Collectivity in large and small systems formed in ultrarelativistic collisions
EEPJ manuscript No. (will be inserted by the editor)
Collectivity in large and small systems formedin ultrarelativistic collisions
Rajeev S. Bhalerao , a Department of Physics, Indian Institute of Science Education and Research (IISER), HomiBhabha Road, Pune 411008, India
Abstract.
Collective flow of the final-state hadrons observed in ultra-relativistic heavy-ion collisions or even in smaller systems formed inhigh-multiplicity pp and p/d/ He-nucleus collisions is one of the mostimportant diagnostic tools to probe the initial state of the system andto shed light on the properties of the short-lived, strongly-interactingmany-body state formed in these collisions. Limited, in the initial years,to the study of mainly the directed and elliptic flows – the first twoFourier harmonics of the single-particle azimuthal distribution – thisfield has evolved in recent years into a much richer area of activity. Thisincludes not only higher Fourier harmonics and multiparticle cumu-lants, but also a variety of other related observables, such as the ridgeseen in two-particle correlations, flow decorrelation, symmetric cumu-lants and event-plane correlators which measure correlations betweenthe magnitudes or phases of the complex flows in different harmonics,coefficients that measure the nonlinear hydrodynamic response, statis-tical properties, such as the non-Gaussianity of the flow fluctuations,etc. We present a Tutorial Review of the modern flow picture and thevarious aspects of the collectivity – an emergent phenomenon in quan-tum chromodynamics. a e-mail: [email protected] a r X i v : . [ nu c l - t h ] O c t Will be inserted by the editor “The aim of theory really is, to a great extent, that of systematically organizingpast experience in such a way that the next generation, our students and their studentsand so on, will be able to absorb the essential aspects in as painless a way as possible,and this is the only way in which you can go on cumulatively building up any kindof scientific activity without eventually coming to a dead end.”— Michael Atiyah, Fields Medalist, in
How research is carried out ill be inserted by the editor 3
Contents1. Introduction2. Modern Flow Picture2.1 Characterization of the initial state2.2 Characterization of the final state2.3 Flow measurements2.4 Probability density function (PDF)2.5 Ridge2.6 Flow decorrelation and factorization breaking3. Observables with Mixed Harmonics3.1 Symmetric cumulants3.2 Event-plane correlators3.3 Nonlinear flow modes and mode coupling or mixing3.3.1 Connections between seemingly unrelated observables4. Collectivity4.1 Origin of collectivity4.1.1 Hydrodynamics4.1.2 Anisotropic parton-escape models4.1.3 CGC effective field theory5. Conclusions6. Acknowledgements7. Appendix A7.1 Moments and cumulants of a probability distribution7.2 Moment-generating function M ( t )7.3 Cumulant-generating function K ( t )8. Appendix B8.1 Correlation functions and cumulants9. Appendix C9.1 Gaussian or normal distribution in 1D9.2 Gaussian or normal distribution in 2DReferences The physics of heavy-ion collisions or even of relativistic heavy-ion collisions has along history, see e.g. [1,2]. However, the modern era in this field began with theadvent of the Relativistic Heavy-Ion Collider (RHIC) at BNL, USA in 2000. Thefield was further enriched when the Large Hadron Collider (LHC) at CERN becameoperational in 2010. The big ideas driving this field are: to test the predictions of thenonperturbative Quantum Chromodynamics (QCD), to study the equilibrium andnonequilibrium (transport) properties of the quark-gluon plasma (QGP), to map outthe QCD phase diagram qualitatively as well as quantitatively, etc. The relativisticnucleus-nucleus collisions (and arguably even smaller systems being studied at RHICand LHC) constitute the only available tool to produce QGP in the laboratory.This pedagogical review is restricted to the issue of collectivity as seen in the largeand small systems formed in the nucleus-nucleus, hadron-nucleus or hadron-hadroncollisions at high energies. The emphasis will be on explaining the mathematicalframework underlying the discussion of the various aspects of the collective flow,which although quite straightforward, might appear intriguing to the uninitiated.Although attempt will be made to display the relevant experimental data, this is not
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Fig. 1. (colour online) Geometry of the collision. Filled circles denote the participants andopen circles the spectators. x (cid:48) and y (cid:48) are the principal axes of inertia of the participant zone.Figure adapted from [3]. a comprehensive review of the phenomenology of the wealth of data on collectivityaccumulated over the years.In the next section we present the modern flow picture. Flow observables withmixed harmonics are discussed in Sec. 3. Collectivity in small and large systems isdiscussed in Sec. 4, which is followed by Conclusions in Sec. 5. Mathematical prelim-inaries are presented in Appendices A to C. Reader unfamiliar with these is advisedto go through the appendices before reading Sec. 2. We begin by describing the geometry of the collision (of two identical spherical nuclei)and the various planes that go with it. Two nuclei are taken to approach each otherparallel to the z (or longitudinal or beam) axis. The origin is taken at the midpointof the impact parameter vector. The standard convention for the x and y axes is asshown in Fig. 1. Due to quantum fluctuations in the wave functions of the incomingnuclei, the participant zone may be shifted and/or tilted with respect to the xy frame. x (cid:48) , y (cid:48) , z (cid:48) are the principal axes of inertia of the participant zone. x (cid:48) , in particular, isthe direction of the short axis of the participating nucleon distribution. – Reaction Plane (RP): xz plane. It is the plane determined by the impact param-eter vector and the beam axis. Orientation of the reaction plane is not directlymeasurable. – Participant Plane (PP): x (cid:48) z (cid:48) plane. Same as the RP if there are no initial-statefluctuations. Participant plane is also not directly measurable. – Event Plane (EP): Estimate of the PP, obtained by measuring the direction ofmaximum final-state particle density. If there are no fluctuations, EP is an estimateof RP. It is established by hydrodynamic calculations and is also expected naturally thatthe initial state of collision governs the final state on an event-by-event basis. In thenext two subsections, we describe how the geometries of the initial and final states,in the coordinate and momentum spaces respectively, are captured in terms of a few See Eqs. (1) and (4) for definitions of PP ( Φ n ) and EP ( Ψ n ) specific to harmonic n .ill be inserted by the editor 5 parameters. One then expects the empirically extracted parameters of the final state(“flow coefficients”) to depend on the theoretically calculated parameters of the initialstate (“eccentricities”), in a hopefully simple way. Recall the old definition of the eccentricity ε ≡ (cid:10) y − x (cid:11) / (cid:10) y + x (cid:11) , which is real.In modern parlance, the complex eccentricity vector E n is defined as [4,5] E n ≡ ε n e inΦ n ≡ − (cid:104) z n (cid:105)(cid:104)| z | n (cid:105) = − (cid:10) r n e inϕ (cid:11) (cid:104) r n (cid:105) = − (cid:82) z n e ( z, τ ) r dr dϕ (cid:82) | z | n e ( z, τ ) r dr dϕ , n ≥ , (1)where ε n is the magnitude and Φ n determines the phase of the complex vector E n ,the complex variable z ≡ x + iy = re iϕ refers to a point in the transverse or xy plane, e ( z, τ ) is the energy (or entropy) density at midrapidity at the initial time τ , i.e., shortly after the collision and (cid:104)· · ·(cid:105) denotes the e ( z, τ )-weighted average overthe transverse plane in a single event after centering: (cid:104) z (cid:105) = (cid:10) re iϕ (cid:11) = 0. Note that0 ≤ ε n ≤ Φ n is called the participant-plane angle in the n -th harmonic. ε n and Φ n are given by ε n = (cid:113) (cid:104) r n cos nϕ (cid:105) + (cid:104) r n sin nϕ (cid:105) (cid:104) r n (cid:105) ,Φ n = 1 n tan − (cid:104) r n sin nϕ (cid:105)(cid:104) r n cos nϕ (cid:105) , (2)modulo 2 π/n , because the n -th order harmonic has an n -fold symmetry in azimuth.It is easy to check that for n = 2, with Φ oriented along the x axis, both Eqs.(1) and (2) yield the old definition of the eccentricity ε ≡ (cid:10) y − x (cid:11) / (cid:10) y + x (cid:11) .Eccentricity ε n cannot be determined experimentally. However, given a model for theinitial state (e.g., Glauber model), ε n can be calculated from the positions ( r, ϕ ) ofthe participating nucleons (or partons) in the transverse plane. The magnitude ε n and the phase Φ n fluctuate from event to event.It is sometimes useful to replace the above definition of E n based on momentsby a definition based on cumulants, because cumulants subtract off the lower-ordercorrelations and retain only the irreducible correlations [6]. The spatial azimuthalanisotropies then become: E = − (cid:10) z (cid:11) (cid:104) r (cid:105) , E = − (cid:10) z (cid:11) (cid:104) r (cid:105) , E = − (cid:104) r (cid:105) (cid:104)(cid:10) z (cid:11) − (cid:10) z (cid:11) (cid:105) , E = − (cid:104) r (cid:105) (cid:2)(cid:10) z (cid:11) − (cid:10) z (cid:11) (cid:10) z (cid:11)(cid:3) , E = − (cid:104) r (cid:105) (cid:104)(cid:10) z (cid:11) − (cid:10) z (cid:11) (cid:10) z (cid:11) − (cid:10) z (cid:11) + 30 (cid:10) z (cid:11) (cid:105) . (3) n = 1 forms a special case [5]. Will be inserted by the editor These will be needed later in Sec. 3.3.
Exercise 1:
Understand Eqs. (3) in the light of the expressions of cumulants givenin Appendix A.
