Internal Cumulants for Femtoscopy with Fixed Charged Multiplicity

Abstract

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-𝑁 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 𝑁 or, if necessary, charged-multiplicity classes. Firstly, the physics of multiparticle correlations will evidently change with 𝑁, and indeed the multiplicity dependence of various quantities such as the intercept parameter and radii associated with Gaussian parametrisations is under constant scrutiny [3–12]. 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-𝑁 are the building blocks which are combined into multiplicity-class and inclusive correlations [14].

However, such fixed-𝑁 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-𝑁 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-𝑁 correlations, but the inconsistencies inherent in the underlying fixed-𝑁 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 [26–29], leptonic [30–33], and nuclear collisions [34–37]. They play a central role in direct QCD-based calculations [38–40], in some recent theory, and in experiment of azimuthal and jet-like correlations [41–45]. Net-charge and other charge combinations are considered probes of the QCD phase diagram [46–48]. Cumulants of order 4 or higher are, of course, increasingly difficult to measure, and so early investigations were largely confined to their scale dependence [49–52]. The large event samples now available have, however, made feasible measurements of fourth- and higher-order cumulants in other variables as proposed in [13, 53–56] as, for example, recently measured by ALICE [57]. Reviews of femtoscopy theory range from [58–60] 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 𝑄=−(𝑝₁−𝑝₂)² for fixed charged multiplicity 𝑁. The method may be implemented for other variables without much fuss.

A second cornerstone of the present paper is the recognition that the 𝑛 particles which enter a correlation analysis are usually only a subset of the 𝑁 charged pions. While in the case of charged-particle correlations, all 𝑁 particles are used in the analysis, Bose-Einstein correlations, for example, would use only the 𝑛≡𝑛₊ positive pions (and, in a separate analysis, only the 𝑛₋=𝑁−𝑛₊ negatives). In addition, there may be reasons to restrict the analysis itself to subregions of the total acceptance Ω in which 𝑁 was measured, as exemplified in this paper by restriction to a “good azimuthal region” subinterval around the beam axis, 𝒜⊂[0,2𝜋], in which detection efficiency is high. 𝒜 can, however, be reinterpreted generically as any restriction in momentum space compared to Ω and/or as a selection such as charge or particle species. Even when setting 𝒜=Ω, that is, doing the femtoscopy analysis in the full acceptance, 𝑛 still does not equal 𝑁 but fluctuates around 𝑁/2. The trivial observation that 𝑛≠𝑁 fundamentally changes the analysis: identical-particle correlations at fixed 𝑁 and charged-particle correlations at fixed 𝑁 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 𝑁 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-𝑁 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-𝑁 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 𝒮, made up of ℰ events 𝑎=1,…,ℰ. Each event consists of a varying number of final-state elementary particles and photons; for our purposes, we consider only the 𝑁(𝑎) charged pions of event 𝑎 in Ω, the maximal acceptance region used. Each pion 𝑖=1,…,𝑁(𝑎) is characterised by a data vector (𝐏^𝑎_𝑖,𝐬^𝑎_𝑖,𝐞^𝑎_𝑖) containing its measured information, including the three components of its momentum, 𝐏^𝑎_𝑖=(𝑝^𝑎_𝑖𝑥,𝑝^𝑎_𝑖𝑦,𝑝^𝑎_𝑖𝑧)or(𝑦^𝑎_𝑖,𝜙^𝑎_𝑖,𝑝^𝑎_𝑡𝑖), while its discrete attributes such as mass and charge are captured in a data vector 𝐬^𝑎_𝑖 of discrete values; for the moment, we will consider only the charge, 𝐬^𝑎_𝑖→𝑐^𝑎_𝑖. From the sample's raw data, we can immediately find derived quantities such as the total charged multiplicity 𝑁(𝑎) 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 𝐞^𝑎_𝑖 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 𝒮={𝐏^𝑎,𝐬^𝑎}^ℰ_𝑎=1 consisting of lists of vectors in continuous and discrete spaces 𝐏^𝑎=𝐏^𝑎₁,𝐏^𝑎₂,…,𝐏^𝑎_𝑁(𝑎),𝐬^𝑎=𝐬^𝑎₁,𝐬^𝑎₂,…,𝐬^𝑎_𝑁(𝑎). (1)

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, 𝒮 is subdivided into a set of fixed-𝑁 subsamples 𝒮_𝑁, each of which contains only events 𝑎 whose measured multiplicity 𝑁(𝑎) is equal to the constant 𝑁 characterising 𝒮_𝑁={𝐏^𝑎,𝐬^𝑎∣𝛿(𝑁,𝑁(𝑎))}, 𝑁=0,1,2,… We use the vertical bar ∣ here and everywhere in the usual sense of “conditioning” whereby the events in sample 𝒮_𝑁 must satisfy the condition that their charged multiplicity must equal the specified constant 𝑁, denoted in this case by the Kronecker delta 𝛿(𝑁,𝑁(𝑎)). 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 𝒮_𝑁 equals the 𝛿-restricted sum over the inclusive sample, ℰ_𝑁=^ℰ_𝑎=1𝛿(𝑁,𝑁(𝑎)),^∞_𝑁=0ℰ_𝑁=ℰ. (2) The usual multiplicity distribution is the list of relative frequencies,¹ℛ_𝑁=ℰ_𝑁ℰ,^∞_𝑁=0ℛ_𝑁=1. (3) 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 𝑁=𝑁(Ω) 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 𝑁 to be a reasonable measure of the multiplicity dependence of the physics. Alternatively, the multiplicity density in pseudorapidity at central rapidities 𝑑𝑁/𝑑𝜂 can be used as a model-dependent proxy for 𝑁.

(ii) Azimuthal Cut. While 𝑁(𝑎) is the charged multiplicity measured in Ω, there is no a priori reason why the correlation analysis itself may not be conducted within a restricted part 𝒜⊂Ω 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, 𝒜 refers to azimuthal regions within which measurement efficiency was high, and pions found in the low-efficiency azimuthal regions were excluded. Correspondingly, the multiplicity 𝑛(𝑎) which enters the correlation analysis itself differs from 𝑁(𝑎) and will, for a given fixed-𝑁, fluctuate with relative frequency ℛ_𝑛𝑁=ℰ_𝑛𝑁ℰ_𝑁foreachfixed𝑁=1,2,…, (4) where ℰ_𝑛𝑁 is the number of events for which 𝑁(𝑎)=𝑁 and 𝑛(𝑎)=𝑛. The outcome space for 𝑛(𝑎) will depend on its definition; in the present case where only positive (or only negative) pions within 𝒜 are used in the analysis, it will be [0,1,…,𝑁] so that the relative frequency is normalised by ^𝑁_𝑛=0ℛ_𝑛𝑁=1∀𝑁=1,2,…. (5)  With approximate charge conservation 𝑛₊≃𝑛₋, we expect the fixed-𝑁 average for positive (or negative) pions in 𝒜 to hover around ⟨𝑛⟩_𝑁≃𝑁2volumeof𝒜.volumeofΩ (6)  An example of the resulting relative frequencies (conditional normalised multiplicity distributions) is shown in Figure 1. Since 𝑛≤𝑁, these conditional multiplicity distributions are almost always sub-Poissonian, that is, narrower than a Poisson distribution with the same ⟨𝑛⟩_𝑁 would be.

Figure 1 

Conditional normalised multiplicity distribution (relative frequency) of the number of positive pions 𝑛 in restricted azimuthal region 𝒜={20∘≤|𝜙|≤160∘} for, respectively, fixed charged multiplicity 𝑁= 3, 5, 10, 20, 30, 40, and 50, for the UA1 dataset used in [65]. In accordance with (6), ⟨𝑛⟩𝑁≃0.39𝑁.

(a)

(b)

Figure 2 

(a) Correction factors 𝐹2𝑁=⟨𝑛(𝑛−1)⟩𝑁/⟨𝑛⟩2𝑁 (red squares) and 𝐹3𝑁=⟨𝑛(𝑛−1)(𝑛−2)⟩𝑁/⟨𝑛⟩3𝑁 (blue circles) as defined in (57) as well as 𝐺3𝑁 (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 (1/𝐹2𝑁)−1 (red filled squares), (1/𝐹3𝑁)−1 (blue filled circles), and both averages over 𝑛 for fixed 𝑁, with prefactors [𝑁2/𝑁(𝑁−1)]−1 (red open squares) and [𝑁3/𝑁(𝑁−1)(𝑁−2)]−1 (blue open circles). The Poissonian limit is represented by zero on the 𝑦-axis; that is, 100 times the 𝑦-scale represents the percentage deviation from the Poissonian limit.

(iii) Generalisation. While in this paper the analysis will be carried out for the 𝑛(𝑎) positive pions of event 𝑎 falling into 𝒜, 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 𝑁. There is no a priori connection between the definitions of 𝑁 and 𝑛.

(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 𝛿(𝑐,𝑐^𝑎_𝑖), where 𝑐 is the desired charge and 𝑐^𝑎_𝑖 the measured charge of track 𝑖 in event 𝑎, into the definitions of the counters in Section 2.3.

2.3. Counters and Densities for Fixed 𝑁

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 𝜌 for a given event 𝑎 are averaged to give sample counters 𝜌.

We take the simple case where event 𝑎 contains 𝑁(𝑎) tracks with three-momenta 𝐏^𝑎={𝐏^𝑎₁,…,𝐏^𝑎_𝑁(𝑎)}, no discrete attributes 𝐬 and no further cuts or selection. For each point 𝐩₁ in momentum space, only that particle 𝑖 (if any) whose momentum 𝐏^𝑎_𝑖 happens to coincide with 𝐩 is to be counted, 𝜌𝐩₁∣𝐏^𝑎=^𝑁(𝑎)_𝑖=1𝛿𝐩₁−𝐏^a_𝑖. (7) Such counters always appear under an integral over some region of the 𝐏 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 Ω yields _Ω𝑑𝐩₁𝜌𝐩₁∣𝐏^𝑎=𝑁(𝑎), (8) while an integral over some subspace or bin Ω_𝑏⊂Ω will yield the number of particles of event 𝑎 in bin Ω_𝑏. The second-order eventwise counter for event 𝑎 is²𝜌𝐩₁,𝐩₂∣𝐏^𝑎=^𝑁(𝑎)_{𝑖≠𝑗=1}𝛿𝐩₁−𝐏^𝑎_𝑖𝛿𝐩₂−𝐏^𝑎_𝑗, (9) with the inequality 𝑖≠𝑗 ensuring that a single particle is not counted as a “pair.” The counter integrates to _Ω𝑑𝐩₁𝑑𝐩₂𝜌𝐩₁,𝐩₂∣𝐏^𝑎=𝑁(𝑎)(𝑁(𝑎)−1)=𝑁(𝑎)², (10) using the falling factorial notation 𝑁^𝑟=𝑁(𝑁−1)⋯(𝑁−𝑟+1)𝑟=1,2,…, (11) as contrasted to the rising factorial (Pochhammer symbol) 𝑁^𝑟=𝑁(𝑁+1)⋯(𝑁+𝑟−1). The single-particle counter is a projection of 𝜌(𝐩₁,𝐩₂∣𝐏^𝑎) because _Ω𝑑𝐩₂𝜌𝐩₁,𝐩₂∣𝐏^𝑎=[𝑁(𝑎)−1]𝜌𝐩₁∣𝐏^𝑎. (12) The most general eventwise counter which enters the exclusive cross-section for events with charged multiplicity 𝑁(𝑎)𝜌𝐩₁,𝐩₂,…,𝐩_𝑁(𝑎)∣𝐏^𝑎=^𝑁(𝑎)_{𝑖1≠𝑖2≠⋯≠𝑖𝑁(𝑎)=1}^𝑁(𝑎)_𝑑=1𝛿𝐩_d−𝐏^𝑎_𝑖𝑑 (13) fully describes the event, including any and all correlations between its particles. It integrates to the factorial of the event multiplicity _Ω𝑑𝐩₁𝑑𝐩₂⋯𝑑𝐩_𝑁(𝑎)𝜌𝐩₁,𝐩₂,…,𝐩_𝑁(𝑎)∣𝐏^𝑎=𝑁(𝑎)! (14) and contains all counters of lower order by projection. An 𝑟th-order counter 𝜌(𝐩₁,𝐩₂,…,𝐩_𝑟|𝐏^𝑎) is zero whenever there are more observation points than particles being observed, 𝑟>𝑁(𝑎).³

To distinguish eventwise counters for nonfixed-𝑁 from eventwise counters for fixed 𝑁, we define the separate eventwise counter for fixed 𝑁 by specifying an additional Kronecker delta, 𝜌𝐩₁,…,𝐩_𝑟∣𝐏^𝑎((𝑎,𝑁=𝛿𝑁,𝑁))^𝑁_{𝑖1≠⋯≠𝑖𝑟𝑑}𝛿𝐩_𝑑−𝐏^𝑎_𝑖𝑑𝑟=1,2,…,𝑁. (15)

