Advances in High Energy Physics

Volume 2015, Article ID 790646, 13 pages

http://dx.doi.org/10.1155/2015/790646

## Azimuthally Integrated HBT Parameters for Charged Pions in Nucleus-Nucleus Interactions versus Collision Energy

National Research Nuclear University “MEPhI” (Moscow Engineering Physics Institute), Kashirskoe Shosse 31, Moscow 115409, Russia

Received 23 June 2014; Revised 9 September 2014; Accepted 13 October 2014

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2015 V. A. Okorokov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The energy dependence of spatiotemporal characteristics of particle emission region is studied for charged pions produced in nuclear collisions. No dramatic change is observed for the HBT parameters with increasing of the center-of-mass (c.m.) energy per nucleon-nucleon pair, , for of a few GeV to a few TeV. The emission duration is obtained to be almost independent of the c.m. energy within the measurement uncertainties. The analytic function is suggested for a smooth approximation of the energy dependence of the main HBT parameters. The fits demonstrate reasonable agreement with the experimental data. Predictions are made for future LHC and FCC experiments.

#### 1. Introduction

At present, two-particle interferometry analysis (often referred to as HBT) in particular that is based on Bose-Einstein correlations is a unique experimental method for determination of sizes and lifetime of particle source in high energy and nuclear physics. Space-time characteristics for emission region of secondary particles created in (heavy) ion collisions are important for study of deconfinement state of strongly interacting matter, strong-coupling quark-gluon plasma (sQGP). Furthermore, the study of energy dependence of HBT observables can be useful for understanding in detail the transition from sQGP produced at higher energies to confined hadronic resonance matter created in final state at lower energies. HBT analysis allows studying dynamic features of interaction process at late, that is, soft, stage of space-time evolution of multiparticle final state. Therefore, the study of nucleus-nucleus collisions in wide energy domain by HBT correlations seems important for better understanding of both the equation of state (EOS) of strongly interacting matter and general dynamic features of soft processes.

The paper is organized as follows. In Section 2, definitions of main observables for analysis of two-pion correlations are briefly described. Section 3 is devoted to discussion of experimental energy dependence for space-time extent of source of charged pions and corresponding fits. Also estimations for HBT observables are shown for the LHC and the future circular collider (FCC) project energies. Some final remarks and conclusions are presented in Section 4.

#### 2. Method and Variables

In general, phenomenological parameterization of correlation function (CF) for two identical particles with 4-momenta , taking into account different forms of corrections on Coulomb final state interaction (FSI) can be written as follows [1]:where corresponds to the standard Coulomb correction, the dilution procedure, and the Bowler-Sinyukov correction; is the relative 4-momentum and the average 4-momentum of particles in pair (pair 4-momentum), for the standard simplest (Gaussian) case:Here and are the matrices and is transposed vector , , , where are parameters that characterized the linear scales of homogeneity region [2]; the products are taken on space components of vectors; , , is the parameter which characterizes the degree of source chaoticity. Different types of Coulomb correction for two-pion correlations are compared in [1]. The space component of pair 4-momentum () is decomposed on longitudinal and transverse parts of pair momentum. In the paper, the decomposition of Pratt-Bertsch [3, 4] is used for as well as the longitudinal comoving system (LCMS) frame. The volume of source can be written as follows:Comparison of (3) with definition from [5] is discussed in detail in [6]. One of the important additional observables is the following difference [7, 8]:If the emission function features no position-momentum correlation, then is finite at nonzero only due to explicit -dependence (resulting from the mass-shell constraint ) [7]. In this casewhere is the transverse velocity of pair of particles with mass , , and is the emission duration for the particle type under discussion. It should be stressed that the last relation is valid in some specific cases of 1D hydrodynamics while it is violated in both the cascade approaches and multidimensional hydrodynamic models. Thus, in the framework of some assumptions, gives direct access to the emission duration of the source and allows us to partially disentangle the spatial and time information contained in radii parameters [7]. The sensitivity to the is the main advantage of the observable (4).

In the paper, the following set of main HBT observables is under consideration as well as the set of some important additional observables which can be calculated with help of HBT radii . The set of parameters characterizes the chaoticity of source and its 4-dimensional geometry at freeze-out stage completely. Scaled parameters , , , and are calculated as follows [1]:Here and are radius and volume of spherically symmetric nucleus, fm [9, 10]. The change is made in the relation (6) in the case of asymmetric nucleus-nucleus collisions [1]. One needs to emphasize that the most central collisions are usually used for studying the space-time characteristics of final-state matter and, in particular, for discussion of global energy dependence of HBT observables (see Section 3). Thus, the using of radius of all the nuclei in (6) seems reasonable. In general case the scale factor in (6) for calculation of scaled HBT radii, and volume should takes into account the centrality of nucleus-nucleus collisions. The normalization procedure suggested in [1] allows us to consider two data samples, namely, (i) only (quasi) symmetric heavy ion collisions and (ii) all available data for nucleus-nucleus collisions. Experimental data sets analyzed here are discussed in detail elsewhere [1, 6].

#### 3. Energy Dependence of Space-Time Extent of Emission Region

Dependencies of HBT parameters , , and are shown in Figures 1(a)–1(d) and Figure 1(e), respectively. The chaoticity parameter decreases with increasing rapidly at lower (AGS) energies and shows the weak changing at GeV (Figure 1(a)). HBT radii of source in transverse plane with respect to the beam direction, (Figure 1(b)) and (Figure 1(c)), show little change over a wide range of energies GeV which corresponds to the highest AGS, SPS, and RHIC beam collision energies. On the other hand, the value of source size in longitudinal direction, (Figure 1(d)), appears to reach a minimum around GeV, rising in energy domain available at RHIC. As seen there is increasing of HBT radii (Figures 1(b)–1(d)) at growth of collision energy from GeV up to the maximum available LHC energy TeV. The significant increasing of HBT radii is seen for much broader energy range (on about two orders of magnitude TeV) only than was expected early at the beginning of RHIC operation. Therefore, the space-time extent of emission region at freeze-out changes slowly with increasing of collision energy. The transverse radius reflects the spatial extent of particle source, whereas is also affected by dynamics [12, 13] and is believed to be related to the duration of particle emission [14]. As indicated, for example, in [15], the ratio was predicted to increase with beam energy by hydrodynamical calculations and might show a significant enhancement if the lifetime of the collision evolution (and, within these models, the duration of particle emission as a result) was to be extended by entrance into a different phase [14]. There is no significant increasing of ratio in all experimentally available energy domains (Figure 1(e)). Recent developments, in particular in viscous hydrodynamics, allow us to get reasonable agreement between experimental and model values of at top RHIC energy and demonstrate that the behavior of experimental dependencies of on kinematic variables can be explained in particular by realistic EoS with crossover phase transition and sQGP at high temperatures [16–21]. Therefore, the soft HBT observables confirm the phase transition and creation of deconfinement state of strongly interacting matter in collider experiments.