# Initial fluctuation effects on harmonic flows in high-energy heavy-ion collisions

###### Abstract

Within the framework of a multi-phase transport model, harmonic flows (n = 2, 3 and 4) are investigated for Au + Au collisions at = 200 GeV and Pb + Pb collisions at = 2.76 TeV. The event-by-event geometry fluctuations significantly contribute to harmonic flows. Triangular flow () originates from initial triangularity () and is developed by partonic interactions. The conversion efficiency () decreases with harmonic order and increases with partonic interaction cross section. A mass ordering in the low region and number of constitute quark scaling in the middle region seem to work roughly for -th harmonic flows at both energies. All features of harmonic flows show similar qualitative behaviors at RHIC and LHC energies, which implies that the formed partonic matters are similar at the two energies.

###### pacs:

12.38.Mh, 11.10.Wx, 25.75.Dw^{†}

^{†}thanks: Author to whom all correspondence should be addressed. Email: ygma@sinap.ac.cn

## I Introduction

Results from the Brookhaven Relativistic Heavy-Ion Collider (RHIC) indicate that a strongly-interacting partonic matter has been created in relativistic nucleus-nucleus collisions RHIC_white_paper (). A powerful probe exposing the characteristics of new matter, elliptic flow, has been measured via the second Fourier coefficient () in the azimuthal distribution of final particles. It is translated from an early stage coordinate space asymmetry, which can reflect how the hot matter evolves hydrodynamically RHIC_white_paper (); Voloshin1 (); Jinhui4 (). The data show remarkable hydrodynamical behaviors, which implies the formed matter is thermalized in a very short time and expands collectively as a perfect-like liquid with a very small shear viscosity over entropy density ratio ) eta_s_L (); eta_s_R (); eta_s_H (); eta_s_E (). Elliptic flow () has been studied widely as functions of centrality, transverse momentum () and pseudorapidity () etc. A mass-ordering at low and a Number of Constituent Quark (NCQ) scaling at intermediate for have been observed, which suggests that a thermalized partonic matter is formed and a collective motion is developed prior to hadronization strangeBaryonmassorder5 (); STARMassOrder6 (); PHENIXNCQ7 (); PHENIXphiNCQ8 (); tianjianNCQ9 (). On the other hand, a geometry (participant eccentricity) scaling was observed for fluctuations, which implies not only participant eccentricity is responsible for elliptic flow, but also the event-by-event initial state geometry fluctuations contribute to harmonic flow Paul (); partie2v210 (); partie2v2fluc11 ().

It has been recently found that the triangular flow () is not zero in the azimuthal distribution of final particles. In fact, because of the non-smooth profile, coming from the event-by-event fluctuations of participant nucleons, it shows a triangular initial geometry shape can be transferred into momentum space by hydrodynamical evolution. In recent studies, it has been demonstrated that triangular flow significantly contributes on the near-side ridge and away-side double bumps in two-particle azimuthal correlations triangularflow12 (); kotriangularflow13 (). As a new probe, triangular flow is believed to provide more information about the formed hot and dense matter. It has been studied as functions of centrality, transverse momentum, pseudorapidity (), as well as the relations with the initial triangularity () and shear viscosity over entropy density ratio triangularflow12 (); kotriangularflow13 (); ampthotspots14 (); Petersenv3 (); Qinv3 (). However, the dependence of triangular flow on the elastic two-body partonic scattering cross section is absent. In addition, a possible NCQ-scaling, which has been found held by the elliptic flow v4NCQ160 (), have not been studied in details for other (=3,4…) when the initial fluctuations are taken into account .

This work presents the initial deformation scaling of elliptic (), triangular () and quadrangular flows () for different cross sections within the framework of the AMPT model AMPT2160 (); AMPT216 (). The mass ordering at low and constituent quark number scaling at higher for the are investigated after considering the event-by-event initial state geometry fluctuations at RHIC and LHC energies. Meanwhile, a special care is discussed for -quark and meson for -scaling.

The paper is organized in the following way. A brief description of the AMPT model is introduced in Sec. II. The results and discussions are presented in Sec. III. Finally, a summary is given in Sec. IV.

