New Theoretical Results on
Several theoretical results concerning event-by-event
fluctuations are discussed:
(1) a role of the global conservation laws and concept of statistical ensembles;
(2) strongly intensive measures for physical systems with volume fluctuations;
(3) identity method for chemical fluctuations in a case of incomplete particle identification;
(4) the example of particle number fluctuations in a vicinity of the critical point.
New Theoretical Results on
Mark I. Gorenstein††thanks: Speaker.
Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
The study of event-by-event (e-by-e) fluctuations in high-energy nucleus-nucleus (A+A) collisions opens new possibilities to investigate properties of strongly interacting matter (see, e.g., Refs.  and  and references therein). Specific fluctuations can signal the onset of deconfinement when the collision energy becomes sufficiently high to create the quark-gluon plasma (QGP) at the initial stage of A+A collision [3, 4]. By measuring the fluctuations, one may also observe effects caused by the dynamical instabilities when the expanding system goes through the 1 order transition line between the QGP and the hadron resonance gas . Furthermore, the critical point (CP) of strongly interacting matter may be signaled by characteristic fluctuation pattern [6, 7, 8]. Therefore, e-by-e fluctuations are an important tool for the study of properties of the onset of deconfinement and the search for the CP of strongly interacting matter. However, mostly due to the incomplete acceptance of detectors, difficulties to control e-by-e the number of interacting nucleons, and also not well adapted data analysis tools, the results on e-by-e fluctuations are not yet mature. Even the simplest tests of statistical and dynamical models at the level of fluctuations are still missing.
In this presentation the theoretical progress in several areas related to the study of e-by-e fluctuations is reported. A role of the global conservation laws is discussed in Sec. 2. The strongly intensive measures of e-by-e fluctuations are introduced in Sec. 3. They give a possibility to study e-by-e fluctuations in a physical system when its average size and size fluctuations can not be controlled experimentally. In Sec. 4 a novel procedure, the identity method, is described for analyzing fluctuations of identified hadrons under typical experimental conditions of incomplete particle identification. Finally, in Sec. 5 using the van der Waals equation of state adopted to the grand canonical ensemble formulation we discuss particle number fluctuations in a vicinity of the CP. Most part of our discussion concerns the e-by-e fluctuations of hadron multiplicities. However, many of our physical conclusions can be applied to more general cases.
2 Global Conservations Laws
In this section we illustrate the role of global conservation laws in calculating of e-by-e fluctuations within statistical mechanics. Successful applications of the statistical model to description of mean hadron multiplicities in high energy collisions (see, e.g., Refs. [9, 10, 11] and references therein) has stimulated investigations of properties of the statistical ensembles. Whenever possible, one prefers to use the grand canonical ensemble (GCE) due to its mathematical convenience. The canonical ensemble (CE) should be applied [12, 13] when the number of carriers of conserved charges is small (of the order of 1), such as strange hadrons , antibaryons , or charmed hadrons . The micro-canonical ensemble (MCE) has been used [17, 18, 19] to describe small systems with fixed energy, e.g., mean hadron multiplicities in proton-antiproton annihilation at rest. In all these cases, calculations performed in different statistical ensembles yield different results. This happens because the systems are ‘small’ and they are ‘far away’ from the thermodynamic limit (TL). The multiplicities of hadrons produced in relativistic heavy ion collisions are typically much larger than 1. Thus, their mean values obtained within GCE, CE, and MCE approach each other. One refers here to the thermodynamical equivalence of statistical ensembles in the TL and uses the GCE, as a most convenient one, for calculating the hadron yields.
A statistical system is characterized by the extensive quantities: volume , energy , and conserved charge(s)111 These conserved charges are usually the net baryon number, strangeness, and electric charge. In non-relativistic statistical mechanics the number of particles plays the role of a conserved ‘charge’. . The MCE is defined by the postulate that all micro-states with given , , and have equal probabilities of being realized. This is the basic postulate of statistical mechanics. The MCE partition function just calculates the number of microscopic states with given fixed values. In the CE the energy exchange between the considered system and ‘infinite thermal bath’ is assumed. Consequently, a new parameter, temperature , is introduced. To define the GCE, one makes a similar construction for the conserved charge : an ‘infinite chemical bath’ and the chemical potential are introduced. The CE introduces the energy fluctuations. In the GCE, there are additionally the charge fluctuations. Therefore, the global conservation laws of and are treated in different ways: in the MCE both and are fixed in each microscopic state, whereas only average value of is fixed in the CE, and average values of and in the GCE.
