Triangular flow in hydrodynamics and transport theory
Burak Han Alver, Clement Gombeaud, Matthew Luzum, Jean-Yves Ollitrault
aa r X i v : . [ nu c l - t h ] S e p Triangular flow in hydrodynamics and transport theory
Burak Han Alver
Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
Cl´ement Gombeaud, Matthew Luzum, and Jean-Yves Ollitrault
CNRS, URA2306, IPhT, Institut de physique theorique de Saclay, F-91191 Gif-sur-Yvette, France (Dated: September 24, 2018)In ultrarelativistic heavy-ion collisions, the Fourier decomposition of the relative azimuthal an-gle, ∆ φ , distribution of particle pairs yields a large cos(3∆ φ ) component, extending out to largerapidity separations ∆ η >
1. This component captures a significant portion of the ridge and shoul-der structures in the ∆ φ distribution, which have been observed after contributions from ellipticflow are subtracted. An average finite triangularity due to event-by-event fluctuations in the initialmatter distribution, followed by collective flow, naturally produces a cos(3∆ φ ) correlation. Usingideal and viscous hydrodynamics, and transport theory, we study the physics of triangular ( v ) flowin comparison to elliptic ( v ), quadrangular ( v ) and pentagonal ( v ) flow. We make quantitativepredictions for v at RHIC and LHC as a function of centrality and transverse momentum. Ourresults for the centrality dependence of v show a quantitative agreement with data extracted fromprevious correlation measurements by the STAR collaboration. This study supports previous re-sults on the importance of triangular flow in the understanding of ridge and shoulder structures.Triangular flow is found to be a sensitive probe of initial geometry fluctuations and viscosity. I. INTRODUCTION
Correlations between particles produced in ultrarela-tivistic heavy-ion collisions have been thoroughly studiedexperimentally. Correlation structures previously iden-tified in proton-proton collisions have been observed tobe modified and patterns which are specific to nucleus-nucleus collisions have been revealed. The dominant fea-ture in two-particle correlations is elliptic flow, one of theearly observations at RHIC [1]. Elliptic flow leads to acos(2∆ φ ) term in the distribution of particle pairs withrelative azimuthal angle ∆ φ . More recently, additionalstructures have been identified in azimuthal correlationsafter accounting for contributions from elliptic flow. [2–7].An excess of correlated particles are observed in a narrow“ridge” near ∆ φ = 0 and the away side peak at ∆ φ = π is wider in comparison to proton-proton collisions. Forcentral collisions and high transverse momentum triggers,the away side structure develops a dip at ∆ φ = π withtwo “shoulders” appearing. These ridge and shoulderstructures persist for large values of the relative rapidity∆ η , which means that they are produced at a very earlytimes [8].It has been recently argued [9] that both the ridge andthe shoulder are natural consequences of the triangularflow ( v ) produced by a triangular fluctuation of the ini-tial distribution. The purpose of this paper is to carryout a systematic study of v using relativistic viscous hy-drodynamics, which is the standard model for ultrarela-tivistic heavy-ion collisions [10]. We also perform trans-port calculations [11], because they allow us to check therange of validity of viscous hydrodynamics, and also be-cause they provide further insight into the physics. Alongwith v , we also investigate v (quadrangular flow) and v (pentagonal flow). In Sec. II, we recall why odd momentsof the azimuthal distributions, such as v , are relevant. In Sec. III, we study the general properties of anisotropicflow induced by a harmonic deformation of the initialdensity profile using hydrodynamics and kinetic theory.In Sec. IV, we present our predictions for v and v atRHIC and LHC. The contribution of quadrangular fluc-tuations to v is difficult to evaluate because v also hasa large contribution from elliptic flow [12]: this will bestudied in a forthcoming publication [13]. II. CORRELATIONS FROM FLUCTUATIONS
A fluid at freeze-out emits particles whose azimuthaldistribution f ( φ ) depends on the distribution of the fluidvelocity [12]. f ( φ ) can generally be written as a Fourierseries f ( φ ) = 12 π + ∞ X n =1 v n cos( nφ − nψ n ) ! (1)where v n are the coefficients of anisotropic flow [14] whichare real and positive, and ψ n is defined modulo 2 π/n (for v n = 0). Equivalently, one can write h e inφ i ≡ Z π e inφ f ( φ ) dφ = v n e inψ n , (2)where angular brackets denote an average value over out-going particles.Generally, v n is measured using the event-planemethod [15]. However, two-particle correlation measure-ments are also sensitive to anisotropic flow. Consider apair of particles with azimuthal angles φ , φ = φ + ∆ φ .Assuming that the only correlation between the particlesis due to the collective expansion, Eq. (2) gives h e in ∆ φ i = h e inφ e − inφ i = h e inφ ih e − inφ i = ( v n ) . (3)The left-hand side can be measured experimentally, and v n can thus be extracted from Eq. (3) [16]. Experimen-tally, one averages over several events. v n fluctuatesfrom one event to the other, and the observable mea-sured through Eq. (3) is the average value of ( v n ) . Itcan be shown that the event-plane method also measuresthe RMS, p v n , unless the “reaction plane resolution” isextremely good [17, 18].Most fluid calculations of heavy-ion collisions are donewith smooth initial profiles [19–23]. These profiles aresymmetric with respect to the reaction plane ψ R , so thatall ψ n in Eq. (1) are equal to ψ R (with this convention,all v n are not necessarily positive). For symmetric colli-sions at midrapidity, smooth profiles are also symmetricunder φ → φ + π , so that all odd harmonics v , v , etc.are identically zero. However, it has been shown thatfluctuations in the positions of nucleons within the col-liding nuclei may lead to significant