Correlation between initial spatial anisotropy and final momentum anisotropies in relativistic heavy ion collisions
Sanchari Thakur, Sumit Kumar Saha, Pingal Dasgupta, Rupa Chatterjee, Subhasis Chattopadhyay
aa r X i v : . [ nu c l - t h ] J a n Correlation between initial spatial anisotropy and final momentum anisotropies inrelativistic heavy ion collisions
Sanchari Thakur, ∗ Sumit Kumar Saha, † Pingal Dasgupta, ‡ Rupa Chatterjee, § and Subhasis Chattopadhyay ¶ Variable Energy Cyclotron Centre, HBNI, 1/AF, Bidhan Nagar, Kolkata-700064, India Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),Institute of Modern Physics, Fudan University, Shanghai 200433, China
The particle momentum anisotropy ( v n ) produced in relativistic nuclear collisions is consideredto be a response of the initial geometry or the spatial anisotropy ǫ n of the system formed in thesecollisions. The linear correlation between ǫ n and v n quantifies the efficiency at which the initialspatial eccentricity is converted to final momentum anisotropy in heavy ion collisions. We studythe transverse momentum, collision centrality, and beam energy dependence of this correlation fordifferent charged particles using a hydrodynamical model framework. The ( ǫ n − v n ) correlation isfound to be stronger for central collisions and also for n=2 compared to that for n=3 as expected.However, the transverse momentum ( p T ) dependent correlation coefficient shows interesting featureswhich strongly depends on the mass as well as p T of the emitted particle. The correlation strengthis found to be larger for lighter particles in the lower p T region. We see that the relative fluctuationin anisotropic flow depends strongly in the value of η/s specially in the region p T < η/s . I. INTRODUCTION
The anisotropic flow of hadrons is known as one of thekey observables produced in relativistic heavy ion colli-sions that provides a strong indication of the formationof hot and dense Quark Gluon Plasma (QGP) phase andits collective behaviour [1–4]. The spatial asymmetry inthe initial energy density distribution on the overlappingzone between two colliding nuclei gives rise to anisotropicflow where the magnitude of the flow parameters dependon several factors such as particle mass, beam energy,collision centrality, transverse momentum.It is well known that the relativistic hydrodynamicsis one of the most successful model frameworks whichhas been used extensively to study the evolution of theQGP medium in order to estimate the several final stateobservables [1, 2, 5–13]. The simultaneous explanation ofthe experimental data of the elliptic flow and the chargedparticle spectra by hydrodynamical model calculations atRHIC energy was one of the initial milestones in this fieldof research which confirms an early thermalization andcollective behaviour of the system produced in relativisticheavy ion collisions [14].The initial spatial anisotropy ( ǫ n ), specially the ellip-ticity increases significantly from central to mid-centralcollisions and consequently the magnitude of the ellip-tic flow coefficient increases towards peripheral collisions.On the other hand, the rise in initial spatial triangularity( ǫ ) with collision centrality is relatively slower comparedto that of ǫ . The efficiency of conversion of the initial ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] spatial eccentricity to the final momentum anisotropy de-pends on the initial state as well as on the evolution of theproduced hot and dense matter. Hydrodynamic modelcalculation can be quite useful to know the initial states(obtained by tuning the model parameter to reproducethe experimental data) and also the space time evolutionas we cannot get direct information of the initial statefrom experimental data.The relation between the initial spatial anisotropy andthe final state anisotropic flow parameters has been stud-ied by several groups earlier [1, 12, 