The MCE, CE, and GCE are the most familiar statistical ensembles. In several textbooks (see, e.g., Ref. [20, 21]), the pressure (or isobaric) ensemble has been also discussed. An ‘infinite bath of the fixed external pressure’ is then introduced. This leads to volume fluctuations around the average value (see Ref. ). In general, there are 3 pairs of variables – – and, thus, the 8 statistical ensembles222For several conserved charges the number of possible ensembles is larger, as each charge can be treated either canonically or grand canonically. can be constructed.
Measurements of hadron multiplicity distributions in A+A collisions, open a new field of applications of the statistical models. The particle multiplicity fluctuations are usually quantified by a ratio of the variance to mean value, the scaled variance,
and are a subject of current experimental activities. In statistical models there is a qualitative difference in properties of a mean multiplicity and a scaled variance of multiplicity distribution. It was recently found [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] that even in the TL corresponding results for the scaled variance are different in different ensembles. Hence the equivalence of ensembles holds for mean values in the TL, but does not extend to fluctuations. Several examples below illustrate this statement.
2.1 Canonical Ensemble
Let us consider a system which consists of one sort of positively +1 and negatively -1 charged particles (e.g., and mesons) with total charge equal to zero . For the relativistic Boltzmann ideal gas in the volume at temperature the GCE partition function reads:
In Eq. (2.1) is a single particle partition function
where is a particle mass, and is the modified Hankel function. Parameters and are auxiliary parameters introduced in order to calculate mean numbers and fluctuations of the positively and negatively charged particles. They are set to one in the final formulas. The chemical potential equals zero to satisfy the condition .
The CE partition function is obtained by an explicit introduction of the charge conservation constrain, , for each microscopic state of the system and it reads:
where the integral representations of the -Kronecker symbol and the modified Bessel function were used. The average number of and can be calculated as :
The exact charge conservation leads to the CE suppression, , of the charged particle multiplicities relative to the results for the GCE (2.0). The ratio of calculated in the CE and GCE is plotted as a function of in Fig. 1 left.
The corresponding scaled variances are :
In the large volume limit ( corresponds also to ) one can use an asymptotic expansion of the modified Bessel function,
The dependence of the scaled variance calculated within the CE and GCE on is shown in Fig. 1 right. The scaled variance shows a very different behavior than the mean multiplicity: in the large limit the mean multiplicity ratio approaches one and the scaled variance ratio 1/2. Thus, in the case of fluctuations the CE and GCE are not equivalent.
2.2 Micro-Canonical Ensemble
Our second example is the ideal gas of massless neutral Boltzmann particles. We consider the same volume and energy in the MCE and GCE and compare these to formulations. The fixed MCE energy and the mean GCE energy are connected via equation (the degeneracy factor is ):
The mean multiplicity in the MCE is approximately equal to the GCE value:
The approximation is valid for and reflects the thermodynamic equivalence of the MCE and GCE. The scaled variances for the multiplicity fluctuations are, however, different in the GCE and MCE :
Thus, despite of thermodynamic equivalence of the MCE and GCE the value of is four times smaller than the scaled variance of the GCE (Poisson) distribution, .
Figure 2(a) shows a comparison of the MCE and GCE results for the multiplicity distributions . Different values of the scaled variances (2.0) and (2.0) signify the different widths of distributions in the GCE abd MCE clearly seen in Fig. 2.
Figure 2(b) shows the single particle momentum spectra in the GCE and MCE ( and MeV). A single particle momentum spectrum in the GCE reads:
The MCE spectrum is very close to the Boltzmann distribution (2.0) at all momenta up to 10 GeV/c. At this momentum, the spectrum is already dropped in a comparison to its value by about a factor of . The MCE spectrum decreases faster than the GCE one at high momenta. Close to the threshold momentum, GeV, where the MCE spectrum goes to zero, large deviations from (2.0) are observed. In order to demonstrate these deviations the MCE and GCE momentum spectra are shown in Fig. 2 (b) over 90 orders of magnitude.
