# Lévy Distributions for One-Dimensional Analysis of the Bose–Einstein Correlations

Authors: V. A. Okorokov
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

A general study of relations between the parameters of two centrally symmetric Lévy distributions, often used for one-dimensional investigation of Bose–Einstein correlations, is given for the first time. These relations of the strength of correlations and of the radius of the emission region take into account possible various finite ranges of the Lorentz invariant four-momentum difference for two centrally symmetric Lévy distributions. In particular, special cases of the relations are investigated for Cauchy and normal (Gaussian) distributions. The mathematical formalism is verified using the recent measurements given that a generalized centrally symmetric Lévy distribution is used. The reasonable agreement is observed between estimations and experimental results for all available types of strong interaction processes and collision energies.


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## Body

## 1. Introduction

Correlations between two identical bosons, called Bose–Einstein correlations (BEC), are a well-known phenomenon in high-energy and nuclear physics. These correlations play an important role in the studies of multiparticle production and soft physics. Constructive interference affects the joint probability for the emission of a pair of identical bosons with four-momentump1 and p2. Experimentally, the one-dimensional BEC effect is observed as an enhancement at low values of the Lorentz invariant quantity q=-(p1-p2)2≥0 in the two-particle correlation function (CF): (1)C2q=ρqρrefq.Here ρ is the two-particle density function and ρref is a reference two-particle density function that by construction is expected to include no BEC. The detailed shape analysis of the peak of CF is an important topic on theoretical and experimental points of view because this shape carries information about the possible features of space-time structure of particle source [1, 2]. For instance, the detail investigations have to do for shape of correlation peak in modern experiments with high statistics for verification of hypothesis of possible self-affine fractal-like geometry of emission region [3, 4]. The BEC effect in one dimension is usually described by a few-parameter function for which several different functional forms have been proposed. The power-law parametrization C2(q)~q-β is the important signature for fractal-like source extending over a large volume [5, 6]. The quite reasonable fit is achieved with this parametrization of two-pion CF in various types of multiparticle production processes [1]. But unfortunately power-law fits are absent for high-statistics modern experimental data so far. On the other hand the stable (on Lévy) distributions [7] are one of the most promising tools for studies of fractal-like space-time extent of emission region. These distributions are a rich class of probability distributions that allow skewness and heavy tails and have many important physical applications. As shown in [3, 4] the subclass of nonisotropic centrally symmetric Lévy distributions [8, 9] is most useful for studies of Bose–Einstein CF. Therefore this subclass of centrally symmetric Lévy distributions is considered regarding of BEC measurements in the present paper.For low-dimensional (1D) analysis the centrally symmetric Lévy distribution results in the most general parametrization of the experimental Bose–Einstein CF(2)C2q∝1+Ωα,λ,z,Ωα,λ,z≡λexp⁡-zα,z≡qR.Here λ is the strength of correlations called also chaoticity, R is the 1D BEC radius, and 0<α≤2 is the Lévy index called also index of stability. As known for a static source with no final state interactions [10, 11], there is the relation C2(q)∝|f~(q)|2, f~(q)=∫dxexp⁡(iqx)f(x); that is, Bose–Einstein CF C2(q) measures the absolute value squared Fourier transformed source density in the coordinate space f(x), ∫dxf(x)=1, called also coordinate-space distribution function of the particle emission points. The various experiments use the different forms of (2) which correspond to the various hypotheses with regard of f(x). For example, most of the earliest experiments with particle beams used the specific case of (2) at α=2; the Gaussian parametrization corresponded to the normal (Gaussian) distribution function fG(x)=(2πRG2)-1/2exp⁡[-(x-x0)2/2RG2], where the Gaussian scale parameter is RG2=〈x2〉-x02, the standard deviation; then another specific case of (2) at α=1 is used widely, especially, for particle (not nuclear) collisions. Equation (2) at α=1 is called exponential parametrization for Bose–Einstein CF C2(q) and it corresponds to the Cauchy (Lorentzian) distribution function fC(x)=π-1RC/[RC2+(x-x0)2] with scale parameter RC [12, 13]. Furthermore the recent studies at the LHC [14–17] demonstrate that general view of (2) allows the reasonable description of experimental CF, particular for proton-proton (p+p) collisions but for centrally symmetric Lévy distribution with α∈(0;2), α≠1 the corresponding source density in coordinate space f(x) can be written analytically for α=3/2,2/3,1/2,1/3 only [9]. It is often difficult to compare results from different experiments because of the many different data analysis methods [11], in particular due to various parameterizations for 1D Bose–Einstein CF C2(q). Therefore the derivation of the relations between the sets of BEC parameters for two centrally symmetric Lévy distributions is the important task for correct comparison of the results from different experiments, creation of the global kinematic (energy, pair transverse momentum, etc.) dependencies of BEC parameters, and so on. Such studies are important for investigations of common features of soft-stage dynamics in various multiparticle production processes as well as for equation of state (EoS) of strongly interacting matter, in particular, search for phase transition to the quark-gluon deconfined matter. It would be noted the study of energy dependence of pion BEC parameters in heavy ion collisions [18] was one of the main causes and drivers for hypothesis of cross-over transition from strongly coupled quark-gluon phase to hadronic one at Relativistic Heavy Ion Collider (RHIC) energies sNN~100 GeV. Furthermore some results for deconfinement in small system [19, 20] indicate remarkable similarity of both the bulk and the thermodynamic properties of strongly interacting matter created in high-energy p+p/p¯+p and A+A collisions. The BEC can provide new knowledge about collectivity and possible creation of droplets of quark-gluon matter in small system collisions. For these studies the correct comparison can be crucially important for BEC parameters in various multiparticle processes for wide energy range. But as mentioned above Bose–Einstein CF C2(q) is often described by different view of (2) depending on type of reaction, collision energy, and features of experimental analysis. Therefore the study of centrally symmetric Lévy distributions and search for relations between parameters for corresponding CF has scientific interest for physics of strong interactions.The paper is organized as follows. In Section2, mathematical formalism is described for case of two general view centrally symmetric Lévy distributions. Dependencies of desired 1D BEC observables on q and α are studied fora priori known parameters for second centrally symmetric Lévy distributions. Section 3 is devoted to the detailed discussion of specific case of these distributions, namely, Cauchy and Gaussian ones most used in experimental investigations of 1D CF C2(q). Database of experimental results for set of 1D BEC parameters {λ,R} for charged pion source in strong interaction processes is created within the framework of the paper in order to verify the mathematical formalism. Section 4 demonstrates the comparison between the estimations calculated for 1D BEC parameters with help of mathematical formalism under discussion and available experimental results for various reactions and in wide energy range (it should be noted that in Sections 3 and 4 the 1D BEC parameters are supplied with the subindexes according to the names of corresponding source distribution function; namely, “L” is for the general view of centrally symmetric Lévy distribution, “C” is for the Cauchy source distribution function, and “G” is for Gaussian one. Otherwise the notations {λE,RE} are often used in papers for second case due to relation between Cauchy distribution for f(x) and exponential parametrization for Bose–Einstein CF C2(q) discussed above. As a consequence the mathematically rigorous terminology is used over full manuscript: the term “Cauchy distribution” corresponds to the source function in coordinate space fC(x) and the term “exponential function/parametrization” is used for the related parametrization of correlation function C2,E(q); for the case of arbitrary 0<α<2, α≠1, the term “centrally symmetric Lévy” is suitable for both the source function in coordinate space fL(x) and the parametrization of correlation function C2,L(q); the similar situation is for α=2: the term “Gaussian” is applicable for both the source function in coordinate space fG(x) and the corresponding parametrization of correlation function C2,G(q)). In Section 5 some final remarks are presented. The experimental database is shown in Appendix for 1D BEC parameters.


## 2. Relations between BEC Parameters in General Case

Let some experimental CFC2(q) be described by two parameterizations (2) with Ω1≡Ω(α1,λ1,z1) and Ω2≡Ω(α2,λ2,z2). Then relations between parameters of Ω1 and Ω2 can be deduced on the basis that both parameterizations describe one experimental CF C2(q), that is one sample of experimental points (in general the approximations of C2(q) are characterized by different qualities for various parameterizations  (2)  with Ω1 and Ω2. The influence of this difference is not studied in present work and can be considered as separate task). Thus one can assume that the areas under fit curves for two parameterizations (2) with Ω1 and Ω2 are approximately equal to each other as well as the first moments of the corresponding centrally symmetric Lévy distributions.


### 2.1. Mathematical Formalism

The relations between two sets of parameters{λ1,R1} and {λ2,R2} of the particle source can be derived from the following system of equations:(3a)S1=S2,∀i=1,2:Si≡∫JiΩαi,λi,zidq;(3b)q1=q2,∀i=1,2:qi≡Si-1∫JiqΩαi,λi,zidq.The first equation (3a) corresponds to the equality of the areas under fit curves and the (3b) is the equality of the first moments of the Ω(αi,λi,zi) distributions (as discussed above the approximate equalities are expected for areas and first moments in general case. This softer condition is enough for applicability of the formalism suggested in the paper. But the exactly equal signs are used in  (3a) and (3b)  as well as in the text below in order to get the mathematically correct forms for the systems of equations). System  (3a) and (3b)  contains the equations allowing the estimation of unknown strength of correlations and 1D BEC radius based on the available values of these parameters but it supposes the Lévy indexes αi are knowna priori for both parameterizations Ωi, i=1,2. In equations (3a) and (3b) the integrals are taken over full fit ranges Ji, i=1,2, for corresponding parameterizations with Ωi of experimental CF. It should be noted that in general case (i) the ranges of integration can be different for parameterizations with Ω1 and Ω2; (ii) the full fit range can be the set of subranges due to possible exception of some intervals of the relative 4-momentum (regions of resonance contributions, etc.), that is, ∀i=1,2:Ji=⋃k=1NiJik≡⋃k=1Ni[qk,min,qk,max] and consequently for all types of integrals and ∀i=1,2 in system (3a) and (3b): ∫Ji→∑k=1Ni∫Jik. But usually the fit ranges Ji are identical for both Ωi, i=1,2, in experimental studies (see, e.g., [15]). In general case of the centrally symmetric Lévy distributions and finite fit ranges the system equations under consideration cannot be solved analytically. The numerical procedure should be used in order to get the relations between two sets of parameters {λ1,R1} and {λ2,R2} of the particle source in this case. Without loss of generality Ω1 and α2 are considered as known and values of BEC parameters {λ2,R2} are supposed as desired below. Then for specific case of semi-infinite ranges for integration ∀i=1,2:Ji=[0;∞) system (3a) and (3b) can be solved analytically and one can derive the following ultimate relations between two sets {λ1,R1} and {λ2,R2} of BEC parameters for corresponding centrally symmetric Lévy parameterizations with Ωi, i=1,2(4a)λ1u=λ2α1Γ2α1-1Γ2α2-1α2Γ2α1-1Γ2α2-1-1;(4b)R1u=R2Γ2α1-1Γα2-1Γα1-1Γ2α2-1-1and vice versa. Here Γ(x)=∫0∞tx-1exp⁡(-t)dt,Rex>0 is the gamma function.In the point of view of data analysis the absence of the general analytic relations between{λ1,R1} and {λ2,R2} leads to the following approach for estimations of the errors of the unknown parameters. Without the loss of generality let us suppose that the values are known for set of parameters {α1,λ1,R1} with its errors {Δ±α1,Δ±λ1,Δ±R1} for centrally symmetric Lévy parametrization with Ω1 as well as for α2 with Δ±α2 for parametrization with Ω2. The two sets of values for unknown BEC parameters {λ2,R2} can be obtained with the help of suitable system of equations: the input values {α1+Δ+α1,λ1+Δ+λ1,R1+Δ+R1} and α2+Δ+α2 produce the output set {λ2+,R2+} and {α1-Δ-α1,λ1-Δ-λ1,R1-Δ-R1,α2-Δ-α2}→{λ2-,R2-}. Then the error estimations for set {λ2,R2} of BEC parameters for parametrization with Ω2 can be calculated as follows:(5)Δ±Y2=Y2±-Y2,Y2≡λ2,R2.One can use the errors (5) which are asymmetric in general case or make the averaging of up and low uncertainties and then to use the symmetric errors ΔY2=(Δ+Y2+Δ-Y2)/2.