deviations from thesmooth profiles event-by-event [24, 25]. They result inlumpy initial conditions which have no particular sym-metry, and this lumpiness should be taken into accountin fluid dynamical calculations [26–29]. More precisely,one should calculate the azimuthal distribution for eachinitial condition, then average over initial conditions.Initial geometry fluctuations are a priori important forall v n , as anticipated in Ref. [30]. Their effect on flowmeasurements has already been considered for ellipticflow v [31, 32] and quadrangular flow v [33]. Event-by-event elliptic flow fluctuations have been measuredand found to be significantly large, consistent with thefluctuations in the nucleon positions [34]. Directed flow, v , is constrained by transverse momentum conservationwhich implies P p t v ( p t ) = 0 and will not be consideredhere. In this paper, we study triangular flow v [9], andpentagonal flow v , which arise solely due to initial ge-ometry fluctuations. III. FLOW FROM HARMONICDEFORMATIONS ψ FIG. 1: (Color online) Contour plots of the energy density(4) for n = 3 and ε = 0 . Elliptic flow is the response of the system to an ini-tial distribution with an elliptic shape in the transverseplane ( x, y ) [35]. In this article, we study the response tohigher-order deformations. For sake of simplicity, we as-sume in this section that the initial energy profile in the transverse plane ( x, y ) is a deformed Gaussian at t = t : ǫ ( x, y ) = ǫ exp (cid:18) − r (1 + ε n cos( n ( φ − ψ n ))2 ρ (cid:19) , (4)where we have introduced polar coordinates x = r cos φ , y = r sin φ . In Eq. (4), n is a positive integer, ε n is themagnitude of the deformation, ψ n is a reference angle,and ρ is the transverse size. Convergence at infinity im-plies 0 ≤ ε n <
1. Fig. 1 displays contour plots of theenergy density for n = 3 and ε = 0 .
2. The sign in frontof ε n in Eq. (4) has been chosen such that ψ n is the di-rection of the flat side of the polygon. For n = 2, it is theminor axis of the ellipse, which is the standard definitionof the participant plane [25]For t > t , we assume that the system evolves accord-ing to the equations of hydrodynamics or to the Boltz-mann transport equation, until particles are emitted, andwe compute the azimuthal distribution f ( φ ) of outgoingparticles. The initial profile (4) is symmetric under thetransformation φ → φ + πn , therefore f ( φ ) has the samesymmetry. The only nonvanishing Fourier coefficients are h e inφ i , h e inφ i , h e inφ i , etc. Symmetry of the initial pro-file under the transformation ( φ − ψ n ) → − ( φ − ψ n ) im-plies h e inφ i = v n e inψ n , (5)where v n is real. As we shall see below, v n is usuallypositive for ε n >
0, which means that anisotropic flowdevelops along the flat side of the polygon (see Fig. 1)We now present quantitative results for v n , as definedby Eq. (5), using two models. The first model is rela-tivistic hydrodynamics (see [10] for details). We fix ǫ , t and the freeze-out temperature to the same values as fora central Au-Au collision at RHIC with Glauber initialconditions [10], and ρ = 3 fm, corresponding roughly tothe rms values of x and y . Unless otherwise stated, re-sults are shown for pions at freeze-out. Corrections dueto resonance decays [36] are not included in this section.They are included only in our final predictions in Sec. IV.The second model is a relativistic Boltzmann equation formassless particles in 2+1 dimensions (see [11] for details).The only parameter in this calculation is the Knudsennumber K = λ/R , where the mean free path λ and thetransverse size R are defined as in [11]. ρ in Eq. (4) is therms width of the energy distribution, while R is definedfrom the rms widths σ x and σ y of the particle distributionby R − = σ − x + σ − y . For a two dimensional ideal gas ofmassless particles, the particle density n is related to theenergy density through n ∝ ǫ / , which gives R = √ ρ .Boltzmann transport theory is less realistic than hydro-dynamics for several reasons: • the equation of state is that of an ideal gas, whilethe equation of state used in hydrodynamics istaken from lattice QCD: it is much softer aroundthe transition to the quark-gluon plasma. Al-though transport is equivalent to ideal hydrody-namics when the mean free path goes to zero, ourresults from transport and ideal hydrodynamics dif-fer in this limit, because of the different equationof state. • there is no longitudinal expansion. • particles are massless.The main advantage of transport theory is that it canbe used for arbitrary values of the mean free path, whilehydrodynamics can only be used if the mean free path issmall. Furthermore, the time evolution of the system canbe studied and no modeling is required for the freeze-outprocess using transport approach, since one follows allelastic collisions until the very last one. A. v n versus ε n v n ε n transport v hydro v transport v hydro v transport v hydro v FIG. 2: (Color online) v n versus ε n in transport theory andideal hydrodynamics. The Knudsen number in the transportcalculation is K = 0 . K = 0. Fig. 2 displays v n versus ε n for n = 2 , , v n are smaller in hydrodynamics, which is dueto the softer equation of state [37].As expected from previous studies of v [38] and v [9],we observe that v n is linear for small values of ε n . Nonlinearities are stronger for larger values of n , both intransport theory and hydrodynamics. A possible inter-pretation of these strong nonlinearities is that the thecontour plot of the initial density is no longer convex if ε n > / ( n − . The threshold values for n = 3 , ε = and ε = . If the contour plot is not convex, thestreamlines (which are orthogonal to equal density con-tours) are no longer divergent: shock waves may appear,which hinder the development of anisotropies.The results presented in the remainder of this sectionare obtained in the linear regime where v n ∝ ε n . In thisregime, we find v /ε ≃ .
