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.

In general, phenomenological parameterization of correlation function (CF) for two identical particles with 4-momenta p1, p2 taking into account different forms of corrections on Coulomb final state interaction (FSI) can be written as follows [1]:(1)C2,mphq,K=ϵPcoulmqϵ-1+K2phA,ϵ=λ,at  m=1,2;1,at  m=3,where m=1 corresponds to the standard Coulomb correction, m=2 the dilution procedure, and m=3 the Bowler-Sinyukov correction; q≡(q0,q→)=p1-p2 is the relative 4-momentum and K≡(K0,K→)=(p1+p2)/2 the average 4-momentum of particles in pair (pair 4-momentum), for the standard simplest (Gaussian) case:(2)K2phA=∏i,j=13‍K2phAij=exp-∑i,j=13‍qiRij2qj.Here A≡q→R2q→T and R2 are the matrices 3×3 and q→T is transposed vector q→, ∀i,j:Rij2=Rji  2, Rii  2≡Ri  2, where Ri=Ri(K) are parameters that characterized the linear scales of homogeneity region [2]; the products are taken on space components of vectors; λ(K)=K2(0,K), 0≤λ≤1, 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 (K→) is decomposed on longitudinal k∥=(p∥,1+p∥,2)/2 and transverse k→⊥=(p→⊥,1+p→⊥,2)/2 parts of pair momentum. In the paper, the decomposition of Pratt-Bertsch [3, 4] is used for q→ as well as the longitudinal comoving system (LCMS) frame. The volume of source can be written as follows:(3)V=2π3/2Rs2Rl.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]:(4)δ≡Ro2-Rs2.If the emission function features no position-momentum correlation, then δ is finite at nonzero K→ only due to explicit K→-dependence (resulting from the mass-shell constraint q0=q→K→/K0) [7]. In this case(5)δ≈β⊥2Δτ2,where β⊥=k⊥/m⊥ is the transverse velocity of pair of particles with mass m, m⊥2=k⊥2+m2, and Δτ 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, δ 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 Rij [7]. The sensitivity to the Δτ is the main advantage of the observable (4).

In the paper, the following set of main HBT observables G1≡{G1i}i=14={λ,Rs,Ro,Rl} is under consideration as well as the set of some important additional observables which can be calculated with help of HBT radii G2≡{G2j}j=13={Ro/Rs,Δτ,V}. The set of parameters G1 characterizes the chaoticity of source and its 4-dimensional geometry at freeze-out stage completely. Scaled parameters G1i, i=2–4, δ, and G23 are calculated as follows [1]:(6)Rin=RiRA, i=s,o,l; δn=δRA2; Vn=VVA.Here RA=r0A1/3 and VA=4πRA3/3 are radius and volume of spherically symmetric nucleus, r0=(1.25±0.05) fm [9, 10]. The change RA→〈RA〉=(RA1+RA2)/2 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, δ 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].

Dependencies of HBT parameters G1i(sNN), i=1–4, and Ro/Rs(sNN) are shown in Figures 1(a)–1(d) and Figure 1(e), respectively. The chaoticity parameter λ decreases with increasing sNN rapidly at lower (AGS) energies and shows the weak changing at sNN>4 GeV (Figure 1(a)). HBT radii of source in transverse plane with respect to the beam direction, Rs (Figure 1(b)) and Ro (Figure 1(c)), show little change over a wide range of energies 5≲sNN≲200 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, Rl (Figure 1(d)), appears to reach a minimum around sNN=5 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 sNN~20 GeV up to the maximum available LHC energy sNN=2.76 TeV. The significant increasing of HBT radii is seen for much broader energy range (on about two orders of magnitude sNN~0.02–3 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 Rs reflects the spatial extent of particle source, whereas Ro 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 Ro/Rs 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 Ro/Rs 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 Ro/Rs at top RHIC energy and demonstrate that the behavior of experimental dependencies of Ro/Rs 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.

