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Advances in High Energy Physics
Volume 2013 (2013), Article ID 230515, 22 pages
Research Article

Internal Cumulants for Femtoscopy with Fixed Charged Multiplicity

1Department of Physics, University of Stellenbosch, Stellenbosch 7602, South Africa
2Institut für Hochenergiephysik, Nikolsdorfergasse 18, 1050 Vienna, Austria

Received 14 March 2013; Accepted 20 May 2013

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2013 H. C. Eggers and B. Buschbeck. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A detailed understanding of all effects and influences on higher-order correlations is essential. At low charged multiplicity, the effect of a non-Poissonian multiplicity distribution can significantly distort correlations. Evidently, the reference samples with respect to which correlations are measured should yield a null result in the absence of correlations. We show how the careful specification of desired properties necessarily leads to an average-of-multinomials reference sample. The resulting internal cumulants and their averaging over several multiplicities fulfill all requirements of correctly taking into account non-Poissonian multiplicity distributions as well as yielding a null result for uncorrelated fixed- samples. Various correction factors are shown to be approximations at best. Careful rederivation of statistical variances and covariances within the frequentist approach yields errors for cumulants that differ from those used so far. We finally briefly discuss the implementation of the analysis through a multiple event buffer algorithm.

1. Introduction and Motivation

The understanding of hadronic collisions is now considered an essential baseline for ultrarelativistic heavy-ion collisions. Given the correspondingly low final-state multiplicities, there are significant deviations, even for inclusive samples, from assumptions commonly made both in the general theory and in the definition of experimentally measured quantities such as a non-Gaussian shape of the correlation function and non-Poissonian multiplicity distributions. Constraints such as energy-momentum conservation [1, 2] would also play a role in at least some regions of phase space. Multiplicity-class and fixed-multiplicity analysis differ increasingly from Poissonian and inclusive distributions, and with the good statistics now available, measurements have become accurate enough to require proper understanding and treatment of these assumptions and deviations, which play an ever larger role with increasing order of correlation.

1.1. Correlations as a Function of Charged Multiplicity

There are a number of reasons to study correlations at fixed charged multiplicity or, if necessary, charged-multiplicity classes. Firstly, the physics of multiparticle correlations will evidently change with , and indeed the multiplicity dependence of various quantities such as the intercept parameter and radii associated with Gaussian parametrisations is under constant scrutiny [312]. Measurement of many observables as a function of multiplicity class, regarded a proxy for centrality dependence, has been routine for years. Corresponding theoretical considerations, for example, in the quantum optical approach go back a long time [13]. Secondly, correlations for fixed- are the building blocks which are combined into multiplicity-class and inclusive correlations [14].

However, such fixed- correlations have been beset by an inconsistency in that they are nonzero even when the underlying sample is uncorrelated and do not integrate to zero either. This has been recognised from the start [15], and various attempts have been made to fix the problem.

Combining events from several fixed- subsamples into multiplicity classes does not solve these problems. To quote an early reference [16], “Averaging over multiplicities inextricably mixes the properties of the correlation mechanism with those of the multiplicity distribution. Instead, the study of correlations at fixed multiplicities allows one to separate both effects and to investigate the behaviour of correlation functions as a function of multiplicity.” Under the somewhat inappropriate name of “Long-Range Short-Range correlations” [15, 17], an attempt was made to separate these multiplicity-mixing correlations from the fixed- correlations, but the inconsistencies inherent in the underlying fixed- correlations were not addressed. Building on [18], we propose doing so now.

1.2. Cumulants in Multiparticle Physics

Multiparticle cumulants have entered the mainstream of analysis, as shown by the following incomplete list of topics. In principle, the considerations presented in this paper would apply to any, and all such cumulants to the degree that their reference distribution deviates from a Poisson process or that the type of particle kept fixed differs from the particle being analysed.

