Binomial acceptance corrections for particle number distributions in high-energy reactions

Oleh Savchuk, Roman V. Poberezhnyuk, 3 Volodymyr Vovchenko, 3 and Mark I. Gorenstein 3 Taras Shevchenko National University of Kyiv, 03022 Kyiv, Ukraine Bogolyubov Institute for Theoretical Physics, 03680 Kyiv, Ukraine Frankfurt Institute for Advanced Studies, Giersch Science Center, D-60438 Frankfurt am Main, Germany Institut für Theoretische Physik, Goethe Universität Frankfurt, D-60438 Frankfurt am Main, Germany (Dated: November 11, 2019)


I. INTRODUCTION
Investigation of the phase diagram of strongly interacting matter is today one of the most important topics in nuclear and particle physics. Transitions between different phases of this matter are expected to reveal themselves as specific patterns in particle number fluctuations. In particular, a critical point (CP) should yield large deviations of the conserved charges from their respective baselines in finite regions of the phase space around a CP, showing universal signals in various high-order susceptibilities [1][2][3][4][5][6]. This generally applies not only to the hypothetical chiral QCD CP which has garnered most attention, but also to the better established CP of the nuclear liquid-gas transition [7,8], which entails characteristic patterns in nucleon number fluctuations [9] as well as nuclear fragment distributions [10].
The particle number fluctuations can be characterized by the central moments, (∆N ) 2 ≡ σ 2 , (∆N ) 3 , (∆N ) 4 , etc, where ... denotes the event-by-event averaging and ∆N ≡ N − N . The scaled variance ω, as well as (normalized) skewness Sσ and kurtosis κσ 2 of particle number distribution are defined as the following combinations of the central moments, These can also be expressed through the cumulants κ n of the N -distribution: N = κ 1 , ω = κ 2 /κ 1 , Sσ = κ 3 /κ 2 , κσ 2 = κ 4 /κ 2 . The quantities (1)-(3) are the well known size-independent (intensive) measures of particle number fluctuations. Besides the particle number fluctuations, the susceptibilities of conserved charges such as net baryon number B and electric charge Q are of special interest. In thermodynamic equilibrium, they are connected to the grand canonical partition function and thus contain information about the QCD equation of state. Namely, the cumulants of conserved charge distributions are calculated as the corresponding derivatives of the system pressure p: where V and T are the system's volume and temperature, Q i = B, Q, and µ B , µ Q are, respectively, the baryon and electric chemical potentials. Having generally longer equilibration times [11,12], the fluctuations of conserved charges are also thought to reflect properties of earlier stages of collision [5]. Studies of the higherorder fluctuation measures are motivated by their larger sensitivity to critical phenomena. Cumulants of a higher order are proportional to increasing powers of the correlation length ξ, and they are considerably more sensitive probes to the proximity of the CP than the variance [4,13], as has been illustrated in a number of model calculations for net baryon and/or net charge fluctuations [9,[14][15][16]. Experimental studies of such fluctuation measures are in progress [17]. This motivates our study of acceptance effects for the higher-order fluctuation measures.
Of course, the baryon number and electric charge are globally conserved in high energy collisions, meaning that these quantities do not fluctuate in the full phase space provided that the events under consideration have the same number of participants. Baryon number and electric charge of the entire system are conserved eventby-event. Therefore, actual fluctuations of conserved charges can only be seen by considering finite acceptance regions. The optimal choice of acceptance for comparing the measurements with predictions of equilibrium thermodynamics in the grand-canonical ensemble is not trivial. If acceptance is too small, the trivial Poissonlike fluctuations dominate [3]. The acceptance should be large enough compared to correlation lengths relevant for various physics processes, in particular, those related to the QCD CP [18].
