><mi>cos</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">(</mo><mi>φ</mi><mo>−</mo><msub><mi>ψ</mi><mrow><mtext>RP</mtext></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> where E is the energy of the final particle, p T is the transverse momentum, y is the rapidity, φ is the azimuthal angle of the particle and ψ RP is the reaction plane angle. The Fourier coefficients, v n ( n =1,2,...), can be described by the following equation <math display="block" id="M26"><mrow><msub><mi>v</mi><mi>n</mi></msub><mo>=</mo><mo>〈</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">[</mo><mi>φ</mi><mo>−</mo><msub><mi>ψ</mi><mrow><mtext>RP</mtext></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>〉</mo><mn>.</mn></mrow></math> Similarly, the calculation of harmonic flow, v n , can be relative to the participant plane angle, ψ n { PP } [ 47 47 , 48 48 ]. For the study of the local parton density fluctuation effect on anisotropy flow, the participant plane method may be more straight forward [ 49 49 ]. The participant plane angle can be defined by the following equation <math display="block" id="M27"><mrow><msub><mi>ψ</mi><mi>n</mi></msub><mo>{</mo><mi>P</mi><mi>P</mi><mo>}</mo><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mrow><mo>[</mo><mrow><mi>arctan</mi><mfrac><mrow><mo>〈</mo><msup><mi>r</mi><mn>2</mn></msup><mi>sin</mi><mo stretchy="false">(</mo><mi>n</mi><mi>φ</mi><mo stretchy="false">)</mo><mo>〉</mo></mrow><mrow><mo>〈</mo><msup><mi>r</mi><mn>2</mn></msup><mi>cos</mi><mo stretchy="false">(</mo><mi>n</mi><mi>φ</mi><mo stretchy="false">)</mo><mo>〉</mo></mrow></mfrac><mo>+</mo><mi>π</mi></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where n is n th-order participant plane, r and φ are the coordinates position and azimuthal angle of partons, and ⟨...⟩ represents parton number density weighting. Figure 4 Figure 4 shows the v 2 of Ω and ϕ in Au+Au collisions at <math id="M28"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><msub><mrow/><mrow><mtext>N</mtext><mi>N</mi></mrow></msub></mrow></msub></mrow></msqrt></mrow></math> = 11.5 and 7.7 GeV. Our results from AMPT with string melting versions and the enhanced local parton density fluctuation scenario are consistent with the experimental data, considering the large statistical uncertainty. Comparing the two sets of AMPT model results, the difference on v 2 is small. It may be because the v 2 is mainly developed in the parton cascade stage in the AMPT model [ 36 36 ]. Another reason may be related to the spacial anisotropy. The spacial anisotropy is quantified by the participant eccentricity coefficients ε part [ 51 51 , 52 52 ]: <math display="block" id="M30"><mrow><msub><mi>ε</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mtext>part</mtext><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>4</mn><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>y</mi></mrow><mn>2</mn></msubsup></mrow></msqrt></mrow><mrow><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup></mrow></mfrac><mn>,</mn></mrow></math> where <math display="block" id="M31"><mrow><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><mo>=</mo><mo>{</mo><msup><mi>x</mi><mn>2</mn></msup><mo>}</mo><mo>−</mo><msup><mrow><mo>{</mo><mi>x</mi><mo>}</mo></mrow><mn>2</mn></msup></mrow></math> <math display="block" id="M32"><mrow><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>=</mo><mo>{</mo><msup><mi>y</mi><mn>2</mn></msup><mo>}</mo><mo>−</mo><msup><mrow><mo>{</mo><mi>y</mi><mo>}</mo></mrow><mn>2</mn></msup></mrow></math> <math display="block" id="M33"><mrow><msub><mi>σ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>x</mi><mi>y</mi><mo>}</mo><mo>−</mo><mo>{</mo><mi>x</mi><mo>}</mo><mo>{</mo><mi>y</mi><mo>}</mo><mn>,</mn></mrow></math> and {…} denotes the average over all participants in one event. The results from the AMPT model prior to quark hadronization with different sets of configuration are shown in Fig.5 Fig.5 . It is found that the local parton density fluctuation does not have a large impact on eccentricity. 6 Summary 1 In summary, we have studied the local parton density fluctuation effect on Ω and ϕ production in heavy ion collisions based on the parton phase space information from the AMPT model. We have calculated he transverse momentum spectra, the particle ratio, and the elliptic flow of Ω and ϕ in Au+Au collisions at <math id="M34"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><msub><mrow/><mrow><mtext>N</mtext><mi>N</mi></mrow></msub></mrow></msub></mrow></msqrt></mrow></math> = 11.5 and 7.7 GeV. For the production of Ω and ϕ , it is found that the local parton density fluctuation increases the yield more than normal AMPT calculation. Our new results improves the description of experimental data. The increase is more visible in the Ω / ϕ ratio vs. p T . 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C 82 , 039903 ( 2010 ). doi: 10.1103/PhysRevC.82.039903 http://doi.org/10.1103/PhysRevC.82.039903 Collision-geometry fluctuations and triangular flow in heavy-ion collisions S. Zhang , Y. G. Ma , J. H. Chen , et al ., Nuclear cluster structure effect on elliptic and triangular flows in heavy-ion collisions . Phys. Rev. C 95 , 064904 ( 2017 ). doi: 10.1103/PhysRevC.95.064904 http://doi.org/10.1103/PhysRevC.95.064904 Nuclear cluster structure effect on elliptic and triangular flows in heavy-ion collisions We are grateful to Dr. Zi-Wei Lin for the help to implement the local parton density fluctuation effect in AMPT model. 10.1007/s41365-018-0393-1 This work was supported in part by the Major State Basic Research Development Program in China (Nos. 2014CB845400 and No. 2015CB856904), and the National Natural Science Foundation of China (Nos. 11775288, 11421505, and 11520101004). 2018-01-03 2018-01-28 2018-02-18 2018-03-16 2018-04-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2018 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-018-0393-1 http://dx.doi.org/10.1007/s41365-018-0393-1 . 2018 Nuclear Science and Techniques 04 29