Fluctuations of harmonic and radial flow in heavy ion collisions with principal components
FFluctuations of harmonic and radial flow in heavy ion collisions with principalcomponents
Aleksas Mazeliauskas ∗ and Derek Teaney † Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA (Dated: October 13, 2018)We analyze the spectrum of harmonic flow, v n ( p T ) for n = 0–5, in event-by-event hydrodynamicsimulations of Pb+Pb collisions at the CERN Large Hadron Collider ( √ s NN = 2 .
76 TeV) withprincipal component analysis (PCA). The PCA procedure finds two dominant contributions to thetwo-particle correlation function. The leading component is identified with the event plane v n ( p T ),while the subleading component is responsible for factorization breaking in hydrodynamics. For v , v , and v the subleading flow is a response to the radial excitation of the corresponding eccentricity.By contrast, for v the subleading flow in peripheral collisions is dominated by the nonlinear mixingbetween the leading elliptic flow and radial flow fluctuations. In the v case, the sub-sub-leadingmode more closely reflects the response to the radial excitation of ε . A consequence of this pictureis that the elliptic flow fluctuations and factorization breaking change rapidly with centrality, andin central collisions (where the leading v is small and nonlinear effects can be neglected) thesubsub-leading mode becomes important. Radial flow fluctuations and nonlinear mixing also play asignificant role in the factorization breaking of v and v . We construct good geometric predictorsfor the orientation and magnitudes of the leading and subleading flows based on a linear response tothe geometry, and a quadratic mixing between the leading principal components. Finally, we suggesta set of measurements involving three point correlations which can experimentally corroborate thenonlinear mixing of radial and elliptic flow and its important contribution to factorization breakingas a function of centrality. I. INTRODUCTION
Two-particle correlation measurements are ofparamount importance in studying ultrarelativisticheavy ion collisions, and provide an extraordinarilystringent test for theoretical models. Indeed, themeasured two-particle correlations exhibit elliptic,triangular, and higher harmonics flows, which can beused to constrain the transport properties of the quarkgluon plasma (QGP) [1, 2]. The remarkable precisionof the experimental data as a function of transversemomentum and pseudorapidity has led to new analysesof factorization breaking, nonlinear mixing, event shapeselection, and forward-backward fluctuations [3–8].In this paper we analyze the detailed structure oftwo-particle transverse momentum correlations by usingevent-by-event (boost-invariant) hydrodynamics andprincipal component analysis (PCA) [9, 10]. Specifically,we decompose the event-by-event harmonic flow V n ( p T )into principal components and investigate the physicalorigin of each of these fluctuations. This paper extendsour previous analysis [10] for triangular flow at the LHC(Pb+Pb at √ s NN = 2 .
76 TeV) to the other harmonics, n = 0–5. In particular, we demonstrate the importanceof radial flow fluctuations for subleading flows of higherharmonics.Taking the second harmonic for definiteness, the two-particle correlation matrix of momentum dependent el-liptic flows, C ( p T , p T ) ≡ (cid:104) V ( p T ) V ∗ ( p T ) (cid:105) is tradi- ∗ [email protected] † [email protected] tionally parametrized by r ( p T , p T ) [11], r ( p T , p T ) ≡ (cid:104) V ( p T ) V ∗ ( p T ) (cid:105) (cid:112) (cid:104)| V ( p T ) | (cid:105) (cid:104)| V ( p T ) | (cid:105) . (1)If there is only one source of elliptic flow in the event[for example if in each event V ( p T ) = f ( p T ) ε with ε a complex eccentricity and f ( p T ) a fixed real func-tion of p T ] then the correlation matrix of elliptic flows C ( p T , p T ) factorizes into a product of functions, andthe r parameter is unity. However, if there are multi-ple independent sources of elliptic flow in the event, thenthe correlation matrix does not factorize, and the r pa-rameter is less than unity [11]. The r parameter hasbeen extensively studied both experimentally [3, 12–14]and theoretically [10, 11, 15, 16]. In particular, in ourprior work on triangular flow we showed that factoriza-tion breaking in event-by-event hydrodynamics arises be-cause the simulated triangular flow is predominantly theresult of two statistically uncorrelated contributions—thelinear response to ε [17] and the linear response to thefirst radial excitation of ε [10]. The goal of the cur-rent paper is to extend this understanding of factoriza-tion breaking to the other harmonics. This extension wassurprisingly subtle due to the quadratic mixing betweenthe leading and subleading harmonic flows.Experimentally, it is observed that factorization break-ing is largest for elliptic flow in central collisions (see inparticular Fig. 28 of Ref. [13] and Fig. 1 of Ref. [3]).Indeed, the r parameter decreases rather dramaticallyfrom mid-central to central collisions. This indicates thatthe relative importance of the various initial state fluctu-ations which drive elliptic flow are changing rapidly as afunction of centrality. The current manuscript explains a r X i v : . [ nu c l - t h ] F e b the rapid centrality dependence of factorization breakingin v as an interplay between the linear response to thefluctuating elliptic geometry, and the nonlinear mixing ofthe radial flow and average elliptic flow. This quadraticmixing is similar to the mixing between v and v , v [18–21], and this picture can be confirmed experimentallyby measuring specific three point correlations analogousto the three plane correlations measured in the v , v , v case [4, 5].To understand the linear and nonlinear