Effect of centrality bin width corrections on two-particle number and transverse momentum differential correlation functions
Victor Gonzalez, Ana Marin, Pedro Ladron de Guevara, Jinjin Pan, Sumit Basu, Claude Pruneau
EEffect of centrality bin width corrections on two-particle number and transversemomentum differential correlation functions
Victor Gonzalez,
1, 2, ∗ Ana Marin, Pedro Ladron de Guevara,
4, 5
Jinjin Pan, † Sumit Basu, ‡ and Claude A. Pruneau Universidad Complutense de Madrid, Madrid 28040, Spain GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH,Research Division and Extreme Matter Institute EMMI, 64291 Darmstadt, Germany GSI Helmholtzzentrum f¨ur Schwerionenforschung,Research Division and ExtreMe Matter Institute EMMI, Darmstadt, Germany Universidad Complutense de Madrid, Spain Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, CP 04510, CDMX, Mexico Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA (Dated: March 28, 2019)Two-particle number and transverse momentum differential correlation functions are powerfultools for unveiling the detailed dynamics and particle production mechanisms involved in relativis-tic heavy-ion collisions. Measurements of transverse momentum correlators P and G , in particular,provide added information not readily accessible with better known number correlation functions R . However, it is found that the R and G correlators are somewhat sensitive to the details ofthe experimental procedure used to measure them. They exhibit, in particular, a dependence onthe collision centrality bin width, which may have a rather detrimental impact on their physicalinterpretation. A technique to correct these correlators for collision centrality bin-width averagingis presented. The technique is based on the hypothesis that the shape of single- and pair- proba-bility densities vary slower with collision centrality than the corresponding integrated yields. Thetechnique is tested with Pb-Pb simulations based on the HIJING and ultrarelativistic quantummolecular dynamics models and shown to enable a precision better than 1% for particles in thekinematic range 0 . ≤ p T ≤ . c . PACS numbers: 25.75.Gz, 25.75.Ld, 24.60.Ky, 24.60.-k ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ phy s i c s . d a t a - a n ] M a r I. INTRODUCTION
Measurements of correlation functions enable in-depth exploration of particle production mechanisms in relativisticheavy-ion collisions (HIC). Together with measurements of the nuclear modification factor, R AA , measured numbercorrelation functions have provided strong evidence for the formation of a dense and opaque medium, a quark gluonplasma (QGP), in the midst of high energy heavy-ion collisions [1–9]. Measurements of transverse momentum dif-ferential correlation functions have also been added to the experimental toolset [10, 11]. For these as for differentialnumber correlation functions, precision measurements require to account for a number of experimental conditions andartifacts [12]. In this paper, the sensitivity of two-particle differential correlators to the protocol used to account forcentrality bin width averaging is studied, and correction techniques to account for finite collision centrality bins usedin typical HIC analyses are considered.Whether measuring two-particle number or transverse momentum differential correlation functions towards thestudy of collective behavior, the elucidation of particle production mechanisms, or the analysis of collision systems’evolution towards equilibrium, most data analyses are typically limited by the available statistics. Analyses are thuscarried out with somewhat wide collision centrality bins and aim to report the evolution of the observables withcentrality while accounting for the limited number of events. It is well established [1–9] that both the shape andstrength of two-particle correlations evolve with collision centrality, i.e., the number, N w , of nucleons wounded in agiven collision. Normalized two-particle cumulant based correlators such as R , P , and G , defined below, scale as theinverse of the number of wounded N w , or the number of nucleon participants, for collisions involving no collectivityand no rescattering of secondaries. Both the amplitude and the shape of these correlators thus indeed evolve withcollision centrality. However, measurements of correlation functions are often carried out using fairly wide centralitybins amounting to 5%, 10%, or even larger values of the total interaction cross section. Effectively, if events withina given centrality bin are analyzed indiscriminately, measured values of the correlator R , P , and G will amountto some average across the width of the centrality bins. The issue arises, however, that such averages may not beproperly calculated unless one explicitly accounts for the fact these three correlators are ratios of quantities that mightbehave differently across the bins. The mean value theorem, written as f ( x ) = (cid:82) ba f ( x ) dxb − a , (1)stipulates that the mean value of a function across an interval [ a, b ] is equal to the value of the function evaluated at x , a value of x within the interval [ a, b ]. In general x does not correspond to the