Anisotropy of the azimuthal distribution of the final-state particles was proposedas a signature of the transverse collective flow in Ref. [7]. Characterization of theanisotropy by a Fourier decomposition was proposed in Ref. [8]. The modern flowpicture is a model in which particles in the final state are emitted randomly andindependently according to some underlying single-particle probability distribution P ( φ ) that fluctuates from event to event [9]: P ( φ ) = 12 π ∞ (cid:88) n = −∞ V n e − inφ = 12 π ∞ (cid:88) n = −∞ v n e − in ( φ − Ψ n ) , (4)where φ is the azimuthal angle of the momentum of the outgoing particle and V n isa complex Fourier flow coefficient whose magnitude ( v n ) and phase ( Ψ n ) fluctuatefrom event to event. The angle Ψ n is called the event-plane angle or symmetry-planeangle or reference angle for harmonic n . It is easy to check that V n ≡ v n exp( inΨ n ) = (cid:104) exp( inφ ) (cid:105) , (5)where (cid:104)· · ·(cid:105) denotes an average over the probability density in a single event. As v n is real, this also means v n = (cid:104) cos n ( φ − Ψ n ) (cid:105) and (cid:104) sin n ( φ − Ψ n ) (cid:105) = 0. The sine termvanishes because of the symmetry with respect to the event plane. The event-planeangle Ψ n can be determined using Eq. (5): Ψ n = 1 n tan − (cid:104) sin nφ (cid:105)(cid:104) cos nφ (cid:105) , (modulo 2 π/ n) . (6)Experimentally, the flow vector Q n defined as Q n ≡ (cid:80) i w i exp( inφ i ) / (cid:80) i w i providesan estimate of the vector V n . Here φ i are the azimuthal angles of the momenta ofthe detected particles and w i are the weights that are generally inserted to opti-mize the estimate by accounting for detector nonuniformity and tracking inefficiency.Experimentally, Ψ n is estimated as [10]: Ψ n = 1 n tan − (cid:80) i w i sin nφ i (cid:80) i w i cos nφ i , (modulo 2 π/ n) . (7) Exercise 2 : Recall that the event plane is an estimate of the participant plane.Because the number of particles emitted in an event is finite, the event-plane angle Ψ n determined from Eq. (7) fluctuates from event to event around the participant-planeangle Φ n . Hence the true flow v n would differ from the observed flow v obs n . Show that v n = v obs n /R n , where the ‘resolution factor’ R n is given by R n = (cid:104) cos n ( Ψ n − Φ n ) (cid:105) < V n is sometimes called a flow vector, and ‘flow’ and ‘azimuthal anisotropy’ are often usedsynonymously.ill be inserted by the editor 7 Defining V − n = V ∗ n (equivalently, v n = v − n , Ψ n = Ψ − n ) and using V = v = 1,one can write P ( φ ) also as P ( φ ) = 12 π (cid:32) ∞ (cid:88) n =1 v n cos n ( φ − Ψ n ) (cid:33) , (8)which is real. As a result, in the final state, the azimuthal distribution of particlesthat fluctuates event to event is dNdφ = N π (cid:32) ∞ (cid:88) n =1 v n cos n ( φ − Ψ n ) (cid:33) . (9)The angle-independent leading term is called the radial flow. The coefficients v n parametrize the momentum anisotropy of the final-state particle yield. They go by thenames directed or dipolar ( n = 1), elliptic ( n = 2), triangular ( n = 3), quadrangular( n = 4), pentagonal ( n = 5), · · · flows. Recently, ALICE collaboration has presentedresults on flow harmonics up to n = 9 [12].In general, v n and Ψ n depend on the transverse momentum ( p T ) and pseudora-pidity ( η ). Thus the event-plane angle is not a unique angle for the entire event. It isso only for nonfluctuating smooth initial conditions. The p T dependence of Ψ n mayarise due to differently-oriented and different-sized hot spots in the transverse planeemitting particles in different directions with different (cid:104) p T (cid:105) [13]. The η dependence of Ψ n may arise due to a fireball that is twisted or torqued in the longitudinal direction[14].Note that the ε n coefficients characterize the various shape components of thefluctuating initial profile in the coordinate space, just as the v n coefficients characterizethose of the fluctuating final state in the momentum space. Theoretically, V n ( p T , η ) ≡ v n e inΨ n ( p T , η ) = (cid:82) dφe inφ dNp T dp T dηdφ (cid:82) dφ dNp T dp T dηdφ = (cid:10) e inφ (cid:11) , (10)where dN/p T dp T dηdφ is the particle distribution — usually charged hadron distri-bution — in an event. V n ( p T ) and V n ( η ) can be defined similarly, by carrying outintegrations over η or p T , respectively, over appropriate ranges, besides the integra-tion over φ . Fully integrated flow V n results from the integration over both p T and η . Since the orientations of the reaction plane and the participant plane are un-known, flow v n is usually measured experimentally from event-averaged azimuthalcorrelations between outgoing particles. Determination of the flow coefficients usingmultiparticle correlations proceeds as follows. Two-, four-. six- and eight-particle az-imuthal correlations are defined as (cid:104)(cid:104) (cid:105)(cid:105) = (cid:104)(cid:104) e in ( φ − φ ) (cid:105)(cid:105) , (cid:104)(cid:104) (cid:105)(cid:105) = (cid:104)(cid:104) e in ( φ + φ − φ − φ ) (cid:105)(cid:105) , (cid:104)(cid:104) (cid:105)(cid:105) = (cid:104)(cid:104) e in ( φ + φ + φ − φ − φ − φ ) (cid:105)(cid:105) , (cid:104)(cid:104) (cid:105)(cid:105) = (cid:104)(cid:104) e in ( φ + φ + φ + φ − φ − φ − φ − φ ) (cid:105)(cid:105) , (11) These names are suggestive of the shapes of the polar plots r = 1 + 2 v n cos nφ , for0 < v n (cid:28)
1. Will be inserted by the editor where (cid:104)(cid:104)· · · (cid:105)(cid:105) denotes averaging over all multiplets in a single collision event and thenover all events in a given centrality class. Note that all these ‘observables’ are invariantunder rotation in the azimuthal plane, as they should be. (As a counterexample,consider (cid:104)(cid:104) e in ( φ + φ − φ ) (cid:105)(cid:105) . To make it invariant, we may replace φ by 2 φ , which,however, mixes harmonics n and 2 n . Mixed-harmonic observables will be discussedin Sec. 3.) Multiparticle cumulants are defined as [15] c n { } = (cid:104)(cid:104) (cid:105)(cid:105) ,c n { } = (cid:104)(cid:104) (cid:105)(cid:105) − (cid:104)(cid:104) (cid:105)(cid:105) ,c n { } = (cid:104)(cid:104) (cid:105)(cid:105) − (cid:104)(cid:104) (cid:105)(cid:105)(cid:104)(cid:104) (cid:105)(cid:105) + 12 (cid:104)(cid:104) (cid:105)(cid:105) ,c n { } = (cid:104)(cid:104) (cid:105)(cid:105) − (cid:104)(cid:104) (cid:105)(cid:105)(cid:104)(cid:104) (cid:105)(cid:105) − (cid:104)(cid:104) (cid:105)(cid:105) + 144 (cid:104)(cid:104) (cid:105)(cid:105) (cid:104)(cid:104) (cid:105)(cid:105) − (cid:104)(cid:104) (cid:105)(cid:105) . (12) Exercise 3:
Understand Eqs. (12) in the light of the definitions of cumulants givenin Eq. (B2) of Appendix B.The two-particle correlation can be rewritten as (cid:104)(cid:104) (cid:105)(cid:105) = (cid:104)(cid:104) e in ( φ − φ ) (cid:105)(cid:105) = (cid:104)(cid:104) e in ( φ − Ψ n ) e − in ( φ − Ψ n ) (cid:105)(cid:105) = (cid:104)(cid:104) e in ( φ − Ψ n ) (cid:105)(cid:104) e − in ( φ − Ψ n ) (cid:105)(cid:105) = (cid:10) v n (cid:11) , (13)where in the first line we assumed that Ψ n is a global phase angle for all the particlesselected for averaging, and in the second line we assumed that there are no nonflowcorrelations , or equivalently, that the angles ( φ − Ψ n ) and ( φ − Ψ n ) are statisti-cally independent. Note that any dependence on the symmetry plane is eliminatedby construction. The higher-order correlations (cid:104)(cid:104) (cid:105)(cid:105) , (cid:104)(cid:104) (cid:105)(cid:105) and (cid:104)(cid:104) (cid:105)(cid:105) can be treatedsimilarly. Thus the multiparticle cumulants take the form: 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) . (14)Finally, the flow coefficients are given by v n { } = (cid:112) c n { } = (cid:112) (cid:104) v n (cid:105) ,v n { } = (cid:112) − c n { } = (cid:112) (cid:104) v n (cid:105) − (cid:104) v n (cid:105) ,v n { } = (cid:114) c n { } = (cid:114)
14 ( (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) ) ,v n { } = (cid:114) − c n { } = (cid:114) −
133 ( (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) ) . (15) Nonflow correlations are not related to the initial-state geometry and hence not associ-ated with the symmetry plane Ψ n , but arise due to jets, particle decays, etc. They are ofshort range. v n { m } are also called multiparticle cumulants of order m of the flow v n .ill be inserted by the editor 9 The coefficients −
1, 1/4 and − /
33 appearing in front of c n { } , c n { } and c n { } ,respectively, in Eqs. (15), can be understood as follows: If the magnitude of the flowvector does not fluctuate event-to-event, then (cid:10) v kn (cid:11) = v kn , and it is clear from Eq. (14)that c n { } = v n ( > c n { } = − v n ( < c n { } = 4 v n ( > c n { } = − v n ( < v n { } = v n { } = v n { } = v n { } = v n , as expected. An advantage of multiparticle cumulants v n { m } is that they suppressnonflow contribution.For the sake of completeness, we present the inverse of Eqs. (15): (cid:104) v n (cid:105) = v n { } , (cid:104) v n (cid:105) = − v n { } + 2 v n { } , (cid:104) v n (cid:105) = 4 v n { } − v n { } v n { } + 6 v n { } , (cid:104) v n (cid:105) = − v n { } + 64 v n { } v n { } + 18 v n { } − v n { } v n { } + 24 v n { } . (16)The flow coefficients in Eq. (15) are driven by the corresponding initial-stateeccentricities defined as [16]: ε n { } = (cid:112) (cid:104) ε n (cid:105) ,ε n { } = (cid:112) (cid:104) ε n (cid:105) − (cid:104) ε n (cid:105) ,ε n { } = (cid:114)
14 ( (cid:104) ε n (cid:105) − (cid:104) ε n (cid:105)(cid:104) ε n (cid:105) + 12 (cid:104) ε n (cid:105) ) ,ε n { } = (cid:114) −
133 ( (cid:104) ε n (cid:105) − (cid:104) ε n (cid:105)(cid:104) ε n (cid:105) − (cid:104) ε n (cid:105) + 144 (cid:104) ε n (cid:105) (cid:104) ε n (cid:105) − (cid:104) ε n (cid:105) ) . (17)Consider V = v exp( i Ψ ) , V = v exp( i Ψ ) and V = v exp( i Ψ ). In theforegoing discussion, we implicitly assumed that V and V are to be analysed withrespect to their respective reference angles, Ψ and Ψ . But that is not necessary. Wecould alternately analyse V with respect to the direction of Ψ , leading to exp( i Ψ − Ψ )). Similarly, analysing V with respect to the direction of Ψ or of Ψ would involveexp( i Ψ − Ψ )) or exp( i Ψ − Ψ )). We shall encounter such quantities later whenwe discuss event-plane correlators (Sec. 3.2).For recent theoretical developments about calculating and analyzing multiparticlecumulants to arbitrary order, see [17,18]. As stated above, the flow magnitude ( v n ) and phase ( Ψ n ) (or equivalently, x and y components of V n ) fluctuate event to event. The probability density function P ( v n )has been measured. Figure 2 shows the ATLAS data on P ( v ) , P ( v ) and P ( v ) forvarious centrality bins at √ s NN = 2 .