Integrated cumulants of multiplicity distributions have a long history in multiparticle physics [19]. Second-order differential cumulants, normally termed “correlation functions,” have likewise been ubiquitous for decades [7] both in charged-particles correlations [15] and in femtoscopy since they provide information on spacetime characteristics of the emitting sources, most recently at the LHC [10, 11, 20]. Differential three-particle cumulants generically measure asymmetries in source geometry and exchange amplitude phases [21]. They also provide consistency checks [22] and a tool to disentangle the coherence parameter from other effects [23, 24]. Three-particle cumulants are also sensitive to differences between longitudinal and transverse correlation lengths in the Lund model [25]. Inclusive three-particle cumulants have been measured, albeit with different methodologies, in, for example, hadronic [2629], leptonic [3033], and nuclear collisions [3437]. They play a central role in direct QCD-based calculations [3840], in some recent theory, and in experiment of azimuthal and jet-like correlations [4145]. Net-charge and other charge combinations are considered probes of the QCD phase diagram [4648]. Cumulants of order 4 or higher are, of course, increasingly difficult to measure, and so early investigations were largely confined to their scale dependence [4952]. The large event samples now available have, however, made feasible measurements of fourth- and higher-order cumulants in other variables as proposed in [13, 5356] as, for example, recently measured by ALICE [57]. Reviews of femtoscopy theory range from [5860] to more recent ones such as [8].

1.3. Outline of This Paper

It has long been obvious that the root cause of the problems and inconsistencies set out in Section 1.1 was the reference sample [61]. Insofar as cumulants are concerned, the solution was outlined in [18] as a subtraction of the reference sample cumulant from the measured one; important pieces of the puzzle were, however, still missing at that stage. In this paper, we clarify and extend the basic concept of internal cumulants and consider in detail the case of second- and third-order differential cumulants in the invariant for fixed charged multiplicity . The method may be implemented for other variables without much fuss.

A second cornerstone of the present paper is the recognition that the particles which enter a correlation analysis are usually only a subset of the charged pions. While in the case of charged-particle correlations, all particles are used in the analysis, Bose-Einstein correlations, for example, would use only the positive pions (and, in a separate analysis, only the negatives). In addition, there may be reasons to restrict the analysis itself to subregions of the total acceptance in which was measured, as exemplified in this paper by restriction to a “good azimuthal region” subinterval around the beam axis, , in which detection efficiency is high. can, however, be reinterpreted generically as any restriction in momentum space compared to and/or as a selection such as charge or particle species. Even when setting , that is, doing the femtoscopy analysis in the full acceptance, still does not equal but fluctuates around . The trivial observation that fundamentally changes the analysis: identical-particle correlations at fixed and charged-particle correlations at fixed require different definitions.

As we will show, ad hoc prescriptions such as simply inserting prefactors or implementing event mixing using only events of the same do alleviate the effect of the overall non-Poissonian multiplicity distribution in part but fail to remove them completely. The same issues will, of course, arise in any other correlation type of, for example, nonidentical particles or net charge correlations. The formalism set out here can be easily extended to such cases. A refined version of the abovementioned Long-Range-Short-Range method, which we term “Averaged-Internal” cumulants, will be presented in Section 5. Along the way, we document in Section 2 extended versions of the particle counters [62, 63] which we need as the basis for correlation studies and in Section 4 demonstrate from first principles that statistical errors for cumulants used so far have captured only some of the terms and with partly incorrect prefactors. Section 6 outlines the implementation of event mixing for fixed- analysis. While experimental results will be published elsewhere, preliminary results in Figures 2 and 3 show that, in third and even in second order, corrections due to proper treatment of fixed- reference samples can be large.

