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Levy Distributions for One-Dimensional Analysis of the Bose-Einstein Correlations.

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-momentum [p.sup.1] and [p.sup.2]. Experimentally, the one-dimensional BEC effect is observed as an enhancement at low values of the Lorentz invariant quantity q = [[square root of -([p.sub.1] - [p.sub.2])].sup.2] [greater than or equal to] 0 in the two-particle correlation function (CF):

[C.sup.2] (q) = [rho](q)/[[rho].sub.ref](q). (1)

Here [rho] is the two-particle density function and [[rho].sub.ref] 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 [C.sub.2](q) ~ [q.sup.-[beta]] 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 Levy) 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 non-isotropic centrally symmetric Levy distributions [8,9] is most useful for studies of Bose-Einstein CF. Therefore this subclass of centrally symmetric Leevy distributions is considered regarding of BEC measurements in the present paper.

For low-dimensional (1D) analysis the centrally symmetric Levy distribution results in the most general parametrization of the experimental Bose-Einstein CF

[C.sub.2] (q) [??] 1 + [OMEGA] ([alpha], [lambda], z), [OMEGA] ([alpha], [lambda], z) [equivalent to] [lambda] exp (-[[absolute value of (z)].sup.[alpha]]), z [equivalent to] qR. (2)

Here X is the strength of correlations called also chaoticity, R is the 1D BEC radius, and 0 < [alpha] [less than or equal to] 2 is the Levy index called also index of stability. As known for a static source with no final state interactions [10, 11], there is the relation [C.sub.2](q) [??] [[absolute value of ([??](q))].sup.2], [??](q) = [integral] dx exp(iqx)f(x); that is, Bose-Einstein CF [C.sub.2](q) measures the absolute value squared Fourier transformed source density in the coordinate space f(x), J dx f(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 a = 2; the Gaussian parametrization corresponded to the normal (Gaussian) distribution function [f.sup.G](x) = [(2[pi][R.sup.2.sub.G]).sup.-1/2] exp[-[(x- [x.sub.0]).sup.2]/2[R.sup.2.sub.G]], where the Gaussian scale parameter is [R.sup.2.sub.G] = <[x.sup.2]> - [x.sup.2.sub.0], the standard deviation; then another specific case of (2) at [alpha] = 1 is used widely, especially, for particle (not nuclear) collisions. Equation (2) at [alpha] = 1 is called exponential parametrization for Bose-Einstein CF [C.sub.2](q) and it corresponds to the Cauchy (Lorentzian) distribution function [f.sup.c](x) = [[pi].sup.-1][R.sup.c]/[[R.sup.2.sub.c] + [(x - [x.sup.0]).sup.2]] with scale parameter [R.sub.C] [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 Levy distribution with [alpha] [member of] (0;2), [alpha] [not equal to] 1 the corresponding source density in coordinate space f(x) can be written analytically for [alpha] = 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 [C.sub.2](q). Therefore the derivation of the relations between the sets of BEC parameters for two centrally symmetric Leevy 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 [square root of [s.sub.NN] ~ 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/[bar.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 [C.sub.2](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 Leevy 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 Section 2, mathematical formalism is described for case of two general view centrally symmetric Levy distributions. Dependencies of desired 1D BEC observables on q and [alpha] are studied for a priori known parameters for second centrally symmetric Leevy 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 [C.sub.2](q). Database of experimental results for set of 1D BEC parameters {[lambda],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 Levy distribution, "C" is for the Cauchy source distribution function, and "G" is for Gaussian one. Otherwise the notations [[[lambda].sub.E], 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 [C.sub.2](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 [f.sub.c](x) and the term "exponential function/parametrization" is used for the related parametrization of correlation function [C.sub.2,E] (q); for the case of arbitrary 0 < [alpha] < 2, [alpha] [not equal to] 1, the term "centrally symmetric Leevy" is suitable for both the source function in coordinate space [f.sub.L](x) and the parametrization of correlation function [C.sub.2,L](q); the similar situation is for [alpha] = 2: the term "Gaussian" is applicable for both the source function in coordinate space [f.sub.G](x) and the corresponding parametrization of correlation function [C.sub.2,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 CF [C.sub.2](q) be described by two parameterizations (2) with [[OMEGA].sub.1] [equivalent to] [OMEGA]([[alpha].sub.1], [[lambda].sub.1], [z.sup.1]) and [[OMEGA].sub.2] = [OMEGA]([[alpha].sub.2], [[lambda].sub.2],[z.sub.2]). Then relations between parameters of [[OMEGA].sub.1] and [[OMEGA].sub.2] can be deduced on the basis that both parameterizations describe one experimental CF [C.sub.2](q), that is one sample of experimental points (in general the approximations of [C.sub.2](q) are characterized by different qualities for various parameterizations (2) with [[OMEGA].sub.1] and [[OMEGA].sub.2]. 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 [[OMEGA].sub.1] and [[OMEGA].sub.2] are approximately equal to each other as well as the first moments of the corresponding centrally symmetric Leevy distributions.

2.1. Mathematical Formalism. The relations between two sets of parameters {[[lambda].sub.1], [R.sub.1]} and {[[lambda].sub.2], [R.sup.2]} of the particle source can be derived from the following system of equations:

[mathematical expression not reproducible]; (3a)

[mathematical expression not reproducible]. (3b)

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 [omega]([[alpha].sub.i], [[lambda].sub.i], [z.sub.i]) 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 Levy indexes [[alpha].sub.i] are known a priori for both parameterizations [[OMEGA].sub.i], i = 1,2. In equations (3a) and (3b) the integrals are taken over full fit ranges [J.sub.i], i = 1,2, for corresponding parameterizations with [[OMEGA].sub.i] ; of experimental CF. It should be noted that in general case (i) the ranges of integration can be different for parameterizations with [[OMEGA].sub.1] and [[OMEGA].sub.2]; (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, [mathematical expression not reproducible] and consequently for all types of integrals and [for all]i = 1,2 in system (3a) and (3b): [mathematical expression not reproducible] But usually the fit ranges [J.sub.i] are identical for both [[OMEGA].sub.i] = 1,2, in experimental studies (see, e.g., [15]). In general case of the centrally symmetric Levy distributions and finite fit ranges the system equations under consideration cannot be solved analytically. The numerical procedure should be[summation] used in order to get the relations between two sets of parameters {[[lambda].sub.1], [R.sub.1]} and {[[lambda].sub.2], [R.sub.2]} of the particle source in this case. Without loss of generality [[OMEGA].sub.1] and [[alpha].sub.2] are considered as known and values of BEC parameters {[[lambda].sub.2], [R.sup.2]} are supposed as desired below. Then for specific case of semi-infinite ranges for integration [for all]i = 1,2 : [J.sub.i] = [0; [infinity]) system (3a) and (3b) can be solved analytically and one can derive the following ultimate relations between two sets {[[lambda].sub.1], [R.sub.1]} and {[[lambda].sub.2], [R.sup.2]} of BEC parameters for corresponding centrally symmetric Levy parameterizations with [[OMEGA].sub.i] = 1,2

[mathematical expression not reproducible]; (4a)

[mathematical expression not reproducible] (4b)

and vice versa. Here [GAMMA](x) = [[integral].sup.[infinity].sub.0] [t.sup.x-1] exp(-t)dt, Re x > 0 is the gamma function.

In the point of view of data analysis the absence of the general analytic relations between {[lamba].sub.1], [R.sub.1]} and {[[lambda].sub.2], [R.sup.2]} 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 {[[alpha].sub.1],[lambda].sub.1], [R.sub.1]} with its errors {[[DELTA].sup.[+ or -]][[alpha].sub.1], [[DELTA].sup.[+ or -]] [[lambda].sub.1], [[DELTA].sup.[+ or -]] [R.sub.1]} for centrally symmetric Levy parametrization with [[OMEGA].sub.1] as well as for [[alpha].sub.2] with [[DELTA].sup.[+ or -]][[alpha].sub.2] for parametrization with [[OMEGA].sub.2]. The two sets of values for unknown BEC parameters {[[lambda].sub.2], [R.sub.2]} can be obtained with the help of suitable system of equations: the input values |[[alpha].sub.1] + [[DELTA].sup.+][[alpha].sub.1], [[lambda].sub.1] + [[DELTA].sup.+] [[lambda].sup.1], + [[DELTA].sup.+] [R.sub.1] and [[alpha].sub.2] + [[DELTA].sup.+] [[alpha].sub.2] produce the output set {[[lambda].sup.+.sub.2], [R.sup.+.sub.2]} and {[[alpha].sub.1]- [[DELTA].sup.-][[alpha].sub.1], [[lambda].sub.1]-[DELTA]-[[lambda].sub.1],- [[DELTA].sup.-][[lambda].sub.1], [[alpha].sub.2]-[[DELTA].sup.-][[alpha].sub.2]} [right arrow] {[[lambda].sup.-.sub.2], [R.sup.-.sub.2]}. Then the error estimations for set {[[lambda].sub.2], [R.sub.2]} of BEC parameters for parametrization with [[OMEGA].sub.2] can be calculated as follows:

[[DELTA].sup.[+ or -]][Y.sub.2] = [absolute value of ([Y.sup.[+ or -].sub.2] - [Y.sub.2]], [Y.sub.2] [equivalent to] 2 [[lambda].sub.2], [R.sup.2]. (5)

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 [DELTA][Y.sub.2] = ([[DELTA].sup.+][Y.sub.2] + [[DELTA].sup.-][Y.sup.2])/2.

