#### 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.

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In difference with A + A collisions [15] function (7) for both choices of agrees with experimental data quantitatively and provides reasonable fit qualities even with statistical errors for all BEC parameters from the set with exception of . In the last case one can only conclude that fit curve is similar to the general trend of experimental points (Figure 1(b)) due to poor fit quality. Account for total errors allows statistically acceptable fit qualities for all main BEC parameters in both approaches (i.1) and (i.2). Furthermore is equal to zero within errors for longitudinal BEC radius for (i.2) and consequently can be described by constant with fm and in the case of the accounting for total errors. For energy range from RHIC to the LHC fit curves for approaches (i.1) and (i.2) are close to each other for , especially for radii (Figure 1(c)) and (Figure 1(d)). Nevertheless the fit within approach (i.1) for only confirms for statistical errors but other BEC radii show faster increasing with especially (Table 1). The growth of approaches the linear behavior in for accounting for total errors but longitudinal radius preserves much faster growth with energy increase in this case too. As a consequence the method of sequential approximations for leads to significant improvement of the fit quality with respect to the quantity for approach (i.2) for at statistical errors and especially for at all considered types of errors. It should be noted that the difference between smooth curves obtained within approaches (i.1) and (i.2) for BEC radii can be much more noticeable for higher energy FCC than that in Figure 1 which can have a relevant effect on the estimated BEC parameters at FCC.

Dependencies of additional BEC parameters , for interactions are shown in Figures 2(a)–2(d), respectively. Notations of smooth curves correspond to Figure 1; namely, the solid curves show the results calculated with help of fits of BEC radii within approach (i.1) and dashed lines are for special fits (i.2). As seen curves of both types agree with experimental points reasonably for (Figure 2(a)) and (Figure 2(b)) in total experimentally available energy range. Otherwise approach (i.1) leads to significant overpredictions for (Figure 2(c)) and (Figure 2(d)) at the LHC energy TeV while the curves for special case (i.2) agree with experimental points at this energy.

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Thus there is significant uncertainty in functional behavior of dependence of experimental BEC parameters on collision energy due to very limited ensemble of 3D experimental data for and future experimental results are important for more definitive conclusion with regard to behavior of dependencies .

The pion emission duration for collisions can be estimated with taking into account the results for and kinematic regime for pion pairs under study. for pion pairs with GeV/ as well as for nuclear collisions [15]. Then pion emission duration increases from fm/ at RHIC energy TeV up to fm/ at the LHC energy TeV which is the highest for available experimental BEC results. Thus the pion emission durations in collisions are smaller significantly than that for nuclear interactions [15] in the energy range from RHIC top up to the LHC.

The energy dependence for BEC parameters from the sets , was studied in detail in [15] for nuclear collisions with exception of and corresponding scaled quantity defined in the present paper. Energy dependence of these additional BEC parameters is obtained with experimental database for nuclear collisions from [15]. Figures 3 and 4 show the experimental and , respectively, as well as smooth curves calculated with fit results for BEC radii at GeV from [15]. In Figures 3 and 4 solid curves for and for , respectively, are obtained with help of results from fits of BEC radii by general view of (7) and dashed curves correspond to the calculations with fit results for BEC radii for special case of (7) at . As seen in Figures 3 and 4 the behaviors of smooth curves with respect to each other as well as to the experimental data are quite similar for and . In both cases curves correspond to experimental points reasonably at intermediate energies GeV with excess over experimental points in dip region GeV, and opposite situation is seen for TeV energies (Figures 3 and 4). In Figures 3 and 4 solid curves are close to dashed ones in total energy range considered especially for but onset of the excess of solid curve over dashed one at TeV for (Figure 3) and at TeV for (Figure 4) can lead to a noticeable discrepancy at FCC energy.

As expected, the quantitative comparisons of Figures 1(b)–1(d) with Figure 2(b) for reactions and Figure 3 with results for BEC radii , in nuclear collisions [15] show that at qualitative level.

