Eliminating volume fluctuations in fixed-target heavy-ion experiments
EEliminating volume fluctuations in fixed-target heavy-ion experiments
M. Mackowiak-Pawlowska, M. Naskręt, and M. Gazdzicki
3, 4 Faculty of Physics, Warsaw University of Technology, Warsaw, Poland ∗ University of Wroclaw, Wrocław, Poland Geothe-University Frankfurt am Main, Germany Jan Kochanowski University, Kielce, Poland
Abstract
Experimental and theoretical studies of fluctuations in nucleus-nucleus interactions at high energieshave started to play a major role in understanding of the concept of strong interactions. The elaboratedprocedures have been developed to disentangle different processes happening during nucleus-nucleuscollisions. The fluctuations caused by a variation of the number of nucleons which participated in acollision are frequently considered the unwanted one. The methods to eliminate these fluctuations infixed-target experiments are reviewed and tested. They can be of key importance in the followingongoing fixed-target heavy-ion experiments: NA61/SHINE at the CERN SPS, STAR-FT at the BNLRHIC, BMN 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.
PACS numbers: 25.75.q, 25.75.Nq, 24.60.KyKeywords: heavy-ion collisions, fluctuations, fixed-target experiments ∗ Electronic address: [email protected] a r X i v : . [ h e p - e x ] F e b . INTRODUCTION Measuring event-by-event fluctuations is the focus of numerous experimental programmes onnucleus-nucleus collisions at high energies. Nowadays, the leading motivation is the possibilityto discover the critical point of strongly interacting matter and a need to understand how theonset of deconfinement influences event-by-event fluctuations. The recent reviews can be foundin Refs. [1–3].Fluctuations in high energy collisions are significantly influenced by fluctuations in theamount of matter ( volume ) and energy involved in a collision, as well as global and localconservation laws. These fluctuations are unwanted effects in the search for the critical pointand the study of the onset of deconfinement.In this paper methods to remove the influence of the volume fluctuations in fixed targetexperiments are reviewed and tested. They can be of key importance in the following ongoingfixed-target heavy-ion experiments: NA61/SHINE [4] at the CERN SPS, STAR-FT [5] at theBNL RHIC, BMN [6] at the JINR Nuclotron, HADES [7] at the GSI SIS18, and in the futureexperiments 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 WoundedNucleon Model (WNM) [11] - used here to test the influence of the volume fluctuations. Thissection 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 andcollider experiments with respect to fluctuation measurements and the volume fluctuations aresummarized in Sec. III. Two methods used to eliminate the effect of the volume fluctuations infixed-target experiments are introduced and compared using WNM in Sec. IV. The summaryconcludes 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-tuations of the amount of matter involved in a collision and their impact on the fluctuations2 A W T W P n n n n n n n Figure 1: The sketch of particle production process in nucleus-nucleus collisions according to theWounded Nucleon Model [11]. Projectile and target nuclei with nuclear mass number A P and A T (here A = A P = A T = 4 ) collide. W T (here W T = 4 ) target wounded nucleons and W P (here W P = 3 )projectile wounded nucleons produce N particles, where N is given by the sum over all woundednucleons of particle multiplicities n i from a single wounded nucleon, N = (cid:80) i =1 n i . of produced particles. The model was proposed in 1976 as the late child of the S-matrix pe-riod [14]. It assumes that particle production in nucleon-nucleon and nucleus-nucleus collisionsis an incoherent superposition of particle production from wounded nucleons. The wounded nu-cleons are the ones which interacted inelastically and which number is calculated using straightline trajectories of nucleons. The properties of wounded nucleons are independent of the size ofthe colliding nuclei, e.g., they are the same in p+p and Pb+Pb collisions at the same collisionenergy 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 rep-resented by W . Let a random variable A measured for each collision be defined as a sum of3orresponding random variables a i for wounded nucleons: A = a + a · · · + a W . (1)For example, a i can be particle multiplicity produced