Eliminating $volume$ fluctuations in fixed-target heavy-ion experiments

Experimental and theoretical studies of fluctuations in nucleus-nucleus interactions at high energies have started to play a major role in understanding of the concept of strong interactions. The elaborated procedures have been developed to disentangle different processes happening during nucleus-nucleus collisions. The fluctuations caused by a variation of the number of nucleons which participated in a collision are frequently considered the unwanted one. The methods to eliminate these fluctuations in fixed-target experiments are reviewed and tested. They can be of key importance in the following ongoing fixed-target heavy-ion experiments: NA61/SHINE at the CERN SPS, STAR-FT at the BNL RHIC, BM\@N at JINR Nuclotron, HADES at the GSI SIS18 and in future experiments such as NA60+ at the CERN SPS, CBM at the FAIR SIS100, JHITS at J-PARC-HI MR.


I. INTRODUCTION
Measuring event-by-event fluctuations is the focus of numerous experimental programmes on nucleus-nucleus collisions at high energies. Nowadays, the leading motivation is the possibility to discover the critical point of strongly interacting matter and a need to understand how the onset of deconfinement influences event-by-event fluctuations. The recent reviews can be found in Refs. [1][2][3]. In this paper methods to remove the influence of the volume fluctuations in fixed target experiments are reviewed and tested. They can be of key importance in the following ongoing fixed-target heavy-ion experiments: NA61/SHINE [4] at the CERN SPS, STAR-FT [5] at the BNL RHIC, BMN [6] at the JINR Nuclotron, HADES [7] at the GSI SIS18, and in the future experiments such as NA60+ [8] at the CERN SPS, CBM [9] at the FAIR SIS100, JHITS [10] at J-PARC-HI MR.
The paper is organized as follows: Section II introduces the reference model -the Wounded Nucleon Model (WNM) [11] -used here to test the influence of the volume fluctuations. This section also introduces extensive, intensive and strongly intensive measures of fluctuations [12,13] and their volume dependence within WNM. The main features of typical fixed-target and collider experiments with respect to fluctuation measurements and the volume fluctuations are summarized in Sec. III. Two methods used to eliminate the effect of the volume fluctuations in fixed-target experiments are introduced and compared using WNM in Sec. IV. The summary concludes the paper.

II. WOUNDED NUCLEON MODEL, EXTENSIVE AND INTENSIVE QUANTITIES
Using the the Wounded Nucleon Model [11] is probably the simplest way to introduce fluc- Figure 1: The sketch of particle production process in nucleus-nucleus collisions according to the Wounded Nucleon Model [11]. Projectile and target nuclei with nuclear mass number A P and A T projectile wounded nucleons produce N particles, where N is given by the sum over all wounded nucleons of particle multiplicities n i from a single wounded nucleon, of produced particles. The model was proposed in 1976 as the late child of the S-matrix period [14]. It assumes that particle production in nucleon-nucleon and nucleus-nucleus collisions is an incoherent superposition of particle production from wounded nucleons. The wounded nucleons are the ones which interacted inelastically and which number is calculated using straight line trajectories of nucleons. The properties of wounded nucleons are independent of the size of the colliding nuclei, e.g., they are the same in p+p and Pb+Pb collisions at the same collision energy per nucleon. Within WNM, the number of wounded nucleons plays the role of volume.
These assumptions are graphically illustrated in Fig. 1.
The extensive quantity is proportional to the system volume, which in the WNM is represented by W . Let a random variable A measured for each collision be defined as a sum of corresponding random variables a i for wounded nucleons: For example, a i can be particle multiplicity produced by i-th wounded nucleon n i and then A is collision multiplicity, N = W i=1 n i . The k-th order moment of the probability distribution of A, P (A), is defined as Then the extensive quantities which correspond to A are cumulants of A by . . . , where δA k = (A − A ) k . The first and the second cumulants are referred to as the mean and variance of A, respectively. The third and fourth cumulants are related to skewness, and kurtosis, κ = κ 4 /κ 2 2 , respectively. By definition, cumulants are proportional to W .
An intensive quantity is the quantity which is independent of volume. Clearly, the ratio of two extensive quantities is the intensive quantity. For example, the ratio of the two first cumulants referred to as scaled variance is an intensive quantity: Other frequently used intensive quantities which involve third and fourth moments of A are: sometimes denoted as Sσ and κσ 2 , respectively. For any probability distribution P (W ), the scaled variance calculated within the WNM reads [13]: where ω[N ] W stands for the scaled variance at any fixed number of wounded nucleons and W = W P + W T . The first component of Eq. 9 is considered the wanted one and it is independent of the volume fluctuations. However, the second component is unwanted and it is proportional to the scaled variance of the W distribution. Corresponding expressions for higher order moments are given in Ref. [15].
It is worth noting that similar relations are valid within Statistical Models of an Ideal Boltzmann gas within the Grand Canonical Ensemble SM(IB-GCE) [13]. Then, in the equations above, the number of wounded nucleons W should be replaced by the gas volume V .

