Plane correlations and hydrodynamic simulations of heavy ion collisions
PPlane correlations and hydrodynamic simulations of heavy ioncollisions
D. Teaney ∗ Department of Physics & Astronomy,Stony Brook University, Stony Brook, NY 11794, USA
L. Yan
CNRS, Institut de Physique Th´eorique de Saclay, F-91191 Gif-sur-Yvette, France † (Dated: September 18, 2018) Abstract
We use a nonlinear response formalism to describe the event plane correlations measured by theATLAS collaboration. With one exception ( (cid:104) cos(2Ψ − + 4Ψ ) (cid:105) ), the event plane correlationsare qualitatively reproduced by considering the linear and quadratic response to the lowest cumu-lants. For the lowest harmonics such as (cid:104) cos(2Ψ + 3Ψ − ) (cid:105) , the correlations are quantitativelyreproduced, even when the naive Glauber model prediction has the wrong sign relative to experi-ment. The quantitative agreement for the higher plane correlations (especially those involving Ψ )is not as good. The centrality dependence of the correlations is naturally explained as an averageof the linear and quadratic response. ∗ [email protected] † [email protected] a r X i v : . [ nu c l - t h ] J a n . INTRODUCTION The collective expansion of the deconfined fireball created in high energy heavy-ion col-lisions maps the initial state of the Quark-Gluon Plasma (QGP) to the final state particlespectrum. The measured correlations in this spectrum can clarify the initial conditions andsubsequent expansion dynamics of the QGP [1–3].On an event-event basis the azimuthal distribution of produced particles can be decom-posed into a Fourier series dNdφ p = N π (cid:32) ∞ (cid:88) n =1 v n cos( nφ p − n Ψ n ) (cid:33) , (1.1)and the measured two particle correlation function determines the root mean square of thethese harmonics, (cid:112) (cid:104) v n (cid:105) . The magnitude of these harmonics is reasonably reproduced byevent-by-event viscous hydrodynamics provided the shear viscosity is not too large [1]. Thecorrelations between the harmonics can provide new tests of the hydrodynamic description,constrain the simulation parameters, and provide an estimate of the uncertainties in thecomputation. In this work we will describe the correlations between the observed eventplane angles Ψ n in order to clarify the expansion dynamics, and ultimately to determine theshear viscosity of the QGP with credible systematic error bars.Clearly, an important input to the hydrodynamic simulations is the distribution of energydensity in the transverse plane, which is usually estimated from the known probability distri-bution of nucleons in the incoming nuclei. There is reasonable evidence, both experimental[4] and theoretical [5], that v and v are to a good approximation linearly proportional tothe corresponding angular fluctuations in the transverse energy density. However, event-by-event hydrodynamic simulations have shown that the higher harmonics, v and v , reflectboth the response to corresponding angular harmonics in the initial state, and the non-linearhydrodynamic response which mixes lower order harmonics [5, 6]. For example, the 5-thflow harmonic, v , is determined in part by the medium response to the 5-th harmonic ofthe initial energy density distribution, and in part by the non-linear mixing between v and v . Such mode-mixing is especially important at high p T where the non-linearities of thephase-space distribution play an important role [7]. Indeed, there are indications that thedominant source of mode mixing comes from freezeout as opposed to the hydrodynamic evo-lution [8]. Motivated by these simulation results, and especially the simulation analysis ofRef. [6], we developed a non-linear response formalism to describe the mixing between modesof different order, and we investigated how the response coefficients depend on centrality,shear viscosity, and transverse momentum [9].These theoretical calculations preceded the corresponding experimental studies by theATLAS [10] and ALICE collaborations [11], which qualitatively confirmed the mode mix-ing picture by measuring significant correlations between the event-plane angles of differentorders , e.g. between Ψ ,Ψ , and Ψ . Event-by-event hydrodynamics [13] and AMPT calcu-lations [14] largely reproduce the structure of these correlations. The goal of this paper is tocompare the response formalism outlined in our previous work to the event-plane correlationsmeasured by the ATLAS collaboration [9]. The ATLAS measurement did not precisely measure Ψ n [12]. Ultimately, this important first measurementwill need to be redone, weighting the event averages with the Q vector to provide an unambiguous quantitywhich can be compared to fairly compared to simulations. See below for further discussion.