2. Raw Data, Counters, and Densities

2.1. Raw Data

The starting point for experimental correlation analysis is the inclusive sample , made up of events . Each event consists of a varying number of final-state elementary particles and photons; for our purposes, we consider only the charged pions of event in , the maximal acceptance region used. Each pion is characterised by a data vector containing its measured information, including the three components of its momentum, , while its discrete attributes such as mass and charge are captured in a data vector of discrete values; for the moment, we will consider only the charge, . From the sample's raw data, we can immediately find derived quantities such as the total charged multiplicity and the total transverse energy, and such derived quantities are hence considered part of the raw data. The list of particle attributes should be augmented by an error vector containing the measurement errors for each track, but we will not consider detector resolution errors here. In summary, the inclusive data sample is fully described in terms of consisting of lists of vectors in continuous and discrete spaces

2.2. Data after Conditioning and Cuts

For a particular analysis, the inclusive sample is invariably subdivided and modified through “conditioning,” the statistics terminology for semiinclusive or triggered analysis. From the total sample of events, a subsample is selected according to some restriction or precondition. In our case, this conditioning proceeds in the following steps.

(i) Conditioning into Fixed-N Subsamples. For the fixed-multiplicity analyses that form the subject of this paper, is subdivided into a set of fixed- subsamples , each of which contains only events whose measured multiplicity is equal to the constant characterising , We use the vertical bar here and everywhere in the usual sense of “conditioning” whereby the events in sample must satisfy the condition that their charged multiplicity must equal the specified constant , denoted in this case by the Kronecker delta . Quantities to the right of the vertical bar are generally considered known and fixed, while quantities to the left of the bar are variable or unknown. The number of events in equals the -restricted sum over the inclusive sample, The usual multiplicity distribution is the list of relative frequencies,1 While desirable, it is not easy to measure the total multiplicity of final-state charged pions, a quantity which approximately tracks the variation in the physics. Choosing charged pions measured within the maximal detector acceptance as marker is in any case only an approximation because it excludes charged particles outside the primary cuts and also ignores final-state particles other than charged pions. Nevertheless, we expect to be a reasonable measure of the multiplicity dependence of the physics. Alternatively, the multiplicity density in pseudorapidity at central rapidities can be used as a model-dependent proxy for .

(ii) Azimuthal Cut. While is the charged multiplicity measured in , there is no a priori reason why the correlation analysis itself may not be conducted within a restricted part of momentum space within which the actual analysis is done. In the case of the UA1 detector from which the data used in the examples below was drawn, refers to azimuthal regions within which measurement efficiency was high, and pions found in the low-efficiency azimuthal regions were excluded. Correspondingly, the multiplicity which enters the correlation analysis itself differs from and will, for a given fixed-, fluctuate with relative frequency where is the number of events for which and . The outcome space for will depend on its definition; in the present case where only positive (or only negative) pions within are used in the analysis, it will be so that the relative frequency is normalised by  With approximate charge conservation , we expect the fixed- average for positive (or negative) pions in to hover around  An example of the resulting relative frequencies (conditional normalised multiplicity distributions) is shown in Figure 1. Since , these conditional multiplicity distributions are almost always sub-Poissonian, that is, narrower than a Poisson distribution with the same would be.

Figure 1: Conditional normalised multiplicity distribution (relative frequency) of the number of positive pions in restricted azimuthal region for, respectively, fixed charged multiplicity 3, 5, 10, 20, 30, 40, and 50, for the UA1 dataset used in [65]. In accordance with (6), .
Figure 2: (a) Correction factors (red squares) and (blue circles) as defined in (57) as well as (green triangles) of (70), for the UA1 dataset used in [65]. (b) Inverse factors as used in the normalisation of internal cumulants (74)-(75) and (83)-(84).
Figure 3: Comparison, for the same UA1 data and cuts, of (red filled squares), (blue filled circles), and both averages over for fixed , with prefactors (red open squares) and (blue open circles). The Poissonian limit is represented by zero on the -axis; that is, 100 times the -scale represents the percentage deviation from the Poissonian limit.