Taking into account the view of experimental dependencies in Figures 1(a)–1(d), the following function is suggested:(7)fsNN=a11+a2lnɛa3for smooth approximation of G1i(sNN), i=1–4, where ɛ≡sNN/s0, s0=1 GeV². Also the specific case of (7) at a3=1.0 is under consideration. As seen from Figures 1(b)–1(d), there is indication on change of behavior of energy dependence (inflection point) for {G1i}i=24 at sNN≃5 GeV. This inflection point is seen most clearly for Rl (Figure 1(d)). Therefore, the fit function (7) is used for approximation of the energy dependence of HBT radii in the energy domain sNN≥5 GeV only. Experimental energy dependence of λ 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 λ (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 {G1i}i=14 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 {G1i}i=14 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 a3=1.0 taking into account the statistical errors. In general, fit function described above agrees reasonably with experimental dependence G1i(sNN), i=1–4 (Figures 1(a)–1(d)). But the fit qualities are poor for all the main HBT parameters, especially for λ, with inclusion of the statistical errors only (Table 1). Spread of experimental points leads to the statistically unacceptable values of χ2/n.d.f. In the case of λ inclusion of estimation for systematic uncertainty for Pb+Pb collisions at sNN=2.76 TeV leads to both the dramatic growth of a2 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 v2 [22] at qualitative level the following functional form can be suggested: g(sNN)=a1+a2(ɛ-a3)a4+∑i=5,6‍aiɛai+1+a9(lnɛ)a10, as first approach for description of G1i(sNN), i=1–4, in all experimentally available energy domains. Smooth solid and dashed curves shown in Figure 1(e) are calculated for the ratio Ro/Rs from the fit results for Rs and Ro (Table 1). As seen these curves agree with experimental points reasonably at sNN≥5 GeV. In general, fits by function (7) at free a3 and fixed a3 show close behavior for all the main HBT parameters from G1 with some differences at intermediate (sNN≲10 GeV) and high (sNN>200 GeV) energies. These differences result in more significant discrepancy between fit curves for Ro/Rs (Figure 2(e)) and for other parameters from the set G2 (see discussion below).

Figure 2 demonstrates the energy dependence of Δτ 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 〈β⊥〉≈0.82 for pion pairs with 〈k⊥〉≃0.2 GeV/c. Value Δτ=(0.53±9.15) fm/c at sNN=130 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 δ is about 2–4 fm/c for any energies under consideration. The visible energy dependence of emission duration is absent, and Δτ(sNN) 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 Δτ(sNN) at sNN~10–20 GeV and locate the possible knee in the experimental dependence. Smooth solid and dashed curves shown in Figure 2 are calculated for Δτ from the fit results for Rs and Ro (Table 1). It seems that function (7) at free a3 agrees better with experimental points at sNN≤200 GeV than that at fixed a3. But large error bars do not allow the choice for preferable curve unambiguously. Moreover, the general function (7) underestimates Δτ 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 V 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 V from (3) and the fit results for Rs, Rl (Table 1). Both curves are very close at sNN≤200 GeV but function (7) at a3=1.0 underestimates V in TeV-region significantly. Therefore, the general function (7) is the preferable approximation of the experimental V(sNN) at sNN≥5 GeV.

Predictions for values of the HBT observables from sets Gm, m=1,2, 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 a3 and with fixed a3=1.0. One can expect the volume of homogeneity region V~6000 fm³ at sNN=5.52 TeV (LHC) and V~9000 fm³ at sNN=39.0 TeV (FCC) based on the reasonable agreement between experimental data and solid curve in Figure 3.

