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Deciphering Azimuthal Correlations in Relativistic Heavy-Ion Collisions Tomasz Cetner and Katarzyna Grebieszkow Faculty of Physics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland Stanisl(cid:32)aw Mrówczyn´ski Institute of Physics, Jan Kochanowski University, ul. S´wi¸etokrzyska 15, PL - 25-406 Kielce, Poland and So(cid:32)ltan Institute for Nuclear Studies, ul. Hoz˙a 69, PL - 00-681 Warsaw, Poland (Dated: January 18, 2010) Wediscussvarioussourcesofazimuthalcorrelationsinrelativisticheavy-ioncollisions. Theinte- gralmeasureΦisappliedtoquantifythecorrelations. Wefirstconsiderseparatelythecorrelations caused by the elliptic flow, resonance decays, jets and transverse momentum conservation. An effect of randomly lost particles is also discussed. Using the PYTHIA and HIJING event generators 1 we produce a sample of events which mimic experimental data. By means of kinematic cuts and 1 particle’s selection criteria, the data are analyzed to identify a dominant source of correlations. 0 2 PACSnumbers: 25.75.-q,25.75.Gz n a J I. INTRODUCTION correlationsbyapplyingkinematiccutsandparticle’sse- 4 lection criteria. 2 Particles produced in relativistic heavy-ion collisions A magnitude of Φ of azimuthal correlations measured are correlated in azimuthal angle due to various mecha- in Pb-Pb collisions at 158A GeV is of order of 0.01 ra- ] h nisms. One mentions here extensively studied jets and dian[9]. Sincetheeffects,whicharetheoreticallystudied -t minijets resulting from (semi-)hard parton-parton scat- here, oftenexceedthisorder, arealisticmodelshouldin- l tering and collective flow due to the cylindrically asym- clude all the effects, and consequently, the model has to c u metric pressure gradients, see the review articles [1] and berather complex. For thisreason wedo notattempt to n [2], respectively. More exotic sources of correlations are compare our model calculations to the preliminary data [ also possible. As argued in [3], the plasma instabilities, [9]. Thereisalsoanadditionalreason. Themeasurement which occur at an early stage of collisions, can generate [9] has been performed in an acceptance window which 2 v the azimuthal fluctuations. Except the dynamically in- is not only limited in particle’s rapidity and transverse 1 terestingmechanisms, therearealsorathertrivialeffects momentum but it is also nonuniform in azimuthal an- 3 caused by decays of hadronic resonances or by energy- gle. Theacceptance,whichischaracteristicfortheNA49 6 momentum conservation. detector, has to be properly included in model calcula- 1 There is a variety of methods designed to study fluc- tionsforaquantitativecomparisonwiththeexperimental . 1 tuations on event-by-event basis. In particular, the so- data. Itisbeyondthescopeofthisstudy. Ouraimhereis 1 called measure Φ proposed in [4] was used to measure to understand how the collective flow, resonance decays, 0 the transverse momentum [5, 6] and electric charge fluc- jets and transverse momentum conservation contribute 1 tuations [7]. The measure proved to be very sensitive to to azimuthal correlations and how the contributions can : v dynamical correlations and it was suggested to apply it be disentangled. i to study azimuthal ones [8]. Such an analysis is under- X way using experimental data accumulated by the NA49 r a and NA61 Collaborations and some preliminary results II. MEASURE Φ are already published [9]. The aim of this paper is to presentmodelsimulationstobeusedininterpretationof Let us first introduce the correlation measure Φ. One the experimental data. The fact that the measure Φ is def