(iii) Generalisation. While in this paper the analysis will be carried out for the positive pions of event falling into , the same formalism obviously applies to negative pions and may equally refer to any other particles such as kaons, baryons, and photons, in any combination which depends on . There is no a priori connection between the definitions of and .

(iv) Identical-Particle versus Multispecies Analysis. While we do not develop the formalism for correlations between two or three particles of different species or charge, the methodology developed here can be easily modified to deal with such cases. For example, positive-negative pion combinations and “charge balance correlations” [64] can be handled by inserting delta functions , where is the desired charge and the measured charge of track in event , into the definitions of the counters in Section 2.3.

2.3. Counters and Densities for Fixed

This section is based on an old formalism [53, 62, 63] which must, however, be updated to accommodate the issues being considered here. The basic building block of correlation analysis is the counter; it is a particular projection of the raw data particularly suited to the construction of histograms. Eventwise counters for a given event are averaged to give sample counters .

We take the simple case where event contains tracks with three-momenta , no discrete attributes and no further cuts or selection. For each point in momentum space, only that particle (if any) whose momentum happens to coincide with is to be counted, Such counters always appear under an integral over some region of the space, so that the delta functions fulfill the purpose of counting those particles falling within that region. Alternatively, one can consider the delta functions here and below to represent small nonoverlapping intervals around the specified momenta. The integral over the full momentum space yields while an integral over some subspace or bin will yield the number of particles of event in bin . The second-order eventwise counter for event is2 with the inequality ensuring that a single particle is not counted as a “pair.” The counter integrates to using the falling factorial notation as contrasted to the rising factorial (Pochhammer symbol) . The single-particle counter is a projection of because The most general eventwise counter which enters the exclusive cross-section for events with charged multiplicity fully describes the event, including any and all correlations between its particles. It integrates to the factorial of the event multiplicity and contains all counters of lower order by projection. An th-order counter is zero whenever there are more observation points than particles being observed, .3

To distinguish eventwise counters for nonfixed- from eventwise counters for fixed , we define the separate eventwise counter for fixed by specifying an additional Kronecker delta,

While the counters and densities defined above and below are clearly frame dependent, it is easy to define corresponding Lorentz-invariant versions by supplementing each delta function in 3 momenta with the corresponding energy; thus (7) would become, for example, with the on-shell energy, and in general which are manifestly invariant. Because such counters and densities are, however, always integrated over some by , the additional factors always cancel and play no role on this level of analysis and will be ignored for the time being. The bin boundaries of do, however, remain frame dependent.

Charge-, spin-, or species-specific counters are defined in the same way, that is, by supplying appropriate Kronecker deltas to the counters; for example, the particle counter for pions with charge at for fixed is while the two-particle counter for charge combination at momenta for any is, for example, In contrast to (18), charge counters rather than particle counters would be so that represents the net charge of event at . The two-particle counter for charges at momenta is and “charge flow” correlations can be constructed from this (for rapidities in the case of [66]) such as which can be expressed as and the related “charge balance functions” described in for example [46].

Returning to the fixed- case, eventwise counters will usually be combined with similar events to form event averages. The simplest average is the fixed- density for the subsample of fixed , with the Kronecker delta in (15) ensuring that only events in are considered, so we need not further specify the individual terms or limits of the -sum. Using (2), it is immediately clear that compared to the integral over the corresponding eventwise counter and to the integral (8); similarly The inclusive averaged density is the weighted average over all of the fixed- averages, Using (3), (15), and (23), this can be written as and for general keeping in mind that will be zero whenever . The integral of any th-order inclusive averaged density is the th-order factorial moment of the multiplicity distribution, with simple angle brackets denoting inclusive averaging.

The averaged counters are of course directly related to the traditional definitions in terms of cross-sections. If is the integrated luminosity of incoming particles, the topological cross-section is , the inelastic cross-section is , and the inclusive cross-section is while the relative frequency (multiplicity distribution) can be written as usual as . The relation between the differential cross sections and our counters is and so as usual inclusive and exclusive densities are related by [14] while the semi-inclusive cross sections and counters follow by the usual projections.

