Research Article | Open Access

V. A. Okorokov, "Lévy Distributions for One-Dimensional Analysis of the Bose–Einstein Correlations", *Advances in High Energy Physics*, vol. 2017, Article ID 5465398, 18 pages, 2017. https://doi.org/10.1155/2017/5465398

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

**Academic Editor:**Frank Filthaut

#### 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.

#### 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 and . Experimentally, the one-dimensional BEC effect is observed as an enhancement at low values of the Lorentz invariant quantity in the two-particle correlation function (CF): Here is the two-particle density function and 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 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 CFHere is the strength of correlations called also chaoticity, is the 1D BEC radius, and 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 , ; that is, Bose–Einstein CF measures the absolute value squared Fourier transformed source density in the coordinate space , , 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 . For example, most of the earliest experiments with particle beams used the specific case of (2) at ; the Gaussian parametrization corresponded to the normal (Gaussian) distribution function , where the Gaussian scale parameter is , the standard deviation; then another specific case of (2) at is used widely, especially, for particle (not nuclear) collisions. Equation (2) at is called exponential parametrization for Bose–Einstein CF and it corresponds to the Cauchy (Lorentzian) distribution function with scale parameter [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 () collisions but for centrally symmetric Lévy distribution with , the corresponding source density in coordinate space can be written analytically for 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 . 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 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 and 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 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 Section 2, mathematical formalism is described for case of two general view centrally symmetric Lévy distributions. Dependencies of desired 1D BEC observables on and are studied for* a 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 . Database of experimental results for set of 1D BEC parameters 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, “” is for the general view of centrally symmetric Lévy distribution, “” is for the Cauchy source distribution function, and “” is for Gaussian one. Otherwise the notations are often used in papers for second case due to relation between Cauchy distribution for and exponential parametrization for Bose–Einstein CF 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 and the term “exponential function/parametrization” is used for the related parametrization of correlation function ; for the case of arbitrary , , the term “centrally symmetric Lévy” is suitable for both the source function in coordinate space and the parametrization of correlation function ; the similar situation is for : the term “Gaussian” is applicable for both the source function in coordinate space and the corresponding parametrization of correlation function ). 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 be described by two parameterizations (2) with and . Then relations between parameters of and can be deduced on the basis that both parameterizations describe one experimental CF , that is one sample of experimental points (in general the approximations of are characterized by different qualities for various parameterizations (2) with and . 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 and 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 and of the particle source can be derived from the following system of equations: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 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 are known* a priori* for both parameterizations , . In equations (3a) and (3b) the integrals are taken over full fit ranges , , for corresponding parameterizations with of experimental CF. It should be noted that in general case (i) the ranges of integration can be different for parameterizations with and ; (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, and consequently for all types of integrals and in system (3a) and (3b): . But usually the fit ranges are identical for both , , 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 and of the particle source in this case. Without loss of generality and are considered as known and values of BEC parameters are supposed as desired below. Then for specific case of semi-infinite ranges for integration system (3a) and (3b) can be solved analytically and one can derive the following ultimate relations between two sets and of BEC parameters for corresponding centrally symmetric Lévy parameterizations with , and vice versa. Here is the gamma function.

In the point of view of data analysis the absence of the general analytic relations between and 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 with its errors for centrally symmetric Lévy parametrization with as well as for with for parametrization with . The two sets of values for unknown BEC parameters can be obtained with the help of suitable system of equations: the input values and produce the output set and . Then the error estimations for set of BEC parameters for parametrization with can be calculated as follows: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 .