Predictions for values of the BEC observables from sets are obtained for suggested types of collisions and energies of the LHC and FCC project [4–7] based on the fit results for the main BEC parameters discussed above and in [15]. Estimations are shown in Table 2 for fits by function (7) and its specific case at with inclusion of statistical errors of experimental points, the first column for each type of collisions corresponds to the nominal LHC energy, and the second column corresponds to the energy of FCC project. One notes the fit by constant predicts for the strength of correlations in collisions for both the LHC and FCC energies. Values for all additional BEC parameters , are calculated with help of its definitions (4)–(6a) and estimations for BEC radii at some energy. The pion emission duration is derived from and kinematic regime for pion pairs under study as well as for at lower energies. Results for asymmetric collisions + Pb are obtained with help of fit results for scaled BEC parameters [15] and rough estimation fm. In the case of Pb + Pb collisions the results for main BEC parameters , as well as for , and emission duration are from [15]; furthermore the brief discussion of estimations for these BEC parameters at the LHC and FCC energies can be also found in the previous studies [15, 29]. As seen from Table 2 all BEC parameters have values coincided with each other for two approaches (i.1) and (i.2) within errors for corresponding collision energies and types of strong interaction processes. In general estimations for BEC parameters calculated with approach (i.1) do not change from the LHC up to FCC energies within large uncertainties for all collisions under consideration. Proton-proton and nucleus-nucleus collisions are characterized by similar strength of correlations for approach (i.2) for the LHC and FCC energies, and the large uncertainty for + Pb allows only the qualitative conclusion that is somewhat smaller for this type of collision than that for Pb + Pb in the case of general view of (7) and, for + , Pb + Pb within the framework of approach (i.2). is quite constant for collisions but shows noticeable decrease for heavy ion collisions at increasing of in the energy range LHC–FCC for fit by specific case of (7) at . For + Pb the estimation of obtained with approach (i.1) at TeV agrees rather well with experimental result [31] while approach (i.2) underpredicts the strength of correlations at the LHC energy. Furthermore estimations for BEC radii of pion source produced in + Pb collisions at TeV (Table 2) agree with experimental results [31, 32] within large errors. The space scales are about 2 fm in , about 4-5 fm in + Pb, and 6–9 fm in Pb + Pb collisions for pion source at FCC energies. In the case of estimations for BEC radii obtained with general view of (7) large uncertainties do not allow the definite conclusion and one can see qualitative indication only that is somewhat smaller than other radii in all collision types for both the LHC and FCC energies. For estimations based on the special case of (7) with there is noticeable increase of all BEC radii for transition from the LHC to FCC energy in heavy ion collision (Table 2). As a consequence and are larger at TeV than that at the LHC energy TeV. Spread of values of BEC radii leads to significant uncertainties for estimations of additional space-time parameters especially for extremely asymmetric + Pb collisions for which the large error for increases greatly uncertainties for BEC quantities in Table 2. Consequently the volume of the pion source can be roughly estimated as about fm^{3} in , fm^{3} in + Pb, and fm^{3} in Pb + Pb collisions at FCC energies in comparison with fm^{3} in , fm^{3} in + Pb, and fm^{3} in Pb + Pb at the nominal LHC energies. These estimations indicate the consistent growth of for transition from the small system collisions to the Pb + Pb. For approach (i.1) estimations for all parameters , and for emission duration do not depend on energy in the range LHC–FCC for all types of collisions within errors. This conclusion is also valid for approach (i.2) with exception of and for Pb + Pb discussed above. It should be noted that weak change of main BEC parameters , is qualitatively expected for energy domain from the LHC up to FCC because of general trends of available experimental points and consequent slow logarithmic increase with collision energy for smooth analytic functions used in the present analysis as well as in [15, 29].

