In relativistic heavy ions collisions, strongly-interacting matter is formed at high temperatures and densities. The initial geometry of two nuclei in the collision affects its evolution. The region of the nuclei overlapping after the collision depends on the impact parameter, which is the length of the vector connecting the centers of the two colliding nuclei. Event classes are used to study the geometry of collisions. A more central collision corresponds to a higher multiplicity of the produced particles and a lower energy in spectators. The Glauber model is used to map the event classes to a range of model parameters, i.e. impact parameter, number of participant nucleons and number of nucleon-nucleon collisions which can not be measured in the experiment. In the NA49 experiment, the event classes can be determined using the multiplicity of the produced particles and/or the energy in spectators.
2. NA49 experimental setup
Main components of the NA49 experimental setup are four large-volume time projection chambers (TPC). The vertex TPCs VTPC-1 and VTPC-2, are placed in the magnetic field of two superconducting dipole magnets VTX-1 and VTX-2. The other two TPCs (MTPC-L and MTPC-R) are positioned downstream of the magnets. The TPCs serve to measure multiplicity of produced charged particles. A Veto Calorimeter placed about 20 m downstream of the target behind a collimator measures the energy carried by spectators .
The data on Pb+Pb collisions at 40 GeV were collected within the energy scan program at the CERN SPS. After event selection 120K events remain for further analysis. To ensure good fit quality tracks are required to have a fit in the range . It is further required that tracks contain at least 55% of the maximum possible number of points to avoid split tracks. Tracks are further required to have distances of closest approach at the interaction vertex of , . At least in one of the detectors the number of hits should exceed 20 in the VTPCs and 30 in the MTPC. Kinematic cuts select particle pseudorapidity in and transverse momentum GeV.
3. Centrality determination procedure
In current analysis Modified Wounded Nucleon model (MWN)  was used to map particle yield of event to geometrical quantities of the initial state. MWN originates from Glauber MC model and describes procedure of mapping between MC results and experimental data. According to MWN emitted particles are produced by a set of ancestors with negative binomial distribution. MWN suggests the following form of multiplicity distribution
(1) where , , – free parameters of the distribution; , – mean value and width of negative binomial distribution, respectively. is a number of participants, – number of binary collisions.
Similarly one can write probability density of spectators energy as follows
Assuming and independent one can build 2-dimensional distribution as . Experimental distribution of multiplicity was obtained from TPC data and spectator energy was retrieved from VETO calorimeters. Using 2-dimensional histogram of one can find , , , giving the best match of experimental distribution to MC simulated data.
Determination of event centrality classes involves the following steps :
4. Results and Discussion
The fit ranges were , . 300k MC events for multiplicity distribution and 400k events for spectators energy distribution were generated. Centrality class events separation was performed according to procedure described in previous section and results are shown in fig. 1, 2a. Since not all emitted particles get caught by detector due to ineffectiveness of trigger system, in the region of the most peripheral collisions MC results deviate from the experimental data. A point where deviation starts is called anchor point. For multiplicity distribution it is and for spectators energy distribution.
On the fig. 2a profile histogram is built horizontally to the correlation and is shown as black triangles, then this correlation was fitted with a polynomial function (black line). Profile the correlation along the polynomial fit line is indicated with red triangles and refit with red line.
After mapping procedure one can use MWN model to estimate geometrical parameters of initial state of event. On fig. 2b are shown impact parameter distributions for events in specific centrality class. For each centrality class mean value and width was obtained. The resulting dependency of and on centrality are shown on fig. 3a. Dependency of relative error on centrality is shown on fig. 3b.
The implementation of the procedure within the software framework of the future CBM experiment was adopted for event classification in the NA49 experiment. Event classes in the experiment and parameters of the MWN model in event classes are determined. Impact parameter resolution is similar for centrality determination via spectator energy and track multiplicity. In future, the procedure will be used for analysis of the new Pb-Pb data collected by the NA61/SHINE experiment and for comparison with the results previously obtained by STAR at RHIC and NA49 at the CERN SPS.
This work was partially supported by the Ministry of Science and Education of the Russian Federation, grant N 3.3380.2017/4.6, and by the National Research Nuclear University MEPhI in the framework of the Russian Academic Excellence Project (contract No. 02.a03.21.0005, 27.08.2013).