sensitivetocorrelationsofvariousoriginisadvantageand defines the variable z = x−x, where x is a single par- disadvantage at the same time, as it is difficult to disen- ticle’s characteristics such as the particle transverse mo- tangle different contributions. Therefore, we model the mentum, electric charge or azimuthal angle. The over- azimuthal correlations driven by several processes and linedenotesaveragingoverasingleparticleinclusivedis- welookhowthecorrelationsshowupwhenquantifiedby tribution. In the subsequent sections, x will be identi- the measure Φ. We first consider separately in terms of fied with the particle azimuthal angle φ and the fluctu- toy models, the elliptic flow, resonance decays, jets and ation measure will be denoted as Φφ. The event vari- transverse momentum conservation. An effect of ran- able Z, which is a multiparticle analog of z, is defined domly lost particles is also examined. Then, we analyze as Z d=ef (cid:80)N (x −x), where the summation runs over i=1 i the data provided by the PYTHIA and HIJING event particles from a given event. By construction, (cid:104)Z(cid:105) = 0, generators showing how to identify the main sources of where (cid:104)...(cid:105) represents averaging over events (collisions). 2 The measure Φ is finally defined as We have first verified the effect of second Fourier coefficient v on Φ by Monte Carlo simulations. For this (cid:115) 2 φ (cid:104)Z2(cid:105) (cid:112) purpose we have generated events of particle multiplic- Φd=ef − z2 . (1) ity given by either Poisson or Negative Binomial (NB) (cid:104)N(cid:105) distribution. The latter is defined as It is evident that Φ = 0, when no inter-particle corre- Γ(N +k) (cid:104)N(cid:105)Nkk P = , (5) lations are present. The measure also possesses a less N Γ(N +1)Γ(k)(cid:0)(cid:104)N(cid:105)+k(cid:1)N+k trivial property - it is independent of the distribution of the number of particle sources if the sources are identi- where Γ(k) is the Gamma function, which for positive calandindependentfromeachother. Thus,themeasure integerargumentsequalsΓ(k)=(k−1)!;theparameterk Φ is ‘blind’ to the impact parameter variation as long as canbeexpressedthroughthevarianceofthedistribution the‘physics’doesnotchangewiththecollisioncentrality. Var(N)≡(cid:104)N2(cid:105)−(cid:104)N(cid:105)2 and the average value (cid:104)N(cid:105) as In particular, Φ is independent of the impact parameter (cid:104)N(cid:105)2 if the nucleus-nucleus collision is a simple superposition k = . (6) ofnucleon-nucleoninteractions. Inthefollowingsections Var(N)−(cid:104)N(cid:105) we discuss how various mechanisms responsible for az- The parameter k is chosen in such a way in our all sim- imuthal correlations contribute to Φ . Then, using the (cid:112) φ ulations that Var(N)=(cid:104)N(cid:105)/2. Then, the multiplicity event generators we show how the dominant contribu- distribution approximately obeys the Wróblewski’s for- tions can be identified. mula [10] which is known to hold for proton-proton interactions in a wide collision energy range. The simulations are performed for both the Negative Binomial and III. COLLECTIVE FLOW Poisson distributions as the former distribution is much broader than the latter one for (cid:104)N(cid:105) (cid:29) 1. We note here Particles produced in relativistic heavy-ion collisions that the width of multiplicity distributions in relativis- reveal a collective behavior which is naturally described ticheavy-ioncollisionsstronglydependsoncentralityse- in terms of hydrodynamics where the collective flow is lection criteria. Thus, it is important to see how the causedbypressuregradients. Thecollectiveflowasquan- correlation