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, 𝜌𝐩₁∣𝐏^𝑎=^𝑁(𝑎)_𝑖=1𝐸𝐩₁𝛿𝐩₁−𝐏^𝑎_𝑖, (16) with 𝐸(𝐩₁)=𝐸₁=𝐩²₁+𝑚² the on-shell energy, and in general 𝜌𝐩₁,𝐩₂,…,𝐩_𝑟∣𝐏^𝑎=^𝑁(𝑎)_{𝑖1≠𝑖2≠⋯≠𝑖𝑟=1}^𝑁(𝑎)_𝑑=1𝐸_𝑑𝛿𝐩_𝑑−𝐏^𝑎_𝑖𝑑(𝑎),𝜌𝐩𝑟=1,…,𝑁₁,…,𝐩_𝑟∣𝐏^𝑎((𝑎,𝑁=𝛿𝑁,𝑁))^𝑁_{𝑖1≠⋯≠𝑖𝑟𝑑}𝐸_𝑑𝛿𝐩_𝑑−𝐏^𝑎_𝑖𝑑𝑟=1,…,𝑁, (17) which are manifestly invariant. Because such counters and densities are, however, always integrated over some Ω_𝑏 by ∏_𝑑(𝑑𝐩_𝑑/𝐸_𝑑), the additional factors 𝐸_𝑑 always cancel and play no role on this level of analysis and will be ignored for the time being. The bin boundaries of Ω_𝑏 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 𝑐₁ at 𝐩₁ for fixed 𝑁 is 𝜌𝐩₁∣𝑐₁,𝐏^𝑎,𝑁=𝛿(𝑁,𝑁(𝑎))^𝑁(𝑎)_𝑖=1𝛿𝑐₁,𝑐^𝑎_𝑖𝛿𝐩₁−𝐏^𝑎_𝑖, (18) while the two-particle counter for charge combination (𝑐₁,𝑐₂) at momenta (𝐩₁,𝐩₂) for any 𝑁 is, for example, 𝜌𝐩₁,𝐩₂∣𝑐₁,𝑐₂,𝐏^𝑎=^𝑁(𝑎)_𝑖,𝑗=1𝛿𝑐₁,𝑐^𝑎_𝑖𝛿𝐩₁−𝐏^𝑎_𝑖𝛿𝑐₂,𝑐^𝑎_𝑗𝛿𝐩₂−𝐏^𝑎_𝑗. (19) In contrast to (18), charge counters rather than particle counters would be 𝜌_𝑐𝐩₁∣𝑐₁,𝐏^𝑎=^𝑁(𝑎)_𝑖=1𝑐₁𝛿𝑐₁,𝑐^𝑎_𝑖𝛿𝐩₁−𝐏^𝑎_𝑖, (20) so that 𝜌_𝑐(𝐩₁∣+1,𝐏^𝑎)+𝜌_𝑐(𝐩₁∣−1,𝐏^𝑎) represents the net charge of event 𝑎 at 𝐩₁. The two-particle counter for charges (𝑐₁,𝑐₂) at momenta (𝐩₁,𝐩₂) is 𝜌^𝑐1𝑐2≡𝜌_𝑐𝐩₁,𝐩₂∣𝑐₁,𝑐₂,𝐏^𝑎=^𝑁(𝑎)_𝑖,𝑗=1𝑐₁𝛿𝑐₁,𝑐^𝑎_𝑖𝛿𝐩₁−𝐏^𝑎_𝑖𝑐₂𝛿𝑐₂,𝑐^𝑎_𝑗𝛿𝐩₂−𝐏^𝑎_𝑗, (21) and “charge flow” correlations can be constructed from this (for rapidities (𝑦,𝑦^′) in the case of [66]) such as Φ𝑦,𝑦^′=−_𝑖≠𝑗𝑐^𝑎_𝑖𝑐^𝑎_𝑗𝛿𝑦−𝑌^𝑎_𝑖𝛿𝑦^′−𝑌^𝑎_𝑗, (22) which can be expressed as Φ=⟨𝜌⁺⁻+𝜌⁺⁻−𝜌⁺⁺−𝜌⁻⁻⟩ and the related “charge balance functions” described in for example [46].

Returning to the fixed-𝑁 case, eventwise counters will usually be combined with similar events to form event averages. The simplest average is the fixed-𝑁 density for the subsample of fixed 𝒮_𝑁, 𝜌𝐩₁∣𝒮_𝑁=1ℰ_𝑁𝑎𝜌𝐩₁∣𝐏^𝑎,𝑁, (23) with the Kronecker delta in (15) ensuring that only events in 𝒮_𝑁 are considered, so we need not further specify the individual terms or limits of the 𝑎-sum. Using (2), it is immediately clear that _Ω𝑑𝐩₁𝜌𝐩₁∣𝒮_𝑁=𝑁 (24) compared to the integral over the corresponding eventwise counter 𝑑𝐩𝜌_Ω𝐩∣𝐏^𝑎,𝑁=𝑁𝛿(𝑁,𝑁(𝑎)) (25) and to the integral (8); similarly _Ω𝑑𝐩₁⋯𝑑𝐩_𝑟𝜌𝐩₁,…,𝐩_𝑟∣𝒮_𝑁=𝑁(𝑁−1)⋯(𝑁−𝑟+1)=𝑁^𝑟. (26) The inclusive averaged density 𝜌(𝐩₁∣𝒮) is the weighted average over all 𝑁 of the fixed-𝑁 averages, 𝜌𝐩₁∣𝒮=^∞_𝑁=1ℛ_𝑁𝜌𝐩₁∣𝒮_𝑁. (27) Using (3), (15), and (23), this can be written as 𝜌𝐩₁=1∣𝒮ℰ_𝑎𝑖𝛿𝐩₁−𝐏^𝑎_𝑖, (28) and for general 𝑟=1,2,…𝜌𝐩₁,…,𝐩_𝑟∣𝒮=^∞_𝑁=𝑟ℛ_𝑁𝜌𝐩₁,…,𝐩_𝑟∣𝒮_𝑁=1ℰ_{𝑎𝑖1≠⋯≠𝑖𝑟}^𝑟_𝑑=1𝛿𝐩_𝑑−𝐏^𝑎_𝑖𝑑, (29) keeping in mind that 𝜌(𝐩₁,…,𝐩_𝑟∣𝒮_𝑁) will be zero whenever 𝑁(𝑎)<𝑟. The integral of any 𝑟th-order inclusive averaged density is the 𝑟th-order factorial moment of the multiplicity distribution, _Ω𝑑𝐩₁⋯𝑑𝐩_𝑟𝜌𝐩₁,…,𝐩_𝑟∣𝒮=^∞_𝑁=𝑟ℛ_𝑁𝑁^𝑟=𝑁^𝑟, (30) 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 ℒ is the integrated luminosity of incoming particles, the topological cross-section is 𝜎_𝑁=ℰ_𝑁/ℒ, the inelastic cross-section is 𝜎_𝐼=ℰ/ℒ, and the inclusive cross-section is 𝜎_incl=∑_𝑁𝑁𝜎_𝑁=⟨𝑁⟩𝜎_𝐼 while the relative frequency (multiplicity distribution) can be written as usual as ℛ_𝑁=𝜎_𝑁/𝜎_𝐼. The relation between the differential cross sections and our counters is 𝜌_incl𝐩₁,…,𝐩_𝑁=𝜌𝐩₁,…,𝐩_𝑁=1∣𝒮𝜎_𝐼𝑑^3𝑁𝜎_incl𝑑𝐩₁⋯𝑑𝐩_𝑁,𝜌𝐩₁,…,𝐩_𝑁∣𝒮_𝑁=1𝜎_𝑁𝑑^3𝑁𝜎_excl𝑑𝐩₁⋯𝑑𝐩_𝑁, (31) and so as usual inclusive and exclusive densities are related by [14] 𝜌𝐩₁,…,𝐩_𝑁=∣𝒮_∞𝑁=𝑟ℛ_𝑁()!𝜌𝐩𝑁−𝑟¹,…,𝐩_𝑁∣𝒮_𝑁𝑑𝐩_𝑟+1⋯𝑑𝐩_𝑁, (32) while the semi-inclusive cross sections and counters follow by the usual projections.

2.4. Counters and Densities for Fixed (𝑁,𝑛)

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 𝑁(𝑎) in the entire momentum space Ω but do the differential correlation analysis using only those pions 𝑛 which fall into the restricted space 𝒜 and of a particular charge +1 or −1. This requires the use of “sub-subsamples” for which both 𝑁 and 𝑛 are kept fixed, 𝒮_𝑛𝑁=𝐏^𝑎,with𝑎constrainedby𝛿𝑛,𝑛₊(𝑎)𝛿(𝑁,𝑁(𝑎)), (33) with 𝑛₊(𝑎) being the number of positive pions of event 𝑎 in 𝒜, and eventwise sub-subsample counters 𝜌𝐩₁,…,𝐩_𝑟∣𝑛,𝑁,𝐏^𝑎=𝛿𝑛,𝑛₊(𝑎)𝛿((𝑎×𝑁,𝑁))_{𝑛𝑖1≠⋯≠𝑖𝑟=1}𝛿𝐩₁−𝐏^𝑎_𝑖1⋯𝛿𝐩_𝑟−𝐏^𝑎_𝑖𝑟. (34) As in (2), the number of events in a sub-subsample ℰ_𝑛𝑁=∑_𝑎𝛿(𝑛,𝑛₊(𝑎))𝛿(𝑁,𝑁(𝑎)) enters the relevant event averages 𝜌𝐩₁,…,𝐩_𝑟∣𝒮_𝑛𝑁=1ℰ_𝑛𝑁𝑎𝜌𝐩₁,…,𝐩_𝑟∣𝑛,𝑁,𝐏^𝑎, (35) where once again the double Kronecker deltas in (34) ensure selection of events in 𝒮_𝑛𝑁 only. Integrals of the counters over the good-azimuth region 𝒜 yield, for the eventwise and 𝒮_𝑛𝑁-averaged counters, _𝒜𝑑𝐩₁⋯𝑑𝐩_𝑟𝜌𝐩₁,…,𝐩_𝑟∣𝑛,𝑁,𝐏^𝑎=𝑛^𝑟𝛿𝑛,𝑛₊(𝑎)𝛿(𝑁,𝑁(𝑎))_𝒜,𝑑𝐩₁⋯𝑑𝐩_𝑟𝜌𝐩₁,…,𝐩_𝑟∣𝒮_𝑛𝑁=𝑛^𝑟. (36) Bearing in mind that observation points 𝐩₁,𝐩₂,… refer to positive pions in 𝒜 only, the event-averaged counters for fixed 𝑁 but any 𝑛 are given by the average weighted in terms of the relative frequency ℛ_𝑛𝑁=ℰ_𝑛𝑁/ℰ_𝑁, 𝜌𝐩₁,…,𝐩_𝑟∣𝒮_𝑁=^𝑁_𝑛=𝑟ℛ_𝑛𝑁𝜌𝐩₁,…,𝐩_𝑟∣𝒮_𝑛𝑁=𝜌𝐩₁,…,𝐩_𝑟∣𝒮_𝑛𝑁𝑁, (37) for 𝑟=1,2,3,…, which integrate to _𝒜𝑑𝐩₁⋯𝑑𝐩_𝑟𝜌𝐩₁,…,𝐩_𝑟∣𝒮_𝑁=𝑛^𝑟_𝑁. (38)

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 𝑁 is an important parameter as a measure of possibly different physics, but only the 𝑛 positive pions in 𝒜 are to be considered in the differential analysis.(2)For a given (𝑁,𝑛), the momenta of the reference density 𝜌^ref(𝐩₁,…,𝐩_𝑟∣𝒮_𝑛𝑁) should be mutually independent for any order 1≤𝑟≤𝑛. This and the previous requirement imply that the reference should be a 𝑛 multinomial distributed over continuous momentum space; see Section 3.2.1.(3)Given fixed 𝑁, the reference density 𝜌^ref(𝐩₁,…,𝐩_𝑟∣𝒮_𝑁) must reproduce the 𝑛-multiplicity structure of the sub-subsamples 𝒮_𝑛𝑁 as embodied in ℛ_𝑛𝑁. As set out further in Section 3.2.2, this translates into an average of multinomials, 𝜌^ref𝐩₁,…,𝐩_𝑟∣𝒮_𝑁=^𝑁_𝑛=𝑟ℛ_𝑛𝑁𝜌^mult𝐩₁,…,𝐩_𝑟∣𝜶,𝒮_𝑛𝑁=𝜌^mult𝐩₁,…,𝐩_𝑟∣𝜶,𝒮_𝑛𝑁𝑁. (39)(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 𝜶: 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 𝐩 and 𝐏^𝑎. The reason is that multinomial distributions for continuous arguments 𝐩 can be written only as a limit of the discrete precursor.

