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Elliptic Flow and Its Fluctuations from Transport Models for Au+Au Collisions at $\sqrt{{s}_{NN}}$ = 7.7 and 11.5 GeV

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## Abstract

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## 1. Introduction

## 2. Monte Carlo Models

## 3. Methods of Elliptic Flow Analysis

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Voloshin, S.; Zhang, Y. Flow study in relativistic nuclear collisions by Fourier expansion of Azimuthal particle distributions. Z. Phys. C
**1996**, 70, 665–672. [Google Scholar] [CrossRef][Green Version] - Poskanzer, A.M.; Voloshin, S.A. Methods for analyzing anisotropic flow in relativistic nuclear collisions. Phys. Rev. C
**1998**, 58, 1671. [Google Scholar] [CrossRef][Green Version] - Shuryak, E.V. What RHIC experiments and theory tell us about properties of quark-gluon plasma? Nucl. Phys. A
**2005**, 750, 64–83. [Google Scholar] [CrossRef][Green Version] - Bernhard, J.E.; Moreland, J.S.; Bass, S.A. Bayesian estimation of the specific shear and bulk viscosity of quark-gluon plasma. Nat. Phys.
**2019**, 15, 1113–1117. [Google Scholar] [CrossRef] - Alver, B.; Back, B.B.; Baker, M.D.; Ballintijn, M.; Barton, D.S.; Betts, R.R.; Bindel, R.; Busza, W.; Chetluru, V.; Garcia, E.; et al. Importance of correlations and fluctuations on the initial source eccentricity in high-energy nucleus-nucleus collisions. Phys. Rev. C
**2008**, 77, 014906. [Google Scholar] [CrossRef] - Voloshin, S.A.; Poskanzer, A.M.; Tang, A.; Wang, G. Elliptic flow in the Gaussian model of eccentricity fluctuations. Phys. Lett. B
**2008**, 659, 537–541. [Google Scholar] [CrossRef][Green Version] - Ollitrault, J.Y.; Poskanzer, A.M.; Voloshin, S.A. Effect of flow fluctuations and nonflow on elliptic flow methods. Phys. Rev. C
**2009**, 80, 014904. [Google Scholar] [CrossRef][Green Version] - Borghini, N.; Dinh, P.M.; Ollitrault, J.Y. A New method for measuring azimuthal distributions in nucleus-nucleus collisions. Phys. Rev. C
**2001**, 63, 054906. [Google Scholar] [CrossRef][Green Version] - Borghini, N.; Dinh, P.M.; Ollitrault, J.Y. Flow analysis from multiparticle azimuthal correlations. Phys. Rev. C
**2001**, 64, 054901. [Google Scholar] [CrossRef][Green Version] - Voloshin, S.A.; Poskanzer, A.M.; Snellings, R. Collective phenomena in non-central nuclear collisions. In Landolt-Bornstein; Springer: Berlin/Heidelberg, Germany, 2010; Volume 23, p. 293. [Google Scholar] [CrossRef]
- Bhalerao, R.S.; Borghini, N.; Ollitrault, J.Y. Analysis of anisotropic flow with Lee–Yang zeroes. Nucl. Phys. A
**2003**, 727, 373–426. [Google Scholar] [CrossRef][Green Version] - Borghini, N.; Bhalerao, R.S.; Ollitrault, J.Y. Anisotropic flow from Lee–Yang zeroes: A Practical guide. J. Phys. G
**2004**, 30, S1213–S1216. [Google Scholar] [CrossRef][Green Version] - Adamczyk, L.; Agakishiev, G.; Aggarwal, M.M.; Ahammed, Z.; Alakhverdyants, A.V.; Alekseev, I.; Alford, J.; Anderson, B.D.; Anson, C.D.; Arkhipkin, D.; et al. Inclusive charged hadron elliptic flow in Au+Au collisions at $\sqrt{{s}_{NN}}$ = 7.7–39 GeV. Phys. Rev. C
**2012**, 86, 054908. [Google Scholar] [CrossRef] - Adamczyk, L.; Adkins, J.K.; Agakishiev, G.; Aggarwal, M.M.; Ahammed, Z.; Alekseev, I.; Alford, J.; Anson, C.D.; Aparin, A.; Arkhipkin, D.; et al. Elliptic flow of identified hadrons in Au+Au collisions at $\sqrt{{s}_{NN}}$ = 7.7–62.4 GeV. Phys. Rev. C
**2013**, 88, 014902. [Google Scholar] [CrossRef] - Golovatyuk, V.; Kekelidze, V.; Kolesnikov, V.; Rogachevsky, O.; Sorin, A. The Multi-Purpose Detector (MPD) of the collider experiment. Eur. Phys. J. A
**2016**, 52, 212. [Google Scholar] [CrossRef] - Abgaryan, V.; Kado, R.A.; Afanasyev, S.V.; Agakishiev, G.N.; Alpatov, E.; Altsybeev, G.; Hernandez, M.A.; Andreeva, S.V.; Andreeva, T.V.; Andronov, E.V.; et al. Status and Initial Physics Performance Studies of the MPD Experiment at NICA. arXiv
**2022**, arXiv:2202.08970. [Google Scholar] [CrossRef] - Kekelidze, V.D. Heavy Ion Collisions: Baryon Density Frontier. Phys. Part. Nucl.