2.4. Counters and Densities for Fixed

Our choice of a basic counter is motivated by the experimental situation set out in Section 1: we wish to work in event subsamples of fixed total charged multiplicity in the entire momentum space but do the differential correlation analysis using only those pions which fall into the restricted space and of a particular charge or . This requires the use of “sub-subsamples” for which both and are kept fixed, with being the number of positive pions of event in , and eventwise sub-subsample counters As in (2), the number of events in a sub-subsample enters the relevant event averages where once again the double Kronecker deltas in (34) ensure selection of events in only. Integrals of the counters over the good-azimuth region yield, for the eventwise and -averaged counters, Bearing in mind that observation points refer to positive pions in only, the event-averaged counters for fixed but any are given by the average weighted in terms of the relative frequency , for , which integrate to

3. Construction of Correlation Quantities

3.1. Criteria

Correlation measurements of any sort are only meaningful if a reference baseline signifying “independence” or “lack of correlation” is defined quantitatively; indeed, many different kinds of correlations may be defined and measured on the same data, depending on which particular physical and mathematical scenario is considered to be known or trivial and taken to be the baseline [61]. In our case, we require the reference distribution to have the following properties.(1)The number of charged pions in all phase space is an important parameter as a measure of possibly different physics, but only the positive pions in are to be considered in the differential analysis.(2)For a given , the momenta of the reference density should be mutually independent for any order . This and the previous requirement imply that the reference should be a multinomial distributed over continuous momentum space; see Section 3.2.1.(3)Given fixed , the reference density must reproduce the -multiplicity structure of the sub-subsamples as embodied in . As set out further in Section 3.2.2, this translates into an average of multinomials, (4)The reference density should reproduce the measured one-particle density in momentum space. This can in principle be satisfied by three different expressions for the multinomial's parameters : see Section 3.2.3.(5)Measures of correlation must reduce to zero even on a differential basis whenever the data is, in fact, uncorrelated. While this may seem self-evident, this requirement is often ignored or not satisfied in the literature. We address the resulting proper baseline through the use of internal cumulants in Section 3.3.(6)The measure of correlation should be insensitive to the one-particle distribution. This is addressed as usual by normalisation; see Section 3.3.

3.2. The Reference Distribution
3.2.1. Multinomials in Discrete and Continuous Spaces

Before (39) can be developed further, it is necessary to take a detour into discrete outcome spaces before tackling the continuous outcome space defined by and . The reason is that multinomial distributions for continuous arguments can be written only as a limit of the discrete precursor.

Let there be bins with the corresponding set of Bernoulli probabilities of a single particle falling into bin , normalised by . Independent tossing of particles into these bins results in the multinomial for the bin counts , with normalisation where the sum must be taken over the “universal set” The multivariate factorial moment generating function (FMGF) for this multinomial for the set of source parameters can be solved in closed form, The FMGF can generally be used to find multivariate factorial moments and factorial cumulants for any selection of bins , including repeated indices, by differentiation For the multinomial case (43), the factorial moments and cumulants are therefore The multinomial for variable in continuous outcome space is derived by keeping constant while taking the limit with bin sizes tending to zero and changing to a Bernoulli probability density normalised by . The result is the point process where the probability for the count in the infinitesimal “bin” around any to be larger than 1 becomes negligible; that is, we have at most one particle at a given . While the multinomial probability itself can be written only as a limit, the FMGF can be written analytically as the functional [67] Factorial moments and factorial cumulants are found generically from functional derivatives [14] which for the multinomial of (46) yield

3.2.2. Multinomial Reference for Fixed

Applying the above general case to our reference distribution (39), we must rewrite (46) to make provision for the fact that may in general depend not only on but also on , Inserting (50) into (39), we find the FMGF for the reference distribution of subsample to be Using (48), the reference factorial moments are therefore with corresponding expressions for the reference factorial cumulants.