Let there be bins Ω_𝑏,𝑏=1,…,𝐵 with the corresponding set of Bernoulli probabilities 𝜶={𝛼(𝑏)}^𝐵_𝑏=1 of a single particle falling into bin Ω_𝑏, normalised by ∑_𝑏𝛼(𝑏)=1. Independent tossing of 𝑛 particles into these bins results in the multinomial for the bin counts 𝐧={𝑛_𝑏}^𝐵_𝑏=1, 𝑝(𝐧∣𝜶,𝑛)=𝑛!^𝐵_𝑏=1𝛼(𝑏)^𝑛𝑏𝑛_𝑏!, (40) with normalisation _𝑈(𝐧)𝑝(𝐧∣𝜶,𝑛)=1, (41) where the sum must be taken over the “universal set” 𝑈(𝐧)=𝐧∣𝑛_𝑏≥0;_𝑏𝑛_𝑏=𝑛. (42) The multivariate factorial moment generating function (FMGF) for this multinomial for the set of source parameters 𝝀={𝜆(𝑏)}^𝐵_𝑏=1 can be solved in closed form, 𝑄^mult(𝝀∣𝜶,𝑛)=_𝑈𝑝(𝐧∣𝜶,𝑛)_𝑏(1−𝜆(𝑏))^𝑛𝑏=1−_𝑏𝜆(𝑏)𝛼(𝑏)^𝑛. (43) The FMGF 𝑄(𝝀) can generally be used to find multivariate factorial moments 𝜌(𝑏_𝑖1,𝑏_𝑖2,…,𝑏_𝑖𝑟) and factorial cumulants 𝜅(𝑏_𝑖1,𝑏_𝑖2,…,𝑏_𝑖𝑟) for any selection of bins (𝑏_𝑖1,𝑏_𝑖2,…,𝑏_𝑖𝑟)∈(1,…,𝐵), including repeated indices, by differentiation 𝜌𝑏_𝑖1,𝑏_𝑖2,…,𝑏_𝑖𝑟=(−1)^𝑟𝜕^𝑟𝑄(𝝀)𝜕𝜆𝑏_𝑖1𝜕𝜆𝑏_𝑖2⋯𝜕𝜆𝑏_{𝑖𝑟𝝀=0},𝜅𝑏_𝑖1,𝑏_𝑖2,…,𝑏_𝑖𝑟=(−1)^𝑟𝜕^𝑟ln𝑄(𝝀)𝜕𝜆𝑏_𝑖1𝜕𝜆𝑏_𝑖2⋯𝜕𝜆𝑏_{𝑖𝑟𝝀=0}. (44) For the multinomial case (43), the factorial moments and cumulants are therefore 𝜌^mult𝑏_𝑖1,𝑏_𝑖2,…,𝑏_𝑖𝑟∣𝜶,𝑛=𝑛^𝑟𝑟_𝑑=1𝛼𝑖_𝑑𝜅∀𝑟≤𝑛,_mult𝑏_𝑖1,𝑏_𝑖2,…,𝑏_𝑖𝑟∣𝜶,𝑛=(−1)^𝑟−1(𝑟−1)!⋅𝑛^𝑟_𝑑=1𝛼𝑖_𝑑. (45) The multinomial for variable 𝐩 in continuous outcome space ℝ is derived by keeping 𝑛 constant while taking the limit 𝐵→∞ with bin sizes tending to zero and changing to a Bernoulli probability density 𝛼(𝑏)→𝑑𝐩𝛼(𝐩) normalised by ∫_𝒜𝑑𝐩𝛼(𝐩)=1. The result is the point process where the probability for the count 𝑛(𝐩) in the infinitesimal “bin” around any 𝐩 to be larger than 1 becomes negligible; that is, we have at most one particle at a given 𝐩. While the multinomial probability itself can be written only as a limit, the FMGF can be written analytically as the functional [67] 𝑄^mult[𝜆(𝐩)∣𝛼(𝐩),𝑛]=1−_𝒜𝑑𝐩𝜆(𝐩)𝛼(𝐩)^𝑛. (46) Factorial moments and factorial cumulants are found generically from functional derivatives [14] 𝜌𝐩_𝑖1,𝐩_𝑖2,…,𝐩_𝑖𝑟=(−1)^𝑟𝛿^𝑟𝑄[𝜆(𝐩)]𝛿𝜆𝐩_𝑖1𝛿𝜆𝐩_𝑖2⋯𝛿𝜆𝐩_{𝑖𝑟𝜆(𝐩)=0},𝜅𝐩_𝑖1,𝐩_𝑖2,…,𝐩_𝑖𝑟=(−1)^𝑟𝛿^𝑟ln𝑄[𝜆(𝐩)]𝛿𝜆𝐩_𝑖1𝛿𝜆𝐩_𝑖2⋯𝛿𝜆𝐩_{𝑖𝑟𝜆(𝐩)=0}, (47) which for the multinomial 𝑄[𝝀(𝐩)]=𝑄^mult[𝜆(𝐩)∣𝛼(𝐩),𝑛] of (46) yield 𝜌^mult𝐩_𝑖1,𝐩_𝑖2,…,𝐩_𝑖𝑟∣𝛼(𝐩),𝑛=𝑛^𝑟𝑟_𝑘=1𝛼𝐩_𝑖𝑘,𝜅1≤𝑟≤𝑛,_mult𝐩_𝑖1,𝐩_𝑖2,…,𝐩_𝑖𝑟(𝐩)=()∣𝛼,𝑛−1^𝑟−1(𝑟−1)!⋅𝑛^𝑟_𝑘=1𝛼𝐩_𝑖𝑘. (48)(49)

3.2.2. Multinomial Reference for Fixed 𝑁

Applying the above general case to our reference distribution (39), we must rewrite (46) to make provision for the fact that 𝜶 may in general depend not only on 𝑁 but also on 𝑛, 𝑄^mult𝜆(𝐩)∣𝛼𝐩∣𝒮_𝑛𝑁=1−𝑑𝐩𝜆(𝐩)𝛼𝐩∣𝒮_𝑛𝑁^𝑛. (50) Inserting (50) into (39), we find the FMGF for the reference distribution of subsample 𝒮_𝑁 to be 𝑄^ref𝜆(𝐩)∣𝛼(𝐩),𝒮_𝑁=_𝑛ℛ_𝑛𝑁𝑄^mult𝜆(𝐩)∣𝛼𝐩∣𝒮_𝑛𝑁=1−𝑑𝐩𝜆(𝐩)𝛼𝐩∣𝒮_𝑛𝑁^𝑛_𝑁. (51) Using (48), the reference factorial moments are therefore 𝜌^ref𝐩₁,𝐩₂,…,𝐩_𝑟∣𝒮_𝑁=𝑛^𝑟𝑟_𝑘=1𝛼𝐩_𝑘∣𝒮_𝑛𝑁𝑁,∀𝑟≤𝑛, (52) with corresponding expressions for the reference factorial cumulants.

3.2.3. Reproducing the One-Particle Distribution

The set of functions 𝛼(𝐩∣𝒮_𝑛𝑁) are as yet undetermined, apart from the general constraints 𝛼(𝐩∣𝒮_𝑛𝑁)≥0 and ∫_𝒜𝑑𝐩𝛼(𝐩∣𝒮_𝑛𝑁)=1. In multinomials of all kinds, the Bernoulli probabilities 𝜶 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” 𝛼(𝐩∣𝒮_𝑛𝑁) 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 𝜌(𝐩₁,…,𝐩_𝑁∣𝒮_𝑁) “are” the physics, including all correlations, and their first-order projections 𝜌(𝐩₁∣𝒮_𝑁) “are” the one-particle physics. The question immediately arises whether 𝛼(𝐩∣𝒮_𝑛𝑁) should be fixed by 𝜌(𝐩∣𝒮_𝑛𝑁) or the 𝑛-average 𝜌(𝐩∣𝒮_𝑁)=⟨𝜌(𝐩∣𝒮_𝑛𝑁)⟩_𝑁. Three possible choices come to mind.(1) It is tempting to define it in terms of the density for each sub-subsample 𝒮_𝑛𝑁, 𝛼𝐩∣𝒮_𝑛𝑁=𝜌𝐩∣𝒮_𝑛𝑁𝑛∀(𝑁,𝑛), (53) which is correctly normalised since ∫𝑑𝐩𝜌(𝐩∣𝒮_𝑛𝑁)=𝑛. As this choice would attribute physical significance to 𝑛, it would be appropriate whenever 𝑛 is associated with additionally measured experimental information. If, however, 𝑛 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 𝑛 is deemed physically relevant, correlations in terms of 𝜌(𝐩₁,…,𝐩_𝑟∣𝒮_𝑛𝑁) of (35) may be feasible, conditional on the availability of a sufficient number of events ℰ_𝑛𝑁. 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-𝑁 correlations versus inclusive ones but rather of fixed-𝑛-fixed-𝑁 correlations versus fluctuating-𝑛-fixed-𝑁 correlations. See Section 3.6.(2) A second choice 𝛼𝐩∣𝒮_𝑛𝑁=𝜌𝐩∣𝒮_𝑛𝑁𝑛_𝑁 (54) 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 𝑁, 𝛼𝐩∣𝒮_𝑛𝑁=𝜌𝐩∣𝒮_𝑛𝑁𝑁⟨𝑛⟩_𝑁=𝜌𝐩∣𝒮_𝑁⟨𝑛⟩_𝑁, (55) which would be appropriate for samples where ℰ_𝑛𝑁 is too small or physical significance can be attributed only to 𝑁 but not to 𝑛. According to (38), it is also correctly normalised and ensures that the Bernoulli parameters are the same for all events in 𝒮_𝑁, independent of 𝑛. Substituting this into (52), the differential reference factorial moments orders become 𝜌^ref𝐩₁,𝐩₂,…,𝐩_𝑟∣𝒮_𝑁=𝑛^𝑟_𝑁⟨𝑛⟩^𝑟_𝑁^𝑟_𝑑=1𝜌𝐩_𝑑∣𝒮_𝑁=𝐹^𝑟_{𝑟𝑁𝑑=1}𝜌𝐩_𝑑∣𝒮_𝑁, (56) where we identify the prefactor as the normalised factorial moments of the 𝑛-multiplicity distribution for given 𝑁, 𝐹_𝑟𝑁=𝑛^𝑟_𝑁⟨𝑛⟩^𝑟_𝑁, (57) while the generating functional (51) becomes (see also [13]) 𝑄^ref𝜆(𝐩)∣𝛼(𝐩),𝒮_𝑁=𝜌1−𝑑𝐩𝜆(𝐩)𝐩∣𝒮_𝑁⟨𝑛⟩_𝑁^𝑛_𝑁. (58)Taking functional derivatives of the logarithm of (58), the first-, second-, and third-order cumulants of the reference density are 𝜅^ref𝐩₁∣𝒮_𝑁=𝜌^ref𝐩₁∣𝒮_𝑁=𝜌𝐩₁∣𝒮_𝑁,𝜅^ref𝐩₁,𝐩₂∣𝒮_𝑁=𝜌^ref𝐩₁,𝐩₂∣𝒮_𝑁−𝜌𝐩₁∣𝒮_𝑁𝜌𝐩₁∣𝒮_𝑁=𝑛²_𝑁⟨𝑛⟩²_𝑁−1𝜌𝐩₁∣𝒮_𝑁×𝜌𝐩₂∣𝒮_𝑁,𝜅^ref𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁=𝜌^ref𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁−[3]𝜌^ref𝐩₁,𝐩₂∣𝒮_𝑁𝜌𝐩₃∣𝒮_𝑁+2𝜌𝐩₁∣𝒮_𝑁𝜌𝐩₂∣𝒮_𝑁𝜌𝐩₃∣𝒮_𝑁=𝑛³_𝑁⟨𝑛⟩³_𝑁𝑛−3^2𝑁⟨𝑛⟩²_𝑁𝐩+2×𝜌₁∣𝒮_𝑁𝜌𝐩₂∣𝒮_𝑁𝜌𝐩₃∣𝒮_𝑁, (59)(60)(61)(62) where the square bracket [3] 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 𝑛 distribution for a given fixed 𝑁𝐾_𝑟𝑁=1⟨𝑛⟩^𝑟_𝑁(−𝜕)^𝑟𝜕Λ^𝑟ln^𝑁_𝑛=0ℛ_𝑛𝑁(1−Λ)ⁿ_Λ=0, (63) and so generalisation to arbitrary orders is immediate, 𝜅^ref𝐩₁,…,𝐩_𝑟∣𝒮_𝑁=𝐾^𝑟_{𝑟𝑁𝑘=1}𝜌𝐩_𝑘∣𝒮_𝑁. (64) This can be proven generally by defining the functional Λ[𝜆(𝐩)]=∫𝑑𝐩𝜆(𝐩)𝜌(𝐩∣𝒮_𝑁)/⟨𝑛⟩_𝑁 which has only a first nonzero functional derivative 𝛿Λ/𝛿𝜆(𝐩₁)=𝜌(𝐩∣𝒮_𝑁)/⟨𝑛⟩_𝑁 and the multiplicity generating function 𝒵(Λ∣𝒮_𝑁)=∑_𝑛ℛ_𝑛𝑁(1−Λ)^𝑛, in terms of which 𝑄^ref[𝜆]=𝒵[Λ[𝜆]].

