Flow Fluctuations from Early-Time Correlations in Nuclear Collisions

Sean Gavina and George Moschellib

a) Department of Physics and Astronomy, Wayne State University, 666 W Hancock, Detroit, MI, 48202, USAb) Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe University,

Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany(Dated: November 5, 2018)

We propose that flow fluctuations have the same origin as transverse momentum fluctuations.The common source of these fluctuations is the spatially inhomogeneous initial state that driveshydrodynamic flow. Longitudinal correlations from an early Glasma stage followed by hydrodynamicflow quantitatively account for many features of multiplicity and pt fluctuation data. We developa framework for studying flow and its fluctuations in this picture. We then compute elliptic andtriangular flow fluctuations, and study their connections to the ridge.

PACS numbers: 25.75.Gz, 25.75.Ld, 12.38.Mh, 25.75.-q

I. INTRODUCTION

Fluctuations in the early stages of nuclear collisionscontribute to the anisotropic flow measured in RHIC andLHC experiments, particularly the odd harmonics. InRef. [1] we find that this early-time variation can alsoaccount for the multiplicity and transverse momentumfluctuations measured in the same experiments. Never-theless, flow and fluctuation observables reveal differentaspects of the initial state. Harmonic flow is generatedlargely by the global spatial anisotropy of the initial state.In contrast, fluctuation observables probe local spatialcorrelations that are largely independent of the overallgeometry [1].

In this paper we argue that the fluctuations of the har-monic flow coefficients are driven by the very same localcorrelations that give rise to pt and multiplicity fluctu-ations. Moreover, if early time correlations are the onlysource of these fluctuations, then the two measurementsare related with no free parameters.

Experimenters define flow fluctuations by studying thedifference of the flow coefficients vn{2} and vn{4} mea-sured using two and four particle correlations, respec-tively [2–5]. Jets, resonance decays, and HBT effects alsocontribute to that difference. Measurements using parti-cles separated in rapidity by more than 1− 2 units elim-inate these short range effects. Significantly, a differencebetween vn{2} and vn{4} remains [6], suggesting thatflow fluctuations are a long range phenomenon. Causal-ity dictates that such long range correlations originate inthe early stages of the collision [7, 8].

To compute flow fluctuations we build on an approachstarted in Refs. [8, 9] in which long range correlationsresult from the fragmentation of Glasma flux tubes. Inthis formulation local spatial correlations emerge fromfluctuations in the number and distribution of flux tubes.These spatial correlations are then modified by transverseexpansion, giving rise to azimuthal correlations. A simi-lar physical picture motivates studies using a wide rangeof different techniques [10–17].

This paper is organized as follows. In Sec.II, we be-gin with a general description of multi-particle correla-

tions and flow coefficients. We relate the intrinsic correla-tions in the multi-particle system to flow and its fluctua-tions. To begin, we write expressions for vn{2} and vn{4}in terms of correlation functions, adapting the tools in-vented in Refs. [18, 19] to our framework. Further resultsare in the appendix. We next define flow fluctuations interms of vn{2} and vn{4}, and turn to discuss the phys-ical interpretation of this definition.

We discuss the common influence of local correlationson flow and pt fluctuations in Sec.III. This relationship isthe heart of our work. The fluctuations of these quanti-ties are both consequences the spatially inhomogeneouscollision environment modified by hydrodynamic flow.The general arguments in Secs. II and III set the stagefor the more phenomenological analysis that follows.

In Sec.IV we describe initial state fluctuations in aCGC-Glasma picture. The number and distribution ofGlasma flux tubes relative to the geometrical shape of thesystem ultimately determine the energy, projectile-mass,and centrality dependence of flow fluctuations. Next, inSec.V we explain how collective flow and the correlationto the reaction plane modify local correlations.

We calculate the contribution of long range correla-tions to the flow coefficients and their fluctuations inSec.V. Our results are in good agreement with the latestLHC and RHIC data. In Sec.VI, we argue that thesesame flow fluctuations are also responsible for the ridge.The same factors that influence flow fluctuations alsodetermine the multiplicity and momentum fluctuationscomputed in [1]. We emphasize that no new parame-ters are introduced here, so that results can be compareddirectly with [1].

In Sec.VII we discuss the extent to which the flow co-efficients determine the angular distribution of the ridge.We also discuss the possible factorization of the Fouriercoefficients of the pair distribution into products of flowcoefficients [20–23]. We show that flow fluctuations canviolate factorization at low pt, depending on how the flowcoefficients are defined. We then speculate that factor-ization holds for momenta > 1− 2 GeV regardless of thesource of anisotropy.

Lastly, we summarize and discuss the broader implica-

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tions of our results in Sec.VIII.

II. FLOW AND ITS FLUCTUATIONS

Nuclear collisions at non-zero impact parameter b pro-duce anisotropic flow [24, 25]. This anisotropy derivesfrom the change in the shape of the collision volume withrespect to the reaction plane, i.e., the plane spanned byb and the beam direction. If the reaction plane is known,this anisotropy is characterized by the moments

〈vn〉 = 〈cosn(φ− ψRP

)〉, (1)

where φ is the azimuthal angle, ψRP

is the angle of thereaction plane, and the brackets denote an average overparticles and events. While the reaction plane cannotbe observed directly, its influence can be deduced frommulti-particle correlation measurements. Many strate-gies have been employed for measuring these flow coeffi-cients, and they all have strengths and weaknesses.

Experimenters often deduce the flow from the two-particle cumulant

vn{2}2 = 〈cosn(φ1 − φ2)〉, (2)

exploiting the reaction plane information implicit in therelative distribution of particle pairs. Specifically, theymeasure the relative azimuthal angle for each particlepair in each event, and then average over events to obtain

vn{2}2 =〈∑i 6=j cosn(φi − φj)〉〈N(N − 1)〉

(3)

[26]. The cumulant method was developed in Ref. [18, 19]and has seen extensive use [24, 25].