## Ii Brief description of AMPT model

A multi-phase transport (AMPT) model consists of four main components: the initial condition, partonic interactions, conversion from partonic to hadronic matter, and hadronic interactions. The initial condition, which includes the spatial and momentum distributions of minijet partons and soft string excitations, is obtained from the Heavy Ion Jet Interaction Generator (HIJING) model. Scatterings among partons are modeled by Zhang’s Parton Cascade (ZPC) model, which includes only two-body scatterings with cross sections obtained from the pQCD calculations with screening mass. In the default version of AMPT model, partons only include minijet partons, and recombine with their parent strings when they stop interactions, then the resulting strings are converted to hadrons by using the Lund string fragmentation mechanism. While in the version with the string melting mechanism, partons include minijet partons and partons from melted strings. And a quark coalescence model is used to combine partons into hadrons. The dynamics of the subsequent hadronic matter is then described by a relativistic transport (ART) model. Details of the AMPT model can be found in a review AMPT216 (). Previous AMPT calculations have found that elliptic flow can be built by strong parton cascade AMPT216 (); SAMPT17 (); AMPT_v2_e250 (); AMPTLHC () and jet losses energy into partonic medium to excite an away-side double-peak structure ampthotspots14 (); amptdihadron19 (); amptgammajet20 (). Therefore, partonic effect can not be neglected and the string melting AMPT version is much more appropriate than the default version when the energy density is much higher than the predicted critical density . In this work, we use the version of AMPT model with the string melting mechanism to simulate Au+Au collisions at = 200 GeV as well as Pb + Pb collisions at = 2.76 TeV. Since collective flow has been built up after the expansion of partonic stage, we neglect the final hadronic rescattering effects on harmonic flows in this work.

## Iii Results and Discussions

### iii.1 Brief definition of with initial fluctuations

We know harmonic flows are defined as the -th Fourier coefficient of the particle distribution with respect to the reaction plane. However, after considering event-by-event fluctuations in the initial density distribution triangularflow12 (), the particle distribution should be written as

(1) |

where is the momentum azimuthal angle of each hadron. is the -th event plane which varies due to event-by-event fluctuations and can be calculated by

(2) |

where and are the coordinate position and azimuthal angle of each parton and the average is density weighted in the initial state, and and the superscript denotes initial coordinate space. The -th order eccentricity for initial geometric distribution is defined as

(3) |

There is some arbitrariness in the definition of and Oll (), because one could, for instance, replace with in Eq. (2) and (3) Li (). With this replacement, however, and only change little in our calculations (less than 3% for and 12% for , respectively, for 0-80% centrality). In the following calculations, we will use Eq.(2) and Eq.(3) to decide and .

After is determined, the -th harmonic flow can be obtained by

(4) |

In alternative way, and can be also calculated in momentum space as,

(5) |

and

(6) |

where and are the transverse momentum and azimuthal angle of each hadron, respectively, which is selected from pseudorapidity 1 in the final state to avoid autocorrelation, and the superscript denotes final momentum space.

For determined by the final momentum phase space, we can even-by-event correct into by

(7) |

where the superscript denotes ”event-wise”, is event-wise event plane resolution, and is event-wise sin-term harmonic coefficient. We found that the contribution from -term is only approximately 10% , therefore we neglect the -term and correct into by , which is more operable experimentally.

It is essential to check if the calculated by different defined in coordinate space and momentum space is similar or not, because the determination of in coordinate space by the Eq.( 2) is not accessible in experiment. The dependences of and with respect to determined by initial coordinate and final momentum spaces are shown in Figure 1 together with the PHENIX data PHENIXDataPi (). We observed that and determined by the final momentum space is very close to the ones with respect to initial coordinate space. (Note: we check that the differences are due to term contributions in event-by-event resolution corrections.) Also, the values can basically fit the PHENIX data, especially at low .

Based upon the above observations on and with different phase space methods, we conclude that they basically can present the same results. Therefore in our following calculations, we apply the initial coordinate space to calculate and then obtain the corresponding .

### iii.2 Initial fluctuations and ratio of

The ratio of elliptic flow to eccentricity (/) has been found to be sensitive to the freeze-out dynamics, the equation of state (EOS) and viscosity, however, / can give more information triangularflow12 (); ebeqiu (); EOS21 (); hydr3+122 (); Dipole23 (); Alverhydro25 (). Figure 2 shows the initial -th order eccentricity and final harmonic flow ( = 2, 3 and 4) in mid-rapidity as a function of impact parameter for Au + Au collisions at = 200 GeV from the AMPT model simulations. The elastic two-body scattering cross section in the parton cascade process is set to be 3 mb.