### 2.2. Dependencies onq and α Variables

Figure1 shows dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for centrally symmetric Lévy parametrization Ω1 with known α1=1.5 on low q1 (a, c) and high q2 (b, d) limits of integration in system (3a) and (3b) for set {α2,λ2,R2}={0.5,0.5,1.5fm} for Ω2. The solid lines correspond to the indicated values of q2 in GeV/c for q1-dependence (a, c) and to shown values of the q1 in GeV/c for q2-dependence (b, d). Values of BEC parameters λ1 and R1 depend strongly on the fixed second limit of integration (q2(1)) for both the q1- (Figures 1(a) and 1(c)) and the q2-dependence (Figures 1(b) and 1(d)). The dashed lines correspond to the results from system (3a) and (3b) with q2→∞ for q1-dependence (Figures 1(a) and 1(c)) and with q1=0 for q2-dependence (Figures 1(b) and 1(d)). As seen the curves for general case of (3a) and (3b) coincide with dashed lines at q2=10 GeV/c for q1-dependence (Figures 1(a) and 1(c)) and at q1=10-3 GeV/c for q2-dependence (Figures 1(b) and 1(d)). These values for q1 and especially for q2 are far from the corresponding limit in modern experimental CF. Therefore one should use system (3a) and (3b) with finite limits in an integrations in the case of experimentally available q-ranges for two parameterizations with Ω1 and Ω2. The thin dotted lines demonstrate the ultimate levels for R1 (Figures 1(a) and 1(b)) and λ1 (Figures 1(c) and 1(d)) calculated with (4a) and (4b) for given values of α1 and the set of parameters {α2,λ2,R2} for second Lévy parametrization Ω2. One can use the simple relations (4a) and (4b) for calculation {λ1,R1} at q1≲10-2 GeV/c and q2≳10 GeV/c (Figures 1(a) and 1(b)) but as expected the values of BEC parameters {λ1,R1} are far from the ultimate levels at any q1 for q2-dependence in the range q2≤2 GeV/c is considered in Figures 1(b) and 1(d).Figure 1
Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for centrally symmetric Lévy parametrization withα1=1.5 on low q1 (a, c) and high q2 (b, d) limits of integration in the system of (3a) and (3b) for fixed parameter values for second centrally symmetric Lévy parametrization: α2=0.5, λ2=0.5, and R2=1.5 fm. The solid lines correspond to the indicated values of the q2 for q1-dependence (a, c) and to shown values of the q1 for q2-dependence (b, d). The dashed lines correspond to the calculations with q2→∞ for q1-dependence (a, c) and with q1=0 for q2-dependence (b, d). The thin dotted lines are the ultimate levels for R1 (a, b) and λ1 (c, d) calculated with (4a) and (4b) for given values of α1 and the set of parameters {α2,λ2,R2} for second Lévy parametrization.
(a) (b) (c) (d)In Figure2 dependence of 1D BEC radius (a, b) and strength of correlations (c, d) is demonstrated for centrally symmetric Lévy parametrization with Ω1 on α1 at fixed values of α2 (a, c) and on α2 at fixed values of α1 (b, d) for given limits of integration in the system of (3a) and (3b) q1=0.02 GeV/c, q2=2.0 GeV/c and for certain values of the BEC parameters for second centrally symmetric Lévy parametrization with Ω2: λ2=0.5 and R2=1.5 fm. The solid lines correspond to the indicated values of the α2 for α1-dependence (a, c) and to shown values of the α1 for α2-dependence (b, d). The values for limits of integration q1 and q2 in system (3a) and (3b) are similar to those used in modern experiments. As seen dependencies of both BEC parameters on αi, i=1,2, at fixed another Lévy index αjj≠i, j=1,2, change very fast at small value αj=0.2 in narrow range αi≃αj. Such behavior is observed for both the results from system (3a) and (3b) and the estimations for R1 (Figures 2(a) and 2(b)) and λ1 (Figures 2(c) and 2(d)) calculated with (4a) and (4b) for semi-infinite ranges for integration and shown by dotted lines. The dependence R1(α1) shown in Figure 2(a) closes to the analytic one calculated with help of (4b) and presented by dotted line for any α2 under study with exception of the small value α2=0.2. For last case the agreement is obtained in very narrow range α1≈0.2 between results from system (3a) and (3b) and (4b). This feature maps clearly in corresponding dependencies R2(α2) shown in Figure 2(b) for α1=0.2. For large α1>1.0 solid and dotted lines are close to each other in the range α2>0.5 but agreement is poor significantly between results from system (3a) and (3b) and (4b) for α1=0.6 especially in domain α2<0.5 (Figure 2(b)) taking into account the sharp behavior for corresponding dependence R1(α2). In general the behavior of dependencies of λ1 on α1 (Figure 2(c)) and α2 (Figure 2(d)) is similar to the corresponding dependencies of 1D BEC radius R1. But the agreement is poor usually between the results deduced from system (3a) and (3b) and shown by the solid lines and the estimations calculated based on (4a) and presented by the dotted lines. Therefore for values of limits of integration q1,2 under consideration the approximate relations (4a) and (4b) should be used carefully for experimental analysis of dependencies on Lévy indexes and last equations can produce the reasonable estimations for BEC parameters for ranges ∀i=1,2:αi≳1 only.Figure 2
Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for centrally symmetric Lévy parametrization onα1 at fixed values of α2 (a, c) and on α2 at fixed values of α1 (b, d) for given limits of integration in the system of (3a) and (3b) q1=0.02 GeV/c, q2=2.0 GeV/c and for fixed values of the BEC parameters for second centrally symmetric Lévy parametrization: λ2=0.5 and R2=1.5 fm. The solid lines correspond to the indicated values of the α2 for α1-dependence (a, c) and to shown values of the α1 for α2-dependence (b, d). The dotted lines are the ultimate cases for R1 (a, b) and λ1 (c, d) calculated with (4a) and (4b) for given values of α1 (a, c) or α2 (b, d) and the set of BEC parameters {λ2,R2} for second Lévy parametrization.
(a) (b) (c) (d)As seen the mathematical formalism described above and the results in Figures1 and 2 are quantitative basis for choice of the applying of general equations (3a) and (3b) or ultimate relations (4a) and (4b) in data analysis for given experiment. Thus the method suggested in the paper is helpful for experimental and phenomenological studies of BEC in various processes at different parameterizations of CF C2(q) corresponding to the centrally symmetric Lévy source distributions.


## 2.1. Mathematical Formalism

The relations between two sets of parameters{λ1,R1} and {λ2,R2} of the particle source can be derived from the following system of equations:(3a)S1=S2,∀i=1,2:Si≡∫JiΩαi,λi,zidq;(3b)q1=q2,∀i=1,2:qi≡Si-1∫JiqΩαi,λi,zidq.The first equation (3a) corresponds to the equality of the areas under fit curves and the (3b) is the equality of the first moments of the Ω(αi,λi,zi) distributions (as discussed above the approximate equalities are expected for areas and first moments in general case. This softer condition is enough for applicability of the formalism suggested in the paper. But the exactly equal signs are used in  (3a) and (3b)  as well as in the text below in order to get the mathematically correct forms for the systems of equations). System  (3a) and (3b)  contains the equations allowing the estimation of unknown strength of correlations and 1D BEC radius based on the available values of these parameters but it supposes the Lévy indexes αi are knowna priori for both parameterizations Ωi, i=1,2. In equations (3a) and (3b) the integrals are taken over full fit ranges Ji, i=1,2, for corresponding parameterizations with Ωi of experimental CF. It should be noted that in general case (i) the ranges of integration can be different for parameterizations with Ω1 and Ω2; (ii) the full fit range can be the set of subranges due to possible exception of some intervals of the relative 4-momentum (regions of resonance contributions, etc.), that is, ∀i=1,2:Ji=⋃k=1NiJik≡⋃k=1Ni[qk,min,qk,max] and consequently for all types of integrals and ∀i=1,2 in system (3a) and (3b): ∫Ji→∑k=1Ni∫Jik. But usually the fit ranges Ji are identical for both Ωi, i=1,2, in experimental studies (see, e.g., [15]). In general case of the centrally symmetric Lévy distributions and finite fit ranges the system equations under consideration cannot be solved analytically. The numerical procedure should be used in order to get the relations between two sets of parameters {λ1,R1} and {λ2,R2} of the particle source in this case. Without loss of generality Ω1 and α2 are considered as known and values of BEC parameters {λ2,R2} are supposed as desired below. Then for specific case of semi-infinite ranges for integration ∀i=1,2:Ji=[0;∞) system (3a) and (3b) can be solved analytically and one can derive the following ultimate relations between two sets {λ1,R1} and {λ2,R2} of BEC parameters for corresponding centrally symmetric Lévy parameterizations with Ωi, i=1,2(4a)λ1u=λ2α1Γ2α1-1Γ2α2-1α2Γ2α1-1Γ2α2-1-1;(4b)R1u=R2Γ2α1-1Γα2-1Γα1-1Γ2α2-1-1and vice versa. Here Γ(x)=∫0∞tx-1exp⁡(-t)dt,Rex>0 is the gamma function.In the point of view of data analysis the absence of the general analytic relations between{λ1,R1} and {λ2,R2} leads to the following approach for estimations of the errors of the unknown parameters. Without the loss of generality let us suppose that the values are known for set of parameters {α1,λ1,R1} with its errors {Δ±α1,Δ±λ1,Δ±R1} for centrally symmetric Lévy parametrization with Ω1 as well as for α2 with Δ±α2 for parametrization with Ω2. The two sets of values for unknown BEC parameters {λ2,R2} can be obtained with the help of suitable system of equations: the input values {α1+Δ+α1,λ1+Δ+λ1,R1+Δ+R1} and α2+Δ+α2 produce the output set {λ2+,R2+} and {α1-Δ-α1,λ1-Δ-λ1,R1-Δ-R1,α2-Δ-α2}→{λ2-,R2-}. Then the error estimations for set {λ2,R2} of BEC parameters for parametrization with Ω2 can be calculated as follows:(5)Δ±Y2=Y2±-Y2,Y2≡λ2,R2.One can use the errors (5) which are asymmetric in general case or make the averaging of up and low uncertainties and then to use the symmetric errors ΔY2=(Δ+Y2+Δ-Y2)/2.