III. FIXED-TARGET VERSUS COLLIDER EXPERIMENTS
Typically, fixed-target experiments -like NA49 and NA61/SHINE at the CERN SPS -cover mostly the forward hemisphere in the center-of-mass system. An advantage of the fixed-target geometry is that it allows to select collisions using the measured energy of spectators from the beam nucleus independently from measurements of the produced particles, see Fig. 2 for illustration. This selection is referred to as centrality selection. It is important to note that the measurement of target spectators is usually impossible as most of them are fully stopped inside the target material.
On the other hand, a typical collider experiment -like STAR at BNL RHIC and ALICE at CERN LHC -has practically energy-independent rapidity acceptance, but without the low transverse momentum region. The track density in the detector increases only moderately with the collision energy. However, left and right spectator regions are only partly accessible to measurements and the collision selection is usually based on the multiplicity of produced particles, see Fig. 3 for illustration. Thus, quantities used to select events and study the properties of particle production are correlated by the physics of particle production. This fact complicates the interpretation of the results.  In order to simplify, let us assume that only collisions with zero number of projectile spectators were selected and thus W P = A P , where A P is the nuclear mass number of projectile nucleus.
Then, it appears that for collisions of sufficiently large nuclei of similar nuclear mass number, the number of target wounded nucleons is also fixed. This is demonstrated in Fig. 4 where results obtained within the HIJING [17] implementation of the Wounded Nucleon Model [11] are shown. These results agree with the predictions of the HSD and UrQMD models [18].
Thus, the total number of wounded nucleons W = W P + W T is approximately fixed for very Since even for the most central collisions the volume fluctuations cannot be fully eliminated, it is important to minimise their effect further by defining suitable fluctuation measures. It appears that for the WNM and the SM(IB-GCE) models [12,13,19] fluctuation measures independent of the volume fluctuations can be constructed using moments of the distribution of two extensive quantities.
As the simplest example, let us consider multiplicities of two different types of hadrons, A and B. Their mean multiplicities are proportional to W : Obviously the ratio of mean multiplicities is independent of W . Moreover, the ratio A / B is independent of P (W ), where P (W ) is the probability distribution of W for a selected set of collisions. The quantities which have the latter property are called strongly intensive quantities [13]. Such quantities are useful in experimental studies of fluctuations in A+A collisions as they eliminate the influence of a usually poorly known distribution of W .
More generally, A and B can be any extensive event quantities such as the sum of transverse momenta, the net charge or the multiplicity of particles of a given type. The scaled variance of A and B and the mixed second moment AB calculated within the WNM [13] read: where the quantities denoted by * are quantities calculated at a fixed volume.
From Eqs. 11-13 it follows [13,20] that and intensive cumulants allow to measure fluctuations of higher order moments [19]. The first four are defined as: Because of their construction, strongly intensive measures of fluctuations require two extensive quantities. This, in general, hampers a straight-forward interpretation of the experimental results. However, under certain conditions the ∆ quantity can be used to obtain the scaled variance of the extensive quantity A separately.
Let A be an extensive quantity, e.g., selected for its sensitivity to critical fluctuations.
Thus, ∆ B [A, B] is equal to the scaled variance ω[A] for a fixed number of wounded nucleons (see Eq. 11). Similar relations can be found for strongly intensive cumulants of any order: In the derivation of Eqs. 17 and 18 one assumes the validity of Eq. 11 which needs to be investigated case-by-case.  The quantities approach the corresponding value for a fixed number of wounded nucleons with It is important to note that only ≈ 0.0007% of all inelastic collisions have It can be concluded that the selection of collisions with W P = A P significantly reduces the effect of the volume fluctuations, however it is at the cost of reduction of event statistics. The remaining bias can be corrected for using a model-dependent correction. The uncertainty of this correction will contribute to the systematic uncertainty of the final results.

B. Using strongly intensive quantities
Strongly intensive quantities were proposed with the aim to reduce the intrinsic limitation of the method based on the selection of central events which may lead to significant systematic and statistical uncertainties. Figure 6 shows the dependence of intensive and strongly intensive quantities on the ratio of W P /A P . Here, collisions were selected using collision impact pa- Consequently, the method of the event selection based on the number of projectile wounded nucleons needs to be used. The results calculated in bins of W P are shown in Fig. 7. In this case strongly intensive quantities, in general, also deviate from the corresponding values for fixed W . They approach them only for the most central collisions, W P /A P → 1. This is due to the introduced correlation when events are selected on the same quantity used to calculate strongly intensive quantities. This can be solved by defining strongly intensive quantities using two extensive quantities related to particle production properties. Particle multiplicity and transverse momentum [22,23] are the most popular examples of these quantities. However, when the goal is to obtain moments of multiplicity distribution for fixed W there is no significant advantage of using strongly intensive quantities. Similarly, intensive quantities have to be calculated in the most central collisions to approach the unbiased results.