Advances in High Energy Physics

Volume 2016 (2016), Article ID 5972709, 14 pages

http://dx.doi.org/10.1155/2016/5972709

## Geometry and Space-Time Extent of Pion Emission Region at FCC Energies

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

Received 10 August 2016; Revised 1 October 2016; Accepted 20 October 2016

Academic Editor: Burak Bilki

Copyright © 2016 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 is investigated for a wide set of space-time characteristics derived from Bose–Einstein correlations of secondary pion pairs produced in proton-proton and nucleus-nucleus interactions. Analytic functions suggested for smooth approximations of the energy dependence of emission region parameters demonstrate reasonable agreement with all available experimental results for proton-proton collisions while the approximations correspond to most of experimental data for nucleus-nucleus collisions at energies above 5 GeV. Estimations for a wide set of space-time quantities are obtained for energies for the Future Circular Collider (FCC) project based on the smooth approximations. The space particle densities at freeze-out are derived also from estimations for the volume of the emission region and for total multiplicity at FCC energies. Estimations for charged particle density and its critical value allow the possibility of lasing behavior for secondary pions in nucleus-nucleus collisions at FCC energy. The mathematical formalism is presented for study of the peak shape of correlation function for general case of central-symmetrical Lévy–Feldheim distribution.

#### 1. Introduction

When two energetic particles or nuclei collide, some matter is created in finite space-time volume. This matter volume, often called “fireball,” emits particles and space-time extent of the fireball is of fundamental interest for understanding of both the multiparticle production dynamics and the evolution of early Universe. One of the collective effects, namely, particle correlations at low relative momentum, represents a unique tool and sensitive probe of the size and the shape of the fireball at the last stage of its evolution (colorless particle emission region). The space-time geometry of particle source can be determined by using a method of interferometry based on the fundamental relation between spin and statistics. The production of identical bosons that are close together in phase space is enhanced by the presence of quantum statistical effect on Bose–Einstein correlations (BEC). The strength and form of the correlation reflect the space-time structure of the source [1]. The most of the secondary particles produced in the strong interactions are pions. Thus in the paper correlations between two identical bosons called BEC are studied for secondary charged pions produced in various strong interaction processes (in these reactions the BEC are often called HBT correlations due to analogy with Hanbury-Brown and Twiss effect [2, 3] used in radio astronomy to measure the angular sizes of stellar objects).

The international project called Future Circular Collider (FCC) is mostly aimed at hadron collider with a centre-of-mass energy TeV for collisions in a new 100 km tunnel of the CERN accelerator complex and detailed characteristics of various beams for FCC can be found elsewhere [4]. For heavy ion collisions the relation gives the energy in centre-of-mass per nucleon-nucleon collision of TeV for Pb + Pb (, ) and 63 TeV for collisions [5–7]. This project provides a unique opportunity to probe quantum chromodynamics (QCD) in the new energy regime [8]. One of the most distinguishing features of QCD is the mechanism of color confinement, the physics of which are not fully understood, due, in part, to its theoretical intractability [9]. The confinement mechanism has a physical scale of the order of the proton radius and is especially important at low momentum. Therefore study of source geometry in new energy domain with help of BEC seems important for better understanding of both the equation of state (EOS) of strongly interacting matter and general dynamic features of soft processes. The peak of two-particle Bose–Einstein correlation function (CF) contains the unique experimental information about particle source at freeze-out. The peak shape carries information, in particular, about the possible complex highly irregular geometry of the source [10, 11], the (1) symmetry restoration in high energy heavy ion reactions [12], and so forth. Therefore the development of a general formalism for detailed shape analysis of the peak in BEC CF is relevant for future high-statistics studies at FCC. It should be stressed that on the one hand the BEC leads also to Bose–Einstein condensates responsible for laser, superfluids, and superconductors [1]. On the other hand the pion multiplicity at midrapidity is larger than in heavy ion collisions in the TeV-energy domain, in particular, at FCC in Pb + Pb collisions at TeV [5, 6]. Therefore the number of pions in a unit value of phase space may be large enough that these bosons condense into the same quantum state and a pion laser could be created [13, 14]. Thus the paper is focused on the study of azimuthally integrated BEC of secondary charged pions produced in strong interactions, especially, on the space-time extent of pion emission region and the possible novel features of multiparticle production mechanism (pion laser) at FCC energies. Also the general formalism is suggested for study of shape of correlation peak in detail.

The paper is organized as follows. In Section 2, definitions of two-particle CF and BEC parameters are described. Section 3 devotes discussion of energy dependence of pion source extent in and A + A collisions, predictions for wide set of space-time characteristics for pion source in various collisions at FCC energies. The possibility for pion laser in strong interaction processes at FCC energies is considered in Section 4. In Section 5, the generalized parametrization of 3D CF is introduced with help of expansion of central-symmetrical Lévy–Feldheim distribution. Some final remarks are presented in Section 6.