In this section we cast the framework of [18, 19] interms of our formulation of correlations in [1] to studyevent-wise flow fluctuations. Many results in this sectionare implicit in [18, 19]; only our application to the com-putation of fluctuations is new. We start by writing (3)as

vn{2}2 =

∫ρ2(p1,p2)

〈N(N − 1)〉cosn(φ1 − φ2) dp1dp2, (4)

where the integrals are over the momenta of particles pifor i = 1, 2 and the distribution of particle pairs is

ρ2(p1,p2) =dN

dp1dp2. (5)

In the absence of correlations, ρ2(p1,p2) →ρ1(p1)ρ1(p2), where the single particle distributionis ρ1(p) = dN/dp. We stress that the densities ρ1

and ρ2 are event-averaged quantities that respect thereaction plane. The factorization of ρ2 then allows forfactorization of (4) such that vn{2}2 → 〈vn〉2.

Including two-particle correlations, the pair distribu-tion does not factorize. To identify the contributions of

the mean anisotropic flow (1) and genuine two-particlecorrelations to vn{2}, we follow [18] and write (5) interms of a cumulant expansion,

ρ2(p1,p2) = ρ1(p1)ρ1(p2) + r(p1,p2), (6)

where r is the two-particle correlation function. We thenuse (1), (4), and (6) to write

vn{2}2 = 〈vn〉2 + 2σ2n. (7)

This result is standard, and the factor of two in σ isconventional [27]. The contribution to (7) from two-bodycorrelations is

σ2n =

∫dp1dp2

r(p1,p2)

2〈N(N − 1)〉cosn∆φ, (8)

where ∆φ = φ1 − φ2 is the relative azimuthal angle ofthe correlated particles.

Strictly speaking, this quantity measures the event-wise fluctuations of anisotropic flow due to geometry anddynamics together with “non-flow” correlations from res-onance decays, the HBT effect, and jets. We will notaddress non-flow effects here. Observe that all such cor-relation effects are of order 1/N , where N is the numberof particles in the system. We focus on contributions tofluctuations at leading order in N . Accordingly, we omita factor 〈N〉2/〈N(N −1)〉 ≈ 1 that multiplies 〈vn〉2 termin (7); see appendix A for exact formulae.

To reduce the effect of fluctuations on the flow signal,experimenters also measure the four particle cumulantvn{4} defined by the relation

vn{4}4 = 2vn{2}4 − 〈cosn(φ1 + φ2 − φ3 − φ4)〉 (9)

[18, 19]. The four-angle term depends of the four particlecorrelation function ρ4(1, 2, 3, 4) ≡ ρ4(p1,p2,p3,p4). Insystems with large numbers of particles, multi-particlecorrelations are dominated by two-particle correlations.We therefore write

ρ4(1, 2, 3, 4) = ρ1(1)ρ1(2)ρ1(3)ρ1(4)

+ ρ1(1)ρ1(2)r(3, 4) + . . .

+ r(1, 2)r(3, 4) + . . . (10)

where ρ1(1) ≡ ρ1(p1), etc., and the ellipses representsthe distinct permutations of the momenta. Computing〈cosn(φ1 + φ2 − φ3 − φ4)〉, one finds that the second andthird lines in (10) contribute respectively terms of order〈vn〉2σ2

n and σ4n that precisely cancel the σn contributions

from vn{2}4. This cancellation is by design – it dictatesthe form of (9) proposed in Ref. [18, 19]. One finds

vn{4} ≈ 〈vn〉, (11)

plus corrections that are in practice very small. Thefact that vn{4} receives no contribution from σn is usu-ally thought of as a consequence of the Bessel-Gaussianapproximation [27], but we see that it is true whenevertwo-body correlations are dominant. However, note that

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partN

STAR Au-Au 200 GeV v2{4}STAR Au-Au 200 GeV v2{2}

FIG. 1. Measured v2{2} and v2{4} from STAR [2] comparedto calculations. Computed v2{2} (solid) uses (7) and (8),while v2{4} (dashed) uses (1), (11), and (24). The differencebetween the calculated curves is due to flow fluctuations.

definitions of 〈vn〉 other than (1) would yield differentresults.

As an aside, we briefly discuss the corrections to (11).Strictly speaking, calculation of (9) using (10) yields

vn{4}2 ≈ 〈vn〉2 − 2Σ2n, (12)

where

Σ2n =

∫dp1dp2

r(p1,p2)

〈N(N − 1)〉cos 2n(Φ− ψ

RP) (13)

for Φ = (φ1 + φ2)/2 the average angle of the correlatedpairs. Further corrections of higher order in 1/N arediscussed in appendix A. Both Σn and σn are of the sameorder in 1/N , but one expects Σn to be smaller than σnbecause it effectively is a higher harmonic, of order 2nrather than n [18]. Indeed, we find the contribution ofΣn to vn{4} smaller than 1.2% for n = 2 for peripheralcollisions in our model. Note that Σn = 0 in centralcollisions. We comment that Σn represents fluctuationsof harmonics of Φ, which are important in studying theChiral Magnetic Effect [28].

Our aim is to understand the contribution of two-particle correlations to flow fluctuation measurements. Ifthe only correlations are due to the reaction plane thenvn{2} = vn{4}. This requires that no other sources ofcorrelation exist, i.e., r(p1,p2) = 0, allowing ρ2(p1,p2)to factorize. However, experiments measure differencesbetween vn{2} and vn{4} for several harmonic orders,n [2–5]. For example, Fig.1 shows v2{2} and v2{4} asmeasured by the STAR experiment in 200 GeV Au+Aucollisions [2]. One can quantify this difference in terms ofσn by substituting the square of (11) into (7), to obtain

σ2n =

vn{2}2 − vn{4}2

2. (14)

Measurements with rapidity separations should suppresscontributions to σn from jets, resonance decays, and

other short range non-flow correlations, but do not ex-plain the difference between vn{2} and vn{4}, suggestingthat 〈v2

n〉-factorization is not a signature of flow or flowfluctuations.