76 TeV [19]. As one goes from central to periph-eral collisions, the distributions broaden reflecting the gradual increase of v n withcentrality. Centrality dependence of P ( v ) is stronger than that of P ( v ) and P ( v ),because ε which drives v changes more dramatically from central to peripheral col-lisions than do ε and ε . Similar results were reported by ALICE at √ s NN = 5 . v ) p ( v |<2.5 η >0.5 GeV, | T p centrality:0 1%5 10%20 25%30 35%40 45%60 65% ATLAS
Pb+Pb =2.76 TeV NN s b µ = 7 int L v ) p ( v |<2.5 η >0.5 GeV, | T p centrality:0 1%5 10%20 25%30 35%40 45% ATLAS
Pb+Pb =2.76 TeV NN s b µ = 7 int L v ) p ( v |<2.5 η >0.5 GeV, | T p centrality:0 1%5 10%20 25%30 35%40 45% ATLAS
Pb+Pb =2.76 TeV NN s b µ = 7 int L Fig. 2. (colour online) ATLAS data on probability density function P ( v n ) in several cen-trality intervals for n = 2 (left panel), n = 3 (middle panel) and n = 4 (right panel) [19]. Exercise 4 : Define σ v ≡ (cid:10) v (cid:11) − (cid:104) v (cid:105) . (Harmonic index n is suppressed for simplicityof notation.) Note that v { } , v { } and v { } defined in Eq. (15) are various functionsof v , denoted generically by f ( v ). Taylor expand f ( v ) around the mean flow (cid:104) v (cid:105) tothe second order and take the mean to get (cid:104) f ( v ) (cid:105) = f ( (cid:104) v (cid:105) ) + σ v f (cid:48)(cid:48) ( (cid:104) v (cid:105) ) /
2. Show that, v { } = (cid:104) v (cid:105) + σ v and v { } ≈ (cid:104) v (cid:105) − σ v ≈ v { } , and hence v { } ≈ (cid:104) v (cid:105) + σ v / (2 (cid:104) v (cid:105) )and v { } ≈ (cid:104) v (cid:105) − σ v / (2 (cid:104) v (cid:105) ) ≈ v { } . Thus v { } > (cid:104) v (cid:105) > v { } ≈ v { } . In other words,fluctuations, in general, tend to enhance v { } and suppress v { } , v { } compared to (cid:104) v (cid:105) [3,21]. (N.B. Here, we did not assume the fluctuations to be Gaussian.)As a measure of the relative flow fluctuations, one defines: F ( v n ) ≡ (cid:115) v n { } − v n { } v n { } + v n { } = σ v n (cid:104) v n (cid:105) , (18)where the last equality follows from Exercise 4. Note that Eq. (18) is based on twoassumptions: (a) there are no nonflow correlations in v n { } and v n { } , and (b) σ v n (cid:28)(cid:104) v n (cid:105) .Flow coefficients v n { m } , m = 2 , , , n = 2. Figure 3 shows the CMS data [22] on v { m } for m = 2 , , ,
8. Evidently, v { } > v { } ≈ v { } ≈ v { } . It is easy to understand this feature of the databy making a simple Gaussian ansatz [23] for the fluctuations of the vector V n in thetransverse ( xy ) plane. Denoting x and y components of V by v x and v y for simplicityof notation, one assumes the probability distribution to be P ( v x , v y ) = 12 πσ exp (cid:32) − ( v x − ¯ v ) + v y σ (cid:33) , (19)where ¯ v = (cid:104) v x (cid:105) , (cid:104) v y (cid:105) = 0 and σ = σ x = σ y is the variance of the Gaussian distribu-tion. Assume further that v x and v y are uncorrelated. Exercise 5 : Using Eqs. (15) and (19) and the moments of the Gaussian distributiongiven in Table 1 in Appendix C, show that v { } = √ ¯ v + 2 σ and v { } = v { } = v { } = ¯ v . This shows that the data in Fig. 3 is consistent with the Gaussian ansatz[23]. ill be inserted by the editor 11 Centrality % } m { v = 2 m = 4 m = 6 m = 8 m CMS (PbPb 5.02 TeV) -1 b m
26 < 3.0 GeV/c T h | Fig. 3. (colour online) Note that v { } > v { } ≈ v { } ≈ v { } . This is consistent withthe Gaussian ansatz [23]; see Exercise 5. Figure from [22]. A more careful inspection of the data, however, shows that there are small differencesin the magnitudes of v { } , v { } and v { } , indicating non-Gaussian fluctuations inthe data on v (Fig. 4).We next turn to n = 3. Unlike v , which can be nonzero even in the absenceof fluctuations, v (at midrapidity) arises only due to fluctuations. So the Gaussianansatz for v is P ( v x , v y ) = 12 πσ exp (cid:32) − v x + v y σ (cid:33) . (20) Exercise 6 : Using Eqs. (15) and (20) and the moments of the Gaussian distributiongiven in Table 1 in Appendix C, show that v { } = √ σ and v { } = v { } = v { } = 0.Figure 5 shows ATLAS results for nc n { } ≡ c n { } c n { } = − v n { } v n { } = (cid:10) v n (cid:11) (cid:104) v n (cid:105) − , (21)for n = 3. It is clear from the figure that v { } (cid:54) = 0, except for the most centralcollisions. This is an evidence for non-Gaussianity in the triangular flow fluctuations.Note that (cid:10) v n (cid:11) / (cid:10) v n (cid:11) − f ( r ) in Appendix C. Centrality % { } v / { } v CMSATLAS (2.76 TeV)Giacalone et al. (2.76 TeV)
CMS (PbPb 5.02 TeV) -1 b m
26 < 3.0 GeV/c T h | Centrality % { } v / { } v CMSATLAS (2.76 TeV)
CMS (PbPb 5.02 TeV) -1 b m
26 < 3.0 GeV/c T h | Centrality % { } v / { } v CMS (PbPb 5.02 TeV) -1 b m
26 < 3.0 GeV/c T h | Fig. 4. (colour online) Note the small differences in the magnitudes of v { } , v { } and v { } . This is the evidence for non-Gaussianity in the elliptic flow fluctuations. Figure from[22]. Fig. 5. (colour online) nc { } ≡ − v { } /v { } . Data points with various colours cor-respond to different p T ranges. Note the nonzero value of v { } . This is the evidence fornon-Gaussianity in the triangular flow fluctuations; see Exercise 6. Figure from [24]. Exercise 7 : Recall that the cumulants (Eq. (14)) are functions of even moments ofthe distribution of v n . For n = 2 (the most central collisions only) and n = 3 (allcentralities), flow is solely due to fluctuations. If the statistics of these fluctuationsis a 2-dimensional Gaussian (Appendix C), then show that the scaled even moments m ( k ) n satisfy: m ( k ) n ≡ (cid:10) v kn (cid:11)(cid:68) v k − n (cid:69) (cid:104) v n (cid:105) = k, ( k = 2 , , . These expectations are borne out quite well in AMPT model calculations [49].If the flow fluctuations are not Gaussian, then what is the right underlying PDF?A related question is what is the PDF for the fluctuating initial eccentricity? Thisissue has been discussed in the literature: A distribution that was proposed early onwas the Bessel-Gaussian distribution [23]. It worked well for central collisions, butnot so well for peripheral collisions. A new parametrization named the Elliptic Powerdistribution, for the PDF of the initial eccentricity at a fixed centrality, was proposedin Ref. [25]. Unlike the Bessel-Gaussian distribution, this fits several Monte Carlomodels of the initial state for all centralities. ill be inserted by the editor 13
For more recent works that probe the skewness and kurtosis of the non-Gaussianfluctuations in heavy-ion collisions, see [26,27]. In cosmology, the primordial non-Gaussianity is consistent with zero, and the present-day non-Gaussianity is the resultof the evolution of the universe. In heavy-ion collisions, on the other hand, the pri-mordial non-Gaussianity is sizable, and it is partially washed out as a result of thesystem evolution [28]. Future high-statistics measurements of kurtosis of elliptic flowfluctuations would provide deeper insights into the hydrodynamic behaviour and theequation of state of quark-gluon plasma.We discussed in this section the probability density function P ( v n ). Ideally, onewould like to measure the full probability density function P ( V , V , · · · , V n ) = P ( v , v , · · · , v n , Ψ , Ψ , · · · , Ψ n ). We are far from that goal. Equation (9) is the Fourier expansion of the single-particle distribution in the az-imuthal plane. Fourier expansion of the two-particle distribution is similarly givenby dN pair d∆φ = N pair π (cid:32) ∞ (cid:88) n =1 V n∆ cos n∆φ (cid:33) , (22)where ∆φ = φ − φ and V n∆ = (cid:104) exp( in∆φ ) (cid:105) are the two-particle Fourier coefficientswhich fluctuate from event to event. One can similarly define d N pair /d∆η d∆φ , where ∆η = η − η . This quantity is displayed in Figs. 6 and 7.Figure 6 shows two-charged-particle correlations as a function of ∆η and ∆φ ,measured by CMS [29], in PbPb and pPb collisions. The sharp near-side ( ∆φ ≈ ∆η ≈ ∆η ) wave-like structure thatwe see on the near-side is called the ridge. It indicates that particles widely separatedin pseudorapidity experience a collective push in the same azimuthal direction. Pro-duction of a small drop of fluid is a natural explanation of such a correlation presentat all rapidities. The peaking around ∆φ = 0 and π is the signature of the dominantelliptic flow ( v ) with some admixture of other harmonics notably v (see Exercise8). Thus not only the flow harmonics v n , but even the correlations observed betweenparticles separated by a large pseudorapidity gap could be explained by a collectivebehaviour. Causality requires that the long-range structure seen in the ridge origi-nates in the earliest stages of the collision when the transverse geometry is almostrapidity independent [32]. Exercise 8 : Make three different plots of cos(2 ∆φ ), cos(2 ∆φ ) + 0 .
25 cos(3 ∆φ ) andcos(2 ∆φ ) + 0 .
55 cos(3 ∆φ ), in the range ∆φ = − π/ π/
2. Notice how the heightsand widths of the two peaks change as one increases the admixture of v . Notice alsothe appearance of a “shoulder” in the away-side peak. Compare with Fig. 5 of [31]which is similar to our Fig. 6(a), but for ultracentral (0-0.2 %) collisions, and clearlyshows the shoulder formation on the away side.Interestingly, the ridge is also observed in high-multiplicity pp collisions (Fig.7). This begs the question: is quark-gluon plasma formed or is hydrodynamic flowdeveloped also in high-multiplicity pp (and p-nucleus) collisions? This issue is stillbeing debated (see Sec. 4). hD -4 -2 0 2 4 (r ad i an s ) fD fD d hD d pa i r N d t r i g N < 260 offlinetrk N £ = 2.76 TeV, 220 NN s(a) CMS PbPb < 3 GeV/c trigT assocT hD -4 -2 0 2 4 (r ad i an s ) fD fD d hD d pa i r N d t r i g N < 260 offlinetrk N £ = 5.02 TeV, 220 NN s(b) CMS pPb < 3 GeV/c trigT assocT Fig. 6. (colour online) Two-charged-particle correlations in (a) PbPb and (b) pPb colli-sions observed by CMS [29]. Semiperipheral PbPb collisions are selected so that particlemultiplicities are similar to those in the pPb case. In (b), the sharp near-side peak from jetcorrelations is truncated to emphasize the structure outside that region. η Δ − − (r ad i an s ) φ Δ φ Δ d η Δ d pa i r N d t r i g N = 13 TeVsCMS pp < 150 offlinetrk N ≤
105 < 3 GeV/c assocT , p trigT ± - h ± h η Δ − − (r ad i an s ) φ Δ φ Δ d η Δ d pa i r N d t r i g N ± - h K η Δ − − (r ad i an s ) φ Δ φ Δ d η Δ d pa i r N d t r i g N ± - h Λ / Λ Fig. 7. (colour online) Two-charged-particle correlations in (left) low-multiplicity (10-20)and (right) high-multiplicity (105-150) pp collisions observed by CMS [30].