2.3 Hadron Resonance Gas in GCE, CE, and MCE
We present now the results of the hadron resonance gas (HRG) for the e-by-e fluctuations of negatively charged and positively charged hadrons (see details in Ref. ). The corresponding scaled variances and are calculated in the GCE, CE, and MCE along the chemical freeze-out line in central Pb+Pb (Au+Au) collisions for the whole energy range from SIS to LHC. The model parameters are the volume , temperature , baryonic chemical potential , and the strangeness saturation parameter . They are chosen by fitting the mean hadron multiplicities. Once a suitable set of the chemical freeze-out parameters is determined for each collision energy, the scaled variances and can be calculated in different statistical ensembles. The results for the GCE, CE, and MCE are presented in Fig. 3 as functions of the center-of-mass energy of the nucleon pair.
A comparison between the data and predictions of statistical models should be performed for results which correspond to A+A collisions with a fixed number of nucleon participants. In Fig. 4 our statistical model results are compared with the NA49 data  at collision energies 20, 30, 40, 80 and 158 A GeV for the 1% most central Pb+Pb collisions selected by the numbers of projectile participants. In the experimental study of A+A collisions at high energies only a fraction of all produced particles is registered. If detected particles are uncorrelated, the scaled variance for the accepted particles, , can be presented in terms of the full 4 scaled variance as (see, e.g., )
where is a probability of a single particle to be accepted. Figure 4 demonstrates that among three different statistical ensembles the MCE is in a better agreement with the NA49 data. The reasons of this are not yet clear.
2.4 Generalized Statistical Ensembles
A choice of statistical ensemble is crucial in calculating fluctuations. On the other hand, it is clear that GCE, CE, and MCE are only some typical examples. A general concept of the statistical ensembles was suggested in Ref. . The extensive quantities define the MCE. Different statistical ensembles are then constructed using externally given distributions of extensive quantities, . The distribution of any observable in is then obtained in two steps. Firstly, the MCE -distribution, , is calculated at fixed values of the extensive quantities . Secondly, this result is averaged over the external distribution :
Fluctuations of extensive quantities around their average values depend not on the system’s physical properties, but rather on external conditions. One can imagine a huge variety of these conditions, thus, the standard statistical ensembles discussed above are only some special examples. The ensemble defined by Eq. (2.0), the -ensemble, includes the standard statistical ensembles as the particular cases. The generalized statistical mechanics based on Eq. (2.0) can be applied to different tasks of hadron production in high energy collisions. For example, based on Eq. (2.0) and introducing the scaling volume fluctuations , an attempt was made in Ref.  to extend the statistical model to the hard domain of high transverse momenta and/or high hadron masses.
3 Strongly Intensive Measures of E-by-E Fluctuations
A significant increase of transverse momentum and multiplicity fluctuations is expected in the vicinity of the CP. One can probe different regions of the phase diagram by varying the collision energy and the size of colliding nuclei. The possibility to observe signatures of the critical point inspired the energy and system size scan program of the NA61/SHINE Collaboration at the CERN SPS  and the low beam energy scan program of the STAR and PHENIX Collaborations at the BNL RHIC . In these studies one measures and then compares e-by-e fluctuations in collisions of different nuclei at different collision energies. The average sizes of the created physical systems and their e-by-e fluctuations are expected to be rather different (see, e.g., Ref. ). This strongly affects the observed fluctuations, i.e., the measured quantities would not describe the local physical properties of the system but rather reflect the system size fluctuations. For instance, A+A collisions with different centralities may produce a system with approximately the same local properties (e.g., the same temperature and baryonic chemical potential) but with the volume changing significantly from interaction to interaction. Note that in high energy collisions the average volume of created matter and its variations from collision to collision usually cannot be controlled experimentally. Therefore, a suitable choice of statistical tools for the study of e-by-e fluctuations is really important.
Intensive quantities are defined within the GCE of statistical mechanics. They depend on temperature and chemical potential(s), but they are independent of the system volume. Strongly intensive quantities introduced in Ref.  are, in addition, independent of volume fluctuations. They are the appropriate measures for studies of e-by-e fluctuations in A+A collisions and can be defined for two extensive state quantities and . Here, we call and extensive when the first moments of their distributions for the ensemble of possible states are proportional to volume. They are referred to as state quantities as they characterize the states of the considered system, e.g., final states of A+A collisions or micro-states of the GCE.