As presented in Figure 2, the -th order eccentricity ( = 2, 3 and 4) increases with impact parameter. is larger than and , except in very central collisions where looks similar to each other. It is consistent with the trend given by Lacey et al. who applied MC-Glauber model, but gave a little smaller magnitude for peripheral collisions e2fluc24 (). On the other hand, the coefficients of anisotropic flow ( = 2, 3 and 4) show rising and falling with impact parameter. Also, has a larger magnitude for lower harmonic than higher harmonic.

Similarly, Figure 3 shows the initial geometry deformation ( = 2, 3 and 4) for Pb + Pb collisions at TeV, which demonstrates very similar behaviors as the RHIC energy.

Once we have and , we can discuss the ratio ( = 2, 3 and 4). Figure 4 shows impact parameter and partonic cross section dependences of for Au + Au collisions at GeV. The value of ratios decreases with impact parameter, which implies that the conversion from the initial geometry asymmetry to final momentum anisotropy is less efficient for peripheral collisions than for central collisions. And for higher harmonics, there is also less conversion efficiency. The trend for as a function of impact parameter looks similar for the different partonic interaction cross sections of 3, 6 and 10 mb. However, the magnitude of decreases with the cross section, which reveals that the conversion from the initial geometry asymmetry to the final momentum anisotropy becomes weaker for a smaller cross section. This indicates that frequent parton-parton collisions help the system to develop the harmonic collectivity.

From Figure 4, we also saw that the becomes smaller for higher harmonic order, this may reflect the viscous damping. Recently, it was claimed that the relative magnitude of the higher-order harmonics (,) can provide additional constraints on both the magnitude of and the determination of initial condition Shuryak (); e2fluc24 (); Alverhydro25 (). Fig. 5 shows the -dependence of / for Au + Au collisions at GeV and Pb + Pb collisions at TeV for 0-20 centrality and low region ( 0.55 GeV/c) (3 mb) with corresponding exponential fitting functions. Compared with PHENIX data , only the trend can be reproduced.

### iii.3 dependence of with different partonic cross sections and comparisons with the data

Figure 6 presents our simulations of , and as a function of with different parton interaction cross sections together with the PHENIX data PHENIXPIData (). For triangular flow, it totally arises from the event-by-event fluctuations of the initial collision geometry, because it persists zero if without considering the fluctuations. ( = 2, 3 and 4) decreases when parton-parton cross section decreases. Experimental data of can be described by the large cross sections (from 3 mb to 10 mb), after one considers of the initial fluctuations. However, the AMPT model underestimates the data if without taking the initial fluctuations into account. Recently, Xu and Ko adjusted more parameters in the AMPT model, which include not only parton interaction cross section but also the parametrization of the Lund string fragmentation, and found that a smaller cross section of 1.5 mb is good to describe both the charged particle multiplicity and elliptic flow JunV3 (). In our work, we will not focus on how to further improve parameters, but we do find that initial geometry fluctuations significantly affect harmonic flows and should not be ignored.

The transverse momentum dependences of and with different cross sections in four different centrality bins are shown in Figure 7 and 8. The PHENIX data is also accompanied PHENIXData (). For each centrality bin, and increase with the cross section. For elliptic flow (Fig. 7), data seem to prefer a bigger cross section in higher transverse momentum range. In the case of triangular flow similar trend is present in Fig. 8, though shows a less centrality dependence than , which is consistent with the trends shown in Figure 2.

The transverse momentum dependences of and are also calculated for four different centrality bins in Pb + Pb collisions at 2.76 TeV for LHC energy, which are shown in Figure 9 and 10 together with the ATLAS data JYJiaLHC (). Similarly, and increase with the cross section from 3 mb to 10 mb, which can basically describe the ATLAS data.