A crucial question is connecting the quantities (1), (2), and (3) measured in finite regions of the momentum space with predictions of various physical models. In the present paper acceptance effects are modeled by the binomial distribution. Namely, the binomial acceptance corrections (BAC) assume that each particle of the ith type is accepted by detector with a fixed probability x i . This probability 0 ≤ x i = n i / N i ≤ 1 equals the ratio of the mean value n i of accepted particles to that of N i of all particles of the i-th type. The main assumption of the binomial acceptance is that the probability x i is the same for all particles of a given type and independent of any properties of a specific event. This assumption allows to relate the cumulants within a finite acceptance to their values in the larger, encompassing phase space. We will use the method of characteristic functions which was used previously for similar purposes in Ref. [19]. The present formalism does recreate the prior results on the BAC [19][20][21], as one would expect. Our main focus here is on a number of special cases for which the present formalism is found to be most suitable. We will consider both, the fluctuations of the specific particle species and also that of globally conserved charges such as baryon number or electric charge. We analyze the performance of the binomial filter for acceptance corrections in the momentum space as well for constructing net proton fluctuations from net baryon ones. For that we use ultra-relativistic quantum molecular dynamics (UrQMD) model [22,23] simulations of inelastic p+p interactions.
The paper is organized as follows. In Sec. II we present the formulas of the BAC which connect the fluctuations measures in the finite x-acceptances with the corresponding quantities in the full phase space. Sec. III presents the typical multiplicity distributions in grand-canonical and canonical statistical mechanics of relativistic particles. In Sec. IV the BAC performance is confronted with the the UrQMD model simulations. Summary in Sec. V closes the article.

II. BINOMIAL ACCEPTANCE CORRECTIONS
Let the P (N ) function denote a normalized probability distribution for observing N particles of a given type in the full phase space. The BAC for particle number fluctuations assume that the probability p(n, x) to observe n particles detected in the finite x-region of the phase space is given as A. BAC for the particle number fluctuation First, we consider the BAC (5) applied to particles of a given type. The characteristic function of the P (N ) distribution is defined as where κ l [N ] is the l-th cumulant of the distribution P (N ). The corresponding characteristic function for the number of accepted particles reads where φ[k, x] ≡ ln 1 − x + xe ik is the cumulant generating function of binomial distribution and F N is given by Eq. (6). We checked that such procedure is correct for discrete random variables. The acceptance parameter 0 ≤ x ≤ 1 has a simple meaning x = n / N , i.e., it equals to the ratio of the average multiplicities of the accepted and all particles. At x → 1 one finds f n (k, x) ∼ = F N (k, x), i.e., p(n, x) ∼ = P (n). In the opposite limiting case x → 0, one finds from Eq. (7), in the first order of x, which is a characteristic function of the Poisson distribution with a mean equal to x N . The cumulants of the p(n, x) probability distribution are calculated as The scaled variance, skewness, and kurtosis for the distribution (5) of the accepted particles are then presented as follows: where Equation (10) was previously obtained in Refs. [24,25]. At x → 1 in Eqs.
It should be noted, however, that these "inverse" relations are to be used with care.
This distribution may correspond, e.g., to an equilibrium system of non-interacting Maxwell-Boltzmann particles in the grand canonical ensemble. One finds for the fluctuations in the full phase space whereas Eqs. (10)- (12) give for fluctuations within acceptance. The BAC quantities (19) are independent of the x-acceptance parameter and equal to the fluctuation measures (18) in the full phase space. This last property of the BAC procedure is a unique feature of the Poisson distribution (17). As our second example we assume that the number of particles in the full phase space is fixed, i.e., Such a scenario is approximately valid for the number of baryons in p + p and nucleus-nucleus reactions at small and intermediate collision energies where the production of baryon-antibaryon pairs is negligible. One finds, Equations (10)-(12) then correspond to the binomial probability distribution B(n, N, x), giving

B. BAC for conserved charge fluctuations
In this subsection, we consider the BAC for fluctuations of conserved charges. We use notations N + , N − and n + , n − , for positively and negatively charged particles in the full space and in the x-acceptance region, respectively. Here conserved charge may correspond to any integer conserved number carried by hadrons, for instance the electric charge or baryon number. Without loss of generality, we focus here on the net electric charge. The nonzero values of electric charge and baryon number of final state hadrons detected in high energy collisions are ±1. Therefore, the net charge Q is straightforwardly connected to the number of positively and negatively charged particles: Q = N + − N − = const in the full space and q = n + − n − within the acceptance.
