Multivariate cumulants in flow analyses: The Next Generation
Ante Bilandzic, Marcel Lesch, Cindy Mordasini, Seyed Farid Taghavi
MMultivariate cumulants in flow analyses: The Next Generation
Ante Bilandzic, Marcel Lesch, Cindy Mordasini, and Seyed Farid Taghavi Physik Department, Technische Universit¨at M¨unchen, Munich, Germany (Dated: February 17, 2021)We reconcile for the first time the strict mathematical formalism of multivariate cumulants withthe usage of cumulants in anisotropic flow analyses in high-energy nuclear collisions. This recon-ciliation yields to the next generation of observables to be used in flow analyses. We review allfundamental properties of multivariate cumulants and use them as a foundation to establish a newand pragmatic two-steps recipe to determine whether some multivariate observable is a multivariatecumulant in the basis they are expressed in. We argue that properties of cumulants are preservedonly for the stochastic observables on which the cumulant expansion has been performed directly,and if there are no underlying symmetries due to which some terms in the cumulant expansion areidentically zero. We illustrate one possibility of how new multivariate cumulants of azimuthal anglescan be defined which do satisfy all fundamental properties of multivariate cumulants, by definingthem event-by-event and by keeping all non-isotropic terms in the cumulant expansion. We intro-duce new cumulants of flow amplitudes named Asymmetric Cumulants, which generalize recentlyintroduced Symmetric Cumulants for the case when flow amplitudes are raised to different powers.Finally, we present the new concept of Cumulants of Symmetry Plane Correlations and provide thefirst realisation for the lowest orders. All the presented results are supported by Monte Carlo studiesusing state-of-the-art models.
PACS numbers: 25.75.Ld, 25.75.Gz, 05.70.Fh a r X i v : . [ phy s i c s . d a t a - a n ] F e b I. INTRODUCTION
In heavy-ion collisions at ultra-relativistic energies, the extreme state of matter in which quarks are deconfined,dubbed quark-gluon plasma (QGP), can be produced. Studies of its properties provide a very rich area of research,both for theorists and experimentalists. Out of different physics phenomena used to probe QGP properties, we focuson collective anisotropic flow [1]. In heavy-ion collisions QGP undergoes collective expansion, details of which aredetermined both by anisotropies in the initial state collision geometry and by the transport properties of QGP (forthe recent reviews we refer to Refs. [2–4]). After subsequent hadronization, as an overall net effect anisotropic particleemission in the plane transverse to the beam direction is recorded in a detector. Such anisotropic distributions inazimuthal angle ϕ are traditionally quantified with the flow amplitudes v n and symmetry planes Ψ n in the Fourierseries [5] f ( ϕ ) = 12 π (cid:34) ∞ (cid:88) n =1 v n cos[ n ( ϕ − Ψ n )] (cid:35) . (1)Anisotropic flow analyses amount to the measurements of flow amplitudes v n and symmetry planes Ψ n . However,since neither v n nor Ψ n degrees of freedom can be measured directly in each heavy-ion collision, they are estimatedindirectly by using the correlation techniques [6, 7]. The cornerstone of this alternative approach is the followingresult, (cid:68) e i ( n ϕ + ··· + n k ϕ k ) (cid:69) = v n · · · v n k e i ( n Ψ n + ··· + n k Ψ nk ) , (2)which analytically relates multiparticle azimuthal correlators and flow degrees of freedom [8]. As of recently, allmultiparticle azimuthal correlators can be evaluated exactly and efficiently for any number of particles k and anychoice of harmonics n , . . . , n k with the Generic Framework published in Ref. [9].Despite its elegance, Eq. (2) holds only if correlations among produced particles are dominated by contributionsfrom the collective anisotropic flow. In reality, however, other sources of correlations are present which typicallyinvolve only a subset of all particles. All such few-particle correlations are typically named nonflow. To distinguishbetween these two possibilities, we can use the following result: When collective anisotropic flow is present and othersources of correlations absent, the joint multivariate probability density function (p.d.f.) of M particles ( M is amultiplicity of an event) fully factorizes into the product of M single-variate marginal p.d.f.’s: f ( ϕ , . . . , ϕ n ) = f ϕ ( ϕ ) · · · f ϕ n ( ϕ n ) . (3)All single-variate p.d.f.’s f ϕ i ( ϕ i ) have the same functional form [10] which is given by Eq. (1). The factorizationproperty in Eq. (3) was a key ingredient in deriving analytic result in Eq. (2). Nonflow correlations break down theequality in Eq. (3), and as a direct consequence Eq. (2) cannot be used reliably to estimate flow degrees of freedom v n and Ψ n with multiparticle azimuthal correlators. To circumvent this problem, in a series of technical papersmultivariate cumulants of azimuthal angles, which are less sensitive to systematic biases originating from nonflowcorrelations, have been introduced [11–13]. This approach yielded the new flow-specific observables, v n { k } , in termsof which most experimental results and theoretical predictions have been reported in flow analyses in the past twodecades.The traditional usage of multiparticle cumulants in flow analyses hinges on the following idea: Cumulant expansionis performed on azimuthal angles ϕ i and then the resulting multiparticle azimuthal correlators in the final result areall individually re-expressed in terms of flow amplitudes v n and symmetry planes Ψ n by using the analytic result inEq. (2). We refer to this approach as old paradigm . Recently, however, it was demonstrated in Ref. [14] that cumulantexpansion cannot be performed in one set of stochastic observables and then the final result of that expansiontransformed to some new set of observables—after such transformation, all fundamental properties of cumulants arelost in general. Based on this observation the new paradigm has emerged, according to which cumulant expansionhas to be performed directly on the stochastic observabes which are being studied [14]. For instance, in the studies ofcorrelated fluctuations of different flow amplitudes, by following the new paradigm cumulant expansion is performeddirectly on the flow amplitudes squared, while azimuthal angles are used merely to build estimators for all resultingterms in such expansion by using Eq. (2). In general, old and new paradigm yield different results. In the series ofcarefully designed Monte Carlo studies, it was demonstrated in Ref. [14] that only the new paradigm can be usedreliably to obtain multivariate cumulants of flow amplitudes.In this paper, we explore further these new ideas and generalize them for other observables of interest in flowanalyses. For the first time, we reconcile the strict mathematical formalism of multivariate cumulants with the usageof cumulants in flow analyses. This reconciliation alters dramatically the usage and interpretation of cumulants inthe field, and yields to the next generation of observables to be used in flow analyses.The latest important improvements of multivariate cumulants in the context of flow analyses was published recentlyin Ref. [15]. In that paper, the new recursive algorithms have been provided which enable computation of all higher-order cumulants with very compact and efficient code. In addition, the cumulant formalism has been unified acrossdifferent areas of its applicability. However, the definition and interpretation of cumulants of azimuthal angles remainedthe same as before: cumulants are defined in terms of all-event averages, and it is argued that non-isotropic terms inthe cumulant expansion become important only if there are non-uniformities in detector’s acceptance. With respectto both of these aspects, the present work in Sec. IV introduces a radically new approach in the definition andinterpretation of cumulants of azimuthal angles. Other recent studies on multivariate cumulants in the context offlow analyses can be found in Refs. [16, 17].The paper is structured as follows: In Sec. II we review all fundamental mathematical properties of multivariatecumulants, and based on them derive the new and pragmatic two-steps recipe which all multivariate cumulants need tosatisfy in the basis they are expressed in. In Sec. III we check explicitly which observables used in the field and namedcumulants satisfy the fundamental properties of cumulants. For the cases which fail, we introduce in Secs. IV–VIthe alternative new observables which do satisfy all fundamental properties of multivariate cumulants. All results aresupported by detailed calculus and carefully designed Toy Monte Carlo studies. Realistic Monte Carlo studies andfirst predictions for the new observables are presented in Sec. VII. All technical details can be found in appendices. II. REVIEW OF FORMAL MATHEMATICAL PROPERTIES OF MULTIVARIATE CUMULANTSA. Notation
In this section, we introduce the notation and review the most important formal mathematical properties of multi-variate cumulants. This section is the mathematical cornerstone on top of which all physics results discussed in thispaper are built. The presented material is heavily motivated by Kubo’s treatment of many-body systems in [18], andexpanded for clarity sake with more detailed proofs and discussions when necessary (see Appendix A).A general set of N stochastic observables is denoted with X , . . . , X N , and the corresponding multivariate (or joint)probability density function (p.d.f.) with f ( X , . . . , X N ). The knowledge of f ( X , . . . , X N ) determines all statisticalproperties of stochastic observables X , . . . , X N in question. However, in cases of practical interest the functional formof f ( X , . . . , X N ) is frequently unknown. Instead, the statistical properties of observables X , . . . , X N are estimatedfrom multivariate moments ( µ ) and cumulants ( κ ), for which we use the following notations, respectively: µ ν ,...,ν N ≡ µ ( X ν , . . . , X ν N N ) ≡ (cid:104) X ν · · · X ν N N (cid:105) , (4) κ ν ,...,ν N ≡ κ ( X ν , . . . , X ν N N ) ≡ (cid:104) X ν · · · X ν N N (cid:105) c . (5)We use all three versions of the above notation interchangeably throughout the paper, since depending on the context,one version is emerging as more suitable than the others. Multivariate moments are defined directly in terms ofmultivariate p.d.f. f ( X , . . . , X N ) as follows: µ ν ,...,ν N ≡ (cid:90) X ν · · · X ν N N f ( X , . . . , X N ) dX · · · dX N . (6)In this paper moments refer to the raw moments, while the other possibility, the central moments, is not considered. B. Definition of multivariate cumulants
The definition of multivariate moments µ ν ,...,ν N in Eq. (6) can be rewritten very compactly in terms of the momentgenerating function, M ( ξ , . . . , ξ N ), which is defined as: M ( ξ , . . . , ξ N ) ≡ (cid:68) e (cid:80) Nj =1 ξ j X j (cid:69) . (7)It can be easily seen that by doing the formal Taylor expansion of exponential function in auxiliary variables ξ , . . . , ξ N about zero, and after replacing all averages of X , . . . , X N with µ ν ,...,ν N (see Eq. (4)), one obtains all multivariatemoments µ ν ,...,ν N as coefficients of different orders in auxiliary variables ξ , . . . , ξ N : M ( ξ , . . . , ξ N ) = (cid:88) ν ,...,ν N (cid:18) (cid:89) j ξ ν j j ν j ! (cid:19) µ ν ,...,ν N , (8)where all indices ν , . . . , ν N run from zero. The multivariate moments can be therefore obtained directly from theirgenerating function with the following standard expression: µ ν ,...,ν N = ∂ ν ∂ξ ν · · · ∂ ν N ∂ξ ν N N M ( ξ , . . . , ξ N ) (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ = ··· = ξ N =0 . (9)The generating function for cumulants, K ( ξ , . . . , ξ N ), is defined in terms of the moment generating function: K ( ξ , . . . , ξ N ) ≡ ln M ( ξ , . . . , ξ N ) = ln (cid:68) e (cid:80) Nj =1 ξ j X j (cid:69) . (10)By definition, multivariate cumulants κ ν ,...,ν N are coefficients in the formal Taylor expansion of their generatingfunction K ( ξ , . . . , ξ N ) about zero: K ( ξ , . . . , ξ N ) = (cid:88) (cid:48) ν ,...,ν N (cid:18) (cid:89) j ξ ν j j ν j ! (cid:19) κ ν ,...,ν N , (11)where the primed sum (cid:80) (cid:48) ν ,...,ν N means that the term ν = · · · = ν N = 0 is excluded from summation. Analogouslyto moments, cumulants can be obtained directly from their generating function: κ ν ,...,ν N = ∂ ν ∂ξ ν · · · ∂ ν N ∂ξ ν N N K ( ξ , . . . , ξ N ) (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ = ··· = ξ N =0 . (12)The generating function for cumulants was defined in Eq. (10) in terms of the moment generating function, andsolely from this relation it can be shown that all cumulants κ ν ,...,ν N can be uniquely expressed in terms of moments µ ν ,...,ν N , and vice versa. Therefore, the details of the underlying multivariate p.d.f. f ( X , . . . , X N ) can be estimatedequivalently either with moments or with cumulants. Moments have the advantage that they can be measured directlyas the simple averages (cid:104) X ν · · · X ν N N (cid:105) , while the advantage of cumulants stems from the fact that their properties canbe linked more directly with the stochastic nature and physical properties of many-body systems. In practice, onefirst measures moments, then in the next step calculates cumulants from them, and finally from cumulants draws thephysics conclusions and constraints on the many-body problem in question. As an example, the first few cumulantsexpressed in terms of moments read: κ = µ ,κ = µ − µ ,κ , = µ , − µ , µ , ,κ , = 2 µ , µ , − µ , µ , − µ , µ , + µ , ,κ , = 2 µ , µ , − µ , µ , − µ , µ , + µ , ,κ , , = 2 µ , , µ , , µ , , − µ , , µ , , − µ , , µ , , − µ , , µ , , + µ , , . (13)Results of this type can be obtained with straightforward (yet tedious) calculus from Eq. (12), or much easier byusing the specialized tools in the latest versions of Mathematica [19]. For instance, the above expression for κ , , interms of moments can be obtained with the following one-line code snippet: MomentConvert[Cumulant[{1,1,1}],"Moment"] // TraditionalForm
This code can be trivially modified to obtain all other cumulants in terms of moments.