15–29]. The effect of η /s as well as of the fractional contributions of the num-ber of participants ( N part ) and the number of binary col-lisions ( N coll ) to the initial entropy and/or energy densityproduction was studied for the first time in an interestingwork by Niemi et. al. [16] for Au+Au collisions at RHICenergy. The initial state anisotropies and their uncer-tainties in ultrarelativistic heavy ion collisions have beenstudied from the Monte Carlo Glauber model by Alvi-oli et. al. [30]. Recent experimental data have showna correlation between the mean transverse momentumof outgoing particles and the anisotropic flow parame-ter in Pb+Pb collisions at LHC [31]. A theory calcula-tion shows that the magnitude of this correlation can bedirectly predicted from the initial conditions using thespatial anisotropy ǫ n [32]. The correlation between thetransverse momentum and anisotropic flow parameter us-ing hydrodynamical model framework has been studiedby Bozek et. al. [33, 34].These studies suggest that the particle transverse mo-mentum, beam energy, collision centrality, all play cru-cial role in determining the final momentum anisotropy.Thus, in order to understand the correlation between ǫ n and the anisotropic flow better, it is important to knowthe simultaneous effect of all these parameters in detail.In this work we study the (linear) correlation betweeninitial ǫ n and the final momentum anisotropies ( v n ) ofpositively charged hadrons using a state-of-the art hy- v Pb+Pb @2.76A TeV20-40% π (b) ∈ v ∈ FIG. 1: (Color online) Distribution of ǫ and π + v at 2.76ATeV Pb+Pb collisions at four different centrality bins. drodynamical model calculation. We focus on the ellip-tic and triangular flow parameters and the correspondinginitial eccentricities are obtained from a sufficiently largenumber of events. It has already been shown in earlierstudies that the correlation between ǫ and v is signifi-cantly weak [16] and as a result we do not consider thisand other higher order harmonics for the present study.The dependence of the correlation coefficient on collisioncentrality and transverse momentum is studied in detailfor three different types of hadrons. We consider Pb+Pbcollisions at 2.76A TeV at LHC and Cu+Cu collisionsat 200A GeV at RHIC to study the dependence of cor-relation strength on the beam energy and system size.The correlation coefficients and the relative fluctuationsin the anisotropic flow parameters are calculated for twodifferent η/s values to check the sensitivity of the resultsto the shear viscosity coefficient. Additionally, we cal-culate the Normalized Symmetric Cumulant (NSC) be-tween ( v , v ) at different centrality bins for positivelycharged pion, kaon and protons.In the next section we briefly discuss the model frame-work and the initial state produced in heavy ion colli-sions. We calculate the p T integrated and p T dependentcorrelation coefficients between ǫ n − v n and discuss our re-sults from Pb+Pb collisions at LHC energy in section IIIand section IV respectively. The correlation coefficientsfor a small system like Cu+Cu collisions at RHIC energyare studied in the next section in order to understandthe system size and beam energy dependence. We showthe η/s dependence of the correlation coefficient and therelative fluctuations in section VI and the NSC resultsin section VII. In section VIII we give the summary andconclusions of all the results. v Pb+Pb @2.76A TeV0-20%k (a) (b) ∈ v (c) ∈ (d) FIG. 2: (Color online) Distribution of ǫ and K + v at 2.76ATeV Pb+Pb collisions at four different centrality bins. v Pb+Pb @2.76A TeV0-20%p (a) (b) ∈ v (c) ∈ (d) FIG. 3: (Color online) Distribution of ǫ and p v at 2.76ATeV Pb+Pb collisions at four different centrality bins. II. FRAMEWORK
