New estimator for symmetry plane correlations in anisotropic flow analyses
NNew estimator for symmetry plane correlations in anisotropic flow analyses
Ante Bilandzic,
1, 2
Marcel Lesch, and Seyed Farid Taghavi Physik Department, Technische Universit¨at M¨unchen, Munich, Germany Excellence Cluster Universe, Technische Universit¨at M¨unchen, Munich, Germany (Dated: April 10, 2020)Correlations of symmetry planes are important observables used to quantify anisotropic flowphenomenon and constrain independently the properties of strongly interacting nuclear matter pro-duced in the collisions of heavy ions at the highest energies. In this paper, we point out currentproblems of measuring correlations between symmetry planes and elaborate on why the availableanalysis techniques have a large systematic bias. To overcome this problem, we introduce a newapproach to approximate multi-harmonic flow fluctuations via a two-dimensional Gaussian distri-bution. Employing this approximation, we introduce a new estimator, dubbed Gaussian Estimator(GE), to extract pure correlation between symmetry planes. We validate GE by using the realisticevent-generator iEBE-VISHNU and demonstrate that it outperforms all existing estimators. Basedon event-shape engineering, we propose an experimental procedure to improve GE accuracy evenfurther.
PACS numbers: 25.75.Ld, 25.75.Gz, 05.70.Fh a r X i v : . [ nu c l - e x ] A p r I. INTRODUCTION
The past years have witnessed the advent of large statistics heavy-ion datasets at RHIC and LHC facilities, compris-ing events with very large multiplicities. It is therefore becoming feasible to study the details of strongly interactingnuclear matter produced in heavy-ion collisions with unprecedented precision by employing multiparticle correlationtechniques. When two heavy ions collide at ultrarelativistic energies a very rich and non-trivial sequence of stagesemerges in the evolution of the produced fireball. Since each of these stages typically involves different underlyingphysics, ideally they would be described separately in theoretical models and probed one at a time in an experi-ment. To date, however, most of the analyzed heavy-ion observables are final-state observables in the momentumspace, which pick up cumulatively the contributions from all stages in the heavy-ion evolution, starting all the waydown from the details of the initial collision geometry. To leading order, these stages can be divided into the follow-ing categories: initial conditions, deconfined Quark-Gluon Plasma (QGP) stage, hadronization, chemical freeze-out,rescatterings, kinematic freeze-out, and finally free streaming. An important program in the field is the developmentof new observables which would be sensitive only to one particular stage in the heavy-ion evolution [1–3].For an idealized description of the heavy-ion collision geometry the initial volume containing interacting nucleons isellipsoidal in non-central collisions. In such a case, anisotropic flow develops the shapes in the final-state momentumdistribution which can be captured solely with the even Fourier amplitudes v n and only one symmetry plane Ψ RP (the reaction plane, spanned by the impact parameter vector and the beam axis) [4–6]. However, in a more realisticdescription of the collision geometry, the initial energy density profiles fluctuate both in magnitude and in shape fromone heavy-ion collision to another. Such initial-state fluctuations are also transferred into the final-state momentumfluctuations via anisotropic pressure gradients which develop in the fireball. Therefore, the full Fourier series expansionneeds to be employed to quantify the anisotropies in the azimuthal distribution of emitted particles in the planetransverse to the beam axis: f ( ϕ ) = 12 π (cid:34) ∞ (cid:88) n =1 v n cos[ n ( ϕ − Ψ n )] (cid:35) , (1)where v n ’s are anisotropic flow amplitudes, and Ψ n ’s the corresponding symmetry planes [7]. In the past, anisotropicflow studies have been focused mostly on the flow amplitudes v n . These results helped a great deal in establishingthe perfect fluid paradigm about QGP properties [8, 9].From the above Fourier series expansion it can be seen immediately that, due to event-by-event flow fluctuations, v n ’s and Ψ n ’s are independent and equally important degrees of freedom to quantify anisotropic flow phenomenon,and therefore both sets of observables need to be studied and measured. However, the nature of these observables isdifferent, which triggers the development of separate analyses techniques for their measurements. Further independentinformation about QGP and the other stages in the heavy-ion evolution can be extracted from the novel studiesof observables which are sensitive to the intercorrelations between different flow amplitudes or between differentsymmetry planes, or from observables which couple the intercorrelations between both degrees of freedom. Anotheropen question is the connection between v n ’s and Ψ n ’s defined in the final-state momentum distribution, and theircounterparts in the initial coordinate space where they quantify the anisotropies stemming from the fluctuations ofthe initial collision geometry. This is mostly studied by the theorists because it is not feasible to access the initialstages of collision in the experiment.Before starting to discuss the physics of v n and Ψ n observables, we first summarize the most important formalmathematical properties, which are used later in the derivation of our main results (additional details can be foundin Appendix A). Solely from the definition of Fourier series one can prove that v − n = v n and Ψ − n = Ψ n , therefore inthis paper we use them interchangeably. By combining the Fourier decomposition in Eq. (1) and the orthogonalityproperties of trigonometric functions, one can show that v n ’s and Ψ n ’s are related via the following mathematicalidentity: v n = (cid:104) cos[ n ( ϕ − Ψ n )] (cid:105) , (2)where the average goes over all azimuthal angles ϕ reconstructed in an event. Despite its simplicity, Eq. (2) has littlerelevance in experimental high-energy physics, due to difficulties in measuring reliably symmetry planes Ψ n in eachheavy-ion collision. Instead, flow amplitudes v n can be estimated even without knowing the symmetry planes Ψ n byutilizing two- and multiparticle azimuthal correlations [10–14]. Ollitrault et al have derived in [15] the most generalrelation between flow degrees of freedom v n and Ψ n and multiparticle azimuthal correlators, which is valid for anynumber of azimuthal angles ϕ , ϕ , . . . , ϕ k , and for any choice of harmonics n , n , . . . , n k : v a n · · · v a k n k e i ( a n Ψ n + ··· + a k n k Ψ nk ) = (cid:68) e i ( a n ϕ + ··· + a k n k ϕ k ) (cid:69) . (3)The average on the RHS in the above expression goes over all distinct tuples of k azimuthal angles in an event.When compared to the original result from [15], we have used a slightly different notation in the above expressionby introducing a i coefficients which are by definition positive integers. The precise meaning of a i coefficient is thefollowing: a i is the number of appearances of harmonic n i associated with different azimuthal angles in the azimuthalcorrelator on the RHS of Eq. (3) (positive and negative harmonics are counted separately). The advantage of this moregeneral notation is that harmonics n i in Eq. (3) are now all unique by definition. In addition, n i and a i naturally splitoff when associated with flow amplitudes on LHS of Eq. (3), which makes their physical interpretation straightforward.It is straightforward to choose harmonics n , n , . . . , n k in this general result in order to cancel the contribution fromsymmetry planes Ψ n and estimate solely flow amplitudes v n (e.g. the choice n = n, n = − n, n k = 0 for k > a = a = 1, yields the standard formula for 2-particle azimuthal correlation (cid:104) cos[ n ( ϕ − ϕ )] (cid:105) = v n ). However, it ismuch more of a challenge to derive an analogous expression that would express multiparticle azimuthal correlatorsonly in terms of symmetry planes, i.e. the expression from which the prefactors v a n · · · v a k n k in Eq. (3) would cancelout exactly. So far in the literature only approximate relations have been presented in this context, all of whichhave the inherent systematic biases straight from their definitions. Such relations are particularly unreliable whenused in the analyses of heavy-ion datasets characterized by large and correlated flow fluctuations, which is the casetypically encountered in practice. In this paper, and as our main result, we introduce a new set of observablesbased on multiparticle azimuthal correlators, which are more reliable estimators of symmetry planes than the onesused previously. Before the presentation of our new results, we clarify for completeness sake which categories ofobservables depending only on symmetry planes are meaningful to study in practice.The fundamental difference between v n and Ψ n flow degrees of freedom lies in the fact that only v n ’s are invariantwith respect to the arbitrary rotations of laboratory coordinate system in which azimuthal angles ϕ are measured.We also remark that due to periodicity the symmetry plane angle Ψ n is uniquely determined only in the range0 ≤ Ψ n < π/n [16]. On the other hand, starting from Eq. (1) and using the constraint 0 ≤ f ( ϕ ) ≤ | v n | < .
5. Therefore, in order to eliminate trivialperiodicity of each symmetry plane, and to ensure invariance of our observables with respect to random event-by-eventfluctuations of the impact parameter vector, we arrive at the conclusion that the fundamental non-trivial observablesinvolving symmetry planes are the following correlators and constraints [15, 17, 18]: (cid:68) e i ( a n Ψ n + ··· + a k n k Ψ nk ) (cid:69) , (cid:88) i a i n i = 0 . (4)The meaning of a i and n i is clarified in the text following Eq. (3). In the rest of the paper, we call observables inEq. (4) Symmetry Plane Correlations (SPC). They can be estimated precisely only in theoretical models in which itis possible to compute each symmetry plane Ψ n for each heavy-ion collision. The main purpose of this paper is toestablish a reliable experimental way to estimate SPC indirectly by using only the azimuthal angles of reconstructedparticles, since only they can be measured reliably in experiments.We conclude the introduction with a brief review of both experimental and theoretical results on SPC obtained sofar in anisotropic flow analyses. We do not consider the evaluation of resolution factors in the standard event planemethod, when symmetry planes corresponding to two or three subevents are used to correct for the effects of finitemultiplicities, since such symmetry planes correspond to the same harmonic, only estimated in different subevents [16].Also, we do not consider the correlations between symmetry planes and the reaction plane, since the latter cannot beestimated reliably in an experiment.The importance of SPC in anisotropic flow measurements has been fully acknowledged only in the LHC era, eventhough the first results were obtained already 20 years ago in E877 experiment [19]. The first results at RHICfor SPC involving two symmetry planes were published by PHENIX in [20, 21], by using the standard event planemethod with subevent technique [16]. Both NA49 and STAR have analyzed 3-particle azimuthal correlators in mixedharmonics, which by definition do have contributions from symmetry planes, but their contribution has been neglectedin these early analyses [22, 23]. The first SPC studies at LHC provided only the binary statements on whether certainsymmetry planes are correlated or not, without