signal changes with the width of multiplicity tifiedbythemeasureΦφ wasstudiedin[8]. Hereweonly distribution. recapitulate the results derived in [8]. The azimuthal angle of each particle has been gener- Sincetheinclusiveazimuthaldistributionisflat,φ=π ated from the distribution andφ2 = 4π2forφ∈[0,2π],andthusz2 = 1π2. Asingle 3 3 1 (cid:16) (cid:0) (cid:1)(cid:17) particle azimuthal distribution in a given event is P(φ)= 1+2v cos 2(φ−φ ) , (7) 2π 2 R 1 (cid:104) (cid:88)∞ (cid:0) (cid:1)(cid:105) where0≤φ≤2π;thereactionplaneangleφR ofagiven P(φ)= 2π 1+2 vncos n(φ−φR) , (2) event has been generated from the flat distribution. The n=1 results of our simulations are shown in Fig. 1 for both thePoisson(leftpanel)andNB(rightpanel)multiplicity where 0 ≤ φ ≤ 2π; φ is the reaction plane angle and R distributions. As seen, the analytical formula (3) works v denotes an amplitude of the n−th Fourier harmonics. n perfectly well. The N−particle distribution is assumed to be a product Therearelarge(∼40%)event-by-eventfluctuationsof on N distributions (2) multiplied by a multiplicity dis- v observed [2] at BNL RHIC. The v fluctuations are tribution. Consequently, the collective flow is the only 2 2 dominatedbythefluctuationsofeccentricityoftheover- source of azimuthal correlations in the system. Averag- lap region of colliding nuclei, see e.g. [11]. We have in- ingoverparticlesisperformedbyintegratingoverφ with i troducedthev fluctuationsinoursimulationsinthefol- i = 1, 2, ...N and averaging over events is achieved by 2 lowing way. For each event the value of v has been gen- integrating over φ and summing over N. The distribu- 2 R erated from the Gaussian distribution of the dispersion tion of reaction plane angle is obviously flat. Thus, one σ . Thefluctuationshavebeenrestrictedtovarywithin finds the measure Φ of azimuthal correlations caused by v2 2σ around the mean (cid:104)v (cid:105) that is (cid:104)v (cid:105)−2σ ≤ v ≤ the flow as v2 2 2 v2 2 (cid:104)v (cid:105)+2σ . Then, v remains positive unless σ /(cid:104)v (cid:105) 2 v2 2 v2 2 (cid:115) exceeds 0.5. π2 (cid:18)(cid:104)N2(cid:105)−(cid:104)N(cid:105)(cid:19) π Φ = + S− √ . (3) In Fig. 2 we demonstrate the effect of flow fluctua- φ 3 (cid:104)N(cid:105) 3 tions relative the effect of flow. Specifically, we show the difference of the correlation measures computed for the where (cid:104)Nm(cid:105) is the m−th moment of multiplicity distri- fluctuating v and fixed v = (cid:104)v (cid:105). The particle multi- 2 2 2 bution and plicityhasbeengeneratedfromtheNBdistributionwith (cid:104)N(cid:105) = 400. As seen, the flow fluctuations of relative ∞ S ≡2(cid:68)(cid:88)(cid:16)vn(cid:17)2(cid:69). (4) magnitude of ∼40% noticeably increase the value of Φφ n if (cid:104)v (cid:105) is not too small. n=1 2 3 ] 4 ] 4 s s n 0.3 Poisson <N>=50 n 0.3 NB <N>=50, k=4.348 a Poisson <N>=400 a NB <N>=400, k=4.040 di 0.2 Poisson <N>=700 di 0.2 NB <N>=700, k=4.023 a 3 a 3 r r [ [ 0.1 0.1 f f F 0 F 0 2 2 0 0.02 0.04 0 0.02 0.04 1 1 0 0 -1 -1 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 v v 2 2 FIG. 1: (Color online) Φ as a function of the second Fourier coefficient v for the Poisson (left panel) and NB (right panel) φ 2 multiplicity distributions. The lines represent the analytical formula (3). The insets show Φ for small values of v . φ 2 IV. RESONANCE DECAYS other, the calculations presented in Appendix lead to √ √ 2−f − 2 Letusstartthediscussionofeffectsofresonancedecays Φφ = √ π . (9) 6 with the toy model where all produced particles come from heavy resonances which have vanishing transverse As seen, for f = 0 the formula (9) gives, as