An alternative measure of flow fluctuations can be con-structed using event-by-event harmonics [27, 29, 30]. Ifeach event produces a set of harmonics {v̂n}, then onecan compute the variance σ̂2

n = 〈v̂2n〉−〈v̂n〉2 for an ensem-

ble of such events. It is then reasonable to ask how ourσn compares to this variance. Taking the mean square ofv̂n to be 〈v̂2

n〉 = 〈(∑i cosn(φi −Ψ

RP))2〉/〈N〉2, we find

σ̂2n = σ̂2

stat + σ2n + Σ2

n/2, (15)

where σn and Σn are given by (8) and (13), and we keeponly leading order in N . We define the statistical vari-ance

σ̂2stat =

1 + 〈v2n〉2〈N〉

. (16)

These fluctuations arise because a finite number of parti-cles are sampled in each event. In contrast, the dynamicalfluctuations σn and Σn are caused by the correlations be-tween particles [31]. To an excellent approximation thev2n contribution to (16) and the Σ2

n contribution to (15)can be neglected.

We see that the event-wise variance of v̂n is very dif-ferent from the quantity (14) that characterizes the dif-ference between vn{2} and vn{4}. The statistical contri-bution σ̂stat can be comparable in magnitude to σn. Wepoint out that the quantity σ̂n may be measured directlyfrom event-by-event harmonic coefficients.

It is instructive to compare our results to the model ofRef. [27, 30] in which flow fluctuations are exclusively dueto the event-wise variation of the global geometry. In thepurely geometric interpretation, one defines an eccentric-ity for each event ε̂n. If one assumes that the relation be-tween ε̂n and the resulting anisotropy of the fluid flow v̂nis approximately deterministic, then fluctuations of theratio v̂n/ε̂n are negligible. The geometric contributionto the variance is then σ̂2

n/〈vn〉2 = (〈ε̂2n〉 − 〈ε̂n〉2)/〈ε̂n〉2.We mention that Glauber model estimates of the relevanteccentricities are consistent with data for n = 2, wherethe correlation of v̂n with an event-wise eccentricity ishighly plausible and well established [32]. However, theapplicability of such an approach for n 6= 2 is less clear.

We comment that the purely geometrical descriptionof flow fluctuations in Ref. [30] requires 〈v̂n〉 � σ̂n in ad-dition to v̂n ∝ ε̂n. Neither our general discussion here norour specific calculations in Sec. V require these assump-tions. In Sec. V, we include geometric fluctuations in ourestimates of flow fluctuations through the distribution offlux tube sources ρ

FT. Correlations are then modified by

the local transverse expansion of the system.

4

III. LOCAL CORRELATIONS

To understand the source of flow fluctuations, we recallsome lessons from general fluctuation studies [31, 33–35].Such studies typically focus on the variation of bulk ob-servables such as multiplicity or transverse momentumwithin an ensemble of collisions. We draw most heavilyfrom the study of pt fluctuations in Ref. [1], to which thepresent work is essentially a sequel.

We measure pt fluctuations using the covariance〈δpt1δpt2〉, where δpti = pti − 〈pt〉 is the deviation ofeach pt from the average. Pairs in which both particleshave higher than average momentum add to 〈δpt1δpt2〉.Lower-than-average pairs also add to the covariance,while high/low pairs subtract from it. In global equi-librium 〈δpt1δpt2〉 ≡ 0. The presence of hot spots makes〈δpt1δpt2〉 > 0 (as would cold spots) [36]. Motion of thesources further enhances this quantity [10]. It followsthat both jets and flow add to the pt covariance.

Moving hot spots in the fluctuating system are sourcesof local correlations. We distinguish such correlationsfrom those due to the global event-wise variation of theshape and size of the collision volume. We write the ptcovariance as

〈δpt1δpt2〉 =

∫dp1dp2

r(p1,p2)

〈N(N − 1)〉δpt1δpt2. (17)

In [1] we demonstrate that the pt covariance is indepen-dent of the global spatial anisotropy, because (17) is in-tegrated over azimuthal angle. It is also independent ofsize fluctuations by construction [37].

To see that flow fluctuations are also primarily drivenby local correlations, observe that (17) has the same formas (8) with δpt1δpt2 replaced by cosn∆φ. As the pt co-variance probes correlations in momentum, so σn mea-sures correlations of the angular separation ∆φ of pairs.

This analogy makes physical sense because of the com-mon influence of hot spots on ∆φ and pt. Flow pushesparticles from a source into a particular opening angle,depending on the source position and the local fluid ve-locity. The same flow velocity boosts the pt of theseparticles. Therefore multiple random hot spots give riseto both flow and pt fluctuations. Based on this analogy,we expect σn in central collisions to be non-zero for allorders n, while 〈vn〉 defined by (1) vanishes. Further-more, it is not likely that σn is as strongly influenced bythe global geometry as the mean 〈vn〉.

In the next sections we discuss flow fluctuations dueto early time local spatial correlations contributing tor(p1,p2). We employ a Glasma-flux tube model in whichparticles are initially correlated at the point of produc-tion and modified by later stage transverse expansion.These flux tubes produce our ‘hot spots’. Transverseexpansion is modeled with a blast wave scenario whichinherits anisotropy from the average eccentricity 〈ε

RP〉.