providing quantitative details—in [24] ALICE, by using the carefullydesigned 5-particle azimuthal correlator (the technical details can be found in Appendix H of [25]), has demonstratedthat the fluctuations of symmetry planes Ψ and Ψ are independent in all considered centralities. Finally, themost thorough experimental analysis to date, which included also the first measurements of correlations among threesymmetry planes, has been published by ATLAS in [17, 26, 27], by using the analysis technique discussed in [18, 28].Theoretical studies have investigated SPC separately in coordinate (typically by using the Monte Carlo Glaubermodel [29] in combination with event plane method) and momentum space [17, 18, 28, 30–35]. In these studies thevalues of symmetry planes are typically the direct output of the model in each heavy-ion collision, and therefore theydo not need to be estimated indirectly by utilizing the azimuthal angles of produced particles. A notable independentapproach to SPC in terms of conditional probabilities has been established in [36]. Other types of studies involvingsymmetry planes which we do not discuss in our paper have been performed in [37–41]. Finally, for the previousattempts to use azimuthal correlators to estimate SPC indirectly, we refer the reader to [15, 42–45].This paper is organized as follows. After introduction, in Section II we present the key idea behind the newobservables for SPC estimation, and point out the inherent systematic biases that plagued the previous approaches.Section III discusses the concrete realization of our new estimators, dubbed Gaussian Estimator (GE). In Section IVwe present the comparison with the theoretical models, both for new and old SPC estimators, and indicate in whichregime our estimators outperform the existing ones. Finally, in Section V we summarize our results and outline thenext steps. The technical details are provided in Appendices. II. KEY IDEA OF THE NEW ESTIMATOR FOR SYMMETRY PLANE CORRELATIONS
In this section we summarize the systematic biases of previous analyses that used azimuthal correlators to estimateSPC and introduce our new approach which improves on those biases. SPC were estimated previously by using thescalar product (SP) method [5, 46] or event plane method [16], both of which yield the theoretical results for SPConly in the absence of correlated fluctuations of different flow magnitudes. Our new method, which we illustrate inthe next paragraphs and elaborate in detail in Section IV, provides a further step forward in a sense that it yields thetheoretical result for SPC also when such correlated fluctuations of flow magnitudes are present in the data. In fact,at RHIC and LHC energies correlations of event-by-event fluctuations of v and v , and of v and v , are large and ifthey are not taken into account and corrected for, the final results for SPC can exhibit large systematic biases, as wedemonstrate in Monte Carlo studies presented in Section IV.As already indicated in the introduction, correlations between k symmetry planes in unique harmonics n , ..., n k (i.e.correlations between Ψ n , ..., Ψ n k ) can be investigated by the measurement of the correlator (cid:104) cos ( a n Ψ n + · · · + a k n k Ψ n k ) (cid:105) ,where the coefficients a i have to be fixed in such a way, that this expression is invariant in respect to the randomnessof the reaction-plane. In theory, such correlators can be built from an event-by-event ratio of two multiparticleazimuthal correlators. As a concrete example, by using the analytic formula in Eq. (3), one can derive the followingresult: (cid:104) cos(2 ϕ +2 ϕ − ϕ − ϕ − ϕ − ϕ ) (cid:105)(cid:104) cos(2 ϕ − ϕ + ϕ − ϕ + ϕ − ϕ ) (cid:105) = v v cos 4(Ψ − Ψ ) v v = cos 4(Ψ − Ψ ) . (5)This idea, which works only for correlators involving 6 or more azimuthal angles, demonstrates that for general ratiosof this kind, the numerator consists out of both, flow amplitudes and symmetry planes, while the denominator onlyout of the respective flow amplitudes but without any contribution of symmetry planes. The correlators in numeratorand denominator were carefully chosen so that the final expression depends only on the symmetry planes, even in thecase of large correlated fluctuations of flow amplitudes v and v .We now generalize this starting example, and write in the most general case: (cid:104) cos ( a n Ψ n + · · · + a k n k Ψ n k ) (cid:105) EbE = (cid:28) v a n · · · v a k n k cos ( a n Ψ n + · · · + a k n k Ψ n k ) + δv a n · · · v a k n k + δ (cid:48) (cid:29) . (6)As such, this event-by-event ratio exhibits only the symmetry planes. This remains true by definition also if theevent-by-event fluctuations of flow amplitudes are correlated, and it is precisely this point which is not satisfied forthe currently used estimators. Since both the numerator and denominator in Eq. (6) have to be estimated withdifferent k -particle azimuthal correlators ( k ≥ δ and δ (cid:48) , respectively. Such a direct event-by-event approach is at the moment experimentally not feasible due to largestatistical uncertainties which prevent such a per-event ratio.We overcome the limitation of event-by-event estimator in Eq. (6) by introducing a new approximate method toestimate the SPC, which we refer to as the “Gaussian Estimator” (GE), in the next section. By using the samenotation we now clarify the currently existing approximation methods, hereby focusing on the SP method. Theexplicit form of SP estimation is given by [45] (cid:104) cos ( a n Ψ n + · · · + a k n k Ψ n k ) (cid:105) SP = (cid:104) v a n · · · v a k n k cos ( a n Ψ n + · · · + a k n k Ψ n k ) (cid:105) (cid:113) (cid:104) v a n (cid:105) · · · (cid:104) v a k n k (cid:105) . (7)The powers a i are chosen in such a way such that the numerator and the denominator are valid multiparticle cor-relators. We see later, that within the GE approximation the same kind of powers a i appear, and we provide a setof constraints which a i must satisfy in Appendix B. We will see further that in most cases, the SP method is notan accurate estimator of the true SPC, since the denominator in Eq. (7) cannot be written in the factorized form.This statement is supported by experimental evidence of large correlations between the flow amplitudes [47–50], whichtherefore leads in general to the non-negligible bias in the SP method. On the other hand, estimators for SPC from theevent plane method are plagued by finite resolutions in estimating each symmetry plane directly event-by-event [17]. III. GAUSSIAN ESTIMATOR