contributionsquantitatively, we will break up the fluctuations in hy-drodynamics into their principal components, and ana-lyze the linear and nonlinear contributions of each prin-cipal component to the simulated harmonic spectrum.The sample of events and most of the PCA methods arethe same as in our previous paper [10], and thereforein Sects. II A and II B we only briefly review the anal-ysis definitions, and the key features of simulations. InSect. II C we discuss the strategy for constructing thebest linear predictor for leading and subleading flows.The second part of our paper, Sect. III, contains in-dividual discussions for each harmonic flow. First, wediscuss radial flow fluctuations in Sect. III A and thendemonstrate their importance in generating subleadingelliptic flow in Sect. III B 1. In Sect. III C we briefly de-scribe our PCA results for direct and triangular flows.Finally, in Sect. III D we discuss the quadrangular andpentagonal flows and how the nonlinear mixing of lowerorder principal components adds to these flows. We putforward some experimental observables in the discussionin Sect. IV. A catalog of figures showing the main resultsof PCA for each harmonic is given in the Appendix. II. PCA OF TWO-PARTICLE CORRELATIONSIN EVENT-BY-EVENT HYDRODYNAMICSA. Principal components
PCA is a statistical technique for extracting the dom-inant components in fluctuating data. In the contextof flow in heavy ion collisions it was first introduced inRef. [9], and then applied to the analysis of triangularflow in our previous paper [10]. Here we review the es-sential definitions.The event-by-event single particle distribution is cus-tomarily expanded in a Fourier series dNd p = V ( p T ) + ∞ (cid:88) n =1 V n ( p T ) e − inϕ + H . c . , (2)where d p = dy dp T dϕ denotes the phase space, ϕ is theazimuthal angle of the distribution, and H.c. denotesHermitian conjugate. V n ( p T ) is a complex Fourier co-efficient recording the magnitude and orientation of the n th harmonic flow, without the typical normalization bymultiplicity.PCA is done by expanding the covariance matrix oftwo-particle correlations (which is real, symmetric, and r ( p T , p T ) p T = 0 . (a) p T = 1 . (b) p T = 1 . r ( p T , p T ) p T (GeV) p T = 2 . (d) 0 1 2 p T (GeV) p T = 2 . (e) 0 1 2 3 p T (GeV) p T = 3 . (f)full r FIG. 1. Factorization ratio r ( p T , p T ) [Eq. (7)] for trian-gular flow and its approximations with principal components(PCs) in central collisions (0–5%). positive semi-definite) into real orthogonal eigenvectors V ( a ) n ( p T ), C n ( p T , p T ) ≡ (cid:10)(cid:0) V n,p T − (cid:104) V n,p T (cid:105) (cid:1)(cid:0) V ∗ n,p T − (cid:10) V ∗ n,p T (cid:11) (cid:1)(cid:11) = (cid:88) a V ( a ) n ( p T ) V ( a ) n ( p T ) , (3)where V ( a ) n = √ λ a ψ ( a ) ( p T ) is the square root of theeigenvalue times a normalized eigenvector. The eigen-value records the squared variance of a given fluctuation.The principal components V (1) n ( p T ) , V (2) n ( p T ) , . . . of agiven event ensemble can be used as optimal basis forevent-by-event expansion of harmonic flow V n ( p T ) − (cid:104) V n ( p T ) (cid:105) = ξ (1) n V (1) n ( p T ) + ξ (2) n V (2) n ( p T ) + . . . . (4)The complex coefficients ξ ( a ) n are the projections of har-monic flow onto principal component basis and recordthe orientation and event-by-event amplitude of their re-spective flows. Principal components are mutually un-correlated (cid:68) ξ ( a ) n ξ ∗ ( b ) n (cid:69) = δ ab . (5)Typically the eigenvalues of C n ( p T , p T ) are stronglyordered and only the first few terms in the expansion aresignificant. Often the large components have a definitephysical interpretation. We define the scaled magnitudeof the flow vector V ( a ) n ( p T ) as (cid:107) v ( a ) n (cid:107) ≡ (cid:82) (cid:0) V ( a ) n ( p T ) (cid:1) dp T (cid:82) (cid:104) dN/dp T (cid:105) dp T , (6)which is a measure of the size of the fluctuation withouttrivial dependencies on the mean multiplicity in a givencentrality class.The leading flow vector V (1) n ( p T ) corresponds to fluc-tuations with the largest root-mean-square amplitude,while subsequent components maximize the variancein the remaining orthogonal directions. This yields avery efficient description of the full covariance matrix C n ( p T , p T ) and factorization ratio r n ( p T , p T ) ≡ C n ( p T , p T ) (cid:112) C n ( p T , p T ) C n ( p T , p T ) ≤ . (7) r n ( p T , p T ) is bounded by unity within hydrodynam-ics [11]. By truncating series expansion of the covariancematrix [Eq. (3)] at two or three principal components wecan approximate C n ( p T , p T ) and r n ( p T , p T ). Trun-cating at the leading term would constitute complete flowfactorization, i.e., r n = 1. The factorization matrix fortriangular flow is shown in Fig. 1. We see that at low mo-mentum p T < r . Analogous decompositions of two-particlecorrelations into principal components exist for all har-monics and all centralities. Interpreting these large flowcomponents physically is the goal of this paper. B. Simulations
We used boost-invariant event-by-event viscous hydro-dynamics to simulate 5000 Pb-Pb collisions at the CERNLarge Hadron Collider (LHC) ( √ s NN = 2 .
76 TeV) infourteen 5% centrality classes selected by impact param-eter. The initial conditions are based on the PhobosGlauber Monte Carlo [17] with a two-component modelfor the entropy distribution in the transverse plane. Weused a lattice equation of state [22] and a “direct” pionfreeze-out at T fo = 140 MeV. The results presented herewere simulated using a shear viscosity to entropy ratioof η/s = 0 .
08. Qualitatively similar results are obtainedwith η/s = 0 .
16, with the most important differencesdiscussed in Sect. III B 2. The same ensemble of eventswas used in Ref. [10], which provides further simulationdetails.