## 2.2. Dependencies onq and α Variables

Figure1 shows dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for centrally symmetric Lévy parametrization Ω1 with known α1=1.5 on low q1 (a, c) and high q2 (b, d) limits of integration in system (3a) and (3b) for set {α2,λ2,R2}={0.5,0.5,1.5fm} for Ω2. The solid lines correspond to the indicated values of q2 in GeV/c for q1-dependence (a, c) and to shown values of the q1 in GeV/c for q2-dependence (b, d). Values of BEC parameters λ1 and R1 depend strongly on the fixed second limit of integration (q2(1)) for both the q1- (Figures 1(a) and 1(c)) and the q2-dependence (Figures 1(b) and 1(d)). The dashed lines correspond to the results from system (3a) and (3b) with q2→∞ for q1-dependence (Figures 1(a) and 1(c)) and with q1=0 for q2-dependence (Figures 1(b) and 1(d)). As seen the curves for general case of (3a) and (3b) coincide with dashed lines at q2=10 GeV/c for q1-dependence (Figures 1(a) and 1(c)) and at q1=10-3 GeV/c for q2-dependence (Figures 1(b) and 1(d)). These values for q1 and especially for q2 are far from the corresponding limit in modern experimental CF. Therefore one should use system (3a) and (3b) with finite limits in an integrations in the case of experimentally available q-ranges for two parameterizations with Ω1 and Ω2. The thin dotted lines demonstrate the ultimate levels for R1 (Figures 1(a) and 1(b)) and λ1 (Figures 1(c) and 1(d)) calculated with (4a) and (4b) for given values of α1 and the set of parameters {α2,λ2,R2} for second Lévy parametrization Ω2. One can use the simple relations (4a) and (4b) for calculation {λ1,R1} at q1≲10-2 GeV/c and q2≳10 GeV/c (Figures 1(a) and 1(b)) but as expected the values of BEC parameters {λ1,R1} are far from the ultimate levels at any q1 for q2-dependence in the range q2≤2 GeV/c is considered in Figures 1(b) and 1(d).Figure 1
Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for centrally symmetric Lévy parametrization withα1=1.5 on low q1 (a, c) and high q2 (b, d) limits of integration in the system of (3a) and (3b) for fixed parameter values for second centrally symmetric Lévy parametrization: α2=0.5, λ2=0.5, and R2=1.5 fm. The solid lines correspond to the indicated values of the q2 for q1-dependence (a, c) and to shown values of the q1 for q2-dependence (b, d). The dashed lines correspond to the calculations with q2→∞ for q1-dependence (a, c) and with q1=0 for q2-dependence (b, d). The thin dotted lines are the ultimate levels for R1 (a, b) and λ1 (c, d) calculated with (4a) and (4b) for given values of α1 and the set of parameters {α2,λ2,R2} for second Lévy parametrization.
(a) (b) (c) (d)In Figure2 dependence of 1D BEC radius (a, b) and strength of correlations (c, d) is demonstrated for centrally symmetric Lévy parametrization with Ω1 on α1 at fixed values of α2 (a, c) and on α2 at fixed values of α1 (b, d) for given limits of integration in the system of (3a) and (3b) q1=0.02 GeV/c, q2=2.0 GeV/c and for certain values of the BEC parameters for second centrally symmetric Lévy parametrization with Ω2: λ2=0.5 and R2=1.5 fm. The solid lines correspond to the indicated values of the α2 for α1-dependence (a, c) and to shown values of the α1 for α2-dependence (b, d). The values for limits of integration q1 and q2 in system (3a) and (3b) are similar to those used in modern experiments. As seen dependencies of both BEC parameters on αi, i=1,2, at fixed another Lévy index αjj≠i, j=1,2, change very fast at small value αj=0.2 in narrow range αi≃αj. Such behavior is observed for both the results from system (3a) and (3b) and the estimations for R1 (Figures 2(a) and 2(b)) and λ1 (Figures 2(c) and 2(d)) calculated with (4a) and (4b) for semi-infinite ranges for integration and shown by dotted lines. The dependence R1(α1) shown in Figure 2(a) closes to the analytic one calculated with help of (4b) and presented by dotted line for any α2 under study with exception of the small value α2=0.2. For last case the agreement is obtained in very narrow range α1≈0.2 between results from system (3a) and (3b) and (4b). This feature maps clearly in corresponding dependencies R2(α2) shown in Figure 2(b) for α1=0.2. For large α1>1.0 solid and dotted lines are close to each other in the range α2>0.5 but agreement is poor significantly between results from system (3a) and (3b) and (4b) for α1=0.6 especially in domain α2<0.5 (Figure 2(b)) taking into account the sharp behavior for corresponding dependence R1(α2). In general the behavior of dependencies of λ1 on α1 (Figure 2(c)) and α2 (Figure 2(d)) is similar to the corresponding dependencies of 1D BEC radius R1. But the agreement is poor usually between the results deduced from system (3a) and (3b) and shown by the solid lines and the estimations calculated based on (4a) and presented by the dotted lines. Therefore for values of limits of integration q1,2 under consideration the approximate relations (4a) and (4b) should be used carefully for experimental analysis of dependencies on Lévy indexes and last equations can produce the reasonable estimations for BEC parameters for ranges ∀i=1,2:αi≳1 only.Figure 2
Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for centrally symmetric Lévy parametrization onα1 at fixed values of α2 (a, c) and on α2 at fixed values of α1 (b, d) for given limits of integration in the system of (3a) and (3b) q1=0.02 GeV/c, q2=2.0 GeV/c and for fixed values of the BEC parameters for second centrally symmetric Lévy parametrization: λ2=0.5 and R2=1.5 fm. The solid lines correspond to the indicated values of the α2 for α1-dependence (a, c) and to shown values of the α1 for α2-dependence (b, d). The dotted lines are the ultimate cases for R1 (a, b) and λ1 (c, d) calculated with (4a) and (4b) for given values of α1 (a, c) or α2 (b, d) and the set of BEC parameters {λ2,R2} for second Lévy parametrization.
(a) (b) (c) (d)As seen the mathematical formalism described above and the results in Figures1 and 2 are quantitative basis for choice of the applying of general equations (3a) and (3b) or ultimate relations (4a) and (4b) in data analysis for given experiment. Thus the method suggested in the paper is helpful for experimental and phenomenological studies of BEC in various processes at different parameterizations of CF C2(q) corresponding to the centrally symmetric Lévy source distributions.


## 3. Relations between BEC Parameters in Specific Cases

As seen in Figure2 both dependencies of the 1D BEC radius R1 (a, b) and the strength of correlations λ1 (c, d) on Lévy indexes αi, i=1,2, show the weaker changing in the domain ∀i=1,2:αi≳1 in comparison with the range of small values of Lévy indexes. The region ∀i=1,2:αi≳1 includes in particular the specific cases of Cauchy and Gaussian distributions for which corresponding parameterizations of Bose–Einstein CF C2(q) with α=1 and α=2 are used mostly for experimental studies. Therefore these certain views of Ωi are studied in detail below. Let Ω1≡ΩG=Ω(2,λG,RG) for Gaussian parametrization (2) and Ω2≡ΩC=Ω(1,λC,RC) for 1D approximation of experimental CF C2(q) by exponential function.


### 3.1. Mathematical Formalism

Relations (4a) and (4b) are valid at any values of indexes of stability 0<αi≤2 in two centrally symmetric Lévy parameterizations with Ωi, i=1,2. If without loss of generality {λC,RC} are considered asa priori known and values of Gaussian BEC parameters {λG,RG} are supposed as desired then as expected (4a) and (4b) result in the ultimate relations:(6a)λGu=2λCπ;(6b)RGu=RCπand vice versa. Relation (6a) is derived in [21] for the first time while formula (6b) for 1D BEC radii is well-known.The following relations can be obtained from general system (3a) and (3b) for finite ranges of integrations and specific values α1=2 and α2=1:(7a)λGπ2RG∑j=1NGerf⁡z2,j,G-erf⁡z1,j,G=λCRC∑i=1NCexp⁡-z1,i,C-exp⁡-z2,i,C;(7b)1RGπ∑j=1NGexp-z1,j,G2-exp-z2,j,G2∑j=1NGerfz2,j,G-erfz1,j,G=1RC∑i=1NCexp-z1,i,C1+z1,i,C-exp-z2,i,C1+z2,i,C∑i=1NCexp-z1,i,C-exp-z2,i,C.Here erf(x)=(2/π)∫0xexp(-t2)dt is the error integral and z1(2),i,C(G)≡q1(2),i,C(G)RC(G) are the limits for integration over corresponding subranges for Cauchy (Gaussian) distribution. The detailed study of all available experimental results in strong interaction processes for 1D parametrization (2) with α1=2 and α2=1 for experimental CF C2(q) shows the following: (i) the ranges of integration for both the exponential and the Gaussian functions are equal; (ii) usually, the range of integration is not divided into subranges; in any case, such division is identical for both functions under consideration and maximum value of NC(G) is equal 2 for experimental analyses. Thus the general statement with regard of identity of integration ranges for Ω1 and Ω2 is quite confirmed for case of Cauchy and Gaussian distributions and NC(G)≡N in system (7a) and (7b).Further simplification for the system of (7a) and (7b) depends on features of certain experiment; direction of calculations ΩC⇄ΩG, that is, what kind of a set of BEC parameters of the two, {λC,RC} and {λG,RG}, it is regarded asa prioriknown and which set is supposed as desired; and requirement on the accuracy level. For available experimental data for BEC of charged pion pairs produced in strong interaction processes (i) the accuracy for 1D BEC radius is better usually than that for λ parameter and (ii) the accuracy for 1D BEC parameters in modern experiments is not better than ～10-3 so far. Thus one can assume the conservative accuracy level ε=5×10-4. At present the most complex case with N=2 is for analyses of proton-proton collisions at some LHC energies only [15, 22]. For this case all contributions are negligible from the subrange of q values larger than the region of the influence of meson resonances excluded from the experimental fits; that is, all terms for i=j=2 can be omitted at given ε and direction of calculation froma priori known Cauchy parameters to desired Gaussian parameters {λC,RC}→{λG,RG}. But the statement is wrong for opposite direction of calculation froma priori known Gaussian parameters to desired Cauchy parameters {λG,RG}→{λC,RC} at ε=5×10-4. Therefore the sum can be omitted in system (7a) and (7b) and equations can be rewritten as follows:(8a)λGRGπerf⁡z2,G-erf⁡z1,G=2λCπRCexp⁡-z1,C-exp⁡-z2,C;(8b)1RGπexp-z1,G2-exp-z2,G2erfz2,G-erfz1,G=1RCexp-z1,C1+z1,C-exp-z2,C1+z2,Cexp-z1,C-exp-z2,Cfor all experiments in the case of estimation of unknown BEC parameters for Gaussian function based on thea priori known set of corresponding parameters for exponential function and for all experiments with exception of p+p collisions at sNN=0.9, 2.36, and 7 TeV [15, 22] in the case of inverse problem. Accounting the relation RG≤RC/π and properties of the functions exp⁡(-x2),erf⁡(x) allows us to simplify (8a) and (8b) to system(9a)λGRGπerfc⁡z1,G=2λCπRCexp⁡-z1,C-exp⁡-z2,C;(9b)1RGπexp-z1,G2erfcz1,G=1RCexp-z1,C1+z1,C-exp-z2,C1+z2,Cexp-z1,C-exp-z2,C,where erfc⁡(x)=1-erf⁡(x). The last system of equations is valid for remain set of experimental results with exception of the WA98 data [23] at given ε. Also the transition from system (8a) and (8b) to simpler equations (9a) and (9b) is not valid for CPLEAR data [24] for direction of calculation froma priori known Gaussian parameters to desired Cauchy parameters {λG,RG}→{λC,RC} at ε=5×10-4. The simplest view of the system of (9a) and (9b)(10a)λGRGπerfc⁡z1,G=2λCπRCexp⁡-z1,C;(10b)1RGπexp-z1,G2erfcz1,G=1+z1,CRCcorresponds to the range of integration [z1,C(G),∞) and can be used for experimental results from ALICE [25], CMS [14, 15, 17, 26] with exception of the collision energy sNN=2.36 TeV [15] in the case of proton-proton collisions, and WA80 [27] for asymmetric nucleus-nucleus collisions O+C, O+Cu. On the other hand, the using of the range of integration [0.0,z2,C(G)] allows the derivation of the following system from (9a) and (9b):(11a)λGRGπerf⁡z2,G=2λCπRC1-exp⁡-z2,C;(11b)1RGπ1-exp-z2,G2erfz2,G=1RC1-exp-z2,C1+z2,C1-exp-z2,C.As expected one can get the ultimate relations (6a) and (6b) from the any systems of ((10a) and (10b)) or ((11a) and (11b)) at q1→0 or q2→∞, respectively. Therefore system (10a) and (10b) can be replaced by ultimate system of (6a) and (6b) with some accuracy ε′ for finite range of q if q1≤q1h and q2 value is large enough to consider this value as q2→∞. Similarly, system (11a) and (11b) can be replaced by ultimate system of (6a) and (6b) with some accuracy ε′ for finite range of q if q2≥q2l and q1 is small enough to consider it as q1→0 for (11a) and (11b). The high/low boundary values q1h/q2l for variables q1/q2 are dominated by assigned value of accuracy. For instance, at ε′=10-2 the ultimate system of (6a) and (6b) is valid for q1≲2×10-3RC(G) or q2≳1.3RC(G), that is, q1≲q1h=2–4 MeV/c or q2≳q2l=1.3–2.6 GeV/c for proton-proton collisions. The derived estimations are close to the values of q variable used in present experimental analyses of BEC correlations.These qualitative estimations are confirmed by quantitative analysis below for theq1- and q2-dependencies of the Gaussian parameters λG and RG derived for some assigned values of the corresponding BEC parameters for exponential function λC, RC and vice versa.