To begin with the GE approximation, we introduce the following quantities: R = v a n · · · v a k n k , Θ = a n Ψ n + · · · + a k n k Ψ n k . (8)Using the above definitions, one can further define R x and R y in the Cartesian coordinate system as R x = R cos Θ , R y = R sin Θ . (9)Due to the event-by-event flow fluctuations, the quantities R x and R y fluctuate from one event to the other. We canstudy the moments (or equivalently cumulants) of these fluctuations which are, in fact, the moments of probabilitydensity function (p.d.f.) P ( R x , R y ) or equivalently P ( R , Θ). One notes that for positive integers p and positive even q , the moment (cid:104)R px R qy (cid:105) = (cid:104)R p + q cos p Θ sin q Θ (cid:105) is non-vanishing where the angular bracket indicates the average overevents. The moments with odd q are zero because of the presence of sin term with odd power. A simple example ofsuch quantities is the case k = 2, n = − n = n , and a = a = (cid:96) leading to R = v (cid:96)n and Θ = 0. In such a case,the moments (cid:104) v (cid:96)n (cid:105) have been extensively studied over the past years. In general, the non-vanishing (cid:104)R px R qy (cid:105) can beexpanded in the basis spanned by the moments (cid:104) m (cid:105) ( a ,n ) ,..., ( a k ,n k ) = v a n · · · v a k n k e i ( a n Ψ n + ··· + a k n k Ψ nk ) , m = (cid:88) a i , (10)and estimated experimentally by employing multiparticle correlation techniques [11–14] (the above notation wasintroduced in the paragraph below Eq. (3)). In the present study, we specially focus on the following moments, (cid:104)R x (cid:105) = Re (cid:104)(cid:104) m (cid:105) ( a ,n ) ,..., ( a k ,n k ) (cid:105) , (cid:104)R x (cid:105) + (cid:104)R y (cid:105) = (cid:12)(cid:12) (cid:104)(cid:104) m (cid:105) (2 a ,n ) ,..., (2 a k ,n k ) (cid:105) (cid:12)(cid:12) . (11)We now develop the procedure to estimate the SPC, which in the current notation amounts to estimating (cid:104) cos Θ (cid:105) .If we had been able to measure P ( R x , R y ), we could have computed the moment (cid:104) cos Θ (cid:105) immediately. Although themoments (cid:104)R px R qy (cid:105) are accessible by multiparticle correlation techniques, the convergence of the moment expansionto the true value of p.d.f. is not guaranteed [51]. Due to the central limit theorem, however, we are still able toapproximately estimate the distribution as a 2D normal distribution, N ( R x , R y ) = 12 πσ x σ y exp (cid:34) − ( R x − µ x ) σ x − R y σ y (cid:35) , (12)where µ x = (cid:104)R x (cid:105) , σ x = (cid:104)R x (cid:105) − (cid:104)R x (cid:105) , and σ y = (cid:104)R y (cid:105) . Since we are only interested in angular part of the normaldistribution in Eq. (12), we integrate out the radial part. After some algebra, we find, N θ (Θ) = (cid:90) R d R N ( R , Θ) = σ x σ y e − µ x / σ x πσ θ (cid:34) √ πµ x σ y e µ θ σ θ [1 + erf ( µ θ )] (cid:35) , (13)where σ θ (Θ) = σ x (cid:113) σ y cos Θ + 2 σ x sin Θ , µ θ (Θ) = µ x σ y cos Θ σ Θ . (14)As a result, one can straightforwardly compute the average (cid:104) cos Θ (cid:105) by computing the following integral, (cid:104) cos Θ (cid:105) GE = (cid:90) d Θ N Θ (Θ) cos Θ , (15)which is the Gaussian Estimator for the true value of (cid:104) cos Θ (cid:105) . To find an analytical result for our estimator, we stillneed some simplifications as we discuss in what follows.The quantity σ θ has no Θ dependence by considering σ x ∼ σ y ∼ σ r / √ σ r = (cid:113) σ x + σ y . This leads to ananalytical result for integral in Eq. (15) written in terms of two first modified Bessel functions. By expanding theresult in term of µ x /σ r and keeping only the leading term, we obtain (cid:104) cos Θ (cid:105) GE (cid:39) (cid:114) π (cid:18) µ x σ r (cid:19) . (16)Using Eq. (11), the above approximation can be written explicitly as follows, (cid:104) cos ( a n Ψ n + · · · + a k n k Ψ n k ) (cid:105) GE (cid:39) (cid:114) π (cid:104) v a n · · · v a k n k cos ( a n Ψ n + · · · + a k n k Ψ n k ) (cid:105) (cid:113) (cid:104) v a n · · · v a k n k (cid:105) , (17)where for the denominator we have used the fact that σ r = (cid:113) (cid:104)R x (cid:105) − µ x + (cid:104)R y (cid:105) (cid:39) (cid:113) (cid:104)R x (cid:105) + (cid:104)R y (cid:105) . The error we havemade in the second equality is of the order of ( µ x /σ r ) . After comparing the above formula with Eq. (7), one findsthat apart from a numerical factor (cid:112) π/ (cid:39) . IV. VALIDATION OF GAUSSIAN ESTIMATOR AND ITS FURTHER IMPROVEMENTS
A new estimator for SPC has been introduced in the previous section by assuming that the ( R x , R y ) fluctuationis approximately described by a 2D normal distribution. The accuracy and applicability of the method depend onthis assumption. To examine the estimator’s accuracy and in order to study possible ways for its improvements, weemploy the realistic Monte Carlo event generator iEBE-VISHNU [52]. We initiate the events at τ = 0 . c byMC-Glauber model [29] implemented in the iEBE-VISHNU. For the hydrodynamic evolution DNMR [53, 54] causalhydrodynamic is solved at fixed shear viscosity over entropy density η/s = 0 .