##### 2.2. Dependencies on and Variables

Figure 1 shows dependence of 1D BEC radius (a, b) and strength of correlations (c, d) for centrally symmetric Lévy parametrization with known on low (a, c) and high (b, d) limits of integration in system (3a) and (3b) for set for . The solid lines correspond to the indicated values of in GeV/c for -dependence (a, c) and to shown values of the in GeV/c for -dependence (b, d). Values of BEC parameters and depend strongly on the fixed second limit of integration () for both the - (Figures 1(a) and 1(c)) and the -dependence (Figures 1(b) and 1(d)). The dashed lines correspond to the results from system (3a) and (3b) with for -dependence (Figures 1(a) and 1(c)) and with for -dependence (Figures 1(b) and 1(d)). As seen the curves for general case of (3a) and (3b) coincide with dashed lines at GeV/c for -dependence (Figures 1(a) and 1(c)) and at GeV/c for -dependence (Figures 1(b) and 1(d)). These values for and especially for 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 -ranges for two parameterizations with and . The thin dotted lines demonstrate the ultimate levels for (Figures 1(a) and 1(b)) and (Figures 1(c) and 1(d)) calculated with (4a) and (4b) for given values of and the set of parameters for second Lévy parametrization . One can use the simple relations (4a) and (4b) for calculation at GeV/c and GeV/c (Figures 1(a) and 1(b)) but as expected the values of BEC parameters are far from the ultimate levels at any for -dependence in the range GeV/c is considered in Figures 1(b) and 1(d).

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In Figure 2 dependence of 1D BEC radius (a, b) and strength of correlations (c, d) is demonstrated for centrally symmetric Lévy parametrization with on at fixed values of (a, c) and on at fixed values of (b, d) for given limits of integration in the system of (3a) and (3b) GeV/c, GeV/c and for certain values of the BEC parameters for second centrally symmetric Lévy parametrization with : and fm. The solid lines correspond to the indicated values of the for -dependence (a, c) and to shown values of the for -dependence (b, d). The values for limits of integration and in system (3a) and (3b) are similar to those used in modern experiments. As seen dependencies of both BEC parameters on , , at fixed another Lévy index , , change very fast at small value in narrow range . Such behavior is observed for both the results from system (3a) and (3b) and the estimations for (Figures 2(a) and 2(b)) and (Figures 2(c) and 2(d)) calculated with (4a) and (4b) for semi-infinite ranges for integration and shown by dotted lines. The dependence shown in Figure 2(a) closes to the analytic one calculated with help of (4b) and presented by dotted line for any under study with exception of the small value . For last case the agreement is obtained in very narrow range between results from system (3a) and (3b) and (4b). This feature maps clearly in corresponding dependencies shown in Figure 2(b) for . For large solid and dotted lines are close to each other in the range but agreement is poor significantly between results from system (3a) and (3b) and (4b) for especially in domain (Figure 2(b)) taking into account the sharp behavior for corresponding dependence . In general the behavior of dependencies of on (Figure 2(c)) and (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 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 only.

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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 corresponding to the centrally symmetric Lévy source distributions.

#### 3. Relations between BEC Parameters in Specific Cases

As seen in Figure 2 both dependencies of the 1D BEC radius (a, b) and the strength of correlations (c, d) on Lévy indexes , , show the weaker changing in the domain in comparison with the range of small values of Lévy indexes. The region includes in particular the specific cases of Cauchy and Gaussian distributions for which corresponding parameterizations of Bose–Einstein CF with and are used mostly for experimental studies. Therefore these certain views of are studied in detail below. Let for Gaussian parametrization (2) and for 1D approximation of experimental CF by exponential function.

##### 3.1. Mathematical Formalism

Relations (4a) and (4b) are valid at any values of indexes of stability in two centrally symmetric Lévy parameterizations with , . If without loss of generality are considered as* a priori* known and values of Gaussian BEC parameters are supposed as desired then as expected (4a) and (4b) result in the ultimate relations: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 and :Here is the error integral and 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 and for experimental CF 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 is equal 2 for experimental analyses. Thus the general statement with regard of identity of integration ranges for and is quite confirmed for case of Cauchy and Gaussian distributions and in system (7a) and (7b).

