Advances in High Energy Physics
Volume 2015 (2015), Article ID 790646, 13 pages
http://dx.doi.org/10.1155/2015/790646

Review Article

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

V. A. Okorokov

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 SCOAP³.

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 case

where 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.

(a)

(b)

(c)

(d)

(e)

Figure 1: Dependence of chaoticity parameter (a), HBT radii (b–d), and ratio (e) on collision energy for central heavy ion interactions at midrapidity and  GeV/ [6]. Experimental results are demonstrated for pairs of mesons (in the cases of ALICE and STAR at and 200 GeV, for pairs) and for standard Coulomb correction (in the cases of ALICE, NA44, NA45, PHOBOS, and STAR at and 200 GeV, for correction ). Statistical errors are shown (for NA44, total uncertainties). The solid lines (a–d) correspond to the fits by function (7) and dashed lines to the fits by specific case , . Fitted data samples for (a) and for (d) do not include the point of the WA97 experiment [11], while the fits for transverse HBT radii (b, c) are shown for samples with point from [11]. Smooth solid and dashed curves at (e) correspond to the ratio calculated from the fit results for and , and dotted line is the level .

Taking into account the view of experimental dependencies in Figures 1(a)–1(d), the following function is suggested:

for smooth approximation of , , where ,  GeV². Also the specific case of (7) at is under consideration. As seen from Figures 1(b)–1(d), there is indication on change of behavior of energy dependence (inflection point) for at  GeV. This inflection point is seen most clearly for (Figure 1(d)). Therefore, the fit function (7) is used for approximation of the energy dependence of HBT radii in the energy domain  GeV only. Experimental energy dependence of is fitted by general function (7) at all available energies. As seen the point from the WA97 experiment [11] differs significantly from other results at close energies for (Figure 1(a)) and longitudinal radius (Figure 1(d)). Thus, for these parameters fits are made for data sample (i) with exception of the point from [11]. For each main HBT parameter 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 each HBT parameter corresponds to the approximation by specific case of (7). Fit curves are shown in Figure 1 by solid lines for (7) and by dashed lines for specific case of fit function at taking into account the statistical errors. In general, fit function described above agrees reasonably with experimental dependence , (Figures 1(a)–1(d)). But the fit qualities are poor for all the main HBT parameters, especially for , with inclusion of the statistical errors only (Table 1). Spread of experimental points leads to the statistically unacceptable values of . In the case of inclusion of estimation for systematic uncertainty for collisions at  TeV leads to both the dramatic growth of and improvement of the fit quality at transition from statistical errors of experimental points to total errors in data sample (i) (Table 1). Inclusion of total errors allows us to get statistically acceptable fit qualities for HBT radii for both function (7) and its specific case. It seems that a more complex fit function should be used in order to describe energy dependence of HBT radii at all available collision energies. This study is in progress. Taking into account the similar behavior of the energy dependence of HBT radii (Figures 2(b)–2(b)) and elliptic flow [22] at qualitative level the following functional form can be suggested: , as first approach for description of , , in all experimentally available energy domains. Smooth solid and dashed curves shown in Figure 1(e) are calculated for the ratio from the fit results for and (Table 1). As seen these curves agree with experimental points reasonably at  GeV. In general, fits by function (7) at free and fixed show close behavior for all the main HBT parameters from with some differences at intermediate ( GeV) and high ( GeV) energies. These differences result in more significant discrepancy between fit curves for (Figure 2(e)) and for other parameters from the set (see discussion below).

Table 1: Values of fit parameters for approximation of the data sample (i).

HBT parameter
Fit with statistical errors
Fit with total errors

/n.d.f.

/n.d.f.