21, in agreement with other calculations [39]. Note that in our hydrodynamic calcula-tion, chemical equilibrium is maintained until freeze-out.When chemical freeze-out is implemented earlier than ki-netic freeze-out, v /ε is slightly larger [19]. Fig. 2 showsthat v /ε has a magnitude comparable to v /ε , while v /ε is significantly smaller. Our results for v /ε (notshown) are even smaller. B. Time dependence v n / ε n t/ ρ v / ε v / ε v / ε v / ε FIG. 3: (Color online) v n /ε n versus time in transport theory.Each curve is the result of a single Monte-Carlo simulationwith K = 0 .
025 and ε n = 0 .
1. The number of particles in thesimulation is N = 4 × , and the corresponding statisticalerror on v n /ε n is 3 . × − . In the transport approach, one follows all the trajec-tories of the particles, so that v n is well defined at alltimes, which is not the case in hydrodynamics beforefreeze-out. Fig. 3 displays the results for v n versus t/ρ ,where ρ is the width of the initial distribution, Eq. (4).As expected for dimensional reasons [37], anisotropic flowappears for t ∼ ρ . However, v n appears slightly later forlarger n . This can be traced to the behavior of v n atearly times. The transport results presented in Fig. 3 areobtained with a very small value of the Knudsen num-ber, K = 0 . t , and v n involves a n th power ofthe fluid velocity, so that v n scales like t n . In transporttheory, the number of collisions increases like t at earlytimes, which gives an extra power of t , and v n increaseslike t n +1 [11]. In both cases, the behavior of v n at small t is flatter for larger values of n , which is clearly seen inFig. 3.While elliptic flow keeps increasing with time (itslightly decreases at later times, not shown in the fig-ure), v n with n ≥ n :The mechanism producing v n is self quenching. C. Differential flow v n ( p t ) / ε n p t [GeV/c]v / ε v / ε v / ε v / ε FIG. 4: (Color online) v n /ε n versus p t in ideal hydrodynam-ics, with ε n = 0 . Fig. 4 displays the differential anisotropic flow v n ( p t )versus the transverse momentum p t for pions in idealhydrodynamics, scaled by the initial eccentricity ε n . Atlow p t , one generally expects v n to scale like ( p t ) n formassive particles [40] . One clearly sees that v n is muchflatter at low p t for larger values of n . For larger valuesof p t , v n ( p t ) is linear in p t . The arguments that explainthis linear dependence for v [12] can be generalized toarbitrary n [41]. The linear behavior at larger p t is alsoclearly seen in Fig. 4. It has already been noted for v [9].The value of v increases with p t , which explains whythe ridge and shoulder are more pronounced with ahigh p t trigger (“hard” ridge) [42]. Though the relativestrength of v , is smaller at low p t , it is still comparableto v , leading to the smaller“soft” ridge [43]. Predictionsfor v ( p t ) in viscous hydrodynamics for identified parti-cles are presented in Sec. IV. D. Viscous damping of v n We study the effect of viscosity first in the transportapproach, then in viscous hydrodynamics. In trans-port, the degree of thermalization is characterized bythe Knudsen number K . Experimentally, 1 /K scales like(1 /S )( dN/dy ), where dN/dy is the multiplicity per unitrapidity, and S is the overlap area between the collidingnuclei [44]. The dependence of v n on K can be studiedby varying the collision system and the centrality of thecollision [45] There is no such constraint for massless particles where the p t → v n ( p t ) ∝ p t at low p t for all n . v n / ε n K)v / ε v / ε v / ε v / ε FIG. 5: (Color online) v n /ε n versus 1 / ( n K ) in transporttheory. Values of ε n are ε = , ε = ε = 0 . ε = 0 . K = 0 (ideal hydrody-namics limit) using Eq. (6). Transport is equivalent to ideal hydrodynamics in thelimit K →
0. For small K , observables (such as v n , orparticle spectra) deviate from the K = 0 limit by correc-tions which are linear in K . These are the viscous correc-tions: both K and the shear viscosity η are proportionalto the particle mean free path λ . Viscous damping is ex-pected to scale with the wave number k like k . Here, thewavelength of the deformation is 2 πR/n , hence k ∼ n/R .Therefore viscous corrections should scale with K and n approximately like n K [46]. The limit K → ∞ (freestreaming) is also interesting, since v n vanishes in thislimit. For large K , one therefore expects v n to scale like1 /K , which is essentially the number of collisions perparticle [11]. For intermediate values of K ( K ∼ v n /ε n versus the scalingvariable 1 / ( n K ) in the transport calculation. Our nu-merical results can be fitted by smooth rational functions(Pad´e approximants) [47] for all K : v n ( K ) = v ih n B n K + D n K A n + B n ) K + C n K + E n K , (6)where v ih n , A n , B n , C n , D n and E n are fit parameters.This formula has the expected behavior in both K → K → ∞ limits. For n = 2, the lowest-order formula,with B = C = D = E = 0, gives a good fit [11]. For n = 3, we obtain a good fit with using the next-to-leadingorder approximant, with D = E = 0 but free B , C .For n = 4 or 5, we need all 6 parameters to achieve agood fit. Fits are represented as solid lines in Fig. 5,and extrapolations to K = 0 are indicated by arrows.As already noted above, the hydrodynamics limits v ih3 /ε and v ih2 /ε are comparable, while v ih4 /ε is smaller byroughly a factor of 2. v ih5 /ε is found to be further smallerby about a factor 5, with a large theoretical uncertainty.For small K , v n ( K ) ≃ v ih n (1 − A n K ): the parameter A n measures the magnitude of the viscous correction. Our fitgives A = 1 . ± . A = 4 . ± . A = 11 . ± . A is too large to extract a meaningfulvalue. For n = 2 , ,
4, we observe A n ∝ n α with α = 2 . ± .