### 3.2. Dependence onq for Desired Cauchy/Gaussian Parameters

For Figures3 and 4  ΩC is considered asa priori known and set of BEC parameters {λG,RG} are studied for Gaussian parametrization (2). Figure 3 shows the q1- and q2-dependence of 1D BEC radius (Figures 3(a) and 3(b)) and strength of correlations (Figures 3(c) and 3(d)) for parametrization (2) with Gaussian function ΩG at fixed values λC=π/2 and RC=π. As seen both Gaussian parameters show the similar behavior with changing the integration limits, namely, λG and RG growth with decreasing of the q1,2 at fixed another limit of integration. The curves λG(q1), RG(q1) approach the asymptotic dashed lines calculated with help of system (10a) and (10b) with increasing of q2. The similar situation is observed in Figures 3(b) and 3(d) for curves RG(q2), λG(q2) and asymptotic dashed lines calculated with help of system (11a) and (11b) with decreasing of q1. As seen the asymptotic lines are achieved at q1≲1 MeV/c (Figures 3(b) and 3(d)) and q2≳0.8 GeV/c (Figures 3(a) and 3(c)). Furthermore the ultimate values of the Gaussian BEC parameters λGu and RGu are valid with good accuracy for q1<2.0 MeV/c and q2>1.0 GeV/c. The last ranges are in the good agreement with qualitative estimations for proton-proton collisions obtained above. It should be emphasized that for specific case of exponential (ΩC) and Gaussian (ΩG) functions the asymptotic q1-dependence is achieved for both the 1D BEC radius (Figure 3(a)) and the strength of correlations (Figure 3(c)) at q2 which is much smaller than that for case of two some other centrally symmetric Lévy parameterizations (Figures 3(a) and 3(c)). This q2 value for case of ΩC and ΩG is similar to those used in analyses of experimental CF C2(q). In Figure 4 the dependencies of relative BEC parameters, namely, RC/RG (a, b) and λC/λG (c, d), on q1 (a, c) and q2 (b, d) are presented for various assigned values of parameters for Cauchy distribution. The curves are calculated with the simpler system of (10a) and (10b) for q1-dependence (Figures 4(a) and 4(c)) and system (11a) and (11b) for q2-dependence (Figures 4(b) and 4(d)), respectively. As seen the larger values of Cauchy parameters lead to the larger values of relative BEC parameters. The q1-dependence of relative BEC parameters grows faster with increasing of the input values of Cauchy parameters (Figures 4(a) and 4(c)). On the contrary the decrease of the q2-dependence of the RC/RG (Figure 4(b)) and λC/λG (Figure 4(d)) is slower with increasing of the input values of the {λC,RC}. As expected the ultimate levels RC/RG=π (Figures 4(a) and 4(b)) and λC/λG=π/2 (Figures 4(c) and 4(d)) shown by thin dotted lines are valid for the same ranges of q1 and q2 as estimated above for Figure 3.Figure 3
Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for Gaussian parametrization on lowq1 (a, c) and high q2 (b, d) limits of integration in the system of (8a) and (8b) for fixed values of the parameters for exponential parametrization: λC=π/2 and RC=π fm. The solid lines correspond to the indicated values of the q2 for q1-dependence (a, c) and to shown values of the q1 for q2-dependence (b, d). The dashed lines correspond to the calculations based on system (10a) and (10b) for q1-dependence (a, c) and on system (11a) and (11b) for q2-dependence (b, d). The thin dotted lines are the ultimate levels RG=1.0 fm (a, b) and λG=1.0 (c, d) calculated with help of (6a) and (6b) for given values of the set of Cauchy parameters {λC,RC}.
(a) (b) (c) (d)Figure 4
Dependence of relative 1D BEC radius (a, b) and strength of correlations (c, d) onq1 (a, c) and q2 (b, d) for various fixed values of the parameters for exponential parametrization. The calculations are made for simpler range of integration [z1,C(G),∞) with help of system (10a) and (10b) for q1-dependence (a, c) and for [0.0,z2,C(G)] with system (11a) and (11b) for q2-dependence (b, d), respectively. The dashed lines correspond to λC=0.6π, RC=1.2π fm, solid lines λC=0.5π, RC=π fm, and dotted lines λC=0.4π, RC=0.8π fm. The thin dotted lines are the ultimate levels RC/RG=π (a, b) and λC/λG=π/2 (c, d) corresponding to system (6a) and (6b).
(a) (b) (c) (d)Figures5 and 6 show results for opposite direction of calculations; that is, ΩG is supposed to bea priori known and BEC parameters {λC,RC} for exponential parametrization of CF C2(q) are derived. Figure 5 shows the q1- and q2-dependence of 1D BEC radius (Figures 5(a) and 5(b)) and strength of correlations (Figures 5(c) and 5(d)) for exponential function ΩC at fixed values λG=2/π and RG=1/π. The ∀i=1,2:qi-dependencies show the opposite behavior for desired parameters of Cauchy source function RC (Figures 5(a) and 5(b)) and λC (Figures 5(c) and 5(d)) with respect to the corresponding dependencies presented in Figure 3 above for another direction of calculation {λC,RC}→{λG,RG}. These differences are seen in domain of relatively large q1≳10-2 GeV/c for q1-dependence and at relatively small q2≲0.7 GeV/c for q2-dependence of BEC parameters. Furthermore the q1-dependence for parameters from set for ΩC (Figures 5(a) and 5(c)) approaches the constant at q1→0 faster noticeably than that for RG (Figure 3(a)) and λG (Figure 3(c)). The opposite situation is observed for achievement of constants by q2-dependence at q2→∞. It should be noted that dependencies RC(q1) and λC(q1) approach their asymptotic curves calculated with help of system (10a) and (10b) and shown by dashed lines in Figures 5(a) and 5(c) slower than corresponding dependencies for desired Gaussian parameters in Figures 3(a) and 3(c). As a consequence RC(q1) and λC(q1) will achieve the asymptotic curves at higher q2 than that for Figures 3(a) and 3(c). The asymptotic value of q1≃10-3 GeV/c is the same for q2-dependence for both directions of calculations {λC,RC}⇄{λG,RG}. Figure 6 demonstrates the dependence of relative BEC parameters, namely, RC/RG (a, b) and λC/λG (c, d), on q1 (a, c) and q2 (b, d) for various assigned values of parameters for Gaussian parametrization. The simpler system of (10a) and (10b) is used for calculation of q1-dependencies in Figures 6(a) and 6(c) and curves on q2 (Figures 6(b) and 6(d)) are derived with help of system (11a) and (11b). In general ∀i=1,2:qi-dependencies show similar behavior for corresponding relative 1D BEC parameters in both cases, Figure 4 and Figure 6, with some faster changing of qi-dependencies in the second case than that for the first one in domain of relatively large q1≳10-2 GeV/c for q1-dependence and at relatively small q2≲0.7 GeV/c for q2-dependence of RC/RG and λC/λG.Figure 5
Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for Cauchy distribution on lowq1 (a, c) and high q2 (b, d) limits of integration in the system of (8a) and (8b) for fixed values of the parameters for Gaussian parametrization: λG=2/π and RG=1/π fm. The solid lines correspond to the indicated values of the q2 for q1-dependence (a, c) and to shown values of the q1 for q2-dependence (b, d). The dashed lines correspond to the calculations based on system (10a) and (10b) for q1-dependence (a, c) and on system (11a) and (11b) for q2-dependence (b, d). The thin dotted lines are the ultimate levels RC=1.0 fm (a, b) and λG=1.0 (c, d) calculated with help of (6a) and (6b) for given values of the set of Gaussian parameters {λG,RG}.
(a) (b) (c) (d)Figure 6
Dependence of relative 1D BEC radius (a, b) and strength of correlations (c, d) onq1 (a, c) and q2 (b, d) for various fixed values of the parameters for Gaussian parametrization. The calculations are made for simpler range of integration [z1,C(G),∞) with help of system (10a) and (10b) for q1-dependence (a, c) and for [0.0,z2,C(G)] with system (11a) and (11b) for q2-dependence (b, d), respectively. The dashed lines correspond to λG=1.2, RG=1.2 fm, solid lines λG=1.0, RG=1.0 fm, and dotted lines λG=0.8, RG=0.8 fm. The thin dotted lines are the ultimate levels RC/RG=π (a, b) and λC/λG=π/2 (c, d) corresponding to system (6a) and (6b).
(a) (b) (c) (d)Simultaneous consideration of available 1D BEC data analyses for strong interaction processes and Figures3 and 5 allow the assertion that ultimate relations (6a) and (6b) are not acceptable with reasonable accuracy for most of experimental results with exponential/Gaussian parametrization (2) of 1D CF C2(q). As seen from Figures 4 and 6 even the asymptotic values of relative 1D BEC parameters {λC/λG,RC/RG} can differ up to several times from ultimate values calculated with help of the system of (6a) and (6b) in some domains of q1 and q2 variables. Therefore Figures 3–6 confirm the conclusion formulated above for case of two general view centrally symmetric Lévy parameterizations; namely, for desired 1D BEC parameters the finite values for limits of integrations can lead to the significant difference between values of BEC observables calculated on exact equations and asymptotic/ultimate values calculated on simpler relations.It should be emphasized that the results of the present paper shown in Figures3–6 are useful for experimental data analysis as well as for phenomenological studies because it allow, in particular, the quantitative choice between systems of ((6a)–(11b)) for estimations of 1D BEC parameters for specific cases of centrally symmetric Lévy parametrization (2) at α=1,2 depending on some features in given experiment.