0.36 0.02
1.90 0.12
−0.91 0.14

0.008 ± 0.002
104 21
−0.291 0.011

0.570 0.004
−0.021 0.001
1.0 (fixed)

0.621 ± 0.005
−0.032 0.001
1.0 (fixed)

4.77 0.02
(1.3 ± 0.8) 10⁻⁴
2.8 ± 0.2

4.5 ± 0.2
(1 2) 10⁻³
2.3 0.9

4.37 0.03
0.0188 0.0012
1.0 (fixed)

3.88 ± 0.18
0.038 ± 0.008
1.0 (fixed)

0.59 0.06
7.4 1.0
0.097 0.009

0.64 ± 0.11
7 ± 2
0.03 0.03

5.38 0.04
0.0120 0.0012
1.0 (fixed)

5.3 ± 0.2
0.003 ± 0.002
1.0 (fixed)

0.110 0.015
32 4
0.267 0.008

0.03 ± 0.03
87 ± 64
0.33 0.03

4.51 0.04
0.0475 0.0018
1.0 (fixed)

4.0 ± 0.2
0.065 ± 0.010
1.0 (fixed)

Figure 2: Energy dependence of emission duration for secondary charged pions in central heavy ion collisions , , in midrapidity region and at  GeV/. Experimental results are shown for the same particle types and Coulomb corrections as well as in Figure 1. Error bars are only statistical (for NA44, total uncertainties). Smooth curves are derived from (5) and the fit results for , without the point of the WA97 experiment [11]. The solid line corresponds to the fits of HBT radii by function (7) and dashed line to the fits by specific case , .

Figure 2 demonstrates the energy dependence of for (quasi) symmetric heavy ion collisions. The emission duration in these collisions is calculated based on known HBT radii (Figures 1(b)–1(d)), kinematic regime for pion pairs, and (5). The for pion pairs with  GeV/. Value  fm/ at  GeV derived from the PHENIX results at this energy is not shown due to extremely large errors. As seen the emission duration of pions extracted from is about  fm/ for any energies under consideration. The visible energy dependence of emission duration is absent, and is close to flat within large error bars. One can see more interesting behavior for this dependence for the STAR high-statistics data [15] only. But additional precise measurements are necessary in order to confirm the change of at  GeV and locate the possible knee in the experimental dependence. Smooth solid and dashed curves shown in Figure 2 are calculated for from the fit results for and (Table 1). It seems that function (7) at free agrees better with experimental points at  GeV than that at fixed . But large error bars do not allow the choice for preferable curve unambiguously. Moreover, the general function (7) underestimates in TeV-region significantly. Volume of the homogeneity region in various heavy ion collisions is calculated based on (3) and known HBT radii which are shown in Figures 1(b)–1(d). The energy dependence of estimations for volume of emission region is shown in Figure 3. Detailed comparison of the results for various definitions of as well as for different sets of STAR data is discussed elsewhere [6]. Smooth solid and dashed curves shown in Figure 3 are calculated for from (3) and the fit results for , (Table 1). Both curves are very close at  GeV but function (7) at underestimates in TeV-region significantly. Therefore, the general function (7) is the preferable approximation of the experimental at  GeV.

Figure 3: Energy dependence of volume of emission region at freeze-out for secondary charged pions in central heavy ion collisions , , in midrapidity region and at  GeV/. Equation (3) is used for calculation of volume values. Experimental results are shown for the same particle types and Coulomb corrections as well as in Figure 1. Error bars are only statistical (for NA44, total uncertainties). Smooth curves are derived from (3) and the fit results for , taking into account the point of the WA97 experiment [11]. The solid line corresponds to the fits of HBT radii by function (7) and dashed line to the fits by specific case , .

Predictions for values of the HBT observables from sets , , are obtained for heavy ion mode energies of the LHC and the FCC project based on the fit results for the main HBT parameters. Estimations are shown in Table 2 for fits with inclusion of statistical errors, and the second line for each collision energy corresponds to the using of the specific case of (7). Large uncertainties obtained for estimations based on function (7) do not allow us to distinguish predictions from (7) with free and with fixed . One can expect the volume of homogeneity region  fm³ at  TeV (LHC) and  fm³ at  TeV (FCC) based on the reasonable agreement between experimental data and solid curve in Figure 3.

Table 2: Estimations for observables based on fit results for the data sample (i).

