Identity method to study chemical fluctuations in relativistic heavy-ion collisions

PHYSICAL REVIEW C 83 054907 (2011)

Identity method to study chemical fluctuations in relativistic heavy-ion collisions

Marek GazdzickiGoethe-Universitat Frankfurt Max-von-Laue Strasse 1 D-60438 Frankfurt am Main Germany and

Institute of Physics Jan Kochanowski University ul Swietokrzyska 15 PL-25-406 Kielce Poland

Katarzyna Grebieszkow and Maja MackowiakFaculty of Physics Warsaw University of Technology ul Koszykowa 75 PL-00-662 Warszawa Poland

Stanisław MrowczynskiInstitute of Physics Jan Kochanowski University ul Swietokrzyska 15 PL-25-406 Kielce Poland and

Sołtan Institute for Nuclear Studies ul Hoza 69 PL-00-681 Warszawa Poland(Received 24 March 2011 published 17 May 2011)

Event-by-event fluctuations of the chemical composition of the hadronic final state of relativistic heavy-ioncollisions carry valuable information on the properties of strongly interacting matter produced in the collisionsHowever in experiments incomplete particle identification distorts the observed fluctuation signals The effect isquantitatively studied and a new technique for measuring chemical fluctuations the identity method is proposedThe method fully eliminates the effect of incomplete particle identification The application of the identity methodto experimental data is explained

DOI 101103PhysRevC83054907 PACS number(s) 2575Gz

I INTRODUCTION

Event-by-event fluctuations of chemical (particle-type)composition of hadronic final states of relativistic heavy-ioncollisions are expected to be sensitive to properties of stronglyinteracting matter produced in the collisions [1] Specificfluctuations can signal the onset of deconfinement when thecollision energy becomes sufficiently high to create droplets ofquark-gluon plasma [2] At higher collision energies where thequark-gluon phase is abundantly produced at the early collisionstage large chemical fluctuations can occur as the system hitsthe critical point of strongly interacting matter in the courseof its temporal evolution [34] It is thus certainly of interestto study event-by-event chemical fluctuations experimentallyFirst data coming from the CERN SPS [5ndash7] and BNL RHIC[8] were already published The results are not very conclusiveyet and more systematic measurements are needed In additionthe question arises whether data analysis methods can beimproved

In real experiments it is impossible to determine uniquelythe type of every detected particle The identification requiresmeasurements of particle electric charge and mass Precisemass measurements are experimentally difficult and expen-sive For this reason analyses of chemical fluctuations areusually performed in a limited acceptance where particleidentification is relatively reliable However sensitivity tofluctuations of range larger than the acceptance windowis then lost and signals from fluctuations of shorter rangeare usually diluted Furthermore it should be noted thatresults on fluctuations unlike those on single-particle spectracannot be corrected for the limited acceptance Often it ispossible to enlarge the acceptance but at the expense ofa significant contamination of the sample by misidentifiedparticles The effect of particle misidentification can distortmeasured fluctuations Thus incomplete particle identification

is a serious obstacle to the precise measurement of chemicalfluctuations

Although it is usually impossible to identify every detectedparticle one can in general determine with high accuracy thepercentage (averaged over many interactions) of say kaonsamong produced hadrons This information will be shownto be sufficient to fully eliminate the effect of incompleteidentification In this paper we propose a new experimentaltechnique called the identity method which achieves thegoal independently of the specific properties of the chemicalfluctuations under study

In the study of chemical fluctuations the NA49 Collabo-ration [5ndash7] used the measure σdyn which is defined as thedifference between fluctuations measured in real and mixedevents The effect of particle misidentification is accounted forby including it in the mixed events The STAR Collaboration[8] used in addition to the σdyn measure the quantity νdyn Thelatter one assumes that particles are uniquely identified

It was suggested long ago [910] to quantify chemicalfluctuations by the measure [11] which proved to beefficient in experimental studies of event-by-event fluctuationsof particle transverse momentum [1213] electric charge [14]and quite recently of azimuthal angle [15] The measureunlike σdyn and νdyn is a strongly intensive measure offluctuations Namely its magnitude is independent of thenumber and of the distribution (fluctuation) of the number ofparticle sources if the sources are identical and independentfrom each other This feature which is discussed in detail inRef [16] is important in experimental studies of relativisticheavy-ion collisions where the collision centrality is neverfully controllable However up to now it was unclear how tocorrect measurements of chemical for the effect of particlemisidentification

054907-10556-2813201183(5)054907(7) copy2011 American Physical Society

MAREK GAZDZICKI et al PHYSICAL REVIEW C 83 054907 (2011)

The identity method which is developed here uses thefluctuation measure a simple modification of The mea-sure similar to is strongly intensive but the modificationmakes it possible to correct the measurements for the effect ofparticle misidentification Below we show that the measure

can be factorized into a coefficient which represents the effectof misidentification and the quantity CI which correspondsto the value would have for complete identification Themisidentification coefficient can be determined from the datain a model independent way Therefore the identity methodprovides the value of the fluctuation measure as it would beobtained in an experiment in which every particle is uniquelyidentified

Before the identity method is presented we introduce anddiscuss in Sec II the fluctuation measures σdyn νdyn and In Sec III we demonstrate by a Monte Carlo simulation howthe effect of misidentification distorts the chemical fluctuationsas quantified by νdyn and The identity method isformulated in Sec IV Instead of conclusions we present inthe last section the steps required to apply the identity methodto experimental data To simplify the presentation we considerchemical fluctuations of events composed only of kaons andpions Clearly kaons and pions can be replaced by particles ofany other sort

II MEASURES OF FLUCTUATIONS

As mentioned in the Introduction fluctuations of chemicalcomposition of final states of relativistic heavy-ion collisionscan be studied in several ways The NA49 Collaboration [5ndash7]measured event-by-event fluctuations of the particle ratiosKπ Kp pπ and determined the quantity σdyn defined as