2s discussed more technically in Section II B we will use a non-linear response formalismto describe the observed event plane correlations, rather than event-by-event hydrodynamics.In practice, this means that we decompose the initial state into an average event plus smallfluctuations, which are systematically analyzed with cumulants. The linear and quadraticresponse to each cumulant is found by perturbing the average background, and finally theobserved plane correlations are found by weighting the response functions with the spectrumof fluctuations. Thus, the response formulation provide a transparent link between the initialstate and the final state, which contains only the linear and quadratic response through aspecified order in the cumulant expansion. As we will see, this approach reproduces a lotof the observed event plane correlations, suggesting that most of the microscopic details ofthe initial state (beyond the lowest cumulants) are irrelevant. Ideally, a limited numberof initial state parameters can be extracted from experiment, and compared to availabletheoretical frameworks such as the Color Glass Condensate to demonstrate the consistencyand uniqueness of the approach. There are indications that the spectrum of fluctuationsfrom the Color Glass Condensate is consistent with the observed harmonics [15], but theuniqueness of this approach is not obvious.A review of the non-linear flow response formalism will be given in Section II. This hasseveral ingredients. First, the spectrum of initial fluctuations in various Glauber type modelsis described in Section II A, and this spectrum is analyzed with the cumulant and momentexpansions. Then, we describe how the response coefficients are calculated, and how thesecoefficients determine the plane correlations in Section II B and Section II C. Finally, wecompare the response formalism to the ATLAS data in Section III and discuss the results.Throughout the paper Φ n will denote participant plane angle based on the cumulants rather than moments. (The correlations in the Glauber model between the cumulant anglesΦ n are markedly different from the correlations found using the analogous moment basedangles – see Section II A.) Ψ n denotes the event plane angle extracted from the final statemomentum spectra. II. REVIEW OF NONLINEAR FLOW RESPONSE FORMALISMA. Characterizing the initial state with cumulants
As discussed in the introduction, an important input to the hydrodynamic calculationsis the spectrum of initial fluctuations. This spectrum is traditionally [4] quantified with theparticipant plane anisotropy based on moments ε n e in Φ n ≡ − (cid:10) r n e inφ r (cid:11) (cid:104) r n (cid:105) (Not used). (2.1)Here the brackets (cid:104) . . . (cid:105) denote an average over the participating nucleons of a single event,while re iφ r = x + i y notates the transverse coordinates of the participants. It is convenientto use a complex notation z ≡ x + iy so that ε n e in Φ n = − (cid:104) z n (cid:105) / (cid:104) r n (cid:105) . As emphasized in ourprevious work, it is often useful to characterize the fluctuations with cumulants rather than In this formula we are using a moment based definition of ε n and Φ n . For most of the text we will use acumulant based definition. z to describethe irreducible correlations ε n e in Φ n ≡ − r n [ (cid:104) z n (cid:105) − subtractions] . (2.2)For example, the fourth order cumulant is ε e i ≡ − (cid:104) r (cid:105) (cid:104)(cid:10) z (cid:11) − (cid:10) z (cid:11) (cid:105) , (2.3)where the factor of three arises because there are three ways to pair four objects. Hereand below we have assumed that we are working in the center of mass coordinate systemwhere (cid:104) z (cid:105) = 0. The usefulness of cumulants can be understood by considering a Gaussiandistribution, ρ ( x, y ) ∝ e − x (cid:104) x (cid:105) − y (cid:104) y (cid:105) , (2.4)whose fourth order moment anisotropy (cid:104) z (cid:105) is non-zero, and is trivially correlated with theeccentricity, (cid:104) z (cid:105) . The fourth order cumulant takes out these trivial correlations, and for aGaussian distribution we have ε ∝ (cid:104) z (cid:105) − (cid:104) z (cid:105) = 0.The azimuthal anisotropies through ε are E ≡ ε e i ≡ − (cid:104) z (cid:105)(cid:104) r (cid:105) , (2.5) E ≡ ε e i ≡ − (cid:104) z (cid:105)(cid:104) r (cid:105) , (2.6) E ≡ ε e i ≡ − (cid:104) r (cid:105) (cid:104)(cid:10) z (cid:11) − (cid:10) z (cid:11) (cid:105) , (2.7) E ≡ ε e i ≡ − (cid:104) r (cid:105) (cid:2)(cid:10) z (cid:11) − (cid:10) z (cid:11) (cid:10) z (cid:11)(cid:3) , (2.8) E ≡ ε e i = − (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) , (2.9)where E n = ε n e in Φ n denotes the eccentricity and its phase. The ε which drives v is a specialcase, and is given by E = ε e i Φ ≡ − (cid:104) r (cid:105) (cid:10) z z ∗ (cid:11) . (2.10)Given