The distribution function of N + and N − can be presented in the following general form: where N ch ≡ N + + N − . The BAC are introduced as where the binomial distributions are defined in Eq. (5), x + and x − are the acceptance parameters for the positively and negatively charged particles, respectively. The characteristic function for the distribution of net charge q = n + − n − in the acceptance can be calculated as follows: Here and F N ch is the characteristic function of the full space charged multiplicity distribution P (N ch ) The l-th BAC cumulant of the net charge, q, fluctuations reads The leading four cumulants read Here As seen from Eqs. (29)-(32), the BAC net charge cumulants are calculated in terms of the cumulants of the P (N ch ) distribution of charged multiplicity in the full phase space. In the case of equal acceptance parameters, We will use UrQMD simulations further on to analyze to what extent the assumption x + = x − holds in realistic situations.
The above results can be straightforwardly generalized for the case of net baryon number fluctuations. This is achieved through the following substitutions in Eqs. (34) and (36) For sufficiently small collision energies in p + p and nucleus-nucleus reactions one has N B N B , meaning that the number of baryons N B is approximately equal to the net baryon number B. Therefore, ] ∼ = 0 and Eqs. (34), (36) reduce to Eqs. (22).
It should be noted that the scaled variance (34) and the skewness (35) exhibit a special behavior for the case of e + + e − and/or p + p reactions. In these reactions all globally conserved charges are equal to zero, and thus ω x [q] ≡ ∞ and Sσ x [q] ≡ 0. On the other hand, the kurtosis (36) attains non-trivial values for all types of reactions. . We note that the data on p + p reactions suggest that ω[N ch ] is an increasing function of the collision energy with its values smaller than 2 at small collision energies and larger than 2 at large collision energies [26]. Assuming Q = 2, the values of N ch and ω[N ch ] presented in Fig. 1 correspond approximately to the p + p data at √ s ∼ = 2 GeV, 10 GeV, 20 GeV, and 100 GeV (see, e.g., Ref. [27]). Similar arguments can be applied to baryon number fluctuations. Note, however, that the fluctuations of N B + N B are essentially smaller than At the SPS and RHIC energies considered in this paper, one For two statistically correlated types of particles the cumulant generating function reads: as follows from Eq. (7). Here κ n,m [N + , N − ] are joint cumulants of P (N + , N − ). They obtain non-zero values if any correlation between positively and negatively charged particle is present.
Then by taking respective derivatives the cumulants of the charge distribution can be obtained, see Eq. (28)), which makes them linear functions of κ n,m [N + , N − ]. Note that Eq. (38) does not include factorial moments [19].

III. (GRAND-)CANONICAL STATISTICAL MECHANICS
In this section, we analyze a couple of common full space multiplicity distributions in statistical mechanics. We consider grand-canonical and canonical distributions of relativistic particles, which represents two useful baselines in the context of heavy-ion collisions.
The grand-canonical multiplicity distribution of non-interacting Maxwell-Boltzmann particles in equilibrium is given by a Poisson distribution. In a relativistic case studied here, the joint probability distribution of the numbers of particles N + and antiparticles N − is given by a product of two Poisson distributions: Here the quantities z + and z − are defined as where V , µ Q , and T are, respectively, the system volume, charge chemical potential, and temperature. g and m are the particle degeneracy factor and mass. The chemical potential regulates the mean net number of particles and antiparticles, Q = N + − N − . A generalization to a system with multiple particle species carrying a conserved charge Q is achieved by simply adding contributions of these extra species to Eq. (40).