, TeV
HBT parameter

, fm
, fm
, fm

, fm/
, fm³

5.52
0.41 ± 0.03
6.8 ± 1.9
6.3 ± 1.0
7.6 ± 1.5
0.9 ± 0.3
—
6 ± 3

0.362 ± 0.009
5.79 ± 0.10
6.49 ± 0.12
8.20 ± 0.16
1.12 ± 0.03
3.6 ± 0.4
4.33 ± 0.17

39.0
0.40 ± 0.02
8 ± 3
6.4 ± 1.0
8.0 ± 1.6
0.8 ± 0.3
—
9 ± 8

0.315 ± 0.011
6.11 ± 0.12
6.74 ± 0.15
9.04 ± 0.19
1.10 ± 0.03
3.5 ± 0.5
5.3 ± 0.2

Figure 4 shows the energy dependence of (a), scaled HBT radii (b–d), and ratio (e) for both the symmetric and asymmetric collisions of various nuclei. Fits of experimental dependencies for the data sample (ii) are made by (7) in the same energy domains and with the same error types as well as for the data sample (i) and are shown in Table 2. It seems that the value from the WA97 experiment [11] cannot be excluded from the data sample (ii) because there are the STAR results for collisions also (Figure 4(a)). There are no physics reasons in order to exclude the points of these experiments from the fitted data sample (ii) in the case of all available nucleus-nucleus collisions. Furthermore, the scaled value of longitudinal radius from [11] agrees better with results of other experiments at close energies (Figure 4(d)) than that for the data sample (i). Therefore, there is no exception of any experimental point from fitted ensemble for any HBT observable in Figure 4 in contrast with the fitting procedure for the data sample (i). The numerical values of fit parameters are presented in Table 3, where the second line for and each normalized HBT radius correspond to the approximation by specific case of (7). Fit curves are shown in Figure 4 by solid lines for (7) and by dashed lines for specific case of fit function at taking into account statistical errors. Fit qualities are improved for in the case of total errors of experimental point and for at any error types of experimental point with respect to the corresponding fit results for the data sample (i) shown in Table 1. There is dramatic growth of . values for fits of data (Figure 4(a)) despite the qualitative agreement between smooth approximations and experimental values for range  GeV. The fit by (7) at underestimates the value at the LHC energy  TeV significantly. The values for asymmetric nucleus-nucleus collisions at intermediate energies  GeV agree well with values of in symmetric heavy ion collisions at close energies. On the other hand, the for collisions is smaller systematically than in collisions in energy range  GeV (Figure 4(a)). New experimental data are important for verification of the suggestion of separate dependencies for moderate and heavy ion collisions. Also the development of some approach is required in order to account for type of colliding beams in the case of parameter and improve quality of smooth approximation. In this case, significant growth of as well as improvement of the fit quality at transition from statistical errors of experimental points to total errors in the data sample (ii) (Table 3) is dominated by inclusion of estimations for systematic uncertainties for collisions and/or ones at  TeV. Smooth curves for scaled HBT radii and ratio are in reasonable agreement with experimental dependencies in fitted domain of collision energies  GeV (Figures 4(b)–4(e)). Parameter values obtained for fit of with total uncertainties by (7) at are equal within errors with results from [23] accounting for the fact that experimental results studied here are obtained at . Dramatic improvement of the fit qualities for scaled HBT radii at transition from the data sample (ii) with statistical errors to the data sample with total errors (Table 3) is dominated mostly by the uncertainty in that leads to additional errors due to scaling (6). At the same time inclusion of total uncertainties for collisions at  GeV results in significant decreasing of parameter in the case of scaled radius.

Table 3: Values of fit parameters for approximation of the data sample (ii).

HBT parameter
Fit with statistical errors
Fit with total errors

/n.d.f.

/n.d.f.