σdyn = sgn(σ 2

data minus σ 2mixed

)radic∣∣σ 2data minus σ 2

mixed

∣∣ (1)

where σdata and σmixed are the relative width (the width dividedby the mean) of the event-by-event particle ratio distributionin respectively the data and artificially generated mixedevents where every particle comes from a different real eventThe fluctuations present in mixed events are attributable tothe effect of particle misidentification and the statistical noisecaused by the finite number of particles

The STAR Collaboration used [8] the quantity νdyn to mea-sure chemical fluctuations For the case of a two-componentsystem of pions and kaons νdyn is defined as

νdyn = 〈NK (NK minus 1)〉〈NK〉2

+ 〈Nπ (Nπ minus 1)〉〈Nπ 〉2

minus 2〈NKNπ 〉

〈NK〉〈Nπ 〉 (2)

where NK and Nπ are the numbers of kaons and pions ina given event and 〈middot middot middot〉 denotes averaging over events Thequantity νdyn is defined in such a way that in particular itvanishes when the multiplicity distributions of pions and kaonsare both Poissonian (〈Ni(Ni minus 1)〉 = 〈Ni〉2 i = π K) and in-dependent from each other (〈NKNπ 〉 = 〈NK〉〈Nπ 〉) Thus it isconstructed to quantify the deviations of the fluctuations fromthe Poissonian noise For large-enough particle multiplicitiesone finds the approximate relation νdyn asympσ 2

data minus σ 2mixed which

gives νdyn asymp sgn(σdyn) σ 2dyn [8] We note that the quantity νdyn

implicitly assumes unique identification of all particles

As already noted it was advocated long ago [910] toemploy the measure [11] to study chemical fluctuations Themeasure is defined in the following way One introduces thevariable z equiv x minus x where x is a single-particle characteristicsuch as the transverse momentum or azimuthal angle Theoverline denotes averaging over the single-particle inclusivedistribution The event variable Z which is a multiparticleanalog of z is defined as Z equiv sumN

i=1(xi minus x) where the sumruns over the N particles in a given event By construction〈Z〉 = 0 The measure is finally defined as

equivradic

〈Z2〉〈N〉 minus

radicz2 (3)

The measure vanishes in the absence of interparticlecorrelations This situation is discussed in some detail belowfor the case of chemical fluctuations Here we note thatthe measure also possesses another important property It isstrongly intensive which means that it is independent of thenumber and of the distribution (fluctuation) of the number ofparticle sources if the sources are identical and independentfrom each other In particular if a nucleus-nucleus collisionis a simple superposition of nucleon-nucleon interactionsthen AA = NN The strongly intensive property is a veryvaluable feature of because centrality selection in relativisticheavy-ion collisions is never perfect and events of differentnumbers of particle sources are always mixed up The stronglyintensive property is also desirable when different centralitiesor different colliding systems are compared For a discussionof strongly intensive quantities see Ref [16]

The analysis of chemical fluctuations can be performedwith the help of in two different but fully equivalent waysIn the first method [9] using the identity variable chemicalfluctuations are treated in analogy to fluctuations of transversemomentum In the second method is calculated from themoments of the multiplicity distributions [10]

We next describe the first method for the example of atwo-component system of pions and kaons One defines thesingle-particle variable x as x = wK where wK is called thekaon identity and wi

K = 1 if the ith particle is a kaon andwi

K = 0 if the ith particle is a pion This implies uniqueparticle identification One then directly uses the definition(3) to evaluate

Let us now discuss the most important case for which themeasure of chemical fluctuations vanishes Because theinclusive distribution of wK equals

P (wK ) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

〈Nπ 〉〈N〉 for wK = 0

〈NK〉〈N〉 for wK = 1

(4)

one finds

z2 equiv w2K minus wK

2 = 〈NK〉〈N〉

(1 minus 〈NK〉

〈N〉)

(5)

When interparticle correlations are absent the distribution ofparticle identities in events of multiplicity N reads

PN

(w1

Kw2K wN

K

) = PNP(w1

K

)P

(w2

K

) middot middot middotP (wN

K

) (6)

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IDENTITY METHOD TO STUDY CHEMICAL PHYSICAL REVIEW C 83 054907 (2011)

where PN is an arbitrary multiplicity distribution of parti-cles of any type One shows that 〈Z2〉 computed with theevent distribution (6) equals 〈Z2〉 = 〈N〉z2 and consequently = 0

In the second method the measure of chemical fluctu-ations is obtained from the moments of the experimentallymeasured multiplicity distributions of kaons and pions Asshown in Ref [10] one has

z2 = 〈NK〉〈Nπ 〉〈N〉2

(7)

〈Z2〉〈N〉 = 〈Nπ 〉2

〈N〉3

(langN2

K

rang minus 〈NK〉2) + 〈NK〉2

〈N〉3

(langN2

π

rang minus 〈Nπ 〉2)

minus2〈NK〉〈Nπ 〉

〈N〉3(〈NKNπ 〉 minus 〈NK〉〈Nπ 〉) (8)

which substituted in Eq (3) gives the measure The formulas (7) and (8) clearly show that like νdyn

vanishes when the multiplicity distributions of pions andkaons are both Poissonian and independent from each otherHowever more generally vanishes for any multiplicitydistribution provided it satisfies (6) The distribution (6) leadsto the multiplicity distribution of the form

PNKNπ= PNK+Nπ

times(

NK + Nπ

NK

)( 〈NK〉〈N〉

)NK(

1 minus 〈NK〉〈N〉

)Nπ

(9)

with the moments

〈NK (NK minus 1)〉 = 〈NK〉2

〈N〉2〈N (N minus 1)〉 (10)

〈Nπ (Nπ minus 1)〉 = 〈Nπ 〉2

〈N〉2〈N (N minus 1)〉 (11)

〈NKNπ 〉 = 〈NK〉〈Nπ 〉〈N〉2

〈N (N minus 1)〉 (12)

One checks that and νdyn both vanish when these momentsare substituted into Eq (8) and Eq (2) respectively