C. Formal mathematical properties of multivariate cumulants
From the definitions presented in the previous section, all properties of multivariate cumulants can be established,and we elaborate next on the most important properties in the context of many-body physics. These properties applyto any particular choice of stochastic observables X , . . . , X N — if any of the formal properties discussed below isviolated for some multivariate observable, that observable is not a multivariate cumulant.1. Statistical independence.
A multivariate cumulant κ ν ,...,ν N ≡ (cid:104) X ν · · · X ν N N (cid:105) c is zero if the observables X , . . . , X N can be divided into two or more groups which are statistically independent. This means thata cumulant is identically zero if at least one of the observables in it is statistically independent of the others.Therefore, a cumulant can be non-zero if and only if all observables in it are correlated, i.e. if there existsa genuine correlation among all N observables X , . . . , X N in question. The detailed proof is presented inAppendix A 1.2. Reduction.
If in a multivariate cumulant some random observables X , . . . , X N are identified to each other, theresulting expression is also a cumulant, but of lower number of observables: κ ν ,...,ν N = (cid:104) X ν · · · X ν N N (cid:105) c = (cid:104) X ˜ ν · · · X ˜ ν M M (cid:105) c = κ ˜ ν ,..., ˜ ν M , (14)where M < N , and ˜ ν i is the sum of all exponents of X i in the initial cumulant. In particular, if we take allrandom observables X , . . . , X N to be the same and equal to X , then the final expression is univariate cumulantof X of order ν + · · · + ν N : κ ν ,...,ν N = (cid:104) X ν · · · X ν N N (cid:105) c = (cid:10) X ν + ··· + ν N (cid:11) c = κ ν + ··· + ν N . (15)If initially ν = ν · · · = ν N = 1, and if we set X = X = . . . = X N ≡ X , then the multivariate cumulantof N random observables reduces to the N th order univariate cumulant, i.e. κ ,..., = κ N . The detail proof ofreduction property is presented in Appendix A 2.3. Semi-invariance.
If for multivariate cumulant κ ν ,...,ν N we have at least one index ν i ≥ (cid:80) i ν i ≥ κ (( X + c ) ν , . . . , ( X N + c N ) ν N ) = κ ( X ν , . . . , X ν N N ) , (cid:88) i ν i ≥ , (16)where c , . . . , c N are constants. For the special case when there is a unique index ν i = 1 and all other indicesare 0, it follows: κ (1 , . . . , , X i + c i , , . . . ,
1) = c i + κ (1 , . . . , , X i , , . . . , . (17)For univariate case, this requirement translates into the statement that all cumulants of order ν ≥ c : κ (( X + c ) ν ) = κ ( X ν ) , ∀ ν ≥ . (18)Only for the first order cumulant the above equality is not satisfied, and we have instead: κ ( X + c ) = c + κ ( X ).The detailed proof of semi-invariance property is presented in Appendix A 3.4. Homogeneity. If c , . . . , c N are constants, we have that: κ (( c X ) ν , . . . , ( c N X N ) ν N ) = c ν · · · c ν N N κ ( X ν , . . . , X ν N N ) . (19)For univariate case, this requirement reduces to: κ (( cX ) ν ) = c ν κ ( X ν ) . (20)The detailed proof is presented in Appendix A 4.5. Multilinearity.
Multivariate cumulants satisfy the following relation in any linear variable: κ ( (cid:88) i X i , Z ν , ..., Z ν N N ) = (cid:88) i κ ( X i , Z ν , ..., Z ν N N ) . (21)To ease the notation, we have taken that the first variable is linear, and the remaining variables, denoted by Z i , can be either linear ( ν i = 1) or non-linear ( ν i > Additivity. If X i are all statistically independent observables, for univariate cumulants it follows κ (( (cid:88) i X i ) N ) = (cid:88) i κ ( X Ni ) . (22)When two or more random observables are statistically independent, the N th-order cumulant of their sum isthe sum of their N th-order cumulants. The proof of this very important property is remarkably simple and ispresented in Appendix A 6.These are the most important mathematical properties that multivariate cumulants must satisfy.We conclude this section by clarifying the range of applicability of cumulants. Both cases below are encounteredfrequently in practice, and they lead in general to the loss of fundamental properties of cumulants:1. Transformation to a new basis.
The above formal properties of cumulants are valid only for the set of stochasticobservables in which cumulant expansion has been performed directly. If cumulant expansion has been performedin one set of stochastic observables and then the final results were transformed in some other set of stochasticobservables, all properties of cumulants are lost in general after such transformation. For instance, if we havetwo sets of observables, X i and Y i , which are related by some functional dependence Y i = Y i ( X , X , . . . ), thenstraight from the definition of generating function in Eq. (10) we can conclude that in general cumulants of X i are not cumulants of Y i , simply because ln (cid:68) e (cid:80) Nj =1 ξ j X j (cid:69) (cid:54) = ln (cid:68) e (cid:80) Nj =1 ξ j Y j ( X ,X ,... ) (cid:69) . As an elementary example, (cid:104) x (cid:105) − (cid:104) x (cid:105) is a valid 2nd order cumulant of x , but not a cumulant of x at any order; (cid:104) x (cid:105) − (cid:104) x (cid:105) is a valid2nd order cumulant of x , but not a cumulant of x at any order, etc.2. Symmetries.
The above formal properties of cumulants are valid only if there are no underlying symmetriesdue to which some terms in the cumulant expansion would vanish identically. This can be demonstrated easilywith the following example: If f ( x, y ) , x, y ∈ ( −∞ , ∞ ) , is a two-variate p.d.f. which does not factorize, thecorresponding two-variate cumulant is not zero. However, if in addition p.d.f. has the following symmetry f ( x, y ) = f ( − x, y ), the corresponding two-variate cumulant is identically zero, because both (cid:104) x (cid:105) and (cid:104) xy (cid:105) areidentically zero due to this symmetry. Clearly, this does not imply that x and y are independent, becausethe starting p.d.f. f ( x, y ) does not factorize [20]. Based on this example, we conclude that the usage andinterpretation of cumulants are reliable only when all terms in the cumulant expansion are present. D. Practical two-steps recipe for multivariate cumulants
After the detailed review of most important properties of multivariate cumulants in the previous section, supportedby the explicit proofs in Appendix A, we now establish the new and very practical two-steps recipe which can be usedfor most cases of interest to make a negative statement, i.e. to rule out some candidate multivariate observable asa multivariate cumulant. To make the final affirmative statement, one needs additionally to check all requirementsfrom the previous section.In what follows next, we denote the starting candidate multivariate observable as a function λ ( X , . . . , X N ). Inits definition, only the moments (i.e. averages) of subsets of stochastic observables X , . . . , X N can appear (toillustrate the notation and categories of observables we are interested in, we can consider for instance λ ( X , X , X ) ≡(cid:104) X X X (cid:105) + (cid:104) X (cid:105)(cid:104) X X (cid:105) − (cid:104) X (cid:105)(cid:104) X (cid:105)(cid:104) X (cid:105) to be a candidate multivariate observable). If any of the two checks belowis violated, the multivariate observable λ ( X , . . . , X N ) is not a multivariate cumulant κ ( X , . . . , X N ):1. We take temporarily that in the definition of λ ( X , . . . , X N ) all observables X , . . . , X N are statistically inde-pendent and factorize all multivariate averages into the product of single averages ⇒ the resulting expressionmust reduce identically to 0;2. We set temporarily in the definition of λ ( X , . . . , X N ) all observables X , . . . , X N to be the same and equal to X ⇒ for the resulting expression it must hold that λ ( aX + b ) = a N λ ( X ) , (23)where a and b are arbitrary constants, and N is the number of observables in the starting definition of λ ( X , . . . , X N ).The first check above is merely the Statistical independence property from Sec. II C while the second check followsnecessarily from the
Reduction, Semi-invariance and
Homogeneity properties. Both of these checks are rather trivialto perform in practice.As an elementary example, we perform these two checks for the simplest two-variate cumulant, κ ( X , X ) = (cid:104) X X (cid:105) − (cid:104) X (cid:105) (cid:104) X (cid:105) . The first check leads immediately to κ ( X , X ) = (cid:104) X (cid:105) (cid:104) X (cid:105) − (cid:104) X (cid:105) (cid:104) X (cid:105) = 0. Following thesecond check, we have that κ ( X ) = (cid:10) X (cid:11) − (cid:104) X (cid:105) , so that: κ ( aX + b ) = (cid:10) ( aX + b ) (cid:11) − (cid:104) aX + b (cid:105) = a (cid:10) X (cid:11) + 2 ab (cid:104) X (cid:105) + b − a (cid:104) X (cid:105) − ab (cid:104) X (cid:105) − b = a (cid:16)(cid:10) X (cid:11) − (cid:104) X (cid:105) (cid:17) = a κ ( X ) , (24)as it should be for a two-variate cumulant.We now use this two-steps recipe in the next section, and sieve all multivariate observables in the field calledcumulants through it, in the chronological order as they were introduced. We demonstrate that, somewhat surpris-ingly, most of those observables fail to satisfy the above two simple checks, and therefore they are not multivariatecumulants. After that, and as the main result of this paper, in subsequent sections we develop the next genera-tion of multivariate cumulants for anisotropic flow analyses, which do satisfy all defining mathematical properties ofmultivariate cumulants. Since the scope of this paper is to scrutinize the usage of multivariate cumulants in flowanalyses, we do not discuss other areas of their applicability (e.g. univariate cumulants of the net proton and netcharge distributions [21–25]). III. THE CUMULANT THAT SHOULD NOT BE
In this section, we confront various multivariate observables used in the field which were named cumulants, andcheck explicitly if they satisfy all requirements discussed in the previous section. We demonstrate that most of theseobservables are not multivariate cumulants, in the strict mathematical sense, in the basis they are presented in.