center of the bin. As will be shown,the correlator R is a ratio of a two-particle density by the square (essentially) of a single particle density. Both thesingle and pair densities are functions of centrality. Evaluation of the average of these densities across a centralitybin involves integrals of the form given by Eq. (1). The two functions are evaluated across the same interval but mayfeature significantly different dependence on the centrality, i.e., variable x in Eq. (1). The two estimated densitiesend up corresponding to different values of x (centrality). The R correlator, if evaluated as a ratio of these integrals,then amounts to a ratio of quantities evaluated at two distinct and unknown values of x . It is thus intrinsically biasedand does not constitute a proper estimator of the average of R across the centrality bin. A correction must thus bemade to account for or suppress this bias. It will be shown that similar considerations also apply to the correlators P and G . In addition to the width of the centrality bin, the magnitude of the corrections might also depend on therate at which these correlators evolve with collision centrality. It is thus useful to invoke existing heavy-ion collisionmodels to obtain quasirealistic correlation functions and simulate their evolution with collision centrality. These canthen be used to assess the magnitude of the corrections and systematic uncertainties associated with such corrections.Correction procedures similar to those presented in this work were used in prior studies but were never formallypublished. The STAR collaboration used the correction procedure towards the study of differential two-particletransverse momentum correlations in Au-Au collisions [13], whereas the ALICE collaboration for measurements ofthe R and P correlators in p -Pb and Pb-Pb collisions [14]. This work establishes a formal record and documentationof these procedures and presents an estimate of their accuracy based on HIJING and UrQMD, two semirealisticheavy-ion collisions models.This article is organized as follows. Inclusive and conditional particle densities are defined in Sec. II. The impactof the centrality bin width and a correction formula are first derived for normalized two-particle cumulants (two-particle number differential correlations) in Sec. III. The study is then extended to momentum correlation functionsin Sec. IV. In Sec. V, estimates of the precision of the method based on simulations carried out with the HIJING [15]and ultrarelativistic quantum molecular dynamics (UrQMD) [16, 17] particle event generators are presented. Thiswork is summarized in Sec. VI. II. OBSERVABLE DEFINITIONS
Whether analyzing pp , p -Pb, or A - A collisions, it is legitimate to classify events on the basis of the energy, E ref , orthe charged particle multiplicity, m , measured in a reference acceptance Ω ref . Studies [18] have shown that in A - A collisions, this multiplicity maps rather narrowly onto the impact parameter b of the collisions, particularly in thecase of mid- to central-collisions. While such narrow mapping does not exist for p -A or pp interactions, it remainsappropriate to classify collisions based on the multiplicity m as it provides some measure of the momentum transferand projectile kinetic energy dissipated in the collisions. In general, one indeed finds that the single and pair densitymeasured in the fiducial acceptance Ω ref are functions of the reference multiplicity m . Thus single- and two-particleconditional densities (i.e., for a given m ) are defined as ρ ( η, ϕ, p T | m ) = 1 p T dNdϕdηdp T (cid:12)(cid:12)(cid:12)(cid:12) m , (2) ρ ( η , ϕ , p T , , η , ϕ , p T , | m ) = 1 p T , p T , d Ndϕ dη dp T , dϕ dη dp T , (cid:12)(cid:12)(cid:12)(cid:12) m . (3)Typically, correlation analyses are carried out over fixed p T ranges [ p T , min , p T , max ]. One thus also defines conditionalsingle- and two-particle densities integrated over p T as ρ ( η, ϕ | m ) = (cid:90) ρ ( η, ϕ, p T | m ) dp T , (4) ρ ( η , ϕ , η , ϕ | m ) = (cid:90) ρ ( η , ϕ , p T , , η , ϕ , p T , | m ) dp T , dp T , , (5)where (here and in the following) it is understood that the integrals in p T are taken over the fixed range [ p T , min , p T , max ].In practice, it is usually not possible (or meaningful) to measure the densities ρ ( η, ϕ | m ) and ρ ( η , ϕ , η , ϕ | m ) forunit resolution in m . One must then evaluate the densities within finite width bins of multiplicity [ m min ,k , m max ,k ](where k = 1 , . . . , K , represents one of K “centrality” bins used in the analysis) as weighted average of the densitiesacross the bins according to ¯ ρ ( k )1 ( η, ϕ ) = 1 Q k m max ,k (cid:88) m = m min ,k q ( m ) ρ ( η, ϕ | m ) , (6)¯ ρ ( k )2 ( η , ϕ , η , ϕ ) = 1 Q k m max ,k (cid:88) m = m min ,k q ( m ) ρ ( η , ϕ , η , ϕ | m ) (7)with Q k = m max ,k (cid:88) m = m min ,k q ( m ) (8)and q ( m ) representing the probability of events with multiplicity m in the reference acceptance. III. NORMALIZED CUMULANTS: R Two-particle normalized cumulants [12] are defined according to R ( η , ϕ , η , ϕ ) = ρ ( η , ϕ , η , ϕ ) − ρ ( η , ϕ ) ρ ( η , ϕ ) ρ ( η , ϕ ) ρ ( η , ϕ ) (9)and form the basis of many two-particle