1.21 0.09
−0.30 0.04
0.38 0.04

0.60 ± 0.02
−0.014 ± 0.008
1.3 0.2

0.717 0.003
−0.051 0.001
(fixed)

0.631 ± 0.005
−0.034 ± 0.001
1.0 (fixed)

0.656 0.002
(6 ± 3) 10⁻⁵
3.11 0.19

0.63 ± 0.02
(6 5) 10⁻⁴
2.4 0.9

0.599 0.003
0.019 0.001
(fixed)

0.56 ± 0.03
0.029 0.008
1.0 (fixed)

0.10 0.02
6.3 1.7
0.068 0.006

0.019 ± 0.003
30 9
0.12 0.05

0.758 0.004
0.008 0.001
(fixed)

0.67 ± 0.04
0.017 0.008
1.0 (fixed)

0.022 0.002
23 3
0.258 0.005

0.23 ± 0.04
0.8 0.2
0.57 0.05

0.634 0.004
0.043 0.001
(fixed)

0.47 ± 0.03
0.089 0.014
1.0 (fixed)

(a)

(b)

(c)

(d)

(e)

Figure 4: Energy dependence of parameter (a), scaled HBT radii (b–d), and ratio (e) in various nucleus-nucleus collisions at  GeV/ [6]. Experimental results are shown for central collisions (for minimum bias event in the case of E802 for ), for pairs of mesons (in the cases of ALICE and STAR for both and at and 200 GeV for pairs, E802 for , and NA44 for , for pairs of mesons), and for standard Coulomb correction (in the cases of ALICE, NA44, NA45, PHOBOS, and STAR for both and at and 200 GeV, for correction ). Statistical errors are shown (for NA44, total uncertainties). The solid lines (a–d) correspond to the fits by function (7) and dashed lines to the fits by specific case of (7) at fixed . Smooth solid and dashed curves at (e) correspond to the ratio calculated from the fit results for and , and dotted line is the level .

The corresponding dependencies for and are demonstrated in Figures 5 and 6, respectively. And in [1] results for pairs are shown in Figures 4–6 also because HBT parameters from the set depend on sign of electrical charge of secondary pions weakly. Relation is observed for of points in Figure 5. In general the can be possible in the model of opaque source with surface dominated emission [24, 25]. But possibly results should be similar for both the same ion beams and close kinematic regimes in various experiments. Therefore, additional study is required in order to distinguish the physical and technique sources of negative values of the in Figure 5 and to get a more definite explanation. The dependence is almost flat within large error bars in all energy domains under consideration. Taking into account the STAR high-statistics results [15] only one can see the indication on change of behavior of inside the range of collision energies  GeV. This observation is in agreement with features of behavior of emission duration dependence on (Figure 2) discussed above. The estimation of energy range agrees well with results of several studies [15, 26–31] in the framework of the phase-I of BES program at RHIC which indicate the transition from dominance of quark-gluon degrees of freedom to hadronic matter at  GeV. But future precise measurements are crucially important for extraction of more definite physics conclusions. Smooth solid and dashed curves shown in Figure 5 are calculated for from the fit results for and (Table 3). The situation is similar to that for : calculation based on the fit function (7) at free agrees reasonably with experimental points at  GeV but underestimates in TeV-region significantly. The large errors in Figure 6 for strongly asymmetric nuclear collisions are dominated by large difference of radii of colliding moderate and heavy nuclei and corresponding large uncertainty for . Smooth solid and dashed curves shown in Figure 6 are calculated for from (3) and the fit results for , (Table 3). The fit results for scaled HBT radii obtained with general function (7) lead to very good agreement between smooth curve and experimental data in TeV-region in contrast with the curve obtained from corresponding fit results for (7) at .

Figure 5: Dependence of scaled difference of squares of transverse radii on collision energy for emission region of secondary charged pion in various nucleus-nucleus collisions at  GeV/. Experimental results are shown for the same particle types and Coulomb corrections as well as in Figure 4. Error bars are only statistical (for NA44, total uncertainties). Dotted line is the level . Smooth curves are derived from (6) and the fit results for , . The solid line corresponds to the fits of scaled HBT radii by function (7) and dashed line to the fits by specific case , .

Figure 6: Energy dependence of scaled volume of emission region at freeze-out for secondary charged pions in various nucleus-nucleus collisions at  GeV/. Equation (3) is used for calculation of volume values. Experimental results are shown for the same collision, particle, and Coulomb correction types as well as in Figure 4. Error bars are only statistical (for NA44, total uncertainties). Smooth curves are derived from (3) and the fit results for , . The solid line corresponds to the fits of scaled HBT radii by function (7) and dashed line to the fits by specific case , .