These fluctuation measures are shown by the horizontal dashed line in Fig. 2 (a). The charge distribution P Sk (Q) corresponds then to the so-called Skellam distribution [28]. The cumulants k n [Q] for the Skellam distribution P Sk (Q) can be easily found as This gives The skewness Sσ[Q] of the Skellam distribution, given by Eq. (42), is shown in Fig. 2 [24,29], while similar questions related to baryon number conservation were discussed in Ref. [30]. The particle number distribution P(N + , N − ) corresponds to a two-Poisson distribution with a fixed difference: where the parameter z is defined in Eq. (40) and is proportional to the volume of statistical system. It follows from Eq. (43) that numbers N ± and N ch = N + + N − are both described by various forms of the Bessel  distribution [31]. Their characteristic functions are the following: Here I Q is the modified Bessel function of the first kind. The expressions (44) where Simplied expressions can be obtained in certain limits. For large systems, z Q and z 1, one has κ l [N ± ] ≈ (2z)/2 l and κ l [N ch ] = 2 l k l [N ± ]. Explicit expressions for means and variances in this limit read For z √ Q + 1 the cumulant generating functions of N ± and N ch read , l > 1. Using the above equations one calculates the asymptotic behavior at z √ Q + 1 : The net-charge Q is conserved globally and does not fluctuate in the full space, i.e., in the limit x → 1. The scaled variance of net-charge fluctuations in full space is, therefore, vanishing: ω[Q] = 0. The skewness and kurtosis of Q, on the other hand, attain finite values in the limit x → 1, which follow from Eqs. (34)-(36): The behavior of Sσ[Q] and κσ 2 [Q] for the Bessel distribution with Q = 2 is shown in Fig. 2 (b). The above results illustrate the non-trivial behavior of fluctuation measures of positively or negatively charged particle numbers that arise due to the exact conservation of the net charge. This behavior is present already for fluctuations in the full phase space, x = 1. The BAC expressions, Eqs. (10)-(12), for single charge and Eqs. (29)-(32) for net charge, allow then to obtain the corresponding behavior in a finite acceptance, x < 1.

IV. URQMD SIMULATIONS OF p + p REACTIONS
Transport simulations can provide useful information about the acceptance dependence of fluctuations, and test an accuracy of the BAC in various setups. Earlier, box simulations were used to study the net charge fluctuations within transport model [32]. The cuts in coordinate space were applied, i.e., it was assumed that the detection of particles takes place only inside the subsystem with volume v = xV , where V denotes the total volume of a box with periodic boundary conditions and x is the acceptance parameter. It was shown that the acceptance dependencies of the skewness and kurtosis of net charge fluctuations in such a system do satisfy the BAC predictions. Because the multiplicity distribution P(N + , N − ) for hadrons inside the box within transport models appears to be close to the Bessel distribution (43), only convex downward curves for κσ 2 [Q] where obtained.
Actual high-energy collision experiments measure the momenta of final state particles rather than coordinates. Therefore, the BAC should be considered in the momentum space. In this section the BAC predictions for fluctuations of particle numbers, as well as of conserved charges B and Q, are compared with results of the UrQMD transport model [22,23] simulations of inelastic p + p reactions. UrQMD is an event generator producing a list of hadrons and their momenta in the final state of the collision. The generator satisfies the exact conservation of energy-momentum and of all the QCD conserved charges. It also naturally incorporates correlations between particles emerging from resonance decays and string fragmentations. Acceptance cuts in the momentum space can be applied straightforwardly, making UrQMD suitable for direct comparisons to data. This is in contrast to statistical-thermal models where additional assumptions are needed, the BAC being one such possible assumption. The measured hadron multiplicities and momentum spectra calculated within the UrQMD simulations are usually in a fair agreement with the available experimental data. All in all, this makes UrQMD a useful tool to analyze the behavior of fluctuations in various acceptance windows, and to test the performance of the BAC in various setups.
Here we analyze inelastic p + p collisions at the SPS energy of √ s = 6.3 GeV as well as at the one of the top RHIC energies, √ s = 62.4 GeV. In p + p collisions the electric charge and the net baryon number are equal to Q = B = 2 and do not fluctuate in the full phase space. We shall analyze in some detail the acceptance dependence of fluctuations of positively and negatively charged hadron multiplicities, as well as of (net) baryon number and net charge.

A. Rapidity window dependence of acceptance parameters
The same value of BAC x-parameter corresponds to quite different regions in the momentum space at different collision energies. To be definite, we chose the acceptance region as a p T -integrated finite rapidity interval −∆y/2 ≤ y ≤ ∆y/2 in the center of mass of the system. Any particle in this rapidity interval is assumed to be detected with 100% efficiency, therefore the BAC parameter is simply the ratio between the mean number of particles in the acceptance relative to the one in the full phase space. Figure 3 (a) and (b) presents the UrQMD results for binomial acceptances parameters for positively and negatively charged hadrons, x + and x − , for baryons and antibaryons, x B and x B , as well as for net charge and net baryon numbers as functions of ∆y for p+p collisions at √ s = 6.3 GeV and √ s = 62.4 GeV, respectively. As seen from Fig. 3, x − > x + and x B > x B . This is due to the difference in rapidity spectra, dN i /dy, of the negatively and positively charged hadrons in p + p collisions, and similar difference of the rapidity spectra of antibaryons and baryons. Thus, one cannot apply the simplified BAC equations (34) Fig. 5 show a non-monotonic dependence on the corresponding acceptance parameters x q and x b . The behavior of κσ 2 x [q] and κσ 2 x [b] is even more nontrivial: they demonstrate a zigzag-like behavior with a maximum at small acceptance parameter x q,b ∼ = 0.1 and a minimum at x q,b = 0.4 − 0.6 for both collision energies. Such features of the skewness and kurtosis for conserved charges can sometimes lead to their non-monotonic dependencies on the collision energy, even in the absence of any mechanisms for critical fluctuations as is the case for UrQMD.