correlation analyses. For a recent study on the evolution with centrality of thetwo-particle differential number density correlation R in p -Pb and Pb-Pb collisions, by the ALICE collaboration seeRef. [24]. Experimentally, in the study of A - A collisions, it is common to average R across large collision centralitybins. Given the width of such bins may depend on the specificities of a given experiment (e.g., the size of the referenceacceptance), normalized cumulants R are thus bin-width dependent. A correction procedure is then required toaccount for the finite width of the centrality bins.In principle, one would like to set the collision centrality bins in terms of collision impact parameter ranges (alter-natively the number of wounded nucleons). In practice, only proxies of the impact parameter are at best possible andone usually expresses the bins directly in terms of such proxies or in terms of the fractional cross section measuredwithin a centrality (multiplicity) bin. Let m , the multiplicity measured in the reference acceptance Ω ref representsuch a proxy. As for densities, one can thus define conditional normalized cumulants according to R ( η , ϕ , η , ϕ | m ) = ρ ( η , ϕ , η , ϕ | m ) ρ ( η , ϕ | m ) ρ ( η , ϕ | m ) − . (10)However, measurements of R ( η , ϕ , η , ϕ | m ) for m unit resolution are typically not possible due to limitationsassociated with the finite size of datasets, CPU time, or storage considerations. It is also impractical to reportdifferential correlators R ( η , ϕ , η , ϕ | m ) for a very large number of values of m . In practice, one thus seeks toreport correlators R averaged over reference multiplicity bins [ m min ,k , m max ,k ]. As for averages of densities definedin Sec. II, this is nominally achieved according to¯ R ( k )2 ( η , ϕ , η , ϕ ) = 1 Q k m max ,k (cid:88) m = m min ,k q ( m ) R ( η , ϕ , η , ϕ | m ) , (11)where q ( m ) represents the probability of events with multiplicity m in the reference acceptance. One may thendetermine R ( k )2 according to R (Bin ,k )2 ( η , ϕ , η , ϕ ) = ρ (Bin ,k )2 ( η , ϕ , η , ϕ ) ρ (Bin ,k )1 ( η , ϕ ) ρ (Bin ,k )1 ( η , ϕ ) −
1, (12)in which ρ (Bin ,k )1 and ρ (Bin ,k )2 are single and pair densities measured directly using finite width bins in m . In theabsence of biases or event detection inefficiencies, one has ρ (Bin ,k )1 ( η , ϕ ) = ¯ ρ ( k )1 ( η, ϕ ) , (13) ρ (Bin ,k )2 ( η , ϕ , η , ϕ ) = ¯ ρ ( k )2 ( η , ϕ , η , ϕ ) . (14)The quantities R (Bin ,k )2 and ¯ R ( k )2 are thus clearly distinct: R (Bin ,k )2 ( η , ϕ , η , ϕ ) = Q k (cid:80) m max ,k m = m min ,k q ( m ) ρ ( η , ϕ , η , ϕ | m ) (cid:104) Q k (cid:80) m max ,k m = m min ,k q ( m ) ρ ( η , ϕ | m ) (cid:105) (cid:104) Q k (cid:80) m max ,k m (cid:48) = m min ,k q ( m (cid:48) ) ρ ( η , ϕ | m (cid:48) ) (cid:105) −
1, (15)¯ R ( k )2 ( η , ϕ , η , ϕ ) = Q k m max ,k (cid:88) m = m min ,k q ( m ) ρ ( η , ϕ , η , ϕ | m ) ρ ( η , ϕ | m ) ρ ( η , ϕ | m ) −
1. (16)To quantify the difference, the particle densities are written in terms of single- and two-particle probability densities, P ( η, ϕ | m ) and P ( η , ϕ , η , ϕ | m ), according to ρ ( η, ϕ | m ) = (cid:104) n (cid:105) m P ( η, ϕ | m ) , (17) ρ ( η , ϕ , η , ϕ | m ) = (cid:104) n ( n − (cid:105) m P ( η , ϕ , η , ϕ | m ) , (18)where, by definition, P and P respectively satisfy (cid:90) dϕdη P ( η, ϕ | m ) = 1, (19) (cid:90) dϕ dη dϕ dη P ( η , ϕ , η , ϕ | m ) = 1, (20)as probability densities. The quantities (cid:104) n (cid:105) m and (cid:104) n ( n − (cid:105) m are the mean number of particles and the mean numberof pairs of particles in the acceptance of the measurement at a given reference multiplicity m ; P ( η, ϕ | m ) is theprobability of finding a particle at η, ϕ when the multiplicity is m , and P ( η , ϕ , η , ϕ | m ) is the joint probability ofmeasuring particles at η , ϕ and η , ϕ when the multiplicity is m . In general, one expects (cid:104) n (cid:105) m and (cid:104) n ( n − (cid:105) m toscale approximately linearly and quadratically, respectively, with m . Let us assume that the shape of P ( η, ϕ | m ) and P ( η , ϕ , η , ϕ | m ) change little through a centrality bin. One can then write R (Bin ,k )2 ( η , ϕ , η , ϕ ) = α ¯ P ( η , ϕ , η , ϕ )¯ P ( η , ϕ )¯ P ( η , ϕ ) − α = Q k (cid:80) m max ,k m = m min ,k q ( m ) (cid:104) n ( n − (cid:105) m (cid:16) Q k (cid:80) m max ,k m = m min ,k q ( m ) (cid:104) n (cid:105) m (cid:17) (22)and ¯ P ( η , ϕ ) and ¯ P ( η , ϕ , η , ϕ ) are the bin-width averaged values of the single- and two-particle probabilitydensities. Similarly, one can also write¯ R ( k )2 ( η , ϕ , η , ϕ ) = β ¯ P ( η , ϕ , η , ϕ )¯ P ( η , ϕ )¯ P ( η , ϕ ) − β = 1 Q k m max ,k (cid:88) m = m min ,k q ( m ) (cid:104) n ( n − (cid:105) m (cid:104) n (cid:105) m . (24)One thus finds that for sufficiently narrow bins (such that P and P can be considered approximately invariantwithin) ¯ R ( k )2 ( η , ϕ , η , ϕ ) = βα − (cid:16) R (Bin ,k )2 ( η , ϕ , η , ϕ ) + 1 (cid:17) − k . If theevent detection efficiency, ε ( m ), varies across [ m min ,k , m max ,k ], and the densities P and P are not biased by thisdependence, then the coefficients α and β may be written as α = Q k (cid:80) m max ,k m = m min ,k q ∗ ( m ) ε ( m ) (cid:104) n ( n − (cid:105) m (cid:16) Q k (cid:80) m max ,k m = m min ,k q ∗ ( m ) ε ( m ) (cid:104) n (cid:105) m (cid:17) , (26) β = 1 Q k m max ,k (cid:88) m = m min ,k q ∗ ( m ) ε ( m ) (cid:104) n ( n − (cid:105) m (cid:104) n (cid:105) m , (27)where q ∗ ( m ) is now the observed (uncorrected) probability distribution of multiplicity m . IV. TRANSVERSE MOMENTUM CORRELATORS