an initial state Glauber model for the distribution of nucleons such as Glissando [16] orthe Phobos Monte Carlo Glauber model [17] one can calculate the correlations between theangles Φ n . Fig. 1 and Fig. 2 show such a calculation from the Phobos Monte-Carlo model.Here and below the double brackets (cid:104)(cid:104) . . . (cid:105)(cid:105) indicate an average over events, while the singlebrackets (cid:104) . . . (cid:105) denote an average over one event. In the Phobos Glauber the participantcenters are used to define the averages in eq. (2.5), while in the Glissando model a slightlydifferent prescription is used, which is based on the wounding profile of the nucleon [16].It is interesting to compare the correlations between the cumulant and moment basedangles. For example, the (cid:104)(cid:104) cos 4(Φ − Φ ) (cid:105)(cid:105) correlation is strongly negative with the momentbased definitions, while the corresponding correlations with cumulant angles are positive. In4 〈〈 cos4( Φ - Φ ) 〉〉 〈〈 cos8( Φ - Φ ) 〉〉 -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 〈〈 cos12( Φ - Φ ) 〉〉 -0.4-0.2 0 0.2 0.4 0.6 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos6( Φ - Φ ) 〉〉× 〈〈 cos6( Φ - Φ ) 〉〉 -0.4-0.2 0 0.2 0.4 0.6 0.8 〈〈 cos6( Φ - Φ ) 〉〉 -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 〈〈 cos12( Φ - Φ ) 〉〉 -2-1.5-1-0.5 0 0.5 1 1.5 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos10( Φ - Φ ) 〉〉× FIG. 1. Participant 2-plane correlations from Phobos Monte Carlo Glauber model [17] as measuredby the cumulant and moment expansions. The measured event plane correlations [10] are presentedfor reference and as a point of contact, and are not supposed to be directly compared to the Glaubermodel results. 〈〈 cos(2 Φ +3 Φ -5 Φ ) 〉〉 ATLAScumulants.moments.-0.80-0.60-0.40-0.200.000.200.400.600.801.00 〈〈 cos(2 Φ +4 Φ -6 Φ ) 〉〉 -0.20-0.15-0.10-0.050.000.050.100.15 0 50 100 150 200 250 300 350 400 450 〈 N part 〉〈〈 cos(2 Φ -6 Φ +4 Φ ) 〉〉 -0.40-0.30-0.20-0.100.000.100.200.30 〈〈 cos(-8 Φ +3 Φ +5 Φ ) 〉〉 -0.40-0.200.000.200.400.600.80 〈〈 cos(-10 Φ +4 Φ +6 Φ ) 〉〉 -0.15-0.10-0.050.000.050.100.150.200.25 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos(-10 Φ +6 Φ +4 Φ ) 〉〉 FIG. 2. Participant 3-plane correlations from Phobos Monte Carlo Glauber model [17] as measuredby the cumulant and moment expansions. The measured event plane correlations [10] are presentedfor reference and as a point of contact, and are not supposed to be directly compared to the Glaubermodel results. ,Φ correlation arises because the fourth order eccentricity E is trivially correlated with the second order eccentricity E through the average geometry.These trivial geometric correlations are removed with the cumulant definition, and the resid-ual correlation is positive. The correlation between Ψ and Ψ seen in the data is positive,but does not seem to be directly related to the participant plane correlation between Φ andΦ . The interpretation of the data is described in Section III. B. Formulation of flow response
The harmonic flow v n and the corresponding flow angle Ψ n are defined by the Fourierdecomposition of the final state particle spectrum, dNdφ p = N π (cid:34) (cid:88) n (cid:0) v n e − in ( φ p − Ψ n ) + c.c. (cid:1)(cid:35) . (2.11)Here we use a complex expression, with c.c. standing for complex conjugate. For simplicity,we also define a complex flow coefficient which takes into account the flow and its anglesimultaneously, V n ≡ v n e in Ψ n . (2.12)Following the same strategy and notation as in our previous work [9], the magnitude ofthe flow and its corresponding angle is given by the response formula V n = (cid:18) w n ε n (cid:19) E n + (cid:88) quadratic (cid:18) w n ( pq ) ε p ε q (cid:19) E p E q + . . . . (2.13)Here w n is the n − th linear response coefficient to a given E n , and w n ( pq ) are the n − thquadratic response coefficients. The ellipses in eq. (2.13) stand for higher order nonlinearcontributions which are generally neglected in this work. The only exception to this rule isfor V where we included the contribution from E . Even in this case, the E contributionwas found to be numerically small compared to the quadratic E E and the E results. Thecurrent calculation uses the following minimal set of response coefficients w . . . w w , w , w , w , w , w , w . (2.14)We found that additional non-linear terms such as w , w , and w were not numeri-cally important for the current set of correlations. Thus, we reverted the code to the minimalset of response coefficients listed in eq. (2.14). The effects of including additional (radial)modes in the linear response was studied in [18, 19]. While a complete analysis will bepresented in future work, a preliminary investigation shows that these (radial) contributionsare small for the inclusive correlations studied here.The form of