It appears convenient for our further considerations tomodify to the form

= 〈Z2〉〈N〉 minus z2 (13)

which preserves the properties of it vanishes in the absenceof interparticle correlations and it is strongly intensive Themeasure will be used to formulate the identity method forthe study of chemical fluctuations When expressed throughmoments of the multiplicity distribution it equals

= 1

〈N〉3

[langN2

π

rang〈NK〉2 + 〈Nπ 〉2langN2K

rangminus2〈Nπ 〉〈NK〉〈NπNK〉 minus 〈Nπ 〉2〈NK〉 minus 〈Nπ 〉〈NK〉2

]

(14)

Comparing Eq (2)) to Eq (14) one finds that and νdyn areproportional to each other

= 〈Nπ 〉2〈NK〉2

〈N〉3νdyn (15)

We note here that νdyn is not intensive but it becomes evenstrongly intensive when multiplied by 〈N〉〈NK〉 or 〈Nπ 〉

III EFFECT OF MISIDENTIFICATION

As mentioned in the Introduction complete identificationof every particle is impossible In this section we show how theincomplete particle identification influences the magnitudes offluctuation measures For this purpose we considered a simplemodel of chemical fluctuations where the multiplicity of pionsis Poissonian with a mean value of 100 and the number ofkaons is 20 of the number of pions (strict correlation of thenumbers of kaons and pions) Actually NK is taken as theinteger number closest to Nπ5 which is smaller or equalto Nπ5 The fluctuation measures can be easily computedanalytically for the model but our aim here is to simulate theeffect of incomplete particle identification

There are many experimental techniques to measure par-ticle mass We discuss here the effect of misidentificationreferring to measurements of energy loss dEdx in a detectormaterial This method is applied by the experiments NA49NA61 and STAR The detectors are equipped with TimeProjection Chambers in which dEdx is measured The valueof dEdx can be used to identify particles because it dependson both particle mass m and momentum p in the combinationof velocity (β = p

radicm2 + p2) In the case of large separation

of the energy loss distributions of pions and kaons asschematically shown in Fig 1(a) almost unique particleidentification is possible This is however not possible whenthe measured pion and kaon dEdx distributions overlap asillustrated in Fig 1(b)

Performing the Monte Carlo simulations we assumed thatthe dEdx distributions of pions and kaons are Gaussianscentered respectively at 14 and 12 in arbitrary units Theyare normalized to the mean multiplicity of pions and ofkaons respectively To quantify the bias caused by particlemisidentification a simple particle identification scheme isused namely a particle is identified as a pion if dEdx gt 13and as a kaon if dEdx 13 The width σ of both Gaussiansis chosen to be the same but its value is varied from 0 to 008With growing width of the peaks of the dEdx distributionthe fraction of misidentified particles obviously increases Theresults of the simulation are illustrated in Fig 2 where thefluctuation measures and νdyn are shown as a functionof σ As seen the magnitudes of and νdyn decrease asthe fraction of misidentified particles grows and the measuresvanish when particle identification becomes totally randomThis sizable and experimentally unavoidable effect was themain motivation to develop the identity method which fullyeliminates the problem

IV IDENTITY METHOD

The identity method which is described here for a two-component system of pions and kaons utilizes the measure defined by Eq (13) However the kaon identity wK is not

054907-3


dEdx [arb units]

09 1 11 12 13 14 15 16 17 18

dN

d(d

Ed

x)

0

1000

2000

3000

4000

5000

6000

7000

=0006σ

K

π

sum

(a)

dEdx [arb units]

09 1 11 12 13 14 15 16 17 18

dN

d(d

Ed

x)

0

100

200

300

400

500

600

700 =006σ

K

π

sum

(b)

FIG 1 (Color online) The distribution of dEdx with nonoverlapping peaks of pions and kaons (a) which allows unique particleidentification and the distribution with overlapping peaks (b) which does not allow unique identification

limited to either 1 or 0 anymore but can take any value fromthe interval [0 1]

In the previous section we assumed that particles are iden-tified according to the energy-loss distribution To make thepresentation of the identity method more general we assumehere that particle identification is achieved by measurementof particle mass not specifying the particular experimentaltechnique which is used for this purpose Because anymeasurement is of finite resolution we deal with continuousdistributions of observed masses of pions and kaons whichare denoted as ρπ (m) and ρK (m) respectively They arenormalized as

intdm ρπ (m) = 〈Nπ 〉

intdm ρK (m) = 〈NK〉 (16)

The kaon identity is defined as

wK (m) equiv ρK (m)

ρ(m) (17)

where ρ(m) equiv ρπ (m) + ρK (m) and is normalized asintdm ρ(m) = 〈N〉 equiv 〈Nπ 〉 + 〈NK〉 (18)

If the distributions ρπ (m) and ρK (m) do not overlap theparticles can be uniquely identified and wK = 0 for a pion andwK = 1 for a kaon When the distributions ρπ (m) and ρK (m)overlap wK can take the value of any real number from [0 1]Figure 3 illustrates the latter case The mass distributions areshown in Fig 3(a) and the distribution of kaon identity inFig 3(b) The peaks close to 0 and 1 in Fig 3(b) correspond tothe mass regions in which pions respectively kaons are well

σ-002 0 002 004 006 008 01

Ψ

Φ

-03

-02

-01

0

ΦΨ

σ-002 0 002 004 006 008 01

dyn

ν

-006

-004

-002

0

FIG 2 (Color online) The measures of chemical fluctuations and (a) and νdyn (b) as functions of the width of the energy-lossdistribution

054907-4


m [MeV]

100 200 300 400 500 600 700 800

]-1

(m)

[MeV

ρ

0

01

02

03

04

05

06

07

08

09

(m)K

ρ

(m)π

ρ

(m)πρ(m)+K

ρ

(a)

Kw

0 02 04 06 08 1 12

Kd

Nd

w

-610

-510

-410

-310

-210

-110

1

10

210

310

410(b)