08 and the Cooper-Frye freeze-out [55]prescription has been implemented in the package for particleization stage. The evolution in hadronic stage is notconsidered in our simulation. For each centrality bin, 14k events of Pb-Pb collisions ( √ s NN = 2 .
76 TeV) have beengenerated and flow magnitudes v n and symmetry planes Ψ n are computed in each event for π ± , K ± and p/ ¯ p in thefinal state. The SPC obtained from these directly computed event-by-event symmetry planes are referred to as truevalue of SPC in the comparisons which we present next.Our first study in Fig. 1 shows eight different choices for the correlation of two symmetry planes, and it demonstratesthat the true value of SPC can be approximated much better with the GE approach, than with the SP estimator.Especially in cases, where the two symmetry planes are strongly correlated (Fig. 1 (a)-(d)) due to their geometriccorrelations that pre-existed in the initial state (e.g. between Ψ and Ψ ), our new method reproduces the true valuevery well in all centrality classes of interest. This demonstrates clearly that the systematic bias caused by neglectingcorrelations between the flow amplitudes in the SP method is large, and therefore cannot be neglected. Only in afew cases it can be observed that the GE and SP yield comparable results (e.g. for SPC between Ψ and Ψ ). Wewill elaborate on this in more detail later and present a way to improve the GE method even further. The centralitydependence of each SPC in Fig. 1 presents strikingly different features, and therefore provides independent constraintsfor the system properties. Further, we present results for correlations between three symmetry planes (Fig. 2) as wellas between four symmetry planes (Fig. 3). It can be observed clearly that for each SPC the GE approach outperformsthe SP estimator in most of considered centralities, while on the remaining few centralities the accuracy of the methodsis comparable.Although the Gaussian Estimator in Eq. (17) works accurately for almost all cases, in contrast to the SP methodwhich in most cases exhibits large systematic biases, there are still minor discrepancies between our estimator and thetrue value for few cases (see e.g. Θ = 2Ψ + 3Ψ − in Fig. 2 (d)). To investigate the reason more deeply, we focuson an extreme example: R = v v v , Θ = 2Ψ + 4Ψ − at 40% centrality (see Fig. 2 (c)). In this case, there isa clear discrepancy between the true value and the Gaussian approximation Eq. (17). In Fig. 4, the iEBE-VISHNUoutcome for ( R x , R y ) fluctuations is shown. As it can be seen from the figure, there is a sharp peak at the center anda few events distributed around it. The tail is elongated in x -direction. Although there are much fewer events in thetail, it leads to inaccuracy in our Gaussian estimation. Specifically, the events are mostly concentrated symmetricallyaround the center while the long tail in the x -direction leads to a large difference between σ x and σ y . Also, it shiftsthe µ x to the right. The GE would work better if we could fit the Gaussian distribution around the peak and removethe outliers. Centrality Percentile - - æ )] Y - Y c o s [ ( Æ (e) Centrality Percentile - - æ )] Y - Y c o s [ ( Æ (f) Centrality Percentile - - - - - æ )] Y - Y c o s [ ( Æ (g) Centrality Percentile - - - - - æ )] Y - Y c o s [ ( Æ (h) Centrality Percentile æ )] Y - Y c o s [ ( Æ SPGEGE CorrectedTrue Value (a)
Centrality Percentile æ )] Y - Y c o s [ ( Æ (b) Centrality Percentile - - æ )] Y - Y c o s [ ( Æ (c) Centrality Percentile - - æ )] Y - Y c o s [ ( Æ (d) FIG. 1. Comparison of GE and SP method to the true value of SPC between two symmetry planes in iEBE-VISHNU.
Since we have access to the value of flow magnitudes and symmetry planes in each event in the simulation, it is nota challenging task to remove the outliers. One can locate the peak in the histogram and fit a Gaussian distributionaround it by ignoring events away from the peak with a certain criteria. Here, however, we try to introduce criteriathat are model-independent and applicable also in experiments. We first compute σ x and σ y from all events. Afterthat we divide the events into two classes: low R class with condition R ≤ ασ r and the rest as high R class. Wehave found that by ignoring the events at the tail of the distribution starting from twice the width σ r ( α = 2), theGE is corrected very well as we will see shortly. After event classification, we compute ˆ µ x and ˆ σ r at low R class, andestimate (cid:104) cos Θ (cid:105) by using Eq. (16). For the specific case shown in Fig. 4 (a), the ratio σ x /σ y computed from all eventsin the given centrality class is around 3 while if we compute the same ratio by using events in the low R class thisratio reduces to 1 .