**2018**, 49, 457. [Google Scholar] [CrossRef] - Bleicher, M.; Zabrodin, E.; Spieles, C.; Bass, S.A.; Ernst, C.; Soff, S.; Bravina, L.; Belkacem, M.; Weber, H.; Stoecker, H.; et al. Relativistic hadron hadron collisions in the ultrarelativistic quantum molecular dynamics model. J. Phys. G
**1999**, 25, 1859–1896. [Google Scholar] [CrossRef][Green Version] - Bass, S.A.; Belkacem, M.; Bleicher, M.; Brandstetter, M.; Bravina, L.; Ernst, C.; Gerland, L.; Hofmann, M.; Hofmann, S.; Konopka, J.; et al. Microscopic models for ultrarelativistic heavy ion collisions. Prog. Part. Nucl. Phys.
**1998**, 41, 255–369. [Google Scholar] [CrossRef][Green Version] - Petersen, H.; Li, Q.; Zhu, X.; Bleicher, M. Directed and elliptic flow in heavy ion collisions at GSI-FAIR and CERN-SPS. Phys. Rev. C
**2006**, 74, 064908. [Google Scholar] [CrossRef] - Petersen, H.; Bleicher, M. Ideal hydrodynamics and elliptic flow at SPS energies: Importance of the initial conditions. Phys. Rev. C
**2009**, 79, 054904. [Google Scholar] [CrossRef][Green Version] - Zhu, X.l.; Bleicher, M.; Stoecker, H. Elliptic flow analysis at RHIC: Fluctuations vs. non-flow effects. Phys. Rev. C
**2005**, 72, 064911. [Google Scholar] [CrossRef][Green Version] - Weil, J.; Steinberg, V.; Staudenmaier, J.; Pang, L.G.; Oliinychenko, D.; Mohs, J.; Kretz, M.; Kehrenberg, T.; Goldschmidt, A.; Bäuchle, B.; et al. Particle production and equilibrium properties within a new hadron transport approach for heavy-ion collisions. Phys. Rev. C
**2016**, 94, 054905. [Google Scholar] [CrossRef][Green Version] - Lin, Z.W.; Ko, C.M.; Li, B.A.; Zhang, B.; Pal, S. A Multi-phase transport model for relativistic heavy ion collisions. Phys. Rev. C
**2005**, 72, 064901. [Google Scholar] [CrossRef][Green Version] - Karpenko, I.; Huovinen, P.; Bleicher, M. A 3+1 dimensional viscous hydrodynamic code for relativistic heavy ion collisions. Comput. Phys. Commun.
**2014**, 185, 3016–3027. [Google Scholar] [CrossRef][Green Version] - Karpenko, I.A.; Huovinen, P.; Petersen, H.; Bleicher, M. Estimation of the shear viscosity at finite net-baryon density from A+A collision data at $\sqrt{{s}_{NN}}$ = 7.7–200 GeV. Phys. Rev. C
**2015**, 91, 064901. [Google Scholar] [CrossRef][Green Version] - Loizides, C.; Nagle, J.; Steinberg, P. Improved version of the PHOBOS Glauber Monte Carlo. SoftwareX
**2015**, 1–2, 13–18. [Google Scholar] [CrossRef][Green Version] - Parfenov, P.; Idrisov, D.; Luong, V.B.; Geraksiev, N.; Truttse, A.; Demanov, A. Anisotropic Flow Measurements of Identified Hadrons with MPD Detector at NICA. Particles
**2021**, 4, 146–158. [Google Scholar] [CrossRef] - Bilandzic, A.; Snellings, R.; Voloshin, S. Flow analysis with cumulants: Direct calculations. Phys. Rev. C
**2011**, 83, 044913. [Google Scholar] [CrossRef][Green Version] - Giacalone, G.; Noronha-Hostler, J.; Ollitrault, J.-Y. Relative Flow Fluctuations as a Probe of Initial State Fluctuations. Phys. Rev. C
**2017**, 95, 054910. [Google Scholar] [CrossRef] - Giacalone, G.; Yan, L.; Noronha-Hostler, J.; Ollitrault, J.-Y. Skewness of Elliptic Flow Fluctuations. Phys. Rev. C
**2017**, 95, 014913. [Google Scholar] [CrossRef]