FIG 3 (Color online) The distributions of observed masses of pions ρπ (m) of kaons ρK (m) and their sum ρ(m) (a) and the correspondingdistribution of kaon identity wK (b)

identified The wK values around 05 correspond to particlesfor which the measured mass is in the transition region betweenthe kaon and pion peaks (m asymp 280 MeV) in the distributionρ(m) shown in Fig 3(a)

Let us now explain how the fluctuation measure iscalculated once the mass distributions were experimentallyobtained The single-particle variable entering Eq (13) isdefined as in Sec II z equiv wK minus wK The bar denotes theinclusive average which is computed as follows

wK equiv 1

〈N〉int

dm ρ(m) wK (m)

(19)

= 1

〈N〉int

dm ρK (m) = 〈NK〉〈N〉

Analogously one finds w2K and z2 equiv w2

K minus wK2 The quantity

〈Z2〉 is obtained as

〈Z2〉 = 1

Nev

Nevsumn=1

(Nnsumi=1

wiK minus Nn wK

)2

(20)

where Nev is the number of events and Nn is the multiplicityof the nth event Substituting 〈Z2〉 z2 and 〈N〉 into Eq (13)one finds the measure the magnitude of which howeveris biased by the effect of particle misidentification Next wediscuss the correction procedure

As shown in the Appendix the measure can be expressedthrough the moments of the multiplicity distributions of pionsand kaons as

= A

( 〈NK〉〈N〉 minus uK

)2

(21)

where

A equiv 1

〈N〉[lang

N2π

rang 〈NK〉2

〈Nπ 〉2+ lang

N2K

rang minus 〈NK〉 minus 〈NK〉2

〈Nπ 〉minus 2〈NπNK〉〈NK〉

〈Nπ 〉]

(22)

and

uK equiv 1

〈NK〉int

dm ρK (m) wK (m) (23)

In the case of complete particle identification (CI) thedistributions ρπ (m) and ρK (m) do not overlap and thusuK = 1 Then the result for is

CI = A

( 〈NK〉〈N〉 minus 1

)2

(24)

which is equivalent to the expression (14)Although particle-by-particle identification is usually diffi-

cult statistical identification is reliable In the latter case wedo not know whether a given particle is a kaon but we knowthe average numbers of kaons and of pions We introduce theconcept of random identification which assumes that for everyparticle the probability of being a kaon equals 〈NK〉〈N〉 Sucha situation is described by mass distributions of the form

ρi(m) =

⎧⎪⎪⎨⎪⎪⎩

0 for m lt mmin

〈Ni 〉mmaxminusmmin

for mmin m mmax

0 for mmax lt m

where i = πK and mmin and mmax denote lower and upperlimits of the measured mass range respectively With thisdistribution

uK = wK = 〈NK〉〈N〉 (25)

When uK = 〈NK〉〈N〉 is substituted into Eq (21) = 0Thus the measure vanishes when particle identification israndom

Finally we arrive at the crucial point of the considerationsIt appears that the measure can be expressed as

= CI

(1 minus VI

VR

)2

(26)

where CI is the measure for the complete identification asgiven by Eq (24) The quantities VI and VR are the values of

054907-5


the integral

V equivint

dm ρ(m) wK (m) [1 minus wK (m)] (27)

evaluated for the cases of imperfect and random identificationrespectively

One proves the equality (26) by observing that

VI = 〈NK〉(1 minus uK ) VR = 〈NK〉〈Nπ 〉〈N〉 (28)

Substituting Eqs (24) and (28) into the equality (26) oneobtains the formula (21) Actually the relation (26) was firstdiscovered by performing various numerical simulations andonly then was it proven analytically Equation (26) allowsone to experimentally obtain CI from Thus the effect ofmisidentification is fully corrected by the factor (1 minus VIVR)2

which measures the quality of the applied procedure of particleidentification The factor is independent of the correlationsunder study and can be determined from experimental data

V EXPERIMENTAL PROCEDURE

The application of the identity method to experimental datais not difficult We present here a step-by-step procedure toobtain the measure CI of fluctuations of kaons (or any otherselected particle type) with respect to all particles (representedby the sum of kaons and pions in the previous sections) It isimportant to note that if misidentification occurs between morethan two particle types the identity method does not allow tostudy relative fluctuations of two of them for example ofkaons and pions in the presence of protons

We come back to the specific but typical example con-sidered in Sec III of particle identification via measurementsof particle energy loss in the detector material The energyloss dEdx is denoted by X The energy loss of particles ofa given type depends on the particle mass and momentum(via the velocity) and detector characteristics The distributionof X is usually a multidimensional function which can bedetermined experimentally by averaging over particles frommany interactions The energy loss distribution which for agiven particle momentum is typically fitted by a sum of fourGaussians corresponding to electrons pions kaons and (anti-)protons allows one to determine the average multiplicities ofkaons and of all particles Having obtained this informationone should proceed as follows

(i) Extract the energy-loss distribution of kaons ρK (X) fromthe inclusive distribution ρ(X) The distributions should benormalized asint

dX ρ(X) = 〈N〉int

dX ρK (X) = 〈NK〉

(ii) Determine the kaon identity

wK (X) = ρK (X)

ρ(X)

for every registered particle

(iii) Compute the fluctuation measure from the definition(13) This is the raw value of which is not corrected yetfor the effect of misidentification

(iv) Knowing the mean multiplicities compute the quantity

VR = 〈NK〉(〈N〉 minus 〈NK〉)〈N〉

(v) Using all particles calculate the integral

VI =int

dXρ(X) wK (X) [1 minus wK (X)]

(vi) Determine the fluctuation measure CI which is free of theeffect of misidentification as

CI =

(1 minus VIVR)2

The experimental data accumulated by the NA49 Collabo-ration are currently under analysis using the identity methodproposed in this paper [17]

ACKNOWLEDGMENTS

We are very grateful to Peter Seyboth for strong encourage-ment and numerous critical discussions We are also indebtedto Mark Gorenstein for critical reading of the manuscript Thiswork was partially supported by the Polish Ministry of Scienceand Higher Education under Grants No N202 204638 and No667N-CERN20100 and by the German Research Foundationunder Grant No GA 14802-1