A. Multivariate cumulants of azimuthal angles
Multivariate cumulants have been introduced for the first time in anisotropic flow analyses at the turn of millenniumin two seminal and heavily influential papers by Ollitrault et al in Refs. [11, 12]. These two papers solved a lot oflong-standing problems in the field and basically changed the way anisotropic flow analysis is performed in high-energy physics. In Eq. (12) of Ref. [11] (or in Eq. (3) of Ref. [12]), the stochastic observables are chosen to be X ≡ exp[ inϕ ] , X ≡ exp[ inϕ ] , X ≡ exp[ − inϕ ] and X ≡ exp[ − inϕ ], where ϕ labels the azimuthal angles ofreconstructed particles. Four-variate cumulants are defined, after all non-isotropic terms in the cumulant expansionhave been neglected (due to symmetry such terms average out to zero for detectors with uniform azimuthal acceptancewhen the average is performed over all events, due to random event-by-event fluctuations of impact parameter vector),as: c n { } ≡ (cid:104)(cid:104) exp[ in ( ϕ + ϕ − ϕ − ϕ )] (cid:105)(cid:105)− (cid:104)(cid:104) exp[ in ( ϕ − ϕ )] (cid:105)(cid:105)(cid:104)(cid:104) exp[ in ( ϕ − ϕ )] (cid:105)(cid:105)− (cid:104)(cid:104) exp[ in ( ϕ − ϕ )] (cid:105)(cid:105)(cid:104)(cid:104) exp[ in ( ϕ − ϕ )] (cid:105)(cid:105) . (25)We have in the above expression adapted the original notation to the notation which became standard later. Usingthe terminology introduced in the previous section, we now consider the above expression to be a candidate observablefor a multivariate cumulant, and write it as: λ ( X , X , X , X ) ≡ (cid:104) X X X X (cid:105) − (cid:104) X X (cid:105)(cid:104) X X (cid:105) − (cid:104) X X (cid:105)(cid:104) X X (cid:105) . (26)We now perform both checks from Sec. II D. The first check yields: λ ( X , X , X , X ) = (cid:104) X (cid:105)(cid:104) X (cid:105)(cid:104) X (cid:105)(cid:104) X (cid:105) − (cid:104) X (cid:105)(cid:104) X (cid:105)(cid:104) X (cid:105)(cid:104) X (cid:105) − (cid:104) X (cid:105)(cid:104) X (cid:105)(cid:104) X (cid:105)(cid:104) X (cid:105) = −(cid:104) X (cid:105)(cid:104) X (cid:105)(cid:104) X (cid:105)(cid:104) X (cid:105)(cid:54) = 0 . (27)Following the second check, it follows λ ( X ) = (cid:104) X (cid:105) − (cid:104) X (cid:105) , so that: λ ( aX + b ) = (cid:104) ( aX + b ) (cid:105) − (cid:104) ( aX + b ) (cid:105) = a (cid:104) X (cid:105) + 4 a b (cid:104) X (cid:105) + 2 a b ( b − (cid:104) X (cid:105) + 4 ab ( b − (cid:104) X (cid:105)− a b (cid:104) X (cid:105)(cid:104) X (cid:105) − a b (cid:104) X (cid:105) − a (cid:104) X (cid:105) − b + b (cid:54) = a ( (cid:104) X (cid:105) − (cid:104) X (cid:105) ) . (28)Therefore, c n { } observable, as defined in Eq. (25), is not a valid four-variate cumulant of azimuthal angles. Sincethis observable is used widely in the field and most of its properties are well understood (e.g. its sensitivity to event-by-event flow fluctuations, etc.), we advocate its continuous usage in the future, but naming it a cumulant is clearlynot justified. As our new contribution in this direction, in Section IV we introduce the new multivariate cumulantsof azimuthal angles, which do satisfy all defining mathematical properties of cumulants. B. Univariate cumulants of Q -vectors and flow amplitudes Cumulants of Q -vectors and cumulants of flow amplitudes have been defined with essentially identical mathematicalequations, therefore we treat them in parallel in this section. We traced back the first appearance of their definingequations to Sec. B1 of Ref. [11]. For simplicity, we write and discuss below all results only in terms of flow amplitudes v n , but everything applies as well to the amplitudes of Q -vectors | Q n | .Our starting point for discussion here are the following definitions [11, 13, 26–28]: c n { } ≡ (cid:104) v n (cid:105) ,c n { } ≡ (cid:104) v n (cid:105) − (cid:104) v n (cid:105) ,c n { } ≡ (cid:104) v n (cid:105) − (cid:104) v n (cid:105)(cid:104) v n (cid:105) + 12 (cid:104) v n (cid:105) ,c n { } ≡ (cid:104) v n (cid:105) − (cid:104) v n (cid:105)(cid:104) v n (cid:105) − (cid:104) v n (cid:105) + 144 (cid:104) v n (cid:105)(cid:104) v n (cid:105) − (cid:104) v n (cid:105) . (29)We now try to reconcile these expressions with the general mathematical formalism of cumulants. There are twopossibilities: the starting stochastic observable is either v n or v n . For both choices, one can demonstrate straightfor-wardly that Semi-invariance property from Sec. II C is not satisfied in general. As a concrete example, for c n { } wehave that: (cid:104) ( v n + α ) (cid:105) − (cid:104) ( v n + α ) (cid:105) (cid:54) = (cid:104) v n (cid:105) − (cid:104) v n (cid:105) , (cid:104) ( v n + α ) (cid:105) − (cid:104) v n + α (cid:105) (cid:54) = (cid:104) v n (cid:105) − (cid:104) v n (cid:105) , (30)where α is an arbitrary constant. The above failures demonstrate that the observable c n { } is not a valid cumulantneither of v n nor of v n .Later in Appendix B we present the new formalism of univariate cumulants of flow amplitudes. That formalismis aligned towards experimental demands, according to which it is feasible to measure reliably with correlationstechniques only the even powers of flow amplitudes (cid:104) v kn (cid:105) , k, n ∈ N . As a direct consequence, we consider thefundamental stochastic observable of interest to be v n and present only its univariate moments and cumulants. Thischoice has the important advantage that there are no obvious symmetries due to which some terms in the cumulantexpansion would be trivially averaged out to zero, when cumulant properties are lost (see discussion at the end ofSec. II C). The difference between old and new formalism for univariate cumulants of flow amplitudes is most strikingin the studies of non-Gaussian flow fluctuations, for which all higher-order cumulants are zero, but only if they satisfyall fundamental properties of cumulants. C. “Asymmetric cumulants” from ATLAS
Recently, the ATLAS Collaboration has in Refs. [29, 30] introduced the following observableac n { } ≡ (cid:104)(cid:104) e i ( nϕ + nϕ − nϕ ) (cid:105)(cid:105) = (cid:104) v n v n cos[2 n (Ψ n − Ψ n )] (cid:105) , (31)and named it “asymmetric cumulant.” However, we were not able to find any publication where the cumulantproperties have been demonstrated to hold for the this observable, either when fundamental stochastic variablesare azimuthal angles, or flow amplitudes v n and/or symmetry planes Ψ n . We have performed that check now: Asdefined in Eq. (31), the observable ac n { } violates the two-step recipe from Sec. II D, and therefore it is not a validmultivariate cumulant, neither of three azimuthal angles ϕ , ϕ , ϕ , nor of flow degrees of freedom v n , v n , Ψ n andΨ n . In fact, the observable ac n { } violates even the most elementary Theorem 1 on multivariate cumulants fromRef. [18]. The relation in Eq. (31) is a trivial result, in a sense that it is merely one particular example obtained fromthe general analytic expression in Eq. (2) (it suffices to insert k = 3, n = n = n and n = − n ), and therefore perse has nothing to do with multivariate cumulants.Later in Sec. V we introduce a new set of observables named asymmetric cumulants of flow amplitudes, which dosatisfy all formal mathematical properties of multivariate cumulants. D. Subevent cumulants in pseudorapidity
Recently in Ref. [31] an alternative cumulant method based on two or more subevents separated in pseudorapidity(∆ η ) is proposed to suppress the contribution from few-particle correlations, which are unrelated to anisotropicflow. Since the starting mathematical equations in this approach are the same as for the standard cumulant methoddiscussed in Sec. III A, all arguments outlined before apply also here: four-particle cumulant in pseudorapidity is nota valid multivariate cumulant of azimuthal angles, since both requirements from Sec. II D are violated. In addition,the usage of ∆ η -subevents to suppress nonflow in multiparticle azimuthal correlations is an ill-defined concept ingeneral, as the ∆ η separation between 3 or more subevents cannot be made equally spaced, given the fact that thereis only one variable at hand (∆ η ). That means that the level of nonflow suppression between different particles ina given multiparticle azimuthal correlator is completely different, i.e. by changing the pseudorapidity boundaries ofeach subevent, the level of nonflow suppression can be made arbitrary. E. Multiparticle mixed-harmonics cumulants
Recently in Ref. [32] the new set of multivariate cumulants, dubbed multiparticle mixed-harmonic cumulants (MHC),have been introduced for the studies of correlated fluctuations of different flow magnitudes. After careful scrutiny, wenow demonstrate that these observables do not lead to valid cumulants of the flow amplitudes, nor of the azimuthalangles. The authors have performed the cumulant expansion on azimuthal angles, then in the final results they havetransformed all azimuthal correlators in terms of flow amplitudes by using Eq. (2), and concluded that the resultingexpression is a valid cumulant of different (mixed) flow amplitudes, raised in general to different powers. However,after such a transformation, all fundamental properties of cumulants are lost. This is true already for the univariatecase, and as an elementary example, one can demonstrate that cumulants of v n and v n are in general different,as discussed at the end of Sec. II C. For instance, MHC observable (cid:104) v m v n (cid:105) − (cid:104) v m (cid:105)(cid:104) v n (cid:105) − (cid:104) v m v n (cid:105)(cid:104) v m (cid:105) + 4 (cid:104) v m (cid:105) (cid:104) v n (cid:105) (Eqs. (14) and (15) in Ref. [32]) violates the second requirement from Sec. II D, and is therefore not a valid multivariatecumulant of flow amplitudes. Moreover, by following this old paradigm, in general there will be also the contributionfrom symmetry planes Ψ n in the final expressions, which renders unwanted contributions in the studies of correlatedfluctuations of different flow magnitudes (this was demonstrated in a clear-cut toy Monte Carlo study in Sec. IIC ofRef. [14]). Because of this, it is impossible for instance to measure with MHC observables correlated fluctuations of v , v and v amplitudes. Since in their derivation they have dropped all non-isotropic azimuthal correlators, theresulting expressions (e.g. Eq. (A1) in Ref. [32]) are not any longer valid cumulants of azimuthal angles either. Themathematical framework of multiharmonic cumulants, when flow amplitudes are raised to different powers, cannot bederived by following the procedure outlined in Ref. [32]. Indeed, the recursive algorithm described in Ref. [32] relieson Eq. (2.9) from Ref. [18] which is valid only in the special case where all the variables in the cumulant expansionare raised to the same power. IV. NEW CUMULANTS OF AZIMUTHAL ANGLES
In this section, we summarize the main ideas and results behind the first self-consistent framework which reconcilescumulants of azimuthal angles with the general mathematical formalism of cumulants. Due to the length of thematerial, we elaborate in detail in our parallel work [33]. The key new findings can be summarized as follows:1. Multivariate cumulants of azimuthal angles are defined event-by-event. This is a radically new approach, sincein all previous studies in the field cumulants of azimuthal angles were defined in terms of all-event averages.2. All terms in the cumulant expansion must be kept. This is also in sheer contrast with the traditional approach,in which all non-isotropic terms are dropped, because they are trivially averaged to zero when the averaging isperformed over all events. If cumulants are defined instead event-by-event, such trivialization is avoided andall mathematical properties of cumulants can be kept. Here lies the heart of the reason why the mathematicalproperties of cumulants are in general lost in the studies performed so far.3. Independence of cumulants with respect to rotations of a coordinate frame in which azimuthal angles are definedhas to be preserved (i.e. all multivariate cumulants must be isotropic). Somewhat surprisingly, we demonstratethat the isotropy of final expressions for cumulants can be preserved even if all individual non-isotropic termsin the cumulant expansion are kept. This is the key result that renders the whole procedure physical.4. Each term in the cumulant expansion must contain the same number of stochastic observables. In the traditionalapproach, and due to symmetry reasons, some observables were identified to each other. However, we now claimthat by following such a procedure, the mathematical properties of cumulants are lost. One possible realization(by no means the only one) to satisfy this property, it to use as many random subevents as there are stochasticobservables, which also automatically solves the problem of self-correlations.05. The resulting physical interpretation of new event-by-event cumulants of azimuthal angles is completely different:For the cases in which combinatorial background is fully under control, these new cumulants are sensitive onlyto nonflow correlations. They are zero both for random walker and if all correlations are dominated by thecollective anisotropic flow. It is in this sense that they are the first observables that can completely separatethe collective correlations due to anisotropic flow and direct few-particle correlations due to nonflow.We now demonstrate all these new findings for the simplest example of two-variate cumulants of azimuthal angles,while the complete discussion and generalization to higher-orders can be found in Ref. [33]. To large extent, notationand terminology are based on Ref. [20].Using the results in Eqs. 13, we start with the general mathematical expression for the two-variate cumulant [18]: κ , = (cid:104) X X (cid:105) − (cid:104) X (cid:105)(cid:104) X (cid:105) . (32)We first divide event in two random subevents, labeled A and B . With such a notation, X ≡ e inϕ A is then stochasticobservable which corresponds to azimuthal angles in subset A , and X ≡ e − inϕ B corresponds to azimuthal angles insubset B . We define in each event the two-particle cumulant as: κ , ≡ (cid:104) e in ( ϕ A − ϕ B ) (cid:105) − (cid:104) e inϕ A (cid:105)(cid:104) e − inϕ B (cid:105) . (33)The biggest conceptual change to the previous usage of cumulants in flow