7. The corrected histogram with new ˆ µ x , ˆ σ x , ˆ σ y , and ˆ σ r is depicted in Fig. 4 (b). The “corrected”Gaussian estimation (cid:104) cos Θ (cid:105) are shown in Figs. 1 - 3 with open diamond markers indicating an improvement in mostcases. This classification is simple in the simulation while experimentally one needs to employ more sophisticatedtechniques such as event-shape engineering [56]. It is worth mentioning that the low R class contains 94% of all eventsin our simulation. This means by removing 6% of high R events the ratio σ x /σ y is reduced approximately by a factorof two. Experimentally, one is able to classify events into low and high R classes with ∼
6% of events at high R class.The classification percentile can be optimized by comparing the low R class ratio σ x /σ y with that obtained from allevents in the given centrality class. Centrality Percentile - æ ] Y - Y - Y c o s [ Æ (a) Centrality Percentile - æ ] Y + Y - Y c o s [ Æ SPGEGE CorrectedTrue Value (b)
Centrality Percentile - æ ] Y - Y + Y c o s [ Æ (c) Centrality Percentile - æ ] Y - Y + Y c o s [ Æ (d) Centrality Percentile - - - - æ ] Y + Y - Y c o s [ Æ (e) Centrality Percentile - - - - æ ] Y - Y + Y c o s [ Æ (f) Centrality Percentile - - - - æ ] Y - Y - Y c o s [ Æ (g) Centrality Percentile - - - - æ ] Y - Y - Y c o s [ Æ (h) FIG. 2. Comparison of GE and SP method to the true value of SPC between three symmetry planes iEBE-VISHNU.
V. CONCLUSIONS
After introducing the new procedure to correct for the correlated flow fluctuations of different flow magnitudes,we have reduced significantly the systematic biases in the existing experimental techniques for symmetry plane cor-relations. This correction emerged from the modeling of experimentally accessible moments with a 2D Gaussiandistribution. By using this new method, dubbed Gaussian Estimator, we have shown a significant improvement overexisting SPC measurements in most cases of interest. We have demonstrated that in combination with event shapeengineering, this new estimator can be optimized even further.The precision measurements of SPC in the future have to acknowledge the remaining small intercorrelation betweenflow amplitudes and symmetry planes, which can still cause a small bias in all available approximation methods forSPC measurements.
ACKNOWLEDGMENTS
This project has received funding from the European Research Council (ERC) under the European Unions Horizon2020 research and innovation programme (grant agreement No 759257).
Centrality Percentile - æ ] Y + Y - Y - Y c o s [ Æ SPGEGE CorrectedTrue Value (a)
Centrality Percentile - æ ] Y - Y - Y + Y c o s [ Æ (b) Centrality Percentile - - - æ ] Y + Y + Y - Y c o s [ Æ (c) Centrality Percentile - - - æ ] Y - Y - Y + Y c o s [ Æ (d) FIG. 3. Comparison of GE and SP method to the true value of SPC between four symmetry planes iEBE-VISHNU. - - - - · ] Y -6 Y +4 Y cos[2 v v v - - - - · ] Y - Y + Y s i n [ v v v (a) - - - - · ] Y -6 Y +4 Y cos[2 v v v
40% Centrality2 / r s ~ y s ~ x s y s , x s Mean(b)
FIG. 4. Distributions of v v v cos [2Ψ + 4Ψ − ] and v v v cos [2Ψ + 4Ψ − ] before (left) and after (right) correctionby rejecting events bigger than R ≤ ασ r ( α = 2). Appendix A: Basic properties of symmetry planes
In this Appendix we outline in more detail the most important formal properties of symmetry planes. Besides theversion of the Fourier series presented in the introduction in Eq. (1), the alternatively used form is: f ( ϕ ) = 12 π (cid:2) ∞ (cid:88) n =1 ( c n cos nϕ + s n sin nϕ ) (cid:3) , (A1)with c n = (cid:90) π f ( ϕ ) cos( nϕ ) dϕ , (A2) s n = (cid:90) π f ( ϕ ) sin( nϕ ) dϕ , (A3)0The Fourier series parametrizations in Eqs. (1) and (A1) are mathematically equivalent and can be interchanged byusing the following relations: v n ≡ (cid:112) c n + s n , (A4)Ψ n ≡ (1 /n ) arctan s n c n . (A5)The relation (A5) can be used as a definition of symmetry plane Ψ n . We discuss next some physical properties ofsymmetry planes, and establish the connection between them and some commonly used observables in anisotropicflow analyses.The symmetry plane Ψ n has an obvious geometrical interpretation when the anisotropic distribution can be pa-rameterized only with one harmonic n , since then one can show immediately that f (Ψ n + ϕ ) = f (Ψ n − ϕ ) , (A6)i.e. a symmetry plane Ψ n is the plane for which it is equally probable for a particle to be emitted above and belowit. From Eq. (A5) one can see that symmetry planes are meaningful only when c n (cid:54) = 0. If the flow amplitude v n is 0, or if the Fourier series