2.2. Dependencies on q and a Variables. Figure 1 shows dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for centrally symmetric Leevy parametrization [[OMEGA].sub.1] with known [[alpha].sub.1] = 1.5 on low [q.sub.1] (a, c) and high [q.sub.2] (b, d) limits of integration in system (3a) and (3b) for set |[[alpha].sub.2], [[lambda].sub.2], [R.sub.2]} = {0.5,0.5,1.5 fm} for [[OMEGA].sub.2]. The solid lines correspond to the indicated values of [q.sub.2] in GeV/c for [q.sub.1]-dependence (a, c) and to shown values of the [q.sup.1] in GeV/c for [q.sup.2]-dependence (b, d). Values of BEC parameters [[lambda].sub.1] and [R.sup.1] depend strongly on the fixed second limit of integration ([q.sub.2(1)]) for both the (Figures 1(a) and 1(c)) and the [q.sub.1]-dependence (Figures 1(b) and 1(d)). The dashed lines correspond to the results from system (3a) and (3b) with [q.sub.2] [right arrow] [infinity] for [q.sub.1]- dependence (Figures 1(a) and 1(c)) and with = 0 for [q.sub.2]-dependence (Figures 1(b) and 1(d)). As seen the curves for general case of (3a) and (3b) coincide with dashed lines at [q.sub.2] = 10 GeV/c for [q.sup.1]-dependence (Figures 1(a) and 1(c)) and at [q.sup.2] = [10.sup.-3] GeV/c for [q.sub.1]-dependence (Figures 1(b) and 1(d)). These values for [q.sup.1] and especially for [q.sub.2] 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 [[OMEGA].sub.1] and [[OMEGA].sub.2]. The thin dotted lines demonstrate the ultimate levels for [R.sub.1] (Figures 1(a) and 1(b)) and [[lambda].sub.1] (Figures 1(c) and 1(d)) calculated with (4a) and (4b) for given values of [[alpha].sub.1] and the set of parameters [[[alpha].sub.2], [[lambda].sub.2], [R.sub.2]} for second Levy parametrization [[OMEGA].sub.2]. One can use the simple relations (4a) and (4b) for calculation {[[lambda].sub.1], [R.sup.1]} at [q.sub.1] [??] [10.sup.-2] GeV/c and [q.sub.2] > 10 GeV/c (Figures 1(a) and 1(b)) but as expected the values of BEC parameters [[[lambda].sub.1], [R.sub.1]} are far from the ultimate levels at any [q.sub.1] for [q.sub.2]-dependence in the range [q.sub.2] [less than or equal to] 2 GeV/c is considered in Figures 1(b) and 1(d).

In Figure 2 dependence of 1D BEC radius (a, b) and strength of correlations (c, d) is demonstrated for centrally symmetric Levy parametrization with [[OMEGA].sub.1] on [[alpha].sub.1] at fixed values of [[alpha].sub.2] (a, c) and on [[alpha].sub.2] at fixed values of [[alpha].sub.1] (b, d) for given limits of integration in the system of (3a) and (3b) [q.sub.1] = 0.02 GeV/c, [q.sub.2] = 2.0 GeV/c and for certain values of the BEC parameters for second centrally symmetric Leevy parametrization with [[OMEGA].sub.2]: [[lambda].sub.2] = 0.5 and [R.sub.2] = 1.5 fm. The solid lines correspond to the indicated values of the [[alpha].sub.2] for [[alpha].sub.1] -dependence (a, c) and to shown values of the [[alpha].sub.1] for [[alpha].sub.2]-dependence (b, d). The values for limits of integration [q.sub.1] and [q.sub.2] in system (3a) and (3b) are similar to those used in modern experiments. As seen dependencies of both BEC parameters on [[alpha].sub.i], i = 1,2, at fixed another Levy index [[alpha].sub.j][|.sub.j[not equal to]i], j = 1,2, change very fast at small value [[alpha].sub.j] = 0.2 in narrow range [[alpha].sub.i] [equivalent] [a.sub.j]. Such behavior is observed for both the results from system (3a) and (3b) and the estimations for [R.sub.1] (Figures 2(a) and 2(b)) and [[lambda].sub.1] (Figures 2(c) and 2(d)) calculated with (4a) and (4b) for semi-infinite ranges for integration and shown by dotted lines. The dependence [R.sup.1]([[alpha].sub.1]) shown in Figure 2(a) closes to the analytic one calculated with help of (4b) and presented by dotted line for any [[alpha].sub.2] under study with exception of the small value [[alpha].sub.2] = 0.2. For last case the agreement is obtained in very narrow range [[alpha].sub.1] [approximately equal to] 0.2 between results from system (3a) and (3b) and (4b). This feature maps clearly in corresponding dependencies [R.sub.2]([[alpha].sub.2]) shown in Figure 2(b) for [[alpha].sub.1] = 0.2. For large [[alpha].sub.1] > 1.0 solid and dotted lines are close to each other in the range [[alpha].sub.2] > 0.5 but agreement is poor significantly between results from system (3a) and (3b) and (4b) for [[alpha].sub.1] = 0.6 especially in domain [[alpha].sub.2] < 0.5 (Figure 2(b)) taking into account the sharp behavior for corresponding dependence [R.sup.1] ([[alpha].sub.2]). In general the behavior of dependencies of [[lambda].sub.1] on [[alpha].sub.1] (Figure 2(c)) and [[alpha].sub.2] (Figure 2(d)) is similar to the corresponding dependencies of 1D BEC radius 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 [q.sub.1,2] under consideration the approximate relations (4a) and (4b) should be used carefully for experimental analysis of dependencies on Leevy indexes and last equations can produce the reasonable estimations for BEC parameters for ranges [for all]i = 1,2 : [[alpha].sub.i] [??] 1 only.

As seen the mathematical formalism described above and the results in Figures 1 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 [C.sub.2](q) corresponding to the centrally symmetric Leevy source distributions.

3. Relations between BEC Parameters in Specific Cases

As seen in Figure 2 both dependencies of the 1D BEC radius [R.sub.1] (a, b) and the strength of correlations [A.sub.1] (c, d) on Levy indexes [a.sub.i], i = 1,2, show the weaker changing in the domain [for all]i = 1,2 : [??] 1 in comparison with the range of small values of Levy indexes. The region [for all]i = 1,2: [[alpha].sub.i] [??] 1 includes in particular the specific cases of Cauchy and Gaussian distributions for which corresponding parameterizations of BoseEinstein CF [C.sub.2](q) with [alpha] = 1 and [alpha] = 2 are used mostly for experimental studies. Therefore these certain views of Q; are studied in detail below. Let [[OMEGA].sub.1] [equivalent to] [[OMEGA].sub.G] = [OMEGA](2, [[lambda].sub.G], [R.sub.G]) for Gaussian parametrization (2) and [[OMEGA].sub.2] = [[OMEGA].sub.c] = [omega](1, [[lambda].sub.G], [R.sub.G]) for 1D approximation of experimental CF [C.sub.2](q) by exponential function.

3.1. Mathematical Formalism. Relations (4a) and (4b) are valid at any values of indexes of stability 0 < [[alpha].sub.i] [less than or equal to] 2 in two centrally symmetric Levy parameterizations with [[OMEGA].sub.i] = 1,2. If without loss of generality {[A.sup.C], [R.sup.c]} are considered as a priori known and values of Gaussian BEC parameters {[[lambda].sub.G], [R.sup.G]} are supposed as desired then as expected (4a) and (4b) result in the ultimate relations:

[[lambda].sup.u.sub.G] = 2[[lambda].sub.c]/[pi]; (6a)

[R.sup.u.sub.G] = [R.sub.c]/[square root of [pi]] (6b)

and 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 [[alpha].sub.1] = 2 and [[alpha].sub.2] = 1:

[mathematical expression not reproducible]; (7a)

[mathematical expression not reproducible]. (7b)

Here erf(x) = (2/[square root of [pi]]) [[integral].sup.*.sub.0] exp(- [t.sup.2])dt is the error integral and [Z.sub.1(2),i,c(G)] [equivalent to] [q.sub.1(2),i,c(G)],[R.sub.c(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 [[alpha].sub.1] =2 and [[alpha].sub.2] = 1 for experimental CF [C.sub.2](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 [N.sub.c(G)] is equal 2 for experimental analyses. Thus the general statement with regard of identity of integration ranges for [[OMEGA].sub.1] and [[OMEGA].sub.2] is quite confirmed for case of Cauchy and Gaussian distributions and [N.sub.C(G)] [equivalent to] N in system (7a) and (7b).