There are two families of strongly intensive quantities which depend on the second and first moments of and and thus allow to study e-by-e (or state-by-state) fluctuations :
and averaging is performed over the ensemble of multi-particle states. The normalization factors and are required to be proportional to the first moments of any extensive quantities.
In Ref.  a specific choice of the and normalization factors was proposed. It makes the quantities and dimensionless and leads to in the independent particle model, as will be shown below.
From the definition of and it follows that in the case of absence of fluctuations of and , i.e., for . Thus the proposed normalization of and leads to a common scale on which the values of the fluctuation measures calculated for different state quantities and can be compared.
There is an important difference between the and quantities. Namely, in order to calculate one needs to measure only the first two moments: , and , . This can be done by independent measurements of the distributions and . The quantity includes the correlation term, , and thus requires, in addition, simultaneous measurements of and in order to obtain the joint distribution .
3.1 and in the Independent Particle Model
The IPM assumes that:
(1) the state quantities and can be expressed as
where and denote single particle contributions to and , respectively, and is the number of particles;
(2) inter-particle correlations are absent, i.e., the probability of any multi-particle state is the product of probability distributions of single-particle states, and these probability distributions are the same for all and independent of ,
where is an arbitrary multiplicity distribution of particles.
It can be shown  that within the IPM the average values of the first and second moments of and are equal to:
The values of and are proportional to the average number of particles and, thus, to the average size of the system. These quantities are extensive. The quantities , and , , are the first and second moments of the single-particle distribution . Within the IPM they are independent of and play the role of intensive quantities.
Using Eq. (3.0) the scaled variance which describes the state-by-state fluctuations of can be expressed as:
where is the scaled variance of the single-particle quantity , and is the scaled variance of . A similar expression follows from Eq. (3.0) for the scaled variance . The scaled variances and depend on the fluctuations of the particle number via . Therefore, and are not strongly intensive quantities.
Thus, the requirement that
within the IPM leads to:
In the IPM the and quantities are expressed in terms of sums of the single particle variables, and . Thus in order to calculate the normalization and factors one has to measure the single particle quantities and . However, this may not always be possible within a given experimental set-up. For example, and may be energies of particles measured by two calorimeters. Then one can study fluctuations in terms of and but can not calculate the normalization factors which are proposed above.
We consider now two examples of specific pairs of extensive variables and . In the first example, we use the transverse momentum , where is the absolute value of the particle transverse momentum, and the number of particles . The requirement that
for the IPM leads then to the normalization factors 
where describes the single particle -fluctuations.
As the second example let us consider the multiplicity fluctuations. Here and will denote the multiplicities of hadrons of types and , respectively (e.g., kaons and pions). One obtains 
The normalization factors (3.0) and (3.0) are suggested to be used for the calculation for and both in theoretical models and for the analysis of experimental data (see Ref.  for further details of the normalization procedure).
The measure, introduced some time ago , belongs to the family within the current classification scheme. The fluctuation measure was introduced for the study of transverse momentum fluctuations. In the general case, when represents any motional variable and is the particle multiplicity, one gets:
For the multiplicity fluctuations of hadrons belonging to non-overlapping types and the connection between the and measures reads:
The IPM plays an important role as the reference model. The deviations of real data from the IPM results Eq. (3.0) can be used to learn about the physical properties of the system. This resembles the situation in studies of particle multiplicity fluctuations. In this case, one uses the Poisson distribution with as the reference model. The other reference value corresponds to , i.e., the absence of -fluctuations. Values of (or ) correspond to “large” (or “very large”) fluctuations of , and (or ) to “small” (or “very small”) fluctuations.
The fluctuation measures and do not depend on the average size of the system and its fluctuations in several different model approaches, namely, statistical mechanics within the GCE, model of Multiple Independent Sources, Mixed Event Model (see Ref.  for details). Thus, one may expect that and preserve their strongly intensive properties in many real experiments too. An extension of the strongly intensive fluctuation measures for higher order cumulants were suggested in recent paper .