eq. (2.13) indicates the dependence of the n -th order harmonic flow and itsangle on the linear response coefficient w n and the quadratic response coefficients w n ( pq ) .These response coefficients are calculated by perturbing the (smooth) background geometryand determining the resulting flow. The details of this procedure have been given in ourprevious work [9], and here we will simply review the most important features.7inear and nonlinear flow response coefficients are obtained from “single-shot” 2+1Dhydrodynamic simulations. In this approach the average geometry for a given centralityclass is modeled with a cylindrically symmetric Gaussian, i.e. the initial entropy density inthe event at Bjorken time τ o is s ( x, y, τ o ) = C s τ o πR e − r /R . (2.15)The rms radius of the Gaussian is adjusted to match the rms radius of a smooth (or aver-aged) Glauber model for a given centrality. The overall constant of the Gaussian is adjustedas a function of centrality to reproduce the measured dN ch /dy at the LHC [20]. The re-sponse coefficients are calculated by perturbing this radially symmetric Gaussian by smalldeformations; running the perturbed Gaussian through the hydro tool chain; and finallycalculating w n or w n ( pq ) . For example, for we calculate w /(cid:15) by deforming the Gaussian bya tiny ε and calculating v . Similarly we calculate w / ( ε ε ) by deforming the Gaussianby ε and by ε and calculating v , which is proportional to ε ε . To summarize, all of theresponse coefficients and their dependence on centrality are obtained by simulating slightlydeformed cylindrically symmetric Gaussian initial conditions.We have implemented 2nd order BRSSS hydrodynamics, taking the necessary secondorder transport coefficients from the AdS/CFT results. The numerical scheme (but not thecode) is similar to the scheme developed in Ref. [21]. The shear viscosity to entropy ratio η/s is constant throughout the whole evolution, and is set to the canonical value of 1 / π . We usean equation of state that parametrizes the lattice results [22], which was used previously byRomatschke and Luzum [23]. Finally, we use a constant freeze-out temperature T fo = 150MeV, and adopt the widely used quadratic ansatz for the first viscous correction to thefreeze-out distribution function [24]. C. Formulation of plane correlations
The plane correlations are measured by event-plane method [10], and a multi-particlecorrelation method [11, 25]. We will focus on the event plane method which was used by theATLAS collaboration. The details of this method were clarified by Luzum and Ollitraultwho showed that if the event plane method is used, the quantity that is measured dependson the reaction plane resolution of the detector [12].We are interested in describing the correlations involving two and three event plane angles.For definiteness we will present formulas for a specific correlation, (cid:104)(cid:104) cos(4Ψ − )) (cid:105)(cid:105) ,which can be easily generalized to other harmonics. (To aid the reader we have written4Ψ = 2(2Ψ ) to expose the general pattern.) The 4-2 plane correlation is related to V and V through (cid:104)(cid:104) cos(4Ψ − )) (cid:105)(cid:105) = (cid:68)(cid:68) Re ( V V ∗ ) (cid:112) ( V V ∗ )( V V ∗ ) (cid:69)(cid:69) = (cid:68)(cid:68) w cos 4(Φ − Φ ) + w | w e − i + w e − i | (cid:69)(cid:69) . (2.16)Thus, both the linear and nonlinear response coefficients enter this formula for the eventplane correlation.The ATLAS collaboration quantified the event plane correlations by measuring relatedcorrelations between the experimental planes, ˆΨ n , as determined by the Q n -vectors, (cid:126)Q n =8 Q n | e − in ˆΨ n [10]. Further investigation showed that the measured quantity can not be directlyinterpreted as an event plane correlation in the form of eq. (2.16). The measured correlationequals eq. (2.16) when the experimental event plane resolution approaches unity, (cid:104) cos(4 ˆΨ − )) (cid:105){ EP } (cid:39) (cid:68)(cid:68) Re ( V V ∗ ) (cid:112) ( V V ∗ )( V V ∗ ) (cid:69)(cid:69) (high resolution limit) . (2.17)Here we have notated the experimental quantity with { EP } [10], and refer to Ref. [12] wherethe precise definition is carefully examined. The notation for the experimental quantity issomewhat misleading since the experimental definition does not actually correspond to theaverage of a cosine, and can be greater than one. In the limit of low event plane resolution,the measured quantity equals (cid:104) cos(4 ˆΨ − )) (cid:105){ EP } (cid:39) (cid:10)(cid:10) Re ( V V ∗ ) (cid:11)(cid:11)(cid:112) (cid:104)(cid:104) V V ∗ (cid:105)(cid:105) (cid:104)(cid:104) ( V V ∗ ) (cid:105)(cid:105) (low resolution limit) . (2.18)Clearly eq. (2.18) differs from eq. (2.17) by how the events are weighted. The event planemeasurements by the ATLAS collaboration (such as (cid:104) cos(4 