## 3.1. Mathematical Formalism

Relations (4a) and (4b) are valid at any values of indexes of stability 0<αi≤2 in two centrally symmetric Lévy parameterizations with Ωi, i=1,2. If without loss of generality {λC,RC} are considered asa priori known and values of Gaussian BEC parameters {λG,RG} are supposed as desired then as expected (4a) and (4b) result in the ultimate relations:(6a)λGu=2λCπ;(6b)RGu=RCπand vice versa. Relation (6a) is derived in [21] for the first time while formula (6b) for 1D BEC radii is well-known.The following relations can be obtained from general system (3a) and (3b) for finite ranges of integrations and specific values α1=2 and α2=1:(7a)λGπ2RG∑j=1NGerf⁡z2,j,G-erf⁡z1,j,G=λCRC∑i=1NCexp⁡-z1,i,C-exp⁡-z2,i,C;(7b)1RGπ∑j=1NGexp-z1,j,G2-exp-z2,j,G2∑j=1NGerfz2,j,G-erfz1,j,G=1RC∑i=1NCexp-z1,i,C1+z1,i,C-exp-z2,i,C1+z2,i,C∑i=1NCexp-z1,i,C-exp-z2,i,C.Here erf(x)=(2/π)∫0xexp(-t2)dt is the error integral and z1(2),i,C(G)≡q1(2),i,C(G)RC(G) are the limits for integration over corresponding subranges for Cauchy (Gaussian) distribution. The detailed study of all available experimental results in strong interaction processes for 1D parametrization (2) with α1=2 and α2=1 for experimental CF C2(q) shows the following: (i) the ranges of integration for both the exponential and the Gaussian functions are equal; (ii) usually, the range of integration is not divided into subranges; in any case, such division is identical for both functions under consideration and maximum value of NC(G) is equal 2 for experimental analyses. Thus the general statement with regard of identity of integration ranges for Ω1 and Ω2 is quite confirmed for case of Cauchy and Gaussian distributions and NC(G)≡N in system (7a) and (7b).Further simplification for the system of (7a) and (7b) depends on features of certain experiment; direction of calculations ΩC⇄ΩG, that is, what kind of a set of BEC parameters of the two, {λC,RC} and {λG,RG}, it is regarded asa prioriknown and which set is supposed as desired; and requirement on the accuracy level. For available experimental data for BEC of charged pion pairs produced in strong interaction processes (i) the accuracy for 1D BEC radius is better usually than that for λ parameter and (ii) the accuracy for 1D BEC parameters in modern experiments is not better than ～10-3 so far. Thus one can assume the conservative accuracy level ε=5×10-4. At present the most complex case with N=2 is for analyses of proton-proton collisions at some LHC energies only [15, 22]. For this case all contributions are negligible from the subrange of q values larger than the region of the influence of meson resonances excluded from the experimental fits; that is, all terms for i=j=2 can be omitted at given ε and direction of calculation froma priori known Cauchy parameters to desired Gaussian parameters {λC,RC}→{λG,RG}. But the statement is wrong for opposite direction of calculation froma priori known Gaussian parameters to desired Cauchy parameters {λG,RG}→{λC,RC} at ε=5×10-4. Therefore the sum can be omitted in system (7a) and (7b) and equations can be rewritten as follows:(8a)λGRGπerf⁡z2,G-erf⁡z1,G=2λCπRCexp⁡-z1,C-exp⁡-z2,C;(8b)1RGπexp-z1,G2-exp-z2,G2erfz2,G-erfz1,G=1RCexp-z1,C1+z1,C-exp-z2,C1+z2,Cexp-z1,C-exp-z2,Cfor all experiments in the case of estimation of unknown BEC parameters for Gaussian function based on thea priori known set of corresponding parameters for exponential function and for all experiments with exception of p+p collisions at sNN=0.9, 2.36, and 7 TeV [15, 22] in the case of inverse problem. Accounting the relation RG≤RC/π and properties of the functions exp⁡(-x2),erf⁡(x) allows us to simplify (8a) and (8b) to system(9a)λGRGπerfc⁡z1,G=2λCπRCexp⁡-z1,C-exp⁡-z2,C;(9b)1RGπexp-z1,G2erfcz1,G=1RCexp-z1,C1+z1,C-exp-z2,C1+z2,Cexp-z1,C-exp-z2,C,where erfc⁡(x)=1-erf⁡(x). The last system of equations is valid for remain set of experimental results with exception of the WA98 data [23] at given ε. Also the transition from system (8a) and (8b) to simpler equations (9a) and (9b) is not valid for CPLEAR data [24] for direction of calculation froma priori known Gaussian parameters to desired Cauchy parameters {λG,RG}→{λC,RC} at ε=5×10-4. The simplest view of the system of (9a) and (9b)(10a)λGRGπerfc⁡z1,G=2λCπRCexp⁡-z1,C;(10b)1RGπexp-z1,G2erfcz1,G=1+z1,CRCcorresponds to the range of integration [z1,C(G),∞) and can be used for experimental results from ALICE [25], CMS [14, 15, 17, 26] with exception of the collision energy sNN=2.36 TeV [15] in the case of proton-proton collisions, and WA80 [27] for asymmetric nucleus-nucleus collisions O+C, O+Cu. On the other hand, the using of the range of integration [0.0,z2,C(G)] allows the derivation of the following system from (9a) and (9b):(11a)λGRGπerf⁡z2,G=2λCπRC1-exp⁡-z2,C;(11b)1RGπ1-exp-z2,G2erfz2,G=1RC1-exp-z2,C1+z2,C1-exp-z2,C.As expected one can get the ultimate relations (6a) and (6b) from the any systems of ((10a) and (10b)) or ((11a) and (11b)) at q1→0 or q2→∞, respectively. Therefore system (10a) and (10b) can be replaced by ultimate system of (6a) and (6b) with some accuracy ε′ for finite range of q if q1≤q1h and q2 value is large enough to consider this value as q2→∞. Similarly, system (11a) and (11b) can be replaced by ultimate system of (6a) and (6b) with some accuracy ε′ for finite range of q if q2≥q2l and q1 is small enough to consider it as q1→0 for (11a) and (11b). The high/low boundary values q1h/q2l for variables q1/q2 are dominated by assigned value of accuracy. For instance, at ε′=10-2 the ultimate system of (6a) and (6b) is valid for q1≲2×10-3RC(G) or q2≳1.3RC(G), that is, q1≲q1h=2–4 MeV/c or q2≳q2l=1.3–2.6 GeV/c for proton-proton collisions. The derived estimations are close to the values of q variable used in present experimental analyses of BEC correlations.These qualitative estimations are confirmed by quantitative analysis below for theq1- and q2-dependencies of the Gaussian parameters λG and RG derived for some assigned values of the corresponding BEC parameters for exponential function λC, RC and vice versa.


## 3.2. Dependence onq for Desired Cauchy/Gaussian Parameters

For Figures3 and 4  ΩC is considered asa priori known and set of BEC parameters {λG,RG} are studied for Gaussian parametrization (2). Figure 3 shows the q1- and q2-dependence of 1D BEC radius (Figures 3(a) and 3(b)) and strength of correlations (Figures 3(c) and 3(d)) for parametrization (2) with Gaussian function ΩG at fixed values λC=π/2 and RC=π. As seen both Gaussian parameters show the similar behavior with changing the integration limits, namely, λG and RG growth with decreasing of the q1,2 at fixed another limit of integration. The curves λG(q1), RG(q1) approach the asymptotic dashed lines calculated with help of system (10a) and (10b) with increasing of q2. The similar situation is observed in Figures 3(b) and 3(d) for curves RG(q2), λG(q2) and asymptotic dashed lines calculated with help of system (11a) and (11b) with decreasing of q1. As seen the asymptotic lines are achieved at q1≲1 MeV/c (Figures 3(b) and 3(d)) and q2≳0.8 GeV/c (Figures 3(a) and 3(c)). Furthermore the ultimate values of the Gaussian BEC parameters λGu and RGu are valid with good accuracy for q1<2.0 MeV/c and q2>1.0 GeV/c. The last ranges are in the good agreement with qualitative estimations for proton-proton collisions obtained above. It should be emphasized that for specific case of exponential (ΩC) and Gaussian (ΩG) functions the asymptotic q1-dependence is achieved for both the 1D BEC radius (Figure 3(a)) and the strength of correlations (Figure 3(c)) at q2 which is much smaller than that for case of two some other centrally symmetric Lévy parameterizations (Figures 3(a) and 3(c)). This q2 value for case of ΩC and ΩG is similar to those used in analyses of experimental CF C2(q). In Figure 4 the dependencies of relative BEC parameters, namely, RC/RG (a, b) and λC/λG (c, d), on q1 (a, c) and q2 (b, d) are presented for various assigned values of parameters for Cauchy distribution. The curves are calculated with the simpler system of (10a) and (10b) for q1-dependence (Figures 4(a) and 4(c)) and system (11a) and (11b) for q2-dependence (Figures 4(b) and 4(d)), respectively. As seen the larger values of Cauchy parameters lead to the larger values of relative BEC parameters. The q1-dependence of relative BEC parameters grows faster with increasing of the input values of Cauchy parameters (Figures 4(a) and 4(c)). On the contrary the decrease of the q2-dependence of the RC/RG (Figure 4(b)) and λC/λG (Figure 4(d)) is slower with increasing of the input values of the {λC,RC}. As expected the ultimate levels RC/RG=π (Figures 4(a) and 4(b)) and λC/λG=π/2 (Figures 4(c) and 4(d)) shown by thin dotted lines are valid for the same ranges of q1 and q2 as estimated above for Figure 3.Figure 3
Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for Gaussian parametrization on lowq1 (a, c) and high q2 (b, d) limits of integration in the system of (8a) and (8b) for fixed values of the parameters for exponential parametrization: λC=π/2 and RC=π fm. The solid lines correspond to the indicated values of the q2 for q1-dependence (a, c) and to shown values of the q1 for q2-dependence (b, d). The dashed lines correspond to the calculations based on system (10a) and (10b) for q1-dependence (a, c) and on system (11a) and (11b) for q2-dependence (b, d). The thin dotted lines are the ultimate levels RG=1.0 fm (a, b) and λG=1.0 (c, d) calculated with help of (6a) and (6b) for given values of the set of Cauchy parameters {λC,RC}.
(a) (b) (c) (d)Figure 4
Dependence of relative 1D BEC radius (a, b) and strength of correlations (c, d) onq1 (a, c) and q2 (b, d) for various fixed values of the parameters for exponential parametrization. The calculations are made for simpler range of integration [z1,C(G),∞) with help of system (10a) and (10b) for q1-dependence (a, c) and for [0.0,z2,C(G)] with system (11a) and (11b) for q2-dependence (b, d), respectively. The dashed lines correspond to λC=0.6π, RC=1.2π fm, solid lines λC=0.5π, RC=π fm, and dotted lines λC=0.4π, RC=0.8π fm. The thin dotted lines are the ultimate levels RC/RG=π (a, b) and λC/λG=π/2 (c, d) corresponding to system (6a) and (6b).
(a) (b) (c) (d)Figures5 and 6 show results for opposite direction of calculations; that is, ΩG is supposed to bea priori known and BEC parameters {λC,RC} for exponential parametrization of CF C2(q) are derived. Figure 5 shows the q1- and q2-dependence of 1D BEC radius (Figures 5(a) and 5(b)) and strength of correlations (Figures 5(c) and 5(d)) for exponential function ΩC at fixed values λG=2/π and RG=1/π. The ∀i=1,2:qi-dependencies show the opposite behavior for desired parameters of Cauchy source function RC (Figures 5(a) and 5(b)) and λC (Figures 5(c) and 5(d)) with respect to the corresponding dependencies presented in Figure 3 above for another direction of calculation {λC,RC}→{λG,RG}. These differences are seen in domain of relatively large q1≳10-2 GeV/c for q1-dependence and at relatively small q2≲0.7 GeV/c for q2-dependence of BEC parameters. Furthermore the q1-dependence for parameters from set for ΩC (Figures 5(a) and 5(c)) approaches the constant at q1→0 faster noticeably than that for RG (Figure 3(a)) and λG (Figure 3(c)). The opposite situation is observed for achievement of constants by q2-dependence at q2→∞. It should be noted that dependencies RC(q1) and λC(q1) approach their asymptotic curves calculated with help of system (10a) and (10b) and shown by dashed lines in Figures 5(a) and 5(c) slower than corresponding dependencies for desired Gaussian parameters in Figures 3(a) and 3(c). As a consequence RC(q1) and λC(q1) will achieve the asymptotic curves at higher q2 than that for Figures 3(a) and 3(c). The asymptotic value of q1≃10-3 GeV/c is the same for q2-dependence for both directions of calculations {λC,RC}⇄{λG,RG}. Figure 6 demonstrates the dependence of relative BEC parameters, namely, RC/RG (a, b) and λC/λG (c, d), on q1 (a, c) and q2 (b, d) for various assigned values of parameters for Gaussian parametrization. The simpler system of (10a) and (10b) is used for calculation of q1-dependencies in Figures 6(a) and 6(c) and curves on q2 (Figures 6(b) and 6(d)) are derived with help of system (11a) and (11b). In general ∀i=1,2:qi-dependencies show similar behavior for corresponding relative 1D BEC parameters in both cases, Figure 4 and Figure 6, with some faster changing of qi-dependencies in the second case than that for the first one in domain of relatively large q1≳10-2 GeV/c for q1-dependence and at relatively small q2≲0.7 GeV/c for q2-dependence of RC/RG and λC/λG.Figure 5
Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for Cauchy distribution on lowq1 (a, c) and high q2 (b, d) limits of integration in the system of (8a) and (8b) for fixed values of the parameters for Gaussian parametrization: λG=2/π and RG=1/π fm. The solid lines correspond to the indicated values of the q2 for q1-dependence (a, c) and to shown values of the q1 for q2-dependence (b, d). The dashed lines correspond to the calculations based on system (10a) and (10b) for q1-dependence (a, c) and on system (11a) and (11b) for q2-dependence (b, d). The thin dotted lines are the ultimate levels RC=1.0 fm (a, b) and λG=1.0 (c, d) calculated with help of (6a) and (6b) for given values of the set of Gaussian parameters {λG,RG}.
(a) (b) (c) (d)Figure 6
Dependence of relative 1D BEC radius (a, b) and strength of correlations (c, d) onq1 (a, c) and q2 (b, d) for various fixed values of the parameters for Gaussian parametrization. The calculations are made for simpler range of integration [z1,C(G),∞) with help of system (10a) and (10b) for q1-dependence (a, c) and for [0.0,z2,C(G)] with system (11a) and (11b) for q2-dependence (b, d), respectively. The dashed lines correspond to λG=1.2, RG=1.2 fm, solid lines λG=1.0, RG=1.0 fm, and dotted lines λG=0.8, RG=0.8 fm. The thin dotted lines are the ultimate levels RC/RG=π (a, b) and λC/λG=π/2 (c, d) corresponding to system (6a) and (6b).
(a) (b) (c) (d)Simultaneous consideration of available 1D BEC data analyses for strong interaction processes and Figures3 and 5 allow the assertion that ultimate relations (6a) and (6b) are not acceptable with reasonable accuracy for most of experimental results with exponential/Gaussian parametrization (2) of 1D CF C2(q). As seen from Figures 4 and 6 even the asymptotic values of relative 1D BEC parameters {λC/λG,RC/RG} can differ up to several times from ultimate values calculated with help of the system of (6a) and (6b) in some domains of q1 and q2 variables. Therefore Figures 3–6 confirm the conclusion formulated above for case of two general view centrally symmetric Lévy parameterizations; namely, for desired 1D BEC parameters the finite values for limits of integrations can lead to the significant difference between values of BEC observables calculated on exact equations and asymptotic/ultimate values calculated on simpler relations.It should be emphasized that the results of the present paper shown in Figures3–6 are useful for experimental data analysis as well as for phenomenological studies because it allow, in particular, the quantitative choice between systems of ((6a)–(11b)) for estimations of 1D BEC parameters for specific cases of centrally symmetric Lévy parametrization (2) at α=1,2 depending on some features in given experiment.