Several forms of transverse momentum correlators have been proposed and reported in the recent literature [10,11, 19–24]. In this work, the focus is on the G correlator proposed by Gavin et al . [11, 20–22] to study transversemomentum current correlations and the P correlator designed to be sensitive to transverse momentum fluctuations [10,23, 24]. A. G correlator At fixed reference multiplicity, m , the differential transverse momentum correlator G can be written as [20] G ( η , ϕ , η , ϕ | m ) = (cid:82) dp T , (cid:82) dp T , p T , p T , ρ ( η , ϕ , p T , , η , ϕ , p T , | m ) ρ ( η , ϕ | m ) ρ ( η , ϕ | m ) (28) −(cid:104) p T ( η , ϕ | m ) (cid:105)(cid:104) p T ( η , ϕ | m ) (cid:105) ,where (cid:104) p T ( η, ϕ | m ) (cid:105) = (cid:82) dp T p T ρ ( η, ϕ, p T | m ) ρ ( η, ϕ | m ) (29)is the event inclusive average particle transverse momentum at η , ϕ . Evaluation of the centrality-bin averaged¯ G ( k )2 ( η , ϕ , η , ϕ ) proceeds as for R and one writes¯ G ( k )2 ( η , ϕ , η , ϕ ) = 1 Q k m max ,k (cid:88) m = m min ,k q ( m ) G ( η , ϕ , η , ϕ | m ) . (30)Introducing S ( η, ϕ | m ) = (cid:90) p T dp T ρ ( η, ϕ, p T | m ) , (31)= (cid:104) n (cid:105) m P p T ( η, ϕ | m ) , (32) S ( η , ϕ , η , ϕ | m ) = (cid:90) p T , dp T , (cid:90) p T , dp T , ρ ( η , ϕ , p T , , η , ϕ , p T , | m ) , (33)= (cid:104) n ( n − (cid:105) m P p T p T ( η , ϕ , η , ϕ | m ) (34)with P p T ( η, ϕ | m ) = (cid:90) p T dp T P ( η, ϕ, p T | m ) , (35) P p T p T ( η , ϕ , η , ϕ | m ) = (cid:90) p T , dp T , (cid:90) p T , dp T , P ( η , ϕ , p T , , η , ϕ , p T , | m ) , (36)Eq. (28) is written as G ( η , ϕ , η , ϕ | m ) = S ( η , ϕ , η , ϕ | m ) ρ ( η , ϕ | m ) ρ ( η , ϕ | m ) − S ( η , ϕ | m ) ρ ( η , ϕ | m ) S ( η , ϕ | m ) ρ ( η , ϕ | m ) (37)= (cid:104) n ( n − (cid:105) m (cid:104) n (cid:105) m P p T p T ( η , ϕ , η , ϕ | m ) P ( η , ϕ | m ) P ( η , ϕ | m ) − P p T ( η , ϕ | m ) P ( η , ϕ | m ) P p T ( η , ϕ | m ) P ( η , ϕ | m ) . (38)If the ratios P p T / P and P p T p T / P P have a modest dependence on the reference multiplicity (within a bin k ), thenit is legitimate to replace them by averages and one gets G ( η , ϕ , η , ϕ | m ) = (cid:104) n ( n − (cid:105) m (cid:104) n (cid:105) m ¯ P p T p T ( k )2 ( η , ϕ , η , ϕ )¯ P ( k )1 ( η , ϕ ) ¯ P ( k )1 ( η , ϕ ) − ¯ P p T ( k )1 ( η , ϕ )¯ P ( k )1 ( η , ϕ ) ¯ P p T ( k )1 ( η , ϕ )¯ P ( k )1 ( η , ϕ ) . (39)The centrality-bin averaged correlator ¯ G ( k )2 ( η , ϕ , η , ϕ ) may then be written as¯ G ( k )2 ( η , ϕ , η , ϕ ) = β ¯ P p T p T ( k )2 ( η , ϕ , η , ϕ )¯ P ( k )1 ( η , ϕ ) ¯ P ( k )1 ( η , ϕ ) − ¯ P p T ( k )1 ( η , ϕ )¯ P ( k )1 ( η , ϕ ) ¯ P p T ( k )1 ( η , ϕ )¯ P ( k )1 ( η , ϕ ) (40)with β defined as in Eq. (24). However, if it is not possible to carry out the analysis in fine (unit) bins of m , thenumerators and denominators of G must be separately averaged over the range [ m min ,k , m max ,k ]. Assuming the ratios P p T / P and P p T p T / P P have a modest dependence on the reference multiplicity, one gets G (Bin ,k )2 ( η , ϕ , η , ϕ ) = α ¯ P p T p T ( k )2 ( η , ϕ , η , ϕ )¯ P ( k )1 ( η , ϕ ) ¯ P ( k )1 ( η , ϕ ) − ¯ P p T ( k )1 ( η , ϕ )¯ P ( k )1 ( η , ϕ ) ¯ P p T ( k )1 ( η , ϕ )¯ P ( k )1 ( η , ϕ ) (41)with α defined as in Eq. (22). Identifying (cid:104) p T ( η, ϕ ) (cid:105) (Bin ,k ) = ¯ P p T ( k )1 ( η, ϕ ) / ¯ P ( k )1 ( η, ϕ ), one finally gets that the desiredcorrelator ¯ G ( k )2 ( η , ϕ , η , ϕ ) may be determined as¯ G ( k )2 ( η , ϕ , η , ϕ ) = βα − (cid:16) G (Bin ,k )2 ( η , ϕ , η , ϕ ) + (cid:104) p T ( η , ϕ ) (cid:105) (Bin ,k ) (cid:104) p T ( η , ϕ ) (cid:105) (Bin ,k ) (cid:17) − (cid:104) p T ( η , ϕ ) (cid:105) (Bin ,k ) (cid:104) p T ( η , ϕ ) (cid:105) (Bin ,k ) , (42)= βα − G (Bin ,k )2 ( η , ϕ , η , ϕ ) + (cid:0) βα − − (cid:1) (cid:104) p T ( η , ϕ ) (cid:105) (Bin ,k ) (cid:104) p T ( η , ϕ ) (cid:105) (Bin ,k ) . B. P correlator The P correlator is defined as P ( η , ϕ , η , ϕ | m ) = 1 (cid:104) p T (cid:105) (cid:82) ∆ p T , dp T , (cid:82) ∆ p T , dp T , ρ ( η , ϕ , p T , , η , ϕ , p T , | m ) ρ ( η , ϕ , η , ϕ | m ) , (43)where ∆ p T , i ≡ p T , i − (cid:104) p T (cid:105) and the (cid:104) p T (cid:105) normalization insures P is dimensionless. Introducing (44) one next verifiesthat P is insensitive to multiplicity fluctuations [10, 23, 24]: P ∆ p T ∆ p T ( η , ϕ , η , ϕ | m ) = (cid:90) ∆ p T , dp T , (cid:90) ∆ p T , dp T , P ( η , ϕ , p T , , η , ϕ , p T , | m ). (44)Factorizing the two-particle density as in Eq. (18), one gets P ( η , ϕ , η , ϕ | m ) = (cid:18) P P p T (cid:19) (cid:104) n ( n − (cid:105) m (cid:104) n ( n − (cid:105) m P ∆ p T ∆ p T ( η , ϕ , η , ϕ | m ) P ( η , ϕ , η , ϕ | m ) . (45)If the ratios P p T / P and P ∆ p T ∆ p T / P have a modest dependence on the reference multiplicity (within a bin k ), thenit is legitimate to replace them by averages and one gets P ( η , ϕ , η , ϕ | m ) = (cid:32) ¯ P ( k )1 ¯ P p T ( k )1 (cid:33) ¯ P ∆ p T ∆ p T ( k )2 ( η , ϕ , η , ϕ )¯ P ( k )2 ( η , ϕ , η , ϕ ) , (46)which is indeed independent of the number of pairs and thus multiplicity fluctuations, since the functions ¯ P ( η , ϕ , η , ϕ | m )are probability densities. The centrality-bin averaged value of P is calculated according to¯ P ( k )2 ( η , ϕ , η , ϕ ) = (cid:32) ¯ P ( k )1 ¯ P p T ( k )1 (cid:33) ¯ P ∆ p T ∆ p T ( k )2 ( η , ϕ , η , ϕ )¯ P ( k )2 ( η , ϕ , η , ϕ ) , (47)while the ratio of bin averages is P (Bin ,k )2 ( η , ϕ , η , ϕ ) = (cid:32) ¯ P ( k )1 ¯ P p T ( k )1 (cid:33) ¯ P ∆ p T ∆ p T ( k )2 ( η , ϕ , η , ϕ )¯ P ( k )2 ( η , ϕ , η , ϕ ) . (48)One thus finds that ¯ P ( k )2 ( η , ϕ , η , ϕ ) = P (Bin ,k )2 ( η , ϕ , η , ϕ ), (49)featuring a unitary correction factor. The P correlator is thus indeed not affected by the width of the multiplicitybins so long as the probability densities P and P exhibit only modest shape variations within those bins. - h D - · C I R HIJING Pb--Pb 2.76 TeV CI - h D - · C I R - h D - h D - h D - · CD R < 2.0 GeV T p jD | CD - h D - · CD R average corrected - h D avg/corr - h D not corr/corr FIG. 1. Longitudinal projection of the two-particle normalized cumulants R CI2 (left) and R CD2 (right) for the most centralPb-Pb collisions produced with the HIJING event generator. Top panels show uncorrected results for the 0–5 % centralitybin together with those from the 0–1 %, 2–3 %, and 4–5 % centrality bins, middle panels show corrected 0–5 % centrality binresults compared with the weighted average of those from the 0–1 %, 1–2 %, 2–3 %, 3–4 %, and 4–5 % centrality bins, whilebottom panels show their ratio.
V. SIMULATIONS
The accuracy of the centrality bin width correction methods defined by Eqs. (25) and (42) is tested using HIJINGand UrQMD simulations of 5% most central central Pb-Pb collisions at √ s NN = 2 .
76 TeV. Overall, 6 . × and 10 Pb-Pb collisions from HIJING and UrQMD were used for this study. The collision centrality classification used forHIJING events mimics the method used by the ALICE collaboration [25, 26] and is based on the number of chargedparticles in the pseudo-rapidity ranges 2 . < η < . − . < η < − . R , G , and P correlators for charged particle combinations(+ − , − +, −− and ++) are first determined as four-dimensional functions based on generator level charged hadronsin the ranges 0 . ≤ p T ≤ . c and | η | < . - hD - · C I R UrQMD Pb--Pb 2.76 TeV CI - hD - · C I R - hD - hD - hD - · CD R < 2.0 GeV T p jD | CD - hD - · CD R average corrected - hD avg/corr - hD not corr/corr FIG. 2. Longitudinal projection of the two-particle normalized cumulants R CI2 (left) and R CD2 (right) for 5% most centralPb-Pb collisions produced with the UrQMD event generator. Top panels show uncorrected results for the 0–5 % centralitybin together with those from the 0–1 %, 2–3 %, and 4–5 % centrality bins, middle panels show corrected 0–5 % centrality binresults compared with the weighted average of those from the 0–1 %, 1–2 %, 2–3 %, 3–4 %, and 4–5 % centrality bins, whilebottom panels show their ratio. (CI) and charge dependent (CD) particle combinations are calculated according to C CI = 14 (cid:104) C (+ − ) + C ( − +) + C ( −− ) + C (++) (cid:105) (50) C CD = 14 (cid:104) C (+ − ) + C ( − +) − C ( −− ) − C (++) (cid:105) . (51)where C stands for any of the R , G , and P correlators. Projections of these correlators onto ∆ η and ∆ ϕ werecomputed using the integer arithmetic technique described in [12].Figure 1 presents ∆ η projections of the R CI2 (left) and R CD2 (right) correlation functions based on 5% most centralHIJING events. The top panels of the figure present uncorrected results for 0–5 % (blue), 0–1 % (green), 2–3 % (red),and 4–5 % (black). The bottom panels displays 0–5 % centrality results corrected with Eq. (25) (blue) compared tothe weighted mean of the correlators obtained for 0–1 %, 1–2 %, 2–3 %, 3–4 %, and 4–5 % centralities (red) and theirratio. One finds that the results corrected with Eq. (25) agree with those obtained with the weighted mean within1% for both R CI2 and R CD2 .Similar results are presented in Fig. 2 for the UrQMD model. In this case, agreement between the 0–5 % corrected R correlators and the weighted mean of the R correlators obtained in 0–1 %, 1–2 %, 2–3 %, 3–4 %, and 4–5 %centralities is found to be also within 1% for both R CI2 and R CD2 . Thus, based on the HIJING and UrQMD modelsimulations presented, it is concluded that Eq. (25) enables reasonably accurate corrections of the R correlators inthe context of these two models. Given these models provide relatively realistic representations of single and pairparticle spectra, the correction method embodied in Eq. (25) should provide reasonably reliable bin-width corrections0 - h D - · ) ) c (( G e V / C I G HIJING Pb--Pb 2.76 TeV < 2.0 GeV T p CI - h D - · ) ) c (( G e V / C I G - h D - h D - h D - · ) ) c (( G e V / C I G HIJING Pb--Pb 2.76 TeV < 0.8 GeV T p CI - h D - · ) ) c (( G e V / C I G - h D - h D - h D - · ) ) c (( G e V / CD G < 2.0 GeV T p jD | CD - h D - · ) ) c (( G e V / CD G