Consider the two-particle Fourier coefficient V n∆ = (cid:104) exp( in∆φ ) (cid:105) occurring in Eq.(22). Making the same assumptions as in the derivation of Eq. (13), one gets V n∆ = v n .However, as stated in Sec. 2.2, the event-plane angle Ψ n may depend on the transversemomentum ( p T ) and pseudorapidity ( η ), and hence may not be the same in the twobins a and b from which the two particles are taken. In this case, V n∆ in a singleevent is given by V n∆ ( a, b ) = (cid:68) e in ( φ a − φ b ) (cid:69) = (cid:68) e inφ a (cid:69) (cid:68) e − inφ b (cid:69) = V n,a V ∗ n,b = v n,a v n,b e in ( Ψ an − Ψ bn ) , where (cid:104)· · ·(cid:105) denotes the average over one event and the second equality is obtained byassuming that there are no nonflow correlations. Factorization (of two-particle cor-relations) refers to V n∆ ( a, b ) getting factorized into a product of single-particle har-monics, v n,a and v n,b : V n∆ ( a, b ) = v n,a × v n,b . Factorization breaks down ( V n∆ ( a, b ) (cid:54) = v n,a × v n,b ) if Ψ n is not the same in the two bins and/or if there are nonflow correlations.The latter effect can be minimized by introducing a large-enough pseudorapidity gapbetween the two bins. ill be inserted by the editor 15 Experimentally, one further averages over many events to write V n∆ ( a, b ) = (cid:10) V n,a V ∗ n,b (cid:11) = (cid:68) v n,a v n,b e in ( Ψ an − Ψ bn ) (cid:69) , (23)where (cid:104)· · ·(cid:105) denotes the average over many events. Factorization ratio r n ( p aT , p bT ) isdefined as [33] r n ( p aT , p bT ) ≡ V n∆ ( p aT , p bT ) (cid:113) V n∆ ( p aT , p aT ) V n∆ ( p bT , p bT ) = (cid:10) v n ( p aT ) v n ( p bT ) cos n ( Ψ n ( p aT ) − Ψ n ( p bT )) (cid:11)(cid:113) (cid:104) v n ( p aT ) (cid:105) (cid:10) v n ( p bT ) (cid:11) . (24)It measures the flow decorrelation between momentum bins, i.e., p T -dependent event-plane angle fluctuations. If V n∆ factorizes, r n is obviously unity, otherwise it is lessthan unity by the Cauchy—Schwarz inequality. In the most central events initial-state fluctuations play a dominant role, and so r n deviates from unity significantly. r n ( p aT , p bT ) is a sensitive discriminator of the models of the initial-state geometry inthe transverse plane. Specifically, it could probe the granularity or spatial extent ofthe density fluctuations in the transverse plane in the initial state.Assuming V n∆ ( p aT , p bT ) factorizes into the product of single-particle anisotropies, p T dependence of the latter can be determined experimentally from v n ( p aT ) = V n∆ ( p aT , p bT ) (cid:113) V n∆ ( p bT , p bT ) , (25)where the reference momentum p bT is chosen not too large to minimize correlationsfrom jets at higher p T .The longitudinal decorrelation or η -dependent factorization breakdown cannot bestudied by simply replacing p aT and p bT in Eq. (24) by η a and η b , respectively, becausethe denominator would be badly contaminated by short-range nonflow correlations.Instead, one defines the following observable: r n ( η ) ≡ (cid:104) V n ( − η ) V ∗ n ( η ref ) (cid:105)(cid:104) V n (+ η ) V ∗ n ( η ref ) (cid:105) = (cid:104) v n ( − η ) v n ( η ref ) cos n ( Ψ n ( − η ) − Ψ n ( η ref )) (cid:105)(cid:104) v n (+ η ) v n ( η ref ) cos n ( Ψ n (+ η ) − Ψ n ( η ref )) (cid:105) , (26)with a large gap between η and η ref . Experimentally, it is found that v n and Ψ n fluctuate along the longitudinal direction, even in a given event [13,34]. r n ( η ) is unityat η = 0 and decreases as η increases [35]. Hydrodynamic model calculations showthat r n ( η ) is driven mostly by the longitudinal structure of the primordial state. Thestudy of r n ( η ) puts constraints on the complete three-dimensional modelling of theinitial state used in 3+1D models of the system evolution. A significant breakdownof factorization in η is also observed in pPb collisions [13].To conclude, experimentally, factorization breakdown or flow decorrelation hasbeen observed as a function of both p T and η [13,31,34]. For v in central PbPbcollisions at LHC the effect can be as large as 20%. Flow observables discussed so far referred to a single harmonic n . We now discussflow observables with mixed harmonics. These provide new handles on the initial state as well as the properties of the medium. Specifically, they can discriminateamong the various initial-state models as well as among different parametrizations ofthe temperature dependence of the specific shear ( η/s ) and bulk viscosity ( ζ/s ); s isthe entropy density.In Sec. 2.3 we discussed multiparticle correlations involving only a single harmonic( n ). The most general k -particle correlation involving harmonics n , n , · · · , n k canbe written as [36] v { n , n , · · · , n k } = (cid:104)(cid:104) e i ( n φ + ··· + n k φ k ) (cid:105)(cid:105) , (27)where n , n , · · · , n k are integers satisfying n + n + · · · + n k = 0, so that thiscorrelation observable is invariant under rotation in the azimuthal plane. The notation (cid:104)(cid:104)· · · (cid:105)(cid:105) is as in Eq. (11). If there are no nonflow correlations, this becomes v { n , n , · · · , n k } = (cid:104)(cid:104) e in φ (cid:105) · · · (cid:104) e in k φ k (cid:105)(cid:105) = (cid:104) v n · · · v n k e i ( n Ψ n + ··· + n k Ψ nk ) (cid:105) , (28)where in the final expression the average is only over events. Using this, a numberof new flow observables were constructed to study mixed correlations between v , v and v [36,37]. Symmetric cumulants SC ( m, n ) and N SC ( m, n ) measure the correlations betweenevent-by-event fluctuations of the magnitudes of the complex flow vectors V m and V n ( m (cid:54) = n ): SC ( m, n ) ≡ (cid:10) v m v n (cid:11) − (cid:10) v m (cid:11) (cid:10) v n (cid:11) ,N SC ( m, n ) ≡ (cid:10) v m v n (cid:11) − (cid:10) v m (cid:11) (cid:10) v n (cid:11) (cid:104) v m (cid:105) (cid:104) v n (cid:105) , (29)which obviously vanish if v m and v n are uncorrelated or if there are no flow fluctu-ations. N SC ( m, n ) can be looked upon as the mixed kurtosis of v m and v n (see Eq.(A3)). Hydrodynamic calculations show that v n scales approximately linearly with ε n , for n = 2 ,
3. An advantage in defining the ratio as in Eq. (29) is that the constantsof proportionality between v n and ε n drop out, thereby directly connecting experi-mental observables with the properties of the initial state. Symmetric cumulant is afour-particle observable. Exercise 9 : Equation (13) together with the discussion following it, explain how (cid:10) v n (cid:11) and (cid:10) v n (cid:11) can be measured. Explain how you would measure (cid:10) v m v n (cid:11) . Note also that,as in Eq. (13), any dependence on the symmetry plane is eliminated by construction.Figure 8 shows the ALICE results for SC (4 ,
2) and SC (3 ,
2) as a function ofcentrality. SC (4 ,
2) is found to be positive and SC (3 ,
2) negative. This indicatesthat if in an event v is larger than (cid:104) v (cid:105) , then there is an enhanced probabilityof finding v larger than (cid:104) v (cid:105) . Similarly, if in an event v is larger than (cid:104) v (cid:105) , thenthere is an enhanced probability of finding v smaller than (cid:104) v (cid:105) . In other words, v and v are correlated, whereas v and v are anticorrelated, for all centralities. Thedetailed behaviour, however, depends on whether one is in the fluctuation-dominated(most central) regime or the geometry-dominated (midcentral) regime. Figure 8 alsoshows the HIJING model results which are consistent with zero. Now, this model ill be inserted by the editor 17 Centrality percentile S C ( m , n ) - - - · = 2.76 TeV NN s ALICE Pb-Pb SC(4,2)SC(3,2)HIJING SC(4,2)SC(3,2)
Fig. 8. (colour online) Centrality dependence of the symmetric cumulants SC (4 ,
2) (redfilled squares) and SC (3 ,
2) (blue filled circles). Boxes represent systematical errors. HIJINGmodel results are shown with hollow markers. Figure from [38]. includes nonflow azimuthal correlations due to jet production, but has no physicsof collectivity. Evidently, the nonzero ALICE results cannot be explained by nonfloweffects, showing that the symmetric cumulants SC ( m, n ) are robust against systematicbiases originating from nonflow effects. More recently, ALICE have presented dataon SC (4 , SC (5 ,
2) and SC (5 ,
3) [39]. Interestingly, negative SC (3 ,
2) has beenobserved also in the highest-multiplicity pp and pPb events [40,41].Recently, the idea of symmetric cumulants was generalized and a new set of ob-servables, which quantify the correlations of flow fluctuations involving more than two flow amplitudes, was introduced [42].
Event-plane correlators or symmetry-plane correlators measure the correlations be-tween event-by-event fluctuations of the phases of the complex flow vectors V m and V n ( m (cid:54) = n ). They represent higher-order correlations, involving at least three par-ticles (see below). One can construct two-, three-, or higher-plane correlators. Theythus bring in a large number of new observables that provide new, detailed insightinto the hydrodynamic response and the initial state.Suppose we want to construct an observable that measures correlations betweenthe event planes Ψ m and Ψ n . Recall first that Ψ l can be determined only modulo 2 π/l (Eq. (6)). Hence the observable should be invariant under Ψ l → Ψ l + 2 π/l . Moreover,it should also be invariant under a global rotation through an arbitrary angle. Anobservable that satisfies both these requirements is (cid:104) cos k ( Ψ m − Ψ n ) (cid:105) , where k is acommon multiple of m and n . Consider the simplest example of (cid:104) cos 4( Ψ − Ψ ) (cid:105) . Inorder to measure this correlator, we considerRe (cid:10) V V ∗ (cid:11)(cid:112) (cid:104)| V | (cid:105) (cid:104)| V | (cid:105) = (cid:10) v v cos 4( Ψ − Ψ ) (cid:11)(cid:112) (cid:104) v (cid:105) (cid:104) v (cid:105) ≈ (cid:104) cos 4( Ψ − Ψ ) (cid:105) , (30)where V n is the experimentally measured flow vector, often denoted by Q n or q n (seeSec. 2.2.). It is clear that the left-hand side of the above equation involves averaging Recall the discussion in the last paragraph of Sec. 2.3.8 Will be inserted by the editor exp(2 iφ j + 2 iφ k − iφ l ) over triplets of particles in an event and then over manyevents, and thus necessarily involves three-particle correlations. In practice, the threeparticles are taken from two or more different pseudorapidity bins to minimize theshort-range nonflow correlations. Observables for other event-plane correlators can beconstructed similarly [43]. Exercise 10 : Table I in Ref. [43] lists two-, three- and four-plane corre-lators, along with the total number of particles that are involved in eachcase. For example, the two-event-plane correlator (cid:104) cos 12( Ψ − Ψ ) (cid:105) , the three-event-plane correlator (cid:104) cos(10 Ψ − Ψ − Ψ ) (cid:105) and the four-event-plane correlator (cid:104) cos(6 Ψ + 3 Ψ − Ψ − Ψ ) (cid:105) , involve 9, 8 and 6 particles, respectively. Understandthis Table.Figure 9 shows the ATLAS data on eight two-plane correlators as a function ofcentrality for Pb-Pb collisions at √ s NN = 2 .