APPENDIX

We express here the measure through the moments ofmultiplicity distributions of pions and kaons For this purposeone writes 〈Z2〉 as

〈Z2〉 =infinsum

Nπ=0

infinsumNK=0

PNπ NK

intdmπ

1 Pπ

(mπ

1

)intdmπ

2 Pπ

(mπ

2

) middot middot middot

timesint

dmπNπ

Pπ

(mπ

Nπ

)intdmK

1 PK

(mK

1

)times

intdmK

2 PK

(mK

2

) middot middot middotint

dmKNK

PK

(mK

NK

)[wK

(mπ

1

)+wK

(mπ

2

) + middot middot middot + wK

(mπ

Nπ

) + wK

(mK

1

)+wK

(mK

2


(mK

NK

) minus (Nπ + NK )wK ]2

(A1)

where PNπ NKis the multiplicity distribution of pions and

kaons Pπ (m) equiv ρπ (m)〈Nπ 〉 and PK (m) equiv ρK (m)〈NK〉 arethe mass distributions of pions and kaons respectively

Equation (A1) gives

〈Z2〉 = 〈Nπ 〉u2π + 〈NK〉u2

K + 〈N2〉〈Nπ (Nπminus1)〉wK2uπ

2

+〈NK (NK minus 1)〉uK2 + 2〈NπNK〉uπ uK

minus 2〈NNπ 〉wK uπ minus 2〈NNK〉wK uK (A2)

054907-6


where

unπ equiv 1

〈Nπ 〉int

dm ρπ (m) wnK (m)

unK equiv 1

〈NK〉int

dm ρK (m) wnK (m) (A3)

with n = 1 2 Because

〈Nπ 〉〈N〉 un

π + 〈NK〉〈N〉 un

K = wnK (A4)

Eq (A2) provides

〈Z2〉 = 〈N〉w2K +

[〈N2〉 + 〈Nπ (Nπ minus 1)〉〈N〉2

〈Nπ 〉2

minus 2〈NNπ 〉〈N〉〈Nπ 〉

]wK

2

minus 2

[〈Nπ (Nπ minus 1)〉〈N〉〈NK〉

〈Nπ 〉2minus 〈NNπ 〉〈NK〉

〈Nπ 〉+ 〈NNK〉 minus 〈NπNK〉〈N〉

〈Nπ 〉]wK uK

+[〈Nπ (Nπ minus 1)〉〈NK〉2

〈Nπ 〉2+ 〈NK (NK minus 1)〉

minus 2〈NπNK〉〈NK〉〈Nπ 〉

]uK

2 (A5)

Keeping in mind that z2 = w2K minus wK

2 and wK = 〈NK〉〈N〉one finds after somewhat lengthy calculations the formula (21)with A given by Eq (22)

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[nucl-ex]

054907-7


The identity method which is developed here uses thefluctuation measure a simple modification of The mea-sure similar to is strongly intensive but the modificationmakes it possible to correct the measurements for the effect ofparticle misidentification Below we show that the measure

can be factorized into a coefficient which represents the effectof misidentification and the quantity CI which correspondsto the value would have for complete identification Themisidentification coefficient can be determined from the datain a model independent way Therefore the identity methodprovides the value of the fluctuation measure as it would beobtained in an experiment in which every particle is uniquelyidentified

Before the identity method is presented we introduce anddiscuss in Sec II the fluctuation measures σdyn νdyn and In Sec III we demonstrate by a Monte Carlo simulation howthe effect of misidentification distorts the chemical fluctuationsas quantified by νdyn and The identity method isformulated in Sec IV Instead of conclusions we present inthe last section the steps required to apply the identity methodto experimental data To simplify the presentation we considerchemical fluctuations of events composed only of kaons andpions Clearly kaons and pions can be replaced by particles ofany other sort

II MEASURES OF FLUCTUATIONS

As mentioned in the Introduction fluctuations of chemicalcomposition of final states of relativistic heavy-ion collisionscan be studied in several ways The NA49 Collaboration [5ndash7]measured event-by-event fluctuations of the particle ratiosKπ Kp pπ and determined the quantity σdyn defined as

σdyn = sgn(σ 2

data minus σ 2mixed

)radic∣∣σ 2data minus σ 2

mixed

∣∣ (1)

where σdata and σmixed are the relative width (the width dividedby the mean) of the event-by-event particle ratio distributionin respectively the data and artificially generated mixedevents where every particle comes from a different real eventThe fluctuations present in mixed events are attributable tothe effect of particle misidentification and the statistical noisecaused by the finite number of particles

The STAR Collaboration used [8] the quantity νdyn to mea-sure chemical fluctuations For the case of a two-componentsystem of pions and kaons νdyn is defined as

νdyn = 〈NK (NK minus 1)〉〈NK〉2

+ 〈Nπ (Nπ minus 1)〉〈Nπ 〉2

minus 2〈NKNπ 〉

〈NK〉〈Nπ 〉 (2)

where NK and Nπ are the numbers of kaons and pions ina given event and 〈middot middot middot〉 denotes averaging over events Thequantity νdyn is defined in such a way that in particular itvanishes when the multiplicity distributions of pions and kaonsare both Poissonian (〈Ni(Ni minus 1)〉 = 〈Ni〉2 i = π K) and in-dependent from each other (〈NKNπ 〉 = 〈NK〉〈Nπ 〉) Thus it isconstructed to quantify the deviations of the fluctuations fromthe Poissonian noise For large-enough particle multiplicitiesone finds the approximate relation νdyn asympσ 2

data minus σ 2mixed which

gives νdyn asymp sgn(σdyn) σ 2dyn [8] We note that the quantity νdyn

implicitly assumes unique identification of all particles

As already noted it was advocated long ago [910] toemploy the measure [11] to study chemical fluctuations Themeasure is defined in the following way One introduces thevariable z equiv x minus x where x is a single-particle characteristicsuch as the transverse momentum or azimuthal angle Theoverline denotes averaging over the single-particle inclusivedistribution The event variable Z which is a multiparticleanalog of z is defined as Z equiv sumN

i=1(xi minus x) where the sumruns over the N particles in a given event By construction〈Z〉 = 0 The measure is finally defined as

equivradic

〈Z2〉〈N〉 minus

radicz2 (3)