average corrected - h D avg/corr - h D not corr/corr FIG. 3. Longitudinal projection of the two-particle transverse momentum correlations G CI2 for default transverse momentumrange particles (left) and for the reduced (see text) transverse momentum range particles (center) and G CD2 for the defaulttransverse momentum range particles (right) for the 5% most central Pb-Pb collisions produced with the HIJING eventgenerator. Top panels show uncorrected results for the 0–5 % centrality bin together with those from the 0–1 %, 2–3 %, and4–5 % centrality bins, middle panels show corrected 0–5 % centrality bin results compared with the weighted average of thosefrom the 0–1 %, 1–2 %, 2–3 %, 3–4 %, and 4-5% centrality bins, while bottom panels show their ratio. of R correlation functions measured at any heavy ion collider.Figures 3 and 4 present ∆ η projections of the G CI2 and G CD2 correlation functions based on 5% most central eventsfrom HIJING and UrQMD models respectively. Similarly as in previous figures, the top panels display uncorrected G correlators for 0–5 % (blue), 0–1 % (green), 2–3 % (red), and 4–5 % (black) while the bottom panels display 0–5% centrality results corrected with Eq. (42) (blue) compared to the weighted mean of the G correlators obtained for0–1 %, 1–2 %, 2–3 %, 3–4 %, and 4–5 % centralities (red) and their ratio. Comparing the corrected results with theweighted averages for HIJING events one finds them within 7% for G CI2 and within 1% for G CD2 while the comparisonfor UrQMD events leaves them within 1% for both G CI2 and G CD2 .The discrepancy between the correction level achieved for G CI2 for collisions simulated with HIJING compared withthose produced using UrQMD is hardly satisfactory, especially considering the 1% correction precision achieved, withboth models, for the R CI2 correlator. Given the transverse momentum enters explicitly into the expression of the G correlator, but not in R , to determine whether the p T range considered may influence the correction precisionachievable with Eq. (42) the above study is repeated for charged particles in the range 0.2–0.8 GeV/ c . Central panelsof Fig. 3 display ∆ η projections of the G CI2 correlation function based on 5% most central events from HIJING modelfor particle within the transverse momentum range 0.2–0.8 GeV/ c . As before, the top panel shows a comparison ofthe correlators calculated within the 0–5 % centrality bin with those from the 0–1 %, 2–3 %, and 4–5 % centralitybins, while the bottom panel displays 0–5 % centrality results corrected with Eq. (42) compared to the weightedmean of the values obtained in the 0–1 %, 1–2 %, 2–3 %, 3–4 %, and 4–5 % collision centrality bins. It is foundthat the 0–5 % centrality corrected result is now within 1% of the weighted mean of those obtained with finercentrality bins. The centrality dependence of the single- and two-particle probability densities [and their p T weightedcounterparts, Eqs. (35) and (36)] is then examined, and found that those obtained within the 0.2–0.8 GeV/ c rangefeature a quantitatively smaller dependence of collision centrality than those integrated in the 0.2–2.0 GeV/ c range.The former range consequently better satisfies the condition, implicit in Eq. (41), that the densities be approximatelyindependent of collision centrality, within each centrality bin k . That these densities, calculated within the range 0.2–2.0 GeV/ c with UrQMD events, similarly exhibit very little shape dependence on collision centrality is additionally1 - hD - · ) ) c (( G e V / C I G UrQMD Pb--Pb 2.76 TeV CI - hD - · ) ) c (( G e V / C I G - hD - hD - hD - · ) ) c (( G e V / CD G < 2.0 GeV T p jD | CD - hD - · ) ) c (( G e V / CD G average corrected - hD avg/corr - hD not corr/corr FIG. 4. Longitudinal projection of the two-particle transverse momentum correlations G CI2 (left) and G CD2 (right) for the 5%most central Pb-Pb collisions produced with the UrQMD event generator. Top panels show uncorrected results for the 0–5% centrality bin together with those from the 0–1 %, 2–3 %, and 4–5 % centrality bins, middle panels show corrected 0–5 %centrality bin results compared with the weighted average of those from the 0–1 %, 1–2 %, 2–3 %, 3–4 %, and 4–5 % centralitybins, while bottom panels show their ratio. verified. This analysis confirms the relevance of the proper behavior of this magnitudes established in Sec. IV A. Asconclusion then, the finite centrality bin width correction, included in Eq. (42), under the criteria established for itsdeduction, provides a reasonably accurate correction technique for G correlators to account for collision centralityaverage when using finite collision centrality bins.It is worth highlighting that, consistently with what was naively expected, the centrality bin width effect is com-pletely charge independent. Indeed, the amplitude of both R CI2 and G CI2 are substantially affected by the widecentrality bin averaging while R CD2 and G CD2 only experience a minor shift owing (1) to the very small differences be-tween the R LS2 and R US2 correlator dependence on particle densities and on the transverse momentum of the particles,and (2) to the approximately equal number of US and LS