76 TeV [44]. Given a model for the ini-tial state, say the Glauber model, one can calculate participant-plane correlators asa function of centrality; these are also shown in Fig. 9. Here the participant-planecorrelators and event-plane correlators are being compared with each other to illus-trate the role played by the hydrodynamic evolution in converting the initial-statespatial correlations between different harmonics into the corresponding final-statemomentum-space correlations. For the most central events, where fluctuations andnot geometry is the dominant feature, the two-plane (Fig. 9) as well as three-plane[44] correlators vanish (with the sole exception of (cid:104) cos 6( Φ − Φ ) (cid:105) ). For off-centralcollisions, geometry takes over, and the correlators increase in magnitude. This is sim-ilar to the centrality dependence of the symmetric cumulants (Fig. 8). For off-centralcollisions, the participant-plane correlators and event-plane correlators generally differsignificantly, especially for higher harmonics, pointing to the misalignment between Ψ n and Φ n , owing to the contributions of the nonlinear flow modes, to be discussedin the next subsection.ALICE collaboration has measured a few two- and three-plane correlators; theydenote them generically by ρ n,mk [12,45]. A recent paper has pointed out problemswith the current methods of measuring symmetry-plane correlations and has proposeda new estimator as an improvement [46]. Hydrodynamic calculations show that V n scales approximately linearly with the ini-tial eccentricity E n , for n = 2 ,
3. For higher harmonics, however, V n receives contribu-tions from eccentricities in the lower harmonics as well [47]. For complex flow vectors V n ( n ≥
4) one can write the vector sum of the linear and nonlinear response termsas [12,45,48]. V = V + V = V + χ , V ,V = V + V = V + χ , V V ,V = V + V = V + χ , V + χ , V + χ , V V L ,V = V + V = V + χ , V V + χ , V V L + χ , V V L ,V = V + V = V + χ , V + χ , V V + E ( V L , V L , V L ) , (31) ATLAS collaboration denotes the event plane by Φ n , which is different from our conven-tion here. In Ref. [43], we have used the ATLAS convention.ill be inserted by the editor 19 〉 part N 〈 〉 ) Φ Φ c o s ( 〈 ATLAS
Pb+Pb=2.76 TeV NN s b µ = 7 int L data w 〉 ) ΦΣ cos( 〈 data 〉 ) ΦΣ cos( 〈 Glauber w 〉 *) ΦΣ cos( 〈 Glauber 〉 *) ΦΣ cos( 〈 〉 part N 〈 〉 ) Φ Φ c o s ( 〈 ATLAS
Pb+Pb=2.76 TeV NN s b µ = 7 int L data w 〉 ) ΦΣ cos( 〈 data 〉 ) ΦΣ cos( 〈 Glauber w 〉 *) ΦΣ cos( 〈 Glauber 〉 *) ΦΣ cos( 〈 〉 part N 〈 〉 ) Φ Φ c o s ( 〈 ATLAS
Pb+Pb=2.76 TeV NN s b µ = 7 int L data w 〉 ) ΦΣ cos( 〈 data 〉 ) ΦΣ cos( 〈 Glauber w 〉 *) ΦΣ cos( 〈 Glauber 〉 *) ΦΣ cos( 〈 〉 part N 〈 〉 ) Φ Φ c o s ( 〈 ATLAS
Pb+Pb=2.76 TeV NN s b µ = 7 int L data w 〉 ) ΦΣ cos( 〈 data 〉 ) ΦΣ cos( 〈 Glauber w 〉 *) ΦΣ cos( 〈 Glauber 〉 *) ΦΣ cos( 〈 〉 part N 〈 〉 ) Φ Φ c o s ( 〈 ATLAS
Pb+Pb=2.76 TeV NN s b µ = 7 int L data w 〉 ) ΦΣ cos( 〈 data 〉 ) ΦΣ cos( 〈 Glauber w 〉 *) ΦΣ cos( 〈 Glauber 〉 *) ΦΣ cos( 〈 〉 part N 〈 〉 ) Φ Φ c o s ( 〈 ATLAS
Pb+Pb=2.76 TeV NN s b µ = 7 int L data w 〉 ) ΦΣ cos( 〈 data 〉 ) ΦΣ cos( 〈 Glauber w 〉 *) ΦΣ cos( 〈 Glauber 〉 *) ΦΣ cos( 〈 〉 part N 〈 〉 ) Φ Φ c o s ( 〈 ATLAS
Pb+Pb=2.76 TeV NN s b µ = 7 int L data w 〉 ) ΦΣ cos( 〈 data 〉 ) ΦΣ cos( 〈 Glauber w 〉 *) ΦΣ cos( 〈 Glauber 〉 *) ΦΣ cos( 〈 〉 part N 〈 〉 ) Φ Φ c o s ( 〈 ATLAS
Pb+Pb=2.76 TeV NN s b µ = 7 int L data w 〉 ) ΦΣ cos( 〈 data 〉 ) ΦΣ cos( 〈 Glauber w 〉 *) ΦΣ cos( 〈 Glauber 〉 *) ΦΣ cos( 〈 Fig. 9. (colour online) ATLAS data [44] on two-event-plane correlators as a function ofcentrality. Red solid symbols and blue open symbols represent two different analysis methods.Red solid and blue dashed curves represent Glauber-model predictions for two-participant-plane correlators using two different methods. Here Φ n denotes the event-plane and Φ ∗ n theparticipant plane, which differs from our convention. where the first term on the right in each equation is called the linear flow mode andthe remaining terms constitute the nonlinear flow modes. The (real) coefficients χ arecalled the nonlinear flow mode coefficients, and E ( · · · ) in the last equation denotesthe remaining terms in V .Now, suppose the linear terms V L , V L , V L , etc. are defined to be linearly pro-portional to the corresponding cumulant-based eccentricities E , E , E , etc. defined inEq. (3), rather than the moment-based eccentricities defined in Eq. (1). For example, V L will be driven not by ε exp( i Φ ), but by − (cid:10) z (cid:11) − (cid:10) z (cid:11) (cid:104) r (cid:105) = ε e i Φ + 3 (cid:10) r (cid:11) (cid:104) r (cid:105) ε e i Φ . With the above definition, the linear and nonlinear parts in Eqs. (31) are not neces-sarily uncorrelated. On the other hand, if the linear term V nL in each of the Eqs. (31)is defined as the part of V n that is uncorrelated with the nonlinear part V nNL , thenthe linear term may or may not be proportional to the corresponding cumulant-basedeccentricity. Clearly, more work needs to be done here for a better understanding.In what follows we describe a few recent works, so that the reader can follow theliterature.Let us first assume that the coefficients χ are the same for all events in a centralityclass. Now if the linear and nonlinear parts in Eq. (31) are uncorrelated, then it is Note also that (cid:104) V nL (cid:105) = 0, because V nL , like V n , is expected to carry a random phasefactor depending on the reaction-plane angle Φ RP .0 Will be inserted by the editor easy to show that [49] (cid:10) V V ∗ v (cid:11) (cid:104) V V ∗ (cid:105) (cid:104) v (cid:105) = (cid:10) v (cid:11) (cid:104) v (cid:105) (cid:104) v (cid:105) , (cid:10) V V ∗ V ∗ v (cid:11) (cid:104) V V ∗ V ∗ (cid:105) (cid:104) v (cid:105) = (cid:10) v v (cid:11) (cid:104) v v (cid:105) (cid:104) v (cid:105) . (32)The left- and right-hand sides of these and other similar equations are found toagree with each other to a good approximation in AMPT, as well as in perfect andimperfect hydrodynamic calculations [48,49], justifying the hypothesis that the linearand nonlinear contributions are uncorrelated.To isolate the linear and nonlinear terms in Eq. (31), it is usually assumed thatthey are uncorrelated [50]. Consider, for example, the first of the five equations in(31), take its complex conjugate, multiply the two, average over many events in acentrality class, to get (cid:10) v (cid:11) = (cid:10) v (cid:11) − χ , (cid:10) v (cid:11) . This, however, cannot be used to determine the linear part because χ , is an un-known. Consider, however, the observable[Re (cid:10) V V ∗ (cid:11) ] (cid:104) v (cid:105) = χ , (cid:10) v (cid:11) . (33)It allows us to determine the linear part as well as the coefficient χ , [50]: (cid:10) v (cid:11) = (cid:10) v (cid:11) − [Re (cid:10) V V ∗ (cid:11) ] (cid:104) v (cid:105) and χ , = Re (cid:10) V V ∗ (cid:11) (cid:104) v (cid:105) . (34)Similarly, it can be shown that (cid:10) v (cid:11) = (cid:10) v (cid:11) − [Re (cid:104) V V ∗ V ∗ (cid:105) ] (cid:104) v v (cid:105) and χ , = Re (cid:104) V V ∗ V ∗ (cid:105)(cid:104) v v (cid:105) . (35)Other equations in (31) can be treated similarly; see [12,48,50,51] for details.The nonlinear flow mode coefficients ( χ ) are expected to be independent of theinitial density profile in a given centrality class. Centrality dependence of these coef-ficients has been presented in Refs. [45] and [12], for PbPb collisions at 2.76 and 5.02TeV, respectively. We now point out an interesting connection between the two seemingly unrelatedobservables, namely, event-plane correlators (Sec. 3.2) and the nonlinear response(Sec. 3.3): This is evident in the similarity between the observables defined in Eqs. (30)and (33), which allows us to write (cid:104) cos 4( Ψ − Ψ ) (cid:105) in terms of χ , . These equationsillustrate how the event-plane correlations can be understood in the framework ofthe nonlinear response. Correlations between the magnitudes of the flow vectors ofdifferent harmonics also exhibit distinctive signs of the nonlinear mode couplings [52].Consider Eqs. (31). One can write similar nonlinear equations connecting theinitial eccentricity vectors E n ( n ≥
4) with E m of lower harmonics ( m < n ), withcoefficients χ replaced by, say, ˜ χ . An interesting question is how the nonlinear modecoupling coefficients in the two cases, χ and ˜ χ , are related. This has been investigated ill be inserted by the editor 21 offlinetrk N v = 13 TeVspp < 3.0 GeV/c T h |CMS |>2} hD {2, | sub2 v {4} v {6} v {8} v {LYZ} v offlinetrk N v = 2.76 TeV NN sPbPb < 3.0 GeV/c T h | offlinetrk N v = 5 TeV NN spPb < 3.0 GeV/c T h | Fig. 10. (colour online) Observation of multiparticle cumulants in pp, pPb and PbPb col-lisions. Points labelled v { LYZ } refer to the analysis method based on the Lee-Yang zeroes[55,56]. The error bars correspond to the statistical uncertainties, while the shaded areasdenote the systematic uncertainties. Figure from [30]. in [48]. Just as the nonlinear-response terms in the higher-harmonic flow vectors giverise to event-plane correlations in the final state, the nonlinear terms in the higher-harmonic eccentricity vectors cause participant-plane correlations in the initial state.Relationships between χ and ˜ χ are reflected in the relationships between event-planeand participant-plane correlations [48]. Collectivity refers to a large number of particles acting in unison. Multiparticle cumu-lants discussed in Sec. 2 probe just this aspect of the data (Figs. 3, 5). Symmetric cu-mulants SC ( m, n ) are sensitive to four-particle correlations (Exercise 9). Event-planecorrelators also probe multiparticle correlations (Exercise 10). Collectivity is also ev-ident in the ridge phenomenon — the azimuthally collimated, near-side, long-rangerapidity correlations between two hadrons (Figs. 6 and 7). All the above signaturesof collectivity have been seen very clearly in the large systems formed in relativisticheavy-ion collisions.Traditionally, small systems such as those formed in pp or p-nucleus collisionswere considered as a benchmark or reference for studying large systems formed innucleus-nucleus collisions, the assumption being that the quark-gluon plasma wasunlikely to be produced in small systems. However, with the small systems display-ing qualitatively similar behaviour as the large systems, this assumption becomesquestionable.We saw in Sec. II that ridges have been observed even in small systems cre-ated in high-multiplicity pp and p-nucleus collisions. Ridge was a crucial piece ofevidence for the strong collective behaviour comparable to a fluid in heavy-ion col-lisions. Some other observables or features of the data that were thought to be con-sistent with the formation of quark-gluon plasma and were originally observed inlarge systems, have now been observed in small systems as well. Among them are:sizable azimuthal anisotropy — not just v , but also higher harmonics — in thehighest-multiplicity events [30,53], mass ordering of v ( p T ) of identified particles [54],multiparticle cumulants (Fig. 10) [30], system size and shape dependence of the func-tions v ( p T ) and v ( p T ) [57], enhanced strangeness production [58], negative SC (3 , A z i m u t h a l C o rr e l a t i o n I m p o r t a n c e Event multiplicity ( fi xed size) Glasma Collective fl owParton escapeQCD bremsstrahlungMini-jets Fig. 11. (colour online) A schematic diagram depicting various sources of azimuthalanisotropy as a function of event multiplicity. Vertical scale is arbitrary. Figure from [62]. in the highest-multiplicity pp and pPb events [40,41], etc. Azimuthal anisotropy sug-gests geometry-driven pressure gradients, mass ordering signifies a common velocityfield of the expanding medium, ridge and multiparticle cumulants point toward acollective behaviour, strangeness enhancement indicates a deconfined medium — allleading to the proposition that perhaps “One fluid rules them all ... pp, pPb, PbPb”[59]. However, this claim has been contested (“One fluid might not rule them all”)on the grounds that the 2+1D hydrodynamic simulations were unable to describethe multiparticle single and mixed harmonics cumulants [60]. Moreover, some otherobservables such as jet quenching, high- p T hadron suppression, bottomonium sup-pression, etc., seen in large systems have remain elusive in small-system collisions[61]. So the question remains: what is the smallest size of a drop of liquid QGP? Or,can one dial the system size and switch off/on the QGP? Azimuthal anisotropy has long been considered as a signature of transverse collectiveflow [7]. It is generally agreed that the azimuthal anisotropy (for p T (cid:46) Hydrodynamic response converts the initial spatial anisotropy of the nuclear over-lap region into the momentum-space anisotropy of the final-state particles. Event-by-event relativistic second-order dissipative hydrodynamics has been successful inexplaining most of the flow-related data. As an example of the success of the hydro-dynamic paradigm, see Fig. 12. Note also the ordering: v > v > v > v . It arisesbecause, for the 30-40% centrality results shown in Fig. 12, the shape of the initialoverlap area is predominantly elliptic. The higher harmonics are driven mostly by the A recent experiment [12] has shown that the ordering persists up to n = 7, with someenhancement for n = 8 , 〈 v n2 〉 / p T [GeV]ATLAS 30-40%, EP narrow: τ switch = 0.4 fm/cwide: τ switch = 0.2 fm/c η /s =0.2 v v v v Fig. 12. (colour online) Root-mean-square flow (Eq. 15) versus transverse momentum. Hy-drodynamic calculations with two different switching times are compared to the ATLASdata [63] obtained using the event-plane (EP) method. Bands indicate statistical errors.Experimental error bars are smaller than the size of the points. Figure from [64]. eccentricity fluctuations. As they probe finer details of the initial density distribu-tion, they are successively smaller. The ordering is seen even in ideal hydrodynamiccalculations; see, e.g., [65]. Viscous hydrodynamic calculations enhance the orderingbecause the higher harmonics are found to be more sensitive to the viscous atten-uation. For the ultracentral collisions, the flow is driven mainly by the fluctuationsrather than by the geometry, and so v and v are found to be similar in magnitude.Higher harmonics receive contributions from nonlinear flow modes (Sec. 3.3) and sotheir interpretation is not very straightforward.Fluid dynamics (or loosely speaking, hydrodynamics) is an effective (macroscopic)theory that describes the slow, long-wavelength motion of a fluid close to local equi-librium. However, in the context of relativistic heavy-ion collisions, hydrodynamics isfound to work very well even when the pressure in the system is far from isotropiza-tion ( p L (cid:54) = p T ). Secondly, applicability of hydrodynamics requires a clear separationbetween the microscopic and macroscopic length or time scales. Traditionally, thetheory is formulated as a systematic expansion in gradients of the fluid four-velocity.Now, for small colliding systems, there is no clear separation between microscopicand macroscopic scales, or the gradients are large, and yet hydrodynamics seems tobe capable of describing these systems. The above observations suggest the existenceof a new theory of hydrodynamics that is applicable even far from local equilibrium[66]. They have triggered a lot of theoretical activity in recent years [67]. As illustrated in Fig. 11, due to the nonspherical shape of the fireball, the partonescape probability depends on the angle at which it is trying to escape, giving rise toan anisotropic momentum distribution of the detected particles, without the need ofany pressure gradient. This has been demonstrated for p(d)-nucleus collisions, withinthe framework of transport models (e.g., AMPT) which by definition deal with diluteor low-density systems and allow only a few scatterings of partons unlike in the hy-drodynamical picture [68]. Apart from the anisotropic flow, its mass ordering was alsoreproduced in these calculations [69]. This mechanism is expected to play an impor-tant role when the event multiplicities are somewhat lower than those correspondingto the hydrodynamic flow (Fig. 11). Interestingly, an early work that considered anexpanding mixture of several species of massive relativistic particles, in the frameworkof the Boltzmann equation, concluded that a single collision per particle on average
Fig. 13. (colour online) Schematic representation of Glasma formed between two recedingnuclei just after the collision. Figure from [77]. already leads to sizeable elliptic flow, with mass ordering between the species [70].More recently, a simple kinetic theory estimate showed that for small-enough systems,even a single-hit dynamics is able to generate significant elliptic flow [71].