C. Geometrical predictors
We will construct several geometric predictors for theleading and subleading flows following strategy outlinedin Ref. [20]. Keeping the discussion general, let ξ ( a ) n pred bea geometric quantity which predicts the event-by-eventamplitude and phase of the corresponding flow ξ ( a ) n . Forexample, for the leading n = 3 component the triangular-ity ε , (defined below) is an excellent choice for ξ (1)3 pred .The geometric predictors are designed to maximize thecorrelation between a particular flow signal and the ge-ometry. Specifically, the predictors maximize the Pear-son correlation coefficient between the event-by-eventmagnitude and orientation of a th principal component, ξ ( a ) n , and the geometrical predictor ξ ( a ) n pred max Q ( a ) n = (cid:10) ξ ( a ) n ξ ∗ ( a ) n pred (cid:11)(cid:113)(cid:10) ξ ( a ) n ξ ∗ n ( a ) (cid:11)(cid:10) ξ ( a ) n pred ξ ∗ ( a ) n pred (cid:11) . (8)We constructed several predictors for the flow ξ ( a ) n byassuming a linear relation between the flow and the geom-etry. The simplest predictor consists of linear combina-tions of the first two eccentricities of the initial geometry.These are defined as ε n,n ≡ − [ r n e inφ ] R n rms , (9a) ε n,n +2 ≡ − [ r n +2 e inφ ] R n +2rms , (9b)where the square brackets [ ] denote an integral over theinitial entropy density for a specific event, normalized bythe average total entropy ¯ S tot . R rms ≡ (cid:112) (cid:104) [ r ] (cid:105) is theevent averaged root-mean-square radius. Note that ourdefinitions of ε n,n and ε n,n +2 are chosen to make theevent-by-event quantities ε n,n and ε n,n +2 linear in thefluctuations. In this notation, the geometric predictorbased on these eccentricities is ξ ( a ) n pred = ε n,n + c ε n,n +2 , (10)where c is adjusted to maximize the correlation coeffi-cient in Eq. (8), and the overall normalization is irrele-vant. While the first two eccentricities provide an excel-lent predictor for the leading flow, they do not predictthe subleading flow very well. This is in part because theradial weight r n +2 is too strong at large r .More generally, one can define eccentricity as a func-tional of radial weight function ρ ( r ): ε n { ρ ( r ) } ≡ − [ ρ ( r ) e inφ ] R n rms . (11)It is the goal of this paper to find the optimal radialweight function ρ ( r ) for predicting both leading and sub-leading flows. For the subleading modes ρ ( r ) will havea node, and thus ε n { ρ ( r ) } will measure the magnitudeand orientation of the first radial excitation of the geom-etry [10].To find the optimal radial weight we expand ρ ( r ) inradial Fourier modes ρ ( r ) = n k (cid:88) b =1 w b n n ! k nb J n ( k b r ) , (12)where J n ( x ) is a Bessel function of order n , w b are expan-sion coefficients, and k b are definite wavenumbers spec-ified below. The prefactor is chosen so that for a single k mode ( w = 1 , w b> = 0) at small k ( kR rms (cid:28)
1) thegeneralized eccentricity approaches ε n,n lim k → ε n { ρ ( r ) } = ε n,n . (13)At small k , we expand the J n ( kr ) and find ε n { ρ ( r ) } (cid:39) ε n,n + c ε n,n +2 , (14)where c = − ( kR rms / / (1 + n ). Thus the functionalform of ρ ( r ) adopted here yields a tunable linear com-bination of the eccentricities in Eq. (9), but the wavenumber parameter regulates the behavior at large r . Fur-ther motivation and discussion of this basis set for ρ ( r )is given in our previous work [10].We have found that an approximately optimal radialweight can be found by using only two well chosen k b values for the Fourier expansion in Eq. (12). Includingadditional k modes in the functional form of ρ ( r ) doesnot significantly improve the predictive power of the gen-eralized geometric eccentricity. For the two k modes werequired (somewhat arbitrarily) that the ratio of k valueswould be fixed to the ratio of the first two Bessel zeros: k k = j n, j n, . (15)With this choice our basis functions were orthogonal inthe interval [0 , R o ], where k = j n, /R o . We then ad-justed R o to maximize the correlation coefficient between ε n { ρ ( r ) } and the flow ξ ( a ) n . To account for changing sys-tem size with centrality, we used a fixed R o /R rms ratio.In most cases we used R o /R rms ≈ .
0, but for all directedflow components ( ξ (1)1 and ξ (2)1 ) and the second ellipticflow component ( ξ (2)2 ), we found that R o /R rms ≈ . ε n { ρ ( r ) } , is based on linear response.If nonlinear physics becomes important (as in the caseof v and v ) then the predictors should be modified toincorporate this physics (see below and Ref. [20]). Thus,below we will refer to the ε n { ρ ( r ) } (with an optimizedradial weight) as the best linear predictor and incorpo-rate quadratic nonlinear corrections to the predictor asneeded. III. RESULTSA. Radial flow
Radial flow (or V ( p T )) is the first term in the Fourierseries and is by far the largest harmonic. Traditionally,the experimental and theoretical study of the fluctuationsof V ( p T ) (i.e., multiplicity and p T fluctuations) has beendistinct from elliptic and triangular flow. There is noreason for this distinction.Examining the scaled V ( p T ) eigenvalues shown inFig. 2(a), we see that there are two large principal com-ponents. The first principal component is sourced bymultiplicity fluctuations, i.e., the magnitude of V ( p T ) fluctuates (but not its shape) due to the impact parame-ter variance in a given centrality bin. Corroborating thisinference, Fig. 2(a) shows the momentum dependence ofthe leading principal component, which is approximatelyflat. Clearly this principal component is not particu-larly interesting, and the PCA procedure gives a practicalmethod for isolating these trivial geometric fluctuationsin the data set. The second principal component is ofmuch greater interest, and shows a linear rise with p T that is indicative of the fluctuations in the radial flowvelocity of the fluid [9].In early insightful papers [23, 24], the fluctuations inthe flow velocity (or mean p T ) were associated with thefluctuations in the initial fireball radius. These radialfluctuations are well described by both the eccentrici-ties ( ε , , ε , ), Eq. (9), and the optimized eccentricity ε { ρ ( r ) } , Eq. (11). Therefore, as seen in Fig. 2(b), thesubleading flow signal is strongly correlated with theselinear geometric predictors.Also shown in Fig. 2(b) is the correlation between sub-leading radial flow ξ (2)0 and mean transverse momentumfluctuations around the average δp T ≡ [ p T ] − (cid:104) [ p T ] (cid:105) . (16)Indeed, the subleading radial flow correlates very wellwith mean momentum fluctuations in all centrality bins.In the next sections we will study the nonlinear mixingbetween the radial flow ξ (2)0 and all other harmonics. B. Elliptic flow