3.3. Internal Cumulants for Fixed 𝒮_𝑁

Equation (64) shows that the differential cumulants of the reference distribution are directly proportional to the integrated cumulants 𝐾_𝑟𝑁 of 𝑛, which are zero only if ℛ_𝑛𝑁 is Poissonian. For fixed 𝑁, neither the integrated cumulants 𝐾_𝑟𝑁 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 𝑁𝜅𝐩₁,𝐩₂∣𝒮_𝑁=𝜌𝐩₁,𝐩₂∣𝒮_𝑁−𝜌𝐩₁∣𝒮_𝑁𝜌𝐩₂∣𝒮_𝑁,𝜅𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁=𝜌𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁−[3]𝜌𝐩₁,𝐩₂∣𝒮_𝑁𝜌𝐩₃∣𝒮_𝑁+2𝜌𝐩₁∣𝒮_𝑁𝜌𝐩₂∣𝒮_𝑁𝜌𝐩₃∣𝒮_𝑁, (65)(66) 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 𝑁 are best measured using the internal cumulants 𝜅^𝐼, which are defined as the differences between the measured and the reference cumulants of the same order 𝜅^𝐼𝐩₁,…,𝐩_𝑟∣𝒮_𝑁=𝜅𝐩₁,…,𝐩_𝑟∣𝒮_𝑁−𝜅^ref𝐩₁,…,𝐩_𝑟∣𝒮_𝑁. (67) 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 𝜅^𝐼𝐩₁,𝐩₂∣𝒮_𝑁=𝜌𝐩₁,𝐩₂∣𝒮_𝑁−𝐹_2𝑁𝜌𝐩₁∣𝒮_𝑁𝜌𝐩₁∣𝒮_𝑁,𝜅^𝐼𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁=𝜌𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁−[3]𝜌𝐩₁,𝐩₂∣𝒮_𝑁𝜌𝐩₃∣𝒮_𝑁+𝐺_3𝑁𝜌𝐩₁∣𝒮_𝑁𝜌𝐩₂∣𝒮_𝑁𝜌𝐩₃∣𝒮_𝑁, (68)(69) with 𝐺_3𝑁=3𝐹_2𝑁−𝐹_3𝑁⟨𝑛(=3𝑛−1)⟩_𝑁⟨𝑛⟩²_𝑁−⟨𝑛(𝑛−1)(𝑛−2)⟩_𝑁⟨𝑛⟩³_𝑁, (70) and so on for higher orders. These internal cumulants are identically zero if and when the measured densities for fixed 𝒮_𝑛𝑁 are multinomials since then from (56) 𝜌𝐩₁,𝐩₂∣𝒮_𝑁⟶𝜌^ref𝐩₁,𝐩₂∣𝒮_𝑁=𝐹_2𝑁𝜌𝐩₁∣𝒮_𝑁𝜌𝐩₂∣𝒮_𝑁, (71) so that 𝜅^𝐼(𝐩₁,𝐩₂∣𝒮_𝑁)→0 whenever the data is multinomial, while 𝜌𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁⟶𝐹_3𝑁𝜌𝐩₁∣𝒮_𝑁𝜌𝐩₂∣𝒮_𝑁𝜌𝐩₃∣𝒮_𝑁 (72) ensures that 𝜅^𝐼(𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁)→0 in the same case. On another level, the internal cumulants always integrate to zero over the full good-azimuth space 𝒜, irrespective of the presence of correlations, _𝒜𝑑𝐩₁𝑑𝐩₂𝜅^𝐼𝐩₁,𝐩₂∣𝒮_𝑁=_𝒜𝑑𝐩₁𝑑𝐩₂𝑑𝐩₃𝜅^𝐼𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁=0, (73) 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, 𝐹_𝑟𝑁=1 for all 𝑟, 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 𝑁 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 𝐾^𝐼𝐩₁,𝐩₂∣𝒮_𝑁=1𝐹_2𝑁𝜌𝐩₁,𝐩₂∣𝒮_𝑁𝜌𝐩₁∣𝒮_𝑁𝜌𝐩₂∣𝒮_𝑁−1, (74) while in third-order we get 𝐾^𝐼𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁=1𝐹_3𝑁×𝜌𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁−[3]𝜌𝐩₁,𝐩₂∣𝒮_𝑁𝜌𝐩₃∣𝒮_𝑁𝜌𝐩₁∣𝒮_𝑁𝜌𝐩₂∣𝒮_𝑁𝜌𝐩₃∣𝒮_𝑁𝐹+3^2𝑁𝐹_3𝑁−1. (75)

3.4. Correlation Integrals for Momentum Differences

In femtoscopy, correlations are mostly expressed in terms of pair variables 𝐊=(1/2)(𝐩₁+𝐩₂) and difference 𝐪=𝐩₁−𝐩₂ or the invariant four-momenta [70] 𝑄=−(𝑝₁−𝑝₂)²=(𝐩₁−𝐩₂)²−(𝐸₁−𝐸₂)² where the energies are on-shell, 𝐸_𝑟=𝐩²_𝑟+𝑚². As shown in [53], the formulation of eventwise counters as sums and products of Dirac delta functions makes it easy to change variables. Writing 𝜌_𝑟𝑁(𝐩₁,…,𝐩_𝑟) as shorthand for 𝜌(𝐩₁,…,𝐩_𝑟∣𝒮_𝑁) and so forth, the second-order unnormalised internal cumulant in terms of 𝑄 is, for example, found from the identity ∫𝑑𝑄𝜅^𝐼_2𝑁(𝑄)=∫d𝑄∫_𝒜𝑑𝐩₁𝑑𝐩₂𝜅^𝐼_2𝑁(𝐩₁,𝐩₂)𝛿(𝑄−(𝐩₁−𝐩₂)²−(𝐸₁−𝐸₂)²) to be 𝜅^𝐼_2𝑁(𝑄)=_𝑖≠𝑗𝛿𝑄−𝑄^𝑎𝑎_{𝑖𝑗𝑁𝑎}−𝐹_{2𝑁𝑖,𝑗}𝛿𝑄−𝑄^𝑎𝑏_{𝑖𝑗𝑁𝑏𝑁𝑎}=𝜌_2𝑁(𝑄)−𝐹_2𝑁𝜌_1𝑁⊗𝜌_1𝑁(𝑄), (76) where the counters in the second line are defined by the terms in the first, while 𝑄^𝑎𝑎_𝑖𝑗=[−(𝑃^𝑎_𝑖−𝑃^𝑎_𝑗)²]^1/2 and 𝑄^𝑎𝑏_𝑖𝑗=[−(𝑃^𝑎_𝑖−𝑃^𝑏_𝑗)²]^1/2 are four-momentum differences between sibling pairs 𝑎𝑎 and event mixing pairs 𝑎𝑏, respectively. It is easy to show that ∫^∞₀𝑑𝑄𝜌_2𝑁(𝑄)=⟨𝑛²⟩_𝑁 and ∫^∞₀𝑑𝑄𝜌_1𝑁⊗𝜌_1𝑁(𝑄)=⟨𝑛⟩²_𝑁 and hence, as before, ∫^∞₀𝑑𝑄𝜅^𝐼_2𝑁(𝑄)=0, which will be true for any correlation whatsoever. The double event averages in the product term 𝜌_1𝑁⊗𝜌_1𝑁(𝑄)=_𝑖,𝑗𝛿𝑄−𝑄^𝑎𝑏_{𝑖𝑗𝑁𝑏𝑁𝑎} (77) are the theoretical foundations of event mixing [53]; the inner 𝑏-average is usually shortened to a smaller “moving average tail” subsample of 𝒮_𝑁.

In third-order, the “GHP average” invariant is defined as the average of three two-momentum differences over all pairs (with or without the √3), 𝑄_𝑎=−𝑝₁−𝑝₂²−𝑝₂−𝑝₃²−𝑝₃−𝑝₁²√3, (78) and it is related to the invariant mass of three pions 𝑀₃=(𝑝₁+𝑝₂+𝑝₃)² by 𝑄_𝑎²=(1/3)𝑀²₃−𝑚². Other “topologies” such as the “GHP max” 𝑄_𝑚=max[−(𝑝₁−𝑝₂)²,−(𝑝₂−𝑝₃)²,−(𝑝₃−𝑝₁)²] 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 𝜅^𝐼_3𝑁𝑄_𝑎=_{𝑖≠𝑗≠𝑘}𝛿𝑄_𝑎−𝑄^𝑎𝑎𝑎_{𝑖𝑘𝑗𝑁𝑎}−3_{𝑖≠𝑗𝑘}𝛿𝑄_𝑎−𝑄^𝑎𝑎𝑏_{𝑖𝑗𝑘𝑁𝑏𝑁𝑎}+𝐺_{3𝑁𝑖,𝑗,𝑘}𝛿𝑄_𝑎−𝑄^𝑎𝑏𝑐_{𝑖𝑗𝑘𝑁𝑐𝑁𝑏𝑁𝑎}, (79) with 𝑄^𝑎𝑏𝑐_𝑖𝑗𝑘=(1/3)[−(𝑃^𝑎_𝑖−𝑃^𝑏_𝑗)²−(𝑃^𝑏_𝑗−𝑃^𝑐_𝑘)²−(𝑃^𝑐_𝑘−𝑃^𝑎_𝑖)²] and similarly for 𝑄^𝑎𝑎𝑎_𝑖𝑘𝑗 and 𝑄^𝑎𝑎𝑏_𝑖𝑘𝑗. Second- and third-order cumulants are normalised by, respectively, 𝐹_2𝑁𝜌_1𝑁⊗𝜌_1𝑁(𝑄)=𝐹_{2𝑁𝑖,𝑗}𝛿𝑄−𝑄^𝑎𝑏_{𝑖𝑗𝑁𝑏𝑁𝑎},𝐹_3𝑁𝜌_1𝑁⊗𝜌_1𝑁⊗𝜌_1𝑁𝑄_𝑎=𝐹_{3𝑁𝑖,𝑗,𝑘}𝛿𝑄_𝑎−𝑄^𝑎𝑏𝑐_{𝑖𝑗𝑘𝑁𝑐𝑁𝑏𝑁𝑎}. (80) After transforming from momenta to 𝑄, the formulae of Section 3.3 become 𝜅^𝐼_2𝑁(𝑄)=𝜌_2𝑁(𝑄)−𝐹_2𝑁𝜌_1𝑁⊗𝜌_1𝑁(𝑄),𝜅^𝐼_3𝑁𝑄_𝑎=𝜌_3𝑁𝑄_𝑎−[3]𝜌_2𝑁⊗𝜌_1𝑁𝑄_𝑎+𝐺_3𝑁𝜌_1𝑁⊗𝜌_1𝑁⊗𝜌_1𝑁𝑄_𝑎, (81)(82) while the normalised cumulants of Section 3.3 become 𝐾^𝐼₂𝑄∣𝒮_𝑁=𝜌_2𝑁(𝑄)𝐹_2𝑁𝜌_1𝑁⊗𝜌_1𝑁(𝑄)𝐾−1,_𝐼3𝑄_𝑎∣𝒮_𝑁=𝜌_3𝑁𝑄_𝑎−[3]𝜌_2𝑁⊗𝜌_1𝑁𝑄_𝑎𝐹_3𝑁𝜌_1𝑁⊗𝜌_1𝑁⊗𝜌_1𝑁𝑄_𝑎𝐹+3^2𝑁𝐹_3𝑁−1. (83)(84)

3.5. Effect of Fixed-𝑁 Correction Factors

To get a feeling for the size of the corrections involved, we measured the correction factors 𝐹_𝑟𝑁 and 𝐺_3𝑁 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 𝑁<10, normalised internal cumulants are hence larger than their Poissonian counterparts but converge to them with increasing 𝑁. Nevertheless, up to 𝑁≃30 corrections of more than 5% for 𝐾^𝐼₂ and more than 20% for 𝐾^𝐼₃ compared to their uncorrected counterparts can be expected. By contrast, the additive correction 𝐺_3𝑁 does not deviate much from the Poissonian limit of 2 except for very small 𝑁. 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 𝑁, would be just 𝑁(𝑁−1)/𝑁² and 𝑁(𝑁−1)(𝑁−2)/𝑁³. In Figure 3, the Poissonian limit corresponds to zero on the 𝑦-axis. It is clear that the fixed-𝑁 factors go some way to correct for the fixed-𝑁 conditioning; the gap between them is approximately determined by ⟨𝑛^𝑟⟩_𝑁/𝑁^𝑟, that is, by the exact definition and outcome space for 𝑛.