permits only the sin term s n , the corresponding symmetry plane Ψ n does not exist.Similarly, odd symmetry planes Ψ n +1 do not exist in a system in which the exact symmetry is f ( ϕ ) = f ( ϕ + π ),since that symmetry sets all odd c n +1 harmonics to zero. Another symmetry of interest is f ( ϕ ) = f ( − ϕ ) due towhich s n = 0 for all n , and therefore from Eq. (A5) Ψ n = 0 ∀ n , i.e. all symmetry planes are the same and equal to0. Physically, this means that a heavy-ion collision was described in the laboratory frame with the coordinate systemoriented such that the impact parameter vector is aligned with the x -axis. The next symmetry which is to leadingorder satisfied in non-central heavy-ion collisions is f ( ϕ ) = f ( π + ϕ ), due to which c n +1 , s n +1 = 0 and therefore onlythe even symmetry planes Ψ n are well-defined and non-trivial. In principle, one could also consider the symmetry f ( ϕ ) = f ( π − ϕ ) in mid-central collisions, but we were not able to extract any new constraint on the symmetryplanes, which was not already covered by the other symmetries. Finally, since we assign to f ( ϕ ) the probabilisticinterpretation (which implies that f ( ϕ ) must be a positive definite function), we do not consider symmetries like f ( ϕ ) = − f ( − ϕ ), which otherwise could lead to additional constraints.Another important physical interpretation of symmetry planes can be drawn from their relation with the Q -vector [4, 7, 57], which is one of the most important objects in flow analyses. For a set of M azimuthal angles ϕ i , the Q -vector in harmonic n is defined as: Q n ≡ M (cid:88) j = i e inϕ j ≡ | Q n | e in Ψ n . (A7)With such a definition, one can easily demonstrate that the angle of the Q -vector is exactly the same as symmetryplanes Ψ n from Fourier series defined before in Eq. (A5), since:(1 /n ) arctan s n c n = (1 /n ) arctan (cid:104) sin nϕ (cid:105)(cid:104) cos nϕ (cid:105) = (1 /n ) arctan M Im( Q n ) M Re( Q n )= (1 /n ) arctan | Q n | sin n Ψ n | Q n | cos n Ψ n = (1 /n ) arctan tan n Ψ n = (1 /n ) n Ψ n = Ψ n . (A8)This relation is utilized in the standard event plane method, where symmetry planes Ψ n are estimated directly from Q -vectors in each event [16]. Appendix B: Choice of Correlators
In this Section we will start from the most general form of multiparticle correlators with non-unique harmonicsfrom which on we will find constraints such that these correlators are applicable for our GE method (Eq. (17)). We1will see that from there on, constraints for the a i will emerge naturally.Consider two general multi-particle correlators (cid:104) k (cid:105) n ,n ,...,n k ( k -particle correlator with set of non-unique harmonics { n , n , ... n k } ) and (cid:104) l (cid:105) p ,p ,...,p l ( l -particle correlator with set of non-unique harmonics { p , p , ... , p l } ). Focusingon the general form of the GE approximation (Eq. (17)), their ratio can in general be written as (cid:68) (cid:104) k (cid:105) n ,n , ··· ,n k (cid:69)(cid:68) (cid:104) l (cid:105) p ,p , ··· ,p l (cid:69) ∝ (cid:10) v n · · · v n k e i ( n Ψ n + ··· + n k Ψ nk ) (cid:11)(cid:113)(cid:10) v p · · · v p l e i ( p Ψ p + ··· + p l Ψ pl ) (cid:11) . (B1)From this general ansatz the following constraints to achieve the desired SPC emerge k (cid:88) j =1 n j = 0 (B2) l (cid:88) j =1 p j = 0 (B3) k (cid:88) j =1 n j · Ψ n j (cid:54) = 0 (B4) l (cid:88) j =1 p j · Ψ p j = 0 (B5) k (cid:89) i =1 v n i = l (cid:89) i =1 v p i . (B6)Constraints (B2) and (B3) satisfy the isotropy condition which has to hold true for any non-trivial multi-particlecorrelator. Constraints (B4) and (B5) lead to a non-vanishing contribution of symmetry planes in the numerator whilethe denominator does not depend on symmetry planes explicitly. Constraint (B6) ensures that the flow amplitudesin numerator and denominator cancel each other exactly. Further, from Constrain (B6) it follows that l = 2 k .Therefore, while measuring a k -particle correlator in the numerator one has to measure a 2 k -particle correlator in thedenominator, when using GE approximation. To obtain the SPC one has to explicitly choose sets of correlators { n , n , ... n k } and { p , p , ... , p l } which satisfy constraints Eqs. (B2) - (B6). We elaborate on this now explicitly for theSPC between two symmetry planes Ψ m and Ψ n , and demonstrate how the coefficients a i used in the main part, seee.g. Eq. (3), emerge naturally and which constraints a i have to fulfil themselves. This formalism can be generalizedfor correlations between any amount of symmetry planes.