#### 4. Pion Laser at FCC Energies

Results shown above for space-time extent of pion emissions region allow the study of possibility of Bose–Einstein condensation with consequent formation of pion laser in high energy strong interaction processes. The key quantity is the charged particle density which is defined as follows:where is the total charged particle multiplicity and is the source volume at freeze-out stage. The critical density can be calculated with help of (8) and transition to the critical total multiplicity . The last multiplicity parameter was derived for 1D thermal Gaussian distribution in [13]. Within the 3D Gaussian parametrization for source the following relation is suggested:Here is the fraction of the pions to be emitted from a static Gaussian source, GeV/ is a momentum spread and [13], is source temperature supposed to be equal to the parameter value at chemical freeze-out, and it is suggested [33]. It should be emphasized that there are no qualitative studies of for collisions so far but set of results for mean multiplicity and the pseudorapidity density at midrapidity for charged particles in various collisions [17, 34–36] as well as recent results for deconfinement in small system [37, 38] indicate remarkable similarity of both the bulk and the thermodynamic properties of strongly interacting matter created in high energy and A + A collisions. Therefore the hypothesis is suggested for similar energy dependence of in both and A + A interactions with taking into account [17, 34–36] and consequently the analytic energy dependence of from [39, 40] is used for all types of strong interaction processes considered in this section. Thus appropriate analytic function is derived for energy dependence of . Also versus collision energy in (8) is defined by some smooth approximations which are specified below for and A + A collisions. Experimental estimations for available in Figure 2(e) for and in [15] for A + A collisions can be used for calculations of at certain energies. The results of such calculations are called experimental points and marked by symbols in Figures 5 and 6 in the sense that BEC measurements are used in these cases. As seen from Table 1 for and from [15] for heavy ion collisions the smooth energy dependence of source volume can be calculated with help of the fits of BEC radii by (ii.1) general function (7) as well as (ii.2) by specific case Relation (6a) for source volume is used in (8) for experimental estimations as well as for calculations of smooth energy dependence of in both and A + A collisions. On the other hand it seems reasonable to use (6b) for estimation of critical charged density because relation (9) with is only available. Taking into account the qualitative relation between and 3D BEC radii shown above one expects that the difference of numerical factors for two definitions of (6a) and (6b) provides additional uncertainty in the ratio . Therefore theoretical investigations are essential for quantitative account for geometry of source and decrease of uncertainty due to calculation of volume of emission region. As it follows from (3) and discussion in Section 2 relation (8) defines the upper boundary for true value of charged particle densityStrictly speaking and possibly the discrepancy between estimation from (8) and will be increased with growth of collision energy due to intensification of longitudinal and radial collective flows. During the recent years the collectivity in small colliding systems is under intensive theoretical [41–48] and experimental studies [49–51] but the results obtained already mean that the statement above is valid for interactions as well as for nucleus-nucleus collisions in TeV-energy domain at least. In general one can assume the weaker dependence of on space-time parameters of particle source compared to that for due to BEC radius in (9). The uncertainties of calculated by standard way from errors of fit parameters are attributed as statistical errors. The statistical uncertainties for are estimated by standard way at assigned relative error and with taking into account the errors for fits of and used while systematic uncertainties are deduced by varying of within the range only. The statistical errors of propagated from corresponding uncertainties of BEC radii are only taken into account below.

Total charged multiplicity is calculated within various approaches [17–19]. In Figure 5 energy dependence is shown for as well as for critical particle density. It should be noted that in the case of collisions minimum-bias events are used in the BEC analyses [52, 53] and these events correspond to the non-singly diffractive (NSD) collisions at TeV [37] as well as at the LHC energies [54, 55]. Thus experimental points are obtained for calculated with hybrid approximation [17]. The thick solid and dashed lines correspond to the source volume within approach (ii.1) and thin lines correspond to from case (ii.2). In Figure 5 results for are shown by solid lines for hybrid approximation of total charged multiplicity [17] and dashed lines are deduced with within 3NLO pQCD approach [18] at [34–36, 56] and parameters from [57] for number of colors . Available experimental estimations show almost constant . Smooth curves agree with experimental points reasonably for any approximation of and . As seen the differences between various approaches for each of the two parameters are small up to the LHC energy and increase for FCC noticeably. The dependence of critical particle density shown by dotted line decreases with energy. The statistical s.d. band limits are drawn by thin dotted line while this large uncertainty is mostly dominated by the precision of BEC parameters of emission region. The systematic s.d. boundaries are shown by heavy grey lines. As seen from Figure 5 is smaller than its critical value in collisions up to FCC energy TeV for any approaches for total charged multiplicities and under study. This conclusion is valid even with taking into account large statistical uncertainties. Thus one can not expect the kind of lasing behavior for secondary pions in collisions within the present approach.

One can note the coincidence between the experimental values of in and collisions at energies 0.2 and 0.9 TeV and general smooth energy dependence of for NSD events in the interactions under discussion [55]. Also reasonable agreement is observed for pseudorapidity density measured in [54, 55, 58] and [59, 60] interactions at energies indicated above. But there are no 1D BEC analyses with Gaussian model in collisions at and 0.9 TeV. Furthermore the quantitative comparison of BEC results from to those from collisions is difficult for 1D case due to limited ensemble of experimental results in the last case [20] and noticeable difference between collision energies in and for available BEC measurements; there is no BEC analysis with 3D Gaussian model for so far. On qualitative level close values for multiplicity quantities can be expected in and collisions in particular at energies about 2 TeV while the 1D BEC Gaussian radius for pion source in at and 1.96 TeV [61, 62] is significantly larger than that from at TeV [63]. Therefore is expected to be smaller in than that in collisions at least in TeV-energy domain at close values of critical quantity due to its weaker dependence on space-time extent of particle source. Thus the pion lasers seem impossible in high energy collisions within the rough assumptions.

As discussed above the estimations of space-time extent of pion source are characterized by large uncertainties; moreover development of equation for critical parameters for multidimensional (3D) case seems important for improvement of precision of studies and for more certain conclusions. The future quantitative experimental and theoretical investigations are essential for verification of the results shown above and possibility of novel coherent effects in different types of collisions in high energy domain.