## 4. Comparison with Experimental Results

Database is created for 1D BEC results for identical charged pions produced in strong interaction processes in order to verify the mathematical formalisms suggested above. This database is shown in Appendix5 and it is used as input for calculations below. Experimental results for strength of correlations and 1D source radius are considered for all types of the processes and centrally symmetric Lévy parameterizations (2) and for total available energy range in the paper. The results for most central nucleus-nucleus collisions are usually included in the database because these collisions are used for studying of new features of final state matter [28]. The dependence of 1D BEC parameters on the outgoing charged particle multiplicity, Nch, is widely studied for p+p and p¯+p collisions at least. Therefore the additional separation is made on experimental 1D BEC values deduced for minimum bias and for high multiplicity event classes sometimes (this separation will be stipulated additionally if experimental 1D BEC results are available for various multiplicity event classes in p+p, p¯+p collisions.). This consideration seems important for both the additional verification of mathematics above and the more careful comparison with nucleus-nucleus results. As seen the additional information is required about experimental q ranges for systems ((3a) and (3b)) and ((7a) and (7b)) in comparison with the ultimate relations ((4a) and (4b)) and ((6a) and (6b)). Therefore experimental q ranges are estimated based on the available published data. In Tables 1–4 the statistical errors are shown first and available systematic uncertainties second, unless otherwise specifically indicated; the types of uncertainties (statistical/total, symmetric/asymmetric) are chosen just the same as well as input parameters for the sake of simplicity.Table 1
Parameter values for exponential and Gaussian parameterizationsΩC(G).
Collision s N N, Experiment Estimation based on (3a), (3b) Experimental values Ref. GeV λ R, fm λ R, fm p + p 900 CMS 0.62 ± 0.08 1.47 ± 0.24 0.616 ± 0.011 ± 0.029 1.56 ± 0.02 ± 0.15 [14] 0.35 ± 0.06 0.81 ± 0.22 0.32 ± 0.01 0.98 ± 0.03 [15] 7000 ATLAS 0.73 ± 0.03 ± 0.54 2.02 ± 0.11 ± 1.79 0.701 ± 0.006 ± 0.067 2.021 ± 0.012 ± 0.281 [16] 0.39 ± 0.01 ± 0.20 1.07 ± 0.04 ± 0.82 0.302 ± 0.002 ± 0.019 1.046 ± 0.005 ± 0.114 CMS 0.62 ± 0.06 1.81 ± 0.23 0.618 ± 0.009 ± 0.042 1.89 ± 0.02 ± 0.21 [14] 0.35 ± 0.03 0.96 ± 0.12 — — —Table 2
Parameter values for general Lévy parametrizationΩL at given αL.
Collision s N N, Experiment Estimation based on (3a), (3b) Experimental values Ref. GeV λ R, fm λ R, fm p + p 900 CMS 0.85 ± 0.04 ± 0.05 2.33 ± 0.18 ± 0.23 0.85 ± 0.06 2.20 ± 0.17 [14] 0.89 ± 0.10 3.2 ± 0.5 0.93 ± 0.11 2.5 ± 0.4 [15] 7000 ATLAS 0.973 ± 0.012 ± 0.332 2.97 ± 0.06 ± 1.27 1.02 ± 0.03 ± 0.41 2.96 ± 0.09 ± 1.31 [16] 0.774 ± 0.010 ± 0.250 2.86 ± 0.06 ± 1.21 CMS 0.89 ± 0.04 ± 0.07 2.96 ± 0.17 ± 0.34 0.90 ± 0.05 2.83 ± 0.18 [14]Table 3
Parameter values for Gaussian functionΩG in (2).
Collision s N N, Experiment Estimation based on (7a), (7b) Experimental values Ref. GeV λ G R G, fm λ G R G, fm p+p 63.0 AFS 0.47 ± 0.04 0.73 ± 0.07 0.40 ± 0.03 0.82 ± 0.05 [29] 900 ALICE 0.305 ± 0.024 1.00 ± 0.0 9 - 0.18 + 0.06 0.35 ± 0.03 1.00 ± 0.0 6 - 0.20 + 0.10 [25] 0.357 ± 0.025 0.89 ± 0.0 6 - 0.18 + 0.08 0.310 ± 0.026 1.18 ± 0.0 9 - 0.17 + 0.07 ATLAS 0.41 ± 0.01 ± 0.03 1.00 ± 0.03 ± 0.08 0.34 ± 0.01 ± 0.03 1.00 ± 0.03 ± 0.08 [30] CMS 0.351 ± 0.006 ± 0.013 0.83 ± 0.01 ± 0.08 0.32 ± 0.01 0.98 ± 0.03 [15] 2360 0.348 ± 0.030 ± 0.013 1.04 ± 0.08 ± 0.10 0.32 ± 0.01 0.98 ± 0.03 2760 0.366 ± 0.005 ± 0.025 0.915 ± 0.007 ± 0.116 — — — 7000 ALICE 0.719 ± 0.002 ± 0.047 1.148 ± 0.00 7 - 0.02 + 0.04 0.645 ± 0.003 ± 0.047 1.430 ± 0.00 5 - 0.30 + 0.16 [31] ATLAS 0.381 ± 0.003 ± 0.022 1.092 ± 0.005 ± 0.074 0.327 ± 0.002 ± 0.020 1.130 ± 0.005 ± 0.086 [30] 0.266 ± 0.009 ± 0.015 1.25 ± 0.03 ± 0.09 0.251 ± 0.010 ± 0.018 1.38 ± 0.04 ± 0.12 CMS 0.344 ± 0.005 ± 0.018 1.00 ± 0.01 ± 0.10 — — —  p - + p 1.89 CPLEAR 2.332 ± 0.025 0.972 ± 0.014 1.96 ± 0.03 1.04 ± 0.01 [24]  p + P b 5020 CMS 0.358 ± 0.007 ± 0.021 1.70 ± 0.02 ± 0.12 — — —  P b + P b 17.3 WA98 0.327 ± 0.008 6.51 ± 0.10 0.307 ± 0.008 6.83 ± 0.10 [23]  O + C 19.4 WA80 0.44 ± 0.05 2.8 ± 0.3 0.40 ± 0.03 2.90 ± 0.21 [27] O + C u 0.24 ± 0.07 2.53 ± 0.11 0.17 ± 0.03 2.35 ± 0.11 O + A g 0.28 ± 0.10 2.71 ± 0.11 0.17 ± 0.04 2.44 ± 0.11 O + A u 0.110 ± 0.015 1.63 ± 0.05 0.085 ± 0.007 1.68 ± 0.06Table 4
Parameter values for exponential functionΩC in (2).
Collision s N N, Experiment Estimation based on (7a), (7b) Experimental values Ref. GeV λ C R C, fm λ C R C, fm p+p 7.21 E766 1.63 ± 0.10 2.29 ± 0.08 — — — 26.0 NA23 1.4 ± 0.7 2.6 ± 0.7 — — — 63.0 AFS 0.67 ± 0.05 1.50 ± 0.10 0.77 ± 0.07 1.32 ± 0.13 [29] 200 STAR 0.588 ± 0.010 2.43 ± 0.04 ± 0.26 — — — 900 ALICE 0.63 ± 0.06 1.89 ± 0.1 2 - 0.40 + 0.20 0.63 ± 0.05 1.67 ± 0.1 2 - 0.35 + 0.16 [25] 0.57 ± 0.06 2.26 ± 0.1 9 - 0.34 + 0.14 0.55 ± 0.05 1.90 ± 0.1 8 - 0.36 + 0.11 ATLAS 0.62 ± 0.03 ± 0.08 1.84 ± 0.07 ± 0.20 0.74 ± 0.03 ± 0.09 1.83 ± 0.07 ± 0.20 [30] CMS 0.57 ± 0.02 1.85 ± 0.06 0.616 ± 0.011 ± 0.029 1.56 ± 0.02 ± 0.15 [14, 15] 2360 0.60 ± 0.03 1.86 ± 0.07 0.66 ± 0.07 ± 0.05 1.99 ± 0.18 ± 0.24 [15] 7000 ALICE 1.104 ± 0.006 ± 0.112 2.627 ± 0.01 0 - 0.626 + 0.320 1.180 ± 0.005 ± 0.084 2.038 ± 0.01 4 - 0.046 + 0.083 [31] ATLAS 0.627 ± 0.005 ± 0.058 2.163 ± 0.012 ± 0.209 0.718 ± 0.006 ± 0.062 2.067 ± 0.012 ± 0.182 [30] 0.521 ± 0.027 ± 0.055 2.76 ± 0.10 ± 0.29 0.531 ± 0.024 ± 0.046 2.46 ± 0.08 ± 0.22  p - + p 1.89 CPLEAR 4.21 ± 0.09 2.079 ± 0.028 4.79 ± 0.10 1.89 ± 0.04 [24]  A u + A u 4.86 E802 0.86 ± 0.08 12.4 ± 0.6 — — — P b + P b 17.3 NA44 1.06 ± 0.10 15.1 ± 0.9 — — — WA98 0.69 ± 0.02 14.1 ± 0.3 0.718 ± 0.023 13.34 ± 0.26 [23] A u + A u 130 STAR 0.99 ± 0.03 ± 0.08 13.0 ± 0.3 ± 0.9 — — —  S i + A l 5.41 E802 1.23 ± 0.08 8.4 ± 0.3 — — — S i + A u 0.95 ± 0.05 9.4 ± 0.3 — — — S + P b 17.3 NA44 0.81 ± 0.05 7.8 ± 0.6 — — — O + C 19.4 WA80 0.85 ± 0.08 5.9 ± 0.5 0.92 ± 0.13 5.7 ± 0.7 [27] O + C u 0.34 ± 0.06 4.7 ± 0.2 0.49 ± 0.14 5.05 ± 0.25 O + A g 0.34 ± 0.09 4.9 ± 0.2 0.59 ± 0.21 5.46 ± 0.24 O + A u 0.156 ± 0.014 3.19 ± 0.13 0.20 ± 0.03 3.07 ± 0.12


### 4.1. Relations between Parameters for Cauchy/Gaussian Distribution and Lévy One

The general system of (3a) and (3b) allow us to estimate the 1D BEC parameters for exponential/Gaussian function ΩC(G) based on thea priori known parameter values for ΩL≡Ω(αL,λL,RL) corresponding to general view of centrally symmetric Lévy distribution and vice versa.


#### 4.1.1. Direction of CalculationsΩL→ΩC/G

The sets{λC,RC} and {λG,RG} are estimated for experimentally known ΩL [14–16] and finite q-ranges with help of (3a) and (3b). The estimations are shown in Table 1 together with the available experimental results and the data for Cauchy distribution are shown on the first line, for Gaussian parametrization, on the second line for certain experiment at given energy. As seen estimations for strength of correlations and 1D radius calculated with help of (3a) and (3b) agree with experimental values within errors for both the Cauchy and the Gaussian distributions for particle emission points at all energies under study. One can note that estimation for λG coincides with experimental values within total errors only at sNN=7 TeV. Nevertheless the general system of (3a) and (3b) provides rather well estimations of 1D BEC parameters for both the Cauchy and the Gaussian distributions. The possibility is considered for application of ultimate relations (4a) and (4b) for the experimental data under study. All estimations from (4a) and (4b) coincide with results from general system (3a) and (3b) within errors with exception of λG for CMS at sNN=7 TeV. For the last case the estimations from ((3a) and (3b)) and ((4a) and (4b)) coincide with each other within 2σ. Thus the ultimate relations (4a) and (4b) provide reasonable estimations for 1D BEC parameters in both cases of the Cauchy and the Gaussian distributions within features of modern experiments under consideration, that is, at q1~10-2 GeV/c, q2~2 GeV/c, and αL~0.8 which is close to the region of Lévy index values with weaker changing of 1D BEC parameters  (Figure 2).