Further simplification for the system of (7a) and (7b) depends on features of certain experiment; direction of calculations [[OMEGA].sub.c] [??] [[OMEGA].sub.G], that is, what kind of a set of BEC parameters of the two, {[[lambda].sub.c], [R.sup.c]} and {[[lambda].sub.G], [R.sup.G]}, it is regarded as a priori known 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 [lambda] parameter and (ii) the accuracy for 1D BEC parameters in modern experiments is not better than ~[10.sup.-3] so far. Thus one can assume the conservative accuracy level [epsilon] = 5 X [10.sup.-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 [epsilon] and direction of calculation from a priori known Cauchy parameters to desired Gaussian parameters {[[lambda].sub.c], [R.sup.c]} [right arrow] {[[lambda].sub.G], [R.sub.G]}. But the statement is wrong for opposite direction of calculation from a priori known Gaussian parameters to desired Cauchy parameters {[[lambda].sub.G], [R.sup.G]} [right arrow] {[[lambda].sub.c], [R.sup.c]} at [epsilon] = 5 x [10.sup.-4]. Therefore the sum can be omitted in system (7a) and (7b) and equations can be rewritten as follows:

[mathematical expression not reproducible]; (8a)

[mathematical expression not reproducible] (8b)

for all experiments in the case of estimation of unknown BEC parameters for Gaussian function based on the a priori known set of corresponding parameters for exponential function and for all experiments with exception of p + p collisions at [square root of [s.sub.NN]] = 0.9, 2.36, and 7 TeV [15, 22] in the case of inverse problem. Accounting the relation [R.sub.G] [less than or equal to] [R.sup.c]/[square root of [pi]] and properties of the functions exp(-[x.sup.2]), erf (x) allows us to simplify (8a) and (8b) to system

[mathematical expression not reproducible]; (9a)

[mathematical expression not reproducible], (9b)

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 [epsilon]. 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 from a priori known Gaussian parameters to desired Cauchy parameters [[[lambda].sub.G], [R.sub.G]} [right arrow] [[[lambda].sub.c], [R.sup.c]} at [epsilon] = 5 x [10.sup.-4]. The simplest view of the system of (9a) and (9b)

[lambda]G/[R.sub.G][square root of [pi]] erfc([z.sub.1,G]) = 2[[lambda].sub.C]/[pi][R.sub.c] exp(-[z.sup.1,C]); (10a)

1/[R.sub.G][square root of [pi]] exp(-[z.sup.2.sub.1,G/erfc([z.sub.1,G]) = 1 + [z.sub.1,c]/[R.sub.c] (10b)

corresponds to the range of integration [[z.sub.1c(G)], [infinity]) and can be used for experimental results from ALICE [25], CMS [14,15,17, 26] with exception of the collision energy [square root of [s.sub.NN]] = 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, [z.sub.2c(G)]] allows the derivation of the following system from (9a) and (9b):

[[lambda].sub.G]/[R.sub.G][square root of [pi]] erf ([z.sub.2,G]) = 2[[lambda].sub.c]/[pi][R.sub.c] [1 - exp (-z2,c)]; (11a)

[mathematical expression not reproducible]. (11b)

As expected one can get the ultimate relations (6a) and (6b) from the any systems of ((10a) and (10b)) or ((11a) and (11b)) at [q.sub.1] [right arrow] 0 or [q.sub.2] [right arrow] [infinity], respectively. Therefore system (10a) and (10b) can be replaced by ultimate system of (6a) and (6b) with some accuracy [epsilon] for finite range of q if [q.sub.1] < [q.sub.h.sub.1] and [q.sub.2] value is large enough to consider this value as [q.sub.2] [right arrow] [infinity]. Similarly, system (11a) and (11b) can be replaced by ultimate system of (6a) and (6b) with some accuracy [epsilon] for finite range of q if [q.sub.2] [greater than or equal to] [q.sup.l.sub.2] and [q.sub.1] is small enough to consider it as [q.sub.1] [right arrow] 0 for (11a) and (11b). The high/low boundary values [q.sup.h.sub.1]/[q.sup.l.sub.2] for variables [q.sub.1]/[q.sub.2] are dominated by assigned value of accuracy. For instance, at [epsilon] = [10.sup.-2] the ultimate system of (6a) and (6b) is valid for [q.sub.1] [??] 2 x [10.sup.-3] [R.sub.c(G)] or [q.sub.2] > 1.3[R.sub.c(G)0], that is, [q.sub.1] [??] [q.sup.h.sub.1] = 2-4 MeV/c or [q.sub.2] [??] [q.sup.l.sub.2] = 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 the [q.sub.1] - and [q.sub.2]-dependencies of the Gaussian parameters [[lambda].sub.G] and [R.sub.G] derived for some assigned values of the corresponding BEC parameters for exponential function [[lambda].sub.C], [R.sub.c] and vice versa.

3.2. Dependence on q for Desired Cauchy/Gaussian Parameters. For Figures 3 and 4 [[OMEGA].sub.C] is considered as a priori known and set of BEC parameters [[[lambda].sub.G], [R.sup.G]} are studied for Gaussian parametrization (2). Figure 3 shows the [q.sub.1]- and [q.sub.2]-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 [[OMEGA].sub.G] at fixed values [[lambda].sub.c] = [pi]/2 and [R.sub.c] = [square root of [pi]]. As seen both Gaussian parameters show the similar behavior with changing the integration limits, namely, [[lambda].sub.G] and [R.sub.G] growth with decreasing of the [q.sub.1,2] at fixed another limit of integration. The curves [[lambda].sub.G]([q.sub.1]), [R.sup.G]([q.sub.1]) approach the asymptotic dashed lines calculated with help of system (10a) and (10b) with increasing of [q.sub.2]. The similar situation is observed in Figures 3(b) and 3(d) for curves [R.sup.G]([q.sub.2]), [[lambda].sub.G]([q.sub.2]) and asymptotic dashed lines calculated with help of system (11a) and (11b) with decreasing of [q.sub.1]. As seen the asymptotic lines are achieved at [q.sub.1] [??] 1 MeV/c (Figures 3(b) and 3(d)) and [q.sub.2] [??] 0.8 GeV/c (Figures 3(a) and 3(c)). Furthermore the ultimate values of the Gaussian BEC parameters [[lambda].sup.U.sub.G] and [R.sup.u.sub.G] are valid with good accuracy for [q.sub.1] < 2.0 MeV/c and [q.sub.2] > 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 ([Q.sub.C]) and Gaussian ([Q.sub.G]) functions the asymptotic [Q.sub.1]-dependence is achieved for both the 1D BEC radius (Figure 3(a)) and the strength of correlations (Figure 3(c)) at [q.sub.2] which is much smaller than that for case of two some other centrally symmetric Levy parameterizations (Figures 3(a) and 3(c)). This [q.sub.2] value for case of [[OMEGA].sub.C] and [[OMEGA].sub.G] is similar to those used in analyses of experimental CF [C.sub.2](q). In Figure 4 the dependencies of relative BEC parameters, namely, -[R.sub.c]/[R.sub.G] (a, b) and[ [lambda].sub.c]/[[lambda].sub.G] (c, d), on [q.sub.1] (a, c) and [q.sub.2] (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 [q.sub.1]-dependence (Figures 4(a) and 4(c)) and system (11a) and (11b) for [q.sub.2]-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 [q.sup.1]-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 [q.sub.1]-dependence of the -[R.sub.C]/[R.sub.G] (Figure 4(b)) and [[lambda].sub.c]/[[lambda].sub.G] (Figure 4(d)) is slower with increasing of the input values of the {[[lambda].sub.c], [R.sub.c]}. As expected the ultimate levels [R.sub.C]/[R.sub.G] = [square root of [pi]] (Figures 4(a) and 4(b)) and [[lambda].sub.c]/[[lambda].sub.G] = [pi]/2 (Figures 4(c) and 4(d)) shown by thin dotted lines are valid for the same ranges of [q.sub.1] and [q.sub.2] as estimated above for Figure 3.