3.2.3. Reproducing the One-Particle Distribution

The set of functions are as yet undetermined, apart from the general constraints and . In multinomials of all kinds, the Bernoulli probabilities are fixed parameters and therefore are the conveyers of whatever remains constant in the outcomes while the detailed outcomes fluctuate as statistical outcomes do. The “field” can and must therefore be seen as the quantity encompassing the “physics” of the one-particle distributions, which, in the absence of additional external information, is embodied by our experimental data sample: the experimental densities “are” the physics, including all correlations, and their first-order projections “are” the one-particle physics. The question immediately arises whether should be fixed by or the -average . Three possible choices come to mind.(1) It is tempting to define it in terms of the density for each sub-subsample , which is correctly normalised since . As this choice would attribute physical significance to , it would be appropriate whenever is associated with additionally measured experimental information. If, however, fluctuates randomly from event to event based in part on unmeasured or unmeasurable properties such as an event's azimuthal orientation, use of (53) makes no sense.If is deemed physically relevant, correlations in terms of of (35) may be feasible, conditional on the availability of a sufficient number of events . Where sample sizes do not permit this, one could nevertheless attempt to measure what have historically been termed “short-range correlations” but in this case not in the traditional sense of fixed- correlations versus inclusive ones but rather of fixed--fixed- correlations versus fluctuating--fixed- correlations. See Section 3.6.(2) A second choice would be properly normalised but fails to satisfy the crucial relations (71)–(73) below and is hence discarded.(3) While remaining open-minded towards Choice 1, we therefore choose the third possibility, the ratio of the average density divided by the average, all for fixed , which would be appropriate for samples where is too small or physical significance can be attributed only to but not to . According to (38), it is also correctly normalised and ensures that the Bernoulli parameters are the same for all events in , independent of . Substituting this into (52), the differential reference factorial moments orders become where we identify the prefactor as the normalised factorial moments of the -multiplicity distribution for given , while the generating functional (51) becomes (see also [13]) Taking functional derivatives of the logarithm of (58), the first-, second-, and third-order cumulants of the reference density are where the square bracket indicates the number of distinct permutations which must be taken into account. The terms in the rounded brackets are readily recognised as the normalised factorial cumulants of the distribution for a given fixed and so generalisation to arbitrary orders is immediate, This can be proven generally by defining the functional which has only a first nonzero functional derivative and the multiplicity generating function , in terms of which .

3.3. Internal Cumulants for Fixed

Equation (64) shows that the differential cumulants of the reference distribution are directly proportional to the integrated cumulants of , which are zero only if is Poissonian. For fixed , neither the integrated cumulants nor the differential ones are zero. While this has long been recognised in the literature [15], the inevitable consequence was not drawn; namely, that “Poissonian” cumulants for fixed and so forth cannot possibly represent true correlations because they are nonzero even when the momenta are fully independent. It is known that the theory of cumulants needs improvement on a fundamental level which reaches well beyond the scope of this paper [68, 69], but those difficulties are irrelevant here. A first step which does address the above concerns was taken in [18], where it was shown very generally on the basis of generating functionals that correlations for samples of fixed are best measured using the internal cumulants , which are defined as the differences between the measured and the reference cumulants of the same order For our averaged-multinomial reference case, the internal cumulants of second and third-orders are given by the differences between (65) and (60) and between (66) and (62), resulting in with and so on for higher orders. These internal cumulants are identically zero if and when the measured densities for fixed are multinomials since then from (56) so that whenever the data is multinomial, while ensures that in the same case. On another level, the internal cumulants always integrate to zero over the full good-azimuth space , irrespective of the presence of correlations, and so on for all orders. Both properties will remain valid after transformation from three-momentum to invariant four-momentum differences in Section 3.4. In the case of Poissonian statistics, for all , so that the above internal cumulants revert to their usual definitions.