analyses is that the averages in the aboveexpression are single-event averages. Therefore, the single-particle terms, (cid:104) e inϕ A (cid:105) and (cid:104) e − inϕ B (cid:105) , are not triviallyaveraged out to zero, and therefore have to be kept in the cumulant expansion. We now establish the procedure toobtain experimentally from data the estimator for the cumulant defined in Eq. (33).From the set of M reconstructed azimuthal angles in a given event, we define the following statistic: κ , ≡ M A M B M A (cid:88) i M B (cid:88) j e in ( ϕ Ai − ϕ Bj ) − M A M A (cid:88) i e inϕ Ai M B M B (cid:88) j e − inϕ Bj , (34)where M A and M B are multiplicities of two random subevents. All statistical properties of M azimuthal angles aredetermined by specifying multivariate p.d.f. f ( ϕ , . . . , ϕ M ). We now investigate the theoretical properties of statisticdefined in Eq. (34) under the assumption that the joined multivariate p.d.f. f ( ϕ , . . . , ϕ M ) fully factorizes into theproduct of single-variate marginal p.d.f.’s f i ( ϕ i ), all of which are the same and given by Eq. (1) (this property holdsif correlations in particle production are dominated by anisotropic flow). For the true expectation value ( E ) of thereal part of cumulant we have (to ease the notation we have suppressed harmonic n in what follows): E [ R ( κ , )] = E [ (cid:104) cos( ϕ A − ϕ B ) (cid:105) ] − E [ (cid:104) cos ϕ A (cid:105)(cid:104) cos ϕ B (cid:105) ] − E [ (cid:104) sin ϕ A (cid:105)(cid:104) sin ϕ B (cid:105) ] . (35)Since (cid:104) cos( ϕ A − ϕ B ) (cid:105) ≡ M A M B M A (cid:88) i M B (cid:88) j cos( ϕ Ai − ϕ Bj ) , (36)it follows E [ (cid:104) cos( ϕ A − ϕ B ) (cid:105) ] = 1 M A M B M A (cid:88) i M B (cid:88) j E [cos( ϕ Ai − ϕ Bj )]= 1 M A M B M A (cid:88) i M B (cid:88) j v = v . (37)In transition from 1st to 2nd line in the above equation we have used that for each individual pair we have thefollowing result for the fundamental expectation value: E [cos( ϕ i − ϕ j )] = v , ∀ i (cid:54) = j .We now calculate E [ (cid:104) cos ϕ A (cid:105)(cid:104) cos ϕ B (cid:105) ] and E [ (cid:104) sin ϕ A (cid:105)(cid:104) sin ϕ B (cid:105) ]. Since ϕ A and ϕ B are from two random subevents,1we do not need to consider self-correlations. It follows: E [ (cid:104) cos ϕ A (cid:105)(cid:104) cos ϕ B (cid:105) ] = E M A M A (cid:88) i =1 cos( ϕ Ai ) 1 M B M B (cid:88) j =1 cos( ϕ Bj ) = 1 M A M B M A (cid:88) i M B (cid:88) j E [cos( ϕ Ai ) cos( ϕ Bj )]= 1 M A M B M A (cid:88) i M B (cid:88) j v cos Ψ= v cos Ψ . (38)In transition from the 2nd to the 3rd line above, we have used another fundamental result for expectation value ofany pair, namely: E [cos( ϕ i ) cos( ϕ j )] = v cos Ψ , ∀ i (cid:54) = j . By following the same reasoning, we have obtained that: E [ (cid:104) sin ϕ A (cid:105)(cid:104) sin ϕ B (cid:105) ] = v sin Ψ . (39)Putting everything together, we have finally arrived at: E [ R ( κ , )] = v − v cos Ψ − v sin Ψ = 0 . (40)The analogous calculus can be performed also for the imaginary part I ( κ , ), to obtain: E [ I ( κ , )] = E [ (cid:104) sin( ϕ A − ϕ B ) (cid:105) ] + E [ (cid:104) cos ϕ A (cid:105)(cid:104) sin ϕ B (cid:105) ] − E [ (cid:104) sin ϕ A (cid:105)(cid:104) cos ϕ B (cid:105) ]= 0 + ( v cos Ψ)( v sin Ψ) − ( v sin Ψ)( v cos Ψ)= 0 . (41)Therefore, if correlations among produced particles are dominated by anisotropic flow, we have that true expectationvalue for real and imaginary terms of two-variate cumulants are: E [ R ( κ , )] = 0 , (42) E [ I ( κ , )] = 0 . (43)The above procedure can be straightforwardly generalized to all higher-order cumulants [33]. We now illustrate thesenew concepts with example toy Monte Carlo studies. A. Toy Monte Carlo studies
We first set up the toy Monte Carlo model which can be solved analytically for the multivariate cumulants ofazimuthal angles, and demonstrate how the experimental estimators introduced in Sec. IV can be used to recoverthem from the sampled azimuthal angles. We start by defining the following normalized two-variate p.d.f. of azimuthalangles: f ( ϕ , ϕ ; M ) = 9375 (cid:16) ϕ − Mϕ (cid:17) π (3 π M − πM + 12500) , (44)where M is a parameter and it corresponds to multiplicity. The only stochastic variables are azimuthal angles ϕ and ϕ , whose sample space is [0 , π ). One can easily check that: (cid:90) π (cid:90) π f ( ϕ , ϕ ; M ) dϕ dϕ = 1 . (45)for any value of multiplicity M . Defined this way, f ( ϕ , ϕ ; M ) cannot be factorized into the product of two single-variate marginal p.d.f.’s, and therefore will yield to the non-zero values of new cumulants introduced in the previoussection.2 multiplicity - - - c u m u l an t s ] (th) k Re[ ] (exp) k Re[ ] (th) k Im[ ] (exp) k Im[ ] (exp) k Re[ ] (exp) k Re[
200 300 400 500 600 700 800 900 1000 multiplicity - - - - - - - c u m u l an t s FIG. 1. Comparison of theoretical values for two-variate cumulants in Eq. (46) (dashed lines) and the ones obtained from theexperimental estimator in Eq. (34 (markers), for the toy Monte Carlo p.d.f. in Eq. (44), plotted at two different scales: small(LHS) and large (RHS) multiplicites. Experimental results for four- and six-variate cumulants are also shown with green andblack markers, respectively (their theoretical values in this toy Monte Carlo model are identically zero, see Eq. (47)).
Next, we proceed to calculate analytically the two-variate cumulant of fundamental observables X = e inϕ and X = e − inϕ , when azimuthal angles are sampled from the joint two-variate p.d.f. defined in Eq. (44). After somestraightforward calculus, for n = 1 we have obtained the following analytic results: R [ κ , ] = 375 M (cid:0) π (cid:0) π − (cid:1) M − (cid:0) − − π + π (cid:1) M + 6250 π (cid:0) π (cid:1)(cid:1) π ( πM (3 πM − , I [ κ , ] = − M (cid:0)(cid:0) π − (cid:1) M − πM − (cid:1) π ( πM (3 πM − . (46)Both the real and the imaginary parts of two-variate cumulants are non-zero and have non-trivial dependence onmultiplicity. For the higher-order cumulants, after straightforward calculus we have obtained analytically: R [ κ , , , ] = I [ κ , , , ] = 0 , R [ κ , , , , , ] = I [ κ , , , , , ] = 0 . (47)This is expected result because in this toy Monte Carlo model there are only genuine two-particle correlations in thestarting definition in Eq. (44), and therefore all higher-order cumulants must be identically zero.After we have obtained the theoretical results, we now use the sampled azimuthal angles, and from them buildexperimental estimators for the cumulants (see Eq. (34)). Since this is a toy Monte Carlo study, we have full controlover the combinatorial background, which is essential for this prescription to work. The final results are shown inFig. 1. The obtained results from experimental estimators introduced in the previous section reproduce to greatprecision the theoretical results in Eqs. (46) and (47). Further studies can be found in Ref. [33].At the end of this section, we highlight one important experimental difference between cumulants of azimuthalangles on one side, and cumulants of flow amplitudes and cumulants of symmetry planes on the other: Only in thelatter cases combinatorial background plays no role. We now demonstrate how the recently introduced SymmetricCumulants (SCs) of flow amplitudes [9, 14, 34–36] can be extended and generalized also for the more challengingasymmetric combinations of different flow amplitudes, when they are raised to different powers. V. ASYMMETRIC CUMULANTS OF FLOW AMPLITUDES
The main idea in the approach used to generalise the SCs to the higher orders was to identify the flow amplitudessquared as the fundamental observables of the cumulant expansion [14]. As v n are stochastic variables, their SCs arevalid multivariate cumulants in this basis. This formalism can then naturally be expanded to the asymmetric case.However, one important remark is that this generalisation is not trivial. The formalism of the SCs has been built onEq. (2.9) from Ref. [18], which is valid only when all the observables are raised to the same power.3We name the novel observables resulting from the extension of our formalism the Asymmetric Cumulants (ACs).By nature, they can probe the genuine correlations between the different moments of different flow harmonics andtherefore, they have access to new and independent information. We will look here at the four different examples fortwo and three harmonics: AC , ( m, n ) ≡ (cid:104) ( v m ) v n (cid:105) c ≡ (cid:104) v m v n (cid:105) c , (48)AC , ( m, n ) ≡ (cid:104) ( v m ) v n (cid:105) c ≡ (cid:104) v m v n (cid:105) c , (49)AC , ( m, n ) ≡ (cid:104) ( v m ) v n (cid:105) c ≡ (cid:104) v m v n (cid:105) c , (50)AC , , ( k, l, m ) ≡ (cid:104) ( v k ) v l v m (cid:105) c ≡ (cid:104) v k v l v m (cid:105) c . (51)This notation has been chosen as it allows to know at first glance the cumulant expansion on which the consideredobservable is based, e.g. κ , for AC , ( m, n ), κ , for AC , ( m, n ) and so on (see Eq. (13) for κ , ). Furthermore,this formalism can easily be extended to higher moments or more flow amplitudes.Using the general mathematical framework from Section II and identifying the stochastic variables X , X and X with the corresponding flow amplitudes squared leads to the following expressions:AC , ( m, n ) = (cid:104) v m v n (cid:105) − (cid:104) v m (cid:105)(cid:104) v n (cid:105) − (cid:104) v m v n (cid:105)(cid:104) v m (cid:105) + 2 (cid:104) v m (cid:105) (cid:104) v n (cid:105) , (52)AC , ( m, n ) = (cid:104) v m v n (cid:105) − (cid:104) v m (cid:105)(cid:104) v n (cid:105) − (cid:104) v m v n (cid:105)(cid:104) v m (cid:105) − (cid:104) v m v n (cid:105)(cid:104) v m (cid:105) + 6 (cid:104) v m (cid:105)(cid:104) v m (cid:105)(cid:104) v n (cid:105) + 6 (cid:104) v m v n (cid:105)(cid:104) v m (cid:105) − (cid:104) v m (cid:105) (cid:104) v n (cid:105) , (53)AC , ( m, n ) = (cid:104) v m v n (cid:105) − (cid:104) v m (cid:105)(cid:104) v n (cid:105) − (cid:104) v m v n (cid:105)(cid:104) v m (cid:105) − (cid:104) v m v n (cid:105)(cid:104) v m (cid:105) + 6 (cid:104) v m (cid:105) (cid:104) v n (cid:105) − (cid:104) v m v n (cid:105)(cid:104) v m (cid:105) + 8 (cid:104) v m (cid:105)(cid:104) v m (cid:105)(cid:104) v n (cid:105) + 24 (cid:104) v m v n (cid:105)(cid:104) v m (cid:105)(cid:104) v m (cid:105) + 12 (cid:104) v m v n (cid:105)(cid:104) v m (cid:105) − (cid:104) v m (cid:105)(cid:104) v m (cid:105) (cid:104) v n (cid:105) − (cid:104) v m v n (cid:105)(cid:104) v m (cid:105) + 24 (cid:104) v m (cid:105) (cid:104) v n (cid:105) , (54)AC , , ( k, l, m ) = (cid:104) v k v l v m (cid:105) − (cid:104) v k v l (cid:105)(cid:104) v m (cid:105) − (cid:104) v k v m (cid:105)(cid:104) v l (cid:105) − (cid:104) v k (cid:105)(cid:104) v l v m (cid:105) + 2 (cid:104) v k (cid:105)(cid:104) v l (cid:105)(cid:104) v m (cid:105) − (cid:104) v k v l (cid:105)(cid:104) v k v m (cid:105) − (cid:104) v k v l v m (cid:105)(cid:104) v k (cid:105) + 4 (cid:104) v k v l (cid:105)(cid:104) v k (cid:105)(cid:104) v m (cid:105) + 4 (cid:104) v k v m (cid:105)(cid:104) v k (cid:105)(cid:104) v l (cid:105) + 2 (cid:104) v k (cid:105) (cid:104) v l v m (cid:105) − (cid:104) v k (cid:105) (cid:104) v l (cid:105)(cid:104) v m (cid:105) . (55)These expressions are genuine multivariate cumulants, according to the Kubo’s formalism. As a concrete example,we show in Appendix C 1 that all the requirements described in Section II C are met in the case of AC , ( m, n ). Thedemonstration can then be generalised for any other AC.As it has been done for the SCs, we can also normalise the ACs. This procedure is beneficial for two reasons. First,the predictions for ACs in the initial and final state do not have the same scale, so normalising the results allows propercomparisons and determination of the initial state effects and the changes brought by the hydrodynamic evolution.Second, flow amplitudes have a dependence on the transverse momentum p T , leading to similar dependence in anylinear combinations of them, e.g. the SCs and the ACs. The normalisation removes this dependence and permits thecomparisons between models and data with different p T ranges. The normalisation of the ACs is done following thestandard method from Ref. [17]: NAC a, ( m, n ) = AC a, ( m, n ) (cid:104) v m (cid:105) a (cid:104) v n (cid:105) , a = 2 , , , (56)NAC , , ( k, l, m ) = AC , , ( k, l, m ) (cid:104) v k (cid:105) (cid:104) v l (cid:105)(cid:104) v m (cid:105) . (57)Furthermore, Eqs. (52)–(55) are sufficient to get predictions with the help of theoretical models. However, as theflow amplitudes are not directly accessible in experimental analyses, estimators in terms of azimuthal angles must befound to measure the ACs with real data. We give here only the final expressions, while the support studies with toyMonte Carlo simulations are presented in the extension paper [37].As the approach used for the ACs is based on the one developed for the generalisation of the SCs [14], we usehere the knowledge gained in the latter. The transition from the azimuthal angles to the flow amplitudes is given4by Eq. (2). The ambiguity already resolved in the SCs (see Appendix C in Ref. [14]) is avoided by maximising thenumber of particles involved in each azimuthal correlators. This leads to the final experimental expression shown inAppendix C 2.We now present an example study that illustrates one of the properties of our novel ACs. With the use of the realisticMonte Carlo (MC) event generator HIJING [38, 39], we can demonstrate the robustness of the proposed observablesagainst few-particle nonflow correlations. HIJING (for Heavy-Ion Jet INteraction Generator ) is a combination ofmodels describing jet- and nuclear-related mechanisms, like jet production and fragmentation or nuclear shadowingto cite only a few of them. One of the particularities of HIJING is that it does not include any collective effects like
Centrality percentile - - · = 2.76 TeV NN s Pb - HIJING Pb æ v v Æ æ v Ææ v Æ æ v Ææ v v Æ æ v Æ æ v Æ Centrality percentile ( , ) , A C - - - - - · Centrality percentile ( , ) , A C - - - - · Centrality percentile ( , ) , A C - · Centrality percentile ( , , ) , , A C - - · FIG. 2. Predictions of the centrality dependence of the different correlators involved in AC , (2 ,
4) (top left), of AC , (2 , , (2 ,
4) (middle left) and AC , (2 ,
4) (middle right) and AC , , (2 , ,
4) (bottom) in Pb–Pb collisionsat √ s NN = 2 .