This is an effective theory of QCD at high energy. Figure 11 shows where this ap-proach is expected to play a dominant role. Unlike hydrodynamics, this is not aninitial-spatial-geometry-driven approach. Models based on this approach have intrin-sic momentum-space correlations in the initial multi-gluon distribution, and hence,are called initial-state models. These correlations have a similar structure as the ex-perimentally observed correlations. The essential physics idea is that the quarks fromthe small-ion projectile scatter coherently off localized domains of strong chromo-electromagnetic fields in the heavy-ion target. The colour domains are of length scale L ∼ /Q s , where Q s is the saturation scale in the target. This approach is ableto reproduce many of the features of multiparticle azimuthal correlations observedin small systems [72,73,74]. It has been argued that the large v for J/ψ observedat LHC can be naturally explained as the initial-state effect within the CGC for-malism [75,76]. Also, the glasma initial conditions — an ensemble of approximatelyboost-invariant flux tubes of longitudinal colour electric and magnetic fields stretch-ing between the colour charges in the two receding nuclei (Fig. 13) — as predictedby CGC are consistent with the long-range structure seen in the ridge phenomenon[78,79].Thus the study of small systems not only deepens our understanding of (hy-dro)dynamics at play in large systems, but also throws light on the working of QCDin a many-body environment. Apart from [62], here are some more recent reviewsthat cover collectivity in small systems: [80,81,82,83,84,85,86,87].
As stated in the Introduction, this was not meant to be a comprehensive review ofthe phenomenology of the collective flow observed in relativistic collisions. Instead,the intention was to provide the necessary mathematical background and explainkey physics concepts, in a pedagogical way, to someone uninitiated in the field, sothat they can follow the literature. To that end, interspersed throughout the text areseveral exercises, which the reader is urged to attempt. ill be inserted by the editor 25
I am very thankful to Jean-Yves Ollitrault for helpful comments on the manuscript.I also thank him for our long-term collaboration which allowed me to learn manythings. I acknowledge the award of the Core Research Grant, by the Science andEngineering Research Board, Department of Science and Technology, Government ofIndia. I thank Bhavya Bhatt for drawing Fig. 14.
The n -th moment of a (real, continuous) function f ( x ), about a constant a , is definedas µ n ( a ) ≡ (cid:90) ∞−∞ ( x − a ) n f ( x ) dx. (A1)We shall assume f ( x ) to be the probability density function (PDF), normalized tounity. The two most interesting values of a are 0 and µ ≡ (cid:104) x (cid:105) , the mean of thedistribution. Usually one refers to µ n ( a = 0) simply as the “moment” and µ n ( a = µ )as the “central moment”. Henceforth, we denote moments µ n ( a = 0) by µ (cid:48) n andcentral moments µ n ( a = µ ) by µ n . Obviously, µ (cid:48) = 1 = µ and µ (cid:48) n = (cid:104) x n (cid:105) . The firstfour central moments are µ = 0 ,µ = (cid:10) x (cid:11) − µ ≡ variance ( σ ) ≡ (standard deviation = σ ) ,µ = (cid:10) x (cid:11) − µ (cid:10) x (cid:11) + 2 µ ,µ = (cid:10) x (cid:11) − µ (cid:10) x (cid:11) + 6 µ (cid:10) x (cid:11) − µ . (A2)It is often convenient to define standardized central moments which are scale-invariantor dimensionless quantities: µ n /σ n . The first four standardized central moments are µ /σ = 0 ,µ /σ = 1 ,µ /σ ≡ skewness ( γ ) ,µ /σ ≡ kurtosis ( κ ) . (A3)Excess kurtosis is defined as κ −
3. However, it is not uncommon to find the excesskurtosis itself termed as kurtosis. The reason for subtracting 3 will become clear whenwe discuss the Gaussian (or normal) distribution (Appendix C). • Variance is a measure of the spread of the random numbers about their meanvalue. • Skewness is a measure of the lopsidedness or asymmetry of the distribution aboutits mean. It is clear from the definition that a distribution that is symmetric about itsmean has vanishing skewness. In general, skewness can be positive or negative. If theleft (right) tail is drawn out, or in other words, is longer than the right (left) tail, thedistribution is said to be left(right)-skewed and has a negative (positive) skewness;see Fig 14. • Kurtosis is a measure of the heaviness of the tails of the distribution as comparedto the normal distribution with the same variance. It is clear from the definition thatkurtosis is a nonnegative number. (Excess kurtosis may be positive or negative.)Kurtosis is a measure of the “tailedness” and not the “peakedness” of a distribution(Fig 14), because the proportion of the kurtosis that is determined by the central µ ± σ range is usually quite small [88]. Fig. 14. (colour online) (left) Positively and negatively skewed distributions. (right) Gaus-sian (excess kurtosis = 0, blue) and a non-Gaussian distribution with heavier tails (excesskurtosis positive, red). M ( t ) M ( t ) ≡ (cid:10) e tx (cid:11) = 1 + t (cid:104) x (cid:105) + t (cid:10) x (cid:11) + · · · = ∞ (cid:88) n =0 t n n ! (cid:104) x n (cid:105) . (A4)Observe that the moments µ (cid:48) n = (cid:104) x n (cid:105) appear in the above expansion. To isolate the m -th moment (cid:104) x m (cid:105) , for example, one uses (cid:20) d m dt m M ( t ) (cid:21) t =0 = (cid:104) x m (cid:105) . (A5)Differentiating M ( t ) m times removes the first m terms, i.e., the terms containing1 , (cid:104) x (cid:105) , · · · , (cid:10) x m − (cid:11) . Further, setting t = 0 removes terms containing (cid:10) x m +1 (cid:11) , (cid:10) x m +2 (cid:11) , · · · , leaving behind only (cid:104) x m (cid:105) . The moment-generating function provides an alter-native description of the PDF. The central moment generating function is e − µt M ( t ). K ( t ) K ( t ) ≡ ln M ( t ) = ln (cid:10) e tx (cid:11) = tκ + t κ + t κ + · · · = ∞ (cid:88) n =1 t n n ! κ n , (A6)where κ n are the cumulants. To isolate the m -th cumulant κ m , for example, one uses (cid:20) d m dt m K ( t ) (cid:21) t =0 = κ m . (A7)Differentiating K ( t ) m times removes the first m − κ , · · · , κ m − . Further, setting t = 0 removes terms containing κ m +1 , κ m +2 , · · · , ill be inserted by the editor 27 leaving behind only κ m . It is straightforward to show that the first few cumulants are κ = (cid:104) x (cid:105) = mean µ,κ = (cid:10) x (cid:11) − (cid:104) x (cid:105) = (cid:10) ( x − (cid:104) x (cid:105) ) (cid:11) = variance ( σ ) = second central moment ,κ = (cid:10) x (cid:11) − (cid:104) x (cid:105) (cid:10) x (cid:11) + 2 (cid:104) x (cid:105) = (cid:10) ( x − (cid:104) x (cid:105) ) (cid:11) = third central moment ,κ = (cid:10) x (cid:11) − (cid:104) x (cid:105) (cid:10) x (cid:11) − (cid:10) x (cid:11) + 12 (cid:104) x (cid:105) (cid:10) x (cid:11) − (cid:104) x (cid:105) (cid:54) = fourth central moment (cid:10) ( x − (cid:104) x (cid:105) ) (cid:11) ,κ = (cid:10) x (cid:11) − (cid:104) x (cid:105) (cid:10) x (cid:11) − (cid:10) x (cid:11) (cid:10) x (cid:11) + 20 (cid:104) x (cid:105) (cid:10) x (cid:11) + 30 (cid:104) x (cid:105) (cid:10) x (cid:11) − (cid:104) x (cid:105) (cid:10) x (cid:11) + 24 (cid:104) x (cid:105) ,κ = (cid:10) x (cid:11) − (cid:104) x (cid:105) (cid:10) x (cid:11) − (cid:10) x (cid:11) (cid:10) x (cid:11) + 30 (cid:104) x (cid:105) (cid:10) x (cid:11) − (cid:10) x (cid:11) + 120 (cid:104) x (cid:105) (cid:10) x (cid:11) (cid:10) x (cid:11) − (cid:104) x (cid:105) (cid:10) x (cid:11) + 30 (cid:10) x (cid:11) − (cid:104) x (cid:105) (cid:10) x (cid:11) + 360 (cid:104) x (cid:105) (cid:10) x (cid:11) − (cid:104) x (cid:105) . (A8)Fourth- and higher-order cumulants are not identical to the corresponding central mo-ments. Thus cumulants are certain polynomial functions of the moments and providean alternative to the moments of the distribution.The above equations can be inverted to express moments in terms of cumulants: (cid:104) x (cid:105) = κ , (cid:10) x (cid:11) = κ + κ , (cid:10) x (cid:11) = κ + 3 κ κ + κ , (cid:10) x (cid:11) = κ + 6 κ κ + 4 κ κ + 3 κ + κ , (cid:10) x (cid:11) = κ + 10 κ κ + 10 κ κ + 15 κ κ + 5 κ κ + 10 κ κ + κ , (cid:10) x (cid:11) = κ + 6 κ κ + 15 κ κ + 20 κ κ + 15 κ κ + 60 κ κ κ + 45 κ κ + 15 κ κ + 10 κ + 15 κ + κ . (A9) The n -particle correlation function (or simply a correlator) ρ (1 , , , · · · , n ) consistsof terms that represent combinations of lower-order correlations and a term thatrepresents a genuine or “true” n -particle correlation C (1 , , , · · · , n ) which is calleda cumulant. For example, ρ (1 ,