by i -th wounded nucleon n i and then A is collision multiplicity, N = (cid:80) Wi =1 n i .The k-th order moment of the probability distribution of A , P ( A ) , is defined as (cid:104) A k (cid:105) = (cid:88) A A k P ( A ) . (2)Then the extensive quantities which correspond to A are cumulants of A by κ [ A ] = (cid:104) A (cid:105) , (3) κ [ A ] = (cid:104) δA (cid:105) = V ar [ A ] , (4) κ [ A ] = (cid:104) δA (cid:105) , (5) κ [ A ] = (cid:104) δA (cid:105) − (cid:104) δA (cid:105) (6) . . . , where (cid:104) δA k (cid:105) = (cid:104) ( A − (cid:104) A (cid:105) ) k (cid:105) . The first and the second cumulants are referred to as the meanand variance of A , respectively. The third and fourth cumulants are related to skewness, S = κ /κ / and kurtosis, κ = κ /κ , respectively. By definition, cumulants are proportionalto W .An intensive quantity is the quantity which is independent of volume . Clearly, the ratioof two extensive quantities is the intensive quantity. For example, the ratio of the two firstcumulants referred to as scaled variance is an intensive quantity: ω [ A ] = κ [ A ] /κ [ A ] . (7)Other frequently used intensive quantities which involve third and fourth moments of A are: κ [ A ] /κ [ A ] , κ [ A ] /κ [ A ] , (8)sometimes denoted as Sσ and κσ , respectively. For any probability distribution P ( W ) , thescaled variance calculated within the WNM reads [13]: ω [ A ] = ω [ A ] W + (cid:104) A (cid:105) / (cid:104) W (cid:105) · ω [ W ] , (9)4here 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 ofthe volume fluctuations. However, the second component is unwanted and it is proportional tothe scaled variance of the W distribution. Corresponding expressions for higher order momentsare given in Ref. [15].It is worth noting that similar relations are valid within Statistical Models of an IdealBoltzmann gas within the Grand Canonical Ensemble SM(IB-GCE) [13]. Then, in the equationsabove, 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 - covermostly the forward hemisphere in the center-of-mass system. An advantage of the fixed-targetgeometry is that it allows to select collisions using the measured energy of spectators fromthe beam nucleus independently from measurements of the produced particles, see Fig. 2 forillustration. This selection is referred to as centrality selection. It is important to note thatthe measurement of target spectators is usually impossible as most of them are fully stoppedinside the target material.On the other hand, a typical collider experiment – like STAR at BNL RHIC and ALICEat CERN LHC – has practically energy-independent rapidity acceptance, but without the lowtransverse momentum region. The track density in the detector increases only moderatelywith the collision energy. However, left and right spectator regions are only partly accessibleto measurements and the collision selection is usually based on the multiplicity of producedparticles, see Fig. 3 for illustration. Thus, quantities used to select events and study theproperties of particle production are correlated by the physics of particle production. This factcomplicates the interpretation of the results. 5 arget
ProjectileSpectatorDetector
Figure 2: The sketch of a nucleus-nucleus collision as seen by a fixed-target experiment. The incomingbeam particle (marked with thick dashed orange line ) interacts inelastically with a target nucleus.Projectile spectators: protons ( ), neutrons ( ) and fragments ( ) propagate to the forwardcalorimeter. Newly produced hadrons’ trajectories ( ) are bent and hadrons propagate to trackingdetectors.Figure 3: The sketch of a nucleus-nucleus collision as seen by collider experiments. The incomingbeam particles (marked with thick dashed orange line ) interact inelastically with each other. Mea-surements of left and right spectators are possible under the same experimental conditions. However,only free nucleon spectators ( and ) – in central collisions about 50% of all nucleons [16] – canbe measured. Fragments ( ) follow approximately the beam trajectories and they are difficult tomeasure. Newly produced hadrons ( ) propagate to other detectors. V. TWO METHODS TO REMOVE
VOLUME
FLUCTUATIONS
In this section, the following two popular methods to reduce the impact of the volume fluctuations – the unwanted component in Eq. 9, are discussed:(i) selection of the most central collisions,(ii) use of strongly intensive quantities.