76 TeV with the HIJING generator. √ s NN = 2 .
76 TeV. Two kinetic criteria have been applied as well: 0 . < p T < . c and | η | < .
8. Figure 2shows the centrality dependence of the two-harmonic ACs between v and v . The different correlators involvedin the expression of AC , (2 ,
4) are shown in the top left panel. One can see that taken individually, these termsexhibit a systematic bias due to nonflow with the centrality. The centrality dependence of AC , (2 , , (2 ,
4) andAC , (2 ,
4) are visible in the top right, middle left and middle right panels respectively. These three observables arein agreement with zero for the full centrality range, meaning they are robust against few-particle nonflow correlations.Finally, the case of AC , , (2 , ,
4) is illustrated in Fig. 2 in the bottom panel, where one can see that it is robustagainst nonflow as well.In Section VII, we present predictions for the ACs obtained with realistic MC models. More studies about theproperties of the ACs will be shown in Ref. [37].
VI. CUMULANTS OF SYMMETRY PLANE CORRELATIONS
In this section, we elaborate on cumulants which are based on symmetry planes Ψ n . Unlike the flow amplitudes v n , symmetry planes are not rotationally invariant and are therefore affected by random fluctuations of the impactparameter vector. Therefore, final observables which involve symmetry planes have to be rotationally invariant underthe randomness of the reaction plane. Otherwise, such observables would trivially average out to 0. Here, we illustrateonly the main idea, with selected MC study, while all remaining items related to the experimental feasibility usingthe estimator presented in [40] will be discussed in a follow-up paper.To find a multivariate cumulant involving symmetry planes, we require stochastic observables which by themselvesare already rotationally invariant. Otherwise, non-invariant terms in the cumulant expressions (Eq. (13)) would cancelout, and cumulant properties are lost (see the discussion at the end of Sec. II C). For the sake of simplicity, we willnow only consider such stochastic observables where each observable includes two symmetry planes. However, onehas to note, that it is possible to extend this concept to observables with more than two or even different number ofsymmetry planes per stochastic observable.We focus on two observables X and X , each containing two symmetry planes X = e i ( a c c Ψ c − a d d Ψ d ) , (58) X = e i ( a l l Ψ l − a m m Ψ m ) , (59)where we assume for the integers c, d, l, m >
0. Therefore, the symmetry plane Ψ c has been taken a c times, Ψ d a d times and so forth. As we want X and X to be rotationally invariant, we have the following conditions a c c = a d d , (60) a l l = a m m . (61)From now on, we will use the following notation X = e ibδ c,d , (62) X = e ikδ l,m , (63)where we abbreviated b ≡ a c c = a d d , (64) k ≡ a l l = a m m , (65) δ c,d ≡ Ψ c − Ψ d , (66) δ l,m ≡ Ψ l − Ψ m . (67)Using the cumulant expression of κ , from Eq. (13) we thus obtain the simplest two-variate Cumulant of SymmetryPlane Correlations (CSC)CSC ( bδ c,d , kδ l,m ) = (cid:68) e i ( bδ c,d + kδ l,m ) (cid:69) − (cid:10) e ibδ c,d (cid:11) (cid:10) e ikδ l,m (cid:11) with b (cid:54) = k , (68)where we imposed the needed condition b (cid:54) = k which will be explained later.6The cumulant in Eq. (68) is rotationally invariant under the random fluctuations of the reaction plane and it fulfilsall needed cumulant properties (see Appendix D). In fact, Eq. (68) is not a cumulant of symmetry planes, but rather acumulant of symmetry plane correlations as we identify the fundamental stochastic observables δ i,j as the correlationbetween two symmetry planes Ψ i and Ψ j . It states if and how the symmetry plane correlations δ c,d and δ l,m areconnected to each other. It therefore captures more of the “dynamics” between different symmetry planes and theircorrelations. As a further note, using the stochastic observables we defined in Eq. (62), we recover for κ = µ (univariate case) the well-known expression for symmetry-plane correlations between two planes.For the cumulant defined in Eq. (68) there are five basic scenarios:1. All involved symmetry planes fluctuate independent from each other. Thus, also δ c,d and δ l,m fluctuate randomlyand independently in the interval [0 , π ) and the cumulant is zero.2. δ c,d and δ l,m are constant. Equation (68) yields zero.3. δ c,d and δ l,m fluctuate independently from each other. Equation (68) yields zero.4. Ψ c is only correlated to Ψ m and Ψ d is only correlated to Ψ l , while no other correlations between any symmetryplanes is given. Thus, δ c,d and δ l,m fluctuate between [0 , π ). However, if we allow b = k , we end up that thesingle averages lead to zero, while the joined average does not cancel out to zero, as we have isotropic parts ofcorrelated symmetry planes. Therefore, we have to impose the condition b (cid:54) = k to ensure that the cumulant inEq. (68) yields zero also in this case.5. δ c,d and δ l,m are genuinely correlated to each other. The cumulant defined in Eq. (68) yields a non-zero value.In Appendix E, we will test these scenarios in Toy Monte Carlo studies and show that Eq. (68) does have all thedesired properties as we described them in the five scenarios.Additionally, one can simplify Eq. (68) by choosing two symmetry planes to be the same. The cumulant will thenyet hold two symmetry plane correlations as the fundamental stochastic observables, however then the correlation oftwo symmetry planes to a third one will be studied. In special cases, it is even possible to neglect the requirement of b (cid:54) = k , namely in the cases when one chooses Ψ c = Ψ l or Ψ d = Ψ m . However, one must not neglect the requirement ifone, for instance, chooses Ψ d = Ψ l . In this case, this plane would cancel out in the joined mean if one neglects b (cid:54) = k and cumulant properties will be lost. In the next section, we will provide first estimates from realistic Monte Carlomodels for all these new observables. VII. MONTE CARLO STUDIES: MC-GLAUBER + VISH2+1
In this part, we present the realistic hydrodynamic predictions for ACs and CSCs. For that, we simulate the initialenergy density with MC-Glauber model [41–43] and the hydrodynamic evolution part with VISH2+1 [44, 45]. Wesimulate Pb–Pb collisions at √ s NN = 2 .
76 TeV with fixed shear viscosity over entropy density η/s = 0 . , , AC , and AC , for different combinations of flow harmonics. To investigatethe origin of the fluctuations inherited from the initial state, we study the same cumulants by replacing the flowharmonics v n ’s with eccentricities (cid:15) n ’s. The eccentricities are defined as [46], (cid:15) n e in Φ n = − (cid:82) r n e inϕ ρ ( r, ϕ ) rdrdϕ (cid:82) r n ρ ( r, ϕ ) rdrdϕ , n > , (69)where ρ ( r, ϕ ) is the energy density in the transverse direction ( r, ϕ ).The plots on the left side of the figures are dedicated to the eccentricity fluctuations computed directly from theMC-Glauber model while the plots on the right side are showing the cumulants of flow harmonic fluctuation. Ascan be seen from the figures, the initial state correlations are growing monotonically from lower centralities to highercentralities while the final state correlations first increase and then decrease. This is due to the fact that at largercentralities a smaller medium is produced. Therefore, the evolution period is shorter and the medium consequentlyhas less time to transfer the initial state correlation to the final state correlation.In Fig. 6, the Normalized Asymmetric Cumulants NAC , (2 ,
3) and NAC , (2 ,
4) are shown (see Eq. (56)). Bynormalizing the ACs with the fluctuation width (cid:104) v n (cid:105) , we are able to compare the initial and final state fluctuations.This comparison becomes more clear when we consider the hydrodynamic response relation between eccentricities andflow harmonics [46, 47], v n e in Ψ n = k n (cid:15) n e in Φ n + non-linear terms . (70)7 Centrality percentile - ( , n ) , A C = 2.76 TeV NN s Pb - MC-Glauber Pbn = 3n = 4n = 5 0 10 20 30 40 50 60 70
Centrality percentile - - - · ( , n ) , A C VISH2+1n = 3n = 4n = 50 10 20 30 40 50 60 70
Centrality percentile - - - - - - - · ( , n ) , A C Centrality percentile - - · ( , n ) , A C Centrality percentile - - - - - - · ( , n ) , A C Centrality percentile - - · ( , n ) , A C FIG. 3. Asymmetric Cumulants for eccentricities fluctuations extracted from MC-Glauber model (left) and for flow harmonicfluctuations extracted from MC-Glauber + VISH2+1 (right).
In the linear response approximation v n (cid:39) k n (cid:15) n , the response coefficient k n is canceled from numerator and de-nominator in Eq. (56). This means that the NAC extracted from initial and final states should be identical in thisapproximation. As it can be seen from Fig. 6 (left), the NAC , (2 ,
3) from initial and final state are approximatelycompatible with each other up to 25% of centrality. The deviation from linear response approximation starts from10% of centrality in NAC , (2 , Centrality percentile ( m , n ) , A C = 2.76 TeV NN s Pb - MC-Glauber Pbm = 3, n = 4m = 3, n = 5m = 4, n = 5 0 10 20 30 40 50 60 70
Centrality percentile - - - - · ( m , n ) , A C VISH2+1m = 3, n = 4m = 3, n = 5m = 4, n = 50 10 20 30 40 50 60 70
Centrality percentile - - - · ( m , n ) , A C Centrality percentile - - - - - · ( m , n ) , A C Centrality percentile - - - - - - · ( m , n ) , A C Centrality percentile - - - · ( m , n ) , A C FIG. 4. Asymmetric Cumulants for eccentricities fluctuations extracted from MC-Glauber model (left) and for flow harmonicfluctuations extracted from MC-Glauber + VISH2+1 (right). normalization. Here, we concentrate on the following two specific examples. The real part of the following cumulants,CSC(4 δ , , δ , ) = (cid:68) e i − Ψ )+ i − Ψ ) (cid:69) − (cid:68) e i − Ψ ) (cid:69) (cid:68) e i − Ψ ) (cid:69) (71)CSC(4 δ , , δ , ) = (cid:68) e i − Ψ )+ i − Ψ ) (cid:69) − (cid:68) e i − Ψ ) (cid:69) (cid:68) e i − Ψ ) (cid:69) (72)are extracted from the hydrodynamic simulation and shown in Fig. 6. The initial state cumulants are obtained byreplacing Ψ n with Φ n . Given the fact that CSCs only depend on the symmetry plane angles, only the phase of the9 Centrality percentile - - ( , ) , N A C = 2.76 TeV NN s Pb - Pb MC-GlauberMC-Glauber + VISH2+1 0 10 20 30 40 50 60 70
Centrality percentile ( , ) , N A C FIG. 5. Normalized Asymmetric Cumulants for eccentricities fluctuations and flow harmonic fluctuations extracted fromMC-Glauber model and MC-Glauber + VISH2+1 model, respectively.