pairs observed at large √ s NN in the central rapidity range.For the sake of completeness, the behavior of the correction procedures on the ∆ ϕ projections of R CI2 (for thewhole p T range) and G CI2 (for the reduced p T range) correlators are shown in Fig. 5. The correction procedures yieldessentially perfect corrections of these ∆ ϕ projections for both correlators. Additionally, the applicability of Eq. (49)for P CI2 correlators is demonstrated in Fig. 6 where it is shown that the P CI2 correlator determined with a 0–5 %centrality bin is virtually identical to that obtained with a weighted average of the correlators calculated from the0–1 %, 1–2 %, 2–3 %, 3–4 %, and 4–5 % collision centrality bins.Finally, in order to obtain a quantitative assessment of the precision of the centrality bin width correction proceduresembodied by Eqs. (25), (49), (42), the ratios of the centrality bin width corrected correlators to those obtained withweighted averages of correlators obtained in fine width centrality bins, shown in Figs. 1–6, are fitted with a constantpolynomial (POL0 in ROOT [27]). Fitted values of the ratios, corresponding to ratios averaged across the ∆ η and2 (radians) j D - · C I R HIJING Pb--Pb 2.76 TeV < 2.0 GeV T p CI (radians) j D - · C I R (radians) j D (radians) j D (radians) j D - · ) ) c (( G e V / C I G < 0.8 GeV T p CI (radians) j D - · ) ) c (( G e V / C I G average corrected (radians) j D avg/corr (radians) j D not corr/corr FIG. 5. Azimuthal projections of the two-particle transverse momentum correlations R CI2 for the whole p T range (left) and G CI2 for the reduced p T range (right) for the 5% most central Pb-Pb collisions produced with the HIJING event generator. Toppanels show uncorrected results for the 0–5 % centrality bin together with those from the 0–1 %, 2–3 %, and 4–5 % centralitybins, middle panels show corrected 0–5 % centrality bin results compared with the weighted average of those from the 0–1 %,1–2 %, 2–3 %, 3–4 %, and 4–5 % centrality bins, while bottom panels show their ratio.TABLE I. POL0 fit to the ratio 0–5 % centrality corrected versus 0–1 %, 1–2 %, 2–3 %, 3–4 %, and 4–5 % centralities weightedmean.Model proj. R CI2 R CD2 P CI2 P CD2 G CI2 G CD2
HIJING ∆ η . ± .
001 0 . ± .
003 0 . ± .
002 0 . ± .
004 0 . ± .
001 0 . ± . ϕ . ± .
001 0 . ± .
003 0 . ± .
003 0 . ± .
008 0 . ± .
001 0 . ± . η . ± . . ± . . ± . . ± . . ± .
001 1 . ± . ϕ . ± . . ± . . ± . . ± . . ± .
001 0 . ± . p T ∆ η . ± .
001 0 . ± . p T ∆ ϕ . ± .
001 0 . ± . ∆ ϕ fiducial ranges, are listed in Table I.Overall, it is found that the averaged ratios have values consistent with unity within statistical errors (i.e., withina ± σ , 96% confidence interval), thereby implying that within the statistical precision achieved in this work, it canbe concluded that the correction procedures do not introduce a significant bias to the correlation functions. Notableexceptions are the values obtained with both ∆ η and ∆ φ projections for G CI2 when this correlator is calculated forcharged particles in the range 0 . ≤ p T ≤ . c with HIJING. However, the ratios of these projections areconsistent with unity when the correlator is calculated in a narrower p T range in line with the demanded behavior ofthe probability density distributions. A large discrepancy is also seemingly observed for the P CD2 correlator obtainedwith the UrQMD simulation. However, the overall magnitude of P CD2 predicted by UrQMD is a seventh of that3 - h D - · C I P HIJING Pb--Pb 2.76 TeV < 2.0 GeV T p CI - h D - · C I P - h D - h D - hD - · C I P UrQMD Pb--Pb 2.76 TeV| < 1.0 jD | CI - hD - · C I P average corrected - hD avg/corr - hD not corr/corr FIG. 6. Longitudinal projection of the two-particle transverse momentum correlation P CI2 for the 5% most central Pb-Pb colli-sions produced with the HIJING event generator (left) and the UrQMD event generator (right). Top panels show uncorrectedresults for the 0–5 % centrality bin together with those from the 0–1 %, 2–3 %, and 4–5 % centrality bins, middle panels showcorrected 0–5 % centrality bin results compared with the weighted average of those from the 0–1 %, 1–2 %, 2–3 %, 3–4 %, and4–5 % centrality bins, while bottom panels show their ratio. obtained with HIJING events and correlation amplitudes are nearly vanishing, within statistical accuracy, across awide ∆ η range. A proper evaluation of the correction is thus more challenging in this case. VI. SUMMARY
Studies of the centrality or particle multiplicity evolution of the integral and amplitudes of two-particle number andtransverse momentum differential correlations, as tools to gain insights into particle production and transport dynam-ics in heavy-ion collisions, are a growing research field of interest, as recent analyses from the ALICE collaborationshow [23, 24]. The bias introduced on the amplitudes and integrals of R , G , and P correlators when measureddirectly in wide collision centrality bins, was shown. Correction techniques to be applied to those correlators, whenthe need for them to be measured on such centrality bins is in place, were described. These correction techniqueswere tested with HIJING and UrQMD simulations of Pb-Pb collisions at √ s NN = 2 .