1. Nonlinear mixing and elliptic flow
We next study the fluctuations of V ( p T ) as functionof centrality. As seen in Fig. 3, the principal componentspectrum of elliptic flow in central collisions consists oftwo nearly degenerate subleading components in addi-tion to the dominant leading component. This degen-eracy is lifted in more peripheral bins. Comparing the p T dependence of the principal flows shown in Figs. 4(a)and 4(b), we see that going from central (0–5%) to pe-ripheral (45–50%) collisions, the magnitude of the sec-ond principal component increases in size and its momen-tum dependence changes dramatically. By contrast, thegrowth of the third principal component is much moremild. This strongly suggests that the average elliptic ge-ometry is more important for the subleading than thesubsub-leading mode. There is a small upward tending slope in our simulations of thiscomponent, because multiplicity and mean p T fluctuations onlyapproximately factorize into leading and subleading principalcomponents. Using different definitions of centrality bins couldperhaps make this separation cleaner. -0.04-0.020.000.020.040.060.080.100.120.140.16 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)0 ( p T ) v (2)0 ( p T ) v (3)0 ( p T ) (a) | Q ( ) | centrality (%) v subleadingbest lin. ε , + ε , δp T (b) FIG. 2. (a) The p T dependence of the principal components of radial flow normalized by the average multiplicity, v ( a )0 ( p T ) ≡ V ( a )0 ( p T ) / (cid:104) dN/dp T (cid:105) . (b) The Pearson correlation coefficient [Eq. (8)] between the subleading radial flow and various predictorsversus centrality. The best linear predictor is described in Sect. II C. − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading FIG. 3. The magnitudes of the principal components of ellip-tic flow, (cid:107) v ( a )2 (cid:107) , versus centrality [see Eq. (6)]. To find a geometrical predictor for the sub- and sub-sub-leading modes we first tried the best linear predictor ε { ρ ( r ) } . In Fig. 5(a) (the red circles), we see that thecorrelation coefficient between this optimal linear predic-tor and the subleading flow signal drops precipitously asa function of centrality. As we will explain now, thisis because nonlinear mixing becomes important for thesubleading mode.The ellipticity of the almond shaped geometry in pe-ripheral collisions is traditionally parametrized by eccen-tricity ε , and it serves as an excellent predictor for the leading elliptic flow. However, ε , does not com-pletely fix the initial geometry, and the radial size of thefireball can fluctuate at fixed eccentricity. As explainedin Sect. III A, the radial size fluctuations modulate themomentum spectrum of the produced particles, and fora background geometry with large constant eccentricitythis generates fluctuations in the p T dependence of theelliptic flow, i.e., subleading elliptic flow. This sublead-ing flow lies in the reaction plane following the averageelliptic flow, but its sign (which is determined by δp T ) isuncorrelated with ε , .The orientation of the reaction plane in peripheral binsis strongly correlated with the integrated v or the lead-ing elliptic principal component ξ (1)2 , while the mean p T fluctuations are tracked by the subleading radial flowcomponent ξ (2)0 . Therefore we correlated the sub- andsub-sub-leading elliptic flows with the product of theleading elliptic and radial flows, i.e. we computed thecorrelation coefficient in Eq. (8) with ξ (2)2 , pred = ξ (1)2 ξ (2)0 .Examining Fig. 5(a) (the black line), we see see that thecorrelation between the subleading elliptic flow and thenonlinear mixing rises with centrality, as the correlationwith best linear predictor drops. Examining Fig. 5(b)on the other hand, we see that the subsub-leading ellip-tic flow has stronger correlation with the initial geom-etry than the nonlinear mixing. Combining best lineargeometric predictor and quadratic mixing terms in thepredictor, i.e. ξ (2)2 pred = ε { ρ ( r ) } + c ξ (1)2 ξ (2)0 , (17)we achieve consistently high correlations for all centrali-ties [the blue diamonds in Fig. 5(a) and (b)]. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)2 ( p T ) v (2)2 ( p T ) v (3)2 ( p T ) (a) -0.050.000.050.100.150.200.250.30 0 1 2 3 v ( a ) ( p T ) p T (GeV)
45 - 50% v (1)2 ( p T ) v (2)2 ( p T ) v (3)2 ( p T ) (b) FIG. 4. The p T dependence of the principal components of elliptic flow normalized by the average multiplicity, v ( a )2 ( p T ) ≡ V ( a )2 ( p T ) / (cid:104) dN/dp T (cid:105) , for central (0–5%) and peripheral collisions (45–50%). | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)2 ξ (2)0 best lin. ξ (1)2 ξ (2)0 (a) | Q ( ) | centrality (%) v subsub-leadingbest lin. + ξ (1)2 ξ (2)0 best lin. ξ (1)2 ξ (2)0 (b) FIG. 5. Pearson correlation coefficient between the subleading elliptic flows and the best linear predictor [Eq. (11)] with andwithout the nonlinear mixing between the radial and leading elliptic flows, ξ (1)2 ξ (2)0 . (a) and (b) show the correlation coefficientfor v subleading and v subsub-leading flows respectively.
2. Dependence on viscosity
Before leaving this section we will briefly commenton the viscosity dependence of these results. Figure 6shows a typical result for a slightly larger shear viscosity, η/s = 0 .