#### 4.1.2. Direction of CalculationsΩC/G→ΩL

Here the set of 1D BEC parameters{λL,RL} is estimated ata priori given αL with help of the system of (3a) and (3b) for experimentally known ΩC [14, 16], ΩG [15, 16] and finite q-ranges. The estimations for parameters of ΩL are shown in Table 2 together with the available experimental results and the values of λL, RL derived from experimental analysis with Cauchy distribution are shown on the first line; the second one corresponds to the calculations with data for Gaussian distribution for certain experiment at given energy. The relatively large systematic uncertainties for ATLAS are driven by corresponding error for Lévy index [16]. There is a remarkable agreement between results of calculations and experimental analyses (Table 2): estimations for all 1D BEC parameters coincide with corresponding experimental values within statistical errors with exception of the λL at sNN=7 TeV for ATLAS data. For the last case the coincidence between estimations from (3a) and (3b) experiment is achieved within total errors. This conclusion is for both ΩC→ΩL and ΩG→ΩL schemes of calculations. Thus the system of general equations (3a) and (3b) provides the high-quality estimations of 1D BEC parameters for general view centrally symmetric Lévy parametrization ΩL based on thea priori known αL and results for exponential/Gaussian function. One can note the ultimate relations (4a) and (4b) for semi-infinite range of Lorentz invariant quantity q result in reasonable estimations for the set of 1D BEC parameters {λL,RL} with help of results for exponential function ΩC as well as for Gaussian one ΩG. Nevertheless the general system (3a) and (3b) allows the noticeable improvement of the results with respect to (4a) and (4b) for chaoticity λL derived from results for Gaussian distribution. This feature can be expected from Figure 2(d) because curve calculated with (3a) and (3b) differs from the corresponding ultimate α2-dependence at values α1≲0.8 which are close to the experimental data (Table 6).


### 4.2. Relations between Parameters for Cauchy and Gaussian Distributions

The system of (7a) and (7b) derived above is used for estimation of the 1D BEC parameter values for Gaussian parametrization based on thea priori known values for set {λC,RC} of BEC parameters for exponential parametrization and experimental ranges on q and vice versa. In the subsection the separation is used on various multiplicity event classes in p+p, p¯+p collisions for 1D BEC results in some experiments. The results for minimum bias events are shown on the first line and for high multiplicity events they are shown on the second line for certain experiment at given energy in Tables 3 and 4.


#### 4.2.1. Direction of CalculationsΩC→ΩG

Parameters for Gaussian function are calculated with help of system (7a) and (7b) anda priori known set {λC,RC}. The results are shown in Table 3 together with available published experimental results for the Gaussian set {λG,RG}. As seen from Table 3 the estimations for the set of the Gaussian parameters are equal for published results within (total) errors for proton-proton collisions with exception of the value of strength of correlations λG in ATLAS minimum bias events at sNN=7 TeV and CMS result at collision energy sNN=2.36 TeV. In the two last cases the agreement is observed within 2σ. The similar situation is for symmetric nucleus-nucleus collisions; that is, the estimations within the present paper for set of Gaussian parameters {λG,RG} agree with the results of the WA98 experiment [23] within 2σ. But there is qualitative agreement only between results of calculations with help of (8a) and (8b) and experimental data for p¯+p collisions [24]. Perhaps, this discrepancy is dominated by some features of experiment provided unusually large values of chaoticity for both the exponential and the Gaussian parameterizations of 1D CF C2(q). For asymmetric nuclear interactions the agreement between results of calculations in the present paper and available experimental data is achieved mostly within errors for both λG and the 1D BEC radius. Only estimations for Gaussian 1D radius RG in O+Ag and for λG in O+Au coincide with corresponding results of the WA80 experiment [27] within 2σ. It should be emphasized that approximate calculations demonstrate the same results as in Table 3 within errors for all consecutive simplifications ((8a)–(11b)) which are valid and can be applied for certain experiment. One can note in particular that as expected the ultimate relations (6a) and (6b) work rather well for the CMS results at sNN=2.76 TeV with low enough q1≈0.6 MeV/c and high enough q2≈2.0 GeV/c. Thus detailed calculations for case ΩC→ΩG confirm both the correctness of suggestions made above for certain experiments and the validity of corresponding systems of ((6a)–(11b)).


#### 4.2.2. Direction of CalculationsΩG→ΩC

Values of 1D BEC parameters for exponential function are estimated with help of system (7a) and (7b) anda priori known values for Gaussian BEC quantities {λG,RG}. The results are presented in Table 4 together with available published experimental results for the set of BEC parameters {λC,RC} corresponding to the Cauchy distribution function in coordinate space for particle emission points. For p+p collisions there is agreement between estimations for parameters for exponential parametrization of CF C2(q) calculated with (7a) and (7b) and experimental results within (total) errors with exception of the 1D BEC radius RC in CMS at sNN=0.9 TeV. In the last case results from calculation and experiment coincide within 2σ. The similar situation is observed for nucleus-nucleus collisions: estimations for parameters of exponential function ΩC derived with (7a) and (7b) agree with corresponding experimental results mostly within 1σ, but the coincidence is achieved within 2σ for RC in both the Pb+Pb collisions at sNN=17.3 GeV [23] and the O+Ag reactions at sNN=19.4 GeV [27]. The estimations of 1D BEC parameters λC, RC obtained for p¯+p with help of (8a) and (8b) are in qualitative agreement with corresponding experimental results [24] even for the case of unusually large chaoticity. Thus the system of (7a) and (7b) provides quite reasonable estimations for parameters for exponential function based on thea priori known values of 1D BEC observables for Gaussian function in various strong interaction processes at all available experimental energies.In summary of the section, the systems of ((3a) and (3b)) and ((7a) and (7b)) provide correct estimations for both desired BEC parameters, namely, the strength of correlations and the 1D radius in the case of centrally symmetric Lévy distribution ΩL as well as for specific Cauchy and Gaussian ones. In general the estimations show remarkable agreement with available experimental data. Thus the systems of equations suggested in Sections 2 and 3 can be useful in experimental data analysis as well as in phenomenological study for estimation of unknown values of BEC parameters for some parametrization (2) of 1D CF C2(q) based on the available values of λ and R for another centrally symmetric Lévy distribution. As seen from Tables 1–4 the new estimations are obtained for 1D BEC parameters in ΩC, ΩG in many cases for which the corresponding experimental results are absent. Thus the systems of equations derived within the framework of this paper allow the expansion of the available ensemble of values for 1D BEC parameters λ and R which it is useful for future investigations.


## 4.1. Relations between Parameters for Cauchy/Gaussian Distribution and Lévy One

The general system of (3a) and (3b) allow us to estimate the 1D BEC parameters for exponential/Gaussian function ΩC(G) based on thea priori known parameter values for ΩL≡Ω(αL,λL,RL) corresponding to general view of centrally symmetric Lévy distribution and vice versa.


### 4.1.1. Direction of CalculationsΩL→ΩC/G

The sets{λC,RC} and {λG,RG} are estimated for experimentally known ΩL [14–16] and finite q-ranges with help of (3a) and (3b). The estimations are shown in Table 1 together with the available experimental results and the data for Cauchy distribution are shown on the first line, for Gaussian parametrization, on the second line for certain experiment at given energy. As seen estimations for strength of correlations and 1D radius calculated with help of (3a) and (3b) agree with experimental values within errors for both the Cauchy and the Gaussian distributions for particle emission points at all energies under study. One can note that estimation for λG coincides with experimental values within total errors only at sNN=7 TeV. Nevertheless the general system of (3a) and (3b) provides rather well estimations of 1D BEC parameters for both the Cauchy and the Gaussian distributions. The possibility is considered for application of ultimate relations (4a) and (4b) for the experimental data under study. All estimations from (4a) and (4b) coincide with results from general system (3a) and (3b) within errors with exception of λG for CMS at sNN=7 TeV. For the last case the estimations from ((3a) and (3b)) and ((4a) and (4b)) coincide with each other within 2σ. Thus the ultimate relations (4a) and (4b) provide reasonable estimations for 1D BEC parameters in both cases of the Cauchy and the Gaussian distributions within features of modern experiments under consideration, that is, at q1~10-2 GeV/c, q2~2 GeV/c, and αL~0.8 which is close to the region of Lévy index values with weaker changing of 1D BEC parameters  (Figure 2).


### 4.1.2. Direction of CalculationsΩC/G→ΩL

Here the set of 1D BEC parameters{λL,RL} is estimated ata priori given αL with help of the system of (3a) and (3b) for experimentally known ΩC [14, 16], ΩG [15, 16] and finite q-ranges. The estimations for parameters of ΩL are shown in Table 2 together with the available experimental results and the values of λL, RL derived from experimental analysis with Cauchy distribution are shown on the first line; the second one corresponds to the calculations with data for Gaussian distribution for certain experiment at given energy. The relatively large systematic uncertainties for ATLAS are driven by corresponding error for Lévy index [16]. There is a remarkable agreement between results of calculations and experimental analyses (Table 2): estimations for all 1D BEC parameters coincide with corresponding experimental values within statistical errors with exception of the λL at sNN=7 TeV for ATLAS data. For the last case the coincidence between estimations from (3a) and (3b) experiment is achieved within total errors. This conclusion is for both ΩC→ΩL and ΩG→ΩL schemes of calculations. Thus the system of general equations (3a) and (3b) provides the high-quality estimations of 1D BEC parameters for general view centrally symmetric Lévy parametrization ΩL based on thea priori known αL and results for exponential/Gaussian function. One can note the ultimate relations (4a) and (4b) for semi-infinite range of Lorentz invariant quantity q result in reasonable estimations for the set of 1D BEC parameters {λL,RL} with help of results for exponential function ΩC as well as for Gaussian one ΩG. Nevertheless the general system (3a) and (3b) allows the noticeable improvement of the results with respect to (4a) and (4b) for chaoticity λL derived from results for Gaussian distribution. This feature can be expected from Figure 2(d) because curve calculated with (3a) and (3b) differs from the corresponding ultimate α2-dependence at values α1≲0.8 which are close to the experimental data (Table 6).


## 4.1.1. Direction of CalculationsΩL→ΩC/G

The sets{λC,RC} and {λG,RG} are estimated for experimentally known ΩL [14–16] and finite q-ranges with help of (3a) and (3b). The estimations are shown in Table 1 together with the available experimental results and the data for Cauchy distribution are shown on the first line, for Gaussian parametrization, on the second line for certain experiment at given energy. As seen estimations for strength of correlations and 1D radius calculated with help of (3a) and (3b) agree with experimental values within errors for both the Cauchy and the Gaussian distributions for particle emission points at all energies under study. One can note that estimation for λG coincides with experimental values within total errors only at sNN=7 TeV. Nevertheless the general system of (3a) and (3b) provides rather well estimations of 1D BEC parameters for both the Cauchy and the Gaussian distributions. The possibility is considered for application of ultimate relations (4a) and (4b) for the experimental data under study. All estimations from (4a) and (4b) coincide with results from general system (3a) and (3b) within errors with exception of λG for CMS at sNN=7 TeV. For the last case the estimations from ((3a) and (3b)) and ((4a) and (4b)) coincide with each other within 2σ. Thus the ultimate relations (4a) and (4b) provide reasonable estimations for 1D BEC parameters in both cases of the Cauchy and the Gaussian distributions within features of modern experiments under consideration, that is, at q1~10-2 GeV/c, q2~2 GeV/c, and αL~0.8 which is close to the region of Lévy index values with weaker changing of 1D BEC parameters  (Figure 2).