We use the (2+1) dimensional longitudinally boost in-variant hydrodynamical model framework MUSIC [35]with fluctuating initial conditions to calculate the initialspatial anisotropies and the corresponding anisotropicflow parameters from heavy ion collisions at different cen-trality bins at mid-rapidity. The initial formation timeof the plasma is considered as 0.4 fm/ c at 2.76A TeVPb+Pb collisions at the LHC energy.A Monte Carlo Glauber initial condition is consideredand the value of η/s is kept fixed at 0.08 (later we changethis value to check the sensitivity of the results to η/s ).The initial energy density is considered to be dependenton a linear combination of soft ( N part ) and hard ( N coll )contributions with appropriate weight factors. A con-stant temperature freeze out is considered and a latticebased equation of state is used for a cross-over transi-tion between the QGP and hadronic matter phases [36].The centrality bins are selected using an impact param-eter range from Ref [37]. The model parameters areset by simultaneously reproducing the experimental dataof final state charged particle multiplicity, p T spectraand anisotropic flow parameters [38–41]. The standardCooper-Frye formula is used to estimate the productionof the hadrons from freeze-out surface [44].The initial spatial eccentricity is calculated using therelation [16]: ǫ n = − R d x d y r n cos [ n ( φ − ψ n )] ε ( x, y, τ ) R d x d y r n ε ( x, y, τ ) . (1)Where ψ n is the n th order event plane angle.The corresponding anisotropic flow parameters v n canbe obtained [16] from the invariant particle momentumdistribution as : dNd p T dY = 12 π dNp T dp T dY [1 + 2 ∞ X n =1 v n ( p T ) cos n( φ − ψ n )] . (2) III. PB+PB COLLISIONS AT THE LHC
The distributions of initial spatial eccentricity ( ǫ ) andthe corresponding ( p T integrated) elliptic flow coefficient v of positively charged pions for Pb+Pb collisions at √ s NN =2.76 TeV at LHC are shown in Fig. 1. Theresults from hydrodynamical model calculation at mid-rapidity are plotted for centrality bins 0–20%, 20–40%,40–60%, and 60–80%. The total number of events usedare 2000 and 2400 for 0-20% and 20-40% respectivelywhich increase towards more peripheral collisions therebyreducing the statistical errors. The average ǫ ( ǫ ) valuesfor 0–20%, 20–40%, 40–60%, and 60–80% central Pb+Pbcollisions are 0.139 (0.099), 0.280 (0.167), 0.405 (0.245),and 0.527 (0.395) respectively. The h ǫ i is found to beabout 40% smaller than h ǫ i for all four centrality bins.It is to be noted that we consider results upto 80%centrality bin as the hydrodynamic model calculationsfor ultra peripheral collisions (more than 80%) may notgive reliable results. The Figs. 2 and 3 show similar ǫ − v distribution for positively charged kaons and protonsrespectively for 4 different centrality bins.One observes a strong positive linear correlation be-tween pion v and ǫ for all centrality bins and thestrength of correlation reduces towards peripheral col-lisions. The ǫ n − v n distribution for 60–80% collisioncentrality clearly shows that the correlation strength re-duces significantly inspite of higher v and ǫ coefficientsfor that centrality bin. A similar trend is observed for kaon and protons aswell where the correlation is found to be stronger formore central collisions.An estimation of the strength of the linear correlationbetween two variables is obtained by dividing the covari-ance of the variables by the product of their respectivestandard deviations. Thus, the correlation coefficient Cbetween the initial spatial eccentricity and final momen-tum anisotropies can be quantified using the relation [16]: C ( ǫ n , v n ) = (cid:28) ( ǫ n − h ǫ n i av )( v n − h v n i av ) σ ǫ n σ v n (cid:29) av . (3)The quantities σ ǫ n and σ v n are the standard devia-tions of ǫ n and v n respectively. The average is takenusing hadron multiplicity as weight factor. The correla-tion coefficient can take any value between -1 to +1. Twoquantities are strongly linearly (anti-linearly) correlatedwhen the coefficient is close to 1 (-1). On the other hand,the value