2) = ρ (1) ρ (2) + C (1 , C (1 ,
2) = ρ (1 , − ρ (1) ρ (2). If the two particles are statistically independent, ρ (1 ,
2) simply reducesto ρ (1) ρ (2), whereas C (1 ,
2) vanishes by construction. The first few correlation For simplicity of notation, we use the same symbols ρ and C to denote 1-,2-,3- andmultiparticle correlation functions and cumulants, respectively.8 Will be inserted by the editor functions are [89] ρ (1) = C (1) ,ρ (1 ,
2) = ρ (1) ρ (2) + C (1 , ,ρ (1 , ,
3) = ρ (1) ρ (2) ρ (3) + ρ (1) C (2 ,
3) + ρ (2) C (3 ,
1) + ρ (3) C (1 ,
2) + C (1 , , , ≡ ρ (1) ρ (2) ρ (3) + (cid:88) (3) ρ (1) C (2 ,
3) + C (1 , , ,ρ (1 , , ,
4) = ρ (1) ρ (2) ρ (3) ρ (4) + (cid:88) (6) ρ (1) ρ (2) C (3 ,
4) + (cid:88) (4) ρ (1) C (2 , , (cid:88) (3) C (1 , C (3 ,
4) + C (1 , , , ,ρ (1 , , , ,
5) = ρ (1) ρ (2) ρ (3) ρ (4) ρ (5) + (cid:88) (10) ρ (1) ρ (2) ρ (3) C (4 , (cid:88) (10) ρ (1) ρ (2) C (3 , ,
5) + (cid:88) (15) ρ (1) C (2 , C (4 , (cid:88) (5) ρ (1) C (2 , , ,
5) + (cid:88) (10) C (1 , C (3 , ,
5) + C (1 , , , , ,ρ (1 , , , , ,
6) = ρ (1) ρ (2) ρ (3) ρ (4) ρ (5) ρ (6) + (cid:88) (6) ρ (1) C (2 , , , , (cid:88) (15) ρ (1) ρ (2) C (3 , , ,
6) + (cid:88) (20) ρ (1) ρ (2) ρ (3) C (4 , , (cid:88) (15) ρ (1) ρ (2) ρ (3) ρ (4) C (5 ,
6) + (cid:88) (60) ρ (1) C (2 , C (4 , , (cid:88) (45) ρ (1) ρ (2) C (3 , C (5 ,
6) + (cid:88) (15) C (1 , C (3 , , , (cid:88) (10) C (1 , , C (4 , ,
6) + (cid:88) (15) C (1 , C (3 , C (5 , C (1 , , , , , , (B1)where the numbers in parentheses under the summation signs indicate the number ofpossible permutations of the indices. ill be inserted by the editor 29 The above equations can be inverted to get the expressions for the cumulants interms of the correlation functions: C (1) = ρ (1) ,C (1 ,
2) = ρ (1 , − ρ (1) ρ (2) ,C (1 , ,
3) = ρ (1 , , − (cid:88) (3) ρ (1) ρ (2 ,
3) + 2 ρ (1) ρ (2) ρ (3) ,C (1 , , ,
4) = ρ (1 , , , − (cid:88) (4) ρ (1) ρ (2 , , − (cid:88) (3) ρ (1 , ρ (3 , (cid:88) (6) ρ (1) ρ (2) ρ (3 , − ρ (1) ρ (2) ρ (3) ρ (4) ,C (1 , , , ,
5) = ρ (1 , , , , − (cid:88) (5) ρ (1) ρ (2 , , , − (cid:88) (10) ρ (1 , ρ (3 , , (cid:88) (10) ρ (1) ρ (2) ρ (3 , ,
5) + 2 (cid:88) (15) ρ (1) ρ (2 , ρ (4 , − (cid:88) (10) ρ (1) ρ (2) ρ (3) ρ (4 ,
5) + 24 ρ (1) ρ (2) ρ (3) ρ (4) ρ (5) ,C (1 , , , , ,
6) = ρ (1 , , , , , − (cid:88) (6) ρ (1) ρ (2 , , , , − (cid:88) (15) ρ (1 , ρ (3 , , , (cid:88) (15) ρ (1) ρ (2) ρ (3 , , , − (cid:88) (10) ρ (1 , , ρ (4 , , (cid:88) (30) ρ (1) ρ (2 , ρ (4 , , − (cid:88) (20) ρ (1) ρ (2) ρ (3) ρ (4 , , (cid:88) (15) ρ (1 , ρ (3 , ρ (5 , − (cid:88) (90) ρ (1) ρ (2) ρ (3 , ρ (5 , (cid:88) (15) ρ (1) ρ (2) ρ (3) ρ (4) ρ (5 , − (cid:88) (120) ρ (1) ρ (2) ρ (3) ρ (4) ρ (5) ρ (6) . (B2)It is easy to verify that the cumulant vanishes if any one or more particles is statisti-cally independent of the others. Thus the n -particle cumulant measures the statisticaldependence of the entire n -particle set. For this reason, cumulants are also called con-nected correlation functions. Exercise 11 : Compare the above expressions of cumulants to those in Appendix A.
Exercise 12 : Note how the above expressions of correlators and cumulants simplifyconsiderably if the single-particle ρ vanishes. One-dimensional Gaussian (centered at µ ) is f ( x ) = 1 σ √ π exp (cid:20) − ( x − µ ) σ (cid:21) , (C1) where µ = (cid:104) x (cid:105) is the mean, σ is the variance and f ( x ) is normalized to unity. Theodd central moments are obviously zero. The even central moments are given by( n − σ n ( n even). The first few even central moments are given in Table 1. Theskewness ( γ ) is zero, kurtosis ( κ ) is 3 and excess kurtosis ( κ −
3) is zero. A probabilitydistribution with tails heavier (lighter) than those of the normal distribution showsa higher (lower) propensity to produce outliers and has a positive (negative) excesskurtosis (Fig 14),
Table 1.
First few central and noncentral moments of the Gaussian distributionOrder Central moment Noncentral moment1 0 µ σ µ + σ µ + 3 µσ σ µ + 6 µ σ + 3 σ µ + 10 µ σ + 15 µσ σ µ + 15 µ σ + 45 µ σ + 15 σ µ + 21 µ σ + 105 µ σ + 105 µσ σ µ + 28 µ σ + 210 µ σ + 420 µ σ + 105 σ For the normal distribution, the moment-generating function is M ( t ) = exp ( µt +( σ t / K ( t ) = ln M ( t ) = µt + ( σ t / t , only the first two cumulants survive, namely the mean µ and the variance σ . It can be shown that the normal distribution is the only onefor which the third and higher cumulants vanish. Exercise 13 : Using the expressions for moments in terms of cumulants given inAppendix A, show that the first six noncentral moments of the normal distributionare as given in Table 1.
Two-dimensional Gaussian centered at the origin and normalized to unity is f ( x, y ) = 12 πσ x σ y exp (cid:20) − x σ x − y σ y (cid:21) , (C2)where σ x and σ y are the variances. Exercise 14 : For an asymmetric ( σ x (cid:54) = σ y ) 2D Gaussian as in Eq. (C2), let z = x + iy .Show that the fourth moment (cid:104) z (cid:105) (cid:54) = 0, and is trivially correlated with the secondmoment (cid:104) z (cid:105) . However, the fourth-order cumulant (cid:104) z (cid:105) − (cid:104) z (cid:105) = 0.It is often convenient to introduce the symmetric case ( σ x = σ y ) and write it interms of r = (cid:112) x + y : f ( r ) = 1 πσ exp (cid:20) − r σ (cid:21) , (C3)where σ ≡ σ x + σ y . f ( r ) is also centered at the origin and normalized to unity.Note, however, that unlike (cid:104) x (cid:105) and (cid:104) y (cid:105) , (cid:104) r (cid:105) is not zero. The first few moments of f ( r ), ill be inserted by the editor 31 µ n = (cid:104) r n (cid:105) = σ n Γ (( n/
2) + 1), are given in Table 2. Note also that (cid:10) r (cid:11) = (cid:10) x (cid:11) + (cid:10) y (cid:11) ;using (cid:10) r (cid:11) from Table 2, we get σ = σ x + σ y . The kurtosis in this case is 2, unlikethe case of a 1D Gaussian discussed earlier where it was 3. Table 2.
First few moments ( µ n ) of f ( r ), Eq. (C3) n µ n √ π σ/ σ √ π σ / σ √ π σ / σ √ π σ /
16 24 σ References
1. S. A. Chin, Phys. Lett. (1978) 552 doi:10.1016/0370-2693(78)90637-82. F. B. Yano and S. E. Koonin, Phys. Lett. (1978) 556 doi:10.1016/0370-2693(78)90638-X3. S. A. Voloshin, A. M. Poskanzer and R. Snellings, Landolt-Bornstein (2010) 293[arXiv:0809.2949 [nucl-ex]].4. B. Alver and G. Roland, Phys. Rev. C (2010) 054905, Erratum: [Phys. Rev.C (2010) 039903] doi:10.1103/PhysRevC.82.039903, 10.1103/PhysRevC.81.054905[arXiv:1003.0194 [nucl-th]].5. U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci. (2013), 123-151doi:10.1146/annurev-nucl-102212-170540 [arXiv:1301.2826 [nucl-th]].6. D. Teaney and L. Yan, Phys. Rev. C , no. 2 (2014) 024902doi:10.1103/PhysRevC.90.024902 [arXiv:1312.3689 [nucl-th]].7. J. Y. Ollitrault, Phys. Rev. D (1992) 229. doi:10.1103/PhysRevD.46.2298. S. Voloshin and Y. Zhang, Z. Phys. C (1996) 665 doi:10.1007/s002880050141 [hep-ph/9407282].9. M. Luzum, J. Phys. G (2011) 124026 doi:10.1088/0954-3899/38/12/124026[arXiv:1107.0592 [nucl-th]].10. A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C (1998) 1671doi:10.1103/PhysRevC.58.1671 [nucl-ex/9805001].11. J. Y. Ollitrault, nucl-ex/9711003.12. S. Acharya et al. [ALICE Collaboration], arXiv:2002.00633 [nucl-ex].13. V. Khachatryan et al. [CMS Collaboration], Phys. Rev. C (2015) no.3, 034911doi:10.1103/PhysRevC.92.034911 [arXiv:1503.01692 [nucl-ex]].14. P. Bozek, W. Broniowski and J. Moreira, Phys. Rev. C (2011) 034911doi:10.1103/PhysRevC.83.034911 [arXiv:1011.3354 [nucl-th]].15. N. Borghini, P. M. Dinh and J. Y. Ollitrault, Phys. Rev. C (2001) 054901doi:10.1103/PhysRevC.64.054901 [nucl-th/0105040].16. R. S. Bhalerao and J. Y. Ollitrault, Phys. Lett. B (2006) 260doi:10.1016/j.physletb.2006.08.055 [nucl-th/0607009].17. P. Di Francesco, M. Guilbaud, M. Luzum and J. Y. Ollitrault, Phys. Rev. C (2017)no.4, 044911 doi:10.1103/PhysRevC.95.044911 [arXiv:1612.05634 [nucl-th]].18. Z. Moravcova, K. Gulbrandsen and Y. Zhou, [arXiv:2005.07974 [nucl-th]].19. G. Aad et al. [ATLAS Collaboration], JHEP (2013) 183doi:10.1007/JHEP11(2013)183 [arXiv:1305.2942 [hep-ex]].20. S. Acharya et al. [ALICE Collaboration], JHEP (2018) 103doi:10.1007/JHEP07(2018)103 [arXiv:1804.02944 [nucl-ex]].21. J. Y. Ollitrault, A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C (2009) 014904doi:10.1103/PhysRevC.80.014904 [arXiv:0904.2315 [nucl-ex]].22. A. M. Sirunyan et al. [CMS Collaboration], Phys. Lett. B (2019) 643doi:10.1016/j.physletb.2018.11.063 [arXiv:1711.05594 [nucl-ex]].2 Will be inserted by the editor23. S. A. Voloshin, A. M. Poskanzer, A. Tang and G. Wang, Phys. Lett. B (2008) 537doi:10.1016/j.physletb.2007.11.043 [arXiv:0708.0800 [nucl-th]].24. M. Aaboud et al. [ATLAS], JHEP (2020), 051 doi:10.1007/JHEP01(2020)051[arXiv:1904.04808 [nucl-ex]].25. L. Yan, J. Y. Ollitrault and A. M. Poskanzer, Phys. Rev. C (2014) no.2, 024903doi:10.1103/PhysRevC.90.024903 [arXiv:1405.6595 [nucl-th]].26. G. Giacalone, L. Yan, J. Noronha-Hostler and J. Y. Ollitrault, Phys. Rev. C (2017)no.1, 014913 doi:10.1103/PhysRevC.95.014913 [arXiv:1608.01823 [nucl-th]].27. R. S. Bhalerao, G. Giacalone and J. Y. Ollitrault, Phys. Rev. C (2019) no.1, 014907doi:10.1103/PhysRevC.99.014907 [arXiv:1811.00837 [nucl-th]].28. R. S. Bhalerao, G. Giacalone and J. Y. Ollitrault, Phys. Rev. C (2019) no.1, 014909doi:10.1103/PhysRevC.100.014909 [arXiv:1904.10350 [nucl-th]].29. S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B (2013) 213doi:10.1016/j.physletb.2013.06.028 [arXiv:1305.0609 [nucl-ex]].30. V. Khachatryan et al. [CMS Collaboration], Phys. Lett. B (2017) 193doi:10.1016/j.physletb.2016.12.009 [arXiv:1606.06198 [nucl-ex]].31. S. Chatrchyan et al. [CMS Collaboration], JHEP (2014) 088doi:10.1007/JHEP02(2014)088 [arXiv:1312.1845 [nucl-ex]].32. A. Dumitru, F. Gelis, L. McLerran and R. Venugopalan, Nucl. Phys. A (2008) 91doi:10.1016/j.nuclphysa.2008.06.012 [arXiv:0804.3858 [hep-ph]].33. F. G. Gardim, F. Grassi, M. Luzum and J. Y. Ollitrault, Phys. Rev. C (2013) no.3,031901 doi:10.1103/PhysRevC.87.031901 [arXiv:1211.0989 [nucl-th]].34. M. Aaboud et al. [ATLAS Collaboration], Eur. Phys. J. C (2018) no.2, 142doi:10.1140/epjc/s10052-018-5605-7 [arXiv:1709.02301 [nucl-ex]].35. G. Aad et al. [ATLAS Collaboration], arXiv:2001.04201 [nucl-ex].36. R. S. Bhalerao, M. Luzum and J. Y. Ollitrault, Phys. Rev. C (2011), 034910doi:10.1103/PhysRevC.84.034910 [arXiv:1104.4740 [nucl-th]].37. R. S. Bhalerao, M. Luzum and J. Y. Ollitrault, J. Phys. G (2011), 124055doi:10.1088/0954-3899/38/12/124055 [arXiv:1106.4940 [nucl-ex]].38. J. Adam et al. [ALICE Collaboration], Phys. Rev. Lett. (2016) 182301doi:10.1103/PhysRevLett.117.182301 [arXiv:1604.07663 [nucl-ex]].39. S. Acharya et al. [ALICE Collaboration], Phys. Rev. C (2018) no.2, 024906doi:10.1103/PhysRevC.97.024906 [arXiv:1709.01127 [nucl-ex]].40. A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. (2018) no.9, 092301doi:10.1103/PhysRevLett.120.092301 [arXiv:1709.09189 [nucl-ex]].41. S. Acharya et al. [ALICE], Phys. Rev. Lett. (2019) no.14, 142301doi:10.1103/PhysRevLett.123.142301 [arXiv:1903.01790 [nucl-ex]].42. C. Mordasini, A. Bilandzic, D. Karako¸c and S. F. Taghavi, Phys. Rev. C (2020)no.2, 024907 doi:10.1103/PhysRevC.102.024907 [arXiv:1901.06968 [nucl-ex]].43. R. S. Bhalerao, J. Y. Ollitrault and S. Pal, Phys. Rev. C (2013) 024909doi:10.1103/PhysRevC.88.024909 [arXiv:1307.0980 [nucl-th]].44. G. Aad et al. [ATLAS Collaboration], Phys. Rev. C (2014) no.2, 024905doi:10.1103/PhysRevC.90.024905 [arXiv:1403.0489 [hep-ex]].45. S. Acharya et al. [ALICE Collaboration], Phys. Lett. B (2017) 68doi:10.1016/j.physletb.2017.07.060 [arXiv:1705.04377 [nucl-ex]].46. A. Bilandzic, M. Lesch and S. F. Taghavi, Phys. Rev. C (2020) no.2, 024910doi:10.1103/PhysRevC.102.024910 [arXiv:2004.01066 [nucl-ex]].47. D. Teaney and L. Yan, Phys. Rev. C (2012), 044908 doi:10.1103/PhysRevC.86.044908[arXiv:1206.1905 [nucl-th]].48. J. Qian, U. W. Heinz and J. Liu, Phys. Rev. C (2016) no.6, 064901doi:10.1103/PhysRevC.93.064901 [arXiv:1602.02813 [nucl-th]].49. R. S. Bhalerao, J. Y. Ollitrault and S. Pal, Phys. Lett. B (2015), 94-98doi:10.1016/j.physletb.2015.01.019 [arXiv:1411.5160 [nucl-th]].50. L. Yan and J. Y. Ollitrault, Phys. Lett. B (2015), 82-87doi:10.1016/j.physletb.2015.03.040 [arXiv:1502.02502 [nucl-th]].ill be inserted by the editor 3351. J. Qian, U. Heinz, R. He and L. Huo, Phys. Rev. C (2017) no.5, 054908doi:10.1103/PhysRevC.95.054908 [arXiv:1703.04077 [nucl-th]].52. G. Aad et al. [ATLAS], Phys. Rev. C (2015) no.3, 034903doi:10.1103/PhysRevC.92.034903 [arXiv:1504.01289 [hep-ex]].53. M. Aaboud et al. [ATLAS Collaboration], Phys. Rev. C (2017) no.2, 024908doi:10.1103/PhysRevC.96.024908 [arXiv:1609.06213 [nucl-ex]].54. V. Pac´ık [ALICE Collaboration], Nucl. Phys. A (2019) 451doi:10.1016/j.nuclphysa.2018.09.020 [arXiv:1807.04538 [nucl-ex]].55. R. S. Bhalerao, N. Borghini and J. Y. Ollitrault, Phys. Lett. B (2004), 157-162doi:10.1016/j.physletb.2003.11.056 [arXiv:nucl-th/0307018 [nucl-th]].56. R. S. Bhalerao, N. Borghini and J. Y. Ollitrault, Nucl. Phys. A (2003), 373-426doi:10.1016/j.nuclphysa.2003.08.007 [arXiv:nucl-th/0310016 [nucl-th]].57. C. Aidala et al. [PHENIX Collaboration], Nature Phys. (2019) no.3, 214doi:10.1038/s41567-018-0360-0 [arXiv:1805.02973 [nucl-ex]].58. J. Adam et al. [ALICE Collaboration], Nature Phys. (2017) 535doi:10.1038/nphys4111 [arXiv:1606.07424 [nucl-ex]].59. R. D. Weller and P. Romatschke, Phys. Lett. B (2017) 351doi:10.1016/j.physletb.2017.09.077 [arXiv:1701.07145 [nucl-th]].60. Y. Zhou, W. Zhao, K. Murase and H. Song, [arXiv:2005.02684 [nucl-th]].61. S. Acharya et al. [ALICE Collaboration], JHEP (2018) 013doi:10.1007/JHEP11(2018)013 [arXiv:1802.09145 [nucl-ex]].62. M. Strickland, Nucl. Phys. A (2019) 92 doi:10.1016/j.nuclphysa.2018.09.071[arXiv:1807.07191 [nucl-th]].63. G. Aad et al. [ATLAS Collaboration], Phys. Rev. C (2012) 014907doi:10.1103/PhysRevC.86.014907 [arXiv:1203.3087 [hep-ex]].64. C. Gale, S. Jeon, B. Schenke, P. Tribedy and R. Venugopalan, Phys. Rev. Lett. (2013) no.1, 012302 doi:10.1103/PhysRevLett.110.012302 [arXiv:1209.6330 [nucl-th]].65. F. G. Gardim, F. Grassi, M. Luzum and J. Y. Ollitrault, Phys. Rev. Lett. (2012)202302 doi:10.1103/PhysRevLett.109.202302 [arXiv:1203.2882 [nucl-th]].66. P. Romatschke, Phys. Rev. Lett. (2018) no.1, 012301doi:10.1103/PhysRevLett.120.012301 [arXiv:1704.08699 [hep-th]].67. W. Florkowski, M. P. Heller and M. Spalinski, Rept. Prog. Phys. (2018) no.4, 046001doi:10.1088/1361-6633/aaa091 [arXiv:1707.02282 [hep-ph]].68. L. He, T. Edmonds, Z. W. Lin, F. Liu, D. Molnar and F. Wang, Phys. Lett. B (2016) 506 doi:10.1016/j.physletb.2015.12.051 [arXiv:1502.05572 [nucl-th]].69. H. Li, L. He, Z. W. Lin, D. Molnar, F. Wang and W. Xie, Phys. Rev. C (2016) no.5,051901 doi:10.1103/PhysRevC.93.051901 [arXiv:1601.05390 [nucl-th]].70. N. Borghini and C. Gombeaud, Eur. Phys. J. C (2011), 1612 doi:10.1140/epjc/s10052-011-1612-7 [arXiv:1012.0899 [nucl-th]].71. A. Kurkela, U. A. Wiedemann and B. Wu, Eur. Phys. J. C (2019) no.9, 759doi:10.1140/epjc/s10052-019-7262-x [arXiv:1805.04081 [hep-ph]].72. K. Dusling, M. Mace and R. Venugopalan, Phys. Rev. Lett. (2018) no.4, 042002doi:10.1103/PhysRevLett.120.042002 [arXiv:1705.00745 [hep-ph]].73. K. Dusling, M. Mace and R. Venugopalan, Phys. Rev. D (2018) no.1, 016014doi:10.1103/PhysRevD.97.016014 [arXiv:1706.06260 [hep-ph]].74. M. Mace, V. V. Skokov, P. Tribedy and R. Venugopalan, Phys. Rev.Lett. (2018) no.5, 052301 Erratum: [Phys. Rev. Lett. (2019) no.3,039901] doi:10.1103/PhysRevLett.123.039901, 10.1103/PhysRevLett.121.052301[arXiv:1805.09342 [hep-ph]].75. C. Zhang, C. Marquet, G. Y. Qin, S. Y. Wei and B. W. Xiao, Phys. Rev. Lett. (2019) no.17, 172302 doi:10.1103/PhysRevLett.122.172302 [arXiv:1901.10320 [hep-ph]].76. C. Zhang, C. Marquet, G. Y. Qin, Y. Shi, L. Wang, S. Y. Wei and B. W. Xiao,arXiv:2002.09878 [hep-ph].77. L. McLerran, doi:10.3204/DESY-PROC-2009-01/26 arXiv:0812.4989 [hep-ph].78. A. Dumitru, K. Dusling, F. Gelis, J. Jalilian-Marian, T. Lappi and R. Venugopalan,Phys. Lett. B (2011) 21 doi:10.1016/j.physletb.2011.01.024 [arXiv:1009.5295 [hep-ph]].4 Will be inserted by the editor79. B. Schenke, S. Schlichting, P. Tribedy and R. Venugopalan, Phys. Rev. Lett. (2016)no.16, 162301 doi:10.1103/PhysRevLett.117.162301 [arXiv:1607.02496 [hep-ph]].80. A. Ortiz [ALICE and ATLAS and CMS and LHCb Collaborations], PoS LHCP (2019) 091 doi:10.22323/1.350.0091 [arXiv:1909.03937 [hep-ex]].81. S. Morrow [PHENIX Collaboration], arXiv:1810.05321 [nucl-ex].82. J. L. Nagle and W. A. Zajc, Ann. Rev. Nucl. Part. Sci. (2018) 211doi:10.1146/annurev-nucl-101916-123209 [arXiv:1801.03477 [nucl-ex]].83. B. Schenke, Nucl. Phys. A (2017) 105 doi:10.1016/j.nuclphysa.2017.05.017[arXiv:1704.03914 [nucl-th]].84. W. Li, Nucl. Phys. A (2017) 59 doi:10.1016/j.nuclphysa.2017.05.011[arXiv:1704.03576 [nucl-ex]].85. S. Schlichting and P. Tribedy, Adv. High Energy Phys. (2016) 8460349doi:10.1155/2016/8460349 [arXiv:1611.00329 [hep-ph]].86. C. Loizides, Nucl. Phys. A (2016) 200 doi:10.1016/j.nuclphysa.2016.04.022[arXiv:1602.09138 [nucl-ex]].87. K. Dusling, W. Li and B. Schenke, Int. J. Mod. Phys. E (2016) no.01, 1630002doi:10.1142/S0218301316300022 [arXiv:1509.07939 [nucl-ex]].88. P. H Westfall, The American Statistician , no. 3 (2014) 191 doi:10.1080/00031305.2014.917055.89. A. Bzdak and P. Bozek, Phys. Rev. C93