As stated in Section 3.1, the measured correlations may in addition be made insensitive to the one-particle distribution through normalisation. As set out in [18], such normalisation is achieved for fixed by dividing the internal cumulants by the corresponding reference distribution density, which for the case at hand is given by (56). This leads to the second-order normalised internal cumulant while in third-order we get

3.4. Correlation Integrals for Momentum Differences

In femtoscopy, correlations are mostly expressed in terms of pair variables and difference or the invariant four-momenta [70] where the energies are on-shell, . As shown in [53], the formulation of eventwise counters as sums and products of Dirac delta functions makes it easy to change variables. Writing as shorthand for and so forth, the second-order unnormalised internal cumulant in terms of is, for example, found from the identity to be where the counters in the second line are defined by the terms in the first, while and are four-momentum differences between sibling pairs and event mixing pairs , respectively. It is easy to show that and and hence, as before, , which will be true for any correlation whatsoever. The double event averages in the product term are the theoretical foundations of event mixing [53]; the inner -average is usually shortened to a smaller “moving average tail” subsample of .

In third-order, the “GHP average” invariant is defined as the average of three two-momentum differences over all pairs (with or without the ), and it is related to the invariant mass of three pions by . Other “topologies” such as the “GHP max” can also be employed. For large multiplicities, the “Star” topology may be preferred [71], but we will not pursue it here. For the GHP average, the third internal cumulant is given by with and similarly for and . Second- and third-order cumulants are normalised by, respectively, After transforming from momenta to , the formulae of Section 3.3 become while the normalised cumulants of Section 3.3 become

3.5. Effect of Fixed- Correction Factors

To get a feeling for the size of the corrections involved, we measured the correction factors and with the same UA1 dataset and the same cuts as in Figure 1. As shown in Figure 2, the consequence of the clearly sub-Poissonian multiplicity distributions shown in Figure 1 is that these factors are significantly less than 1, in contrast to the usual factorial moments of the charged multiplicity distribution which are super-Poissonian with factors exceeding 1. For very low multiplicities , normalised internal cumulants are hence larger than their Poissonian counterparts but converge to them with increasing . Nevertheless, up to corrections of more than 5% for and more than 20% for compared to their uncorrected counterparts can be expected. By contrast, the additive correction does not deviate much from the Poissonian limit of 2 except for very small . By contrast, unnormalised internal cumulants (81) and (82) are far less sensitive to the multinomial correction.

It is of interest to zoom in on the approach to the Poissonian limit of 1 and to compare these corrections to the equivalent charged-multiplicity-based ones, which for the case of fixed , would be just and . In Figure 3, the Poissonian limit corresponds to zero on the -axis. It is clear that the fixed- factors go some way to correct for the fixed- conditioning; the gap between them is approximately determined by , that is, by the exact definition and outcome space for .

3.6. Eliminating Fluctuations in

We return briefly to the first choice in Section 3.2.3, that is, which would permit different physics for each combination. If we were willing and able to do analyses for each , we would use the fixed- equivalent of (68) and (69) derived from , and normalise by . Where that is not possible, we can still average over the above to form “Averaged Internal” (AI) correlations (see Section 5), but in this case averaging over for fixed , and normalise if necessary by the moment . Given that this involves products of moments in the sub-subsample , event mixing would have to be restricted to the same sub-subsamples also, for example The transformation to pair variables works in the same way as in previous sections.

4. Statistical Errors

While the various versions of internal cumulants constructed above may all be relevant at some point, we concentrate on finding expressions for experimental standard errors for the unnormalised and normalised internal cumulants of (81)–(84). This turns out to be more subtle than merely applying a generic root-mean-square prescription. We will show in this section that standard errors implemented thus far may have been underestimated even in the standard two-particle case.