of C close to zero implies that the quantitiesare not correlated linearly.The correlation coefficients for pion, kaon and protonfrom Pb+Pb collisions at different centrality bins areshown in Table I.The values of the correlation coefficient for pions for0–20% and 20–40% centrality bins remains almost sameat about 0.95 (See table 1). For 40–60% centrality bin,we observe a very small drop in the value of C ( ǫ , v )(to 0.91). Whereas, C ( ǫ , v ) drops to a significantlylower value of 0.60 for 60–80% centrality bin. It is to benoted that we have used a larger number of events forperipheral collisions to reduce the statistical uncertaintyin the calculation.The correlation coefficient C ( ǫ , v ) for kaon and pro-ton is also found to be large and positive (0.94 and 0.93for kaon and proton respectively) for 0–20 and 20–40%centrality bins. The value of C is about 0.9 for 40–60%and at 60–80% centrality bin it drops significantly to avalue of about 0.6 for both the particles.It is well known that the triangular flow of chargedparticles does not show strong dependence on the colli-sion centrality [45]. However, the initial triangularity ofthe medium is found to be sensitive to the system sizewhich makes the estimation of correlation coefficient be-tween ǫ and v important. The Figs. 4, 5, and 6 showthe distribution of ǫ and v for pion, kaon and protonsrespectively for the four different centrality bins at LHC.Many earlier studies have shown that the linear correla-tion between ǫ − v is weaker than the correlation be-tween ǫ − v [2] and the same can be seen from the figs(1-6) as well.The correlation coefficient C( ǫ , v ) is about 0.75, 0.68and 0.4 for 0-20%, 20-40% and 40-60% centrality bins re-spectively for all π + , K + , and p at LHC. For 60–80%centrality bin, we see a complete absence of linear corre-lation as the value of C is found to be close to zero.The correlation strength between v n and ǫ n is summa-rized in the Fig. 7. This figure clearly shows the varia-tion of correlation strength with collision centrality for v π Pb+Pb @2.76A TeV (a) (b) ∈ v (c) ∈ (d) FIG. 4: (Color online) Distribution of ǫ and π + v at 2.76ATeV Pb+Pb collisions at four different centrality bins. v Pb+Pb @2.76A TeV0-20%k (a) (b) ∈ v (c) ∈ (d) FIG. 5: (Color online) Distribution of ǫ and K + v at 2.76ATeV Pb+Pb collisions at four different centrality bins. the three hadrons. The correlation strength for π + , K + ,and p is found to be close to each other although the ( p T integrated) anisotropic flow is different for them. Wealso see that C( ǫ , v ) shows a stronger sensitivity to thecollision centrality as it decreases faster for peripheralcollisions compared to C( ǫ , v ).We also estimate error in the correlation coefficient cal-culation with finite number of events. The probable er-rors in the correlation coefficients for ( ǫ , v ) and ( ǫ , v )are found to be less than 1% and 2% respectively. v Pb+Pb @2.76A TeV0-20%p (a) (b) ∈ v (c) ∈ (d) FIG. 6: (Color online) Distribution of ǫ and p v at 2.76ATeV Pb+Pb collisions at four different centrality bins. Centrality (%)10 20 30 40 50 60 70 C o rr e l a t i on C oe ff i c i en t C ν − ϵ ν − ϵ Pb+Pb @2.76A TeV + π + Kp FIG. 7: (Color online) C( ǫ n , v n ) as a function of centralityfrom 2.76A TeV Pb+Pb collisions at LHC. IV. CORRELATION BETWEEN ǫ n AND v n ( p T ) We understand that the final p T integrated anisotropicflow parameter is a consequence of the initial spatial de-formation of medium formed in heavy ion collisions. Al-though the magnitude of both the elliptic flow and thespatial anisotropy increases while going away from cen-tral collisions, the correlation strength between ǫ and v decreases towards peripheral collisions. We see similarreduction in correlation strength with collision centralitybetween ǫ and v as well.The values of the correlation strength C( ǫ n , v n ) for all π + , K + and p are found to be similar as a function