Centrality percentile - - - - ) ] , d , , d R e [ C S C ( = 2.76 TeV NN s Pb - Pb MC-GlauberMC-Glauber + VISH2+1 0 10 20 30 40 50 60 70
Centrality percentile - - - ) ] , d , , d R e [ C S C ( FIG. 6. Cumulants of Symmetry Plane Correlations for eccentricities fluctuation and flow harmonic fluctuations extracted fromMC-Glauber model and MC-Glauber + VISH2+1 model, respectively. hydrodynamic response relation in Eq. (70) needs to be considered,Ψ n = Φ n + m n πn + non-linear terms , m n = 0 or 1 . (73)The presence of the term m n π/n above is the reflection of the sign of k n . Noting Eq. (70), if k n is positive then m n = 0. The negative response coefficient leads to an extra phase e iπ and consequently to m n = 1. Using thisnotation and assuming CSC (cid:15) as the cumulant of the initial state eccentricities, we findCSC(4 δ , , δ , ) = (cid:68) e i − Ψ )+ i − Ψ ) (cid:69) − (cid:68) e i − Ψ ) (cid:69) (cid:68) e i − Ψ ) (cid:69) (cid:39) e iπ ( m − m +3 m − m ) (cid:104)(cid:68) e i − Φ )+ i − Φ ) (cid:69) − (cid:68) e i − Φ ) (cid:69) (cid:68) e i − Φ ) (cid:69)(cid:105) = e iπ ( m − m +3 m − m ) CSC (cid:15) (4 δ , , δ , ) , (74)where we have substituted Eq. (73) into Eq. (71) and ignored the non-linear terms. In Ref. [17], it has been shown that k and k are positive valued quantities while k and k have negative sign. This equivalently means m = m = 0and m = m = 1 which leads to e iπ ( m − m +3 m − m ) = 1. As a result, up to hydrodynamic linear approximation,we expect CSC(4 δ , , δ , ) (cid:39) CSC (cid:15) (4 δ , , δ , ). Referring to Fig. 6 (left), one observes that this relation is correctwith a reasonable accuracy. This is compatible with former observation in Ref. [17].0We can follow the same procedure with the cumulant in Eq. (72) to findCSC(4 δ , , δ , ) (cid:39) e iπ ( m − m + m − m ) CSC (cid:15) (4 δ , , δ , )= e iπ (1+ m ) CSC (cid:15) (4 δ , , δ , ) , (75)where in the last equality we have used m = m = 0 and m = 1. The initial and final state cumulants inFig. (6) (right), are compatible for centralities below 30%. The observed deviation at higher centralities is relatedto the non-linear hydrodynamic response. The compatibility at lower centralities can be considered as a hint thatCSC(4 δ , , δ , ) (cid:39) CSC (cid:15) (4 δ , , δ , ). Referring to Eq. (75), this is true only of m = 1 which means the responsecoefficient k should be a negative valued quantity. More investigation with higher simulation statistics is needed inthis respect. VIII. SUMMARY AND OUTLOOK
In summary, we have introduced new anisotropic flow observables based on the formalism of multivariate cumulants.These observables manifestly satisfy all mathematical properties of cumulants in the basis they are expressed in. Wehave developed a pragmatic two-step recipe that any multivariate cumulants must satisfy as a necessary condition. Itwas demonstrated that fundamental properties of cumulants are preserved only if there are no underlying symmetriesdue to which some terms in the cumulant expansion are identically zero. New cumulants of azimuthal angles have beendefined event-by-event and by keeping all terms in the cumulant expansion, which have different physical interpretationwhen compared to the ones used so far. We have derived the Asymmetric Cumulants of flow amplitudes as thegeneralization of the widely used Symmetric Cumulants. This extension aims at the probing the correlations presentbetween different moments of the flow amplitudes. With the help of the HIJING Monte Carlo generator, we thenshowed their robustness against nonflow correlations. Next, for the first time cumulants of symmetry plane correlationshave been derived. All these new observables can provide further new and independent constraints to the initialconditions and system properties of matter produced in high-energy nuclear collisions.
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). We would like to thank Jiangyong Jia, DongJo Kim, Matt Luzum and Raimond Snellings for their interest in this work and fruitful discussions.
Appendix A: Detailed derivations
In this Appendix we outline the detailed mathematical proofs for the statements made in the main part. This isessentially a review of well-known results, however, the detailed proofs, especially for multivariate case, are difficultto find in the literature. To ease the notation, we outline derivations either for two- or three-variate cases wheneverthe generalization to multivariate case is trivial and it does not introduce any new conceptual step.
1. Proof of statistical independence
The proof is trivial (see Theorem I and the surrounding discussion in Ref. [18]). If we have N random observablesand two subsets which are independent from each other, we can denote them without loss of generality as X (cid:48) , . . . X (cid:48) K and X (cid:48)(cid:48) K +1 , . . . X (cid:48)(cid:48) N , respectively, where K < N . Then straight from the general definitions given in Eqs. (10) and (12)1we have: κ ν ,...,ν N = ∂ ν ∂ξ ν · · · ∂ ν N ∂ξ ν N N ln (cid:68) e (cid:80) Nj =1 ξ j X j (cid:69)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ = ··· = ξ N =0 = ∂ ν ∂ξ ν · · · ∂ ν N ∂ξ ν N N ln (cid:68) e (cid:80) Kj =1 ξ j X (cid:48) j + (cid:80) Nj = K +1 ξ j X (cid:48)(cid:48) j (cid:69)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ = ··· = ξ N =0 = ∂ ν ∂ξ ν · · · ∂ ν N ∂ξ ν N N ln (cid:68) e (cid:80) Kj =1 ξ j X (cid:48) j (cid:69)(cid:12)(cid:12)(cid:12) ξ = ξ = ··· = ξ N =0 + ∂ ν ∂ξ ν · · · ∂ ν N ∂ξ ν N N ln (cid:68) e (cid:80) Nj = K +1 ξ j X (cid:48)(cid:48) j (cid:69)(cid:12)(cid:12)(cid:12) ξ = ξ = ··· = ξ N =0 = 0 . (A1)We have obtained that the final result is identically zero, simply because N > K and
N > N − K −
1, so that in bothcases we take more derivatives then there are available independent auxiliary variables ξ . If we have more then twosubsets of observables which are independent from each other, the above proof generalizes trivially.
2. Proof of reduction
We use the general definitions in Eqs. (10) and (12) and apply them to the three-variate case: κ ν ,ν ,ν ( X , X , X ) = ∂ ν ∂ξ ν ∂ ν ∂ξ ν ∂ ν ∂ξ ν ln (cid:10) e ξ X + ξ X + ξ X (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ = ξ =0 . (A2)We identify X ≡ X and write: κ ν ,ν ,ν ( X , X , X ) = ∂ ν ∂ξ ν ∂ ν ∂ξ ν ∂ ν ∂ξ ν ln (cid:68) e ξ X +( ξ + ξ ) X (cid:69)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ = ξ =0 . (A3)The key observation now is that we can redefine the sum of two auxiliary observables into the new auxiliary observable η ≡ ξ + ξ . Since ξ and ξ are unrelated, it follows that: ∂∂ξ ln (cid:10) e ξ X + ηX (cid:11) = (cid:10) X e ξ X + ηX (cid:11) (cid:104) e ξ X + ηX (cid:105) = ∂∂ξ ln (cid:10) e ξ X + ηX (cid:11) = ∂∂η ln (cid:10) e ξ X + ηX (cid:11) . Therefore: κ ν ,ν ,ν ( X , X , X ) = ∂ ν ∂ξ ν ∂ ν + ν ∂η ν + ν ln (cid:10) e ξ X + ηX (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ξ = η =0 . (A4)Since in the above equality both ξ and η are auxiliary, it follows immediately: κ ν ,ν ,ν ( X , X , X ) = κ ν ,ν + ν ( X , X ) . (A5)The above proof generalizes trivially for other multivariate cases.2
3. Proof of semi-invariance
We start as follows: κ (( X + c ) ν , ( X + c ) ν ) = ∂ ν ∂ξ ν ∂ ν ∂ξ ν ln (cid:68) e ξ ( X + c )+ ξ ( X + c ) (cid:69)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 = ∂ ν ∂ξ ν ∂ ν ∂ξ ν ln (cid:10) e ξ X e ξ c e ξ X e ξ c (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 = ∂ ν ∂ξ ν ∂ ν ∂ξ ν ln (cid:2) e ξ c e ξ c (cid:10) e ξ X e ξ X (cid:11) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 = ∂ ν ∂ξ ν ∂ ν ∂ξ ν (cid:2) ξ c + ξ c + ln (cid:10) e ξ X e ξ X (cid:11) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 = ∂ ν ∂ξ ν ∂ ν ∂ξ ν (cid:2) ξ c + ξ c (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 + ∂ ν ∂ξ ν ∂ ν ∂ξ ν (cid:2) ln (cid:10) e ξ X e ξ X (cid:11) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 = ∂ ν ∂ξ ν ∂ ν ∂ξ ν (cid:2) ξ c + ξ c (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 + κ ( X ν , X ν ) . (A6)In the transition from 2nd to 3rd line we have used the fact that (cid:104)· · · (cid:105) is an average with respect to X and X , whichinvolves only an integration over X and X , with respect to which all of ξ , ξ , c and c are constants, and thereforecan be pulled out of the average. We have the following final result: κ ( X + c ,
1) = c + κ ( X , ,κ (1 , X + c ) = c + κ (1 , X ) ,κ (( X + c ) ν , ( X + c ) ν ) = κ ( X ν , X ν ) , (cid:88) i ν i ≥ . (A7)The above proof generalizes trivially for more than two random observables.
4. Proof of homogeneity
As in the previous cases, for simplicity we consider here only the two-variate case explicitly, because the general-ization to multivariate case is trivial and it does not involve any new conceptual step. From Eqs. (10) and (12) itfollows: κ (( c X ) ν , ( c X ) ν ) = ∂ ν ∂ξ ν ∂ ν ∂ξ ν ln (cid:10) e ξ c X + ξ c X (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 . (A8)Since ξ and ξ are auxiliary variables, we can redefine them as η ≡ c ξ and η ≡ c ξ . Then: ∂ ν ∂ξ ν = c ν ∂ ν ∂η ν ,∂ ν ∂ξ ν = c ν ∂ ν ∂η ν . (A9)It follows: ∂ ν ∂ξ ν ∂ ν ∂ξ ν ln (cid:10) e ξ c X + ξ c X (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 = c ν c ν ∂ ν ∂η ν ∂ ν ∂η ν ln (cid:10) e η X + η X (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) η = η =0 (A10)In the last equality above η and η are auxiliary, so that indeed: κ (( c X ) ν , ( c X ) ν ) = c ν c ν κ ( X ν , X ν ) . (A11)The above proof generalizes trivially for more than two random observables.3
5. Proof of multilinearity
To demonstrate all conceptual steps needed to prove multilinearity, it suffices to evaluate general multivariatedefinitions in Eqs. (10) and (12) for the following case: κ ( X + X , Z ν ) = ∂∂ξ ∂ ν ∂ξ ν ln (cid:68) e ξ ( X + X )+ ξ Z ) (cid:69)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 . (A12)Doing derivative only with respect to ξ , it follows: κ ( X + X , Z ν ) = ∂ ν ∂ξ ν (cid:10) ( X + X ) e ξ ( X + X )+ ξ Z ) (cid:11)(cid:10) e ξ ( X + X )+ ξ Z ) (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 = ∂ ν ∂ξ ν (cid:10) X e ξ ( X + X )+ ξ Z ) (cid:11)(cid:10) e ξ ( X + X )+ ξ Z ) (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 + ∂ ν ∂ξ ν (cid:10) X e ξ ( X + X )+ ξ Z ) (cid:11)(cid:10) e ξ ( X + X )+ ξ Z ) (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 . (A13)Since the remaining derivatives act only on auxiliary variable ξ , We observe that ∂ ν ∂ξ ν (cid:10) X e ξ ( X + X )+ ξ Z ) (cid:11)(cid:10) e ξ ( X + X )+ ξ Z ) (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 = ∂ ν ∂ξ ν (cid:10) X e ξ X + ξ Z ) (cid:11)(cid:10) e ξ X + ξ Z ) (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 (A14)and analogously ∂ ν ∂ξ ν (cid:10) X e ξ ( X + X )+ ξ Z ) (cid:11)(cid:10) e ξ ( X + X )+ ξ Z ) (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 = ∂ ν ∂ξ ν (cid:10) X e ξ X + ξ Z ) (cid:11)(cid:10) e ξ X + ξ Z ) (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ =0 . (A15)It follows immediately: κ ( X + X , Z ν ) = κ ( X , Z ν ) + κ ( X , Z ν ) . (A16)The above proof generalizes trivially for more that two linear observables X i and more than one non-linear observable Z .