1. Correlators between two symmetry planes
Focussing now on the SPC between two symmetry planes Ψ m and Ψ n and given the constraints Eq. (B4) to Eq. (B6),the general sets of correlators in harmonics m and n (where m (cid:54) = n ) are schematically m (cid:124)(cid:123)(cid:122)(cid:125) a m times , · · · , m, − n (cid:124)(cid:123)(cid:122)(cid:125) a n times , · · · , − n (numerator) (B7) m, − m (cid:124) (cid:123)(cid:122) (cid:125) a m times , · · · , m, − m, n, − n, (cid:124) (cid:123)(cid:122) (cid:125) a n times , · · · , n, − n (denominator) (B8)where a m , a n ∈ N . Given the constraints Eq. (B2) and Eq. (B3) the following constraints for a m and a n are valid a m (cid:88) j =1 m + a n (cid:88) k =1 ( − n ) = a m m − a n n = 0 = ⇒ a m n = a n m , (B9) a m (cid:88) j =1 ( − j · m + a n (cid:88) k =1 ( − k · n = 0 = ⇒ a m ∧ a n even (B10)2where ∧ is the logical AND. This way, the constraints from Eq. (B4) and Eq. (B5) are satisfied as well. We see thatConstraint (B10) will hold true for any a m , a n . Therefore, as the concrete example one can choose a m and a n as a m = l mn m (B11) a n = l mn n (B12)where l mn denotes the least common multiple between m and n . The order of particle correlator in the numerator isgiven as l mn (cid:18) m + 1 n (cid:19) (B13)and for the numerator twice the size respectively. This method of using the least common multiple presents the lowestorder of valid multiparticle correlators for the SPC between two symmetry planes. Any other method exhibits higherorder of correlators. Given by this, the GE approach reads (cid:104) cos [ l mn (Ψ m − Ψ n )] (cid:105) GE ∝ (cid:104) v a m m v a n n cos [ l mn (Ψ m − Ψ n )] (cid:105) (cid:113)(cid:10) v a m m v a n n (cid:11) . (B14)Although the method of the least common multiple exhibits the lowest possible order for a SPC with two planes,any multiple k ∈ N of this method represents a valid correlator as well. We can therefore always expand the set ofcorrelators by changing a m → ka m and a n → ka n and there find in general (cid:104) cos [ kl mn (Ψ m − Ψ n )] (cid:105) GE ∝ (cid:10) v ka m m v ka n n cos [ kl mn (Ψ m − Ψ n )] (cid:11)(cid:114)(cid:68) v ka m m v ka n n (cid:69) . (B15)
2. Correlators between three symmetry planes
A general choice for the set of correlators for three unique harmonics m , n and p are schematically m (cid:124)(cid:123)(cid:122)(cid:125) a m times , · · · , m, − n (cid:124)(cid:123)(cid:122)(cid:125) a n times , · · · , − n, − p (cid:124)(cid:123)(cid:122)(cid:125) a p times , · · · , − p (numerator) (B16) m, − m (cid:124) (cid:123)(cid:122) (cid:125) a m times , · · · , m, − m, n, − n, (cid:124) (cid:123)(cid:122) (cid:125) a n times , · · · , n, − n, p, − p, (cid:124) (cid:123)(cid:122) (cid:125) a p times , · · · , p, − p (denominator) (B17)Following the general constraints presented above we find the following constraints on a m , a n and a pa m (cid:88) j =1 m + a n (cid:88) k =1 ( − n ) + a p (cid:88) l =1 ( − p ) = a m m − a n n − a p p = 0 (B18) a m (cid:88) j =1 ( − j · m + a n (cid:88) k =1 ( − k · n + a p (cid:88) l =1 ( − l · p = 0 = ⇒ a m ∧ a n ∧ a p even (B19)Again the latter constraint is fulfilled trivially. In general these kind of correlators will be of high order, thereforelimiting experimental feasibility. We cannot reduce the problem of a 3-SPC into one single closed formula as ithas been the case for two planes, as now more combinatorial possibilities exist. As a trivial example, in cases that m = n + p we can set trivially a m = a n = a p = 1 (cid:104) cos [ m Ψ m − n Ψ n − p Ψ p ] (cid:105) GE ∝ (cid:104) v m v n v p cos [ m Ψ m − n Ψ n − p Ψ p ] (cid:105) (cid:113)(cid:10) v m v n v p (cid:11) . (B20)3 [1] C. Gale, S. Jeon and B. Schenke, Int. J. Mod. Phys. A (2013) 1340011 doi:10.1142/S0217751X13400113 [arXiv:1301.5893[nucl-th]].[2] U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci. (2013) 123 doi:10.1146/annurev-nucl-102212-170540[arXiv:1301.2826 [nucl-th]].[3] P. Braun-Munzinger, V. Koch, T. Schaefer and J. Stachel, arXiv:1510.00442 [nucl-th].[4] J. -Y. Ollitrault, Phys. Rev. D (1992) 229.[5] S. A. Voloshin, A. M. Poskanzer and R. Snellings, arXiv:0809.2949 [nucl-ex].[6] P. F. Kolb, Phys. Rev. C (2003) 031902 doi:10.1103/PhysRevC.68.031902 [nucl-th/0306081].[7] S. Voloshin and Y. Zhang, Z. Phys. C (1996) 665 [arXiv:hep-ph/9407282].[8] U. W. Heinz, nucl-th/0512051.[9] Y. Akiba et al. , arXiv:1502.02730 [nucl-ex].[10] S. Wang, Y. Z. Jiang, Y. M. Liu, D. Keane, D. Beavis, S. Y. Chu, S. Y. Fung and M. Vient et al. , Phys. Rev. C (1991)1091.[11] J. Jiang, D. Beavis, S. Y. Chu, G. I. Fai, S. Y. Fung, Y. Z. Jiang, D. Keane and Q. J. Liu et al. , Phys. Rev. Lett. (1992)2739.[12] N. Borghini, P. M. Dinh, J. -Y. Ollitrault, Phys. Rev. C64 (2001) 054901.[12] N. Borghini, P. M. Dinh, J. -Y. Ollitrault, Phys. Rev. C64 (2001) 054901 (cumulants in flow analyses, but calculated withapproximate formalism of generatic functions)[12] https://arxiv.org/abs/1312.3572 (currently the state-of-the-art of cumulant formalism in flow analyses)[13] A. Bilandzic, R. Snellings, S. Voloshin, Phys. Rev.
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