Figures 5 and 6 show results for opposite direction of calculations; that is, [[OMEGA].sub.G] is supposed to be a priori known and BEC parameters {[lambda.sub.C], [R.sub.C]} for exponential parametrization of CF [C.sub.2](q) are derived. Figure 5 shows the and dependence of 1D BEC radius (Figures 5(a) and 5(b)) [q.sub.1] and [q.sub.2] strength of correlations (Figures 5(c) and 5(d)) for exponential function [[OMEGA].sub.c] at fixed values [[lambda].sub.G] = 2/[pi] and [R.sub.G] =1/[square root of [pi]]. The [for all]i =1,2: [q.sub.i]-dependencies show the opposite behavior for desired parameters of Cauchy source function [R.sub.C] (Figures 5(a) and 5(b)) and [lambda.sub.C] (Figures 5(c) and 5(d)) with respect to the corresponding dependencies presented in Figure 3 above for another direction of calculation |[lambda.sub.C], [R.sub.C]} [right arrow] {[[lambda].sub.G], [R.sub.G]}. These differences are seen in domain of relatively large [q.sub.1] [??] [10.sup.-2] GeV/c for [q.sub.1]-dependence and at relatively small [q.sub.2] [??] 0.7 GeV/c for [q.sub.2]-dependence of BEC parameters. Furthermore the -dependence for parameters from set for [[OMEGA].sub.C] (Figures 5(a) and 5(c)) approaches the constant at [q.sub.1] [right arrow] 0 faster noticeably than that for [R.sub.G] (Figure 3(a)) and [[lambda].sub.G] (Figure 3(c)). The opposite situation is observed for achievement of constants by [q.sub.2]-dependence at [q.sub.2] [right arrow] [infinity]. It should be noted that dependencies -[R.sub.C]([q.sub.1]) and [lambda.sub.C]([q.sup.1]) 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 -[R.sub.C]([q.sub.1]) and [lambda.sub.C]([q.sub.1]) will achieve the asymptotic curves at higher than that for Figures 3(a) and 3(c). The asymptotic value of [q.sub.1] [equivalent] [10.sup.-3] GeV/c is the same for dependence for both directions of calculations |[lambda.sub.C], [R.sub.C]} [q.sup.2] {[[lambda].sub.G], [R.sub.G]}. Figure 6 demonstrates the dependence of relative BEC parameters, namely, [R.sub.C]/[R.sub.G] (a, b) and [lambda.sub.C]/[[lambda].sub.G] (c, d), on [q.sub.1] (a, c) and [q.sub.2] (b, d) for various assigned values of parameters for Gaussian parametrization. The simpler system of (10a) and (10b) is used for calculation of [q.sub.1]-dependencies in Figures 6(a) and 6(c) and curves on [q.sub.2] (Figures 6(b) and 6(d)) are derived with help of system (11a) and (11b). In general [for all]i = 1,2 : [q.sub.i]-dependencies show similar behavior for corresponding relative 1D BEC parameters in both cases, Figure 4 and Figure 6, with some faster changing of [q.sub.i]-dependencies in the second case than that for the first one in domain of relatively large [q.sub.1] [??] [10.sup.-2] GeV/cfor [q.sub.1]- dependence and at relatively small [q.sub.2] [??] 0.7 GeV/c for [q.sub.2]-dependence of [R.sub.C]/[R.sub.G] and [[lambda].sub.C][[lambda].sub.G].

Simultaneous consideration of available 1D BEC data analyses for strong interaction processes and Figures 3 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 [C.sub.2](q). As seen from Figures 4 and 6 even the asymptotic values of relative 1D BEC parameters [[[lambda].sub.C]/[[lambda].sub.G], [R.sub.C]/[R.sub.G]} can differ up to several times from ultimate values calculated with help of the system of (6a) and (6b) in some domains of and variables. Therefore Figures 3-6 confirm the conclusion formulated above for case of two general view centrally symmetric Leevy 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 Figures 3-6 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 Leevy parametrization (2) at [alpha] = 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 Appendix 5 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 Leevy 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, [N.sub.ch], is widely studied for p + p and [bar.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, [bar.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.

4.1. Relations between Parameters for Cauchy/Gaussian Distribution and Levy One. The general system of (3a) and (3b) allow us to estimate the 1D BEC parameters for exponential/Gaussian function [[OMEGA].sub.c(G)] based on the a priori known parameter values for [[OMEGA].sub.L] [equivalent to] [OMEGA]([[alpha].sub.L], [[lambda].sub.L], [R.sup.L]) corresponding to general view of centrally symmetric Leevy distribution and vice versa.

4.1.1. Direction of Calculations [[OMEGA].sub.L] [right arrow] [Q.sub.C/G]. The sets |[lambda.sub.C], [R.sub.C]} and {[L.sub.G], [R.sup.G]} are estimated for experimentally known [[OMEGA].sub.L] [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 [[lambda].sub.G] coincides with experimental values within total errors only at [square root of [s.sub.NN] = 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 [[lambda].sub.G] for CMS at VNN = 7 TeV. For the last case the estimations from ((3a) and (3b)) and ((4a) and (4b)) coincide with each other within 2a. 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 [q.sup.1] ~ [10.sup.-2] GeV/c, ~ [q.sub.2] GeV/c, and [[alpha].sub.L] ~ 0.8 which is close to the region of Leevy index values with weaker changing of 1D BEC parameters (Figure 2).

4.1.2. Direction of Calculations [[OMEGA].sub.c/G] [right arrow] [[OMEGA].sub.L]. Here the set of 1D BEC parameters {[[lambda].sub.L], [R.sub.L]} is estimated at a priori given [[alpha].sub.L] with help of the system of (3a) and (3b) for experimentally known [[OMEGA].sub.c] [14,16], [[OMEGA].sub.G] [15,16] and finite q-ranges. The estimations for parameters of [[OMEGA].sub.L] are shown in Table 2 together with the available experimental results and the values of [[lambda].sub.L], [R.sub.L] 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 Levy 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 [[lambda].sub.L] at [square root of [s.sub.NN]] = 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 [[OMEGA].sub.c] - [[OMEGA].sub.L] and [[OMEGA].sub.G] [right arrow] [[OMEGA].sub.L] 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 Leevy parametrization [[OMEGA].sub.L] based on the a priori known aL and results for xponential/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 {[[lambda].sub.L], [R.sub.L]} with help of results for exponential function [[OMEGA].sub.c] as well as for Gaussian one [[OMEGA].sub.G].

Nevertheless the general system (3a) and (3b) allows the noticeable improvement of the results with respect to (4a) and (4b) for chaoticity [[lambda].sub.L] 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 [[alpha].sub.2]-dependence at values [[alpha].sub.1] [??] 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 the a priori known values for set [[[lambda].sub.C,] [R.sub.C]} 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, [bar.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 Calculations [[OMEGA].sub.C] [right arrow] [[OMEGA].sub.G]. Parameters for Gaussian function are calculated with help of system (7a) and (7b) and a priori known set [[[lambda].sub.C], [R.sub.C]}. The results are shown in Table 3 together with available published experimental results for the Gaussian set [[[lambda].sub.C], [R.sub.C]}. 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 [[lambda].sub.G] in ATLAS minimum bias events at [square root of [s.sub.NN]] = 7 TeV and CMS result at collision energy [square root of [s.sub.NN]] = 2.36 TeV. In the two last cases the agreement is observed within 2[sigma]. The similar situation is for symmetric nucleus-nucleus collisions; that is, the estimations within the present paper for set of Gaussian parameters [[[lambda].sub.G], [R.sub.G]} agree with the results of the WA98 experiment [23] within [2.sub.[sigma]]. But there is qualitative agreement only between results of calculations with help of (8a) and (8b) and experimental data for [bar.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 [C.sub.2](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 [[lambda].sub.G] and the 1D BEC radius. Only estimations for Gaussian 1D radius [R.sub.G] in O + Ag and for [[lambda].sub.G] in O + Au coincide with corresponding results of the WA80 experiment [27] within 2[sigma]. 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 [square root of [s.sub.NN]] = 2.76 TeV with low enough [q.sub.1] ~ 0.6 MeV/c and high enough [q.sub.2] [approximately equal to] 2.0 GeV/c. Thus detailed calculations for case [Q.sub.c] [right arrow] [[OMEGA].sub.G] 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 Calculations [[OMEGA].sub.G] [right arrow] [[OMEGA].sub.c]. Values of 1D BEC parameters for exponential function are estimated with help of system (7a) and (7b) and a priori known values for Gaussian BEC quantities [[[lambda].sub.G], [R.sub.G]]. The results are presented in Table 4 together with available published experimental results for the set of BEC parameters [[[lambda].sub.c], [R.sub.C]] 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 [C.sub.2](q) calculated with (7a) and (7b) and experimental results within (total) errors with exception of the 1D BEC radius [R.sub.C] in CMS at [square root of [s.sub.NN]] = 0.9 TeV. In the last case results from calculation and experiment coincide within 2a. The similar situation is observed for nucleus-nucleus collisions: estimations for parameters of exponential function Qc derived with (7a) and (7b) agree with corresponding experimental results mostly within 1[sigma], but the coincidence is achieved within 2a for [R.sub.C] in both the Pb + Pb collisions at [square root of [s.sub.NN]] = 17.3 GeV [23] and the O + Ag reactions at [square root of [s.sub.NN]] = 19.4 GeV [27]. The estimations of 1D BEC parameters [lambda.sub.C], [R.sub.C] obtained for [bar.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 the a 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 Levy distribution [[OMEGA].sub.L] 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 [C.sub.2](q) based on the available values of [lambda] and R for another centrally symmetric Levy distribution. As seen from Tables 1-4 the new estimations are obtained for 1D BEC parameters in [[OMEGA].sub.C], [[OMEGA].sub.G] 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 [lambda] 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 Leevy 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 Leevy 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 of q should be taken into account and corresponding systems of equations should be used for derivation of set [[lambda], R] based on the a priori known values of corresponding parameters for both cases of the two general view centrally symmetric Leevy 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 Levy indexes [for all]i = 1,2 : [[alpha].sub.i] [??] 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 Levy 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 the a priori known values of parameters of general Leevy parametrization and vice versa, for estimations of Gaussian parameters based on the a 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 Levy 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.