The measure vanishes in the absence of interparticlecorrelations This situation is discussed in some detail belowfor the case of chemical fluctuations Here we note thatthe measure also possesses another important property It isstrongly intensive which means that it is independent of thenumber and of the distribution (fluctuation) of the number ofparticle sources if the sources are identical and independentfrom each other In particular if a nucleus-nucleus collisionis a simple superposition of nucleon-nucleon interactionsthen AA = NN The strongly intensive property is a veryvaluable feature of because centrality selection in relativisticheavy-ion collisions is never perfect and events of differentnumbers of particle sources are always mixed up The stronglyintensive property is also desirable when different centralitiesor different colliding systems are compared For a discussionof strongly intensive quantities see Ref [16]

The analysis of chemical fluctuations can be performedwith the help of in two different but fully equivalent waysIn the first method [9] using the identity variable chemicalfluctuations are treated in analogy to fluctuations of transversemomentum In the second method is calculated from themoments of the multiplicity distributions [10]

We next describe the first method for the example of atwo-component system of pions and kaons One defines thesingle-particle variable x as x = wK where wK is called thekaon identity and wi

K = 1 if the ith particle is a kaon andwi

K = 0 if the ith particle is a pion This implies uniqueparticle identification One then directly uses the definition(3) to evaluate

Let us now discuss the most important case for which themeasure of chemical fluctuations vanishes Because theinclusive distribution of wK equals

P (wK ) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

〈Nπ 〉〈N〉 for wK = 0

〈NK〉〈N〉 for wK = 1

(4)

one finds

z2 equiv w2K minus wK

2 = 〈NK〉〈N〉

(1 minus 〈NK〉

〈N〉)

(5)

When interparticle correlations are absent the distribution ofparticle identities in events of multiplicity N reads

PN

(w1

Kw2K wN

K

) = PNP(w1

K

)P

(w2

K

) middot middot middotP (wN

K

) (6)

054907-2




z2 = 〈NK〉〈Nπ 〉〈N〉2

(7)

〈Z2〉〈N〉 = 〈Nπ 〉2

〈N〉3

(langN2

K


〈N〉3

(langN2

π






PNKNπ= PNK+Nπ

times(

NK + Nπ

NK

)( 〈NK〉〈N〉

)NK(


)Nπ

(9)

with the moments


〈N〉2〈N (N minus 1)〉 (10)


〈N〉2〈N (N minus 1)〉 (11)

〈NKNπ 〉 = 〈NK〉〈Nπ 〉〈N〉2

〈N (N minus 1)〉 (12)



= 〈Z2〉〈N〉 minus z2 (13)


= 1

〈N〉3

[langN2

π



]

(14)


= 〈Nπ 〉2〈NK〉2

〈N〉3νdyn (15)








IV IDENTITY METHOD


054907-3


dEdx [arb units]

09 1 11 12 13 14 15 16 17 18

dN

d(d

Ed

x)

0

1000

2000

3000

4000

5000

6000

7000

=0006σ

K

π

sum

(a)

dEdx [arb units]

09 1 11 12 13 14 15 16 17 18

dN

d(d

Ed

x)

0

100

200

300

400

500

600

700 =006σ

K

π

sum

(b)








ρ(m) (17)



σ-002 0 002 004 006 008 01

Ψ

Φ

-03

-02

-01

0

ΦΨ

σ-002 0 002 004 006 008 01

dyn

ν

-006

-004

-002

0


054907-4


m [MeV]

100 200 300 400 500 600 700 800

]-1

(m)

[MeV

ρ

0

01

02

03

04

05

06

07

08

09

(m)K

ρ

(m)π

ρ

(m)πρ(m)+K

ρ

(a)

Kw

0 02 04 06 08 1 12

Kd

Nd

w

-610

-510

-410

-310

-210

-110

1

10

210

310

410(b)




wK equiv 1

〈N〉int

dm ρ(m) wK (m)

(19)

= 1

〈N〉int





〈Z2〉 = 1

Nev

Nevsumn=1

(Nnsumi=1

wiK minus Nn wK

)2

(20)



= A


)2

(21)

where

A equiv 1

〈N〉[lang

N2π

rang 〈NK〉2

〈Nπ 〉2+ lang

N2K



〈Nπ 〉]

(22)

and

uK equiv 1

〈NK〉int

dm ρK (m) wK (m) (23)


CI = A


)2

(24)



ρi(m) =

⎧⎪⎪⎨⎪⎪⎩

0 for m lt mmin


for mmin m mmax

0 for mmax lt m


uK = wK = 〈NK〉〈N〉 (25)



= CI

(1 minus VI

VR

)2

(26)


054907-5


the integral

V equivint














wK (X) = ρK (X)

ρ(X)






VI =int



CI =

(1 minus VIVR)2


ACKNOWLEDGMENTS


APPENDIX


〈Z2〉 =infinsum

Nπ=0

infinsumNK=0

PNπ NK

intdmπ

1 Pπ

(mπ

1

)intdmπ

2 Pπ

(mπ

2


timesint

dmπNπ

Pπ

(mπ

Nπ

)intdmK

1 PK

(mK

1

)times

intdmK

2 PK

(mK

2


dmKNK

PK

(mK

NK

)[wK

(mπ

1

)+wK

(mπ

2


(mπ

Nπ

) + wK

(mK

1

)+wK

(mK

2


(mK

NK


(A1)



Equation (A1) gives

〈Z2〉 = 〈Nπ 〉u2π + 〈NK〉u2


2



054907-6


where

unπ equiv 1

〈Nπ 〉int

dm ρπ (m) wnK (m)

unK equiv 1

〈NK〉int





K = wnK (A4)