Even though a general behavior of -dependent can be nicely demonstrated by the comparison of our calculations with the data, we found that the AMPT simulations can only describe the trend of the data qualitatively. Actually, AMPT can not describe the and data simultaneously with a same cross section. For example, the 3mb results describe the and for 0-10% centrality but underpredict other centralities at RHIC energy. The 10mb data describe the for 10-20% centrality but are unable to reproduce for the same centrality. Also, the 10mb results describe the and for 30-40%, but underpredict the high data for 50-60% centrality. For LHC data, no AMPT calculations can describe the and for the 0-10% centrality range.

### iii.4 NCQ-scaling of at RHIC energy

For elliptic flow, a mass ordering (the heavier the hadron mass, the smaller the ) and a NCQ-scaling (baryon versus meson) have been observed at low and intermediate , respectively, in Au + Au collisions at = 200 GeV STARmassord260 (). The observed NCQ-scaling (/ vs /) reveals a universal scaling of for all identified particles over the full transverse kinetic energy () range, which is more pronounced rather than strangeBaryonmassorder5 (); STARMassOrder6 (); PHENIXNCQ7 (); PHENIXphiNCQ8 (); tianjianNCQ9 (); STAR08v2NCQ26 (); Jinhui4 (). Such scaling indicates that the collective elliptic flow has been developed during the partonic stage and the effective constituent quark degree of freedom plays an important role in hadronization process. For higher even-order harmonics, and etc, appear to be scaled as STARData () and their NCQ-scaling has also been suggested in Ref. VoloshinvnNCQ29 (). Even in very low energy heavy ion collisions, the -scaling and the -scaling have been suggested for light nuclear clusters in nucleonic level interaction Yan (). Instead scaling by the number of constituent quarks () for , the measured data , however, seems to be scaled by v4NCQ160 (). It is interesting to check if these scaling relations are still valid for the calculations in which the initial fluctuations are taken into account, including the odd harmonics, such as .

Figure 11 presents , and of different types of hadrons in mid-rapidity for Au + Au collisions (0-80% centrality) at GeV. Figure 11 (a) shows that preserves an obvious mass ordering in relatively low region, and hadron type grouping in intermediate region, even after considering event-by-event fluctuations. Similarly, and , [Figure 11 (b) and (c)] also present a mass ordering in the low region. The study on of different hadron species will give more information about the initial geometry and the viscosity of hot and dense matter Alverhydro25 ().

As shown in Figure 12(a), scaled by the number of constituent quarks () as a function of the transverse kinetic energy () scaled by the number of constituent quarks () shows a universal scaling regardless of the initial geometry fluctuations are taken into consideration or not. The only difference is that the initial fluctuations enhance the value of . Therefore, the initial fluctuations have little effect on the breaking of the NCQ-scaling for elliptic flow. Figure 12 (b) and (c) display and for all hadrons as a function of , respectively, when the initial fluctuations are considered. Form the above results, it seems that can still be roughly scaled by for all hadrons as a function of . Of course, the scaling behavior is not perfect within the present statistics. For example, the amount of spread between different particle species is less than 10% for the -scaling, it is less than 20% for the -scaling, but it can reach 20-30% for the -scaling.

In order to understand possible origin of the NCQ-scaling of for different mesons and baryons, we also check of , and -quarks as a function of or . As expected, there exists similar NCQ-scaling of (=2-4) for all those constituent quarks. Furthermore, we find that the values of of different hadrons are similar to the values of of -quarks, which reflects that the NCQ-scaling of for different hadrons stems from partonic level.

Furthermore, the ratios of and as functions of for three different centrality bins (10-20%, 20-30%, and 30-40%) in Au + Au collisions at GeV are shown in Figure 13. It shows that the both ratios of [Figure 13 (a)] and [Figure 13 (b)] exhibit centrality dependences, i.e. more central collisions result in more larger ratios. However, it is almost independent of for each centrality bin, which is consistent with the scaling of observed in Figure 12. However, it is difficult to obtain the relationship between and , such as between and , in terms of a simple coalescence model in Ref. Coalescence (), because is purely determined by initial geometry fluctuations which is independent of . It is interesting that recently Lacey et al. linked such a scaling to the acoustic nature of anisotropic flow to constrain initial conditions, and viscous horizon NCQ_R (). It further gives more insights on the dynamics of strongly-interacting partonic matter and constituent quark degree of freedom in hadronization process.