**Figure 1.**Squared modulus of the second harmonic product generating function $|{G}^{\theta}{\left(\mathrm{i}r\right)|}^{2}$ of Lee–Yang zeros method as a function of r for $\theta =\pi /5$ for four centrality classes from Au+Au collisions at $\sqrt{{s}_{NN}}$ = 7.7 (

**a**) and 11.5 GeV (

**b**) using AMPT SM model.

**Figure 2.**${p}_{T}$-dependence of ${v}_{2}$ of inclusive charged hadrons from 20–30% centrality Au+Au collisions at $\sqrt{{s}_{NN}}$ = 7.7 GeV (upper row) and 11.5 GeV (lower row) obtained using ${v}_{2}\left\{{\mathrm{\Psi}}_{2,\mathrm{TPC}}\right\}$ (

**a**,

**d**), ${v}_{2}\left\{2\right\}$ (

**b**,

**e**), and ${v}_{2}\left\{4\right\}$ (

**c**,

**f**) methods of flow measurements in comparison with STAR data [13].

**Figure 3.**Centrality dependence of the ${v}_{2}\left\{4\right\}/{v}_{2}\left\{2\right\}$ ratio of charged hadrons from Au+Au collisions at $\sqrt{{s}_{NN}}=$ 39–11.5 GeV (

**a**) measured in STAR [13], and its comparison with vHLLE+UrQMD, AMPT SM, UrQMD and SMASH models at $\sqrt{{s}_{NN}}=$ 7.7–39 GeV (

**b**–

**f**).

**Figure 4.**Centrality dependence of ${v}_{2}$ of inclusive charged hadrons from Au+Au collisions at $\sqrt{{s}_{NN}}$ = 7.7 and 11.5 GeV measured by different methods. Panels (

**a**–

**e**) correspond to different models. Panels (

**f**–

**j**) show the ${v}_{2}\left\{\mathrm{method}\right\}/{v}_{2}\left\{2\right\}$ ratio.

**Figure 5.**${p}_{T}$-dependence of ${v}_{2}$ of inclusive charged hadrons in 10–40% central Au+Au collisions at $\sqrt{{s}_{NN}}$ = 7.7 and 11.5 GeV measured by different methods. Panels (

**a**–

**e**) correspond to different models. Panels (

**f**–

**j**) show the ${v}_{2}\left\{\mathrm{method}\right\}/{v}_{2}\left\{2\right\}$ ratio.

**Figure 6.**Centrality and ${p}_{T}$ dependences of relative elliptic flow fluctuations ${v}_{2}\left\{4\right\}/{v}_{2}\left\{2\right\}$ of identified hadrons from Au+Au collisions at $\sqrt{{s}_{NN}}$ = 7.7 GeV (

**a**–

**d**) and 11.5 GeV (

**e**–

**h**). Columns correspond to different used models.

**Figure 7.**Centrality dependence of ${v}_{2}$ of inclusive charged hadrons from Au+Au collisions at $\sqrt{{s}_{NN}}=$ 7.7 and 11.5 GeV obtained using the high-order Q-cumulants ${v}_{2}\left\{4\right\}$, ${v}_{2}\left\{6\right\}$ methods in different models (

**a**–

**e**). Panels (

**f**–

**j**) show the ratio ${v}_{2}\left\{6\right\}/{v}_{2}\left\{4\right\}$.

**Figure 8.**Comparison of ${v}_{2}\left({p}_{T}\right)$ for pions and protons in 10–40% mid-central Au+Au collisions at $\sqrt{{s}_{NN}}=$ 7.7 GeV and $\sqrt{{s}_{NN}}=$ 11.5 GeV obtained by four-particle cumulants (

**a**,

**f**), two-particle cumulants (

**b**,

**g**), TPC event plane (

**c**,

**h**), TPC scalar product (

**d**,

**i**), FHCal event plane (

**e**,

**j**) methods of fully reconstructed (“reco”) and generated UrQMD events (“true”).

**Figure 9.**Comparison of ${v}_{2}\left({p}_{T}\right)$ of cases with uniform acceptance (open markers) and with a “hole” in the TPC azimuthal acceptance (closed markers) for charged hadrons in 10–40% mid-central Au+Au collisions at $\sqrt{{s}_{NN}}=$ 7.7 GeV obtained by four-particle Q-cumulants (

**a**), two-particle Q-cumulants (

**b**), TPC (

**c**) and FHCal (

**d**) event planes. Lower panels (

**e**–

**h**) show the ratio of the non-uniform case to the uniform one.

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**MDPI and ACS Style**

Luong, V.B.; Idrisov, D.; Parfenov, P.; Taranenko, A.
Elliptic Flow and Its Fluctuations from Transport Models for Au+Au Collisions at *Particles* **2023**, *6*, 17-29.
https://doi.org/10.3390/particles6010002

**AMA Style**

Luong VB, Idrisov D, Parfenov P, Taranenko A.
Elliptic Flow and Its Fluctuations from Transport Models for Au+Au Collisions at *Particles*. 2023; 6(1):17-29.
https://doi.org/10.3390/particles6010002

**Chicago/Turabian Style**

Luong, Vinh Ba, Dim Idrisov, Petr Parfenov, and Arkadiy Taranenko.
2023. "Elliptic Flow and Its Fluctuations from Transport Models for Au+Au Collisions at *Particles* 6, no. 1: 17-29.
https://doi.org/10.3390/particles6010002