IV. SOURCE OF CORRELATIONS

Flow transforms early-time spatial correlations intoobservable momentum correlations. Two-particle corre-lations probe the initial state average pair distributionn2(x1,x2), which encompasses both genuine pair corre-lation as well as correlations due to fluctuating event ec-centricities, shapes, and orientations (with respect to thereaction plane). Define the initial state spatial correla-tion function

c(x1,x2) = n2(x1,x2)− n1(x1)n1(x2), (18)

where n1(xi) is the particle density. In the absenceof genuine two-particle correlations the pair distribu-tion factors into a product of one body distributions, sothat the correlation function vanishes. Correlations comeabout because the position and number of the collidingnucleons fluctuates in individual events. In general, thesefluctuations result in high-density ‘hot spots’ of vary-ing size distributed throughout the collision volume. Itis more likely to find pairs of particles near hot spots.Averaging over an ensemble of events therefore yields anonzero c(x1,x2). We define the correlation strength Ras∫c(x1,x2)d3x1d

3x2 ≡ 〈N〉2R.In Ref.[1] we construct a correlation function (18) to

describe a specific early-time scenario based on a Glasmatheory. In that picture parton production arises from thefragmentation of longitudinal color fields created at themoment of the collision. It is helpful to think of thelongitudinal fields as flux tubes of transverse size ∼ Q−1

s ,where Qs is the saturation scale. The saturation scaledepends on many collision variables including the densityof participant nucleons and the collision energy [38, 39].

Particle production then proceeds via a system of fluxtubes distributed over the collision volume. Given thatthe transverse area of a flux tube is much smaller than thetransverse collision area and taking particles emergingfrom the same tube as correlated, we write

c(x1,x2) = 〈N〉2R δ(rt)ρFT(Rt). (19)

The spatial coordinates are relative, rt = rt1 − rt2, andaverage, Rt = (rt1 + rt2)/2, transverse elliptical coordi-nates. The delta function enforces the correlation dueto the common point of production, and ρ

FT(Rt) is the

average transverse distribution of flux tubes

ρFT

(Rt) ≈2

πR2A

(1− R2

t

R2A

). (20)

Here the shape of (20) resembles the nuclear thicknessfunction and πR2

A is the area of the overlap region. Thedistribution (20) represents an average over all possibleshapes, which we have taken to be a simple ellipse.

To compute the correlation strength R, we imagineeach event produces K flux tubes with transverse size∼ Q−2

s . In the saturation regime K is proportional tothe transverse area R2

A divided by the area per flux tube,

5

Q−2s [38]. Allowing K to fluctuate from event to event

with average 〈K〉, we calculate in Ref.[8] that

R =〈N2〉 − 〈N〉2 − 〈N〉

〈N〉2∝ 〈K〉−1. (21)

Each Glasma flux tube yields an average multiplicity of ∼α−1s (Qs) gluons and as in Ref.[38], the number of gluons

in a rapidity interval ∆y is then

〈N〉 = (dN/dy)∆y ∼ αs−1(Qs)〈K〉. (22)

Finally, the Glasma correlation scale,

RdN/dy = κα−1s (Q2

s), (23)

follows from the multiplication of (21) and (22). Notice(23) is dimensionless and depends only on the saturationscale, Q2

s, which can be calculated from first principles.Measurements of the ridge at various beam energies, tar-get masses, and centralities fix the dimensionless coeffi-cient κ and are in excellent accord with the leading-orderdependence [8, 9].

In the next section, we will see how transverse expan-sion modifies Glasma correlations (19), but it is impor-tant to note that (23) significantly impacts the centralitydependence and determines virtually all of the energydependence.

V. FLOW FLUCTUATIONS FROM GLASMAAND TRANSVERSE EXPANSION

Transverse collective expansion gives rise to momen-tum space correlations in two ways. First, the eventgeometry influences global correlations; collective flowemerges with respect to a reaction plane. Second, thelocal fluctuations, that together determine the globalshape, individually induce local correlations; flow, on av-erage, modifies the momenta from all partons emergingfrom the same fluctuation in a common way.

To understand how flow modifies local correlationswe first model the average anisotropic expansion of thesystem with a blast wave model [8, 9, 40–42]. As inRefs.[8, 9] the invariant single-particle distribution,

ρ1 (p) ≡ dN

dyd2pt=

∫f (x,p) dΓ, (24)

is defined as the particle flux through a surface, σ[43]. At constant proper time, τF , the flux elementis dΓ = pµdσµ = τFmt cosh(y − η)dηd2r, whereη = (1/2) ln((t + z)/(t − z)) is the spatial rapidity.Here f(x,p) = (2π)−3 exp{−uµpµ/T} is the Boltzmannphase-space density for a temperature T and fluid four-velocity uµ. We follow Ref.[40] and write the four ve-locity of the longitudinal-boost invariant blast wave asuµ = γt(cosh η,vt, sinh η), with vt, the transverse veloc-

ity and γt = (1 − v2t )−1/2. The phase space density is

then f ∝ exp{−γtmt cosh(y − η)/T} exp{γtvt · pt/T}.

To account for the initial elliptical shape andanisotropic velocity we require one additional parame-ter, the (average) event eccentricity, ε. The anisotropictransverse flow velocity,

γtvt = λ (εxx + εyy) , (25)

mimics the geometrical parton density gradients. To ac-count for the larger pressure in the direction of the reac-tion plane (x-direction in (25)), we take εx,y =

√1± ε.

Rewriting Eq.(25) as (γtvt)2 = λ2r2(1 + ε cos(2φ)), one

can see how this choice of flow velocity results in an asym-metric flow velocity peaked in the direction of the reac-tion plane. Note, r is now the elliptical polar radius anddepends on ε and the spatial angle.

In this formulation, particles passing through thefreeze out surface have a velocity, vs, that is connected tothe position of production, thus the transverse velocityis constrained via γ2

t v2s = λ2R2

surface. As in [1], we take

the blast wave parameters vs and T from experiment [44]and parameterize ε to fit v2. However, in Ref.[1], we didnot include the impact of fluctuations on the calculationof vn-like correlations - a focus in this work. Hence, inRef.[1], ε is chosen to fit v2{2}. In this work we identify〈v2〉BW

with purely reaction plane angular correlations,more suitably comparable to v2{4}. In Fig.1, the dashedline shows this comparison where the open squares arethe STAR Au-Au 200 GeV v2{4} measurement [2].