Total charged multiplicity is calculated with hybrid equations [17, 19]. Figure 6 demonstrates energy dependence for both and the critical particle density where smooth approximation for is shown by solid line for hybrid approximation of total charged multiplicity [17] and by dashed line for from [19], and experimental points are obtained for calculated with equation from [19]. Both the curves for and experimental points are deduced with mean number of participants which corresponds to the 0–5% central Pb + Pb collisions [19]. This simplest approach seems reasonable because heavy ion collisions are only considered in Figure 6. The source volume is calculated within approach (ii.1) and results for BEC radii in nuclear collisions [15]. Experimental points for increase with and agree reasonably with smooth curves for both parameterizations of total charged multiplicity under consideration. Comparison between particle densities in (Figure 5) and A + A (Figure 6) strong interaction processes indicates the enhancement of over starting with RHIC energy 200 GeV per nucleon-nucleon pair; furthermore this enhancement increases with growth of collision energy. The critical particle density shown by dotted line depends weakly on in nuclear collisions. In Figure 6 the line types for statistical and systematic s.d. band limits are the same as well as for corresponding smooth curves for collisions (Figure 5). The statistical uncertainty driven by the precision of BEC parameters of emission region increases noticeably for multi-TeV region TeV in A + A interactions. The situation changes dramatically with transition from to nuclear collisions at high energies. As seen from Figure 6 the following relation is valid at RHIC and the LHC energies within wide uncertainty band for critical value of charged particle density. Furthermore there is clear enhancement of smooth curves for over at TeV with taking into account large statistical uncertainty for . Thus one can expect the appearance of novel effects dominated by Bose–Einstein condensation in nucleus-nucleus collisions at FCC energy. In particular, Figure 6 indicates the possibility for pion laser effect in heavy ion collisions at TeV within the approach under study.

With theory point of view the conception of the pion laser was intensively studied within the framework of the model of independent factorized sources [13, 14, 64–66] as well as in the model of disoriented chiral condensate (DCC) decay [67]. On the other hand possible experimental signatures of Bose–Einstein condensation, in particular, the pion laser effect in heavy ion collisions at FCC energy, should be the subject of future detailed investigations. Here one notes the following experimental signatures of Bose–Einstein condensation. In general, one can expect enhancement of high-multiplicity events [13, 14] and the decrease of chaoticity parameter derived from two-particle BEC analysis due to amplification of coherent particle production [68]. The effects of multiboson symmetrization regarding isospin fluctuations can manifest itself through enhancement of the events with anomalous isospin imbalance like CENTAURO events in high energy cosmic ray [13, 14]. The shrinkage of the BEC radius is the more specific prediction within the model of the DCC decay when the Bose–Einstein condensation takes place [67]. This effect potentially represents one of the most pronounced features of the pion laser, because the available experimental BEC radii show smooth increase with collision energy both in interactions (Figure 1) and the heavy ion collisions [15, 29].

#### 5. Study of Correlation Peak Shape

The accelerator parameters within the FCC project [4–7] open the new possibility for detailed study of peak structure for two-particle BEC correlation function. The peak of CF is described by . In general there is rich class of random processes with additive stochastic variables for which (i.e., for these processes) there are finite distributions but the Central Limit Theorem (CLT) in the traditional (Gaussian) formulation is not valid. The class of random processes under consideration are characterized by large fluctuations, power-law behavior of distributions in the range of large absolute values of random variables, and nonanalytic behavior of characteristic function of the probability distribution for small values of its arguments [69]. In mathematical statistics and probability theory the class of such distributions are called stable (on Lévy) distributions (in literature for physics and mathematics the multidimensional distributions included in the class are called Lévy–Feldheim distributions) [70, 71]. The general stable distribution is described by four parameters: an index of stability (or Lévy index) , a parameter of skewness , scale , and location . These distributions satisfy requirements of generalized Central Limit Theorem (gCLT) and self-similarity (the applications of stable distributions in the physics of fundamental interactions and, in particular, for correlation femtoscopy are described, e.g., in [26]). Therefore the detailed investigation of the shape of correlation peak has to do with verification of hypothesis of possible self-affine fractal-like geometry of emission region. At present the study of Lévy–Feldheim distributions is the advanced region of mathematics but the specific case of central-symmetrical stable distributions is known in more detail [72]. Just this subclass of stable distributions is most important on the point of view of investigation for BEC. In this case the application of subset of nonisotropic central-symmetrical Lévy–Feldheim distributions [73] seems reasonable because the projections of the 3D relative momentum are independent random variables.