ˆΨ − )) (cid:105){ EP } ) interpolatebetween the high and low resolution limits depending on the reaction plane resolution.As the experimental resolution depends on the harmonic number, the detector accep-tance, and centrality, we will compute both the high and low resolution limits and compareboth curves to the experimental data. In the future, such ambiguities in the measurementdefinition can be avoided by measuring (cid:68) v v cos(4 ˆΨ − )) (cid:69)(cid:113) (cid:104) v (cid:105) (cid:104) v (cid:105) = (cid:10)(cid:10) Re ( V V ∗ ) (cid:11)(cid:11)(cid:113) (cid:104)(cid:104) V V ∗ (cid:105)(cid:105) (cid:104)(cid:104) ( V V ∗ ) (cid:105)(cid:105) , (2.19)as originally suggested in [26], and more recently in [12]. Such angular of correlationshave already been measured by the ALICE collaboration [11], but we will not address thispreliminary data here. Certainly eq. (2.19) is the most natural from the perspective of theresponse formalism developed in this work.Finally, we give one additional example, − +3Ψ +5Ψ of how a three plane correlationfunction is calculated in the high and low resolution limits: (cid:104) cos( − ) + 3 ˆΨ + 5 ˆΨ ) (cid:105){ EP } (cid:39) (cid:68)(cid:68) Re ( V ∗ V V ) (cid:112) ( V V ∗ ) ( V V ∗ )( V V ∗ ) (cid:69)(cid:69) (high resolution) , (2.20) (cid:104) cos( − ) + 3 ˆΨ + 5 ˆΨ ) (cid:105){ EP } (cid:39) (cid:10)(cid:10) Re ( V ∗ V V ) (cid:11)(cid:11)(cid:112) (cid:104)(cid:104) ( V V ∗ ) (cid:105)(cid:105) (cid:104)(cid:104) V V ∗ (cid:105)(cid:105) (cid:104)(cid:104) V V ∗ (cid:105)(cid:105) (low resolution) . (2.21)In the future the quantity which is most easily compared to theoretical calculations is (cid:104) v v v cos( − ) + 3 ˆΨ − ) (cid:105) (cid:113) (cid:104) v (cid:105) (cid:104) v (cid:105) (cid:104) v (cid:105) = (cid:10)(cid:10) Re ( V ∗ V V ) (cid:11)(cid:11)(cid:113) (cid:104)(cid:104) V V ∗ (cid:105)(cid:105) (cid:104)(cid:104) V V ∗ (cid:105)(cid:105) (cid:104)(cid:104) V V ∗ (cid:105)(cid:105) . (2.22)9 II. DISCUSSIONS AND CONCLUSIONS
Figs. 3 and 4 show a comparison of the measured two and three plane correlation func-tions with the response formalism in the high and low resolution limits using the PHOBOSGlauber. To test the sensitivity to the Glauber model in the high resolution limit we com-pare two widely used monte-carlos – the PHOBOS Monte Carlo Glauber [17] and Glissando[16]. In Figs. 5 and 6, the predictions of viscous hydrodynamics based on these two initialstate models are shown by the blue and green lines, respectively. The two Glauber modelsgive similar results, although the correlations from Glissando are somewhat stronger.For the highest harmonics (such as v ), viscous corrections in peripheral collisions canbecome too large to be trusted. In this regime the linear and non-linear response coefficientscan become negative as a result of the first viscous correction to the distribution function [9].Second order corrections to the viscous distribution are positive [24], suggesting that suchnegative response coefficients are artificial. Indeed, kinetic theory simulations have positiveresponse coefficients for all values of the Knudsen parameter [27]. To understand whenviscous corrections to the response coefficients are out of control, we have performed twosimulations. In the first case (un-cut), we blindly allow the response coefficients to becomenegative. In the second case (cut), we set these coefficients to zero (as a function of centrality)when they turn negative. In Figs. 5 and 6 we show the correlation results of the un-cut(solid) and cut (dashed) response coefficients. As seen in these figures, the ambiguity isnoticeable only for peripheral collisions, and for correlations involving the highest harmonic,Ψ . Examining the (cid:104)(cid:104) cos 6(Ψ − Ψ ) (cid:105)(cid:105) correlation, we see that the negative dive in peripheralcollisions is an artifact of out-of-control viscous corrections. A similar negative dive is seenin event-by-event hydro simulations [13].Inspecting these correlations, we make the following observations. First, many of the mostimportant correlation functions are reasonably reproduced, at least if the high resolutionlimit is used. The agreement with the low resolution limit is not as good. The ambiguities inthe measurement can be avoided by taking definite moments as in eq. (2.19) [26]. Examiningthe definitions of the high and low resolution limits (Eqs. 2.17 and 2.18), we see that thedifference between the two measurements can be best quantified by measuring the probabilitydistribution P ( v n ) [28], or the moments of this distribution [29], e.g. for v ( v { } ) ≡ (cid:10) v (cid:11) and ( v { } ) ≡ − (cid:2)(cid:10) v (cid:11) − (cid:10) v (cid:11)(cid:3) . (3.1)It is then a separate and important question whether the response formalism outlined herecan reproduce these probability