16. As discussed above, the subleading ellip-tic flow [i.e., the event-by-event fluctuations in V ( p T )]is a result of the linear response to the first radial exci-tation of the elliptic eccentricity, and a nonlinear mix- ing of radial flow fluctuations and the leading ellipticflow. In Fig. 6 we see that a slightly larger shear vis-cosity tends to preferentially damp the linear responseleaving a stronger nonlinear signal. This is because theinitial geometry driving the linear response has a signif-icantly larger gradients due to the combined azimuthaland radial variations. Thus in Fig. 6 the linear responsedominates the subleading flow only in very central colli-sions. These trends with centrality are qualitatively fa-miliar from previous analyses of the effect of shear vis- | Q ( ) | centrality (%) v subleading0.00.10.20.30.40.50.60.70.80.91.0 0 10 20 30 40 50 60 70 | Q ( ) | centrality (%) v subleading0.00.10.20.30.40.50.60.70.80.91.0 0 10 20 30 40 50 60 70 | Q ( ) | centrality (%) v subleading . η/s = 0 . best lin. + ξ (1)2 ξ (2)0 best lin. ξ (1)2 ξ (2)0 FIG. 6. Pearson correlation coefficients for the subleadingelliptic flow at viscosity over entropy ratio η/s = 0 .
16. Dashedlines repeat η/s = 0 .
08 results from Fig. 5(a) for the ease ofcomparison. cosity on the nonlinear mixing of harmonics [21, 25].
C. Triangular and directed flows
Triangular flow was extensively studied in our previouswork [10]. For the sake of completeness we relegate sev-eral comparative plots to the Appendix. Previously, weconstructed an optimal linear predictor ε { ρ ( r ) } for thesubleading triangular mode which characterizes the radi-ally excited triangular geometry. As shown in Fig. 7(b),adding the nonlinear mixing term ξ (2)0 ξ (1)3 to the best lin-ear predictor marginally improves the already good cor-relation with the subleading flow in peripheral collisions.Directed flow exhibits many similarities to triangularflow. Specifically the subleading directed flow is rea-sonably well correlated with the optimal linear predic-tor, characterizing the radially excited dipolar geometry.Nonlinear mixing between the leading directed flow andthe radial flow is unimportant [see Fig. 7(a)]. D. The n = 4 and n = 5 harmonic flows It is well known that the leading components of the n = 4 and n = 5 harmonics are determined by the non-linear mixing of lower order harmonics in peripheral col-lisions [5, 18–21].For comparison with other works [20, 26], in the Ap-pendix in Figs. 13(d) and 14(d) we construct a predictorbased on a linear combination of the eccentricities ε , + c ε , ε , for n = 4 , (18) ε , + c ε , ε , for n = 5 , (19) where here and below the coefficient c is adjusted tomaximize the correlation with the flow. This predictor iscompared to a linear combination of the optimal eccen-tricity ε n { ρ ( r ) } and the corresponding nonlinear mixingsof the leading principal components ε { ρ ( r ) } + c ξ (1)2 ξ (1)2 for n = 4 , (20a) ε { ρ ( r ) } + c ξ (1)2 ξ (1)3 for n = 5 . (20b)Both sets of predictors perform reasonably well, thoughthe second set has a somewhat stronger correlation withthe flow.Returning to the subleading components, we first cor-related with the best linear predictors, ε { ρ ( r ) } and ε { ρ ( r ) } , with the corresponding subleading flow signals.As seen in Fig. 8 (the red circles), the correlation de-creases rapidly with centrality, especially for v . Moti-vated by Eq. (20) which predicts the event-by-event lead-ing v and v in terms v and v , we construct a predictorfor the subleading v and v in terms of the fluctuationsof v and v (see Secs. III B 1 and III C, respectively).The full predictor reads ε { ρ ( r ) } + c ξ (1)2 ξ (2)2 , for n = 4 , (21a) ε { ρ ( r ) } + c ξ (1)2 ξ (2)3 + c ξ (1)3 ξ (2)2 , for n = 5 . (21b)Including the mixings between the subleading v and v and the corresponding leading components greatly im-proves the correlation in mid-central bins (the blue dia-monds). Finally, in an effort to improve the v predictorin the most peripheral bins we have added additionalnonlinear mixings between the radial flow and the lead-ing principal components ε { ρ ( r ) } + c ξ (1)2 ξ (2)2 + c ξ (1)4 ξ (2)0 , for n = 4 , (22a) ε { ρ ( r ) } + c ξ (1)2 ξ (2)3 + c ξ (1)3 ξ (2)2 + c ξ (1)5 ξ (2)0 , for n = 5 . (22b)As seen in Fig. 8(a) (the grey line) the coupling to theradial flow improves the correlation between the sublead-ing v and the predictor in peripheral collisions. On theother hand, for v , Fig. 8(b), all of the information aboutthe coupling to the radial flow is already included inEq. (21b) and adding v does not improve the correla-tion. IV. DISCUSSION
In this paper we classified the event-by-event fluctu-ations of the momentum dependent Fourier harmonics V n ( p T ) for n = 0–5 by performing a principal componentanalysis of the two-particle correlation matrix in hydro-dynamic simulations of heavy ion collisions. The leading principal component for each harmonic is very stronglycorrelated with the integrated flow, and therefore this | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)1 ξ (2)0 best lin. ε , + ε , (a) | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)3 ξ (2)0 best lin. ε , + ε , (b) FIG. 7. Pearson correlation coefficient between the subleading (a) directed and (b) triangular flows and the best linearpredictor with and without radial flow mixing. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)2 ξ (2)2 + ξ (1)4 ξ (2)0 best lin. + ξ (1)2 ξ (2)2 best lin. (a) | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)2 ξ (2)3 + ξ (2)2 ξ (1)3 + ξ (1)5 ξ (2)0 best lin. + ξ (1)2 ξ (2)3 + ξ (2)2 ξ (1)3 best lin. (b) FIG. 8. Pearson correlation coefficient between the subleading v and v flows and the best linear predictor with and withoutseveral nonlinear terms [see Eqs. (21) and (22)]. component is essentially the familiar v n ( p T ) measured inthe event plane. The subleading components describe ad-ditional p T dependent fluctuations of the magnitude andphase of v n ( p T ). This paper focuses on the physical ori-gins of the subleading flows, which are the largest sourceof factorization breaking in hydrodynamics.Our systematic study started by placing radial flow(the n = 0 harmonic) in the same framework as theother harmonic flows. We identified the subleading n = 0principal component with mean p T fluctuations and con-firmed (as is well known [23, 24]) that these fluctuations are predicted by the variance of the radial size of thefireball.The subleading directed and triangular flows wereshown to be a linear response to the