6. Proof of additivity
We start with the general multivariate definitions in Eqs. (10) and (12) and apply them for the univariate case. Itfollows: κ ( X N ) = ∂ N ∂ξ N ln (cid:10) e ξX (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ξ =0 , (A17)so that κ (( (cid:88) i X i ) N ) = ∂ N ∂ξ N ln (cid:68) e ξ (cid:80) i X i (cid:69)(cid:12)(cid:12)(cid:12)(cid:12) ξ =0 . (A18)If all X i are statistically independent, we can factorize the average in the above expression, therefore κ (( (cid:88) i X i ) N ) = ∂ N ∂ξ N ln (cid:89) i (cid:10) e ξX i (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ξ =0 = (cid:88) i ∂ N ∂ξ N ln (cid:10) e ξX i (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ξ =0 = (cid:88) i κ ( X Ni ) . (A19)4 ( v ) ( v ) ( v ) FIG. 7. Toy Monte Carlo study to compare variance of moments and cumulants, for the Bessel-Gaussian p.d.f. defined inEq. (B5). In each study parameters a and b were constrained by a + 2 b = 0 . v ∼ a = b (left panel); b) a = b/
10 (middle panal); c) a = 10 b (right panel).For all 3 cases we have obtained that the variance of moments is larger than the variance of cumulants. In particular, for a = b we have obtained Var[ κ ] / Var[ µ ] = 0 . κ ] / Var[ µ ] = 0 . κ ] / Var[ µ ] = 0 . a = b/
10 wehave obtained Var[ κ ] / Var[ µ ] = 0 . κ ] / Var[ µ ] = 0 . κ ] / Var[ µ ] = 0 . a = 10 b we haveobtained Var[ κ ] / Var[ µ ] = 0 . κ ] / Var[ µ ] = 0 . κ ] / Var[ µ ] = 0 . v , variance of both moments and cumulants was obtained with bootstrap technique using 10subsamples. Appendix B: Univariate moments and cumulants of flow amplitudes
For completeness sake, in this appendix we present univariate moments and cumulants of flow amplitudes. Weformulate the problem from an experimental point of view, according to which flow amplitudes can be estimatedreliably only with correlation techniques. Therefore, the simplest observable which can be measured with two- andmultiparticle azimuthal correlations is v n . We consider v n to be the fundamental observable of interest, and calculateits moments and cumulants for a few selected p.d.f.’s of flow magnitude fluctuations f ( v n ). Since there is 1-to-1mapping between moments and cumulants, in practice it suffices to measure only one set.The k -th moment of v n , µ k , can be defined and obtained as: µ k ≡ E [ v kn ] ≡ (cid:10) v kn (cid:11) ≡ (cid:90) v kn f ( v n ) dv n , (B1)where k = 1 , , ... . We remark that we can in the above expression use the p.d.f. f ( v n ) even though we are interestedin the stochastic properties of v n , because straight from the conservation of probability it follows that the expectationvalue is: E [ a ( x )] = (cid:90) ag ( a ) da = (cid:90) a ( x ) f ( x ) dx , (B2)for any function a ( x ) of the starting stochastic variable x [20].On the other hand, the first univariate cumulants of v n are: κ = (cid:10) v n (cid:11) ,κ = (cid:10) v n (cid:11) − (cid:10) v n (cid:11) ,κ = (cid:10) v n (cid:11) − (cid:10) v n (cid:11) (cid:10) v n (cid:11) + 2 (cid:10) v n (cid:11) ,κ = (cid:10) v n (cid:11) − (cid:10) v n (cid:11) (cid:10) v n (cid:11) − (cid:10) v n (cid:11) + 12 (cid:10) v n (cid:11) (cid:10) v n (cid:11) − (cid:10) v n (cid:11) . (B3)The above cumulants manifestly satisfy all mathematical properties of cumulants if the fundamental stochastic ob-servable is v n . For instance, for the shift-invariance we have ( α is an arbitrary constant): (cid:104) ( v n + α ) (cid:105) − (cid:104) v n + α (cid:105) = (cid:104) v n (cid:105) − (cid:104) v n (cid:105) , (B4)and similarly for other properties discussed in Sec. II. In this respect, these univariate cumulants are different fromthe ones used so far in the field and which were discussed in Sec. III B.We now calculate these new univariate moments and cumulants for a widely studied Bessel-Gaussian p.d.f. of flowfluctuations, which is defined as (we suppress harmonic n in the subscript to ease the notation): f ( v ) ≡ vb e − v a b I ( vab ) , (B5)5 Centrality percentile ] m ]/ V [ k V [ n = 2n = 3 FIG. 8. Realistic Monte Carlo study to compare variances of moments and cumulants, using MC-Glauber + VISH2+1 modeldescribed in Sec. VII. The study has been performed independently for elliptic flow v (blue curve) and triangular flow v (redcurve). For both harmonics, and in all centralities, we have that variance of cumulants, V [ κ ], is smaller than the variance ofmoments V [ µ ]. The variances shown here correspond to the cumulant (cid:10) v n (cid:11) − (cid:10) v n (cid:11) and to the moment (cid:10) v n (cid:11) , respectively. Inboth cases, variances have been calculated using the bootstrap technique with 10 subsamples. where a and b are constants, and I is the modified Bessel function of the first kind. To make all integrals analyticallytractable, we have used for the domain of v the interval [0 , ∞ ) (this is justified by the fact that f ( v ) ≈ v (cid:29) (cid:82) ∞ f ( v ) dv = 1, for any choice of constants a and b . For the first few moments of v we have obtained: µ = a + 2 b ,µ = a + 8 a b + 8 b ,µ = a + 18 a b + 72 a b + 48 b ,µ = a + 32 a b + 288 a b + 768 a b + 384 b , (B6)while for the first few cumulants we have: κ = a + 2 b ,κ = 4 b ( a + b ) ,κ = 8 b (3 a + 2 b ) ,κ = 96 b (2 a + b ) . (B7)We observe immediately one striking difference: for the case when a > a (cid:29) b , all higher-order momentsare dominated by the terms which depend only on a , while higher-order cumulants exhibit the same leading orderdependence on that parameter. Therefore, even by using this simple example, we demonstrate that the measurementsof higher-order cumulants can reveal independent information, i.e. the importance of sub-leading parameter b in thisconcrete example.While it was easy for the multivariate case to find many general advantages of cumulants over moments straightfrom definitions, for the univariate case it is not that straightforward. We have nevertheless accumulated a fewobservations which may be of relevance:1. An undesirable property of moments is that lower-order moments can dominate higher-order moments [48].Since higher-order cumulants are less degenerate, it can be therefore easier to extract new and independentinformation from higher-order cumulants than from higher-order moments, in the cases of practical interest;2. The sample statistics of cumulants typically have lower variance than the one of moments [48]. This is of greatrelevance for flow analyses in high-energy physics, because most of these studies are still dominated by largestatistical uncertainties, both among the theorists and experimentalists. We confirm this observation in a toyMonte Carlo study using Bessel-Gaussian p.d.f. defined in Eq. (B5) (see Fig. 7), and using realistic Monte Carlomodel MC-Glauber + VISH2+1 (see Fig. 8, the model description can be found in Sec. VII);63. It is a well known fact that for the Gaussian p.d.f. parameterized with mean µ and variance σ , the cumulantsare κ = µ , κ = σ , and κ k = 0 ∀ k ≥
3. Therefore, only higher-order cumulants can be naturally utilizedto quantify the difference between a distribution and its Gaussian approximation. We remark that this is trueonly if cumulants are given by Eqs. (B3), while the previous formulas used in the field for cumulants of flowamplitudes (see Eqs. (29) in Sec. III B) do not satisfy this property. For the recent studies on non-Gaussianityin this context we refer to Refs. [49, 50].
Appendix C: Demonstrations and estimators for the Asymmetric Cumulants
This Appendix presents the demonstrations of the properties required for AC , ( m, n ) to be a valid multivariatecumulant of the flow amplitudes squared according to the formalism of Kubo. In a second section, the experimentalestimators for the proposed ACs are summarised.
1. Demonstrations of the requirements a. Statistical independence
We consider the fluctuations of v m and v n to be completely uncorrelated. Equation (52) becomes thenAC , ( m, n ) = (cid:104) v m (cid:105)(cid:104) v n (cid:105) − (cid:104) v m (cid:105)(cid:104) v n (cid:105) − (cid:104) v m (cid:105)(cid:104) v n (cid:105)(cid:104) v m (cid:105) + 2 (cid:104) v m (cid:105) (cid:104) v n (cid:105) = 0 , (C1)as expected in absence of genuine correlations between the two observables. b. Reduction We now set the two flow amplitudes to the same quantity, i.e. v m = v n ≡ v . This implies thatAC , ( m, n ) = (cid:104) v (cid:105) − (cid:104) v (cid:105)(cid:104) v (cid:105) − (cid:104) v (cid:105)(cid:104) v (cid:105) + 2 (cid:104) v (cid:105) (cid:104) v (cid:105) = (cid:104) v (cid:105) − (cid:104) v (cid:105)(cid:104) v (cid:105) + 2 (cid:104) v (cid:105) . (C2)This is the expansion for κ with v as the fundamental stochastic variable, and therefore, it is a valid univariatecumulant. It has to be noted that SC( k, l, m ), which is also of order three, leads to the same cumulant when reducedas well. c. Semi-invariance Let us consider two constants c m and c n . We can now express the semi-invariance property as (cid:104) ( v m + c m ) ( v n + c n ) (cid:105) c = (cid:104) ( v m + c m ) ( v n + c n ) (cid:105) − (cid:104) ( v m + c m ) (cid:105)(cid:104) v n + c n (cid:105)− (cid:104) ( v m + c m )( v n + c n ) (cid:105)(cid:104) v m + c m (cid:105) + 2 (cid:104) v m + c m (cid:105) (cid:104) v n + c n (cid:105) = (cid:104) ( v m + 2 c m v m + c m )( v n + c n ) (cid:105)− (cid:104) v m + 2 c m v m + c m (cid:105)(cid:104) v n + c n (cid:105)− (cid:0) (cid:104) v m v n + c m v n + c n v m + c m c n (cid:105)(cid:104) v m + c m (cid:105) (cid:1) + 2 (cid:0) ( (cid:104) v m (cid:105) + 2 c m (cid:104) v m (cid:105) + c m ) (cid:104) v n + c n (cid:105) (cid:1) = AC , ( m, n )+ 2 c m (cid:0) (cid:104) v m v n (cid:105) − (cid:104) v m (cid:105)(cid:104) v n (cid:105) − (cid:104) v m v n (cid:105) + 2 (cid:104) v m (cid:105)(cid:104) v n (cid:105) (cid:1) + c n (cid:0) (cid:104) v m (cid:105) − (cid:104) v m (cid:105) + 2 (cid:104) v m (cid:105) − (cid:104) v m (cid:105) (cid:1) + c m (cid:0) (cid:104) v n (cid:105) − (cid:104) v n (cid:105) (cid:1) + 2 c m c n (cid:0) (cid:104) v m (cid:105) − (cid:104) v m (cid:105) (cid:1) + c m c n (3 − , ( m, n ) . (C3)7 d. Homogeneity With the two constants c m and c n , the homogeneity can be shown with (cid:104) ( c m v m ) ( c n v n ) (cid:105) c = (cid:104) c m v m c n v n (cid:105) − (cid:104) c m v m (cid:105)(cid:104) c n v n (cid:105)− (cid:104) c m v m c n v n (cid:105)(cid:104) c m v m (cid:105) + 2 (cid:104) c m v m (cid:105) (cid:104) c n v n (cid:105) = c m c n AC , ( m, n ) . (C4) e. Multilinearity We consider now the three flow amplitudes v m , v n and v k . The multilinearity of the linear moment becomesAC , ( m, n + k ) = (cid:104) v m ( v n + v k ) (cid:105) − (cid:104) v m (cid:105)(cid:104) v n + v k (cid:105)− (cid:104) v m ( v n + v k ) (cid:105)(cid:104) v m (cid:105) + 2 (cid:104) v m (cid:105) (cid:104) v n + v k (cid:105) = AC , ( m, n ) + AC , ( m, k ) . (cid:3) (C5)With the help of the demonstrations above, we have shown that the expressions of AC , ( m, n ) given by Eq. (52)is a valid multivariate cumulant with v m and v n the fundamental observables.