3.2 and Evaluated in Specific Models
The UrQMD  calculations of and measures were performed in Refs. [47, 48]. The Monte Carlo simulations and analytical model results for and were presented in Ref. . These measures were also studied in Ref.  for the ideal Bose and Fermi gases within the GCE. The GCE for the Boltzmann approximation satisfies the conditions of the IPM, i.e., Eq. (3.0) is valid. The following general relations were found :
i.e., the Bose statistics makes smaller and larger than unity, whereas the Fermi statistics works in the opposite way. The Bose statistics of pions appears to be the main source of quantum statistics effects in a hadron gas with a temperature typical for the hadron system created in A+A collisions. It gives about 20% decrease of and 10% increase of , at MeV with respect to the IPM results (3.0). The Fermi statistics of protons modifies insignificantly and for typical values of and . Note that UrQMD takes into account several sources of fluctuations and correlations, e.g., the exact conservation laws and resonance decays. On the other hand, it does not include the effects of Bose and Fermi statistics. First experimental results on and in p+p and Pb+Pb collisions have been reported in Refs. [51, 52, 53].
Now we consider the and measures for two particle multiplicities and . Resonance decays, when particle species 1 and 2 appear simultaneously among the decay products, lead to the (positive) correlations between and numbers. Let and are the multiplicities of positively and negatively charged pions, respectively. A presence of two components is assumed: the correlated pion pairs coming from decays, , and the uncorrelated and from other sources. The and numbers are then equal to:
where is the number of resonances decaying into pairs, while and are the numbers of uncorrelated and , respectively.
Using approximate relations,
one obtains two alternative expressions for the number of -resonances :
i.e., can be calculated using the measurable quantities of e-by-e fluctuations. As a result, these resonance abundances, which are difficult to be measured by other methods, can be estimated by measuring the fluctuations and correlations of the numbers of stable hadrons. Note that an idea to use the e-by-e fluctuations of particle number ratios to estimate the number of hadronic resonances was suggested for the first time in Ref. .
The analysis of and fluctuations and correlations from resonance decays is done in Ref. . It is based on the HRG model, both in the GCE and CE, and on the relativistic transport model UrQMD. This analysis illustrates a role of the centrality selection, limited acceptance, and global charge conservation in A+A collisions. We present now the results for Eq. (3.0) and (3.0) in Pb+Pb and p+p collisions. Note that in Eq. (3.0) is an intensive but not strongly intensive quantity. Thus, it is expected to be sensitive to the system size fluctuations. On the other hand, the in Eq. (3.0) is the strongly intensive measure and it should keep the same value in a presence of the system size fluctuations.
The samples of 5% central Pb+Pb collision events at and 17.3 GeV are considered. These UrQMD results are compared with those for most central Pb+Pb collisions at zero impact parameter, fm, and for reactions at the same collision energies. Several mid-rapidity windows for final and particles are considered.
A width of the rapidity window is an important parameter. The two hadrons which are the products of a resonance decay have, in average, a rapidity difference of the order of unity. Therefore, while searching for the effects of resonance decays one should choose to enlarge a probability for simultaneous hit into the rapidity window of both correlated hadrons (e.g., and ) from resonance decays. Thus, should be large enough. However, should be small in comparison to the whole rapidity interval accessible for final hadron with mass . Only for one can expect a validity of the GCE results. Considering a small part of the statistical system, one does not need to impose the restrictions of the exact global charge conservations: the GCE which only regulates the average values of the conserved charges is fully acceptable. For large , when the detected hadrons correspond to an essential part of the whole system, the effects of the global charge conservation become more important. In the HRG this should be treated within the CE, where the conserved charges are fixed for all microscopic states. The global charge conservation influences the particle number fluctuations and introduces additional correlations between numbers of different particle species.
The UrQMD values of correlation parameter in 5% central Pb+Pb collision events are shown by full circles in Fig. 5(a) and full boxes in 5(b) as functions of the acceptance windows . The UrQMD results for the most central Pb+Pb collision events with zero impact parameter, fm, are shown by open symbols. The triangles show the results of the UrQMD simulations in reactions. The collision energy is taken as GeV in Fig. 5(a) and GeV in Fig. 5(b). The windows at the center of mass mid-rapidity are taken as , 1, 2, 3, 4, and . A symbol denotes the case when all final state particles are detected (i.e., a full -acceptance). Note that the UrQMD model does not assume any, even local, thermal and/or chemical equilibration. Therefore, a connection between the UrQMD and HRG results for particle number fluctuations and correlations is a priori unknown.
For the 5% most central Pb+Pb collisions the correlation increases with . As seen from Fig. 5(b), this increase is rather strong at high collision energy: at large , the value of becomes much larger than the HGM results in both the CE and GCE. The behavior of and is rather similar to that of .