ofcentrality and do not depend on the particle mass signifi-cantly as shown in Fig. 7. We know that the mass order-ing of differential anisotropic flow ( v n ( p T )) parametersis a signature of the collective behaviour of the mediumformed in heavy ion collisions and it is well explainedby hydrodynamical model calculation. Thus, it is impor-tant to know if there also exists any mass dependencein the p T dependent correlation coefficients for the dif-ferent hadrons and the underlying mechanism does notdepend on the particle mass significantly which results ina similar correlation strength for them.It is to be noted that the freeze-out temperature playsa crucial role in determining the p T dependent correlationcoefficient considering our simplified assumption of con-stant temperature freeze-out for all hadrons. Some ear-lier studies have shown that the differential anisotropicflow parameter is sensitive to the freeze-out temperaturemostly towards larger p T ( > − . ǫ n , v n ( p T )) for π + , K + , p at different centrality bins is shown in Figs 8, 9, and 10respectively. A clear mass dependence in the correlationcoefficient between ǫ − v can be seen for all the centralitybins. The value of C ( ǫ , v ( p T )) is found to be larger forlighter particles in the p T region 0.1 to 2 GeV shown inthe figs.The strength of ǫ − v correlation for pions remainclose to 0.9 in the p T regain 0.2 to 2 GeV for all threecentrality bins and then drops slowly for large p T values.At very low p T ( < . π + issmaller, as those may be emitted from the initial fewfm time period when the build up of transverse flow isnot very strong and the elliptic flow v ( p T ) is also small.We see a relatively stronger p T dependent correlation for K + and p where the coefficient for them falls sharplywith smaller p T values in the region p T < p T > ǫ − v as a function of p T also shows a similar behaviour to ǫ − v although themagnitude is much smaller. The strength of correlationfor protons is found to be very small below p T = 0.5 GeVfor all centrality bins. We see that C( ǫ n , v n ( p T )) dropsfaster towards peripheral collisions for n=3 compared ton=2 at higher p T values.These results clearly show that the p T dependent cor-relation coefficient strongly depends on the mass of theparticle and p T region that contributes maximum to thecorrelation strength is also different for different parti-cles. (a) 0-20% π + K + p C( ǫ , v ) 0.95 0.95 0.93C( ǫ , v ) 0.74 0.75 0.75(b) 20-40%C( ǫ , v ) 0.94 0.94 0.93C( ǫ , v ) 0.71 0.71 0.71(c) 40-60%C( ǫ , v ) 0.91 0.90 0.90C( ǫ , v ) 0.51 0.52 0.53(d) 60-80%C( ǫ , v ) 0.60 0.59 0.59C( ǫ , v ) 0.07 0.08 0.09TABLE I: C( v n , ε n ) of π + , K + , and p from (a) 0-20%, (b) 20-40%, (c) 40-60%, and (d) 60-80% Pb+Pb collisions at 2.76ATeV at the LHC . (GeV) T p0.0 0.5 1.0 1.5 2.0 2.5 3.0 C o rr e l a t i on C oe ff i c i en t C ) + π ( ∈ - v ) + π ( ∈ - v ) + (K ∈ - v ) + (K ∈ - v (p) ∈ - v (p) ∈ - v Pb+Pb @2.76A TeV
FIG. 8: (Color online) (modified) Correlation between ǫ n and v n ( p T ) for pion at √ s NN =2.76 TeV Pb+Pb collisions fordifferent centrality bins. V. CU+CU COLLISIONS AT RHIC
The C( ǫ n , v n ) of hadrons as a function of p T fromPb+Pb collisions at LHC is found to exhibit interest-ing features which depend strongly on the mass of theparticles. Cu+Cu collisions at 200A GeV at RHIC areexpected to produce a system with relatively smaller tem-perature and energy density as well as smaller transversedimension compared to Pb+Pb collisions at LHC. Onthe other hand, the initial state density fluctuations (in-creases anisotropic flow for smaller systems and lowerbeam energies) are expected to be higher for Cu+Cu col-lisions than for Pb+Pb collisions. Thus, a comparison ofthe correlation coefficients from Cu+Cu and Pb+Pb sys-tems at RHIC and LHC energies respectively is expectedto provide better understanding of the beam energy and (GeV) T p0.0 0.5 1.0 1.5 2.0 2.5 3.0 C o rr e l a t i on C oe ff i c i en t C − ) + π ( ∈ - v ) + π ( ∈ - v ) + (K ∈ - v ) + (K ∈ - v (p) ∈ - v (p) ∈ - v Pb+Pb @2.76A TeV