Now that our blast wave formalism is tuned to re-produce the average transverse expansion in heavy ioncollisions, we are in a position to calculate the momen-tum space two-particle correlation function. As discussedby other authors, see e.g. [45–47], the two-particle jointprobability distribution (5) which we rewrite as

r(p1,p2) ≡ ρ2(p1,p2)− ρ1(p1)ρ1(p2) (26)

to highlight the fact that any genuine correlations, in-cluding those from local initial state fluctuations (18),break the factorization ρ2(p1,p2)→ ρ1(p1)ρ1(p2).

The spatial correlation function, (19), embodies thecorrespondence between the two particle joint probabil-ity distribution and transverse expansion. Intrinsicallycorrelated partons originate from the same transverse po-sition and experience, on average, the same transversemomentum modulation from flow. Generalizing (24), werepresent the genuine two-particle correlation as

r(p1,p2) =

∫c(x1,x2)f(x1,p1)f(x2,p2)dΓ1dΓ2. (27)

Notice that full integration of (27) yields 〈N〉2R as does(18), tying azimuthal correlations to multiplicity andtransverse momentum fluctuations [1].

To compute flow fluctuations using this correlationfunction, we combine (8) with (19) and (27). We then use(7), (11), and (8) to calculate v2{2} and v2{4} in Fig.1,where the flow coefficient relative to the reaction planeis 〈vn〉 =

∫ρ1(p) cosn(φ−Ψ

RP)dp. It is significant that

6

σ2n ∝ R; this is also true for the long range contribution

to other fluctuation quantities [1].The influence of event shapes and eccentricities en-

ters (27) in two ways. First, the correlation functionc(x1,x2) includes a probability distribution of sources(20) that implicitly accounts for event-wise variation ofthe volume and eccentricity. Integration over spatial co-ordinates effectively represents an average over all possi-ble event shapes. Second, the azimuthal distribution ofparticles from each source is modified by flow. A pairemerging from a source that experiences a greater pushwill have a narrower opening angle. The magnitude ofthe push follows (25) and depends on position.

It is important to note that any radial expansion willinduce correlations contributing to all orders of vn. Par-ticular choices of the flow velocity affect their relativecontributions, i.e. an elliptical flow velocity profile in-duces a large v2 and a triangular flow velocity profilewould induce a large v3. Equation (25) is chosen to mimicthe average (elliptical) flow profile.

The long range behavior introduced by Glasma corre-lations provides key centrality and collision energy de-pendence on the saturation scale Q2

s through R in (23).Although, the blast wave parameters T , vs, and ε alsoprovide significant centrality dependence, their changewith collision energy is minimal. The average vs andfreeze out temperature in 62 GeV and 200 GeV Au+Aucollisions are the same as those used in [1, 8, 9] and arebased on an analysis in [44]. At 2.76 TeV the velocityis scaled up from the 200 GeV values by 6% and thetemperature is scaled up by 7% as presented in [1].

In Fig.2 we show comparisons of our calculated v2{2}(top panel) and v2{4} (bottom panel) to measured data[2, 4, 5] and find the agreement concerning change in col-lision energy is quite good. Similarly in Fig.3 we calcu-late v4{2} and v4{4}, represented as the solid and dashedlines, respectively. Comparisons to vn{4} in both figuresshow how the eccentricity ε affects the results. Recallthat ε is tuned to fit v2 in Au+Au 200 GeV collisions.The effect of eccentricity is to reduce the even harmonicsas the collision area becomes circular and ε → 0. Thenon-zero vn{2} in central collisions are due to fluctua-tions (14), which in our case are due to Glasma correla-tions (27). The effect of the change in blast wave param-eters with collision energy is apparent from the separa-tion of the solid lines in the bottom panel of Fig.2 andthe dashed lines in Fig. 3. The importance of the Glasmacontribution is similarly evident by the change is separa-tion between vn{4} and vn{2} with collision energy. Theincrease in growth of flow fluctuations (8) is coupled tothe growth in the saturation scale.

STAR has also measured v2 fluctuations in the form of

σvn〈vn〉

=

√vn{2}2 − vn{4}2vn{2}2 + vn{4}2

(28)

resembling the so-called “coefficient of variation” definedas the standard deviation divided by the mean. Care

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STAR Au-Au 62.4 GeV v2{4}

STAR Au-Au 200 GeV v2{4}ALICE Pb-Pb 2.76 TeV v2{4}v2{4}

FIG. 2. Measured v2{2} (top) from STAR [2] and ALICE[4, 5] compared to calculations using (7) and (8). Same forv2{4} (bottom) computed from (1), (11), and (24)

.

0 50 100 150 200 250 300 350 4000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.024v

partN

Au-Au 62.4 GeV

Au-Au 200 GeV

Pb-Pb 2.76 TeV

ALICE Pb-Pb 2.76 TeV v4{2}

FIG. 3. Same calculation as in Fig.2 but for the fourth orderflow coefficients v4{2} (solid) and v4{4} (dashed). Data isfrom the ALICE collaboration [4].

should be taken here since the definition of σ2n, Eq.(14),

is not strictly the variance. In Fig.4 we compare our cal-culation of (28) to measurement. Calculations at RHICenergies seem to agree reasonably well and we include thecalculation for Pb+Pb 2.76 GeV as a prediction.

Observe that local correlations that contribute to σnlead to fluctuations of all harmonic orders in n includingv3. As a result, Glasma correlations generate a measur-able v3{2} even in the absence of v3{4} or any triangular

7

0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1>2<v

2vσ

partN

Au-Au 62.4 GeV

Au-Au 200 GeV

Pb-Pb 2.76 TeV

STAR Au-Au 62.4 GeVSTAR Au-Au 200 GeV

FIG. 4. Flow fluctuations in terms of the coefficient of varia-tion for elliptic flow (28) compared to STAR data [2].