3.6. Eliminating Fluctuations in 𝑛

We return briefly to the first choice in Section 3.2.3, that is, 𝛼(𝐩∣𝒮_𝑛𝑁)=𝜌(𝐩∣𝒮_𝑛𝑁)/𝑛 which would permit different physics for each (𝑁,𝑛) combination. If we were willing and able to do analyses for each 𝒮_𝑛𝑁, we would use the fixed-(𝑁,𝑛) equivalent of (68) and (69) derived from 𝜅^ref(𝐩₁,…,𝐩_𝑟∣𝒮_𝑛𝑁)=(−1)^𝑟−1(𝑟−1)!𝑛∏^𝑟_𝑘=1𝜌(𝐩_𝑘∣𝒮_𝑛𝑁), 𝜅^𝐼𝐩₁,𝐩₂∣𝒮_𝑛𝑁=𝜌𝐩₁,𝐩₂∣𝒮_𝑛𝑁−11−𝑛𝜌𝐩₁∣𝒮_𝑛𝑁𝜌𝐩₂∣𝒮_𝑛𝑁,𝜅^𝐼𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑛𝑁=𝜌𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑛𝑁−[3]𝜌𝐩₁,𝐩₂∣𝒮_𝑛𝑁𝜌𝐩₃∣𝒮_𝑛𝑁1+21−𝑛²𝜌𝐩₁∣𝒮_𝑛𝑁×𝜌𝐩₂∣𝒮_𝑛𝑁𝜌𝐩₃∣𝒮_𝑛𝑁, (85) and normalise by 𝜌^ref(𝐩₁,…,𝐩_𝑟∣𝒮_𝑛𝑁)=(𝑛^𝑟/𝑛^𝑟)∏^𝑟_𝑘=1𝜌(𝐩_𝑘|𝒮_𝑛𝑁). 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 𝑛 for fixed 𝑁, 𝜅^SRC𝐩₁,𝐩₂∣𝒮_𝑁=𝜅^𝐼𝐩₁,𝐩₂∣𝒮_𝑛𝑁𝑁,𝜅^SRC𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁=𝜅^𝐼𝐩₁,𝐩₂,𝐩₃∣𝒮_𝑁𝑁, (86) and normalise if necessary by the moment 𝜌^SRC(𝐩₁,…,𝐩_𝑟∣𝒮_𝑁)=⟨(𝑛^𝑟/𝑛^𝑟)∏^𝑟_𝑘=1𝜌(𝐩_𝑘∣𝒮_𝑛𝑁)⟩_𝑁. Given that this involves products of moments in the sub-subsample 𝒮_𝑛𝑁, event mixing would have to be restricted to the same sub-subsamples also, for example 11−𝑛𝜌𝐩₁∣𝒮_𝑛𝑁𝜌𝐩₂∣𝒮_𝑛𝑁𝑁=^𝑁_𝑛=2ℰ_𝑛𝑁ℰ_𝑁11−𝑛1ℰ_𝑛𝑁²×_{𝑎,𝑏∈𝒮𝑛𝑁}𝜌𝐩₁∣𝑛,𝑁,𝐏^𝑎𝜌𝐩₂∣𝑛,𝑁,𝐏^𝑏. (87) 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 𝜌_𝑟(𝑄∣𝒮_𝑁)→𝜌_𝑟𝑁 and 𝜌₁⊗𝜌₁(𝑄∣𝒮_𝑁)→𝜌²_1𝑁 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 var𝜅^𝐼₂𝑄∣𝒮_𝑁=var𝜌_2𝑁−𝐹_2𝑁𝜌²_1𝑁=var𝜌_2𝑁+𝐹²_2𝑁var𝜌²_1𝑁−2𝐹_2𝑁cov𝜌_2𝑁,𝜌²_1𝑁,var𝐾^𝐼₂𝑄∣𝒮_𝑁1=var𝐹_2𝑁𝜌_2𝑁𝜌²_1𝑁=1−1𝐹_2𝑁𝜌_2𝑁𝜌²_1𝑁²×var𝜌_2𝑁𝜌²_2𝑁+var𝜌²_1𝑁𝜌⁴_1𝑁𝜌−2cov_2𝑁,𝜌²_1𝑁𝜌_2𝑁⋅𝜌²_1𝑁, (88)(89)(90) under the assumption that var(𝐹_2𝑁) is much smaller than the other variances, so that 𝐹_2𝑁 can be treated as a constant; this is the case if there are many bins for 𝑄. Similarly, from (69), the variance of the unnormalised internal cumulant is, assuming 𝐺₃ of (70) to be constant, var𝜅^𝐼₃𝑄_𝑎∣𝒮_𝑁=var𝜌_3𝑁−3𝜌_2𝑁𝜌_1𝑁+𝐺₃𝜌³_1𝑁=var𝜌_3𝑁+9var𝜌_2𝑁𝜌_1𝑁+𝐺²₃var𝜌³_1𝑁−6cov𝜌_3𝑁,𝜌_2𝑁𝜌_1𝑁+2𝐺₃cov𝜌_3𝑁,𝜌³_1𝑁−6𝐺₃cov𝜌_2𝑁𝜌_1𝑁,𝜌³_1𝑁, (91)(92) while the normalised version has variance (again assuming var(𝐺₃)≪var𝜌_𝑟) var𝐾^𝐼₃𝑄_𝑎∣𝒮_𝑁𝜌=var^3𝑁−3𝜌_2𝑁𝜌_1𝑁+𝐺₃𝜌³_1𝑁𝐹_3𝑁𝜌³_1𝑁=1𝐹²_3𝑁𝜌var^3𝑁−3𝜌_2𝑁𝜌_1𝑁𝜌³_1𝑁+𝐺₃=𝜌_3𝑁−3𝜌_2𝑁𝜌_1𝑁𝐹_3𝑁𝜌³_1𝑁²×var𝜌_3𝑁−3𝜌_2𝑁𝜌_1𝑁𝜌_3𝑁−3𝜌_2𝑁𝜌_1𝑁²+var𝜌³_1𝑁𝜌⁶_1𝑁−2cov𝜌_3𝑁−3𝜌_2𝑁𝜌_1𝑁,𝜌³_1𝑁𝜌_3𝑁−3𝜌_2𝑁𝜌_1𝑁𝜌³_1𝑁=𝜌_3𝑁−3𝜌_2𝑁𝜌_1𝑁𝐹_3𝑁𝜌³_1𝑁²×var𝜌_3𝑁+9var𝜌_2𝑁𝜌_1𝑁−6cov𝜌_3𝑁,𝜌_2𝑁𝜌_1𝑁𝜌_3𝑁−3𝜌_2𝑁𝜌_1𝑁²+var𝜌³_1𝑁𝜌⁶_1𝑁+6cov𝜌_2𝑁𝜌_1𝑁,𝜌³_1𝑁−2cov𝜌_3𝑁,𝜌³_1𝑁𝜌_3𝑁−3𝜌_2𝑁𝜌_1𝑁𝜌³_1𝑁. (93)(94) The unnormalised cumulants (81) and (82) and their variances (88) and (92) require knowledge of the multiplicity factorial moments 𝐹_2𝑁,𝐹_3𝑁, 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 𝑓(𝐩) of the momenta, the theoretical expectation value 𝐸[𝑓] is defined as the integral over the entire outcome space Ω of 𝑓 weighted by a “parent distribution” 𝑃(𝐩), an abstract entity supposedly containing everything there is to know on this level, 𝐸𝑓(𝐩)=_Ω𝑑𝐩𝑃(𝐩)𝑓(𝐩). (95) Purely theoretical concepts such as 𝑃(𝐩) and 𝐸[𝑓] 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 𝑥,𝑦 are statistically independent, 𝐸[𝑥𝑦]=∫𝑑𝑥𝑑𝑦𝑃(𝑥,𝑦)𝑥𝑦=𝐸[𝑥]𝐸[𝑦].

Expectation values for pairwise variables such as the four-momentum difference 𝑄 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: 𝐶₂(𝑄)≡𝐾₂(𝑄)𝜌+1=^sibling2(𝑄)𝜌^reference₂(𝑄)=𝜌𝑄^𝑎𝑎𝜌𝑄^𝑎𝑏, (96) which is a function of two entirely different quantities: the four-momentum differences 𝑄^𝑎𝑎 of “sibling” tracks taken from the same event 𝑎 and one constructed from the mixed-event sample using tracks from different events, written as 𝑄^𝑎𝑏, 𝑄^𝑏𝑐, and so forth. For second-order correlations, the parent distribution is therefore necessarily a two-variable probability⁴  𝑃(𝑄^𝑎𝑎,𝑄^𝑏𝑐) which, depending on whether the cases 𝑏=𝑎 and 𝑐=𝑎 occur, may or may not factorise into “sibling” and “mixed” marginal probabilities 𝑃𝑄^𝑎𝑎,𝑄^𝑏𝑐=𝑃_𝑠𝑄^𝑎𝑎𝑃_𝑚𝑄^𝑏𝑐iff𝑎≠𝑏≠𝑐, (97) but (unless 𝑎=𝑏=𝑐), the marginals will always be 𝑃_𝑠𝑄^𝑎𝑎=𝑑𝑄^𝑏𝑐𝑃𝑄^𝑎𝑎,𝑄^𝑏𝑐,𝑃_𝑚𝑄^𝑏𝑐=𝑑𝑄^𝑎𝑎𝑃𝑄^𝑎𝑎,𝑄^𝑏𝑐. (98) The shapes of 𝑃_𝑠(𝑄) and 𝑃_𝑚(𝑄) 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-𝑁, or fixed-𝑛 cases) as 𝐸𝜌𝑄^𝑎𝑎=_Ω𝑑𝑄^𝑎𝑎𝑑𝑄^𝑏𝑐𝑃𝑄^𝑎𝑎,𝑄^𝑏𝑐𝜌_𝑎𝑎=_{𝑖≠𝑗Ω}𝑑𝑄^𝑎𝑎𝑃_𝑠𝑄^𝑎𝑎𝛿𝑄^𝑎𝑎−𝑄^𝑎𝑎_𝑖𝑗=_𝑖≠𝑗𝑃_𝑠𝑄^𝑎𝑎_𝑖𝑗,𝐸𝜌𝑄^𝑏𝑐=_Ω𝑑𝑄^𝑎𝑎𝑑𝑄^𝑏𝑐𝑃𝑄^𝑎𝑎,𝑄^𝑏𝑐𝜌_𝑏𝑐=_𝑖,𝑗Ω𝑑Q^𝑏𝑐𝑃_𝑚𝑄^𝑏𝑐𝛿𝑄^𝑏𝑐−𝑄^𝑏𝑐_𝑖𝑗=_𝑖,𝑗𝑃_𝑚𝑄^𝑏𝑐_𝑖𝑗. (99) Later, we will meet expectation values for cases such as𝑎=𝑐, 𝐸𝜌𝑄^𝑎𝑎𝜌𝑄^𝑎𝑏=_{𝑖≠𝑗𝑘,ℓΩ}𝑑𝑄^𝑎𝑎𝑑𝑄^𝑎𝑏𝑃𝑄^𝑎𝑎,𝑄^𝑎𝑏×𝛿𝑄^𝑎𝑎−𝑄^𝑎𝑎_𝑖𝑗𝛿𝑄^𝑎𝑏−𝑄^𝑎𝑏_𝑘ℓ=_{𝑖≠𝑗𝑘,ℓ}𝑃𝑄^𝑎𝑎_𝑖𝑗,𝑄^𝑎𝑏_𝑘ℓ, (100) 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 𝑖,𝑗,𝑘, and ℓ𝑃𝑄^𝑎𝑎_𝑖𝑗,𝑄^𝑏𝑐_𝑘ℓ=𝑃𝑄^𝑎𝑎,𝑄^𝑏𝑐∀𝑖,𝑗,𝑘,ℓ (101) and similarly 𝑃_𝑠(𝑄^𝑎𝑎_𝑖𝑗)=𝑃_𝑠(𝑄^𝑎𝑎) and 𝑃_𝑚(𝑄^𝑏𝑐_𝑘ℓ)=𝑃_𝑚(𝑄^𝑏𝑐). For the event-averaged counters, this results in 𝐸𝜌_𝑎𝑎(𝑄)=𝑛²_𝑎N𝑃_𝑠𝑄^𝑎𝑎,𝐸𝜌_𝑏𝑐(𝑄)=𝑛_𝑏𝑁𝑛_𝑐𝑁𝑃_𝑚𝑄^𝑏𝑐, (102) or in terms of the notation of Section 3.4, 𝐸𝜌_2𝑁(𝑄)=𝑛²_𝑁𝑃_𝑠𝑄^𝑎𝑎,𝐸𝜌_1𝑁⊗𝜌_1𝑁(𝑄)=⟨𝑛⟩²_𝑁𝑃_𝑚𝑄^𝑏𝑐. (103) As mentioned, we do not need the factorisation (97) of 𝑃(𝑄^𝑎𝑎,𝑄^𝑏𝑐) as long as we keep careful track of the equal-event-indices cases. Whenever 𝑎≠𝑏 or 𝑎≠𝑐, independence of the events ensures that expectation values of products of any functions 𝑓(𝑄^𝑎𝑎) and 𝑔(𝑄^𝑎𝑏) of the pair variables do factorise, 𝐸𝑓𝑄^𝑎𝑎𝑔𝑄^𝑎𝑏=𝐸𝑓𝑄^𝑎𝑎𝐸𝑔𝑄^𝑎𝑏𝑎≠𝑏. (104) For third-order correlations, the parent distribution is a function of three different variables 𝑄^𝑎𝑎𝑎, 𝑄^𝑏𝑏𝑐, and 𝑄^𝑑𝑒𝑓 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 𝐸[⋅] 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 𝑥 be any generic eventwise counter or any other eventwise statistic. Since the formulae in this section remain true for inclusive and fixed-𝑁 samples, we omit any notation related to 𝑁 in this derivation. In this simplified notation, the well-known standard error of the sample mean ⟨𝑥⟩ is given by (simplifying ℰ−1→ℰ) 𝜎(⟨𝑥⟩)=var(⟨𝑥⟩)=1ℰ𝑥²−⟨𝑥⟩², (105) which follows from the combinatorics of equal and unequal event indices by the above artificial use of expectation values, reverting from expectation values 𝐸[⋅] to sample means ⟨⋅⟩ in the last step: var(⟨𝑥⟩)=𝐸⟨𝑥⟩²−𝐸[⟨𝑥⟩]²=1ℰ²_𝑎,𝑏𝐸𝑥_𝑎𝑥_𝑏−𝐸𝑥_𝑎𝐸𝑥_𝑏=1ℰ²_𝑎=𝑏𝐸𝑥_𝑎𝑥_𝑏−𝐸𝑥_𝑎𝐸𝑥_𝑏+1ℰ²_𝑎≠𝑏𝐸𝑥_𝑎𝐸𝑥_𝑏−𝐸𝑥_𝑎𝐸𝑥_𝑏=1ℰ²_𝑎𝐸𝑥²−𝐸[𝑥]²1+0=ℰ𝑥²−⟨𝑥⟩², (106)(107) where we have used the fact that 𝐸[𝑥_𝑎𝑥_𝑏]=𝐸[𝑥_𝑎]𝐸[𝑥_𝑏] for all 𝑎≠𝑏 and assumed that all 𝑥 are identically distributed, 𝐸[𝑥_𝑎]=𝐸[𝑥] for all 𝑎. Equality or inequality of event indices is thus crucial. We will follow the same approach below, keeping careful track of equal and unequal event indices, factorising expectation values for unequal event indices, and reverting to sample means in the last step.