A. The selection of the most central collisions
To limit the unwanted component in fixed-target experiments, collisions with the smallestnumber of projectile spectators are selected. This is done with collision-by-collision measure-ment of a forward energy that is predominantly the energy of projectile spectators, see Fig. 2.In order to simplify, let us assume that only collisions with zero number of projectile spectatorswere 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 whereresults 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 central collisions and its scaled variance is close to zero so the unwanted component in Eq. 9 iseliminated. 7
20 40 60 80 100 120 140 160 180 200 220 P W ] T [ W w Be+BeAr+ArXe+XePb+Pb
Figure 4: Scaled variance of the distribution of the number of target wounded nucleons W T as afunction of the number of projectile wounded nucleons W P calculated within the HIJING [17] imple-mentation of the Wounded Nucleon Model [11]. The results for Be + Be , Ar + Ar , Xe + Xe and P b + P b collisions at 19 A GeV /c are presented. . The use of strongly intensive quantities 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. Itappears that for the WNM and the SM(IB-GCE) models [12, 13, 19] fluctuation measuresindependent of the volume fluctuations can be constructed using moments of the distributionof 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 : (cid:104) A (cid:105) ∼ W , (cid:104) B (cid:105) ∼ W. (10)Obviously the ratio of mean multiplicities is independent of W . Moreover, the ratio (cid:104) A (cid:105) / (cid:104) B (cid:105) is independent of P ( W ) , where P ( W ) is the probability distribution of W for a selected set ofcollisions. The quantities which have the latter property are called strongly intensive quanti-ties [13]. Such quantities are useful in experimental studies of fluctuations in A+A collisions asthey 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 transversemomenta, the net charge or the multiplicity of particles of a given type. The scaled varianceof A and B and the mixed second moment (cid:104) AB (cid:105) calculated within the WNM [13] read: ω [ A ] = ω ∗ [ A ] + (cid:104) A (cid:105) / (cid:104) W (cid:105) · ω [ W ] , (11) ω [ B ] = ω ∗ [ B ] + (cid:104) B (cid:105) / (cid:104) W (cid:105) · ω [ W ] , (12) (cid:104) AB (cid:105) = (cid:104) AB (cid:105) ∗ (cid:104) W (cid:105) + (cid:104) A (cid:105)(cid:104) B (cid:105)(cid:104) W (cid:105) · ( (cid:104) W (cid:105) − (cid:104) W (cid:105) ) , (13)where the quantities denoted by ∗ are quantities calculated at a fixed volume .From Eqs. 11-13 it follows [13, 20] that ∆[ A, B ] = 1 C ∆ (cid:104) (cid:104) B (cid:105) ω [ A ] − (cid:104) A (cid:105) ω [ B ] (cid:105) (14)9nd Σ[ A, B ] = 1 C Σ (cid:104) (cid:104) B (cid:105) ω [ A ] + (cid:104) A (cid:105) ω [ B ] − (cid:104) AB (cid:105) − (cid:104) A (cid:105)(cid:104) B (cid:105) ) (cid:105) (15)are independent of P ( W ) in the WNM. Here, the normalisation factors C ∆ and C Σ are requiredto be proportional to the first moments of any extensive quantity. In Ref. [20] a specific choiceof the C ∆ and C Σ normalisation factors was proposed which makes the quantities ∆[ A, B ] and Σ[ A, B ] dimensionless and leads to ∆[ A, B ] = Σ[
A, B ] = 1 in the independent particle model(IPM) [20]. This normalisation is referred to as the IPM normalisation and it is used here.Thus, ∆[ A, B ] and Σ[ A, B ] are strongly intensive quantities which measure fluctuations of A and B , i.e. they are sensitive to second moments of the distributions of the quantities A and B . The results on ∆[ A, B ] and Σ[ A, B ] are referred to as the results on A − B fluctuations,e.g., transverse momentum - multiplicity fluctuations. The analogous quantities called stronglyintensive cumulants allow to measure fluctuations of higher order moments [19]. The first fourare defined as: κ ∗ [ A, B ] = (cid:104) A (cid:105)(cid:104) B (cid:105) κ ∗ [ A, B ] = (cid:104) A (cid:105)(cid:104) B (cid:105) − (cid:104) A (cid:105)(cid:104) AB (cid:105)(cid:104) B (cid:105) κ ∗ [ A, B ] = (cid:104) A (cid:105)(cid:104) B (cid:105) − (cid:104) A (cid:105)(cid:104) AB (cid:105) + (cid:104) A (cid:105)(cid:104) A B (cid:105)(cid:104) B (cid:105) + 2 (cid:104) A (cid:105)(cid:104) AB (cid:105) (cid:104) B (cid:105) (16) κ ∗ [ A, B ] = (cid:104) A (cid:105)(cid:104) B (cid:105) − (cid:104) A (cid:105)(cid:104) AB (cid:105) + (cid:104) A (cid:105)(cid:104) A B (cid:105)(cid:104) B (cid:105) − (cid:104) A (cid:105)(cid:104) A B (cid:105)(cid:104) B (cid:105) +6 (cid:104) A (cid:105)(cid:104) AB (cid:105) + 6 (cid:104) A (cid:105)(cid:104) A B (cid:105)(cid:104) AB (cid:105)(cid:104) B (cid:105) − (cid:104) A (cid:105)(cid:104) AB (cid:105) (cid:104) B (cid:105) Because of their construction, strongly intensive measures of fluctuations require two exten-sive quantities. This, in general, hampers a straight-forward interpretation of the experimentalresults. However, under certain conditions the ∆ quantity can be used to obtain the scaledvariance of the extensive quantity A separately.Let A be an extensive quantity, e.g., selected for its sensitivity to critical fluctuations. Thenchoose a quantity B such that B ∼ W and denote it as B W . It is easy to show [1] that thestrongly intensive measures ∆ B [ A, B ] and Σ B [ A, B ] (equal to ∆[ A, B ] and Σ B [ A, B ] with thenormalisation C ∆ = (cid:104) B (cid:105) ∼ (cid:104) W (cid:105) ) obey the relation: ∆ B [ A, B ] = Σ B [ A, B ] = ω ∗ [ A ] . (17)10hus, ∆ 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: κ ∗ n κ ∗ k [ A, B W ] = κ n [ A ] κ k [ A ] . (18)In the derivation of Eqs. 17 and 18 one assumes the validity of Eq. 11 which needs to beinvestigated case-by-case. V. NUMERICAL TESTS
Numerical tests of the methods to eliminate the volume fluctuations introduced above arepresented in this section. The simulations were performed using the HIJING [17] implementa-tion of the Wounded Nucleon Model:(i) Ar+ Ar collisions at 150 A GeV /c were generated. This reaction closely corresponds todata recorded by NA61/SHINE at the CERN SPS [21].(ii) for each collision, number of projectile and target wounded nucleons and impact parameter b are stored.(iii) number of particles produced by a given wounded nucleon N is drawn from the binomialdistribution with N max = 2 and p = 0 . . Moments of this distribution are: (cid:104) N (cid:105) = 1 , ω [ N ] = 0 . , κ [ N ] /κ [ N ] = 0 and κ [ N ] /κ [ N ] = − . . A. Selecting the most central collisions
Figure 5 shows the dependence of ω [ N ] , κ [ N ] /κ [ N ] and κ [ N ] /κ [ N ] on the ratio W P /A P .The quantities approach the corresponding value for a fixed number of wounded nucleons with W P /A P → . It is important to note that only ≈ . of all inelastic collisions have W P = A P .It can be concluded that the selection of collisions with W P = A P significantly reduces theeffect of the volume fluctuations, however it is at the cost of reduction of event statistics. Theremaining bias can be corrected for using a model-dependent correction. The uncertainty ofthis correction will contribute to the systematic uncertainty of the final results.11 P /A P W [ N ] w Ar+Ar at 150A GeV/c P /A P W - [ N ] k [ N ] / k Ar+Ar at 150A GeV/c P /A P W - - [ N ] k [ N ] / k Ar+Ar at 150A Ge/c
Figure 5: The dependence of ω [ N ] , κ [ N ] /κ [ N ] and κ [ N ] /κ [ N ] on the ratio W P /A P within theWounded Nucleon Model with input defined in Sec. V. The reference values for any fixed numberof wounded nucleons W = const are shown by dashed lines. The calculations were performed for Ar+ Ar collisions at 150 A GeV /c . B. Using strongly intensive quantities