0 50 100 150 200 250 300 350 4000

0.005

0.01

0.015

0.02

0.0253vσ

partN

σv3 = 0.5*(v3{2}2-v3{4}2)

Au-Au 62.4 GeV

Au-Au 200 GeV

Pb-Pb 2.76 TeVSTAR Au-Au 200 GeV

ALICE Pb-Pb 2.76 TeV

FIG. 5. Triangular flow fluctuations, σv3 , calculated from(8) and compared to data computed from v3{2} and v3{4}measurements by the STAR and ALICE collaborations [3, 4].

flow. In Fig.5 we show v3 fluctuations from Glasma. Wefurther emphasize that the energy dependence in Fig.5 isin good accord with data, supporting the Glasma scalingwith Q2

s. The shape with centrality reflects contributionsnot only from Glasma, but also our parameterizations ofthe average reaction plane eccentricity ε and our choicethe flux tube distribution ρ

FT.

To use (8) and (14) to calculate v3{2}, we must cometo grips with the fact that our blast wave parametrizationassumes v3{4} = 0. While this seems to be the case forthe STAR measurements, ALICE has measured a non-zero v3{4} for Pb+Pb collisions at 2.76 TeV. To correctfor this possible discrepancy, we can provide an ad-hocparameterization of v3{4} and use (8) and (14) to calcu-late v3{2}. Agreement shown in Fig.6 is reasonable. Asa preferable alternative, we compare our calculated σn tofluctuations extracted from ALICE and STAR measure-ments in Fig.5.

0 50 100 150 200 250 300 350 4000

0.005

0.01

0.015

0.02

0.025

0.03

0.0353v

partN

Au-Au 62.4 GeV

Au-Au 200 GeV

Pb-Pb 2.76 TeV

STAR Au-Au 200 GeV v3 Preliminary ALICE Pb-Pb 2.76 TeV v3{2}

FIG. 6. The triangular flow coefficient, v3{2} compared toSTAR and ALICE data [3, 4].

VI. THE RIDGE

Early analyses of the soft or untriggered ridge, includ-ing our own, focus on the idea that the ridge is com-posed of correlations in excess of momentum conserva-tion and elliptic flow. The novel observation of Alver andRoland that correlations resulting from geometrical fluc-tuations result in odd flow harmonics such as triangularflow, v3, suggests that v3, together with flow harmonics ofall higher orders, explains this excess [48]. In this paperwe have studied how angular correlations tied to com-mon production points influence two-particle correlationmeasurements of flow harmonics vn{2} through flow fluc-tuations σ2

n. Most notably, as discussed in Sec.V, we findsignificant contributions to v3{2} even in the absence oftriangular flow. In fact, the same correlations contribut-ing to flow fluctuations were initially proposed as expla-nations of the ridge [7–12, 14–17, 49, 50], although theirrole in reproducing the full ∆φ azimuthal structure wasnot understood.

Flow and its fluctuations both contribute to the two-particle correlation landscape. The pair distribution is

ρ2(∆φ) = ρref

(1 + 2

∞∑n=1

〈vn〉2 cosn∆φ

)+ r(∆φ), (29)

where ρref is the experimental mixed-event background.Observe that the Fourier coefficients of (29) reproduce(7). We take ρref = 1

2π

∫ρ1

∫ρ1 where the overbar indi-

cates an event plane average, to mimick the experimentalmixed event technique for constructing ρref . The quan-tity ∆ρ(∆φ) = ρ2(∆φ) − ρref characterizes the angularcorrelations. STAR measures the ratio

∆ρ(∆φ)√ρref

=1

2π

dN

dη

(2

∞∑n=1

〈vn〉2 cosn∆φ+r(∆φ)

ρref

),

(30)where the 〈vn〉 terms follow from (1).

STAR performs a multi-parameter fit to this distribu-tion to separate the ridge peak from the elliptic flow con-tributions (along with momentum conservation and HBT

8

1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3∆ρ ρ

ν=2*Nbin/Npart

Au-Au 62.4 GeV

Au-Au 200 GeV

Pb-Pb 2.76 TeV

STAR Au-Au 200 GeVSTAR Au-Au 62.4 GeV

ALICE Pb-Pb 2.76 TeV Preliminary

FIG. 7. Near side ridge amplitude calculation from Glasmasource correlations. Experimental data is from (STAR) [51–53] and (ALICE) [54]

contributions , which we omit for clarity). They report aflow-subtracted ridge amplitude (∆ρ(∆φ,∆η)/

√ρref)|FS

on the near side, centered at ∆φ = ∆η = 0 [51–53]. Weuse (23) and (27) to calculate the flow-subtracted ridgeamplitude

∆ρ√ρref

∣∣∣∣FS

= RdNdy

F (∆φ), (31)

where F (∆φ) ∝ r(∆φ) is the angular correlation functionnormalized so that

∫F (∆φ)d∆φ = 1. The energy and

centrality independent scale constant κ in (23) is fixedby Au-Au 200 GeV data as in [8, 9].

The blast wave parameters have little energy depen-dence and the Glasma factor (23) allows for strong agree-ment with the 62 GeV data without adjustment of κ. Ad-ditionally, the ALICE collaboration has measured (31)in Pb+Pb collisions at

√s =2.76 TeV [54], providing

a further test of the CGC-Glasma energy dependence.We confront the ALICE data in the same way, withoutadjusting κ, however the ALICE [54] measurement pro-cedure differs slightly than that from STAR [53]. Thisdifference in fitting procedures is mostly insignificant, es-pecially in central collisions, but not necessarily so in pe-ripheral collisions to which we normalize our calculation(at 200 GeV). To adjust for this possible effect, we mul-tiply (31) by an additional scale factor of ∼ 1.5 to returnagreement in peripheral collisions; see note [55]. In Fig.7we compare calculated ridge amplitudes from (31) withmeasured data for 200 and 62.4 GeV Au+Au collisionsfrom STAR as well as 2.76 TeV Pb+Pb collisions fromALICE. In this paper we only compare to measurementsfrom which elliptic flow has been subtracted, because wefind that fluctuations dominate v3 at STAR.