## 4.1.2. Direction of CalculationsΩC/G→ΩL

Here the set of 1D BEC parameters{λL,RL} is estimated ata priori given αL with help of the system of (3a) and (3b) for experimentally known ΩC [14, 16], ΩG [15, 16] and finite q-ranges. The estimations for parameters of ΩL are shown in Table 2 together with the available experimental results and the values of λL, RL derived from experimental analysis with Cauchy distribution are shown on the first line; the second one corresponds to the calculations with data for Gaussian distribution for certain experiment at given energy. The relatively large systematic uncertainties for ATLAS are driven by corresponding error for Lévy index [16]. There is a remarkable agreement between results of calculations and experimental analyses (Table 2): estimations for all 1D BEC parameters coincide with corresponding experimental values within statistical errors with exception of the λL at sNN=7 TeV for ATLAS data. For the last case the coincidence between estimations from (3a) and (3b) experiment is achieved within total errors. This conclusion is for both ΩC→ΩL and ΩG→ΩL schemes of calculations. Thus the system of general equations (3a) and (3b) provides the high-quality estimations of 1D BEC parameters for general view centrally symmetric Lévy parametrization ΩL based on thea priori known αL and results for exponential/Gaussian function. One can note the ultimate relations (4a) and (4b) for semi-infinite range of Lorentz invariant quantity q result in reasonable estimations for the set of 1D BEC parameters {λL,RL} with help of results for exponential function ΩC as well as for Gaussian one ΩG. Nevertheless the general system (3a) and (3b) allows the noticeable improvement of the results with respect to (4a) and (4b) for chaoticity λL derived from results for Gaussian distribution. This feature can be expected from Figure 2(d) because curve calculated with (3a) and (3b) differs from the corresponding ultimate α2-dependence at values α1≲0.8 which are close to the experimental data (Table 6).


## 4.2. Relations between Parameters for Cauchy and Gaussian Distributions

The system of (7a) and (7b) derived above is used for estimation of the 1D BEC parameter values for Gaussian parametrization based on thea priori known values for set {λC,RC} of BEC parameters for exponential parametrization and experimental ranges on q and vice versa. In the subsection the separation is used on various multiplicity event classes in p+p, p¯+p collisions for 1D BEC results in some experiments. The results for minimum bias events are shown on the first line and for high multiplicity events they are shown on the second line for certain experiment at given energy in Tables 3 and 4.


### 4.2.1. Direction of CalculationsΩC→ΩG

Parameters for Gaussian function are calculated with help of system (7a) and (7b) anda priori known set {λC,RC}. The results are shown in Table 3 together with available published experimental results for the Gaussian set {λG,RG}. As seen from Table 3 the estimations for the set of the Gaussian parameters are equal for published results within (total) errors for proton-proton collisions with exception of the value of strength of correlations λG in ATLAS minimum bias events at sNN=7 TeV and CMS result at collision energy sNN=2.36 TeV. In the two last cases the agreement is observed within 2σ. The similar situation is for symmetric nucleus-nucleus collisions; that is, the estimations within the present paper for set of Gaussian parameters {λG,RG} agree with the results of the WA98 experiment [23] within 2σ. But there is qualitative agreement only between results of calculations with help of (8a) and (8b) and experimental data for p¯+p collisions [24]. Perhaps, this discrepancy is dominated by some features of experiment provided unusually large values of chaoticity for both the exponential and the Gaussian parameterizations of 1D CF C2(q). For asymmetric nuclear interactions the agreement between results of calculations in the present paper and available experimental data is achieved mostly within errors for both λG and the 1D BEC radius. Only estimations for Gaussian 1D radius RG in O+Ag and for λG in O+Au coincide with corresponding results of the WA80 experiment [27] within 2σ. It should be emphasized that approximate calculations demonstrate the same results as in Table 3 within errors for all consecutive simplifications ((8a)–(11b)) which are valid and can be applied for certain experiment. One can note in particular that as expected the ultimate relations (6a) and (6b) work rather well for the CMS results at sNN=2.76 TeV with low enough q1≈0.6 MeV/c and high enough q2≈2.0 GeV/c. Thus detailed calculations for case ΩC→ΩG confirm both the correctness of suggestions made above for certain experiments and the validity of corresponding systems of ((6a)–(11b)).


### 4.2.2. Direction of CalculationsΩG→ΩC

Values of 1D BEC parameters for exponential function are estimated with help of system (7a) and (7b) anda priori known values for Gaussian BEC quantities {λG,RG}. The results are presented in Table 4 together with available published experimental results for the set of BEC parameters {λC,RC} corresponding to the Cauchy distribution function in coordinate space for particle emission points. For p+p collisions there is agreement between estimations for parameters for exponential parametrization of CF C2(q) calculated with (7a) and (7b) and experimental results within (total) errors with exception of the 1D BEC radius RC in CMS at sNN=0.9 TeV. In the last case results from calculation and experiment coincide within 2σ. The similar situation is observed for nucleus-nucleus collisions: estimations for parameters of exponential function ΩC derived with (7a) and (7b) agree with corresponding experimental results mostly within 1σ, but the coincidence is achieved within 2σ for RC in both the Pb+Pb collisions at sNN=17.3 GeV [23] and the O+Ag reactions at sNN=19.4 GeV [27]. The estimations of 1D BEC parameters λC, RC obtained for p¯+p with help of (8a) and (8b) are in qualitative agreement with corresponding experimental results [24] even for the case of unusually large chaoticity. Thus the system of (7a) and (7b) provides quite reasonable estimations for parameters for exponential function based on thea priori known values of 1D BEC observables for Gaussian function in various strong interaction processes at all available experimental energies.In summary of the section, the systems of ((3a) and (3b)) and ((7a) and (7b)) provide correct estimations for both desired BEC parameters, namely, the strength of correlations and the 1D radius in the case of centrally symmetric Lévy distribution ΩL as well as for specific Cauchy and Gaussian ones. In general the estimations show remarkable agreement with available experimental data. Thus the systems of equations suggested in Sections 2 and 3 can be useful in experimental data analysis as well as in phenomenological study for estimation of unknown values of BEC parameters for some parametrization (2) of 1D CF C2(q) based on the available values of λ and R for another centrally symmetric Lévy distribution. As seen from Tables 1–4 the new estimations are obtained for 1D BEC parameters in ΩC, ΩG in many cases for which the corresponding experimental results are absent. Thus the systems of equations derived within the framework of this paper allow the expansion of the available ensemble of values for 1D BEC parameters λ and R which it is useful for future investigations.


## 4.2.1. Direction of CalculationsΩC→ΩG

Parameters for Gaussian function are calculated with help of system (7a) and (7b) anda priori known set {λC,RC}. The results are shown in Table 3 together with available published experimental results for the Gaussian set {λG,RG}. As seen from Table 3 the estimations for the set of the Gaussian parameters are equal for published results within (total) errors for proton-proton collisions with exception of the value of strength of correlations λG in ATLAS minimum bias events at sNN=7 TeV and CMS result at collision energy sNN=2.36 TeV. In the two last cases the agreement is observed within 2σ. The similar situation is for symmetric nucleus-nucleus collisions; that is, the estimations within the present paper for set of Gaussian parameters {λG,RG} agree with the results of the WA98 experiment [23] within 2σ. But there is qualitative agreement only between results of calculations with help of (8a) and (8b) and experimental data for p¯+p collisions [24]. Perhaps, this discrepancy is dominated by some features of experiment provided unusually large values of chaoticity for both the exponential and the Gaussian parameterizations of 1D CF C2(q). For asymmetric nuclear interactions the agreement between results of calculations in the present paper and available experimental data is achieved mostly within errors for both λG and the 1D BEC radius. Only estimations for Gaussian 1D radius RG in O+Ag and for λG in O+Au coincide with corresponding results of the WA80 experiment [27] within 2σ. It should be emphasized that approximate calculations demonstrate the same results as in Table 3 within errors for all consecutive simplifications ((8a)–(11b)) which are valid and can be applied for certain experiment. One can note in particular that as expected the ultimate relations (6a) and (6b) work rather well for the CMS results at sNN=2.76 TeV with low enough q1≈0.6 MeV/c and high enough q2≈2.0 GeV/c. Thus detailed calculations for case ΩC→ΩG confirm both the correctness of suggestions made above for certain experiments and the validity of corresponding systems of ((6a)–(11b)).


## 4.2.2. Direction of CalculationsΩG→ΩC

Values of 1D BEC parameters for exponential function are estimated with help of system (7a) and (7b) anda priori known values for Gaussian BEC quantities {λG,RG}. The results are presented in Table 4 together with available published experimental results for the set of BEC parameters {λC,RC} corresponding to the Cauchy distribution function in coordinate space for particle emission points. For p+p collisions there is agreement between estimations for parameters for exponential parametrization of CF C2(q) calculated with (7a) and (7b) and experimental results within (total) errors with exception of the 1D BEC radius RC in CMS at sNN=0.9 TeV. In the last case results from calculation and experiment coincide within 2σ. The similar situation is observed for nucleus-nucleus collisions: estimations for parameters of exponential function ΩC derived with (7a) and (7b) agree with corresponding experimental results mostly within 1σ, but the coincidence is achieved within 2σ for RC in both the Pb+Pb collisions at sNN=17.3 GeV [23] and the O+Ag reactions at sNN=19.4 GeV [27]. The estimations of 1D BEC parameters λC, RC obtained for p¯+p with help of (8a) and (8b) are in qualitative agreement with corresponding experimental results [24] even for the case of unusually large chaoticity. Thus the system of (7a) and (7b) provides quite reasonable estimations for parameters for exponential function based on thea priori known values of 1D BEC observables for Gaussian function in various strong interaction processes at all available experimental energies.In summary of the section, the systems of ((3a) and (3b)) and ((7a) and (7b)) provide correct estimations for both desired BEC parameters, namely, the strength of correlations and the 1D radius in the case of centrally symmetric Lévy distribution ΩL as well as for specific Cauchy and Gaussian ones. In general the estimations show remarkable agreement with available experimental data. Thus the systems of equations suggested in Sections 2 and 3 can be useful in experimental data analysis as well as in phenomenological study for estimation of unknown values of BEC parameters for some parametrization (2) of 1D CF C2(q) based on the available values of λ and R for another centrally symmetric Lévy distribution. As seen from Tables 1–4 the new estimations are obtained for 1D BEC parameters in ΩC, ΩG in many cases for which the corresponding experimental results are absent. Thus the systems of equations derived within the framework of this paper allow the expansion of the available ensemble of values for 1D BEC parameters λ and R which it is useful for future investigations.


## 5. Summary

The case is investigated for smooth approximation of the one experimental 1D Bose–Einstein correlation function by two various centrally symmetric Lévy parameterizations. It is suggested that lowest moments of corresponding distributions are equal approximately. Then the relations are derived between sets of 1D BEC observables, namely, strength of correlations and source radius, for two general view centrally symmetric Lévy parameterizations under consideration for the first time. The relations obtained in the paper take into account the finiteness of range of Lorentz invariant four-momentum difference in experimental studies. Detailed analysis results in the systems of transcendental equations for various finite ranges of the Lorentz invariant four-momentum difference in the specific case of the exponential and Gaussian parameterizations for correlation function. It is shown that finite range ofq should be taken into account and corresponding systems of equations should be used for derivation of set {λ,R} based on thea priori known values of corresponding parameters for both cases of the two general view centrally symmetric Lévy parameterizations and the two specific functions (exponential and Gaussian) most used in experimental studies. The ultimate relations derived for semi-infinite range of q can be utilized carefully for experimental analysis and these equations can produce the reasonable estimations for 1D BEC parameters for ranges of Lévy indexes ∀i=1,2:αi≳1 only. Furthermore it is demonstrated that the corresponding ultimate relations for specific case of Cauchy and Gaussian distributions for source in coordinate space produce the reasonable estimations of 1D BEC parameters for few modern experimental analyses only. The mathematical formalism suggested within the framework of the preset paper is verified with help of experimental results obtained for wide set of strong interaction processes in all available energy range. The two pairs of distributions are considered: general view centrally symmetric Lévy one with specific case (Cauchy/Gaussian); two specific Cauchy and Gaussian distributions. For both cases verifications are made for both directions of calculations. Namely, the calculations have been made for estimation of 1D BEC observables for Cauchy/Gaussian function based on thea priori known values of parameters of general Lévy parametrization and vice versa, for estimations of Gaussian parameters based on thea priori known values of observables for Cauchy distribution and vice versa. Comparison shows the quantitative agreement between estimations derived with help of mathematical formalism developed in the paper and most of available experimental results for both pairs consisting of the general view centrally symmetric Lévy parametrization and specific (exponential/Gaussian) function and the two specific source distributions (Cauchy and Gaussian) most used in experimental studies for any direction of calculations.



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