Estimations for , , and scaled HBT parameters at the LHC and the FCC energies are shown in Table 4 for fits of various nucleus-nucleus collisions with inclusion of statistical errors, and the second line for each collision energy corresponds to the using of the specific case of (7) at . All the smooth approximations discussed above predict amplification of coherent pion emission with significant decreasing of . Uncertainties are large for estimations obtained on the basis of results of fits by function (7) at free . Thus, values of HBT observables in Table 4 are equal within errors for general and specific case of (7) at  TeV (LHC) and  TeV (FCC) as well as for estimations obtained on basis of the data sample (i) above.

Table 4: Estimations for observables based on fit results for the data sample (ii).

, TeV
HBT parameter

5.52
0.16 0.19
0.9 0.2
0.8 0.3
1.06 0.16
0.9 0.4
−0.2 0.6
3.5 1.6

0.091 0.004
0.792 0.009
0.860 0.010
1.099 0.013
1.086 0.018
0.11 0.02
2.59 0.07

39.0
0.07 0.21
1.2 0.4
0.9 0.3
1.11 0.16
0.7 0.3
−0.7 1.1
6 4

—
0.836 0.011
0.883 0.012
1.205 0.015
1.06 0.02
0.08 0.03
3.17 0.09

The energy dependencies for sets , , of HBT parameters taking into account the scaling relation (6) and the high-statistics STAR data [15] demonstrate reasonable agreement between values of parameters obtained for interactions of various ions (shown in Figures 4–6). The observation confirms the suggestion [1] that scaled HBT parameters allow us to unite the study of both the symmetric and asymmetric nucleus-nucleus collisions in the framework of united approach. This qualitative suggestion is confirmed indirectly by recent study of two-pion correlations in the collisions of the lightest nucleus () with heavy ion () at RHIC. Estimations of space-time extent of the pion emission source in collisions at top RHIC energy [32] in dependence on kinematic observables (collision centrality, the mean transverse momentum for pion pairs) indicate similar patterns with corresponding dependencies in collisions and indicate similarity in expansion dynamics in collisions of various systems ( and at RHIC, , and at LHC). The scaling results for some radii indicate that hydrodynamic-like collective expansion is driven by final-state rescattering effects [32]. On the other hand, the scaled HBT parameters allow us to get the common kinematic dependencies only without any additional information about possible general dynamic features in different collisions. Thus, the hypothesis discussed above is qualitative only. The future quantitative theoretical and phenomenological studies are essential for verification of general features of soft stage dynamics for different collisions at high energies.

4. Summary

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

Energy dependence is investigated for a range of all experimentally available initial energies and for estimations of the main HBT parameters from the set ( and radii) derived in the framework of Gauss approach as well as for the set of important additional observables included ratio of transverse radii, emission duration (or ), and HBT volume. There is no dramatic change of values of HBT parameters with increasing of in domain of collision energies  GeV. The estimation of emission duration of pions is about  fm/ for any energies under consideration. The energy dependence is almost flat for both the emission duration and the parameter within large error bars. The indication on possible curve knee at  GeV obtained in the STAR high-statistics data agrees with other results in the framework of the phase-I of the beam energy scan program at RHIC. But additional precise measurements are crucially important at various in order to confirm this feature in energy dependence of additional parameters ().

Analytic function is suggested for approximation of energy dependence of main HBT parameters. The fit curves demonstrate qualitative agreement with experimental data for at all available collision energies and for both the absolute and scaled HBT radii in energy domain  GeV. Reasonable fit qualities are obtained for HBT radii at approximation of experimental points with total errors. Smooth curves calculated for energy dependence of the set of additional HBT parameters agree reasonably with corresponding experimental data in most cases. Estimations of HBT observables are obtained on the basis of the fit results for energies of the LHC and the FCC project. For multi-TeV energy domain the emission region of pions will be characterized by decreased chaoticity parameter, linear sizes about 8.5–9.5 fm in longitudinal direction and 7-8 fm in transverse plane, and volume of about  fm³.

Conflict of Interests

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

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