2, closer to n than to the expected n . The fact thatviscous corrections are larger for larger n also implies thatthe range of validity of viscous hydrodynamics is smallerfor v n with n ≥ v . Even after rescaling K by n , corrections are linear in K only for very small K ,which is why higher-order Pad´e approximants are needed. v n / ε n η /s v / ε v / ε v / ε v / ε FIG. 6: (Color online) v n /ε n versus η/s in hydrodynamics.The initial and freeze-out temperature are T i = 340 MeV and T f = 140 MeV, respectively. The magnitude of viscous effects can be seen more di-rectly by varying the shear viscosity η in viscous hydro-dynamics [48]. For each value of n , we have performedthree calculations with η ≃ η/s = 0 . ≃ / π [49], and η/s = 0 .
16, where s isthe entropy density. The result is presented in Fig. 6.The variation of v n with η is found to be linear for all n for this range of viscosities, which is a hint that viscoushydrodynamics (which addresses first-order deviations tolocal equilibrium) is a reasonable description. Interest-ingly, the lines are almost parallel, which means that theabsolute viscous correction to v n /ε n depends little on n .However, since v n /ε n is smaller for larger n , the rela-tive viscous correction is larger for larger n . From thetransport calculation, we expect that the relative viscouscorrection is 3 times larger for v than for v , and 8 timeslarger for v than for v . The increase in Fig 6 is moremodest. Note that we keep the freeze-out temperatureconstant for all values of η/s . Strictly speaking, this isinconsistent. Freeze-out is defined as the point whereviscous corrections become so large that hydrodynam-ics breaks down: when the viscosity goes to zero, so doesthe freeze-out temperature [12]. By varying only η/s andkeeping T f constant, we only capture part of the viscous correction . Since triangular flow, like elliptic flow, de-velops at early times, v is sensitive to the value of η/s at the high-density phase of the collision. IV. PREDICTIONS FOR v AT RHIC AND LHCA. Triangularity fluctuations
We now give realistic predictions for v at RHIC andLHC. The transport calculations in Ref. [9] show thateven with lumpy initial conditions, v in a given eventscales like the triangularity ε . We define ε n as in [9]: ε n e inψ n ≡ − R ǫ ( x, y ) r e inφ dxdy R ǫ ( x, y ) r dxdy , (7)where ǫ ( x, y ) is the initial energy density and ( r, φ ) arethe usual polar coordinates, x = r cos φ , y = r sin φ .Following the discussion in Sec. II, experiments mea-sure the average value of ( v n ) , so that v exp n = p h ( v n ) i . (8)Assuming v n = κε n in each event, the measured v n scaleslike the root mean square ε n defined by ε rms n ≡ p h ( ε n ) i (9)We compute ε rms n using two different models. The firstmodel is the PHOBOS Monte-Carlo Glauber model [50],where it is assumed that the initial energy is distributedin the transverse plane in the same way as nucleonswithin colliding nuclei. We modify the initial modelslightly [33] by giving each nucleon a weight w = 1 − x + xN coll , where N coll is the number of binary collisionsof the nucleon. We take x = 0 .