4.4. Variances and Covariances for Multiple Event Averages

4.4.1. Statistical Errors for Second-Order Cumulants

According to (88)–(94), we must handle variances and covariances of products of several event averages. To derive these, we will use the following shortened notation: letting 𝛿^𝑎𝑏_𝑖𝑗≡𝛿(𝑄−𝑄^𝑎𝑏_𝑖𝑗) and so forth, then 𝜌_𝑎𝑎=∑_𝑖≠𝑗𝛿(𝑄−𝑄^𝑎𝑎_𝑖𝑗)=∑_𝑖≠𝑗𝛿^𝑎𝑎_𝑖𝑗 is the eventwise pair counter for event 𝑎, while 𝜌_𝑏𝑐=∑_𝑖,𝑗𝛿(𝑄−𝑄^𝑏𝑐_𝑖𝑗)=∑_𝑖,𝑗𝛿^𝑏𝑐_𝑖𝑗 is the mixed-event counter of events 𝑏 and 𝑐 (with 𝑏≠𝑐≠𝑎 assumed), so that 𝜌₂(𝑄)=⟨𝜌_𝑎𝑎⟩_𝑎 while 𝜌₁⊗𝜌₁(𝑄)=⟨𝜌_𝑎𝑏⟩_𝑎𝑏 is a double event average. We reserve the event index 𝑎 for the “sibling” event whose correlations are currently being analysed and use indices 𝑏,𝑐,…,𝑡,𝑢,𝑣,𝑤,… for events entering the event-mixing parts.⁵ All quantities are assumed to be measured within a particular subsample 𝒮_𝑁, but we omit the 𝑁-subscript and the argument. The event-index subscripts such as ⟨⋅⟩_𝑏𝑐 above are included or omitted depending on whether they convey relevant information on the specific averaging.

In this notation, the method that led to (107) reads var𝜌₂=var𝜌_𝑎𝑎𝑎=1ℰ²_𝑎,𝑏𝐸𝜌_𝑎𝑎𝜌_𝑏𝑏−𝐸𝜌_𝑎𝑎𝐸𝜌_𝑏𝑏=1ℰ𝜌_𝑎𝑎²−𝜌_𝑎𝑎². (108) The same method of disentangling the combinatorics of equal and unequal event indices is applied consistently to all variances and covariances below. The 𝜌²₁ term in second-order cumulants has variance var𝜌₁⊗𝜌₁=var𝜌_{𝑏𝑐𝑏𝑐}=1ℰ²²_{𝑏≠𝑐𝑑≠𝑒}𝐸𝜌_𝑏𝑐𝜌_𝑑𝑒−𝐸𝜌_𝑏𝑐𝐸𝜌_𝑑𝑒. (109) The case 𝑏≠𝑐≠𝑑≠𝑒 yields zero, but the cases 𝑏=𝑑≠𝑐≠𝑒 and three other equivalent combinations yield var𝜌₁⊗𝜌₁=4ℰ³ℰ²²𝐸𝜌_𝑏𝑐𝜌_𝑏𝑒−𝐸𝜌_𝑏𝑐𝐸𝜌_𝑏𝑒⟶4ℰ³ℰ²²𝜌_𝑏𝑐𝜌_{𝑏𝑒𝑏𝑐e}−𝜌_𝑏𝑐𝜌_𝑏𝑒⟶4ℰ𝜌_𝑏𝑐𝜌_𝑏𝑒−𝜌₁⊗𝜌₁², (110) where in the second step we reverted 𝐸[⋅]→⟨⋅⟩ and in the third⁶ assumed ℰ≫1. Note that the requirement 𝑏≠𝑐≠𝑒 implies that ⟨𝜌_𝑏𝑐𝜌_𝑏𝑒⟩_𝑏𝑐𝑒=⟨⟨∑_𝑖≠𝑗𝛿^𝑏𝑐_𝑖𝑗⟩_𝑐⟨∑_𝑘≠𝑙𝛿^𝑏𝑒_𝑘ℓ⟩_𝑒⟩_𝑏 cannot be simplified to the square of a single counter ⟨⟨∑_𝑖≠𝑗𝛿^𝑏𝑐_𝑖𝑗⟩²_𝑐⟩_𝑏: the event mixing involves three different events, not two. Secondly, the combinations of two equalities 𝑏=𝑑≠𝑐=𝑒 and 𝑏=𝑒≠𝑐=𝑑 in (109) yield another term of order ℰ⁻², 2ℰ²𝜌_𝑏𝑑𝜌_{𝑏𝑑𝑏𝑑}−𝜌₁⊗𝜌₁², (111) which we can safely neglect when ⟨𝑛²⟩/ℰ≪1 except when there are few bins or large multiplicities even in small bins. It is worth emphasising that the extra factor 4 which appears in (110) arises from the same method that has been used for decades to justify use of (105). We find, by the same method, that the covariance between 𝜌₂ and 𝜌₁⊗𝜌₁ is given by cov𝜌₂,𝜌₁⊗𝜌₁=cov𝜌_𝑑𝑑,𝜌_𝑏𝑐=1ℰℰ²_{𝑑𝑏≠𝑐}𝐸𝜌_𝑑𝑑𝜌_𝑏𝑐−𝐸𝜌_𝑑𝑑𝐸𝜌_𝑏𝑐=2ℰ𝜌_𝑑𝑑𝜌_𝑑𝑏−𝜌_𝑑𝑑𝜌_𝑑𝑐=2ℰ𝜌_𝑑𝑑𝜌_𝑑𝑐−𝜌₂𝜌₁⊗𝜌₁, (112) so that we must, in addition, accumulate, for every event 𝑑, the product of the counters 𝜌_𝑑𝑑𝜌_𝑑𝑐=_𝑖≠𝑗𝛿^𝑑𝑑_{𝑖𝑗𝑘ℓ}𝛿^𝑑𝑐_𝑘ℓ𝑐. (113) Note that there is no restriction on track indices 𝑘≠𝑖 or 𝑘≠𝑗 in the 𝑑-event, meaning that events with 𝑛(𝑎)=2 contribute to this counter which would otherwise not be the case. Combining these, we find, to leading order in ℰ⁻¹ and renaming mixed-event indices, var𝜅^𝐼₂=1ℰ𝜌_𝑎𝑎−2𝐹_2𝑁𝜌_𝑎𝑐𝜌_𝑎𝑎−2𝐹_2𝑁𝜌_𝑎𝑑𝑎−𝜌₂−2𝐹_2𝑁𝜌₁⊗𝜌₁², (114) with all event indices strictly unequal and 𝑐- and 𝑑 event averages understood where appropriate.⁷ Contrasting this with the traditional way to calculate the same variance, var𝜅^𝐼₂=1ℰ𝜌_𝑎𝑎−𝐹_2𝑁𝜌_𝑎𝑐𝜌_𝑎𝑎−𝐹_2𝑁𝜌_𝑎𝑑−𝜌₂−𝐹_2𝑁𝜌₁⊗𝜌₁², (115) it is clear that in previous analyses the two possible ways to set 𝑎 equal to 𝑏 or 𝑐 were overlooked, while normal (non-internal) cumulants also omit the 𝐹_2𝑁.

4.4.2. Statistical Errors for Third-Order Cumulants

In third-order, we will need 𝛿^𝑎𝑏𝑐_𝑖𝑗𝑘≡𝛿(𝑄_𝑎−𝑄^𝑎𝑏𝑐_𝑖𝑗𝑘) and similar quantities and the notation for counters 𝜌_𝑎𝑎𝑎, 𝜌_𝑎𝑎𝑏, and 𝜌_𝑎𝑏𝑐 corresponding to the event averages 𝜌₃(𝑄_𝑎)=⟨𝜌_𝑎𝑎𝑎⟩, 𝜌₂⊗𝜌₁(𝑄_𝑎)=⟨𝜌_𝑎𝑎𝑏⟩, and 𝜌₁⊗𝜌₁⊗𝜌₁(𝑄_𝑎)=⟨𝜌_𝑎𝑏𝑐⟩, respectively. Clearly, 𝑎≠𝑏≠𝑐 must hold in the third-order case. We obtain for the necessary third-order quantities (shuffling and/or renaming indices if necessary) var𝜌₃=1ℰ𝜌_𝑎𝑎𝑎²_𝑎−𝜌²₃cov𝜌₃,𝜌₂⊗𝜌₁=1ℰℰ²_{𝑟𝑠≠𝑡}𝐸𝜌_𝑟𝑟𝑟𝜌_𝑠𝑠𝑡−𝐸𝜌_𝑟𝑟𝑟𝐸𝜌_𝑠𝑠𝑡=1ℰ𝜌_𝑟𝑟𝑟𝜌_𝑟𝑟𝑠+𝜌_𝑟𝑟𝑟𝜌_𝑟𝑠𝑠−2𝜌₃𝜌₂⊗𝜌₁, (116) with 𝜌_𝑟𝑟𝑟𝜌_𝑟𝑟𝑠=∑_{𝑖≠𝑗≠𝑘}𝛿^𝑟𝑟𝑟_𝑖𝑗𝑘∑_ℓ≠𝑚⟨∑_𝑛𝛿^𝑟𝑟𝑠_ℓ𝑚𝑛⟩_𝑠 and 𝜌_𝑟𝑟𝑟𝜌_𝑟𝑠𝑠=∑_{𝑖≠𝑗≠𝑘}𝛿^𝑟𝑟𝑟_𝑖𝑗𝑘∑_ℓ⟨∑_𝑚≠𝑛𝛿^𝑟𝑠𝑠_ℓ𝑚𝑛⟩_𝑠. The remaining variances and covariances needed for third-order correlations with GHP topology are, after renaming of indices, var𝜌₂⊗𝜌₁=ℰ³ℰ^{22𝑔≠𝑒}_𝑐≠𝑑𝐸𝜌_𝑔𝑔𝑒𝜌_𝑐𝑐𝑑−𝐸𝜌_𝑔𝑔𝑒𝐸𝜌_𝑐𝑐𝑑=ℰ³ℰ²²𝜌_𝑔𝑔𝑑𝜌_𝑔𝑔𝑐+𝜌_𝑔𝑐𝑐+𝜌_𝑑𝑑𝑐+𝜌_𝑑𝑐𝑐−4𝜌₂⊗𝜌₁², (117) while we neglect 1ℰ²𝜌_𝑔𝑔𝑐𝜌_𝑔𝑔𝑐+𝜌_𝑔𝑔𝑐𝜌_𝑔𝑐𝑐−2𝜌₂⊗𝜌₁². (118) The next term is simpler, cov𝜌₃,𝜌³₁=1ℰℰ³_{𝑡𝑢≠𝑣≠𝑤}𝐸𝜌_𝑡𝑡𝑡𝜌_𝑢𝑣𝑤−𝐸𝜌_𝑡𝑡𝑡𝐸𝜌_𝑢𝑣𝑤=3ℰ𝜌_𝑡𝑡𝑡𝜌_𝑡𝑢𝑣−𝜌₃𝜌³₁, (119) but the following is not, cov𝜌₂⊗𝜌₁,𝜌³₁=1ℰ²ℰ³_{𝑢≠𝑣𝑥≠𝑦≠𝑧}𝐸𝜌_𝑢𝑢𝑣𝜌_𝑥𝑦𝑧−𝐸𝜌_𝑢𝑢𝑣𝐸𝜌_𝑥𝑦𝑧=3ℰ⁴ℰ²ℰ³𝜌_𝑤𝑤𝑥𝜌_𝑤𝑦𝑧+𝜌_𝑤𝑥𝑥𝜌_𝑤𝑦𝑧−2𝜌₂𝜌₁𝜌³₁+6ℰ²𝜌_𝑤𝑤𝑥𝜌_𝑤𝑥𝑦−𝜌₂𝜌₁𝜌³₁, (120) and the large number of combinations makes the variance of 𝜌³₁ particularly complicated, var𝜌₁⊗𝜌₁⊗𝜌₁=1ℰ³²_{𝑑≠𝑒≠ℎ𝑏≠𝑝≠𝑞}𝐸𝜌_𝑑𝑒ℎ𝜌_𝑏𝑝𝑞−𝐸𝜌_𝑑𝑒ℎ𝐸𝜌_𝑏𝑝𝑞=9ℰ⁵ℰ³²𝜌_𝑏𝑒ℎ𝜌_𝑏𝑝𝑞−𝜌³₁𝜌³₁+18ℰ⁴ℰ³²𝜌_𝑏𝑒ℎ𝜌_𝑏𝑒𝑞−𝜌³₁𝜌³₁+6ℰ³𝜌_𝑏𝑒ℎ𝜌_𝑏𝑒ℎ−𝜌³₁𝜌³₁. (121) For large ℰ, the leading order terms will usually dominate, so that we can neglect the subleading terms.⁸ To leading order, we therefore obtain after substitution in (91) and again omitting brackets for non-𝑎 event averages var𝜅^𝐼₃=1ℰ𝜌²_𝑎𝑎𝑎+9𝜌_𝑎𝑎𝑏𝜌_𝑎𝑎𝑐+𝜌_𝑎𝑐𝑐+𝜌_𝑏𝑏𝑐+𝜌_𝑏𝑐𝑐+9𝐺²₃𝜌_𝑎𝑎𝑎𝜌_𝑎𝑏𝑐𝜌_𝑎𝑑𝑒−6𝜌_𝑎𝑎𝑎𝜌_𝑎𝑎𝑏+𝜌_𝑎𝑏𝑏+6𝐺₃𝜌_𝑎𝑎𝑎𝜌_𝑎𝑏𝑐−18𝐺₃𝜌_𝑎𝑎𝑏𝜌_𝑎𝑐𝑑+𝜌_{𝑏𝑐𝑑𝑎}−𝜌²₃+36𝜌₂⊗𝜌₁²+9𝐺²₃𝜌³₁²−12𝜌₃𝜌₂⊗𝜌₁+6𝐺₃𝜌₃𝜌₁³−36𝐺²₃𝜌₂⊗𝜌₁𝜌₁³,=1ℰ𝜌_𝑎𝑎𝑎−3𝜌_𝑎𝑎𝑏−3𝜌_𝑎𝑏𝑏+3𝐺₃𝜌_𝑎𝑏𝑐×(𝜌_𝑎𝑎𝑎−3𝜌_𝑎𝑎𝑑−3𝜌_𝑎𝑑𝑑+3𝐺₃𝜌_𝑎𝑑𝑒)_𝑎−𝜌₃−6𝜌₂⊗𝜌₁+3𝐺₃(𝜌₁)³². (122) While the factorised form is again instructive, it cannot be calculated in this form within the 𝑎-loop in the analysis since the 𝐺_3𝑁 constants are known only on completion of the entire sample analysis. Rather, the full palette of product counters 𝜌𝜌 has to be accumulated and averaged and combined only in the final phase of the analysis.