Figure 4 shows the energy dependence of λ (a), scaled HBT radii (b–d), and Ro/Rs 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 λ value from the WA97 experiment [11] cannot be excluded from the data sample (ii) because there are the STAR results λ~0.3 for Cu+Cu 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 Rln 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 λ 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 a3=1.0 taking into account statistical errors. Fit qualities are improved for Rsn in the case of total errors of experimental point and for Ron 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 χ2/n.d.f. values for fits of λ data (Figure 4(a)) despite the qualitative agreement between smooth approximations and experimental λ values for range 10≲sNN≲200 GeV. The fit by (7) at a3=1.0 underestimates the λ value at the LHC energy sNN=2.76 TeV significantly. The λ values for asymmetric nucleus-nucleus collisions at intermediate energies sNN≲20 GeV agree well with values of λ in symmetric heavy ion collisions at close energies. On the other hand, the λ for Cu+Cu collisions is smaller systematically than λ in Au+Au collisions in energy range sNN=62–200 GeV (Figure 4(a)). New experimental data are important for verification of the suggestion of separate dependencies λ(sNN) 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 λ parameter and improve quality of smooth approximation. In this case, significant growth of a2 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 Cu+Cu collisions and/or Pb+Pb ones at sNN=2.76 TeV. Smooth curves for scaled HBT radii and ratio Ro/Rs are in reasonable agreement with experimental dependencies in fitted domain of collision energies sNN≥5 GeV (Figures 4(b)–4(e)). Parameter values obtained for fit of Rln with total uncertainties by (7) at a3=1.0 are equal within errors with results from [23] accounting for the fact that experimental results studied here are obtained at 〈m⊥〉≃1.75mπ. 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 r0 that leads to additional errors due to scaling (6). At the same time inclusion of total uncertainties for Au+Au collisions at sNN=19.6 GeV results in significant decreasing of a2 parameter in the case of Rln scaled radius.

The corresponding dependencies for δn and Vn are demonstrated in Figures 5 and 6, respectively. And in [1] results for π+π+ pairs are shown in Figures 4–6 also because HBT parameters from the set G1 depend on sign of electrical charge of secondary pions weakly. Relation Ro<Rs is observed for ≈11% of points in Figure 5. In general the δ<0 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 δn in Figure 5 and to get a more definite explanation. The dependence δn(sNN) 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 δn(sNN) inside the range of collision energies sNN=11.5–19.6 GeV. This observation is in agreement with features of behavior of emission duration dependence on sNN (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 sNN≲19.6 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 δn from the fit results for Rsn and Ron (Table 3). The situation is similar to that for Δτ: calculation based on the fit function (7) at free a3 agrees reasonably with experimental points at sNN≤200 GeV but underestimates δn 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 〈RA〉. Smooth solid and dashed curves shown in Figure 6 are calculated for Vn from (3) and the fit results for Rsn, Rln (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 a3=1.0.

Estimations for λ, Ro/Rs, 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 a3=1.0. All the smooth approximations discussed above predict amplification of coherent pion emission with significant decreasing of λ. Uncertainties are large for estimations obtained on the basis of results of fits by function (7) at free a3. Thus, values of HBT observables in Table 4 are equal within errors for general and specific case of (7) at sNN=5.52 TeV (LHC) and sNN=39.0 TeV (FCC) as well as for estimations obtained on basis of the data sample (i) above.

The energy dependencies for sets Gm, m=1,2, 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 (d) with heavy ion (Au) at RHIC. Estimations of space-time extent of the pion emission source in d+Au 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 Au+Au collisions and indicate similarity in expansion dynamics in collisions of various systems (d+Au and Au+Au at RHIC, p+Pb, and Pb+Pb 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.

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 G1 (λ and radii) derived in the framework of Gauss approach as well as for the set of important additional observables G2 included ratio of transverse radii, emission duration (or δ), and HBT volume. There is no dramatic change of values of HBT parameters with increasing of sNN in domain of collision energies sNN≥5 GeV. The estimation of emission duration of pions is about 2–4 fm/c for any energies under consideration. The energy dependence is almost flat for both the emission duration and the δn parameter within large error bars. The indication on possible curve knee at sNN~10–20 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 sNN in order to confirm this feature in energy dependence of additional parameters (Ro/Rs,Δτ,δn).

Analytic function is suggested for approximation of energy dependence of main HBT parameters. The fit curves demonstrate qualitative agreement with experimental data for λ at all available collision energies and for both the absolute and scaled HBT radii in energy domain sNN≥5 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 G2 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 104 fm³.