Comparison between classification using impact parameter and using number of participants in relativistic nuclear collisions

Comparison between classification using impact parameter and using number of participants in relativistic nuclear collisions

C.E. Aguiar^I; R. Andrade^II; F. Grassi^II; Y. Hama^II; T. Kodama^I; T.Osada^III; O. Socolowski Jr.^II

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

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

^IIIUniversidade Ochnomizu, Tóquio, Japão

ABSTRACT

Using the hydrodynamical code NeXSPheRIO, we compare predictions as usually done in hydrodynamics, using centrality windows defined through the impact parameter, and as obtainable experimentally, using windows in participant number.

1 Introduction

Relativistic heavy-ion collisions as performed at the AGS, SPS and RHIC, can be classified as central (nuclei overlap), peripheral (nuclei pass grazing each other) or semi-peripherical. Quantitatively, one can classify these collisions using the impact parameter b: zero for central collisions, sum of the radii for peripheral, etc. More precisely, one can define a centrality window as incorporating the n % most central collisions i.e. all b's solution of pb²/s_inel = n %). This criteria to define centrality window is purely geometrical.

When the impact parameter is chosen, one can determine the initial conditions and run a hydrodynamical code to obtain predictions for such quantities as transverse momentum and rapidity spectrum, particle abundances, elliptic flow, etc, for various windows in impact parameter.

Experimentally, however, one does not have access to direct measurement of the impact parameter. So experimental results are presented in windows of energy deposited in a zero-degree calorimeter, number of participants, multiplicity, etc. One expects that the n % most central collisions in term of impact parameter are also the n % collisions with lower energy in the zero-degree calorimeter, higher number of participants and higher multiplicity.

However fluctuations are expected, for example the same value for the impact parameter may lead to somewhat different values of the number of participants due to the probabilistic nature of nucleon-nucleon collisions. The aim of this paper is to compare predictions using a classification of events using the impact parameter, the standard approach in hydrodynamics, and using the number of participants, as obtained experimentally. We also confront these two approaches to experimental data obtained at SPS by the NA49 collaboration, to turn the comparison more realistic.

All predictions are made using the NexSPheRIO code. The initial conditions are given by the NeXus code[1] for nuclear collisions and so fluctuate from event to event. These initial conditions are then used as input for SPheRIO, a 3+1 hydrodynamical code[2] based on a technique called Smoothed Particle Hydrodynamics.

2 Determination of the centrality windows

Using NeXus, 800 Pb+Pb collisions at beam energy of 158 GeV A were simulated. In Fig. 1a, we show the resulting distribution of participating nucleons. Events were binned in six classes that contain 5%, 9%, 9%, 8%, 17% and 52% of the total number of collisions (going from the most central to peripheral), respectively. We explain later why we made this particular choice of classes.

As indicated in the previous section, a fixed value of b can lead to various values of N_part and vice versa. In Fig. 1b, we show for each of our 800 events, its precise value of b and N_part. The horizontal lines are the windows in term of N_part as defined above and in Fig. 1a. The vertical lines are the windows in term of the impact parameter. They are defined by solving in term of b the equation pb²/s_inel = 0–5%, 5–14%, 14–23%,23–31%,31–48%, 48–100% with s_inel = p(2R)² and R the nuclear radius (this neglects boundary effects). In Fig. 1b, we see that the rectangles delimited by a continuous line include events that belong to a same window, independently of the classification used, N_part or b. However if we used a N_part classification for e.g. window 1, we add to the rectangle (1,1), the events in the (2,1) rectangle while if use the b classification, we would add instead the events in the (1,2) rectangle. In this case, the number of events in (1,2) or (2,1) is small. However if we do the same for rectangle (3,3), this is no more the case. As a consequence, the average number of participants is modified when going from one classification to the other. The average number of participants for each window and each classification is shown in the table below. In the next section, we explore consequences of this.

We now explain why we made our particular choice of centrality windows. To classify our collisions in a realistic way, we use data obtained by the NA49 collaboration, shown in Fig. 2. In this case, the amount of energy deposited in a zero-degree calorimeter was used to select events with a given centrality[3].

In an ideal situation, we expect that spectator nucleons free-stream towards the zero-degree calorimeter and deposit their energy there. Then the number of spectators would be the ratio between the deposited energy E_ZDC by the beam energy for the nucleon-nucleon collision (158 GeV in our case) and the number of participating nucleons would N_part = A – E_ZDC/158. In practice, this is an approximation because there may be other collisions than Pb+Pb depositing energy (such as Pb+air occurring because the zero-degree calorimeter is very far from the target and which can be corrected for to a certain extent), the beam energy fluctuates, etc. However as a guideline, we may consider that the 5 % collisions with lower E_ZDC are also the 5 % collisions with higher number of participants, and so on. With this assumption, the NA49 classification which bins the events in Fig. 2 in six classes that contain 5%, 9%, 9%, 8%, 17% and 52% of the total distribution of E_ZDC would be the same as our classification of events in six classes of N_part. In the table below, for each NA49 window in E_ZDC, the mean number of participants is shown. This number of participating nucleons was estimated by NA49 in a way independent of the E_ZDC measurement, by integrating rapidity distributions. As a result, it may be wiser not to compare the < N_part > of our two classifications, with that of NA49.

3 Particle distributions

The number and distribution of particles in a window depend on the average number of participants, therefore they depend on the classification used. To quantify this, we run the hydrodynamical code NeXSPheRIO for 180 Pb+Pb collisions at 153 GeV A and computed the rapidity and transverse momentum distribution for p – and charged pions in the various windows for our two classifications.

As explained above, in window 1, we do not expect large differences using one classification or another. This can indeed be checked in Fig. 3 for p – . However in window 3, we expect larger differences and this is shown in Fig. 4 for charged pions.

In our code, the initial conditions are fixed running NeXus, but we still have the freedom to adjust the freeze out temperature. In Figs. 3 and 4, we made the canonical choice T_f.out = 140 MeV, i.e. we did not try to adjust T_f.out to best fit the data. However, we see that the hydrodynamical results are reasonably close to the data.

4 Conclusion

In this paper, we compared predictions as usually done in hydrodynamics, using centrality windows defined through the impact parameter, and as obtainable experimentally, using windows in participant number. We computed rapidity and transverse mass distributions for p – and charged pions in various windows, and found no significant difference between the two classifications. This result corroborates those of Broniowski and Florkowski[6]. We feel however that a more detailed study should be done for other quantities such as the elliptic flow and for other experimental classification, for example using multiplicity.

Acknowledgements

This work was partially supported by CAPES, CNPq, FAPERJ and FAPESP (2000/04422-7, 2000/05769-0, 2001/09861-1).

References

[1] H.J. Drescher, M. Hladik, S. Ostapchenko, T. Pierog, and K. Werner, Phys. Rep. 350, 93 (2001); H.J. Drescher, F.M. Liu, 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); Nucl. Phys. A698, 639c (2002); Proceedings of the 6th International Workshop on RANP, World Scientific Ed., 2000.

[3] J. Bächler et al., Nucl. Phys. A661, 45c (1999).

[4] H. Appelshäuser et al., Phys. Rev. Lett. 82, 2471 (1999).

[5] G.E. Cooper, Ph. D. thesis, LBNL-45467 (2000).

[6] W. Broniowski and W. Florkowski, Phys. Rev. C 65, 024905 (2002).

Received on 15 August, 2003.

Publication Dates

• Publication in this collection
  11 May 2004
• Date of issue
  Mar 2004

History

• Accepted
  15 Aug 2003
• Received
  15 Aug 2003