145 at RHIC and x = 0 . ε rms n as a function of the number ofparticipants. ε rms2 is larger than ε rms3 , , for non-centralcollisions, which is due to the almond shape of the over-lap area. The eccentricity is somewhat larger with CGCthan Glauber [54]. ε rms3 is very close to ε rms5 . Both varywith N Part essentially like ( N Part ) − / , as generally ex-pected for statistical fluctuations [55]. Unlike ε rms2 , theyare slightly smaller for CGC than for Glauber. Sincethe only source of fluctuations that is considered in both We have checked that v /ε is larger with a lower freeze-out tem-perature T f = 100 MeV. In particular, we find v /ε > v /ε ,in agreement with the transport calculation. < ε n2 > / N Part n=2n=3n=4n=5
FIG. 7: (Color online) Root mean square eccentricities ε rms n for n = 2 , , , N Part . N Part isused as a measure of the centrality in nucleus-nucleus colli-sions: it is largest for central collisions, with zero impact pa-rameter [53]. Thick lines: Monte-Carlo Glauber model [50];Thin lines: Monte-Carlo KLN model [52]. models is the position of the nucleons in the collidingnuclei, this difference may be due to the technical im-plementation of the Monte-Carlo KLN model. Finally, ε rms4 is slightly larger than odd harmonics for peripheralcollisions because the almond shape induces a nonzero ε as a second order effect. Fig. 7 only displays results forAu-Au collisions at RHIC. Results for Pb-Pb collisions atthe LHC are similar, except for a different range in N part ,and a somewhat larger difference between Glauber andCGC for ε . B. Method for obtaining v in hydrodynamics In order to make predictions for v , we start from asmooth initial energy profile ǫ ( r, φ ), possessing the usualsymmetries φ → − φ and φ → φ + π . We then put byhand a cos(3 φ ) deformation through the transformation,inspired by Eq. (4), ǫ ( r, φ ) → ǫ (cid:18) r q ε ′ cos(3( φ − ψ ′ )) , φ (cid:19) , (10)where ε ′ is the magnitude of the deformation, and ψ ′ the flat axis of the triangle. We choose ε ′ = ε rms3 . Thechoice of ψ ′ is arbitrary. The initial profile has a nonzeroeccentricity for noncentral collisions, due to the almondshape of the overlap area. Through Eq. (10), we add atriangular deformation to an ellipse. Since the originalprofile has φ → φ + π symmetry, ψ ′ is equivalent to ψ ′ + π . Furthermore, ψ ′ is equivalent to − ψ ′ due to φ → − φ symmetry. Therefore, one need only vary ψ ′ between 0 and π . We choose the values 0, π and π .We then compute ε and ψ defined by Eq. (7). Withthe gaussian profile (4), the input and output values are identical: ε ′ = ε , ψ ′ = ψ . Our predictions use twosets of profiles which both describe RHIC data well [10]:optical Glauber and (fKLN) CGC. With both profiles, ε ′ differs from ε by a few percent. ψ is essentially identi-cal to ψ ′ , which means that the elliptic deformation doesnot interfere with the triangular deformation. Accordingto the previous discussion, we should tune ε ′ in such away that ε = ε rms3 in order to make predictions for v .It is however easier to use the proportionality between v and ε : one can then do the calculation for an arbi-trary ε ′ , and rescale the final results by ε rms3 /ε . We use ε rms3 from the Monte-Carlo Glauber model with Glauberinitial conditions, and from the Monte-Carlo KLN modelwith CGC initial conditions.There is some arbitrariness in the definition of the tri-angularity ε : one could for instance replace r by r inEq. (7) [56]. With this replacement, both ε and ε rms3 (from the Monte-Carlo calculations) increase, but theratio ε rms3 /ε — and therefore also our predicted v —changes little (less than 7% for all centralities and bothsets of initial conditions) .Finally, we compute v in viscous hydrodynamics. Ithas been shown that RHIC data are fit equally well withGlauber initial conditions and η/s = 0 .
08 or with CGCinitial conditions and η/s = 0 .
16 [10]. The larger eccen-tricity of CGC (which should produce more elliptic flow)is compensated by the larger viscosity (larger dampingand less flow), so that the final values of v are very simi-lar. For LHC energies, details are as in Ref. [57] (with v calculated from a Cooper-Frye freeze-out prescription).In all cases, v is found to be independent of the orien-tation of the triangle ψ ′ . In the case of Glauber initialconditions, we perform calculations of v with and with-out resonance decays at freeze out [36]. Resonance decaysroughly amount to multiplying v by 0 .
75 at RHIC, andby 0 .
83 at LHC. Our CGC results are computed withoutresonance decays, and multiplied by the same factor atthe end of the calculation.
C. Results and comparison with data
Results are displayed in Fig. 8 for both sets of initialconditions. CGC initial conditions have both a smallertriangularity, and a larger viscosity, so that they pre-dict a much smaller v . The change in viscosity explainsroughly 70% of the difference between CGC and Glauberat RHIC, and about half at LHC. The centrality depen-dence is much flatter in Fig. 8 than in Fig. 7. The de-crease of ε rms3 with increasing N Part is compensated bythe increase of the system size and lifetime, which leads If one replaces r by r k in Eq. (7), ε n scales with k like k + 2 fora smooth, symmetric density profile ǫ ( r ) deformed according toEq. (10). Therefore, ε is larger by if defined with a factor r instead of r . v N Part
LHC, η /s=0.08RHIC, η /s=0.08LHC, η /s=0.16RHIC, η /s=0.16 FIG. 8: (Color online) Average v of pions as a function of thenumber of participants for Au-Au collisions at 200 GeV pernucleon (RHIC) and Pb-Pb collisions at 5.5 TeV per nucleon(LHC). Hydrodynamic predictions are for Glauber initial con-ditions with η/s = 0.08, and CGC initial conditions with η/s = 0.16, which best fit v data at RHIC [10]. to a smaller effective Knudsen number K or, equivalently,a smaller viscous correction. We predict values of v sig-nificantly larger at LHC than at RHIC. This is becauseviscous damping is less important due to the larger life-time of the fluid at LHC [57].Although experimental data for triangular flow are notyet available, both v and v can be extracted fromthe measured two-particle azimuthal correlation usingEq. (3) [9]. Figs. 9 and 10 display a comparison betweenexperimental data from STAR [4] and our hydrodynamiccalculations. The STAR data is obtained from correla-tions between particles at midrapidity ( | η | <
1) and in-termediate transverse momentum (0 . < p t < . . < ∆ η < . v . As explained above,our hydrodynamic model has smooth initial conditions,and does not include the effect of eccentricity fluctuationsfor v . Since v ∝ ε to a good approximation, we haverescaled our result for v by the rms ε from Fig. 7 (againusing the Monte-Carlo Glauber for the Glauber initialconditions and the Monte-Carlo KLN for CGC). Thisrescaling significantly improves the agreement with data,compared to [10], for the most central bin. As shownin Fig. 9, the agreement between theory and data is ex-cellent with both sets of initial conditions. The smallerviscosity associated with Glauber initial conditions re-sults in a somewhat steeper centrality dependence thanfor CGC initial conditions.Results for v are shown in Fig. 10. The larger magni-tude, compared to Fig. 8, is due to the low p t cutoff. Thecutoff also enhances the effect of viscosity, resulting in alarger difference between Glauber and CGC. With a low p t cutoff, the viscous correction is mostly due to the dis-tortion of the momentum distribution at freeze-out [58]. v N Part
GlauberCGCSTAR
FIG. 9: (Color online) v for charged particles with 0 .
