A study of <FONT FACE=Symbol>á</FONT>v2<FONT FACE=Symbol>ñ</FONT> with NeXSPheRIO

Abstract

Elliptic flow at RHIC is computed event by event with NeXSPheRIO. Reasonable agreement with experimental data on v2(eta) and v2(pt) is obtained. Various effects are studied as well: equation of state (with or without critical point), emission mechanism (Cooper-Frye prescription or continuous emission), type of the initial conditions (average or fluctuating initial conditions).

Elliptic flow; Equation of state; Continuous emission; Hydrodynamics

A study of áv₂ñ with NeXSPheRIO

R. P. G. Andrade^I; Y. Hama^I; F. Grassi^I; O. Socolowski Jr.^II; T. Kodama^III

^IInstituto de Física, Universidade de São Paulo, C.P. 66318, 05315-970, São Paulo-SP, Brazil

^IICTA/ITA, Praça Marechal Eduardo Gomes 50, CEP 12228-900, São José dos Campos-SP, Brazil

^IIIInstituto de Física, Universidade Federal do Rio de Janeiro, C.P. 68528, 21945-970, Rio de Janeiro-RJ, Brazil

ABSTRACT

Elliptic flow at RHIC is computed event by event with NeXSPheRIO. Reasonable agreement with experimental data on v₂(h) and v₂(pt) is obtained. Various effects are studied as well: equation of state (with or without critical point), emission mechanism (Cooper-Frye prescription or continuous emission), type of the initial conditions (average or fluctuating initial conditions).

Keywords: Elliptic flow; Equation of state; Continuous emission; Hydrodynamics

I. INTRODUCTION

The elliptic flow parameter v₂ is defined as the second Fourier coefficient of the azimuthal distribution of particles dN/df. The average value of v₂, over N_ev events, is given by

Here, N_j is the particle number of the j-th event and F_b is the angle between the impact parameter and the Ox axis[8]. The index b indicates that the elliptic flow is calculated with respect to the impact parameter. We can understand as a measure of the stretch of ádN/dfñ in the direction of . In order to compute dN/df, in each event, we use the NeXSPheRIO code.

NeXSPheRIO is the tool which we use to do hydrodynamical calculations. It is a junction of two codes: Nexus+SPheRIO.

The Nexus code is used to compute the initial conditions T^µn, j^m and u^µ on a proper time hypersurface [1]. In figure 1 we show an example of initial condition for one random event (in this case we show the initial energy density at mid-rapidity plane).

The SPheRIO code is used to compute the hydrodynamical evolution. It is based on Smoothed Particle Hydrodynamics, a method originally developed in astrophysics and adapted to relativistic heavy ion collisions [2]. Its main advantage is that any geometry in the initial conditions can be incorporated.

NeXSPheRIO is run many times, corresponding to many different events or initial conditions. At the end, an average over final results is performed. Such process mimics the experimental conditions. This is different from the canonical approach in hydrodynamics where initial conditions are adjusted to reproduce same selected data and are very smooth (see figure 2).

Summarizing, we can compute elliptic flow from fluctuating initial conditions (event by event) or from average initial conditions.

II. RESULTS

A. centrality and h dependence of

In Fig. 3 we show , as a function of h, in three centrality windows. We have checked that reproduces the characteristic shape of the experimental data, with a maximum at h = 0 and decreasing as |h| increases. Moreover, decreases as the centrality increases.

B. Effect of the continuous emission on

In Fig. 4 we compare results obtained from Cooper-Frye prescription [3] and from continuous emission [6]. We have checked that decreases when we use the continuous emission mechanism. Indeed, in this mechanism, some particles are emited earlier and these particles present a more isotropic distribution as a function of azimuthal angle.

In the case of , as a function of p_t (figure 5), we observe a considerable reduction of the elliptic flow in the region of high p_t. However, such reduction does not characterize a saturation (observed in experimental data for p_t > 1.5 GeV).

C. Effect of the EoS with critical point on

In Figs. 6 (T_f= 150MeV) and 7 (T_f = 140MeV) we show the effect of the EoS with critical point on . We observe that is greater when we use the EoS with critical point [7]. Indeed, we expect a larger elliptic flow for the cross over region, since the matter is always accelerated in that region. Note that the effect is better observed when we use T_f = 140MeV, i.e., when the time of expansion of the fluid is increased.

D. Effect of the type of initial conditions on

Finally, in Fig. 8 we compare computed event by event (solid line) and computed from average initial conditions (dashed line) [4]. We observe that the latter is greater and presents three peaks. The central peak is related to the central rapidity region where the temperature is very high. The left and right peaks are probably related, to the participant nucleons after the collision.

It seems that smooth initial conditions favor longer expansion as compared to fluctuating initial conditions.

III. CONCLUSION

In this work, we calculated the elliptic flow parameter v₂, as a function of the pseudorapidity h, the transverse momentum p_t and the centrality of the collision. We found that v₂, as a function of h, reproduces the characteristic shape of the experimental data, with a maximum at h = 0 and decreasing as |h| increases. We also observed that v₂ increases linearly as a function of p_t. It does not show the saturation observed in experimental data, for p_t > 1.5 GeV (when we use the Cooper-Frye prescription). Using the continuous emission mechanism, we observed a considerable reduction of the elliptic flow in the region of high p_t. However, such a reduction does not characterize a saturation in the present computation, probably because of a two rough approximation used for continuous emission [4]. In the case of v₂ as a function of centrality, the results are consistent with the experimental data. We also verified that the effect of a equation of state with critical point is of little importance to the elliptic flow. On the other side, we found a strong dependence of v₂ on the type of initial conditions used (average or fluctuating initials conditions).

We acknowledge financial support by FAPESP (04/10619-9, 04/15560-2, 04/13309-0), CAPES/PROBRAL, CNPq, FAPERJ and PRONEX.

[1] H. J. Drescher, S. Ostapchenko, T. Pierog, and K. Werner, Phys. Rev. C 65, 054902 (2002).

[2] C. E. Aguiar, T. Kodama, T. Osada, and Y. Hama, J. Phys. G 27, 75 (2001).

[3] F. Cooper and G. Frye, Phys. Rev. D 10, 186 (1974).

[4] Y. Hama, T. Kodama, and O. Socolowski Jr., Braz. J. Phys. 35, 24 (2005).

[5] B. B. Back et al., Phys. Rev. C 72, 051901 (2005).

[6] F. Grassi, Y. Hama, and T. Kodama, Phys. Lett. B 355, 9 (1995); Z. Phys. C 73, 153 (1996).

[7] Y. Hama, R. P. G. Andrade, F. Grassi, O. Socolowski Jr., T. Kodama, B. Tavares, S. S. Padula, Nucl. Phys. A 774, 169 (2006).

[8] In this case the z axis is parallel to beam axis.

Received on 29 September, 2006

Publication Dates

• Publication in this collection
  24 Apr 2007
• Date of issue
  Mar 2007

History

• Received
  29 Sept 2006