Eq (A2) provides

〈Z2〉 = 〈N〉w2K +

[〈N2〉 + 〈Nπ (Nπ minus 1)〉〈N〉2

〈Nπ 〉2


]wK

2

minus 2




〈Nπ 〉]wK uK




]uK

2 (A5)


















[nucl-ex]

054907-7




z2 = 〈NK〉〈Nπ 〉〈N〉2

(7)

〈Z2〉〈N〉 = 〈Nπ 〉2

〈N〉3

(langN2

K


〈N〉3

(langN2

π






PNKNπ= PNK+Nπ

times(

NK + Nπ

NK

)( 〈NK〉〈N〉

)NK(


)Nπ

(9)

with the moments


〈N〉2〈N (N minus 1)〉 (10)


〈N〉2〈N (N minus 1)〉 (11)

〈NKNπ 〉 = 〈NK〉〈Nπ 〉〈N〉2

〈N (N minus 1)〉 (12)



= 〈Z2〉〈N〉 minus z2 (13)


= 1

〈N〉3

[langN2

π



]

(14)


= 〈Nπ 〉2〈NK〉2

〈N〉3νdyn (15)








IV IDENTITY METHOD


054907-3


dEdx [arb units]

09 1 11 12 13 14 15 16 17 18

dN

d(d

Ed

x)

0

1000

2000

3000

4000

5000

6000

7000

=0006σ

K

π

sum

(a)

dEdx [arb units]

09 1 11 12 13 14 15 16 17 18

dN

d(d

Ed

x)

0

100

200

300

400

500

600

700 =006σ

K

π

sum

(b)








ρ(m) (17)



σ-002 0 002 004 006 008 01

Ψ

Φ

-03

-02

-01

0

ΦΨ

σ-002 0 002 004 006 008 01

dyn

ν

-006

-004

-002

0


054907-4


m [MeV]

100 200 300 400 500 600 700 800

]-1

(m)

[MeV

ρ

0

01

02

03

04

05

06

07

08

09

(m)K

ρ

(m)π

ρ

(m)πρ(m)+K

ρ

(a)

Kw

0 02 04 06 08 1 12

Kd

Nd

w

-610

-510

-410

-310

-210

-110

1

10

210

310

410(b)




wK equiv 1

〈N〉int

dm ρ(m) wK (m)

(19)

= 1

〈N〉int





〈Z2〉 = 1

Nev

Nevsumn=1

(Nnsumi=1

wiK minus Nn wK

)2

(20)



= A


)2

(21)

where

A equiv 1

〈N〉[lang

N2π

rang 〈NK〉2

〈Nπ 〉2+ lang

N2K



〈Nπ 〉]

(22)

and

uK equiv 1

〈NK〉int

dm ρK (m) wK (m) (23)


CI = A


)2

(24)



ρi(m) =

⎧⎪⎪⎨⎪⎪⎩

0 for m lt mmin


for mmin m mmax

0 for mmax lt m


uK = wK = 〈NK〉〈N〉 (25)



= CI

(1 minus VI

VR

)2

(26)


054907-5


the integral

V equivint














wK (X) = ρK (X)

ρ(X)






VI =int



CI =

(1 minus VIVR)2


ACKNOWLEDGMENTS


APPENDIX


〈Z2〉 =infinsum

Nπ=0

infinsumNK=0

PNπ NK

intdmπ

1 Pπ

(mπ

1

)intdmπ

2 Pπ

(mπ

2


timesint

dmπNπ

Pπ

(mπ

Nπ

)intdmK

1 PK

(mK

1

)times

intdmK

2 PK

(mK

2


dmKNK

PK

(mK

NK

)[wK

(mπ

1

)+wK

(mπ

2


(mπ

Nπ

) + wK

(mK

1

)+wK

(mK

2


(mK

NK


(A1)



Equation (A1) gives

〈Z2〉 = 〈Nπ 〉u2π + 〈NK〉u2


2



054907-6


where

unπ equiv 1

〈Nπ 〉int

dm ρπ (m) wnK (m)

unK equiv 1

〈NK〉int





K = wnK (A4)

Eq (A2) provides

〈Z2〉 = 〈N〉w2K +

[〈N2〉 + 〈Nπ (Nπ minus 1)〉〈N〉2

〈Nπ 〉2


]wK

2

minus 2




〈Nπ 〉]wK uK




]uK

2 (A5)


















[nucl-ex]

054907-7


dEdx [arb units]

09 1 11 12 13 14 15 16 17 18

dN

d(d

Ed

x)

0

1000

2000

3000

4000

5000

6000

7000

=0006σ

K

π

sum

(a)

dEdx [arb units]

09 1 11 12 13 14 15 16 17 18

dN

d(d

Ed

x)

0

100

200

300

400

500

600

700 =006σ

K

π

sum

(b)








ρ(m) (17)



σ-002 0 002 004 006 008 01

Ψ

Φ

-03

-02

-01

0

ΦΨ

σ-002 0 002 004 006 008 01

dyn

ν

-006

-004

-002

0


054907-4


m [MeV]

100 200 300 400 500 600 700 800

]-1

(m)

[MeV

ρ

0

01

02

03

04

05

06

07

08

09

(m)K

ρ

(m)π

ρ

(m)πρ(m)+K

ρ

(a)

Kw

0 02 04 06 08 1 12

Kd

Nd

w

-610

-510

-410

-310

-210

-110

1

10

210

310

410(b)




wK equiv 1

〈N〉int

dm ρ(m) wK (m)

(19)

= 1

〈N〉int





〈Z2〉 = 1

Nev

Nevsumn=1

(Nnsumi=1

wiK minus Nn wK

)2

(20)



= A


)2

(21)

where

A equiv 1

〈N〉[lang

N2π

rang 〈NK〉2

〈Nπ 〉2+ lang

N2K



〈Nπ 〉]

(22)

and

uK equiv 1

〈NK〉int

dm ρK (m) wK (m) (23)


CI = A


)2

(24)