Appendix

Data for 1D BEC Parameters in Strong Interactions

In this Appendix experimental database is shown in Tables 5 and 6 for 1D BEC parameters for identical charged pions produced in p + p, [bar.p] + p, and [A.sup.1] + [A.sub.2]s interactions (in Table 5 total uncertainties are shown for exponential parametrization in CPLEAR [24] and for NA44 experiment [42]). Some of the numerical values are used in Section 4. The pion pairs with low average transverse momentum, <[k.sub.T]>, are considered for all types of strong interaction processes. The results for most central nucleus-nucleus collisions are used and as a consequence additional separation is made for p + p collisions on minimum bias and high multiplicity events if corresponding experimental 1D BEC values are available. In the last case the results for minimum bias events are shown on the first line, for high multiplicity events, and on the second line, for certain experiment at given energy.

http://dx.doi.org/10.1155/2017/5465398

Competing Interests

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

knowledgments

The author is grateful to Professor G. Alexander for useful discussions.

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[22] G. Aad, B. Abbott, J. Abdallah et al., "Two-particle Bose-Einstein correlations in pp collisions at [square root of s] = 0.9 and 7 TeV measured with the ATLAS detector," The European Physical Journal C, vol. 75, article 466, 2015.

[23] M. M. Aggarwal, A. Agnihotri, Z. Ahammed et al., "Central Pb + Pb collisions at 158 A GeV/c studied by [[pi].sup.-][[pi].sup.-] interferometry," The European Physical Journal C, vol. 16, no. 3, pp. 445-451, 2000.

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[27] R. Albrecht, T. C. Awes, P. Beckmann et al., "Bose-einstein correlations in the target fragmentation region in 200A Ge[V.sup.16]O+ nucleus collisions," Zeitschrift fur Physik C, vol. 53, no. 2, pp. 225-237, 1992.

[28] V. A. Okorokov, "Azimuthally integrated HBT parameters for charged pions in nucleus-nucleus interactions versus collision energy," Advances in High Energy Physics, vol. 2015, Article ID 790646, 13 pages, 2015.

[29] T. Akesson, M. G. Albrow, S. Almehed et al., "Pion interferometry in jet events at the CERN Intersecting Storage Rings," Zeitschrift fur Physik C, vol. 36, no. 4, pp. 517-526, 1987.

[30] I. Sykora, "Measurements of Bose-Einstein correlations with the ATLAS detector," in Proceedings of the 23rd International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS '15), p. 157, Dallas, Tex, USA, 2015.

[31] B. Abelev, J. Adamak, D. Adamova et al., "Freeze-out radii extracted from three-pion cumulants in pp, p-Pb and Pb-Pb collisions at the LHC," Physics Letters B, vol. 739, pp. 139-151, 2014.

[32] J. Uribe, E. P. Hartouni, D. A. Jensen et al., "Pion-pion correlations at low relative momentum produced in p - p collisions at 27.5 GeV/c," Physical Review D, vol. 49, no. 9, article 4373, 1994.

[33] J. L. Bailly, C. Caso, Y. Chiba et al., "Bose-Einstein correlations for pions produced in pp collisions at 360 GeV/c," Zeitschrift fur Physik C, vol. 43, no. 3, pp. 341-348, 1989.

[34] M. Aguilar-Benitez, W. W. M. Allison, A. A. Batalov et al., "Bose-Einstein correlations in pp collisions at 400 GeV/c," Zeitschriftfur Physik C, vol. 54, no. 1, pp. 21-32, 1992.

[35] A. Breakstone, R. Campanini, H. B. Crawley et al., "Multiplicity dependence of the average transverse momentum and of the particle source size in p - p interactions at [square root of s] = 62, 44 and 31 GeV Zeitschrift fur Physik C, vol. 33, no. 3, pp. 333-338, 1987.

[36] M. M. Aggarwal, Z. Ahammed, A. Alakhverdyants et al., "Pion femtoscopyin p[bar.p] collisions at [square root of s] = 200 GeV" Physical Review C, vol. 83, no. 6, Article ID 064905, 2011.

[37] T. Alexopoulos, C. Allen, E. W. Anderson et al., "Study of source size in p[bar.p] collisions at [square root of s] = 1.8 TeV using pion interferometry," Physical Review D, vol. 48, no. 5, pp. 1931-1942, 1931.

[38] L. Lovas, The Bose-Einstein correlations in CDF II experiment [Ph.D. thesis], Comenius University in Bratislava, Bratislava, Slovakia, 2008.

[39] G. Aad, B. Abbott, J. Abdallah et al., "Charged-particle distributions in [square root of s] = 13 TeV pp interactions measured with the ATLAS detector at the LHC," Physics Letters B, vol. 758, pp. 67-88, 2016.

[40] L. Ahle, Y. Akiba, K. Ashktorab et al., "System, centrality, and transverse mass dependence of two-pion correlation radii in heavy ion collisions at 11.6A and 14.6A GeV/c," Physical Review C, vol. 66, no. 5, Article ID 054906,15 pages, 2002.

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V. A. Okorokov

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

Correspondence should be addressed to V. A. Okorokov; vaokorokov@mephi.ru

Received 16 May 2016; Revised 25 October 2016; [lambda.sub.C]cepted 17 November 2016; Published 2 January 2017

Academic Editor: Frank Filthaut

Caption: FIGURE 1: Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for centrally symmetric Levy parametrization with [[alpha].sub.1] = 1.5 on low [q.sub.1] (a, c) and high [q.sub.2] (b, d) limits of integration in the system of (3a) and (3b) for fixed parameter values for second centrally symmetric Levy parametrization: [[alpha].sub.2] = 0.5, [[lambda].sub.2] = 0.5, and [R.sub.2] = 1.5 fm. The solid lines correspond to the indicated values of the [q.sub.2] for [q.sub.1]-dependence (a, c) and to shown values of the [q.sub.1] for [q.sub.1]-dependence (b, d). The dashed lines correspond to the calculations with [q.sub.2] [right arrow] [infinity] for [q.sub.1]-dependence (a, c) and with [q.sub.1] = 0 for [q.sub.1]-dependence (b, d). The thin dotted lines are the ultimate levels for [R.sub.1] (a, b) and [[lambda].sub.1] (c, d) calculated with (4a) and (4b) for given values of [[alpha].sub.1] and the set of parameters [[[alpha].sub.2], [[lambda].sub.2], [R.sub.2]] for second Levy parametrization.

Caption: FIGURE 2: Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for centrally symmetric Levy parametrization on [[alpha].sub.1] at fixed values of [[alpha].sub.2] (a, c) and on [[alpha].sub.2] at fixed values of [[alpha].sub.1] (b, d) for given limits of integration in the system of (3a) and (3b) [q.sub.1] = 0.02 GeV/c, [q.sub.2] = 2.0 GeV/c and for fixed values of the BEC parameters for second centrally symmetric Levy parametrization: [[lambda].sub.2] = 0.5 and [R.sub.2] = 1.5 fm. The solid lines correspond to the indicated values of the [[alpha].sub.2] for [[alpha].sub.1] -dependence (a, c) and to shown values of the [[alpha].sub.1] for [[alpha].sub.2]-dependence (b, d). The dotted lines are the ultimate cases for [R.sub.1] (a, b) and [[lambda].sub.1] (c, d) calculated with (4a) and (4b) for given values of [[alpha].sub.1] (a, c) or [[alpha].sub.2] (b, d) and the set of BEC parameters [[[lambda].sub.2], [R.sub.2]] for second Levy parametrization.

Caption: FIGURE 3: Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for Gaussian parametrization on low [q.sub.1] (a, c) and high [q.sub.2] (b, d) limits of integration in the system of (8a) and (8b) for fixed values of the parameters for exponential parametrization: [[lambda].sub.c] = [pi]/2 and [R.sub.c] = [square root of [pi]] fm. The solid lines correspond to the indicated values of the [q.sup.2] for [q.sub.1]-dependence (a, c) and to shown values of the for independence (b, d). The dashed lines correspond to the calculations based on system (10a) and (10b) for [q.sub.1]-dependence (a, c) and on system (11a) and (11b) for [q.sub.1]-dependence (b, d). The thin dotted lines are the ultimate levels [R.sub.G] = 1.0 fm (a, b) and [[lambda].sub.G] = 1.0 (c, d) calculated with help of (6a) and (6b) for given values of the set of Cauchy parameters {[A.sub.c], [R.sub.c]}.