The characterization of the ridge in terms of Fouriercoefficients can prove a valuable tool for analysis of suchdiscrepancies as well as relate the magnitude its effectwith respect to other phenomena. Viewing (14) in terms

of (8) and (31), one can write

2dN

dyσ2n ≈

∫∆ρ(∆φ)√ρref

cos(n∆φ) d∆φ. (32)

A similar equation was pointed out by Sorensen et. al.[56, 57], and discussed in terms of event plane eccentricityfluctuations.

We stress, that (32) is a general result that is modelindependent. Direct measurement of this quantity canisolate the effects of flow fluctuations on the ridge by com-paring the results to independent flow fluctuation mea-surements. Furthermore, flow fluctuation measurementscan be an effective way to quantify the ridge. Devia-tions from (32) at large ∆η. can indicate the degree atwhich other phenomena contribute to long range corre-lation measurements. At smaller ∆η shorter range phe-nomenon such as jets or diffusion also come into play.Combined flow fluctuation and ridge studies analyzedwith increasing lower pt limits as done in [58] can revealthe emergence of jets and their influence. For example,one could test contributions to vn{2} from correlationsinduced by jet quenching as suggested in [9, 59].

VII. FACTORIZATION

We now ask whether the azimuthal dependence of thepair distribution is entirely determined by the flow co-efficients. This is somewhat of a circular question, be-cause flow coefficients are themselves derived from cor-relation analyses. A meaningful answer depends on howthe flow coefficients are defined. Experimentalists expandthe transverse-momentum-dependent pair distribution as

ρ2 ∝ 1 + 2∑n

Vn∆(pt1, pt2) cos(n∆φ). (33)

The term “factorization” is often applied when one canexpress the Fourier coefficients Vn∆(pt1, pt2) as a productof flow coefficients vn(pti). Authors often cite factoriza-tion as a signature of collective flow [20–22, 60]; see also[23] for a theoretical perspective. Problems stem fromthe fact that Vn∆ ≡ vn{2}2 as an exact consequence ofthe definition (4). In other words, Vn∆ always factorizesas vn{2} × vn{2}, regardless of the source of anisotropy.

To define factorization in a useful way, one must com-pare Vn∆ to the coefficients 〈vn(pti)〉 defined by the reac-tion plane as in (1). Proxies like vn{4} can also be used.To see why this is true, we define rn to be the Fouriercoefficient of the two particle correlation function r(∆φ)in (6). We then compare (33) to (29) to find

Vn∆(pt1, pt2) = 〈vn(pt1)〉〈vn(pt2)〉+rn(pt1, pt2)

ρref(pt1, pt2), (34)

where, for simplicity, we keep only the leading order inthe number of particles in the relevant pt ranges. Thefirst term in (34) measures the effect of global geometric

9

anisotropy, while the second term is due to local fluc-tuations. Integrating over momenta and using (7), weindeed find

Vn∆ = 〈vn〉2 + 2σ2n ≡ vn{2}2, (35)

as noted above. However, (34) shows that the Fouriercoefficients of the correlation function rn(pt1, pt2) breakfactorization in terms of 〈vn(pt2)〉.

We see that hydrodynamic flow does not generally im-ply factorization, as follows from (27) and (34). Specifi-cally, (32) shows that flow fluctuations violate factoriza-tion. In addition, jets, HBT, and resonances contributeto rn along with the long range correlations we have con-sidered here.

Flow coefficients at the LHC are found to factorize forpairs of high pt particles, but not when high and low ptparticles are mixed [60]. This may seem puzzling if youview factorization as a hydrodynamic signature. To un-derstand this result, observe that the contribution rn/ρref

in (34) can be large compared to anisotropy only whena substantial fraction of pairs in the selected momentumrange are correlated. RHIC energy measurements showthat the overall magnitude of the correlation functiondrops precipitously relative to mixed-event pairs when alower pt cutoff is increased above ∼ 1 GeV [58]. Our cal-culations in [9], which include both flow and jet-mediuminteractions, agree with that trend; see, e.g., fig. 9 of[9]. We therefore do not expect significant violations offactorization from the components rn due to long rangecorrelations from flow or jets above pti > 1− 2 GeV. Ananalogous trend at higher beam energy may explain theLHC results.

VIII. SUMMARY

In this paper we study the connection between flowfluctuations and initial state correlations. Flow fluctua-tions are defined by (14) using two and four-particle cor-relation measurements of the harmonic flow coefficients.Section II provides a general discussion of the calculationof flow observables from multiparticle momentum spacedistributions. Following the cumulant expansion methoddiscussed in [18, 19], multiparticle distributions can beexpanded into factorized and correlated parts, e.g. (5)and (10); correlated parts contribute directly to the fluc-tuation observable σn.

We study the contribution of local long range correla-tions to σn based on parton production at common trans-verse positions. Section V presents calculations of σn inour Glasma flux tube model. We mention that our modelincorporates much of the same physics as used by event-by-event hydro in [17, 61–65]. Those authors obtain theinitial state in individual events by sampling a probabilitydistribution analogous to our ρ

FT. In all cases, Cooper

Fry freeze out is used. We use a blast wave constrainedby data to approximate the velocity distribution and thefreeze-out surface. The key differences are that 1) our

correlated regions are point-like in the transverse plane,while theirs may be larger and 2) they evolve their ini-tial distributions using deterministic hydrodynamics ineach event. We expect that they will eventually achievea better description of the vn at each energy. Our com-plementary aim is to study the big picture by exploringthe energy range with a variety of variables.

The Glasma formulation provides key collision sys-tem, energy, and centrality dependences resulting in rea-sonable agreement with experimental measurements ofvn{2} from 62.4 GeV Au+Au to 2.76 TeV Pb+Pb colli-sions for n=2, 3, and 4. The presence of σn provides twokey results: non-zero values of even harmonics in central(circular) collisions, and the existence of v3{2} withouttriangular flow.