Further simplification for the system of (7a) and (7b) depends on features of certain experiment; direction of calculations , that is, what kind of a set of BEC parameters of the two, and , 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 parameter and (ii) the accuracy for 1D BEC parameters in modern experiments is not better than *～* so far. Thus one can assume the conservative accuracy level . At present the most complex case with 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 values larger than the region of the influence of meson resonances excluded from the experimental fits; that is, all terms for can be omitted at given and direction of calculation from* a priori* known Cauchy parameters to desired Gaussian parameters . But the statement is wrong for opposite direction of calculation from* a priori* known Gaussian parameters to desired Cauchy parameters at . Therefore the sum can be omitted in system (7a) and (7b) and equations can be rewritten as follows: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 collisions at , 2.36, and 7 TeV [15, 22] in the case of inverse problem. Accounting the relation and properties of the functions allows us to simplify (8a) and (8b) to systemwhere . The last system of equations is valid for remain set of experimental results with exception of the WA98 data [23] at given . 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 at . The simplest view of the system of (9a) and (9b)corresponds to the range of integration and can be used for experimental results from ALICE [25], CMS [14, 15, 17, 26] with exception of the collision energy TeV [15] in the case of proton-proton collisions, and WA80 [27] for asymmetric nucleus-nucleus collisions , . On the other hand, the using of the range of integration allows the derivation of the following system from (9a) and (9b):As expected one can get the ultimate relations (6a) and (6b) from the any systems of ((10a) and (10b)) or ((11a) and (11b)) at or , respectively. Therefore system (10a) and (10b) can be replaced by ultimate system of (6a) and (6b) with some accuracy for finite range of if and value is large enough to consider this value as . Similarly, system (11a) and (11b) can be replaced by ultimate system of (6a) and (6b) with some accuracy for finite range of if and is small enough to consider it as for (11a) and (11b). The high/low boundary values for variables are dominated by assigned value of accuracy. For instance, at the ultimate system of (6a) and (6b) is valid for or , that is, MeV/c or GeV/c for proton-proton collisions. The derived estimations are close to the values of variable used in present experimental analyses of BEC correlations.

These qualitative estimations are confirmed by quantitative analysis below for the - and -dependencies of the Gaussian parameters and derived for some assigned values of the corresponding BEC parameters for exponential function , and vice versa.

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

For Figures 3 and 4 is considered as* a priori* known and set of BEC parameters are studied for Gaussian parametrization (2). Figure 3 shows the - and -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 at fixed values and . As seen both Gaussian parameters show the similar behavior with changing the integration limits, namely, and growth with decreasing of the at fixed another limit of integration. The curves , approach the asymptotic dashed lines calculated with help of system (10a) and (10b) with increasing of . The similar situation is observed in Figures 3(b) and 3(d) for curves , and asymptotic dashed lines calculated with help of system (11a) and (11b) with decreasing of . As seen the asymptotic lines are achieved at MeV/c (Figures 3(b) and 3(d)) and GeV/c (Figures 3(a) and 3(c)). Furthermore the ultimate values of the Gaussian BEC parameters and are valid with good accuracy for MeV/c and 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 () and Gaussian () functions the asymptotic -dependence is achieved for both the 1D BEC radius (Figure 3(a)) and the strength of correlations (Figure 3(c)) at which is much smaller than that for case of two some other centrally symmetric Lévy parameterizations (Figures 3(a) and 3(c)). This value for case of and is similar to those used in analyses of experimental CF . In Figure 4 the dependencies of relative BEC parameters, namely, (a, b) and (c, d), on (a, c) and (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 -dependence (Figures 4(a) and 4(c)) and system (11a) and (11b) for -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 -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 -dependence of the (Figure 4(b)) and (Figure 4(d)) is slower with increasing of the input values of the . As expected the ultimate levels (Figures 4(a) and 4(b)) and (Figures 4(c) and 4(d)) shown by thin dotted lines are valid for the same ranges of and as estimated above for Figure 3.

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

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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 . As seen from Figures 4 and 6 even the asymptotic values of relative 1D BEC parameters 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 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 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 Lévy parametrization (2) at 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 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, , is widely studied for and 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 , 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 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 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.