08, orCGC initial conditions with η/s = 0 .
16. Theoretical calcula-tions are for pions with the same p t cut as data, and scaledby the rms eccentricity from the corresponding Monte-Carlomodel. See text for details. v N Part η /s=0.08 η /s=0.16STAR FIG. 10: (Color online) Same as Fig. 9 for v . The momentum dependence of this distortion is stronglymodel-dependent [59]. The present calculation uses thestandard quadratic ansatz, which may overestimate theviscous correction at large p t [60]. The magnitude andthe centrality dependence of v observed by STAR arerather well reproduced by our calculation with Glauberinitial conditions, except for peripheral collisions wherehydrodynamics is not expected to be valid.Fig. 11 displays our predictions for v ( p t ) of identifiedparticles at RHIC. As anticipated in Ref. [41], the well-known mass ordering of elliptic flow [61] is also expectedfor v . At high p t , a strong viscous suppression is ob-served. As explained above, the p t dependence of theviscous correction is model dependent, and it is likelythat the quadratic ansatz used here overestimates theviscous corrections at large p t [60]. Note that effects of v p t [GeV/c] η /s=0.08 η /s=0.16pionskaonsprotons FIG. 11: (Color online) Differential triangular flow for iden-tified particles in central (0 − resonance decays are not included in Fig. 11. Resonancedecays only change the results slightly in the low- p t re-gion.Finally, we have also computed v along the same linesas v . The driving force for v is the rms ε , which isvery close to ε (see Fig. 7). However, the hydrodynamicresponse is much smaller, and viscous damping is alsomuch larger as discussed in Sec. III. We find that theaverage integrated v is smaller than v by at least afactor of 10. Results for differential v are presented inFig. 12. v varies more strongly with p t than v and v ,and becomes as large as 0 .
02 at p t = 1 . η/s is as small as 0 .
08. For larger viscosity, however, v may be too small to measure even with a high p t trigger. -0.04-0.02 0 0.02 0.04 0 0.5 1 1.5 2 2.5 3 v p t [GeV/c] η /s=0.08 η /s=0.16pionskaonsprotons FIG. 12: (Color online) Differential pentagonal flow for iden-tified particles in central (0 − V. CONCLUSIONS
We have presented a systematic study of triangularflow in ideal and viscous hydrodynamics, and transporttheory. Triangular flow is driven by the average event-by-event triangularity in the transverse distribution ofnucleons, in the same way as elliptic flow is driven by theinitial eccentricity of this distribution. The physics of v is in many respects similar to the physics of v . In idealhydrodynamics, the response to the initial deformation isalmost identical in both harmonics: v /ε ≃ v /ε ≃ . v /ε is smaller, typically by afactor 2. For pentagonal flow, v /ε is so small that v is unlikely to be measurable, even though ε and ε arealmost equal. v develops slightly more slowly than v ,though over comparable time scales. The dependence ontransverse momentum p t is similar for v and v , but v /v increases with p t . Hydrodynamics predicts a simi-lar mass ordering for v ( p t ) and v ( p t ): v at fixed p t issmaller for more massive particles. These results can bechecked experimentally by a differential measurement oftriangular flow.We have also made predictions for triangular flow, v ,at RHIC and LHC, using viscous hydrodynamics. Usingas input the triangularity from a standard Monte-CarloGlauber model, and a viscosity η/s = 0 .