FIG. 9: (Color online) (modified) Correlation between ǫ n and v n ( p T ) for kaon at √ s NN = 2.76 TeV Pb+Pb collisions fordifferent centrality bins. (GeV) T p0.0 0.5 1.0 1.5 2.0 2.5 3.0 C o rr e l a t i on C oe ff i c i en t C − ) + π ( ∈ - v ) + π ( ∈ - v ) + (K ∈ - v ) + (K ∈ - v (p) ∈ - v (p) ∈ - v Pb+Pb @2.76A TeV
FIG. 10: (Color online)(modified results) Correlation between ǫ n and v n ( p T ) for proton at √ s NN = 2.76 TeV Pb+Pb colli-sions for different centrality bins. system size dependence of the correlation strength.We study the correlation between v n and ǫ n for threedifferent centrality bins of Cu+Cu collisions at RHIC andcompare with the results obtained from Pb+Pb collisionsat LHC. Similar to the Pb+Pb collisions, the initial pa-rameters are tuned to reproduce the experimental dataof charged particle multiplicity and particle spectra formost central Cu+Cu collisions at RHIC.We consider τ as 0.4 fm/c and and η /s=0.08 for thiscase. A sufficiently large number of events have beengenerated for all 0–20%, 20–40%, and 40–60% centralitybins of Cu+Cu collisions to calculate the ǫ n and v n and the corresponding correlation coefficients between them.Again, for Cu+Cu collisions we see that the correlationbetween v n and ǫ n is relatively stronger for more centralcollisions and also for n=2 compared to n=3 (see TableII).The Fig. 11 shows the p T dependent C( ǫ n , v n ) for π + , K + and p for different centrality bins. The C( ǫ n , v n )shows non-monotonic behavior as a function of p T specif-ically for protons. We see a mass ordering of the corre-lation coefficient in a relatively narrower p T range forCu+Cu collisions compared to Pb+Pb collisions. For0–20% and 20-40% centrality bins it can be seen in therange p T < π + and K + the correlation strength as a function of p T is foundto be maximum in the p T region 0.1 to 0.5 GeV and thenit drops slowly for larger p T values. On the other hand,for heavier proton the strength of correlation is found tobe maximum around p T ∼ v n for Cu+Cu collisions is smallerthan the same for Pb+Pb collisions for a particular cen-trality bin although the spatial anisotropy is slightlyhigher for the smaller system. In addition, the build upof transverse flow velocity is much weaker for Cu+Cu col-lisions and as a result the efficiency at which the spatialanisotropy is converted to momentum anisotropy is alsorelatively weaker for them compared to Pb+Pb collisions.Thus, for Pb+Pb collisions we see that even at 2 GeV p T value the correlation between ( ǫ n , v n ) is still strongerwhereas for Cu+Cu collisions the maximum contributionto the correlation strength comes from much smaller p T values.These results show that the strength of correlation islarger for higher beam energies and is relatively weakerfor Cu+Cu collisions than for Pb+Pb collisions. The esti-mation of correlation coefficient from same type of systemat different beam energies would provide more conclusiveinformation about the dependence of C on particle massand beam energy.Fig. 12 shows the h v n i / h ǫ n i as a function of central-ity for Pb+Pb collisions. Results from Cu+Cu collisionsat RHIC for two centrality bins are also shown in thesame plot for a comparison. The slope ( C n ) between twolinearly correlated variables ǫ n and v n can be written as v n = C n ǫ n + δ . After averaging over large numberof events the slope is simply C n = h v n i / h ǫ n i as h δ i iszero [16]. The C for Pb+Pb collisions falls faster than C towards peripheral collisions. The slope for Cu+Cucollisions is found to be much smaller than Pb+Pb colli-sions. However, the slope at peripheral Pb+Pb collisionresembles the slope at the central Cu+Cu collisions, fur-ther assuring similar viscous effects from two differentcollision systems with multiplicities