ρi(m) =

⎧⎪⎪⎨⎪⎪⎩

0 for m lt mmin


for mmin m mmax

0 for mmax lt m


uK = wK = 〈NK〉〈N〉 (25)



= CI

(1 minus VI

VR

)2

(26)


054907-5


the integral

V equivint














wK (X) = ρK (X)

ρ(X)






VI =int



CI =

(1 minus VIVR)2


ACKNOWLEDGMENTS


APPENDIX


〈Z2〉 =infinsum

Nπ=0

infinsumNK=0

PNπ NK

intdmπ

1 Pπ

(mπ

1

)intdmπ

2 Pπ

(mπ

2


timesint

dmπNπ

Pπ

(mπ

Nπ

)intdmK

1 PK

(mK

1

)times

intdmK

2 PK

(mK

2


dmKNK

PK

(mK

NK

)[wK

(mπ

1

)+wK

(mπ

2


(mπ

Nπ

) + wK

(mK

1

)+wK

(mK

2


(mK

NK


(A1)



Equation (A1) gives

〈Z2〉 = 〈Nπ 〉u2π + 〈NK〉u2


2



054907-6


where

unπ equiv 1

〈Nπ 〉int

dm ρπ (m) wnK (m)

unK equiv 1

〈NK〉int





K = wnK (A4)

Eq (A2) provides

〈Z2〉 = 〈N〉w2K +

[〈N2〉 + 〈Nπ (Nπ minus 1)〉〈N〉2

〈Nπ 〉2


]wK

2

minus 2




〈Nπ 〉]wK uK




]uK

2 (A5)


















[nucl-ex]

054907-7


m [MeV]

100 200 300 400 500 600 700 800

]-1

(m)

[MeV

ρ

0

01

02

03

04

05

06

07

08

09

(m)K

ρ

(m)π

ρ

(m)πρ(m)+K

ρ

(a)

Kw

0 02 04 06 08 1 12

Kd

Nd

w

-610

-510

-410

-310

-210

-110

1

10

210

310

410(b)




wK equiv 1

〈N〉int

dm ρ(m) wK (m)

(19)

= 1

〈N〉int





〈Z2〉 = 1

Nev

Nevsumn=1

(Nnsumi=1

wiK minus Nn wK

)2

(20)



= A


)2

(21)

where

A equiv 1

〈N〉[lang

N2π

rang 〈NK〉2

〈Nπ 〉2+ lang

N2K



〈Nπ 〉]

(22)

and

uK equiv 1

〈NK〉int

dm ρK (m) wK (m) (23)


CI = A


)2

(24)



ρi(m) =

⎧⎪⎪⎨⎪⎪⎩

0 for m lt mmin


for mmin m mmax

0 for mmax lt m


uK = wK = 〈NK〉〈N〉 (25)



= CI

(1 minus VI

VR

)2

(26)


054907-5


the integral

V equivint














wK (X) = ρK (X)

ρ(X)






VI =int



CI =

(1 minus VIVR)2


ACKNOWLEDGMENTS


APPENDIX


〈Z2〉 =infinsum

Nπ=0

infinsumNK=0

PNπ NK

intdmπ

1 Pπ

(mπ

1

)intdmπ

2 Pπ

(mπ

2


timesint

dmπNπ

Pπ

(mπ

Nπ

)intdmK

1 PK

(mK

1

)times

intdmK

2 PK

(mK

2


dmKNK

PK

(mK

NK

)[wK

(mπ

1

)+wK

(mπ

2


(mπ

Nπ

) + wK

(mK

1

)+wK

(mK

2


(mK

NK


(A1)



Equation (A1) gives

〈Z2〉 = 〈Nπ 〉u2π + 〈NK〉u2


2



054907-6


where

unπ equiv 1

〈Nπ 〉int

dm ρπ (m) wnK (m)

unK equiv 1

〈NK〉int





K = wnK (A4)

Eq (A2) provides

〈Z2〉 = 〈N〉w2K +

[〈N2〉 + 〈Nπ (Nπ minus 1)〉〈N〉2

〈Nπ 〉2


]wK

2

minus 2




〈Nπ 〉]wK uK




]uK

2 (A5)


















[nucl-ex]

054907-7


the integral

V equivint














wK (X) = ρK (X)

ρ(X)






VI =int



CI =

(1 minus VIVR)2


ACKNOWLEDGMENTS


APPENDIX


〈Z2〉 =infinsum

Nπ=0

infinsumNK=0

PNπ NK

intdmπ

1 Pπ

(mπ

1

)intdmπ

2 Pπ

(mπ

2


timesint

dmπNπ

Pπ

(mπ

Nπ

)intdmK

1 PK

(mK

1

)times

intdmK

2 PK

(mK

2


dmKNK

PK

(mK

NK

)[wK

(mπ

1

)+wK

(mπ

2


(mπ

Nπ

) + wK

(mK

1

)+wK

(mK

2


(mK

NK


(A1)



Equation (A1) gives

〈Z2〉 = 〈Nπ 〉u2π + 〈NK〉u2


2



054907-6


where

unπ equiv 1

〈Nπ 〉int

dm ρπ (m) wnK (m)

unK equiv 1

〈NK〉int





K = wnK (A4)

Eq (A2) provides

〈Z2〉 = 〈N〉w2K +

[〈N2〉 + 〈Nπ (Nπ minus 1)〉〈N〉2

〈Nπ 〉2


]wK

2

minus 2




〈Nπ 〉]wK uK




]uK

2 (A5)


















[nucl-ex]

054907-7


where

unπ equiv 1

〈Nπ 〉int

dm ρπ (m) wnK (m)

unK equiv 1

〈NK〉int





K = wnK (A4)

Eq (A2) provides

〈Z2〉 = 〈N〉w2K +

[〈N2〉 + 〈Nπ (Nπ minus 1)〉〈N〉2

〈Nπ 〉2


]wK

2

minus 2




〈Nπ 〉]wK uK




]uK

2 (A5)


















[nucl-ex]

054907-7

Documents

Identity method to study chemical fluctuations in relativistic heavy-ion collisions