#### 2. Method and Variables

The BEC effect is observed as an enhancement in the two-particle CF at low values of some difference constructed from 4 momentum , or its components of the entering particles, , where is the two-particle density function and is a reference two-particle density function that by construction is expected to include no BEC. Recent study [20] shows that BEC 1D experimental data samples are not enough for study of energy dependence of source parameters in proton-nucleus and nucleus-nucleus collisions. Therefore the present paper is focused on the 3D analysis of BEC in strong interaction processes.

In general phenomenological parametrization of CF with taking into account different forms of corrections on Coulomb final state interaction (FSI) can be written as follows [11]:where is the cumulant correlation function (cCF), at , and at while corresponds to the standard Coulomb correction, corresponds to the dilution procedure, and corresponds to the Bowler–Sinyukov correction, is the relative 4-momentum, is the average 4-momentum of particles in pair (pair 4-momentum), and are the matrices , and is transposed vector , and , where are parameters characterized the linear scales of the region of homogeneity [21, 22]; the products are taken on space components of vectors, , is the parameter which characterizes the strength of correlations called also chaoticity. Different types of Coulomb correction for two-pion correlations are compared in [11]. 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 [23, 24] is used for as well as the longitudinal comoving system (LCMS) frame. The parametrization of depends on type of distribution which was chosen for emission region [11]. For instance, the lowest order cCF can be written asfor specific case of Gaussian distribution which is one of the most used ones in BEC study. As known the study of BEC allows the estimation of space-time extent for region of homogeneity which is only some part of whole source. Therefore the BEC parameters are smaller* a priori* than corresponding scales of whole emission region and consequently the experimental BEC dimensions can be considered as low boundary for corresponding true linear scales of sourceFor this reason the BEC parameters are called BEC radii and it is assumed that correlation analysis for pairs of identical particles with low provides which are adequate experimental estimations for space-time extent of whole emission region within the simplest approach at least. It should be noted that azimuthally integrated BEC analysis allows rougher estimations for space-time scales of whole source with increase of collision energy because more intensive collective expansions reduce the sizes of the region of homogeneity more significantly at higher energies. Thus in the present paper , are considered as source BEC radii with taking into account relation (3) and influence of collective flows on the quality of this approximation.

In the 3D case and the Pratt–Bertsch coordinate system the space-time extents of the region of homogeneity or, with taking into account the discussion above, whole source is described by the following dimensions: is the source size along the beam axis, is extent along , and is the source size along the axis perpendicular to those two. Then one can define the geometric mean BEC radiusas well as the difference which is an important observable especially for some specific cases of 1D hydrodynamics (static, nonflowing source) due to its relation with particle emission duration [25, 26], where is the transverse velocity of pair of particles with mass , . Here the scaled geometric mean BEC radius is defined as follows: in accordance with approach suggested in [11, 15], where is the mean radius for beam nuclei, is radius of spherically symmetric nucleus, and fm [27, 28]. The volume of source can be written as follows: where the first case is the standard relation for BEC while the second case corresponds to the simplest approach of spherically symmetric source and it can be useful for future study of pion laser. Thus in the paper the following set of main BEC observables is under consideration as well as the set of important additional observables which can be calculated with help of BEC radii . The set of parameters characterizes the chaoticity of source and its 4-dimensional geometry at freeze-out stage completely.

#### 3. Space-Time Extent of Pion Source

In this study experimental BEC data sets are from [29] for and from [15] for A + A collisions.

Dependencies of BEC parameters , for high energy collisions are shown in Figures 1(a)–1(d), respectively. As seen for energy range from Relativistic Heavy Ion Collider (RHIC) to the Large Hadron Collider (LHC) the experimental is close to the constant (Figure 1(a)) while some decrease is observed for experimental deduced from 1D two-pion BEC analyses [30]. The BEC radii increase with collision energy (Figures 1(b)–1(d)) more significantly in transverse plane with respect of the beam direction compared to that for longitudinal direction. Taking into account the view of experimental , in as well as the detailed study of energy dependence of azimuthally integrated main BEC parameters , for charged pions in nucleus-nucleus interactions [15] the following function is used:for smooth approximation of experimental dependencies , in interactions. Here for or for A + A reactions, where GeV^{2}. The very limited ensemble of experimental points from 3D Gaussian analyses of does not allow the fit by (7) with all parameters to be free. In general the various quantities from the set can show the different behavior as function of [15]. Thus the following two views of (7) are used for approximation of experimental points: (i.1) function (7) at fixed value of which is defined by method of sequential approximations and (i.2) the specific case of (7) at . Only statistical uncertainties are available for strength of correlations while for each of the BEC radii fits are made for both the statistical and total errors, where total errors of experimental points include available clear indicated systematic errors added in quadrature to statistical ones. The numerical values of fit parameters are presented in Table 1, where the second line for chaoticity parameter corresponds to the simplest fit by constant and for each of the BEC radii to the approximation by specific case of (7). Approximation curves are shown in Figure 1(a) by solid line for specific case of (7) and by dashed line for fit by constant. Figures 1(b)–1(d) show the fit results for BEC radii by solid lines for approach (i.1) and by dashed lines for specific case (i.2) with taking into account the statistical errors of experimental points.