The calculations performed in this section belong to the “frequentist” view of probability; a proper Bayesian analysis, which can be expected to rest on more solid foundations, is beyond the scope of this paper. The two viewpoints can reasonably be expected to yield similar results in the limit of large bin contents and sample sizes.

In this section, we often simplify notation by writing and and so forth, since the formulae apply to samples and variables of any kind.

4.1. Propagation of Statistical Errors

Because cumulants can be measured only through the moments that enter their definitions, the first task is to identify which moment variances and covariances are needed. By means of standard error propagation [68], we find the sample variances for second-order internal cumulants of (81) and (83) to be under the assumption that is much smaller than the other variances, so that can be treated as a constant; this is the case if there are many bins for . Similarly, from (69), the variance of the unnormalised internal cumulant is, assuming of (70) to be constant, while the normalised version has variance (again assuming ) The unnormalised cumulants (81) and (82) and their variances (88) and (92) require knowledge of the multiplicity factorial moments , so that the individual terms must be accumulated until the entire sample has been analysed. By contrast, the normalised cumulants (83) and (84) and their variances (90) and (94) contain the multiplicity moments only as prefactors.

4.2. Expectation Values of Counters

While we will not make direct use of the results in this section, it is nevertheless useful briefly to consider what we might mean by an “expectation value of experimental counters and densities.” For any scalar function of the momenta, the theoretical expectation value is defined as the integral over the entire outcome space of weighted by a “parent distribution” , an abstract entity supposedly containing everything there is to know on this level, Purely theoretical concepts such as and should be given little or no room in a strongly experimentally-oriented study. In calculating standard errors on counters below, we will, however, make use of the exact factorisation that expectation values provide whenever two variables are statistically independent, .

Expectation values for pairwise variables such as the four-momentum difference we are considering here must be based on the underlying physics. We can deduce some properties of the parent distribution based on the usual definition of the femtoscopic correlation function: which is a function of two entirely different quantities: the four-momentum differences of “sibling” tracks taken from the same event and one constructed from the mixed-event sample using tracks from different events, written as , , and so forth. For second-order correlations, the parent distribution is therefore necessarily a two-variable probability4   which, depending on whether the cases and occur, may or may not factorise into “sibling” and “mixed” marginal probabilities but (unless ), the marginals will always be The shapes of and must necessarily be different since it is precisely this difference that leads to a nontrivial signal in (96). In terms of this joint probability, we can write expectation values of eventwise counters (separately for inclusive, fixed-, or fixed- cases) as Later, we will meet expectation values for cases such as, which definitely do not factorise. The above expressions can be simplified because we know that the parent distribution is not a function of the individual track indices , and and similarly and . For the event-averaged counters, this results in or in terms of the notation of Section 3.4, As mentioned, we do not need the factorisation (97) of as long as we keep careful track of the equal-event-indices cases. Whenever or , independence of the events ensures that expectation values of products of any functions and of the pair variables do factorise, For third-order correlations, the parent distribution is a function of three different variables , , and containing, respectively, three, two, or one track from the same event, and corresponding considerations regarding equal and unequal event indices apply there, too.

4.3. Statistical Error Calculation from First Principles

It was shown in Section 4.2 that expectation values would have well-defined meanings in terms of underlying parent distributions and their marginals if their parent distributions were known, which, however, they are not. We are therefore forced to revert from expectation values to sample averages after completing a calculation. The real use of such expectation values in frequentist statistics has been in the form of a gedankenexperiment which we now reproduce from Kendall [68]. Let be any generic eventwise counter or any other eventwise statistic. Since the formulae in this section remain true for inclusive and fixed- samples, we omit any notation related to in this derivation. In this simplified notation, the well-known standard error of the sample mean is given by (simplifying ) which follows from the combinatorics of equal and unequal event indices by the above artificial use of expectation values, reverting from expectation values to sample means in the last step: where we have used the fact that for all <