In Sec.VI we turn to discuss two particle correlationsand, in particular, the ridge. We see that flow fluctua-tions and the ridge are facets of the same phenomenon,related by a Fourier transform (32). The computed peakridge amplitude in Fig. 7 is in reasonable agreement withcalculations from 62.4 GeV Au+Au to 2.76 TeV Pb+Pb,consistent with our flow results. Further investigation isneeded to understand the full correlation landscape. Inparticular, we did not discuss the n = 1 harmonic, inwhich momentum conservation plays an important role.Characterization of the ridge in terms of flow fluctuations(i.e., harmonics) combined with earlier fit procedures asin Ref. [53] will surely prove a useful tool.

We discuss the question of whether the Fourier coef-ficients of the correlation function factorize into a prod-uct of flow coefficients in Sec.VII. In general, irreducibletwo-particle correlations from any mechanism violate thisfactorization [20–22]. In particular, the long range corre-lations that we compute violate factorization at low pt,as would jets, resonance decays, and HBT effects. We ex-pect any such correlations to become negligible for pairsabove pt ∼ 1 − 2 GeV, due to the rapid decrease of thenumber of correlated pairs relative to the mixed-eventbackground that is observed experimentally [58].

Finally, we observe that early-time fluctuations havebroad implications beyond the flow fluctuations and az-imuthal correlations studied here. In Ref. [1] we stud-ied the impact of these correlations on multiplicity andpt fluctuations. We argued in Sec. III that pt and flowfluctuations are intimately related because they are bothdriven by local hydrodynamic fluctuations. While theaverage flow coefficients are primarily determined by theglobal event shape, the fluctuations of these coefficientshave more in common with other fluctuation observables.Importantly, the overall magnitude of the contributionfrom long range correlations to all of these fluctuations –flow, multiplicity, and pt – is set by a single scale factorR.In Glasma theory, the dependence of R on energy, cen-trality, and projectile mass are fixed its variation with thesaturation scale Qs. Results here and in Ref. [1] providea survey of the ridge, and multiplicity, momentum andflow fluctuations that reveals a common energy and cen-trality dependence that we attribute to the production

10

mechanism. Glasma calculations are consistent with thisdependence.

ACKNOWLEDGMENTS

We thank M. Bleicher, H. Caines, C. DeSilva, J. Jia,A. Majumder, P. Mota, C. Pruneau, P. Sorensen, R.Snellings, A. Timmins and S. Voloshin for discussions.S.G. thanks the Institute for Nuclear Theory in Seattle,Washington for hospitality during the completion of thiswork. S.G. thanks the Institute for Nuclear Theory inSeattle, Washington for hospitality during the comple-tion of this work. This work was supported in part bythe U.S. NSF grant PHY-0855369 (SG) and The AllianceProgram of the Helmholtz Association (HA216/EMMI)(GM)

Appendix A: Four-Particle Flow Coefficients

In this appendix we calculate corrections to the four-particle flow coefficient measurement vn{4} from two-particle correlation sources. Trying to keep the notationas general as possible we follow [18] using the cumulantexpansion method to write

〈eın(φ1−φ2)〉 =

∫ρ2(p1,p2)eın(φ1−φ2)dp1dp2∫

ρ2(p1,p2)dp1dp2(A1)

=〈N〉2〈vn〉2

〈N(N − 1)〉+

∫r(p1,p2)eın∆φdp1dp2

〈N(N − 1)〉, (A2)

where vn{2}2 = 〈eın(φ1−φ2)〉. Notice that since sin(n∆φ)is odd, the second term in (A2) is equivalent to (8) whenthe pair distribution is symmetric about the normal tothe reaction plane. Based on the definition (21) the fac-tor 〈N〉2/〈N(N−1)〉 in the first term of (A2) contributesa correction of 1/(1 +R) ≈ 1 as long as the multiplicity,N , is large. For example, in central 200 GeV Au+Aucollision we calculate R ≈ 0.003. This number will de-crease with increasing collision energy but will increaseas collisions become more peripheral. This behavior isdiscussed in [1].

Our goal is to calculate the four-particle flow coefficient

vn{4}4 = 2〈eın(φ1−φ2)〉2 − 〈eın(φ1+φ2−φ3−φ4)〉, (A3)

where the final term is calculated analogously to (A1) butfrom the four-particle cumulant expansion (10). Keepingonly contributions from two-particle correlations we have

〈N(N − 1)(N − 2)(N − 3)〉〈eın(φ1+φ2−φ3−φ4)〉 = (A4a)

= 〈N〉4〈vn〉4 (A4b)

+ 〈N〉2〈vn〉2 · 2Re

{∫r(p1,p2)eı2n(Φ−ψ

RP)dp1dp2

}(A4c)

+ 4〈N〉2〈vn〉2∫r(p1,p2)eın∆φdp1dp2 (A4d)

+

∣∣∣∣∫ r(p1,p2)eı2n(Φ−ψRP

)dp1dp2

∣∣∣∣2 (A4e)

+ 2

∣∣∣∣∫ r(p1,p2)eın∆φdp1dp2

∣∣∣∣2 . (A4f)

where ∆φ = φ1−φ2 and Φ = (φ1 +φ2)/2 are the relativeand average coordinates. To reduce the equations furtherit is necessary to make the approximation that 〈N(N −1)(N − 2)(N − 3)〉 ≈ 〈N(N − 1)〉2 then the terms (A4d)and (A4f) will cancel with corresponding terms emergingfrom the twice the square of (A2), leaving the corrections

vn{4}4 = 〈vn〉4 − 〈vn〉2 · 2Re{

Σ2n

}−∣∣Σ2n

∣∣2 , (A5)

where

Σ2n =

∫r(p1,p2)eı2n(Φ−ψ

RP)dp1dp2

〈N(N − 1)〉(A6)

which is equivalent to (13). Notice that these correctionsare effectively of order 2n and depend on the reactionplane. As discussed in the text, the corrections in (A5)maximally modify (11) by 1.2% in our model.

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