08, we repro-duce both the magnitude (within 20%) and the central-ity dependence of v extracted from STAR correlationmeasurements, without any adjustable parameter. Ourresults support the hypothesis made in Ref. [9] that trian-gular flow explains most of the ridge and shoulder struc-tures observed in the two-particle azimuthal correlation.Triangular flow is a sensitive probe of viscosity. Vis-cous effects drive the energy and centrality dependence of v . More central collisions have less fluctuations, hencesmaller triangularity. This decrease is to a large extentcompensated by the increase in the system size and life-time, resulting in a very slow decrease of v with cen-trality (except for peripheral collisions where viscous hy-drodynamics is unlikely to be valid). Comparison withexisting data favors a low value of η/s . At LHC, smallerviscous corrections are expected due to the increased life-time of the fluid: we predict that v should be larger thanat RHIC, typically by a factor .The absolute value of v scales linearly with the aver-age initial initial triangularity. We have used two modelsof initial geometry which incorporate fluctuations, theMonte-Carlo Glauber model and the Monte-Carlo KLNmodel. The underlying source of fluctuations is the samein both of these models. More work is needed to constraininitial fluctuations on the theoretical side. More work isalso needed to incorporate these fluctuations more read-ily into hydrodynamic calculations. Although triangularflow is expected to be created by lumpy initial conditions,our predictions are based on smooth initial conditions,in the same spirit as the study of transverse momentumfluctuations of Ref. [22]. The underlying assumption isthat v /ε is the same for lumpy initial conditions and forsmooth initial conditions. The validity of this assump-tion should eventually be checked.Triangular flow is a new observable which should beused to constrain models of heavy-ion collisions, alongwith elliptic flow. Elliptic flow depends on initial ec-centricity, fluctuations, and viscosity, which are poorlyconstrained theoretically. Triangular flow solely dependson fluctuations and viscosity, with a stronger sensitivityto viscosity than v . Two different sets of initial condi-tions, which fit v data equally well, give very differentresults for v . Experiments could measure v as a func-tion of transverse momentum, system size and centrality.As shown in this paper, theoretical predictions for the de-pendence of v on these parameters are very specific. Ifexperiments confirm our predictions, simultaneous anal-yses of v and v can be used to improve our understand-ing of the initial geometry of heavy-ion collisions, and pin down the viscosity of hot QCD. Acknowledgments
This work is funded by “Agence Nationale de laRecherche” under grant ANR-08-BLAN-0093-01 and byU.S. DOE grant DE-FG02-94ER40818. We thank S.Gavin, T. Hirano, P. Huovinen and A. Poskanzer forstimulating discussions, and W. Zajc for useful commentson the manuscript. M. L. and J.-Y. O. thank the orga-nizers of the program “Quantifying the properties of hotQCD matter” and the Institute for Nuclear Theory at theUniversity of Washington, where part of this work wasdone, for its hospitality, and the Department of Energyfor partial support. [1] K. H. Ackermann et al. [STAR Collaboration], Phys.Rev. Lett. , 402 (2001) [arXiv:nucl-ex/0009011].[2] A. Adare et al. [PHENIX Collaboration], Phys. Rev. C , 014901 (2008) [arXiv:0801.4545 [nucl-ex]].[3] B. I. Abelev et al. [STAR Collaboration], Phys. Rev. Lett. , 052302 (2009) [arXiv:0805.0622 [nucl-ex]].[4] B. I. Abelev et al. [STAR Collaboration],arXiv:0806.0513 [nucl-ex].[5] B. Alver et al. [PHOBOS Collaboration], Phys. Rev. C , 024904 (2010) [arXiv:0812.1172 [nucl-ex]].[6] B. Alver et al. [PHOBOS Collaboration], Phys. Rev.Lett. , 062301 (2010) [arXiv:0903.2811 [nucl-ex]].[7] B. I. Abelev et al. [STAR Collaboration], Phys. Rev. C , 064912 (2009) [arXiv:0909.0191 [nucl-ex]].[8] A. Dumitru, F. Gelis, L. McLerran and R. Venugopalan,Nucl. Phys. A , 91 (2008) [arXiv:0804.3858 [hep-ph]].[9] B. Alver and G. Roland, Phys. Rev. C , 054905 (2010)[arXiv:1003.0194 [nucl-th]].[10] M. Luzum and P. Romatschke, Phys. Rev. C ,034915 (2008) [Erratum-ibid. C , 039903 (2009)][arXiv:0804.4015 [nucl-th]].[11] C. Gombeaud and J. Y. Ollitrault, Phys. Rev. C ,054904 (2008) [arXiv:nucl-th/0702075].[12] N. Borghini and J. Y. Ollitrault, Phys. Lett. B , 227(2006) [arXiv:nucl-th/0506045].