expected, velocity and decay back to back into pairs of particles. Φ=0 and for f =1 we get the value (8). Theparticlemultiplicityisarbitrarybutfixedevennum- Wehavecheckedtheformula(9)byMonteCarlosimu- ber. Then, as shown in Appendix, we have lationsandthenwehaveconsideredthemodelwherethe particle multiplicity is not fixed but it is given by either √ Poisson or NB distribution with the average multiplicity 1− 2 Φφ = √ π ≈−0.531. (8) equal 50, 400, or 700 particles. For a given fraction f 6 of particles coming from the back-to-back decays of resonances, thenumberofparticlescomingfromthedecays When only a fraction f of all produced particles comes in the event of multiplicity N has been the even number from the back-to-back decays of resonances while the re- which is the nearest to fN. The measure Φ as a func- maining particles are produced independently from each φ tionoff isshowninFig.3. Asseen,theformula(9)still works very well. When a resonance, which is at rest, decays back to ] s back, the difference of azimuthal angles of the decay n Neg. Bin. <N>=400, k=4.04 a products is ∆φ = π. When the resonance has a finite adi 0.2 <v2>=0.025 velocity, the difference of azimuthal angles of the decay r <v >=0.05 [ 2 productsissmallerthanπ. Whentheresonance’skinetic st. <<vv22>>==00..115 energy is much larger than the energy released in its de- n cay, ∆φ is zero. Therefore, we consider a model where o c 0.1 = a fraction of particles comes from the resonance decays v2 , and the particles are emitted in pairs with the difference f F- of their azimuthal angles ∆φ varying from 0 to π. We f first assume that all particles are emitted in pairs and F 0 the relative azimuthal angle of two correlated particles equals ∆φ. As shown in Appendix, we then have (cid:114) 2 1 π Φ = π2−∆φπ+ (∆φ)2− √ . (10) -0.1 φ 3 2 3 0 10 20 30 40 50 60 AsseenthatΦ changesitssignfrompositivetonegative s /v [%] φ v2 2 with growing ∆φ; Φφ vanishes when FIG.2: (Coloronline)ThedifferenceofΦφ computedforthe (cid:16) 1 (cid:17) fluctuating and fixed v . ∆φ=π 1− √ ≈1.328 (11) 2 3 4 0 0 ] ] s s n Poisson <N>=50 n Neg. Bin. <N>=50, k=4.348 adia PPooiissssoonn <<NN>>==470000 adia NNeegg.. BBiinn.. <<NN>>==470000,, kk==44..004203 r r [ [ f f -0.2 -0.2 F F -0.4 -0.4 -0.6 -0.6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 f f FIG. 3: (Color online) Φ as a function of fraction of particles coming from the back-to-back resonance decays. The particle φ multiplicity is distributed according to the Poisson (left panel) or NB (right panel) distribution. The dashed and solid lines represent the analytical formulas (8) and (9), respectively. and for ∆φ = π we deal with the model described by fraction of particles emitted in pairs equals 0.3, 0.5, 0.7 the formula (8). Further on, we have considered a model or 1.0. The remaining particles, which are not emitted where only a fraction f of particles is emitted in corre- in pairs, carry no correlations. The particle multiplicity (cid:112) lated pairs. Then, as explained in Appendix, Eq. (10) is generated from the NB distribution with Var(N) = gets the form (cid:104)N(cid:105)/2 = 50. As seen, the formula (12) works perfectly well. (cid:114) π2 (cid:16)π2 1 (cid:17) π We next discuss the combined effect of resonance de- Φ = +f −∆φπ+ (∆φ)2 − √ . (12) φ 3 3 2 3 caysandellipticflow. Themultiplicityofeventsisgener- ated from the Poisson or NB binomial distribution. For As previously, Φ changes its sign and Φ = 0 for ∆φ each event 30% of particles is assumed to originate from φ φ given by Eq. (11). heavy resonances which decay back to back into pairs of InFig.4wecomparetheformula(12)withtheresults particles. Neither