close to each other.These results also clearly show that a smaller increase inspatial anisotropy results in a larger anisotropic flow forbigger system as well as for higher beam energies. (GeV) T p0.0 0.5 1.0 1.5 2.0 C o rr e l a t i on C oe ff i c i en t C ) ∈ - (v + π ) ∈ - (v + K ) ∈ - p (v ) ∈ - (v + π ) ∈ - (v + K ) ∈ - p (v Cu+Cu @200A GeV0-20% (a) (GeV) T p0.0 0.5 1.0 1.5 2.0 C o rr e l a t i on C oe ff i c i en t C Cu+Cu @200A GeV20-40% ) ∈ - (v + π ) ∈ - (v + K ) ∈ - p (v ) ∈ - (v + π ) ∈ - (v + K ) ∈ - p (v (b) (GeV) T p0.0 0.5 1.0 1.5 2.0 C o rr e l a t i on C oe ff i c i en t C Cu+Cu @200A GeV40-60% ) ∈ - (v + π ) ∈ - (v + K ) ∈ - p (v ) ∈ - (v + π ) ∈ - (v + K ) ∈ - p (v (c) FIG. 11: (Color online) Correlation coefficient C( ǫ n , v n ) for π + , K + , and p as function of p T for (a) 0–20% (new re-sult) and (b) 20–40% centrality bins from Cu+Cu collisionsat RHIC. Centrality (%)
10 20 30 40 50 60 70 〉 n ∈ 〈 / 〉 n v 〈 n = 2n = 3 n = 2n = 3 Pb+Pb @2.76A TeV Cu+Cu @200A GeV
FIG. 12: (Color online) The ratio of h v n i and h ǫ n i as a func-tion of collision centrality.TABLE II: C( ǫ n , v n ) for 200A GeV Cu+Cu collisions at RHIC0–20% π + K + p C( ǫ , v ) 0.91 0.90 0.90C( ǫ , v ) 0.72 0.71 0.7120–40%C( ǫ , v ) 0.84 0.83 0.82C( ǫ , v ) 0.51 0.50 0.5040–60%C( ǫ , v ) 0.68 0.66 0.64C( ǫ , v ) 0.11 0.10 0.10 VI. η/s
DEPENDENCE
The the dependence of the ǫ n − v n correlation on thevalue of η/s has been studied in detail in the literature.It has been shown in Ref. [16] that the higher order cor-relation coefficients are more sensitive to the value of the η/s . In Fig 13 we show the correlation coefficients fortwo different η/s values for 20–40% and 40–60% Pb+Pbcollisions at the LHC. The value of C for all the threeparticles are found to vary only marginally when η/s ischanged from 0.08 to 0.16.A better understanding of the initial state from finalstate flow observables has always been a primary goal forflow analysis in heavy ion collisions. Due to the varyingrelation of the linear response parameter with multiplic-ity, it is challenging to relate the initial anisotropy tothe final state momentum anisotropy in a linear fashion.The relative fluctuation in the anisotropic flow parame-ters σ v n / h v n i is considered to be a potential observable,reflecting the ratio of the first two moments of the ini- ) , v ∈ C ( + π + Kp + π + KpPb+Pb @2.76A TeV /s=0.08 η /s=0.16 η ( GeV ) T p0.0 0.5 1.0 1.5 2.0 2.5 3.0 ) , v ∈ C ( Pb+Pb @2.76A TeV
Pb+Pb @2.76A TeV (c) ) , v ∈ C ( ( GeV ) T p0.0 0.5 1.0 1.5 2.0 2.5 3.0 ) , v ∈ C ( FIG. 13: (Color online) p T dependent correlation coefficientsat the LHC considering two different η/s values. > / < v v σ + π + Kp + π + Kp/s=0.08 η /s=0.16 η Pb+Pb @2.76A TeV20-40% (a) ( GeV ) T p0.0 0.5 1.0 1.5 2.0 2.5 3.0 > / < v v σ (c) > / < v v σ (b) ( GeV ) T p0.0 0.5 1.0 1.5 2.0 2.5 3.0 > / < v v σ (d) FIG. 14: (Color online) Relative fluctuations in theanisotropic flow parameters at the LHC considering two dif-ferent η/s values. tial state eccentricity distribution (i.e, σ ǫ n ). The relativefluctuations in the anisotropic flow parameters for thesame set of collisions are shown in Fig. 14. Interestingly,the relative fluctuation σ v n / h v n i as a function of p T isfound to be quite sensitive to the value of η/s . One cansee from the figures that the sensitivity to the value of η/s is much stronger for protons than for pions and alsoin the low p T ( < VII. NORMALIZED SYMMETRIC CUMULANT
The correlation between different order of anisotropicflow coefficients is studied in heavy ion experiments us-ing cumulant method and is considered to be an efficientmethod to reduce the non-flow effects in the measure-ments [48, 49].
Centrality (%)0 10 20 30 40 50 60 70 N S C ( , ) − − Pb+Pb @2.76A TeVCu+Cu @200A GeV + π + Kp + π + Kp FIG. 15: (Color online) NSC(2,3) for pion, kaon and protonsas a function of centrality from Pb+Pb collisions at LHC.