[13] M. Luzum, C. Gombeaud and J.-Y. Ollitrault, in prepa-ration.[14] S. Voloshin and Y. Zhang, Z. Phys. C , 665 (1996)[arXiv:hep-ph/9407282].[15] A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C ,1671 (1998) [arXiv:nucl-ex/9805001].[16] S. Wang et al. , Phys. Rev. C , 1091 (1991).[17] J. Y. Ollitrault, A. M. Poskanzer and S. A. Voloshin,Phys. Rev. C , 014904 (2009) [arXiv:0904.2315 [nucl-ex]].[18] B. Alver et al. , Phys. Rev. C , 014906 (2008)[arXiv:0711.3724 [nucl-ex]].[19] P. Huovinen, Eur. Phys. J. A , 121 (2008)[arXiv:0710.4379 [nucl-th]].[20] T. Hirano and Y. Nara, Phys. Rev. C , 064904 (2009)[arXiv:0904.4080 [nucl-th]]. [21] H. Song and U. W. Heinz, Phys. Rev. C , 024905(2010) [arXiv:0909.1549 [nucl-th]].[22] W. Broniowski, M. Chojnacki and L. Obara, Phys. Rev.C , 051902 (2009) [arXiv:0907.3216 [nucl-th]].[23] P. Bozek, Phys. Rev. C , 034909 (2010)[arXiv:0911.2397].[24] M. Miller and R. Snellings, arXiv:nucl-ex/0312008.[25] S. Manly et al. [PHOBOS Collaboration], Nucl. Phys. A , 523 (2006) [arXiv:nucl-ex/0510031].[26] M. Gyulassy, D. H. Rischke and B. Zhang, Nucl. Phys.A , 397 (1997) [arXiv:nucl-th/9609030].[27] O. J. Socolowski, F. Grassi, Y. Hama andT. Kodama, Phys. Rev. Lett. , 182301 (2004)[arXiv:hep-ph/0405181].[28] H. Holopainen, H. Niemi and K. J. Eskola,arXiv:1007.0368.[29] K. Werner, I. Karpenko, T. Pierog, M. Bleicher andK. Mikhailov, arXiv:1004.0805.[30] A. P. Mishra, R. K. Mohapatra, P. S. Saumia andA. M. Srivastava, Phys. Rev. C , 064902 (2008)[arXiv:0711.1323 [hep-ph]].[31] R. Andrade, F. Grassi, Y. Hama, T. Kodama andO. J. Socolowski, Phys. Rev. Lett. , 202302 (2006)[arXiv:nucl-th/0608067].[32] B. Alver et al. [PHOBOS Collaboration], Phys. Rev.Lett. , 242302 (2007) [arXiv:nucl-ex/0610037].[33] C. Gombeaud and J. Y. Ollitrault, Phys. Rev. C ,014901 (2010) [arXiv:0907.4664 [nucl-th]].[34] B. Alver et al. [PHOBOS Collaboration], Phys. Rev. C , 034915 (2010) [arXiv:1002.0534 [nucl-ex]].[35] J. Y. Ollitrault, Phys. Rev. D , 229 (1992).[36] J. Sollfrank, P. Koch and U. W. Heinz, Z. Phys. C ,593 (1991).[37] R. S. Bhalerao, J. P. Blaizot, N. Borghini andJ. Y. Ollitrault, Phys. Lett. B , 49 (2005)[arXiv:nucl-th/0508009].[38] H. Sorge, Phys. Rev. Lett. , 2048 (1999)[arXiv:nucl-th/9812057].[39] P. Huovinen, T. Hirano, private communications.[40] P. M. Dinh, N. Borghini and J. Y. Ollitrault, Phys. Lett.B , 51 (2000) [arXiv:nucl-th/9912013]. [41] A. P. Mishra, R. K. Mohapatra, P. S. Saumia andA. M. Srivastava, Phys. Rev. C , 034903 (2010)[arXiv:0811.0292 [hep-ph]].[42] J. Putschke, J. Phys. G , S679 (2007)[arXiv:nucl-ex/0701074].[43] M. Daugherity [STAR Collaboration], J. Phys. G ,104090 (2008) [arXiv:0806.2121 [nucl-ex]].[44] H. J. Drescher, A. Dumitru, C. Gombeaud and J. Y. Olli-trault, Phys. Rev. C , 024905 (2007) [arXiv:0704.3553[nucl-th]].[45] S. A. Voloshin and A. M. Poskanzer, Phys. Lett. B ,27 (2000) [arXiv:nucl-th/9906075].[46] Sean Gavin, private communication.[47] J. L. Nagle, P. Steinberg and W. A. Zajc, Phys. Rev. C , 024901 (2010) [arXiv:0908.3684 [nucl-th]].[48] U. W. Heinz, J. S. Moreland and H. Song, Phys. Rev. C , 061901 (2009) [arXiv:0908.2617 [nucl-th]].[49] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev.Lett. , 111601 (2005) [arXiv:hep-th/0405231].[50] B. Alver, M. Baker, C. Loizides and P. Steinberg,arXiv:0805.4411 [nucl-ex].[51] P. Bozek, M. Chojnacki, W. Florkowski and B. Tomasik,arXiv:1007.2294 [Unknown].[52] H. J. Drescher and Y. Nara, Phys. Rev. C , 041903 (2007) [arXiv:0707.0249 [nucl-th]].[53] M. L. Miller, K. Reygers, S. J. Sanders and P. Stein-berg, Ann. Rev. Nucl. Part. Sci. , 205 (2007)[arXiv:nucl-ex/0701025].[54] T. Lappi and R. Venugopalan, Phys. Rev. C , 054905(2006) [arXiv:nucl-th/0609021].[55] R. S. Bhalerao and J. Y. Ollitrault, Phys. Lett. B ,260 (2006) [arXiv:nucl-th/0607009].[56] Yan Li and D.Teaney, “Triangle and Dipole Flow in IdealHydrodynamics”, in preparation; D. Teaney, contribu-tion to Strong and Electroweak Matter 2010, June 29 -July 2, 2010, Montreal, Canada.[57] M. Luzum and P. Romatschke, Phys. Rev. Lett. ,262302 (2009) [arXiv:0901.4588 [nucl-th]].[58] D. Teaney, Phys. Rev. C , 034913 (2003)[arXiv:nucl-th/0301099].[59] K. Dusling, G. D. Moore and D. Teaney, Phys. Rev. C , 034907 (2010) [arXiv:0909.0754 [nucl-th]].[60] M. Luzum and J. Y. Ollitrault, Phys. Rev. C , 014906(2010) [arXiv:1004.2023 [nucl-th]].[61] P. Huovinen, P. F. Kolb, U. W. Heinz, P. V. Ruuska-nen and S. A. Voloshin, Phys. Lett. B503