resonances nor their decay products ofMonteCarlosimulationofΦ asafunctionof∆φ. The experience any flow, but the remaining 70% of particles φ manifest the collective elliptic flow according to Eq. (7). The results of the simulation are shown in Fig. 5. When ] 1 the particle multiplicity or v2 are sufficiently small, the s n f=0.3 effects of resonance decays dominates and Φφ is nega- dia ff==00..57 tive. It becomes positive when the effect elliptic flow a takes over. r f=1.0 [ f 0.5 F V. ‘DIJETS’ 0 We call a dijet the two groups (jets) of particles flying inexactlyoppositedirections. Particlesfromeachjetare distributed within a cone of the azimutal angle σφ. We have considered the dijets of 2+2, 5+5 and 10+10 parti- -0.5 cles. The total particle multiplicity is correspondingly 4, 10, 20 as there is exactly one dijet per event and there arenootherparticles. Theresultsofdijetsimulationare 0 0.5 1 1.5 2 2.5 3 3.5 shown in Fig. 6. D f [radians] We have here two sources of azimuthal correlations which counteract each other. As we already know, the FIG. 4: (Color online) Φφ as a function of the correlation back-to-backemissionofparticlesgeneratesnegativecor- angle ∆φ for varying fraction f of particles emitted in pairs. relations while the collinear emission leads to positive The particle multiplicity is given by the NB distribution. ones. When the particle’s multiplicity of dijets is suf- 5 ] ] s s n Poisson <N>=50 n Neg. Bin. <N>=50, k=4.348 dia 1.5 PPooiissssoonn <<NN>>==470000 dia 1.5 NNeegg.. BBiinn,. <<NN>>==470000,, kk==44..004203 a a r r [ [ f = 0.3 f = 0.3 f f F 1 F 1 0.5 0.5 0 0 -0.5 -0.5 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 v v 2 2 FIG. 5: (Color online) Φ resulting from the combined effect of elliptic flow and resonance decays. The particle multiplicity is φ distributed according to the Poisson (left panel) or NB (right panel) distribution. ficiently high and σφ is sufficiently small, the effect of computedasp =p cosφandp =p sinφ. Tomakethe x T y T collinear emission wins and Φ is positive. total transverse momentum of N particles vanish, the x φ andy componentofmomentumofeachparticlehasbeen shifted as VI. MOMENTUM CONSERVATION N N 1 (cid:88) 1 (cid:88) p →p − pi, p →p − pi. (14) The momentum conservation obviously leads to inter- x x N x y y N y i=1 i=1 particle correlations. We have studied the effect on Φ , φ generatingthesetsofparticlesofmultiplicityN. Theaz- The simulation showing the effect of transverse mo- imuthal angle distribution of a single particle is assumed mentum conservation is illustrated in Fig. 7. The parti- to be flat while the transverse momentum distribution is cle multiplicity has been generated according to NB dis- (cid:112) chosen in the form tribution with Var(N) = (cid:104)N(cid:105)/2. As seen, the effect of momentum conservation is sizable and it survives to P(pT)=β2pTe−βpT (13) large multiplicities. Inrealexperimentsonlyafractionofallproducedpar- with the slope parameter β−1 =200 MeV. For each particles is observed due to a finite detector efficiency and ticlethexandy componentsofitsmomentumhavebeen acceptance. We model the effect of detector efficiency by randomly loosing particles independently of their az- ] s n 2+2 imuthal angle. In Fig. 8 we show how the effect of de- dia 3 51+0+510 tector efficiency modifies the correlations caused by the a transversemomentumconservation. Asseen,therandom r [ losses of particles lead to the dilution of the correlation f that is Φ monotonically goes to zero as the fraction of F φ 2 registered particles f →0. reg 1 VII. PROTON-PROTON COLLISIONS IN PYTHIA 0 After the discussion of various mechanisms