Caption: FIGURE 4: Dependence of relative 1D BEC radius (a, b) and strength of correlations (c, d) on (a, c) and [q.sub.2] (b, d) for various fixed values of the parameters for exponential parametrization. The calculations are made for simpler range of integration [[z.sub.1,c(G)], [infinity]) with help of system (10a) and (10b) for [q.sub.1]-dependence (a, c) and for [0.0, [z.sub.2]c(g)] with system (11a) and (11b) for [q.sub.1]-dependence (b, d), respectively. The dashed lines correspond to [[lambda].sub.c] = 0.6re, [R.sub.c] = 1.2 [square root of [xi]] fm, solid lines [[lambda].sub.c] = 0.5re, [R.sub.c] = [square root of [xi]] fm, and dotted lines [[lambda].sub.c] = 0.4re, [R.sub.c] = 0.8 [square root of [xi]] fm. The thin dotted lines are the ultimate levels [R.sub.c]/[R.sub.G] = [square root of [pi]] (a, b) and [[lambda].sub.c]/[[lambda].sub.G] = [pi]/2 (c, d) corresponding to system (6a) and (6b).

Caption: FIGURE 5: Dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for Cauchy distribution on low [q.sub.1] (a, c) and high [q.sub.2] (b, d) limits of integration in the system of (8a) and (8b) for fixed values of the parameters for Gaussian parametrization: [[lambda].sub.G] = 2/[pi] and [R.sub.G] = 1/[square root of [pi]] fm. The solid lines correspond to the indicated values of the [q.sub.2] for [q.sub.1]-dependence (a, c) and to shown values of the for independence (b, d). The dashed lines correspond to the calculations based on system (10a) and (10b) for -dependence (a, c) and on system (11a) and (11b) for [q.sub.2]-dependence (b, d). The thin dotted lines are the ultimate levels [R.sub.C] = 1.0 fm (a, b) and [[lambda].sub.G] = 1.0 (c, d) calculated with help of (6a) and (6b) for given values of the set of Gaussian parameters {[[lambda].sub.G], [R.sub.G]}.

Caption: FIGURE 6: Dependence of relative 1D BEC radius (a, b) and strength of correlations (c, d) on [q.sub.1] (a, c) and [q.sub.2] (b, d) for various fixed values of the parameters for Gaussian parametrization. The calculations are made for simpler range of integration [[z.sub.1,c(G),] [infinity]) with help of system (10a) and (10b) for [q.sub.1]-dependence (a, c) and for [0.0, [z.sub.2c(G)]] with system (11a) and (11b) for [q.sub.2]-dependence (b, d), respectively. The dashed lines correspond to [[lambda].sub.G] = 1.2, [R.sub.G] = 1.2 fm, solid lines [[lambda].sub.G] = 1.0, [R.sub.G] = 1.0 fm, and dotted lines [[lambda].sub.G] = 0.8, [R.sub.G] = 0.8 fm. The thin dotted lines are the ultimate levels [R.sub.C]/[R.sub.G] = [square root of [pi]] (a, b) and [[lambda].sub.c]/[[lambda].sub.G] = [pi]/2 (c, d) corresponding to system (6a) and (6b).
TABLE 1: Parameter values for exponential and Gaussian
parameterizations [[OMEGA].sub.C(G)].

Collision   [square root    Experiment
            [s.sub.NN]],
                 GeV

P + P            900           CMS

                              ATLAS
                7000
                               CMS

Collision    Estimation based on (3a), (3b)

                        [lambda]

P + P              0.62 [+ or -] 0.08
                   0.35 [+ or -] 0.06
            0.73 [+ or -] 0.03 [+ or -] 0.54
            0.39 [+ or -] 0.01 [+ or -] 0.20
                   0.62 [+ or -] 0.06
                   0.35 [+ or -] 0.03

Collision    Estimation based on (3a), (3b)

                          R, fm

P + P              1.47 [+ or -] 0.24
                   0.81 [+ or -] 0.22
            2.02 [+ or -] 0.11 [+ or -] 1.79
            1.07 [+ or -] 0.04 [+ or -] 0.82
                   1.81 [+ or -] 0.23
                   0.96 [+ or -] 0.12

Collision           Experimental values

                          [lambda]

P + P       0.616 [+ or -] 0.011 [+ or -] 0.029
                     0.32 [+ or -] 0.01
            0.701 [+ or -] 0.006 [+ or -] 0.067
            0.302 [+ or -] 0.002 [+ or -] 0.019
            0.618 [+ or -] 0.009 [+ or -] 0.042
                             --

Collision           Experimental values            Ref.

                           R, fm

P + P         1.56 [+ or -] 0.02 [+ or -] 0.15     [14]
                     0.98 [+ or -] 0.03            [15]
            2.021 [+ or -] 0.012 [+ or -] 0.281    [16]
            1.046 [+ or -] 0.005 [+ or -] 0.114    [16]
              1.89 [+ or -] 0.02 [+ or -] 0.21     [14]
                             --                     --

TABLE 2: Parameter values for general Levy parametrization
[[OMEGA].sub.L at given [[alpha].sub.L].

Collision   [square root    Experiment
            [s.sub.NN]],
                 GeV

P + P            900           CMS

                7000          ATLAS

                               CMS

Collision      Estimation based on (3a), (3b)

                          [lambda]

P + P         0.85 [+ or -] 0.04 [+ or -] 0.05
                     0.89 [+ or -] 0.10
            0.973 [+ or -] 0.012 [+ or -] 0.332
             0.774 [+ or -] 0.010 [+ or -] 0.250
              0.89 [+ or -] 0.04 [+ or -] 0.07

Collision     Estimation based on (3a), (3b)

                          R, fm

P + P        2.33 [+ or -] 0.18 [+ or -] 0.23
                     3.2 [+ or -] 0.5
            2.97 [+ or -] 0.06 [+ or -] 1.27
             2.86 [+ or -] 0.06 [+ or -] 1.21
             2.96 [+ or -] 0.17 [+ or -] 0.34

Collision          Experimental values

                        [lambda]

P + P              0.85 [+ or -] 0.06
                   0.93 [+ or -] 0.11
            1.02 [+ or -] 0.03 [+ or -] 0.41

                   0.90 [+ or -] 0.05

Collision          Experimental values          Ref.

                          R, fm

P + P              2.20 [+ or -] 0.17           [14]
                    2.5 [+ or -] 0.4            [15]
            2.96 [+ or -] 0.09 [+ or -] 1.31    [16]

                   2.83 [+ or -] 0.18           [14]

TABLE 3: Parameter values for Gaussian function
[[OMEGA].sub.G] in (2).

Collision     [square root of   Experiment
               [s.sub.NN]],
                    GeV

P + P              63.0            AFS
                                  ALICE
                    900
                                  ATLAS

                   2360            CMS
                   2760
                                  ALICE
                   7000           ATLAS

                                   CMS

[bar.P] + P        1.89           CPLEAR

p + Pb             5020            CMS

Pb + Pb            17.3            WA98

O + C
O + Cu             19.4            WA80
O + Ag
O + Au

Collision       Estimation based on (7a), (7b)

                        [[lambda].sub.G]

P + P                  0.47 [+ or -] 0.04
                      0.305 [+ or -] 0.024
                      0.357 [+ or -] 0.025
                0.41 [+ or -] 0.01 [+ or -] 0.03
              0.351 [+ or -] 0.006 [+ or -] 0.013
              0.348 [+ or -] 0.030 [+ or -] 0.013
              0.366 [+ or -] 0.005 [+ or -] 0.025
              0.719 [+ or -] 0.002 [+ or -] 0.047
              0.381 [+ or -] 0.003 [+ or -] 0.022
              0.266 [+ or -] 0.009 [+ or -] 0.015
              0.344 [+ or -] 0.005 [+ or -] 0.018

[bar.P] + P           2.332 [+ or -] 0.025

p + Pb        0.358 [+ or -] 0.007 [+ or -] 0.021

Pb + Pb               0.327 [+ or -] 0.008

O + C                  0.44 [+ or -] 0.05
O + Cu                 0.24 [+ or -] 0.07
O + Ag                 0.28 [+ or -] 0.10
O + Au                0.110 [+ or -] 0.015

Collision           Estimation based on (7a), (7b)

                             [R.sub.G], fm

P + P                     0.73 [+ or -] 0.07
               1.00 [+ or -] [0.09.sup.+0.06.sub.-0.18]
               0.89 [+ or -] [0.06.sup.+0.18.sub.-0.18]
                   1.00 [+ or -] 0.03 [+ or -] 0.08
                   0.83 [+ or -] 0.01 [+ or -] 0.08
                   1.04 [+ or -] 0.08 [+ or -] 0.10
                  0.915 [+ or -] 0.007 [+ or -] 0.116
              1.148 [+ or -] [0.007.sup.+0.04.sub.-0.02]
                  1.092 [+ or -] 0.005 [+ or -] 0.074
                   1.25 [+ or -] 0.03 [+ or -] 0.09
                   1.00 [+ or -] 0.01 [+ or -] 0.10

[bar.P] + P              0.972 [+ or -] 0.014

p + Pb             1.70 [+ or -] 0.02 [+ or -] 0.12

Pb + Pb                   6.51 [+ or -] 0.10

O + C                      2.8 [+ or -] 0.3
O + Cu                    2.53 [+ or -] 0.11
O + Ag                    2.71 [+ or -] 0.11
O + Au                    1.63 [+ or -] 0.05

Collision             Experimental values

                        [[lambda].sub.G]

P + P                  0.40 [+ or -] 0.03
                       0.35 [+ or -] 0.03
                      0.310 [+ or -] 0.026
                0.34 [+ or -] 0.01 [+ or -] 0.03
                       0.32 [+ or -] 0.01
                       0.32 [+ or -] 0.01
                               --
              0.645 [+ or -] 0.003 [+ or -] 0.047
              0.327 [+ or -] 0.002 [+ or -] 0.020
              0.251 [+ or -] 0.010 [+ or -] 0.018
                               --

[bar.P] + P            1.96 [+ or -] 0.03

p + Pb                         --

Pb + Pb               0.307 [+ or -] 0.008

O + C                  0.40 [+ or -] 0.03
O + Cu                 0.17 [+ or -] 0.03
O + Ag                 0.17 [+ or -] 0.04
O + Au                0.085 [+ or -] 0.007

Collision                Experimental values                Ref.