D. BAC inside a limited phase space
A comparison of the BAC formulas with the actual results of the UrQMD simulations demonstrate rather large differences. For the fluctuation measures ω x , Sσ x , and κσ 2 x calculated in the finite rapidity regions |y| ≤ ∆y/2 using the BAC with corresponding acceptance x-parameters an agreement with the actual UrQMD results is found at the limits x → 0 and x → 1. However, at 0 < x < 1 the BAC and the actual UrQMD results are essentially different, even qualitatively. This means that certain assumptions behind the BAC procedure are not fulfilled in UrQMD. The key BAC assumption is that a probability for a given particle to be within acceptance is independent of all other particles. However, there is an evident reason for event-by-event correlations between the shapes of rapidity distributions and total event multiplicities. From kinematical arguments one expects more final particles just at small center of mass rapidities in events with larger total hadron multiplicities. More special interparticle rapidity correlations emerge in the UrQMD simulations from decays of resonances and strings. For these reasons, the inaccuracy of the BAC predictions is not all that surprising. On the other hand, at smaller momentum scales the BAC assumptions might be more reasonable. Let us take the fixed rapidity interval ∆y = 2. We will treat now this rapidity interval as a 'full phase space' region, and the BAC values will be calculated at smaller parts of the rapidity interval ∆y = 2. The acceptance x i -parameters is now defined as As an example, the UrQMD results for ω x [n − ] and κσ 2 x [q] in p + p reactions at √ s = 62.4 GeV inside the rapidity interval ∆y = 2 are shown in Fig. 6 (a) and (b), respectively. As seen from Fig. 6 (a) an agreement of the BAC procedure and direct UrQMD results for ω x [n − ] inside the rapidity interval ∆y = 2 is almost perfect. The BAC formula (10) is used with n − ∆y=2 ≡ N − and ω[N − ] = 2.4. Similar results were obtained for other fluctuation measures. In Fig. 6 (b) the same is done for κσ 2 x [q] treating all UrQMD quantities at the rapidity interval ∆y as the 'full phase space' values in the BAC formulas. Thus, a basic assumption of the binomial distribution becomes valid for these rapidity intervals and the BAC procedure leads to the results consistent with the actual UrQMD simulations as presented in Fig. 6.
It should be noted that the basic question mentioned in Sec. I -how to define a suitable acceptance region to observe the statistical fluctuations of conserved charges within the grand canonical ensemble -remains beyond the scope of the present study. It can be wrong to search for the statistical fluctuations in the framework of non-equilibrium transport model. This is especially clear in p + p reactions at large collision energy. Both the UrQMD results and the p + p reactions data demonstrate large values of particle number fluctuations, e.g., ω[N ch ] ∝ N ch 1, much above of the standard statistical estimates [27]. Applicability of the BAC procedure inside the central rapidity interval, as in Fig. 6, is by no means the argument in favor of the statistical character of particle number fluctuations inside this region. In the statistical system treated within the grand canonical ensemble all intensive fluctuation measures remain unchanged in their subsystems, if only these subsystems are not too small compared to the correlation length. The BAC procedure is fully consistent with such systems only in the simplest case -a mixture of non-interacting Boltzmann (classical statistics) particles at fixed volume V and temperature T .