76 TeV. They enable a precisionof the order of 1%, or better, when using 5% wide centrality bins. Clearly, the reached precision has a direct impacton both the amplitude and the integral of the correlation functions which will allow a better description or constrainof the underlying physics.4
ACKNOWLEDGEMENTS
The authors thank the ALICE collaboration for access to a sample of simulated HIJING events used in thisanalysis and the GSI Helmholtzzentrum f¨ur Schwerionenforschung for providing the computational resources neededfor producing the UrQMD events used in this analysis. This work was supported in part by the United StatesDepartment of Energy, Office of Nuclear Physics (DOE NP), United States of America under Award No. DE-FG02-92ER-40713 and by Consejo Nacional de Ciencia y Tecnolog´ıa (CONACYT), through Fondo de Cooperaci´onInternacional en Ciencia y Tecnolog´ıa (FONCICYT) and Direcci´on General de Asuntos del Personal Acad´emico(DGAPA), Universidad Nacional Aut´onoma de M´exico, Mexico. [1] J. Adams et al. (STAR), Nucl. Phys.
A757 , 102 (2005), arXiv:nucl-ex/0501009 [nucl-ex].[2] K. Adcox et al. (PHENIX), Nucl. Phys.
A757 , 184 (2005), arXiv:nucl-ex/0410003 [nucl-ex].[3] I. Arsene et al. (BRAHMS), Nucl. Phys.
A757 , 1 (2005), arXiv:nucl-ex/0410020 [nucl-ex].[4] B. B. Back et al. , Nucl. Phys.
A757 , 28 (2005), arXiv:nucl-ex/0410022 [nucl-ex].[5] K. Aamodt et al. (ALICE), Phys. Rev. Lett. , 252301 (2010), arXiv:1011.3916 [nucl-ex].[6] K. Aamodt et al. (ALICE), Phys. Rev. Lett. , 252302 (2010), arXiv:1011.3914 [nucl-ex].[7] G. Aad et al. (ATLAS Collaboration), Phys. Rev. Lett. , 252303 (2010).[8] J. Adam et al. (ALICE), Phys. Rev. Lett. , 222302 (2016), arXiv:1512.06104 [nucl-ex].[9] J. Adam et al. (ALICE), Phys. Rev. Lett. , 132302 (2016), arXiv:1602.01119 [nucl-ex].[10] M. Sharma and C. A. Pruneau, Phys. Rev.
C79 , 024905 (2009), arXiv:0810.0716 [nucl-ex].[11] H. Agakishiev et al. (STAR), Phys. Lett.
B704 , 467 (2011), arXiv:1106.4334 [nucl-ex].[12] S. Ravan, P. Pujahari, S. Prasad, and C. A. Pruneau, Phys. Rev. C , 024906 (2014).[13] J. Adams et al. (STAR), Phys. Rev. C72 , 044902 (2005), arXiv:nucl-ex/0504031 [nucl-ex].[14] B. B. Abelev et al. (ALICE), Eur. Phys. J.
C74 , 3077 (2014), arXiv:1407.5530 [nucl-ex].[15] X.-N. Wang and M. Gyulassy, Phys. Rev. D , 3501 (1991).[16] S. A. Bass et al. , Prog. Part. Nucl. Phys. , 255 (1998), [Prog. Part. Nucl. Phys.41,225(1998)],arXiv:nucl-th/9803035 [nucl-th].[17] M. Bleicher et al. , J. Phys. G25 , 1859 (1999), arXiv:hep-ph/9909407 [hep-ph].[18] R. Rogly, G. Giacalone, and J.-Y. Ollitrault, (2018), arXiv:1804.03031 [nucl-th].[19] J. Adams et al. (STAR), J. Phys.
G32 , L37 (2006), arXiv:nucl-ex/0509030 [nucl-ex].[20] M. Abdel-Aziz and S. Gavin, Acta Phys. Hung.
A25 , 515 (2006).[21] S. Gavin, G. Moschelli, and C. Zin,
Proceedings, 32th Winter Workshop on Nuclear Dynamics (WWND 2016): Guadeloupe,French West Indies , J. Phys. Conf. Ser. , 012020 (2016), arXiv:1608.05389 [nucl-th].[22] S. Gavin, G. Moschelli, and C. Zin, Phys. Rev.
C94 , 024921 (2016), arXiv:1606.02692 [nucl-th].[23] S. Acharya et al. (ALICE Collaboration), Phys. Rev. Lett. , 162302 (2017), arXiv:1702.02665 [nucl-ex].[24] S. Acharya et al. (ALICE), (2018), arXiv:1805.04422 [nucl-ex].[25] B. Abelev et al. (ALICE), Phys. Rev.
C88 , 044909 (2013), arXiv:1301.4361 [nucl-ex].[26] B. B. Abelev et al. (ALICE), Int. J. Mod. Phys.
A29 , 1430044 (2014), arXiv:1402.4476 [nucl-ex].[27] R. Brun and F. Rademakers,
New computing techniques in physics research V. Proceedings, 5th International Workshop,AIHENP ’96, Lausanne, Switzerland, September 2-6, 1996 , Nucl. Instrum. Meth.