radial excitationsof the corresponding eccentricity of the initial geometry.In these cases a generalized eccentricity ε n { ρ ( r ) } withan optimized radial weight (describing the radial excita-tion) provides a good predictor for the subleading flows(Fig. 7). This extends our previous analysis of v to v [10].Next, we investigated the nature of the subleading el-liptic flows. The principal component analysis revealsthat in central collisions there are two comparable sourcesof subleading elliptic flow, but they have strikingly dif-ferent centrality dependence (see Figs. 3 and 4). Inmid-peripheral collisions the first subleading componentmainly reflects a nonlinear mixing between elliptic andradial flows, and this component is only weakly corre-lated with the radially excitations of the elliptic geome-try. The second subleading component in this centralityrange is substantially smaller and more closely reflectsthe radial excitations. In more central collisions, how-ever, the nonlinear mixing with the average elliptic flowbecomes small, and the sub and subsub-leading principalcomponents become comparable in size. Thus, the rapidcentrality dependence of factorization breaking in v isthe result of an interplay between the linear response tothe fluctuating elliptic geometry, and the nonlinear mix-ing of the radial and average elliptic flows.This nonlinear mixing can be confirmed experimen-tally by measuring the correlations between the princi-pal components (cid:68) ξ (2)2 ( ξ (1)2 ξ (2)0 ) ∗ (cid:69) which is predicted inFig. 5. The prediction is that three point correlation be-tween the subleading elliptic event plane, the mean p T fluctuations, and the leading elliptic event plane definedby the Q vector, i.e., (cid:68) ξ (2)2 δp T Q ∗ (cid:69)(cid:112) (cid:104) ( δp T ) (cid:105) (cid:104)| Q | (cid:105) , (23) changes rapidly from central to midperipheral collisions.This correlation is analogous to the three plane correla-tions such as (cid:104) V ( V V ) ∗ (cid:105) measured previously [5].Finally, we studied factorization breaking in v and v . With the comprehensive understanding of the fluc-tuations of v and v described above, the correspondingfluctuations in v and v were naturally explained as thenonlinear mixing of subleading v and v with their lead-ing counterparts, together with linear response to thequadrangular and pentagonal geometries (see Fig. 8).The study of the fluctuations in the harmonics spec-trum presented here shows the power of the principalcomponent method in elucidating the physics which drivethe event-by-event flow. We hope that this motivatesa comprehensive experimental program measuring theprincipal components and their correlations for n = 0 − ACKNOWLEDGMENTS
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APPENDIX: LIST OF FIGURES
Here we present a comprehensive catalog of plots foreach harmonic n = 0-5. Centrality dependence of flowmagnitudes for n = 0, appearing as Fig. 3 in the textabove, is repeated here as Fig. 9(a), and analogous plotsfor other harmonics are given in Figs. 10-14(a). The p T dependence of normalized principal components for ra-dial and elliptic flows in central (0-5%) collisions shown inFigs. 2(a) and 4(a) are reproduced as Figs. 9(c) and 11(c)and complemented with Figs. 10(c) and 12-14(c). Addi-tionally, Figs. 9-14(b) depict the same principal compo-nents, but without normalization by average multiplicity (cid:104) dN/dp T (cid:105) . Finally, in the paper we showed the Pearsoncorrelation coefficients for the subleading flows for eachharmonic n = 0-5 in Figs. 2(b),5(a),7(a), 7(b), 8(a) and8(b), while in this appendix we show results for both lead-ing and subleading flows in the series of figures Figs. 9-14(d) and Figs. 9-14(e).110 − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )0 (cid:107) . -10-50510152025 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)0 ( p T ) V (2)0 ( p T ) V (3)0 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )0 ( p T )in central collisions. -0.04-0.020.000.020.040.060.080.100.120.140.16 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)0 ( p T ) v (2)0 ( p T ) v (3)0 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )0 ( p T ) ≡ V ( a )0 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. ε , + ε , δp T (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 9. n = 0 − − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )1 (cid:107) . -8-7-6-5-4-3-2-1012 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)1 ( p T ) V (2)1 ( p T ) V (3)1 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )1 ( p T )in central collisions. -0.04-0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)1 ( p T ) v (2)1 ( p T ) v (3)1 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )1 ( p T ) ≡ V ( a )1 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)1 ξ (2)0 best lin. ε , + ε , (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 10. nn
Here we present a comprehensive catalog of plots foreach harmonic n = 0-5. Centrality dependence of flowmagnitudes for n = 0, appearing as Fig. 3 in the textabove, is repeated here as Fig. 9(a), and analogous plotsfor other harmonics are given in Figs. 10-14(a). The p T dependence of normalized principal components for ra-dial and elliptic flows in central (0-5%) collisions shown inFigs. 2(a) and 4(a) are reproduced as Figs. 9(c) and 11(c)and complemented with Figs. 10(c) and 12-14(c). Addi-tionally, Figs. 9-14(b) depict the same principal compo-nents, but without normalization by average multiplicity (cid:104) dN/dp T (cid:105) . Finally, in the paper we showed the Pearsoncorrelation coefficients for the subleading flows for eachharmonic n = 0-5 in Figs. 2(b),5(a),7(a), 7(b), 8(a) and8(b), while in this appendix we show results for both lead-ing and subleading flows in the series of figures Figs. 9-14(d) and Figs. 9-14(e).110 − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )0 (cid:107) . -10-50510152025 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)0 ( p T ) V (2)0 ( p T ) V (3)0 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )0 ( p T )in central collisions. -0.04-0.020.000.020.040.060.080.100.120.140.16 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)0 ( p T ) v (2)0 ( p T ) v (3)0 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )0 ( p T ) ≡ V ( a )0 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. ε , + ε , δp T (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 9. n = 0 − − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )1 (cid:107) . -8-7-6-5-4-3-2-1012 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)1 ( p T ) V (2)1 ( p T ) V (3)1 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )1 ( p T )in central collisions. -0.04-0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)1 ( p T ) v (2)1 ( p T ) v (3)1 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )1 ( p T ) ≡ V ( a )1 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)1 ξ (2)0 best lin. ε , + ε , (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 10. nn = 1 − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )2 (cid:107) . -1012345678 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)2 ( p T ) V (2)2 ( p T ) V (3)2 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )2 ( p T )in central collisions. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)2 ( p T ) v (2)2 ( p T ) v (3)2 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )2 ( p T ) ≡ V ( a )2 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)2 ξ (2)0 best lin. ξ (1)2 ξ (2)0 (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 11. nn
Here we present a comprehensive catalog of plots foreach harmonic n = 0-5. Centrality dependence of flowmagnitudes for n = 0, appearing as Fig. 3 in the textabove, is repeated here as Fig. 9(a), and analogous plotsfor other harmonics are given in Figs. 10-14(a). The p T dependence of normalized principal components for ra-dial and elliptic flows in central (0-5%) collisions shown inFigs. 2(a) and 4(a) are reproduced as Figs. 9(c) and 11(c)and complemented with Figs. 10(c) and 12-14(c). Addi-tionally, Figs. 9-14(b) depict the same principal compo-nents, but without normalization by average multiplicity (cid:104) dN/dp T (cid:105) . Finally, in the paper we showed the Pearsoncorrelation coefficients for the subleading flows for eachharmonic n = 0-5 in Figs. 2(b),5(a),7(a), 7(b), 8(a) and8(b), while in this appendix we show results for both lead-ing and subleading flows in the series of figures Figs. 9-14(d) and Figs. 9-14(e).110 − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )0 (cid:107) . -10-50510152025 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)0 ( p T ) V (2)0 ( p T ) V (3)0 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )0 ( p T )in central collisions. -0.04-0.020.000.020.040.060.080.100.120.140.16 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)0 ( p T ) v (2)0 ( p T ) v (3)0 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )0 ( p T ) ≡ V ( a )0 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. ε , + ε , δp T (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 9. n = 0 − − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )1 (cid:107) . -8-7-6-5-4-3-2-1012 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)1 ( p T ) V (2)1 ( p T ) V (3)1 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )1 ( p T )in central collisions. -0.04-0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)1 ( p T ) v (2)1 ( p T ) v (3)1 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )1 ( p T ) ≡ V ( a )1 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)1 ξ (2)0 best lin. ε , + ε , (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 10. nn = 1 − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )2 (cid:107) . -1012345678 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)2 ( p T ) V (2)2 ( p T ) V (3)2 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )2 ( p T )in central collisions. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)2 ( p T ) v (2)2 ( p T ) v (3)2 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )2 ( p T ) ≡ V ( a )2 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)2 ξ (2)0 best lin. ξ (1)2 ξ (2)0 (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 11. nn = 2 − − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )3 (cid:107) . -2-1012345 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)3 ( p T ) V (2)3 ( p T ) V (3)3 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )3 ( p T )in central collisions. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)3 ( p T ) v (2)3 ( p T ) v (3)3 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )3 ( p T ) ≡ V ( a )3 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)3 ξ (2)0 best lin. ε , + ε , (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 12. nn
Here we present a comprehensive catalog of plots foreach harmonic n = 0-5. Centrality dependence of flowmagnitudes for n = 0, appearing as Fig. 3 in the textabove, is repeated here as Fig. 9(a), and analogous plotsfor other harmonics are given in Figs. 10-14(a). The p T dependence of normalized principal components for ra-dial and elliptic flows in central (0-5%) collisions shown inFigs. 2(a) and 4(a) are reproduced as Figs. 9(c) and 11(c)and complemented with Figs. 10(c) and 12-14(c). Addi-tionally, Figs. 9-14(b) depict the same principal compo-nents, but without normalization by average multiplicity (cid:104) dN/dp T (cid:105) . Finally, in the paper we showed the Pearsoncorrelation coefficients for the subleading flows for eachharmonic n = 0-5 in Figs. 2(b),5(a),7(a), 7(b), 8(a) and8(b), while in this appendix we show results for both lead-ing and subleading flows in the series of figures Figs. 9-14(d) and Figs. 9-14(e).110 − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )0 (cid:107) . -10-50510152025 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)0 ( p T ) V (2)0 ( p T ) V (3)0 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )0 ( p T )in central collisions. -0.04-0.020.000.020.040.060.080.100.120.140.16 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)0 ( p T ) v (2)0 ( p T ) v (3)0 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )0 ( p T ) ≡ V ( a )0 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. ε , + ε , δp T (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 9. n = 0 − − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )1 (cid:107) . -8-7-6-5-4-3-2-1012 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)1 ( p T ) V (2)1 ( p T ) V (3)1 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )1 ( p T )in central collisions. -0.04-0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)1 ( p T ) v (2)1 ( p T ) v (3)1 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )1 ( p T ) ≡ V ( a )1 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)1 ξ (2)0 best lin. ε , + ε , (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 10. nn = 1 − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )2 (cid:107) . -1012345678 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)2 ( p T ) V (2)2 ( p T ) V (3)2 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )2 ( p T )in central collisions. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)2 ( p T ) v (2)2 ( p T ) v (3)2 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )2 ( p T ) ≡ V ( a )2 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)2 ξ (2)0 best lin. ξ (1)2 ξ (2)0 (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 11. nn = 2 − − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )3 (cid:107) . -2-1012345 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)3 ( p T ) V (2)3 ( p T ) V (3)3 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )3 ( p T )in central collisions. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)3 ( p T ) v (2)3 ( p T ) v (3)3 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )3 ( p T ) ≡ V ( a )3 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)3 ξ (2)0 best lin. ε , + ε , (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 12. nn = 3 − − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )4 (cid:107) . -1-0.500.511.522.5 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)4 ( p T ) V (2)4 ( p T ) V (3)4 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )4 ( p T )in central collisions. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)4 ( p T ) v (2)4 ( p T ) v (3)4 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )4 ( p T ) ≡ V ( a )4 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. + ξ (1)2 ξ (1)2 ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)2 ξ (2)2 + ξ (1)4 ξ (2)0 best lin. + ξ (1)2 ξ (2)2 best lin. (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 13. nn
Here we present a comprehensive catalog of plots foreach harmonic n = 0-5. Centrality dependence of flowmagnitudes for n = 0, appearing as Fig. 3 in the textabove, is repeated here as Fig. 9(a), and analogous plotsfor other harmonics are given in Figs. 10-14(a). The p T dependence of normalized principal components for ra-dial and elliptic flows in central (0-5%) collisions shown inFigs. 2(a) and 4(a) are reproduced as Figs. 9(c) and 11(c)and complemented with Figs. 10(c) and 12-14(c). Addi-tionally, Figs. 9-14(b) depict the same principal compo-nents, but without normalization by average multiplicity (cid:104) dN/dp T (cid:105) . Finally, in the paper we showed the Pearsoncorrelation coefficients for the subleading flows for eachharmonic n = 0-5 in Figs. 2(b),5(a),7(a), 7(b), 8(a) and8(b), while in this appendix we show results for both lead-ing and subleading flows in the series of figures Figs. 9-14(d) and Figs. 9-14(e).110 − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )0 (cid:107) . -10-50510152025 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)0 ( p T ) V (2)0 ( p T ) V (3)0 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )0 ( p T )in central collisions. -0.04-0.020.000.020.040.060.080.100.120.140.16 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)0 ( p T ) v (2)0 ( p T ) v (3)0 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )0 ( p T ) ≡ V ( a )0 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. ε , + ε , δp T (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 9. n = 0 − − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )1 (cid:107) . -8-7-6-5-4-3-2-1012 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)1 ( p T ) V (2)1 ( p T ) V (3)1 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )1 ( p T )in central collisions. -0.04-0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)1 ( p T ) v (2)1 ( p T ) v (3)1 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )1 ( p T ) ≡ V ( a )1 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)1 ξ (2)0 best lin. ε , + ε , (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 10. nn = 1 − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )2 (cid:107) . -1012345678 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)2 ( p T ) V (2)2 ( p T ) V (3)2 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )2 ( p T )in central collisions. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)2 ( p T ) v (2)2 ( p T ) v (3)2 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )2 ( p T ) ≡ V ( a )2 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)2 ξ (2)0 best lin. ξ (1)2 ξ (2)0 (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 11. nn = 2 − − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )3 (cid:107) . -2-1012345 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)3 ( p T ) V (2)3 ( p T ) V (3)3 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )3 ( p T )in central collisions. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)3 ( p T ) v (2)3 ( p T ) v (3)3 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )3 ( p T ) ≡ V ( a )3 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)3 ξ (2)0 best lin. ε , + ε , (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 12. nn = 3 − − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )4 (cid:107) . -1-0.500.511.522.5 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)4 ( p T ) V (2)4 ( p T ) V (3)4 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )4 ( p T )in central collisions. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)4 ( p T ) v (2)4 ( p T ) v (3)4 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )4 ( p T ) ≡ V ( a )4 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. + ξ (1)2 ξ (1)2 ε , + ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)2 ξ (2)2 + ξ (1)4 ξ (2)0 best lin. + ξ (1)2 ξ (2)2 best lin. (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 13. nn = 4 − − − − k v ( a ) k centrality (%) leadingsubleadingsubsub-leading (a) Centrality dependence of the (scaled) magnitudes of flows (cid:107) v ( a )5 (cid:107) . -0.4-0.200.20.40.60.811.2 0 1 2 3 V ( a ) ( p T ) p T (GeV) V (1)5 ( p T ) V (2)5 ( p T ) V (3)5 ( p T ) (b) Momentum dependence of principal flow vectors V ( a )5 ( p T )in central collisions. -0.020.000.020.040.060.080.10 0 1 2 3 v ( a ) ( p T ) p T (GeV) v (1)5 ( p T ) v (2)5 ( p T ) v (3)5 ( p T ) (c) Principal flow vectors divided by the average multiplicity, v ( a )5 ( p T ) ≡ V ( a )5 ( p T ) / (cid:104) dN/dp T (cid:105) . | Q ( ) | centrality (%) v leadingbest lin. + ξ (1)2 ξ (1)3 ε , + ε , ε , (d) Pearson correlation coefficient between the leading flow(zero suppressed for clarity) and several predictors. | Q ( ) | centrality (%) v subleadingbest lin. + ξ (1)2 ξ (2)3 + ξ (2)2 ξ (1)3 + ξ (1)5 ξ (2)0 best lin. + ξ (1)2 ξ (2)3 + ξ (2)2 ξ (1)3 best lin. (e) Pearson correlation coefficient between the subleading flowand several predictors. FIG. 14. nn