distributions. This will be addressed in future work.There are a few correlations which are seemingly not well reproduced even in the highresolution limit. First, one could hope for better agreement with the correlations involvingΨ such as cos(6Ψ − ) and cos(6Ψ − ). v is a relatively high harmonic, andviscous corrections are not in perfect control in peripheral collisions [24]. This is clearlyevident in Fig. 5 which estimates the contributions of higher order viscous corrections to thedistribution function (see above). For the Ψ -correlations (and no others), these correctionsare large in peripheral collisions.The most troubling correlation function, which is not qualitatively reproduced by theresponse formulation, is cos(2Ψ − + 4Ψ ). It is possible that that this discrepancystems from an underestimate of the mixing of v with other modes, which naturally mixes10 〈〈 cos4( Ψ - Ψ ) 〉〉 ATLASlow res.high res. 0 0.2 0.4 0.6 0.8 1 〈〈 cos8( Ψ - Ψ ) 〉〉 〈〈 cos12( Ψ - Ψ ) 〉〉 -0.2 0 0.2 0.4 0.6 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos6( Ψ - Ψ ) 〉〉× 〈〈 cos6( Ψ - Ψ ) 〉〉 -0.2 0 0.2 0.4 0.6 0.8 〈〈 cos6( Ψ - Ψ ) 〉〉 -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 〈〈 cos12( Ψ - Ψ ) 〉〉 -2-1.5-1-0.5 0 0.5 1 1.5 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos10( Ψ - Ψ ) 〉〉× FIG. 3. Two plane correlations using the non-linear response formalism. Here η/s = 1 / π forPHOBOS Monte-Carlo Glauber initial conditions. The data are from the ATLAS collaboration[10]. The solid lines indicate the high resolution limit, eq. (2.17), while the dashed lines indicatethe low resolution limit, eq. (2.18). .000.200.400.600.801.001.20 〈〈 cos(2 Ψ +3 Ψ -5 Ψ ) 〉〉 ATLASlow res.high res.-0.500.000.501.00 〈〈 cos(2 Ψ +4 Ψ -6 Ψ ) 〉〉 -0.25-0.20-0.15-0.10-0.050.000.05 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos(2 Ψ -6 Ψ +4 Ψ ) 〉〉 -0.40-0.30-0.20-0.100.000.100.200.30 〈〈 cos(-8 Ψ +3 Ψ +5 Ψ ) 〉〉 -0.200.000.200.400.600.801.001.201.401.60 〈〈 cos(-10 Ψ +4 Ψ +6 Ψ ) 〉〉 -0.20-0.15-0.10-0.050.000.050.100.150.200.25 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos(-10 Ψ +6 Ψ +4 Ψ ) 〉〉 FIG. 4. Three plane correlations using the non-linear response formalism. Here η/s = 1 / π forPHOBOS Monte-Carlo Glauber initial conditions. The data are from the ATLAS collaboration[10]. The solid lines indicate the high resolution limit, eq. (2.17), while the dashed lines indicatethe low resolution limit, eq. (2.18). 〈〈 cos4( Ψ - Ψ ) 〉〉 ATLASGlissandoPHOBOS 0 0.2 0.4 0.6 0.8 1 〈〈 cos8( Ψ - Ψ ) 〉〉 〈〈 cos12( Ψ - Ψ ) 〉〉 -0.4-0.2 0 0.2 0.4 0.6 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos6( Ψ - Ψ ) 〉〉× 〈〈 cos6( Ψ - Ψ ) 〉〉 -0.4-0.2 0 0.2 0.4 0.6 0.8 〈〈 cos6( Ψ - Ψ ) 〉〉 -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 〈〈 cos12( Ψ - Ψ ) 〉〉 -2-1.5-1-0.5 0 0.5 1 1.5 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos10( Ψ - Ψ ) 〉〉× ⇔ solid/dash FIG. 5. (Color online) A comparison of the two-plane correlations in the high resolution limitfor two different Glauber models, Glissando [16] and the PHOBOS Glauber [17]. The solid lines(un-cut) include the negative response in peripheral collisions due to a large δf , while the dashedlines (cut) truncate the negative response – see Section III. .000.200.400.600.801.001.20 〈〈 cos(2 Ψ +3 Ψ -5 Ψ ) 〉〉 uncut/cut ⇔ solid/dash-0.40-0.200.000.200.400.600.801.00 〈〈 cos(2 Ψ +4 Ψ -6 Ψ ) 〉〉 -0.25-0.20-0.15-0.10-0.050.000.05 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos(2 Ψ -6 Ψ +4 Ψ ) 〉〉 -0.40-0.30-0.20-0.100.000.100.200.30 〈〈 cos(-8 Ψ +3 Ψ -5 Ψ ) 〉〉 ATLASGlissandoPHOBOS-0.200.000.200.400.600.80 〈〈 cos(-10 Ψ +4 Ψ +6 Ψ ) 〉〉 -0.20-0.15-0.10-0.050.000.050.100.150.200.25 0 50 100 150 200 250 300 350 400 〈 N part 〉〈〈 cos(-10 Ψ +6 Ψ +4 Ψ ) 〉〉 FIG. 6. (Color online) A comparison of the three-plane correlations in the high resolution limitfor two different Glauber models, Glissando [16] and the PHOBOS Glauber [17]. The solid lines(un-cut) include the negative response in peripheral collisions due to a large δf , while the dashedlines (cut) truncate the negative response – see Section III. -0.5 0 0.5 1 100 200 300 400 〈 c o s ( Ψ - Ψ ) 〉 N part Linear resp. onlyNon-linear resp. onlyGlauber expect.linear+nonlin. resp. ATLAS -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 100 200 300 400 〈 c o s ( Ψ + Ψ - Ψ ) 〉 N part Linear resp. onlyNon-linear resp. onlyGlauber expect.linear+nonlin. resp. ATLAS