E. Net proton fluctuations from net baryon fluctuations
Let us now discuss the BAC procedure in the case of incomplete detector efficiency. A notable example is a connection of (anti-)proton fluctuations with those of (anti-)baryons. In the experiment, the skewness and kurtosis of the net proton number fluctuations are measured, but not of the net baryon number ones due to the problems with detecting neutral baryons and anti-baryons, mainly neutrons and anti-neutrons. One can consider now BAC by assuming that a randomly chosen baryon is within an acceptance if it is a proton and outside of it otherwise. In p + p reactions at √ s = 62.4 GeV the UrQMD results in the full phase space provide: where N p and N p denote the numbers of protons and antiprotons, respectively. Figure 7 presents the skewness and kurtosis of the net baryon and the net proton number fluctuations in p + p reactions at √ s = 62.4 GeV. The UrQMD results, depicted by lines with symbols, show sizable differences between net baryon and net proton fluctuation measures, suggesting that the latter may not be a particularly good direct proxy for the former. The red line shows the net proton fluctuations constructed out of the net baryon fluctuations by applying the BAC using the x p and xp acceptance parameters listed above. One observes a quite good agreement of the net-baryon fluctuations calculated by the BAC procedure from the net-baryon values using acceptance parameters (58) with their exact UrQMD values. The results suggest that reconstructing the net baryon fluctuations from net proton ones using a binomial filter, as suggested in Refs. [5,20], might be a reasonable procedure. This observation is important in the context of attempts to relate the net proton fluctuation measurements in heavy-ion collisions with QCD net baryon number susceptibilities computed e.g. using first-principle lattice simulations [33]. We note that corrections for other types of detector efficiencies are sometimes needed as well and these might require procedures which are more involved than the simple binomial filter [34,35].

V. SUMMARY
We studied the binomial acceptance corrections which relate distributions of various particle number and/or conserved charge distributions in a region of phase space to the corresponding distributions in the full phase space. The binomial acceptance corrections are derived under the assumption that each particle is accepted with a certain probability independently from all other particles. Based on this, we derive explicit formulas that connect high order cumulants and their various ratios such as scaled variance, skewness and kurtosis of fluctuations within a given acceptance (sub-system) with those in a broader acceptance (system). Where applicable, our formalism reproduces earlier results on the binomial acceptance [19,20].
The BAC transform a Poisson distribution into another Poisson distribution. Therefore BAC cancels out in all cumulant ratios if the underlying particle number distribution in the full space is Poissonian. The behavior is less trivial in other cases. Particularly the fluctuations of conserved charges, i.e. quantities which are conserved globally, are studied in some detail in the present work. Exact conservation induces correlations between positive and negative particles, in contrast to the Poisson baseline that entails no correlations. We show that fluctuations of these quantities within a given acceptance are expressed through fluctuations of a sum of positively and negatively charged particles N ch = N + + N − in the full phase space, Eqs. (29)-(32). As a particular example, we explore N ch and N ± distributions within canonical relativistic statistical mechanics. In contrast to the grand-canonical ensemble where all these fluctuations are the trivial Poisson ones, in the canonical ensemble, these are given by a more involved Bessel distribution. These observations will be useful for future measurements and analysis of net electric charge fluctuations.
UrQMD simulations of inelastic p+p interactions were then used to explore the performance of the BAC when applied to a momentum space acceptance, as is appropriate for high-energy collision experiments. It was found that actual UrQMD fluctuations of various particle numbers in a given rapidity window deviate considerably from those predicted using the BAC procedure applied to fluctuations in full phase space. This indicates that the BAC assumption of an uncorrelated acceptance probability is not fulfilled in UrQMD simulations, which can take place if particle rapidities are correlated on scales smaller than beam rapidities. We do find that the BAC procedure is found to be significantly more accurate when applied to relate fluctuations between various smaller rapidity windows which span no more than two units. We plan to address system-size systematics of the BAC accuracy in a future work.
The BAC can also be used to account for detector efficiency, in particular to take into consideration the inability to measure cumulants of neutral particles such as neutrons. Our UrQMD analysis of p+p collisions shows that net proton fluctuations obtained by applying the BAC to net baryon fluctuations agrees quite well with actual net proton fluctuations. The BAC can thus be used to reconstruct the net baryon fluctuations from the measured net proton ones [5,20] or to estimate the net proton fluctuations in a framework where their explicit calculation is problematic but where baryon and antibaryon fluctuations are tractable by theoretical models.