2. Experimental estimators
In this section, we give the final combinations of azimuthal correlators used to estimate experimentally the ACsproposed in Sec. V. AC , ( m, n ) = (cid:104)(cid:104) e i ( mϕ + mϕ + nϕ − mϕ − mϕ − nϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( mϕ + mϕ − mϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( nϕ − nϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( mϕ + nϕ − mϕ − nϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) + 2 (cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) (cid:104)(cid:104) e i ( nϕ − nϕ ) (cid:105)(cid:105) (C6)AC , ( m, n ) = (cid:104)(cid:104) e i ( mϕ + mϕ + mϕ + nϕ − mϕ − mϕ − mϕ − nϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( mϕ + mϕ + mϕ − mϕ − mϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( nϕ − nϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( mϕ + mϕ − mϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ + nϕ − mϕ − nϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( mϕ + mϕ + nϕ − mϕ − mϕ − nϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) + 6 (cid:104)(cid:104) e i ( mϕ + mϕ − mϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( nϕ − nϕ ) (cid:105)(cid:105) + 6 (cid:104)(cid:104) e i ( mϕ + nϕ − mϕ − nϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) − (cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) (cid:104)(cid:104) e i ( nϕ − nϕ ) (cid:105)(cid:105) (C7)8AC , ( m, n ) = (cid:104)(cid:104) e i ( mϕ + mϕ + mϕ + mϕ + nϕ − mϕ − mϕ − mϕ − mϕ − nϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( mϕ + mϕ + mϕ + mϕ − mϕ − mϕ − mϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( nϕ − nϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( mϕ + nϕ − mϕ − nϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ + mϕ + mϕ − mϕ − mϕ − mϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( mϕ + mϕ + nϕ − mϕ − mϕ − nϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ + mϕ − mϕ − mϕ ) (cid:105)(cid:105) + 6 (cid:104)(cid:104) e i ( mϕ + mϕ − mϕ − mϕ ) (cid:105)(cid:105) (cid:104)(cid:104) e i ( nϕ − nϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( mϕ + mϕ + mϕ + nϕ − mϕ − mϕ − mϕ − nϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) + 8 (cid:104)(cid:104) e i ( mϕ + mϕ + mϕ − mϕ − mϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( nϕ − nϕ ) (cid:105)(cid:105) + 24 (cid:104)(cid:104) e i ( mϕ + nϕ − mϕ − nϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ + mϕ − mϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) + 12 (cid:104)(cid:104) e i ( mϕ + mϕ + nϕ − mϕ − mϕ − nϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) − (cid:104)(cid:104) e i ( mϕ + mϕ − mϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) (cid:104)(cid:104) e i ( nϕ − nϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( mϕ + nϕ − mϕ − nϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) + 24 (cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) (cid:104)(cid:104) e i ( nϕ − nϕ ) (cid:105)(cid:105) (C8)AC , , ( k, l, m ) = (cid:104)(cid:104) e i ( kϕ + kϕ + lϕ + mϕ − kϕ − kϕ − lϕ − mϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( kϕ + kϕ + lϕ − kϕ − kϕ − lϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( kϕ + kϕ + mϕ − kϕ − kϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( lϕ − lϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( kϕ + kϕ − kϕ − kϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( lϕ + mϕ − lϕ − mϕ ) (cid:105)(cid:105) + 2 (cid:104)(cid:104) e i ( kϕ + kϕ − kϕ − kϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( lϕ − lϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( kϕ + lϕ − kϕ − lϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( kϕ + mϕ − kϕ − mϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( kϕ + lϕ + mϕ − kϕ − lϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( kϕ − kϕ ) (cid:105)(cid:105) + 4 (cid:104)(cid:104) e i ( kϕ + lϕ − kϕ − lϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( kϕ − kϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) + 4 (cid:104)(cid:104) e i ( kϕ + mϕ − kϕ − mϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( kϕ − kϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( lϕ − lϕ ) (cid:105)(cid:105) + 2 (cid:104)(cid:104) e i ( kϕ − kϕ ) (cid:105)(cid:105) (cid:104)(cid:104) e i ( lϕ + mϕ − lϕ − mϕ ) (cid:105)(cid:105)− (cid:104)(cid:104) e i ( kϕ − kϕ ) (cid:105)(cid:105) (cid:104)(cid:104) e i ( lϕ − lϕ ) (cid:105)(cid:105)(cid:104)(cid:104) e i ( mϕ − mϕ ) (cid:105)(cid:105) (C9) Appendix D: Cumulant properties for the Cumulants of Symmetry Plane Correlations
In this section we demonstrate the cumulant properties ofCSC ( bδ c,d , kδ l,m ) = (cid:68) e i ( bδ c,d + kδ l,m ) (cid:69) − (cid:10) e ibδ c,d (cid:11) (cid:10) e ikδ l,m (cid:11) with b (cid:54) = k . (D1)
1. Statistical independence
Assuming that δ c,d and δ l,m are independent from each other we getCSC ( bδ c,d , kδ l,m ) = (cid:68) e i ( bδ c,d + kδ l,m ) (cid:69) − (cid:10) e ibδ c,d (cid:11) (cid:10) e ikδ l,m (cid:11) = (cid:10) e ibδ c,d (cid:11) (cid:10) e ikδ l,m (cid:11) − (cid:10) e ibδ c,d (cid:11) (cid:10) e ikδ l,m (cid:11) = 0 . (D2)
2. Reduction
If we consider that e ibδ c,d = e ikδ l,m , we obtainCSC ( bδ c,d , bδ c,d ) = (cid:10) e i bδ c,d (cid:11) − (cid:0)(cid:10) e ibδ c,d (cid:11)(cid:1) (D3)which is nothing else than κ .9
3. Semi-invariance
Consider two constants c and c (cid:10)(cid:0) e ibδ c,d + c (cid:1) (cid:0) e ikδ l,m + c (cid:1)(cid:11) − (cid:10)(cid:0) e ibδ c,d + c (cid:1)(cid:11) (cid:10)(cid:0) e ikδ l,m + c (cid:1)(cid:11) = (cid:68) e i ( bδ c,d + kδ l,m ) (cid:69) + c (cid:10) e ikδ l,m (cid:11) + c (cid:10) e ibδ c,d (cid:11) + c c − (cid:0)(cid:10) e ibδ c,d (cid:11) (cid:10) e ikδ l,m (cid:11) + c (cid:10) e ikδ l,m (cid:11) + c (cid:10) e ibδ c,d (cid:11) + c c (cid:1) = (cid:68) e i ( bδ c,d + kδ l,m ) (cid:69) − (cid:10) e ibδ c,d (cid:11) (cid:10) e ikδ l,m (cid:11) = CSC ( bδ c,d , kδ l,m ) . (D4)Thus, semi-invariance is fulfilled.
4. Homogeneity
Considering two constants c and c ,we see (cid:10)(cid:0) c e ibδ c,d (cid:1) (cid:0) c e ikδ l,m (cid:1)(cid:11) − (cid:10)(cid:0) c e ibδ c,d (cid:1)(cid:11) (cid:10)(cid:0) c e ikδ l,m (cid:1)(cid:11) = c c CSC ( bδ c,d , kδ l,m ) , (D5)and thus that the homogeneity requirement is fulfilled.
5. Multilinearity
Consider an additional stochastic observable e ixδ y,z with x (cid:54) = b . Then, we see that the multi-linearity condition isfulfilled as (cid:10) e ibδ c,d (cid:0) e ikδ l,m + e ixδ y,z (cid:1)(cid:11) − (cid:10) e ibδ c,d (cid:11) (cid:10)(cid:0) e ikδ l,m + e ixδ y,z (cid:1)(cid:11) = (cid:68) e i ( bδ c,d + kδ l,m ) (cid:69) + (cid:68) e i ( bδ c,d + xδ y,z ) (cid:69) − (cid:0)(cid:10) e ibδ c,d (cid:11) (cid:10) e ikδ l,m (cid:11) + (cid:10) e ibδ c,d (cid:11) (cid:10) e ixδ y,z (cid:11)(cid:1) = (cid:16)(cid:68) e i ( bδ c,d + kδ l,m ) (cid:69) − (cid:10) e ibδ c,d (cid:11) (cid:10) e ikδ l,m (cid:11)(cid:17) + (cid:16)(cid:68) e i ( bδ c,d + xδ y,z ) (cid:69) − (cid:10) e ibδ c,d (cid:11) (cid:10) e ixδ y,z (cid:11)(cid:17) = CSC ( bδ c,d , kδ l,m ) + CSC ( bδ c,d , xδ y,z ) . (D6) Appendix E: Toy Monte Carlo studies for the Cumulants of Symmetry Plane Correlations
We set up a Toy Monte Carlo study to check the behaviour of the cumulant defined in Eq. (68). In detail, we willstudy the following, exemplary cumulantCSC (4 δ , , δ , ) = (cid:68) e i (4 δ , +6 δ , ) (cid:69) − (cid:10) e i δ , (cid:11) (cid:10) e i δ , (cid:11) = (cid:68) e i (4(Ψ − Ψ )+6(Ψ − Ψ )) (cid:69) − (cid:68) e i − Ψ ) (cid:69) (cid:68) e i − Ψ ) (cid:69) . (E1)We study this cumulant for six different multiplicities M = 50 , , , , , N ev = 10 events for each multiplicity. In each event, and for each studied multiplicity, we sample M azimuthalangles ϕ from the following p.d.f. f ( ϕ ) = 12 π [1 + 2 ( v cos [( ϕ − Ψ )] + v cos [2 ( ϕ − Ψ )] + v cos [3 ( ϕ − Ψ )]+ v cos [4 ( ϕ − Ψ )] + + v cos [5 ( ϕ − Ψ )] + v cos [6 ( ϕ − Ψ )])] . (E2)As there is no exact way to experimentally measure the needed expressions of symmetry plane correlations, weuse the following trick in the Toy Monte Carlo study: We set constant values for the flow amplitudes, in particular0 v = v = 0 and v = v = v = v = 0 .
1. As we cannot measure single symmetry planes (or their correlations), weuse Eq. (2) and employ multi-particle azimuthal correlators to measure the following expressions (cid:68) v v v v e i (4 δ , +6 δ , ) (cid:69) , (E3) (cid:10) v v e i δ , (cid:11) , (E4) (cid:10) v v e i δ , (cid:11) , (E5)which will then have a prefactor of the corresponding flow amplitudes. Afterwards, we divide the obtained resultsby the flow amplitude prefactor, which we are allowed to do as they are constant and uncorrelated to the symmetryplanes. Thus, we obtain the terms needed for Eq. (E1).We study the following five set-ups, which represented the five scenarios presented before:1. All symmetry planes Ψ , Ψ , Ψ and Ψ are uncorrelated and fluctuate within [0 , π ). Therefore, δ , and δ , fluctuate randomly and independent in the interval [0 , π ).2. Ψ fluctuating with Ψ ∈ (0 , π ] and Ψ = Ψ + π . Therefore, δ , = π = const. Ψ fluctuating with Ψ ∈ [0 , π ) and Ψ = Ψ + π . Therefore, δ , = π = const.
3. Ψ fluctuating with Ψ ∈ [0 , π ) and Ψ = Ψ + a with a fluctuating a ∈ (cid:2) , π (cid:3) . Ψ fluctuating with Ψ ∈ [0 , π )and Ψ = Ψ + b with b fluctuating b ∈ (cid:2) , π (cid:3) .4. δ , and δ , fluctuate randomly in the interval [0 , π ), but we have the following correlations:Ψ fluctuating with Ψ ∈ [0 , π ) and Ψ = Ψ + a with a fluctuating a ∈ (cid:2) , π (cid:3) . Ψ fluctuating withΨ ∈ [0 , π )and Ψ = Ψ + b with b fluctuating b ∈ (cid:2) , π (cid:3) .5. Ψ fluctuating with Ψ ∈ [0 , π ) and Ψ = Ψ + a with a fluctuating a ∈ (cid:2) , π (cid:3) . Ψ fluctuating withΨ ∈ [0 , π )and Ψ = Ψ + a + π √ with a having the same value per event as for δ , .We conclude this list with the following remark: If we sample a symmetry plane Ψ j = Ψ i + h with h as an additiveterm and Ψ i sample randomly from (0 , π ], and Ψ j turns out to be greater than 2 π , we bring it back to the intervalbetween 0 and 2 π .The results of this Toy Monte Carlo study can be seen in Fig. 9. They show that the studied CSC observable iscompatible with zero in the cases 1-4 and non-zero for case 5, as it was expected from the described scenarios. Thus,this observable is a valid cumulant of symmetry plane correlations.1 Multiplicity - - - - Case 1
Re (exp.)Im (exp.) Re (theo.)Im (theo.) 0 200 400 600 800 1000
Multiplicity - - - - Case 2
Multiplicity - - - - Case 3
Multiplicity - - - - Case 4
Multiplicity - - - - Case 5
FIG. 9. Results for the TMC for the 5 scenarios of CSC.[1] J.-Y. Ollitrault, Phys. Rev.
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