responsible for azimuthal correlations, let us now consider more realistic situation where several mechanisms of -1 0 0.2 0.4 0.6 0.8 azimuthal correlations are present at the same time. We used the PYTHIA generator [12] to simulate p-p s f [radians] collisions at several collision energies accessible at √ SPS ( s = 6.27, 7.62, 8.73, 12.3, 17.3 GeV) NN √ FIG. 6: (Color online) Φφ as a function of the jet opening and RHIC ( sNN = 19.6, 62.4, 130, 200 GeV). angle σφ for several numbers of particles in a jet. For every energy a set of minimum bias events was 6 -0.25 ] ] s s n n <N>=10 a S p = 0 in every event a <N>=50 di T di <N>=500 a a r r 0 [ [ f f -0.3 F F -0.2 -0.35 -0.4 -0.4 10 102 103 0.2 0.4 0.6 0.8 1 <N> freg FIG. 7: Φ as a function of average multiplicity in events FIG. 8: (Color online) Φ as a function of fraction of regis- φ φ where total transverse momentum exactly vanishes. tered particles f . The correlation results from the trans- reg verse momentum conservation. collected. We treated as stable the following particles: for such events are shown in Fig. 10. As seen, the val- µ−, π0, π+, K0, K+, K0, K0, Λ, Σ+, Σ−, Ξ0, Ξ−, Ω− ues of Φ for the PYTHIA events agree quite well with L S φ andtheirantiparticles. Noacceptancecutswereapplied. the results of our toy-model simulations which take into For every energy we computed Φ separately for pos- account only the effect of transverse momentum conser- φ itive, negative and all charged particles. The results are vation. It is somewhat surprising that the agreement for shown in Fig. 9. As seen, Φ is negative and weakly de- same-sign particles is not much better that that for all φ pends on collision energy. To understand why Φ is so chargedparticles. Itmeansthattheresonancedecaysdo φ different for negative and for positive particles, we ex- not generate strong correlations in the PYTHIA events. cluded protons from all charged and from positive parti- We note, however, that the effect of transverse momen- cles. The corresponding values of Φ are also shown in tum conservation overshoots the correlations of the like- φ Fig. 9. After excluding protons, the correlations among sign particles and it undershoots the correlations of all negativeparticlesandamongpositiveareverysimilarto charged particles. The latter results presumably signals each other. presenceofresonancesdecayingintopairsofonepositive and one negative particle. Whatisthemechanismresponsiblefornegativevalues of Φ ? We first checked that high p particles play no φ T role here, as Φ does not significantly change when par- φ ticles with p >1.5 GeV are excluded. The correlations VIII. NUCLEUS-NUCLEUS COLLISIONS IN T among charged particles can be caused by the effect res- HIJING onance decays but the effect is certainly very minor for same-sign particles, as there are very a few resonances Wehavealsoperformedsimulationsofnucleus-nucleus decaying into two positive or two negative particles. collisions using the HIJING [13] event generator. We The transverse momentum conservation, which is dis- have simulated the collisions of p-p, C-C, Si-Si and Pb- √ cussed in Sec. VI, is another possible source of neg- Pb at s = 17.3 GeV and Φ has been computed NN φ ative values of Φ . We checked that the PHYTHIA separatelyforpositive,negativeandallchargedparticles φ events indeed obey the transverse momentum conserva- comingfromminimumbiasevents. Theresultsareshown tion. Specifically, we proved vanishing of the total mo- inFig.11. Asseen,Φ isalmostindependentofthemass φ mentum in x and in y directions of all particles (charged number of colliding nuclei and the values of Φ are very φ and neutral) from