### iii.5 NCQ-scaling of at LHC energy

At the same time, the mass ordering in the low region and the NCQ-scaling in intermediate region of are also investigated at LHC energy in AMPT simulations. Fig. 14 presents the results of , , and for different hadron species in Pb + Pb collisions (0-80% centrality) at TeV in mid-rapidity from the AMPT calculations (3 mb), with considering initial fluctuations. It displays that the mass ordering is satisfied, i.e. decreases from , , , to in the lower region (note that is very close to in the figure. However, the strict mass-ordering needs ’s is little less than ’s ). The baryon-meson typing is also evident above 1.2 GeV. By transformation of to as well as to , the results of , and as a function of are shown in Fig. 15. Again, the NCQ-scaling of is roughly kept except for meson whose is a little larger. Of course, the amount of spread between different particle species for -scaling keeps similar as RHIC energy. Comparing with the above results at RHIC, LHC results are very similar but they reveal larger values than RHIC’s due to stronger partonic interactions at higher energy. But in general, the partonic matter formed at LHC energy is very similar to that created at RHIC energy.

As mentioned above, -meson shows a little larger value of as compared to those of other hadrons in Fig. 15. Keep in mind that -meson is always a very interesting hadron in previous studies since its mass is close to proton but it is a multi-strange meson Ma-phi (); Jinhui4 (); Chen-PRL (); Beganda (); Chen-phialign (); MaGL-phi (). This could be understood from the parton’s in the same condition: of -quark displays a slight deviation from the ()-quarks (not shown here). The reason could be that the of heavier strange quarks has a smaller value at low but a larger value at high , i.e. the mass ordering of partonic flow. However, a larger collective radial flow at LHC energy could push heavier -quark to have stronger . The effect is of course more distinct at LHC energy because of larger initial partonic pressure. In contrast, in low energy RHIC run, such as 11.5 GeV/c Au + Au collision, -quark may not reach full thermalization and therefore result in a less of -quark as compared to ()-quarks, which can lead to a smaller of , i.e. the violation of the -scaling for the -mesons relative to other hadrons as observed in the STAR data Beganda () as well as in a simulation tianjianNCQ9 (). Considering that the -meson is coalesced by in the present AMPT model calculation, it will certainly induce a larger for in comparison with other hadrons, as shown in Fig. 15. Unfortunately, the data of ’s is not available yet at LHC energy, which is worth waiting for checking.

Before closing the discussions on the NCQ-scaling of in this subsection, we remind that the hadronic rescattering process is not yet taken into account in our calculation. Recently, the ALICE data shows that proton’s and seem to deviate from the NCQ-scaling of of charged and Krzewicki (). The reason could be stronger final-state interaction for protons. Detailed model investigations are underway.

## Iv Summary

Within the framework of a multi-phase transport model, we investigated the different orders of harmonic flows, namely elliptic flow, triangular flow and quadrangular flow for Au + Au collisions at GeV as well as Pb + Pb collisions at TeV when the initial geometry fluctuations are taken into account. Basically, the harmonic flow is converted from initial geometry shape via parton cascade process, and its conversion efficiency () decreases with the increasing of harmonic order as well as the decreasing of the partonic cross section at both RHIC and LHC energies. Dependences of transverse momentum, centrality and partonic cross section of the (n=2, 3 and 4) have been studied and compared with data. For each centrality bin, and increases with cross section, especially at higher transverse momentum.

Triangular and quadrangular flows also roughly present a mass ordering in low region and the number of constitute quark scaling in intermediate region, similar to the behaviors of elliptic flow. Form our results, a NCQ-scaling of versus for different hadrons holds for harmonic flow (, = 2, 3 and 4), which can be related to -scaling in partonic level. From all above results, it implies that the formed partonic matter should be very similar for RHIC and LHC energies.

## Acknowledgements

The authors thank Dr. B. Johnson, and Dr. S. Esumi for providing data kindly. The authors also thank Dr. M. Luzum, Dr. J. Y. Ollitrault, Dr. Y. Zhou and Dr. R. Lacey for helpful discussion. This work was supported in part by the National Natural Science Foundation of China under contract Nos. 11035009, 11005140, 11175232, 10979074, 10875159 and the Knowledge Innovation Project of the Chinese Academy of Sciences under Grant No. KJCX2-EW-N01, and the Project-sponsored by SRF for ROCS, SEM.

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