5. Averaged Internal Cumulants

As 𝑁 is only an approximation for the true total event multiplicity anyway, and for cases of small sample statistics, it may be necessary or desirable to group subsamples of fixed 𝑁 into multiplicity classes 𝑁∈[𝐴,𝐵]. It is important, however, not to simply lump all events within this multiplicity class into a single “half-inclusive” subsample, because, as has long been known [15], that results in terms entering the cumulants which arise solely to “multiplicity mixing” (MM) of events of different 𝑁. Given the arbitrary choice of [𝐴,𝐵], such MM correlations are spurious and avoided in favour of “Averaged-Internal” (AI) correlations⁹ defined as follows. Using the renormalised multiplicity distribution ℛ^′_𝑁=ℛ_𝑁∑^𝐵_𝑁=𝐴ℛ_𝑁=ℰ_𝑁∑^𝐵_𝑁=𝐴ℰ_𝑁, (123) the AI unnormalised cumulants, reference distributions, and normalised AI cumulants are 𝜅^AI₂𝑄∣𝒮_𝐴𝐵=^𝐵_𝑁=𝐴ℛ^′_𝑁𝜅^𝐼₂𝑄∣𝒮_𝑁,𝜌₁⊗𝜌₁𝑄∣𝒮_𝐴𝐵=^𝐵_𝑁=𝐴ℛ^′_𝑁𝐹_2𝑁𝜌₁⊗𝜌₁𝑄∣𝒮_𝑁,𝐾^AI₂𝑄∣𝒮_𝐴𝐵=𝜅^AI₂𝑄∣𝒮_𝐴𝐵𝜌₁⊗𝜌₁𝑄∣𝒮_𝐴𝐵=∑^𝐵_𝑁=𝐴ℛ^′_𝑁𝜌₂𝑄∣𝒮_𝑁∑^𝐵_𝑁=𝐴ℛ^′_𝑁𝐹_2𝑁𝜌₁⊗𝜌₁𝑄∣𝒮_𝑁−1. (124)(125)(126) Note that the correction factors 𝐹_2𝑁, which are normalised factorial moments of 𝑛 for fixed 𝑁, are part of the summed normalisations.¹⁰ In third-order, we have correspondingly 𝜅^AI₃𝑄_𝑎∣𝒮_𝐴𝐵=^𝐵_𝑁=𝐴ℛ^′_𝑁𝜅^𝐼₃𝑄_𝑎∣𝒮_𝑁,𝜌₁⊗𝜌₁⊗𝜌₁𝑄_𝑎∣𝒮_𝐴𝐵=^𝐵_𝑁=𝐴ℛ^′_𝑁𝐹_3𝑁𝜌₁⊗𝜌₁⊗𝜌₁𝑄_𝑎∣𝒮_𝑁,𝐾^AI₃𝑄_𝑎∣𝒮_𝐴𝐵=𝜅^AI₃𝑄_𝑎∣𝒮_𝐴𝐵𝜌₁⊗𝜌₁⊗𝜌₁𝑄_𝑎∣𝒮_𝐴𝐵. (127)(128)(129) Note also that (128) holds for the normalisation only and not for the last term in 𝜅^𝐼₃, which is (3𝐹_2𝑁−𝐹_3𝑁)𝜌₁⊗𝜌₁⊗𝜌₁— but that is already taken care of in the formula (69) for 𝜅^𝐼₃ itself. Expressions for an inclusive (all-𝑁) multiplicity summation of internal cumulants are obtained from the above by setting 𝐴=0 and 𝐵=∞. The AI (Averaged Internal) correlations (124) and (127) represent refined versions of what has traditionally been termed “Short-Range Correlations,” differing from the original formulae [15, 17] by the 𝐹_2𝑁 and 𝐺_3𝑁 factors, respectively. This was originally pointed out in [18] but only for multinomials in 𝑁.

Regarding variances and standard errors for AI correlations, we first note that, since subsamples 𝒮_𝑁 are strictly mutually independent, a variance over the [𝐴,𝐵] range is simply the weighted sum of the corresponding fixed-𝑁 variances. From (124) and (129), we have for all orders 𝑟𝜅^AI_𝑟𝑄∣𝒮_𝐴𝐵=^𝐵_𝑁=𝐴ℛ^′_𝑁𝜅^𝐼_𝑟𝑄∣𝒮_𝑁𝑟=2,3,…, (130) and given the independence of any functions 𝑓 and 𝑔 of different multiplicity subsamples, 𝐸[𝑓(𝒮_𝑁)⋅𝑔(𝒮_𝑁′)]=𝐸[𝑓(𝒮_𝑁)]⋅𝐸[𝑔(𝒮_𝑁′)] for all 𝑁≠𝑁^′, we conclude that var𝜅^AI_𝑟𝑄∣𝒮_𝐴𝐵=_𝑁ℛ^′_𝑁²var𝜅^𝐼_𝑟𝑄∣𝒮_𝑁,var𝜌^𝑟₁𝑄∣𝒮_𝐴𝐵=_𝑁ℛ^′_𝑁²𝐹_𝑟𝑁²var𝜌^𝑟₁𝑄∣𝒮_𝑁,cov𝜅^AI_𝑟𝑄∣𝒮_𝐴𝐵,𝜌^𝑟₁𝑄∣𝒮_𝐴𝐵=_𝑁ℛ^′_𝑁²𝐹_𝑟𝑁cov𝜅^𝐼_𝑟𝑄∣𝒮_𝑁,𝜌^𝑟₁𝑄∣𝒮_𝑁, (131)(132)(133) which are known functions in terms of Sections 4.4.1 and 4.4.2, while for the normalised cumulants in [𝐴,𝐵], covariances between numerator and denominator are (omitting the 𝑄) cov𝜅^𝐼₂𝒮_𝑁,𝜌²₁𝒮_𝑁=cov𝜌₂𝒮_𝑁,𝜌₁⊗𝜌₁𝒮_𝑁−𝐹_2𝑁var𝜌₁⊗𝜌₁𝒮_N,cov𝜅^𝐼₃𝒮_𝑁,𝜌³₁𝒮_𝑁=cov𝜌₃𝒮_𝑁,𝜌³₁𝒮_𝑁−3cov𝜌₂⊗𝜌₁𝒮_𝑁,𝜌³₁𝒮_𝑁+𝐺₃var𝜌³₁𝒮_𝑁, (134) so that the normalised range cumulants have variances var𝐾^AI_𝑟𝑄∣𝒮_𝐴𝐵=𝐾^AI_𝑟²⋅var𝜅_𝑟𝒮_𝐴𝐵𝜅_𝑟𝒮_𝐴𝐵²+var𝜌^𝑟₁𝒮_𝐴𝐵𝜌^𝑟₁𝒮_𝐴𝐵²𝜅−2cov_𝑟𝒮_𝐴𝐵,𝜌^𝑟₁𝒮_𝐴𝐵𝜅_𝑟𝒮_𝐴𝐵⋅𝜌^𝑟₁𝒮_𝐴𝐵, (135) and standard errors are given by¹¹𝜎𝐾^AI_𝑟𝒮_𝐴𝐵=var𝐾^AI_𝑟𝑄∣𝒮_𝐴𝐵. (136)

6. Event Mixing Algorithms

“Event mixing” [72] is widely used to simulate uncorrelated or semicorrelated quantities such as 𝜌³₁ and 𝜌₂⊗𝜌₁. The idea has always been to use the experimental sample at hand to simulate the baseline of independence referred to in Section 3.1 in such a way that criteria 2 (independence of momenta), 4 (reproducing the one-particle momentum space distribution), and 6 (normalisation) are all addressed simultaneously. Ideally, all effects bar the desired correlation are elegantly removed in this way.

For the internal cumulants and their variances and covariances derived above, event mixing requires keeping track of counters of all orders in each of the subsamples 𝒮_𝑁. A count of event indices in Section 4 shows that in a brute-force calculation we would need, for each subsample, a minimum of five independent event averages or 𝑂(ℰ⁵) event combinations; furthermore, caution would advise not to use the same event in calculating related counters, so that selection and use of more than five events in mixing are advisable. The resulting excessive number of full event averages, mixing every (sub) sample event with every other one, is therefore not feasible.

If the order of events in the sample is random, the multiple event averages can be simplified by the use of the following multiple event buffer algorithm. (1)A single overall event loop equivalent to the event index 𝑎 runs over the entire inclusive sample 𝒮. A given event 𝑎 will have a multiplicity 𝑁=𝑁(𝑎), so for that particular 𝑎 correlation analysis for subsample 𝒮_𝑁 is advanced by one event while the others remain dormant.¹² In this way, 𝑎, which always refers to the sibling event, runs over all ℰ_𝑁 events of every subsample 𝒮_𝑁. There is no need to either explicitly sort the inclusive sample into subsamples or to run multiple jobs for fixed 𝑁.(2)The first ℰ_𝐵 events¹³ of a given multiplicity 𝑁 are used solely to fill up the buffer without doing any analysis. Once a given buffer has been filled, event mixing analysis proceeds for that subsample as follows.(a) A newly read 𝑎-event is assigned to the 𝑁=𝑁(𝑎) buffer, the earliest event in that buffer is discarded, and sums for averages entering 𝐹_2𝑁 and 𝐹_3𝑁 as well as the sibling counters 𝜌_𝑎𝑎, 𝜌_𝑎𝑎, (𝜌_𝑎𝑎)², and (𝜌_𝑎𝑎𝑎)² are updated.(b) Event combinations for mixed-event counters are built up by picking any one of the ℰ_𝐵−1 other events in that buffer and calling it 𝑏, thereafter picking any one of the remaining ℰ_𝐵−2 events in the buffer, calling it 𝑐 and so on. While for third-order, only five events (including the sibling event) are needed to construct all the counters required; in practice it is better to use different mixing events for different counters to root out even traces of unwanted correlations between different mixing counters. The random selection of events rather than tracks for mixing is necessary to ensure that more than one track per event can be used as required for counters of Sections 3 and 4 such as 𝜌_𝑏𝑏𝑐.(c) For a given event set 𝑏,𝑐,𝑑,…, the mixing counters are incremented using all possible combinations of the 𝑛_𝑎 tracks in event 𝑎 together with all the 𝑛_𝑏,𝑛_𝑐,… tracks in the selected events 𝑏,𝑐,… mixing events. For example, 𝜌_𝑏𝑏𝑐 would use all possible 𝑛_𝑏(𝑛_𝑏−1) pairs of 𝑏-tracks¹⁴ together with all possible 𝑛_𝑐 single 𝑐-tracks. The mixing of all tracks of a given event rather than just selected ones ensures that the fluctuations in 𝑛 for given fixed 𝑁 are automatically contained in the counters.(d) For constant 𝑎, the process of selecting events 𝑏,𝑐,… is repeated 𝐶_mix= 10–100 times to reduce the statistical errors, avoiding events that have been used in previous selections. Efficiency can be maximised by tuning of both the number of events ℰ_𝐵 stored in each buffer and by the number of resamples 𝐶_mix.(3)Once the entire sample has been processed via the 𝑎-event loop, the 𝑏- 𝑐- 𝑑-event averages are normalised by 𝐶_mix(ℰ_𝑁−ℰ_𝐵), while the primary 𝑎-average is normalised by (ℰ_𝑁−ℰ_𝐵).(4)Unnormalised and normalised correlation quantities, their standard errors, and correction factors are constructed by appropriate combinations of averaged counters.(5)Results from fixed-𝑁 subsamples can then be combined into AI correlations over partial ranges of 𝑁 or the entire inclusive sample at the end of the event loop using the methods outlined in Section 5.