                             [R.sub.G], fm

P + P                     0.82 [+ or -] 0.05                [29]
               1.00 [+ or -] [0.06.sup.+0.10.sub.-0.20]     [25]
               1.18 [+ or -] [0.09.sup.+0.07.sub.-0.17]
                   1.00 [+ or -] 0.03 [+ or -] 0.08         [30]
                          0.98 [+ or -] 0.03                [15]
                          0.98 [+ or -] 0.03
                                  --                         --
              1.430 [+ or -] [0.005.sup.-0.16.sub.-0.30]    [31]
                  1.130 [+ or -] 0.005 [+ or -] 0.086       [30]
                   1.38 [+ or -] 0.04 [+ or -] 0.12
                                  --                         --

[bar.P] + P               1.04 [+ or -] 0.01                [24]

p + Pb                            --                         --

Pb + Pb                   6.83 [+ or -] 0.10                [23]

O + C                     2.90 [+ or -] 0.21
O + Cu                    2.35 [+ or -] 0.11                [27]
O + Ag                    2.44 [+ or -] 0.11
O + Au                    1.68 [+ or -] 0.06

TABLE 4: Parameter values for exponential function
[[OMEGA].sub.C] in (2).

Collision     [square root of   Experiment
               [s.sub.NN]],
                    GeV

P + P              7.21            E766
                   26.0            NA23
                   63.0            AFS
                    200            STAR
                                  ALICE
                    900
                                  ATLAS
                                   CMS
                   2360
                                  ALICE
                   7000           ATLAS

[bar.P] + P        1.89           CPLEAR

Au + Au            4.86            E802
Pb + Pb            17.3            NA44
                                   WA98
Au + Au             130            STAR

Si + Al            5.41            E802
Si + Au
S + Pb             17.3            NA44
O + C
O + Cu             19.4            WA80
O + Ag
O + Au

Collision       Estimation based on (7a), (7b)

                        [[lambda].sub.C]

P + P                  1.63 [+ or -] 0.10
                        1.4 [+ or -] 0.7
                       0.67 [+ or -] 0.05
                      0.588 [+ or -] 0.010
                       0.63 [+ or -] 0.06
                       0.57 [+ or -] 0.06
                0.62 [+ or -] 0.03 [+ or -] 0.08
                       0.57 [+ or -] 0.02
                       0.60 [+ or -] 0.03
              1.104 [+ or -] 0.006 [+ or -] 0.112
              0.627 [+ or -] 0.005 [+ or -] 0.058
              0.521 [+ or -] 0.027 [+ or -] 0.055

[bar.P] + P            4.21 [+ or -] 0.09

Au + Au                0.86 [+ or -] 0.08
Pb + Pb                1.06 [+ or -] 0.10
                       0.69 [+ or -] 0.02
Au + Au         0.99 [+ or -] 0.03 [+ or -] 0.08

Si + Al                1.23 [+ or -] 0.08
Si + Au                0.95 [+ or -] 0.05
S + Pb                 0.81 [+ or -] 0.05
O + C                  0.85 [+ or -] 0.08
O + Cu                 0.34 [+ or -] 0.06
O + Ag                 0.34 [+ or -] 0.09
O + Au                0.156 [+ or -] 0.014

Collision           Estimation based on (7a), (7b)

                             [R.sub.C], fm

P + P                      2.29 [+ or -] 0.08
                            2.6 [+ or -] 0.7
                           1.50 [+ or -] 0.10
                    2.43 [+ or -] 0.04 [+ or -] 0.26
                1.89 [+ or -] [0.12.sup.+0.20.sub.0.40]
                2.26 [+ or -] [0.19.sup.+0.14.sub.0.34]
                    1.84 [+ or -] 0.07 [+ or -] 0.20
                           1.85 [+ or -] 0.06
                           1.86 [+ or -] 0.07
              2.627 [+ or -] [0.010.sup.+0.320.sub.0.626]
                  2.163 [+ or -] 0.012 [+ or -] 0.209
                    2.76 [+ or -] 0.10 [+ or -] 0.29

[bar.P] + P               2.079 [+ or -] 0.028

Au + Au                    12.4 [+ or -] 0.6
Pb + Pb                    15.1 [+ or -] 0.9
                           14.1 [+ or -] 0.3
Au + Au              13.0 [+ or -] 0.3 [+ or -] 0.9

Si + Al                     8.4 [+ or -] 0.3
Si + Au                     9.4 [+ or -] 0.3
S + Pb                      7.8 [+ or -] 0.6
O + C                       5.9 [+ or -] 0.5
O + Cu                      4.7 [+ or -] 0.2
O + Ag                      4.9 [+ or -] 0.2
O + Au                     3.19 [+ or -] 0.13

Collision             Experimental values

                        [[lambda].sub.C]

P + P                          --
                               --
                       0.77 [+ or -] 0.07
                               --
                       0.63 [+ or -] 0.05
                       0.55 [+ or -] 0.05
                0.74 [+ or -] 0.03 [+ or -] 0.09
              0.616 [+ or -] 0.011 [+ or -] 0.029
                0.66 [+ or -] 0.07 [+ or -] 0.05
              1.180 [+ or -] 0.005 [+ or -] 0.084
              0.718 [+ or -] 0.006 [+ or -] 0.062
              0.531 [+ or -] 0.024 [+ or -] 0.046

[bar.P] + P            4.79 [+ or -] 0.10

Au + Au                        --
Pb + Pb                        --
                      0.718 [+ or -] 0.023
Au + Au                        --

Si + Al                        --
Si + Au                        --
S + Pb                         --
O + C                  0.92 [+ or -] 0.13
O + Cu                 0.49 [+ or -] 0.14
O + Ag                 0.59 [+ or -] 0.21
O + Au                 0.20 [+ or -] 0.03

Collision                 Experimental values                 Ref.

                             [R.sub.C], fm

P + P                              --                          --
                                   --                          --
                           1.32 [+ or -] 0.13                 [29]
                                   --                          --
                1.67 [+ or -] [0.12.sup.+0.16.sub.0.35]       [25]
                1.90 [+ or -] [0.18.sup.+0.11.sub.0.36]
                    1.83 [+ or -] 0.07 [+ or -] 0.20          [30]
                    1.56 [+ or -] 0.02 [+ or -] 0.15         [14,15]
                    1.99 [+ or -] 0.18 [+ or -] 0.24          [15]
              2.038 [+ or -] [0.014.sup.+0.083.sub.0.046]     [31]
                  2.067 [+ or -] 0.012 [+ or -] 0.182         [30]
                    2.46 [+ or -] 0.08 [+ or -] 0.22

[bar.P] + P                1.89 [+ or -] 0.04                 [24]

Au + Au                            --                          --
Pb + Pb                            --                          --
                          13.34 [+ or -] 0.26                 [23]
Au + Au                            --                          --

Si + Al                            --                          --
Si + Au                            --                          --
S + Pb                             --                          --
O + C                       5.7 [+ or -] 0.7
O + Cu                     5.05 [+ or -] 0.25                 [27]
O + Ag                     5.46 [+ or -] 0.24
O + Au                     3.07 [+ or -] 0.12

TABLE 5: Experimental results for
special cases of parametrization (2).