FIG. 7. The separate contributions of the linear and non-linear response to a two-plane correlation, (cid:104) cos(4Ψ − ) (cid:105) , and a three-plane correlation, (cid:104) cos(2Ψ + 3Ψ − ) (cid:105) . The dashed lines showthe naive Glauber expectation (see text). The data is from Ref. [10]. v with v . Indeed, a preliminary analysis suggests that this correlation is closely relatedto the transverse shift from the geometrical center to the center of participants. The 2 , , (cid:104) cos(4(Ψ − Ψ )) (cid:105) and (cid:104) cos(2Ψ + 3Ψ − ) (cid:105) respectively. For definiteness, we study the2,3,5 combination shown in Fig. 7(b). The naive expectation of the Glauber model (where v n is proportional to the n − th order moment based eccentricity) is shown by the dottedline, and has the wrong sign. In the naive approach the observed correlation between theevent plane angles 2 , , v is produced through a combination of the linear and non-linearresponse. • In linear response, v is proportional to the 5-th cumulant (cid:15) , and the correlationbetween the event plane angles Ψ , Ψ , Ψ reflects the initial state correlation betweenthe associated cumulant angles, Φ , Φ , Φ . The predictions of linear response areshown in Fig. 7, and fail to reproduce the observed correlations in non-central collisions. • In non-linear response, v is determined through the mode mixing of v and v . If v was determined entirely by this mechanism, the Ψ event plane would be entirelydetermined by Ψ and Ψ , leading to a perfect 2 , , v is determined by a weighted average of the linear and non-linear responsecurves. The relative size of these two contributions is determined by viscous hydrodynamicswhich predicts the magnitude of these response coefficients as a function of centrality. Evi-dently, hydrodynamics and the response formalism reproduces the centrality dependence of15he observed correlation functions. It is satisfying to see how the data transition between thelinear response curves in central collisions, and the non-linear response curves in peripheralcollisions.Finally, we conclude by discussing the importance of higher order terms in the responseformalism. First, we have neglected the third order mixing of harmonics. The most im-portant third order term is proportional to E , and we have found that this term is smallcompared to the E and E E terms. Thus, the response formalism seems to converge, andincluding the mixing of higher harmonics will not change the results of this study signifi-cantly.In the future it will be important to characterize the fluctuations around the responseformalism. For any given initial state characterized by a few macroscopic cumulants such as E , E , E . . . , the observed v n will on average be given by the response formalism. However,additional fluctuations (which leave the macroscopic cumulants fixed) will reduce the perfectcorrelation between v , v , v , . . . and the predictions of non-linear response. Thus, in general,the response formalism will overestimate the strength of the correlations that are observed.Ideally, the fluctuations around the response formalism can be parametrized by universalGaussian noise, which will be independent of the microscopic details of the initial state. Thestudy of fluctuations around the response formalism is left for future work. Acknowledgments:
We thank J. Y. Ollitrault, Z. Qiu, U. Heinz, J. Jia, and S. Mohapatra for many con-structive and insightful comments. D. Teaney is a RIKEN-RBRC fellow. This work issupported by the Department of Energy, DE-FG-02-08ER4154. Li Yan is also funded bythe European Research Council under the Advanced Investigator Grant ERC-AD-267258. [1] Ulrich W Heinz and Raimond Snellings, “Collective flow and viscosity in relativistic heavy-ioncollisions,” (2013), arXiv:1301.2826 [nucl-th].[2] Boris Hippolyte and Dirk H. Rischke, “Global variables and correlations: Summary of theresults presented at the Quark Matter 2012 conference,” Nucl.Phys.
A904-905 , 318c–325c(2013), arXiv:1211.6714 [nucl-ex].[3] Derek A. Teaney, “Viscous Hydrodynamics and the Quark Gluon Plasma,” (2009), invitedreview for ’Quark Gluon Plasma 4’. Editors: R.C. Hwa and X.N. Wang, World Scientific,Singapore., arXiv:0905.2433 [nucl-th].[4] B. Alver and G. Roland, “Collision geometry fluctuations and triangular flow in heavy-ioncollisions,” Phys.Rev.
C81 , 054905 (2010), arXiv:1003.0194 [nucl-th].[5] Zhi Qiu and Ulrich W. Heinz, “Event-by-event shape and flow fluctuations of relativisticheavy-ion collision fireballs,” Phys.Rev.
C84 , 024911 (2011), arXiv:1104.0650 [nucl-th].[6] Fernando G. Gardim, Frederique Grassi, Matthew Luzum, and Jean-Yves Ollitrault, “Map-ping the hydrodynamic response to the initial geometry in heavy-ion collisions,” Phys.Rev. , 024908 (2012), arXiv:1111.6538 [nucl-th].[7] Nicolas Borghini and Jean-Yves Ollitrault, “Momentum spectra, anisotropic flow, and idealfluids,” Phys.Lett. B642 , 227–231 (2006), arXiv:nucl-th/0506045 [nucl-th].[8] Stefan Floerchinger, Urs Achim Wiedemann, Andrea Beraudo, Luca Del Zanna, GabrieleInghirami, et al. , “How (non-) linear is the hydrodynamics of heavy ion collisions?” (2013),arXiv:1312.5482 [hep-ph].[9] Derek Teaney and Li Yan, “Non linearities in the harmonic spectrum of heavy ion collisionswith ideal and viscous hydrodynamics,” Phys.Rev.