7. Discussion and Conclusions

(1) Correlations are only defined properly if the null case or reference distribution is defined on the same level of sophistication as the correlation itself. Translated into statistics, the six criteria set out in Section 3.1 for a reference sample for correlations at fixed charged multiplicity 𝑁 lead straight to the definition of the reference sample as the average of multinomials given in (51), weighted by the conditional multiplicity distribution ℛ_𝑛𝑁. Assigning Bernoulli probabilities 𝛼(𝐩∣𝒮_𝑛𝑁)=⟨𝜌(𝐩∣𝒮_𝑛𝑁)⟩_𝑁/⟨𝑛⟩_𝑁 yields the reference density (56) and generating functional (58).

(2) From the theorem that internal cumulants are given by the difference between measured and reference cumulants, we obtain normalised and unnormalised internal cumulants which satisfy every stated criterion for proper correlations for fixed-𝑁 samples.

(3) We have highlighted the distinction between 𝑛, the particles entering the correlation analysis itself, and 𝑁, the particles determining the event selection criterion for a particular semi-inclusive subsample. Various correction factors are shown to be fair to good approximations of these exact results in some cases but far off the mark in others. Surprisingly, normalised cumulants are far more sensitive to these corrections, through the normalisation prefactor, than their unnormalised counterparts. To belabour the point, for any variables (𝑥₁,𝑥₂), different definitions for correction factor 1/𝐹 for fixed-𝑁 correlations of positive pions, 𝐾₂𝑥₁,𝑥₂∣𝒮_𝑁=1𝐹𝜌𝑥₁,𝑥₂∣𝒮_𝑁𝜌𝑥₁∣𝒮_𝑁𝜌𝑥₂∣𝒮_𝑁−1, (137) can be very important at low multiplicities, with 𝐹=1 (Poisson) being the worst approximation, 𝐹=𝑁(𝑁−1)/𝑁² being a fair one, and ⟨𝑛(𝑛−1)⟩_𝑁/⟨𝑛⟩²_𝑁 being the best.

For inclusive correlations, correction factors such as ⟨𝑛⟩²/⟨𝑛(𝑛−1)⟩_incl were proposed early on in [73, 74] in an approach based on probabilities rather than densities. Reference [75] specifically calls the inclusion of these correction factors meaningless because the theory then requires that the emission function be identically zero. We note that the argument in all those references relates to inclusive samples, while for the samples of fixed 𝑁 considered in this paper the prefactor, which is an average of 𝑛 at fixed 𝑁, is a necessity. Either way, the arbitrariness of the use or nonuse of the prefactor has been eliminated here based solely on considerations related to the reference distribution.

(4) The problem posed in this paper, namely, the relation between charged multiplicity on the one hand and correlations based on the conditional 𝑛-distribution ℛ_𝑛𝑁 has attracted little attention in the literature. Indeed, almost all theoretical work on multipion correlations, as for example summarised in [76], starts from the projection of final-state events, with all their different particle species, onto the single-species subspace of either 𝑛₊ (positive pions) or 𝑛₋ (negative pions) correlations, to the exclusion of the other charge. It would be interesting to see a combined theory for both 𝑛₊and 𝑛₋, which would encompass all the work done so far plus correlations between unlike-sign pions and, of course, the issue raised by us here.

(5) As shown in Figures 2 and 3, correction factors for third-order are larger than the second-order ones. For higher 𝑟th-order correlations, the effect of using a fixed-𝑁 subsample is suppressed by approximately 1/⟨𝑛^𝑟−1⟩_𝑁 for unnormalised cumulants but actually worsens for normalised cumulants due to the normalisation prefactors ⟨𝑛⟩^𝑟_𝑁/⟨𝑛^𝑟⟩_𝑁 for small 𝑛. The importance of accurate correction of normalised quantities therefore rises with order of correlation.

(6) The difference between Poissonian and internal cumulants is largest at small multiplicities 𝑛. The mixed-multinomial prescription will therefore be required for any correlation analysis involving small 𝑛, independently of the magnitude of 𝑁. Apart from the usual suspects of leptonic, hadronic, and low-energy collisions, the low-𝑛 case occurs both for very restricted phase space (such as in spectrometer experiments) and for correlations of rare particles such as kaons and baryons, even for large 𝑁.

(7) Since 𝑛 fluctuates according to the conditional multiplicity distribution ℛ_𝑛𝑁, the degree to which fixed-𝑁 correlations differ from inclusive ones is strongly coupled to the character of ℛ_𝑛𝑁. In general, ℛ_𝑛𝑁 is sub-Poissonian and so 𝐹_𝑟𝑁 falls below the Poisson limit of 1. The correction from fixed-𝑁 Poissonian to internal normalised cumulants is hence upward, not downward as in the case of multiplicity-mixing corrections.

(8) While not the main subject of the present paper, some light is cast on the relationship between three levels of correlation, namely, correlations inherent in the overall multiplicity distribution, multiplicity-mixing correlations, and the true internal correlations for fixed 𝑁. Each of these can and should be treated separately. The averaged-internal correlations of Section 5 are a compromise solution which may be useful both for physics reasons and for small datasets.

(9) The fixed-𝑁 corrections discussed here are separate and complementary to other important effects at low multiplicity. References [76, 77] highlight, for example, possible effects of “residual correlations” resulting from projecting from multipion to two-pion correlations.

Energy-momentum conservation would also play a role. Borghini [78, 79] has, for example, calculated the effect of momentum conservation for normalised two- and three-particle cumulants in momenta and 𝐩_𝑡. However, the saddlepoint method used applies to the large-𝑁 limit, and the results cannot be directly applied to the low-𝑁 (and hence low-𝑛) samples under discussion here. Indeed, momentum conservation will be near-irrelevant for cases of large 𝑁 and small 𝑛 as discussed above, but the small-𝑛 corrections of this paper will remain important. For the specific case of like-sign pion femtoscopy, the fact that only 𝑛∼𝒜𝑁/2Ω out of the 𝑁 charged pions are used and that momentum conservation constraints include all other final-state particles both imply that momentum conservation constraints may be less important than the internal-cumulant correction introduced here.

For the specific choice of correlation variables 𝑄 and 𝑄_𝑎 for two- and three-particle cumulants, the contribution of momentum conservation to cumulants at small 𝑄 will be small since the counts will be dominated by pairs at small (Δ𝜙,Δ𝑦) and intermediate (𝑝_𝑡1,𝑝_𝑡2). As pointed out by [1], momentum conservation exerts the greatest influence at large pair or triplet momenta and hence mostly at large 𝑄, where it may lead to moments and cumulants which do not converge to a constant as presupposed in most fits. The ad hoc method of multiplying fit parametrisations by a prefactor 1+𝑐𝑄 with 𝑐 a free parameter does not adequately address the problem.

Regarding the multiplicity dependence of the influence of energy-momentum conservation, [2] calculates the effect of energy-momentum conservation on single-particle differential observables and finds significant systematic effects. No doubt this must also be the case for multiparticle observables, although as we have pointed out above, the effect of conservation laws will be diluted by the fact that fewer than half of the final-state particles of any given event are actually used in the present analysis. Detailed investigations are beyond the scope of this paper.

(10) We have also recalculated statistical errors for products of event averages starting from the original prescription which forms the basis of frequentist statistical error calculations. Compared to conventional statistical error calculations, new prefactors appear in our calculations — see for example (110) and the results in Section 4.4.2—which have surprisingly been missed so far.

(11) We note that the present formalism is still in the frequentist statistics mindset, which may be inaccurate for low multiplicities and should be supplanted by a proper Bayesian analysis. The final word has certainly not been spoken about correlation analysis of small-𝑛 datasets.

Acknowledgment

This work was supported in part by the National Research Foundation of South Africa.

Endnotes

1. These event ratios are not probabilities in the strict sense; in the frequentist definition of probability, the two are equal only in the limit ℰ→∞. We therefore avoid the use of the symbol 𝑃_𝑁 for such and similar data ratios.
2. Pairs are ordered, for example, a particular pair is counted twice. Unordered pair counting is possible but unnecessarily complicates sum limits.
3. A counter of order 𝑁(𝑎) can be made to behave like one of order 𝑁(𝑎)+1 by defining an additional dummy data point 𝐏_𝑁(𝑎)+1 which lies outside the normal domain Ω, but we will not pursue this here.
4. We could argue that there are three different variables 𝑄^𝑎𝑎, 𝑄^𝑎𝑏, and 𝑄^𝑏𝑐, where the last two differ in the sense that 𝑄^𝑎𝑏 contains a track from the “current” event while 𝑄^𝑏𝑐 does not. As shown below, this distinction is unnecessary as long as we keep careful track of possible occurrences of equal event indices.
5. While it is irrelevant whether event 𝑎 is included or excluded in theoretical calculations of event mixing, it should never be used in actual implementations of mixing.
6. Due to the factorisation of the expectation values earlier on, the fact that index 𝑏 appears in two separate sample averages does not prevent us from replacing ⟨𝜌_𝑏𝑐⟩ and ⟨𝜌_𝑏𝑒⟩ by (𝜌₁⊗𝜌₁)².
7. While this factorised form is instructive, it cannot be used directly since 𝐹_2𝑁 can be determined only on completion of the entire sample analysis. Each of the counter products in (114) must hence be implemented separately.
8. If and when large bins are used and the sixth power of the measured positive-pion multiplicity becomes comparable to ℰ, subleading terms will have to be included. This requirement is less trivial than it may sound, since for subsamples of fixed multiplicity 𝑁, the number of events ℰ_𝑁 is much smaller than ℰ, while of course 𝑛 may be substantial when 𝑁 is large. For UA1, ℰ_𝑁=𝑂(10⁴) while ℰ=𝑂(10⁶).
9. Historically, this issue was discussed under the name “Short-Range Correlations” and “Long-Range Correlations” [15]. Since current usage of the term “Short-Range Correlations” refers to correlations over small scales in momentum space, we rather define them more accurately as “Averaged Internal” (AI) correlations and “Multiplicity-Mixing” (MM) correlations, noting also that the correction factors in (124)–(129) do not appear in the earlier literature.
10. An expression with a correction factor outside the sums such as 𝐾₂=⟨𝑛⟩²⟨𝑛(⋅∑𝑛−1)⟩^𝑁ℛ^′_𝑁𝜅^𝐼₂𝑄∣𝒮_𝑁∑_𝑁ℛ^′_𝑁𝜌₁⊗𝜌₁𝑄∣𝒮_𝑁 (138) is inconsistent with AI correlation averaging if the single-particle spectra or some other physical effect change significantly within the range [𝐴,𝐵]. We also note that the above differs from the formula used in [65] for second-order correlations in 𝐪. In the present notation, the cumulant used in [65] reads 𝐾₂(𝐪)=∑_𝑁ℛ^′_𝑁𝜌₂𝐪∣𝒮_𝑁∑_𝑁ℛ^′_𝑁((𝑁−1)/𝑁)𝜌₁⊗𝜌₁𝐪∣𝒮_𝑁−1, (139) that is a correction for an 𝑁-multinomial rather than the weighted sum of 𝑛-multinomials used in (124)–(126).
11. One might expect 𝜎(𝐾^AI_𝑟(𝒮_𝐴𝐵)) to include a prefactor of the sort seen in (105) that is something like [var(𝐾^AI_𝑟(𝑄∣𝒮_𝐴𝐵)]/[𝐵−𝐴], but this would be incorrect. The reason in that the formulae (131)–(133) for range 𝐴𝐵 can be considered as an average, so that we can apply the methods of Section 4.3 to obtain the same results. For example, considering 𝜅^AI₂(𝑄∣𝒮_𝐴𝐵)≡𝜅₂ of (124) as an average and writing 𝜅^𝐼₂(𝑄∣𝒮_𝑁)≡𝜅_2𝑁, the variance on this average is var𝜅₂=𝐸𝜅₂²−𝐸𝜅₂²=𝐸_𝑁ℛ^′_𝑁𝜅_2𝑁⋅_𝑁′ℛ^′_𝑁′𝜅_2𝑁′−𝐸_𝑁ℛ^′_𝑁𝜅_2𝑁𝐸_𝑁′ℛ^′_𝑁′𝜅_2𝑁′=_𝑁=𝑁′ℛ^′_𝑁²𝐸𝜅²_2𝑁−𝐸𝜅^𝐼₂𝑄∣𝒮_𝑁²=_𝑁ℛ^′_𝑁²var𝜅_2𝑁, (140) which is identical with (131). Therefore, division by 𝐵−𝐴 is incorrect.
12. Inevitably, there are very few events in the high-multiplicity tail of the entire sample. These must be treated separately, for example, by putting all events with 𝑁 greater than some threshold into a single buffer.
13. The number of events in a buffer ℰ_𝐵 is usually kept the same for each buffer.
14. As in the definition of the counters, each pair is counted twice: these are ordered pairs.