every event. To quantitatively study close to those found using PYTHIA. It is by no means the effect of transverse momentum conservation we pro- accidental. When a nucleus-nucleus collision is a simple ceeded as follows. For every collision energy we deter- superposition of nucleon-nucleon interactions, the value mined the average multiplicity of positive, negative and ofΦisexactlythesameforp-pinteractionsandnucleus- neutral particles. Then, we performed the simple simu- nucleuscollisionsatanycentrality. IntheHIJINGmodel lations described in Sec. VI, generating events of a given a nucleus-nucleus collision is not exactly a superposition total multiplicity which satisfy the transverse momen- of nucleon-nucleon collisions but it is almost so. And tum conservation. Then, a fraction of particles was ran- thetreatmentofproton-protoninteractionsisessentially domly eliminated to get the multiplicity of charged, pos- the same in PYTHIA and HIJING. For these reasons itive or negative particles. The values of Φ computed ouranalysisofPYTHIAeventspresentedintheprevious φ 7 0.1 s] ] 0.2 n all particles no protons s Pythia mom. cons. adia anlel gcahtaivreged dian aclhl a(wrgiethd neutral) r positive a negative [ 0 [r positive f f 0 F F -0.1 -0.2 -0.2 -0.4 -0.3 10 102 10 102 s [GeV] NN s [GeV] NN FIG. 9: (Color online) The energy dependence of Φ for φ FIG. 10: (Color online) Φ for the PYTHIA events (open positive, negative and all charged particles in the PYTHIA φ symbols) compared to the results of toy-model simulations simulations of p-p collisions. The pale symbols correspond (full symbols) which take into account only the effect of to the results with protons excluded from the positive and transverse momentum conservation. The asterisks show the all charged particles. toy-model results for all (neutral and charged) particles. section fully applies here. proved to be very sensitive to various dynamical correlations,isusedintheanalysis. Tointerprettheexperimen- tal results it should be understood how different sources of correlations manifest themselves when measured by IX. SUMMARY AND OUTLOOK means of Φ . This was the aim of our study. We per- φ formed several simulations to analyze separately the az- Azimuthal correlations of final state particles from imuthalcorrelationscausedbytheellipticflow,resonance high-energy collisions carry valuable information on the decays, jetsandtransversemomentumconservation. We collision dynamics. It motivates the analysis of experi- also discussed how the correlations are diluted due to mentaldatacollectedbytheNA49andNA61Collabora- randomly lost particles. Finally we used the PYTHIA tions which is in progress with some preliminary results and HIJING event generators to produce a big sample already published [9]. The integral measure Φ , which φ of events which mimic experimental data from p-p and nucleus-nucleus collisions at the SPS and RHIC collision energies. Φ appeared to be surprisingly independent of φ ] 0.1 the collision energy and of the size of colliding systems. s n all charged Applying some kinematic cuts and selection criteria of a negatively charged di positively charged particles, we showed that the azimuthal correlations are ra dominated by rather trivial effect of transverse momen- [ 0 tum conservation which appeared to be almost indepen- f F dentofparticle’smultiplicitywhichchangesdramatically for collision energies and system’s sizes under considera- -0.1 tion. -0.2 The experience gathered in the course of this theoreti- calstudywillbeusedtobetterunderstandexperimental -0.3 p+p C+C Si+Si Pb+Pb data. Quantitativeanalysisofseveralsimplemechanisms of azimuthal correlations we discussed will facilitate an system observation of possible new phenomena like critical fluc- FIG.11: TheHijingsimulationofthesystemsizedependence tuationsatphaseboundariesofstronglyinteractingmat- √ ofΦφfornucleus-nucleuscollisionsat sNN =17.3GeVwith ter or plasma instabilities from the early stage of rela- no acceptance cuts. tivistic heavy-ion collisions. 