Collision   [square root of   Experiment
             [s.sub.NN]],
                  GeV

P + P            7.21            E766
                 26.0            NA23
                 27.4            NA27
                 31.0
                 44.0           ABCDHW
                 62.0
                 63.0            AFS
                  200            STAR
                                ALICE

                  900
                                ATLAS

                 2360            CMS
                 2760
                                ALICE
                 7000           ATLAS

                                 CMS

P + P            1.89           CPLEAR
                 1800            E735
                 1960            CDF

p + Pb           5020           ALICE
                                ATLAS
                                 CMS

Au + Au          4.86            E802
Au + Pb                          NA49
Pb + Pb          17.3            NA44
                                 WA98
Au + Au           130           PHENIX
                                 STAR
Pb + Pb          2760           ALICE

Si + Al          5.41            E802
Si + Au
S + Pb           17.3            NA44
O + C
O + Cu                           WA80
O + Ag           19.4
O + Au
                                 NA35

Collision       Estimation based on [[OMEGA].sub.C]

                          [[lambda].sub.C]

P + P                            --
                                 --
                                 --
                                 --
                                 --
                                 --
                         0.77 [+ or -] 0.07
                                 --
                         0.63 [+ or -] 0.05
                         0.55 [+ or -] 0.05
                  0.74 [+ or -] 0.03 [+ or -] 0.09
                0.616 [+ or -] 0.011 [+ or -] 0.029
                  0.66 [+ or -] 0.07 [+ or -] 0.05
                0.808 [+ or -] 0.017 [+ or -] 0.062
                1.180 [+ or -] 0.005 [+ or -] 0.084
                0.718 [+ or -] 0.006 [+ or -] 0.062
                0.531 [+ or -] 0.024 [+ or -] 0.046
                0.618 [+ or -] 0.009 [+ or -] 0.042

P + P                     4.79 [+ or -] 10
                                 --
                         0.89 [+ or -] 0.03

p + Pb      1.230 [+ or -] [0.016.sup.+0.088.sub.-0141]
                                 --
                  0.81 [+ or -] 0.02 [+ or -] 0.07

Au + Au                          --
Au + Pb                          --
Pb + Pb                          --
                        0.718 [+ or -] 0.023
Au + Au                          --
                                 --
Pb + Pb         1.830 [+ or -] 0.003 [+ or -] 0.156

Si + Al                          --
Si + Au                          --
S + Pb                           --
O + C                    0.92 [+ or -] 0.13
O + Cu                   0.49 [+ or -] 0.14
O + Ag                   0.59 [+ or -] 0.21
O + Au                   0.20 [+ or -] 0.03
                                 --

Collision        Estimation based on [[OMEGA].sub.C]

                            [R.sub.C], fm

P + P                             --
                                  --
                                  --
                                  --
                                  --
                                  --
                          1.32 [+ or -] 0.13
                                  --
               1.87 [+ or -] [0.12.sup.+0.16.sub.-0.35]
               1.90 [+ or -] [0.18.sup.+0.11.sub.-0.36]
                   1.83 [+ or -] 0.07 [+ or -] 0.20
                   1.56 [+ or -] 0.02 [+ or -] 0.15
                   1.99 [+ or -] 0.18 [+ or -] 0.24
                   2.35 [+ or -] 0.07 [+ or -] 0.31
            2.038 [+ or -] [0.014+.sup.+0.083.sub.-0.046]
                 2.067 [+ or -] 0.012 [+ or -] 0.182
                   2.46 [+ or -] 0.08 [+ or -] 0.22
                   1.89 [+ or -] 0.02 [+ or -] 0.21

P + P                     1.89 [+ or -] 0.04
                                  --
                          1.67 [+ or -] 0.05

p + Pb         4.82 [+ or -] [0.05.sup.+0.25.sub.-0.72]
               5.32 [+ or -] [0.06.sup.+0.50.sub.-0.14]
                   3.55 [+ or -] 0.05 [+ or -] 0.30

Au + Au                           --
Au + Pb                           --
Pb + Pb                           --
                         13.34 [+ or -] 0.26
Au + Au                           --
                                  --
Pb + Pb           19.85 [+ or -] 0.02 [+ or -] 0.28

Si + Al                           --
Si + Au                           --
S + Pb                            --
O + C                      5.7 [+ or -] 0.7
O + Cu                    5.05 [+ or -] 0.25
O + Ag                    5.46 [+ or -] 0.24
O + Au                    3.07 [+ or -] 0.12
                                  --

Collision     Gaussian function [[OMEGA.sub.G]

                      [[lambda].sub.G]

P + P               0.466 [+ or -] 0.015
                     0.32 [+ or -] 0.08
                             -
                     0.35 [+ or -] 0.04
                     0.42 [+ or -] 0.04
                     0.42 [+ or -] 0.08
                     0.40 [+ or -] 0.03
                    0.345 [+ or -] 0.005
                     0.35 [+ or -] 0.03
                    0.310 [+ or -] 0.026
              0.34 [+ or -] 0.01 [+ or -] 0.03
                     0.32 [+ or -] 0.01
                     0.32 [+ or -] 0.01
                             --
            0.645 [+ or -] 0.003 [+ or -] 0.047
            0.327 [+ or -] 0.002 [+ or -] 0.020
            0.251 [+ or -] 0.010 [+ or -] 0.018
                             --

P + P                1.96 [+ or -] 0.03
                     0.24 [+ or -] 0.02
                     0.50 [+ or -] 0.04

p + Pb      0.603 [+ or -] 0.006 [+ or -] 0.056
                             --
                             --

Au + Au              0.44 [+ or -] 0.03
Au + Pb             0.560 [+ or -] 0.023
Pb + Pb              0.52 [+ or -] 0.04
                    0.307 [+ or -] 0.008
Au + Au                      --
            0.450 [+ or -] 0.009 [+ or -] 0.027
Pb + Pb     0.689 [+ or -] 0.001 [+ or -] 0.096

Si + Al              0.68 [+ or -] 0.04
Si + Au             0.511 [+ or -] 0.026
S + Pb               0.42 [+ or -] 0.02
O + C                0.40 [+ or -] 0.03
O + Cu               0.17 [+ or -] 0.03
O + Ag               0.17 [+ or -] 0.04
O + Au              0.085 [+ or -] 0.007
                     0.29 [+ or -] 0.03

Collision          Gaussian function [[OMEGA.sub.G]           Ref.

                            [R.sub.G], fm

P + P                     0.95 [+ or -] 0.03                  [32]
                          1.02 [+ or -] 0.20                  [33]
                          1.20 [+ or -] 0.03                  [34]
                          1.01 [+ or -] 0.08
                          1.13 [+ or -] 0.07                  [35]
                          1.69 [+ or -] 0.25
                          0.82 [+ or -] 0.05                  [29]
                   1.32 [+ or -] 0.02 [+ or -] 0.13           [36]
               1.00 [+ or -] [0.06.sup.+0.10.sub.-0.20]       [25]
            1.184 [+ or -] [0.092.sup.+0.067.sub.-0.0168]
                   1.00 [+ or -] 0.03 [+ or -] 0.08           [30]
                          0.98 [+ or -] 0.03                 [14,15]
                          0.98 [+ or -] 0.03                  [15]
                                  --                          [17]
             1.430 [+ or -] [0.005.sup.+0.158.sub.-0.300]     [31]
                 1.130 [+ or -] 0.005 [+ or -] 0.086          [30]
                   1.38 [+ or -] 0.04 [+ or -] 0.12
                                  --                          [14]

P + P                     1.04 [+ or -] 0.01                  [24]
                   1.46 [+ or -] 0.10 [+ or -] 0.23           [37]
                          1.79 [+ or -] 0.08                  [38]

p + Pb       2.780 [+ or -] [0.018.sup.+0.158.sub.-0.300]     [31]
                                  --                          [39]
                                  --                          [26]

Au + Au                   6.32 [+ or -] 0.29                  [40]
Au + Pb                           --                          [41]
Pb + Pb                    7.6 [+ or -] 0.4                   [42]
                          6.83 [+ or -] 0.10                  [23]
Au + Au                    6.0 [+ or -] 0.3                   [43]
                   6.30 [+ or -] 0.12 [+ or -] 0.38           [18]
Pb + Pb            9.70 [+ or -] 0.06 [+ or -] 1.17           [31]

Si + Al                   4.42 [+ or -] 0.16                  [40]
Si + Au                   4.91 [+ or -] 0.15
S + Pb                    4.00 [+ or -] 0.27                  [42]
O + C                     2.90 [+ or -] 0.21
O + Cu                    2.35 [+ or -] 0.11                  [27]
O + Ag                    2.44 [+ or -] 0.11
O + Au                    1.68 [+ or -] 0.06
                          4.00 [+ or -] 0.20                  [44]

TABLE 6: Experimental results for general
centrally symmetric Levy parametrization (2).

Collision   [square root   Experiment
            [s.sub.NN]],
                GEV

P + P           900           CMS

                7000         ATLAS
                              CMS

Collision   General view function [[OMEGA].sub.L]

                       [[lambda].sub.L]

P + P                 0.93 [+ or -] 0.11
                      0.85 [+ or -] 0.06
               1.02 [+ or -] 0.03 [+ or -] 0.41
                      0.90 [+ or -] 0.05

Collision   General view function [[OMEGA].sub.L]

                         [R.sub.L] fm

P + P                  2.5 [+ or -] 0.4
                      2.20 [+ or -] 0.17
               2.96 [+ or -] 0.09 [+ or -] 1.31
                      2.83 [+ or -] 0.18

Collision   General view function [[OMEGA].sub.L]    Ref.

                       [[alpha].sub.L]

P + P                 0.76 [+ or -] 0.06             [15]
                      0.81 [+ or -] 0.03             [14]
               0.81 [+ or -] 0.01 [+ or -] 0.18      [16]
                     0.792 [+ or -] 0.024            [14]
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Title Annotation:Research Article
Author:Okorokov, V.A.
Publication:Advances in High Energy Physics
Date:Jan 1, 2017
Words:14729
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