Selecting within the UrQMD simulations the most central Pb+Pb collision events with zero impact parameter fm, one finds essentially smaller values of (open symbols in Fig. 5). This means that in the 5% centrality bin of Pb+Pb collision events large fluctuations of the number of nucleon participants (i.e., the volume fluctuations) are present. These volume fluctuations produce large additional contributions to the scaled variances of pions and to the correlation parameter . They become more and more important with increasing collision energy. This is due to an increase of the number of pions per participating nucleon with increasing collision energy. However, one hopes that these volume fluctuations will be canceled out to a large extent when they are combined in the strongly intensive measures. Note also that the UrQMD results for in inelastic collisions, shown in Fig. 5 by triangles, are qualitatively similar to those in Pb+Pb collisions at fm.
The UrQMD results for in Pb+Pb collisions at GeV and 17.3 GeV are presented in Fig. 6(a) and (b), respectively, as a function of the acceptance window at mid-rapidity. In contrast to the results shown in Fig. 5, both centrality selections in Pb+Pb collisions (5% centrality bin and fm) lead to very similar results for shown in Fig. 6. This means that the measure has the strongly intensive properties, at least in the UrQMD simulations. The UrQMD results in p+p reactions are close to those in Pb+Pb ones.
The GCE and CE results of the HRG are presented in Figs. 5 and 6 by the horizontal dashed and dashed-dotted lines, respectively. The UrQMD results for , presented in Fig. 6, demonstrate a strong dependence on the size of rapidity window . At these results are close to those of the GCE HRM. On the other hand, with increasing the role of exact charge conservation becomes more and more important. From Fig. 6, one observes that the UrQMD values of at large are close to the CE results.
As seen from Fig. 5, a similar correspondence between the UrQMD results for in Pb+Pb collisions at fm and their GCE and CE values is approximately valid. However, this is not the case for the 5% most central Pb+Pb events. In that centrality bin the volume fluctuations give the dominant contributions to for large .
For very small acceptance, , one expects an approximate validity of the Poisson distribution for any type of the detected particles. Their scaled variances are then close to unity, i.e., . Particle number correlations, due to both the resonance decays and the global charge conservation, become negligible, i.e., . These expectations are indeed supported by the UrQMD results at presented in Fig. 5. Therefore, one expects at . This expectation is also valid, as seen from the UrQMD results at presented in Fig. 6.
4 Incomplete Particle Identification
Fluctuations of the chemical (particle-type) composition of hadronic final states in A+A collisions are expected to be sensitive to the phase transition between hadronic and partonic matter. First experimental results on e-by-e chemical fluctuations have already been published from the CERN SPS and BNL RHIC, and more systematic measurements are in progress.
Studies of chemical fluctuations in general require to determine the number of particles of different hadron species (e.g., pions, kaons, and protons) e-by-e. A serious experimental problem in such measurements is incomplete particle identification, i.e., the impossibility to identify uniquely the type of each detected particle. The effect of particle misidentification distorts the measured fluctuation quantities. For this reason the analysis of chemical fluctuations is usually performed in a small acceptance, where particle identification is relatively reliable. However, an important part of the information on e-by-e fluctuations in full phase space is then lost.
Although it is usually impossible to identify each detected particle, one can nevertheless determine with high accuracy the average multiplicities (averaged over many events) for different hadron species.
4.1 The Identity Method
The identity variable was introduced in Ref. , and in Ref.  a new experimental technique called the identity method was proposed. It solved the misidentification problem for one specific combination of the second moments in a system of two hadron species (‘kaons’ and ‘pions’). In Refs. [58, 59] this method was extended to show that all the second moments as well as the higher moments of the joint multiplicity distribution of particles of different types can be uniquely reconstructed in spite of the effects of incomplete identification. Notably, the results [58, 59] can be used for an arbitrary number of hadron species. It is assumed that particle identification is achieved by measuring the particle mass . Since any measurement is of finite resolution, we deal with continuous distributions of observed masses denoted as and normalized as ()
Note that for experimental data the functions for particles of type are obtained from the inclusive distribution of the -values for all particles from all collision events. The identity variables are defined as
Complete identification (CI) of particles corresponds to distributions which do not overlap. In this case, for all particle species and for the th species. When the distributions overlap, can take the value of any real number from . We introduce the quantities