C86 , 044908 (2012), arXiv:1206.1905 [nucl-th].[10] The ATLAS Collaboration, “Measurement of reaction plane correlations in Pb-Pb collisions at √ s NN =2.76 TeV,” (May, 2012), ATLAS-CONF-2012-049. See also https://cdsweb.cern.ch/record/1451882 .[11] Ante Bilandzic (ALICE Collaboration), “Anisotropic flow measured from multi-particle az-imuthal correlations for Pb-Pb collisions at 2.76 TeV by ALICE at the LHC,” Nucl.Phys.A904-905 , 515c–518c (2013), arXiv:1210.6222 [nucl-ex].[12] Matthew Luzum and Jean-Yves Ollitrault, “The event-plane method is obsolete,” Phys.Rev. C87 , 044907 (2013), arXiv:1209.2323 [nucl-ex].[13] Zhi Qiu and Ulrich Heinz, “Hydrodynamic event-plane correlations in Pb+Pb collisions at √ s = 2 . B717 , 261–265 (2012), arXiv:1208.1200 [nucl-th].[14] Rajeev S. Bhalerao, Jean-Yves Ollitrault, and Subrata Pal, “Event-plane correlators,” (2013),arXiv:1307.0980 [nucl-th].[15] Charles Gale, Sangyong Jeon, Bjorn Schenke, Prithwish Tribedy, and Raju Venugopalan,“Event-by-event anisotropic flow in heavy-ion collisions from combined Yang-Mills and viscousfluid dynamics,” Phys.Rev.Lett. , 012302 (2013), arXiv:1209.6330 [nucl-th].[16] Wojciech Broniowski, Maciej Rybczynski, and Piotr Bozek, “GLISSANDO: Glauber initial-state simulation and more..” Comput.Phys.Commun. , 69–83 (2009), arXiv:0710.5731[nucl-th].[17] B. Alver, M. Baker, C. Loizides, and P. Steinberg, “The PHOBOS Glauber Monte Carlo,”(2008), arXiv:0805.4411 [nucl-ex].[18] Stefan Floerchinger and Urs Achim Wiedemann, “Mode-by-mode fluid dynamics for relativis-tic heavy ion collisions,” (2013), arXiv:1307.3453.[19] Stefan Floerchinger and Urs Achim Wiedemann, “Kinetic freeze-out, particle spectra andharmonic flow coefficients from mode-by-mode hydrodynamics,” (2013), arXiv:1311.7613 [hep-ph].[20] Li Yan, “A Hydrodynamic Analysis of Collective Flow in Heavy-Ion Collisions,” Ph.D. thesis,Stony Brook University (2013).
21] Kevin Dusling, Guy D. Moore, and Derek Teaney, “Radiative energy loss and v(2) spectrafor viscous hydrodynamics,” Phys.Rev.
C81 , 034907 (2010), arXiv:0909.0754 [nucl-th].[22] Mikko Laine and York Schroder, “Quark mass thresholds in QCD thermodynamics,”Phys.Rev.
D73 , 085009 (2006), arXiv:hep-ph/0603048 [hep-ph].[23] Matthew Luzum and Paul Romatschke, “Conformal Relativistic Viscous Hydrodynamics: Ap-plications to RHIC results at √ s = 200 GeV,” Phys. Rev. C78 , 034915 (2008), arXiv:0804.4015[nucl-th].[24] Derek Teaney and Li Yan, “Second order viscous corrections to the harmonic spectrum inheavy ion collisions,” (2013), arXiv:1304.3753 [nucl-th].[25] Rajeev S. Bhalerao, Matthew Luzum, and Jean-Yves Ollitrault, “Understandinganisotropy generated by fluctuations in heavy-ion collisions,” Phys.Rev.
C84 , 054901 (2011),arXiv:1107.5485 [nucl-th].[26] Rajeev S. Bhalerao, Matthew Luzum, and Jean-Yves Ollitrault, “Determining initial-statefluctuations from flow measurements in heavy-ion collisions,” Phys.Rev.
C84 , 034910 (2011),arXiv:1104.4740 [nucl-th].[27] Burak Han Alver, Clement Gombeaud, Matthew Luzum, and Jean-Yves Ollitrault, “Tri-angular flow in hydrodynamics and transport theory,” Phys.Rev.
C82 , 034913 (2010),arXiv:1007.5469 [nucl-th].[28] Georges Aad et al. (ATLAS Collaboration), “Measurement of the distributions of event-by-event flow harmonics in lead–lead collisions at √ s NN =2.76 TeV with the ATLAS detector atthe LHC,” (2013), arXiv:1305.2942 [hep-ex].[29] B. Alver et al. (PHOBOS Collaboration), “Event-by-Event Fluctuations of Azimuthal ParticleAnisotropy in Au + Au Collisions at √ s NN = 200 GeV,” Phys.Rev.Lett. , 142301 (2010),arXiv:nucl-ex/0702036 [nucl-ex]., 142301 (2010),arXiv:nucl-ex/0702036 [nucl-ex].