The multidimensional generalized parametrization of th order for CF (1a) and (1b) can be written as follows [11]:where is phenomenological parametrization of th order for cCF (2) and functions take into account formally all corrections on degree of source chaoticity, final state interactions, and so forth. The experimental and theoretical investigations in the field of BEC allow us to derive some approach for cumulant two-particle function (2) in the lowest order. Within the framework of the subset of nonisotropic central-symmetrical Lévy–Feldheim distributions the most general parametrization of can be given byHere were take into account that , for correlation femtoscopy, the products are on the space components of vectors. is the closed system of orthogonal polynomials in the Hilbert space , . The system for exponential weight function can be derived with the help of the following recurrent relations , [73–77], and moments [78]. Here , and and are the following determinants:where and , and is defined by normalization which is chosen for system under consideration. The clear view of can be found, for instance, in [79] for few lowest orders at normalization . The specific cases and correspond to Cauchy and Gauss distributions, respectively, which are mostly used in the correlation femtoscopy. For the first case the Laguerre polynomials, , are used as ; the Hermite polynomials, , are chosen as the closed system of orthogonal polynomials for the second specific case [80].

The generalized parametrization of (11a) and (11b) contains the important physical information with regard to the possible high irregular geometry of emission region and dynamics of its creation and it is additional with respect to information derived for sets of space-time parameters based on traditional Gaussian parametrization. At present there are a few studies [81–84] which used the formalism outlined above for specific case of Gaussian distribution only. But these investigations confirm already the importance of detailed analysis of peak shape of CF. Thus high statistics and parameters of multiparticle final state for FCC energies allow us the qualitative study of complex geometry of emission region for secondary pions.

#### 6. Summary

The following conclusions can be obtained by summarizing the results of the present study.

Energy dependence is investigated for main BEC parameters from the set derived in the framework of 3D Gaussian approach in collisions as well as for the set of important additional observables . Analytic function is suggested for approximation of energy dependence of main BEC parameters. The fits demonstrate statistically acceptable qualities for and for most radii even for taking into account statistical errors of experimental points. Smooth curves calculated for energy dependence of the set of additional BEC parameters agree with corresponding experimental data at least of qualitative level. The estimation of emission duration of pions in collisions increases from about 0.4 fm/ at RHIC energy up to 1.3 fm/ at the LHC energy. Estimations are obtained for wide set of space-time characteristics of pion source at FCC energies on the basis of the fit results for and nucleus-nucleus collisions. The pion source is characterized by linear sizes about 2 fm in , about 4-5 fm in + Pb, and 6–9 fm in Pb + Pb collisions at FCC energies. Volume of pion source at freeze-out is estimated from few hundreds of fm^{3} in through few thousands of fm^{3} in + Pb up to fm^{3} in Pb + Pb collisions at FCC energies.

The charged particle density and its critical value are investigated for high energy and A + A collisions. The experimental dependence is almost flat and it is described by smooth curves reasonably for various parameterizations of total charged particle multiplicity. decreases with energy; nevertheless the estimations of critical value are larger significantly than charged particle density up to FCC energy. Therefore one can not expect the Bose–Einstein condensation and appropriate effects for secondary pions in collisions at FCC within the present approach. The charged particle density is noticeably larger in heavy ion collisions than that in at similar collision energies. The experimental dependence increases with energy in difference with reactions. Smooth curves calculated for various approaches of total charged particle multiplicity agree with experimental points. The situation is dramatically different in high energy nucleus-nucleus collisions with respect to case. The charged particle density is in the range of estimations of critical parameter within its large uncertainties from the RHIC energy TeV up to the LHC TeV which is highest for experimentally available BEC results in nucleus-nucleus collisions. Furthermore there is clear enhancement of values estimated for over values for critical density at TeV even with taking into account large statistical uncertainty for . Thus there is possibility for Bose–Einstein condensation and novel effects, in particular, pion laser in nucleus-nucleus collisions at FCC energy. It seems that the theoretical and experimental developments are essential for future progress in this field as well as for more definite conclusions due to improvement of precision for appropriate quantities.

The generalized parametrization for two-particle BEC correlation function is suggested and it takes into account the expansion in closed system of orthogonal polynomials for general case of nonisotropic central-symmetrical Lévy–Feldheim distribution. Possibly, the view of BEC CF can be useful for detailed study of correlation peak shape at FCC.

#### Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author is grateful to Professor E. K. G. Sarkisyan for fruitful discussions and helpful comments.