8 Acknowledgments to back into two particles. When one particle is emitted at the azimuthal angle φ and 0 ≤ φ < π, the second 1 1 We are grateful to Maciej Rybczyn´ski for providing us particle is emitted at φ2 = φ1+π. When π ≤ φ1 < 2π, √ withthesample ofnucleus-nucleus collisionsat sNN = thenφ2 =φ1−π. Therefore,thetwo-particledistribution 17.3 GeV simulated with the HIJING event generator. of azimuthal angles reads We thank to Marek Ga´zdzicki for critical reading of the manuscript and his numerous comments. This work 1 P (φ ,φ ) = Θ(π−φ )δ(φ −φ +π) (A.2) was partially supported by Polish Ministry of Science 2 1 2 2π 1 1 2 and Higher Education under grants N N202 204638 and 1 667/N-CERN/2010/0. + 2π Θ(φ1−π)δ(φ1−φ2−π). One observes that Appendix: Toy model of resonance decays (cid:90) (cid:90) The inclusive distribution of azimuthal angle is as- dφ P (φ ,φ)= dφ P (φ,φ )=P (φ), (A.3) 1 2 1 2 2 2 inc sumed to be flat that is 1 P (φ)= Θ(φ)Θ(2π−φ), (A.1) and computes inc 2π which gives φ¯= π and φ¯2 = 4π2/3. Consequently, z¯2 = (cid:90) 5 π2/3. dφ1dφ2φ1φ2P2(φ1,φ2)= 6π2 . (A.4) Let us first assume that all produced particles come from heavy resonances which are at rest and decay back We further assume that the particle multiplicity is arbitrary but fixed even number N. Then, the N−particle distribution of azimuthal angles is P (φ ,φ ,...,φ )=P (φ ,φ )P (φ ,φ ) ··· P (φ ,φ ). (A.5) N 1 2 N 2 1 2 2 3 4 2 N−1 N The variable Z is defined as Z =φ +φ +...+φ −Nπ and one computes (cid:104)Z2(cid:105) in the following way 1 2 N (cid:90) (cid:104)Z2(cid:105) = dφ dφ ...dφ (φ +φ +...+φ −Nπ)2P (φ ,φ ,...,φ ) (A.6) 1 2 N 1 2 N N 1 2 N (cid:90) N = Nφ¯2+N dφ dφ φ φ P (φ ,φ )+N(N −2)φ¯2−2N2φ¯π+N2π2 = π2 . 1 2 1 2 2 1 2 6 Using the result (A.6) and keeping in mind that z¯2 =π2/3, we find the formula (8) from the definition (1). LetusnowassumethatN particlescomefromresonancesandadditionalN particlesareproducedindependently 1 2 from each other and from resonances. Then, we still have z¯2 =π2/3 and (cid:104)Z2(cid:105) is computed as (cid:90) (cid:104)Z2(cid:105) = Nφ¯2+N dφ dφ φ φ P (φ ,φ )+N (N −2)φ¯2 (A.7) 1 1 2 1 2 2 1 2 1 1 π2 + 2N N φ¯2+N (N −1)φ¯2−2N2φ¯π+N2π2 = (2N −N ), 1 2 2 2 6 1 where N ≡N +N . The result (A.7) with f ≡N /N gives the formula (9). 1 2 1 Themodelcanbeeasilygeneralizedtothesituationwhentheparticlesfromacorrelatedpairarenotemittedback to back but their relative azimuthal angle is fixed and equal ∆φ. Then, the two-particle distribution of azimuthal angles is 1 1 P (φ ,φ )= Θ(2π−∆φ−φ )δ(φ −φ +∆φ)+ Θ(φ −2π+∆φ)δ(φ −φ +∆φ−2π), (A.8) 2 1 2 2π 1 1 2 2π 1 1 2 and instead of Eq. (A.4) we have Substituting the result (A.9) to Eq. (A.6) or Eq. (A.7), (cid:90) 4 1 dφ dφ φ φ P (φ ,φ )= π2−∆φπ+ (∆φ)2 . 1 2 1 2 2 1 2 3 2 (A.9) 9 one finds (cid:104)Z2(cid:105) which finally gives the formula (10) or (12), respectively. [1] J. Casalderrey-Solana and C. A. Salgado, Acta Phys. 064903 (2004). Polon. B 38, 3731 (2007). [8] St. Mro´wczyn´ski, Acta Phys. Polon. B 31, 2065 (2000). [2] S. A. Voloshin, A. M. Poskanzer and R. Snellings, [9] T. Cetner and K. Grebieszkow [for the NA49 Collabora- arXiv:0809.2949 [nucl-ex]. tion], arXiv:1008.3412 [nucl-ex]. [3] St. Mro´wczyn´ski, J. Phys. 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