The Normalized Symmetric Cumulant method hasgathered a lot of attention in recent times which focuseson the correlation strength between different orders ofanisotropic flow harmonics by removing the dependenceon the magnitude of the harmonics.We calculate NSC(2,3) between v and v using therelation, NSC(2 ,
3) = h v v i − h v ih v ih v ih v i . (4)Fig. 15 shows the NSC between v , v as a function ofcollision centrality for π + , K + and p from TeV Pb+Pbcollisions at LHC and Cu+Cu collisions at RHIC. Asexpected, a clear anti-correlation between v and v canbe observed for all the particles for collision centralitymore than 20% [47]. Although the NSC(2,3) values arefound to be close to each other for all π + , K + and p itis found to be slightly higher for heavier particles as afunction of centrality.It is to be noted that the NSC(2,3) is found to besmall and positive for Pb+Pb 0–20% centrality bin andcontrary to the peripheral collisions the value is found tobe larger for lighter particles there. A similar observationhas been reported in earlier studies as well where it wasshown that the centrality dependence of NSC(2,3) differsfrom most central to peripheral collisions [47]. It hasbeen shown in Ref. [29] that the value of NSC is sensitiveto the size of the centrality bin specially for the centralcollisions. We see that the NSC as a function of centralitydoes not change significantly with change in the valueof η/s from 0.08 to 0.16 for both Cu+Cu and Pb+Pbcollisions. VIII. SUMMARY AND CONCLUSIONS
We calculate the correlation between the initial spa-tial anisotropy and the final momentum anisotropy forpositively charged pion, kaon and proton from 2.76ATeV Pb+Pb collisions at LHC and at different cen-trality bins using an event-by-event viscous hydrody-namic model framework with fluctuating initial condi-tions. The ǫ n − v n correlation from a relatively smallersystem at lower beam energy (Cu+Cu collisions at 200AGeV at RHIC) is also calculated for a comparison withthe Pb+Pb collisions.The linear correlation is found to be stronger for cen-tral collisions than for peripheral collisions for all theparticles. In addition, the correlation between ǫ − v isfound to be weaker than ǫ − v . However, the correla-tion between v n ( p T ) and ǫ n as a function of p T showsinteresting behaviour where the correlation coefficient Cis found to depend strongly on the mass of the particles.We see a clear ordering of the correlation coefficient inthe lower p T region depending on the particle mass wherethe correlation strength is found to be larger for lighterparticles. The p T range for the ordering depends on thecollision centrality and also on the beam energy. The p T dependent correlation strength is found to rise with p T ,reach maximum, and then drop slowly beyond 2 GeV p T value for the Pb+Pb collisions. The correlation strengthfor π + reaches maximum at a relatively smaller p T valuethan for K + and protons. Although the strength of thecorrelation between ǫ and v is found to be relativelyweaker compared to ǫ and v , we see a similar quali-tative p T dependent behaviour of correlation co-efficientfor both of them.The Cu+Cu collisions at RHIC produce a relativelysmaller system than for Pb+Pb collisions for a particu-lar centrality bin and the correlation coeffient C( ǫ n , v n )is found to be slightly smaller for Cu+Cu collisions. However, the p T dependent correlation coefficient fromCu+Cu collisions provide valuable insight about the sys-tem size and beam energy dependence of the correlationstrength. The C( ǫ n , v n ( p T )) also shows mass ordering,however the p T range is much smaller (0.3 – 0.8 GeV)compared to Pb+Pb collision. This could be due to arelatively weaker development of the transverse flow ve-locity for Cu+Cu collisions at lower beam energy thanfor Pb+Pb collisions at LHC. In addition, the correla-tion strength is found to be strongest at a much smaller p T value for Cu+Cu collisions than for Pb+Pb collisions.The study of correlation strength for same system at dif-ferent beam energies would give a more quantitative es-timation of the beam energy dependence.The correlation coefficient is found to depend onlymarginally on the value of η/s . However, the rela-tive fluctuations in the anisotropic flow parameter showstrong sensitivity to the value of η/s . The value of σ v n / h v n i is found to be significantly larger for larger η/s for heavier particle and in the region p T < π + , K + , and p for different central-ity bins of Pb+Pb and Cu+Cu collisions. The NSC(2,3)as a function of collision centrality shows a clear anti-correlation between v and v for peripheral collisionsand also does not show a strong dependence on the massof the particles. IX. ACKNOWLEDGMENT