Strongly intensive quantities were proposed with the aim to reduce the intrinsic limitationof the method based on the selection of central events which may lead to significant systematicand statistical uncertainties. Figure 6 shows the dependence of intensive and strongly intensivequantities on the ratio of (cid:104) W P (cid:105) /A P . Here, collisions were selected using collision impact pa-rameter. As expected, strongly intensive quantities are equal or are close to the correspondingvalues for fixed W . Unlike strongly intensive quantities their intensive partners also shown12n Fig. 6 significantly depend on the impact parameter selection. Thus, it can be concludedthat strongly intensive quantities together with the impact parameter selection of collisionsfully eliminates the effect of volume fluctuations. Unfortunately, this is not the solution ofthe problem. The collision impact parameter is not a measurable quantity. So, calculatingstrongly intensive quantities for all inelastic collisions should be considered. Within the WNMand SM(IB-GCE) models, strongly intensive quantities for those collisions are equal to thecorresponding quantities for fixed W . However, in general, the models are not valid in the fullrange of the impact parameter.Consequently, the method of the event selection based on the number of projectile woundednucleons needs to be used. The results calculated in bins of W P are shown in Fig. 7. In thiscase strongly intensive quantities, in general, also deviate from the corresponding values forfixed W . They approach them only for the most central collisions, W P /A P → . This is dueto the introduced correlation when events are selected on the same quantity used to calculatestrongly intensive quantities. This can be solved by defining strongly intensive quantities usingtwo extensive quantities related to particle production properties. Particle multiplicity andtransverse 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 significantadvantage of using strongly intensive quantities. Similarly, intensive quantities have to becalculated in the most central collisions to approach the unbiased results.13 P /A æ P W Æ - Ar+Ar at 150A GeV/c b = 10% D W [N] w [N] w ] P *[N,W w P /A æ P W Æ - Ar+Ar at 150A GeV/c b = 10% D W [N] k / k [N] k / k ] P *[N,W k */ k P /A æ P W Æ - - - Ar+Ar at 150A GeV/c b = 10% D W [N] k / k [N] k / k ] P *[N,W k */ k Figure 6: The dependence of ω ∗ [ N, W P ] , κ ∗ [ N, W P ] /κ ∗ [ N, W P ] and κ ∗ [ N, W P ] /κ ∗ [ N, W P ] (full circles)as well as ω [ N ] , κ [ N ] /κ [ N ] and κ [ N ] /κ [ N ] (open circles) on the ratio (cid:104) W P (cid:105) /A P within the WoundedNucleon Model with input defined in Sec. V. Results are obtained in bins of b . The most right twopoints which correspond to ∆ b equal to and . The values for W = const are shown by dashedlines. P /A æ P W Æ Ar+Ar at 150A GeV/c = 4 P W D W [N] w [N] w ] P *[N,W w P /A æ P W Æ Ar+Ar at 150A GeV/c = 4 P W D W [N] k / k [N] k / k ] P *[N,W k */ k P /A æ P W Æ - - Ar+Ar at 150A GeV/c = 4 P W D W [N] k / k [N] k / k ] P *[N,W k */ k Figure 7: The dependence of ω ∗ [ N, W P ] , κ ∗ [ N, W P ] /κ ∗ [ N, W P ] and κ ∗ [ N, W P ] /κ ∗ [ N, W P ] (full circles)as well as ω [ N ] , κ [ N ] /κ [ N ] and κ [ N ] /κ [ N ] (open circles) on the ratio (cid:104) W P (cid:105) /A P within the WoundedNucleon Model with input defined in Sec. V. Results are obtained in bins of W P . The values for W = const are shown by dashed lines. I. SUMMARY
The paper addresses a currently important question of measuring event-by-event particlenumber fluctuations in nucleus-nucleus collisions unbiased by fluctuations of the collision vol-ume . Two methods to remove the influence of the volume fluctuations in fixed target experi-ments are reviewed and tested. Some of the limitations of strongly intensive quantities in thesetypes of analysis are shown. The results indicate the need to select the most central collisionsusing the number of projectile spectators.
Acknowledgments
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