Relative flow fluctuations as a probe of initial state fluctuations
Giuliano Giacalone, Jacquelyn Noronha-Hostler, Jean-Yves Ollitrault
RRelative flow fluctuations as a probe of initial state fluctuations
Giuliano Giacalone, Jacquelyn Noronha-Hostler, and Jean-Yves Ollitrault Institut de physique th´eorique, Universit´e Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France Department of Physics, University of Houston, Houston TX 77204, USA
Elliptic flow, v , and triangular flow, v , are to a good approximation linearly proportional tothe corresponding spatial anisotropies of the initial density profile, ε and ε . Using event-by-event hydrodynamic simulations, we point out when deviations from this linear scaling are to beexpected. When these deviations are negligible, relative v n fluctuations are equal to relative ε n fluctuations, and one can directly probe models of initial conditions using ratios of cumulants, forinstance v n { } /v n { } . We argue that existing models of initial conditions tend to overestimateflow fluctuations in central Pb+Pb collisions, and to underestimate them in peripheral collisions.We make predictions for v { } in noncentral Pb+Pb collisions, and for v { } and v { } in high-multiplicity p+Pb collisions. I. INTRODUCTION
Anisotropic flow is the key observable providing evi-dence for the creation of a collective medium in ultra-relativistic heavy-ion collisions. In the current paradigmof bulk particle production [1], anisotropic flow emergesfrom the hydrodynamical response of the created mediumto the anisotropies of its initial energy density profile [2].Hydrodynamic simulations [3–5] show that elliptic flow, v , and triangular flow, v , correlate almost linearly withthe initial eccentricity, ε , and triangularity, ε , of thesystem. Since the initial energy density profile is shapedout of stochastic nucleon-nucleon interactions, both ini-tial anisotropies and flow coefficients fluctuate on a event-by-event basis [6]. To the extent that v n is proportionalto ε n , the probability distribution of v n [7] coincides, upto a global rescaling, with the probability distributionof ε n [8, 9]. The latter is provided by models of initialconditions.Many models of initial conditions have been proposedfor proton-nucleus and nucleus-nucleus collisions. Someare based on variations of the Glauber Monte Carlomodel [10–14], others are more directly inspired fromhigh-energy QCD, and involve, in particular, the ideaof gluon saturation [15–20]. The initial anisotropies ε n probe the geometrical shape of the initial density pro-file, and, thus, provide information which is independentof the final multiplicity distribution, which is the typ-ical observable to which models are tuned. Therefore,observables which can be linked to initial anisotropies al-low one to further constrain initial condition models, andto eventually obtain new insight into the early dynamicsof the collision.In this paper, we analyze the relative fluctuations of v and v in p+Pb and Pb+Pb collisions at CERN LargeHadron Collider (LHC) energies. The observables wechoose for this analysis are ratios of cumulants of thedistribution of v n , whose definition is recalled in Sec. II.In Sec. III, we compute the lowest non-trivial ratiosof cumulants, v { } /v { } and v { } /v { } , in event-by-event hydrodynamic simulations of Pb+Pb collisions,and we determine in which centrality intervals they are compatible with the ratios of cumulants of the corre-sponding initial anisotropies, ε n . In these centrality in-tervals, we compute ratios of cumulants using models ofinitial conditions, that can in this way be tested directlyagainst experimental data on v n { } /v n { } . To make ouranalysis as inclusive as possible, we test a wide varietyof initial condition models, thus covering the spectrumof models typically used in hydrodynamic calculations.Eventually, we employ these initial state parametriza-tions to predict v { } /v { } in Pb+Pb collisions. Asimilar study is carried over to high-multiplicity p+Pbcollisions, in Sec. IV. Specifically, we employ the state-of-the-art Monte Carlo model of initial conditions forp+Pb collisions to make predictions for v { } /v { } , and v { } /v { } . II. CUMULANTS AND RELATIVEFLUCTUATIONS
Anisotropic flow is the observation of a full spectrum ofnonzero Fourier coefficients characterizing the azimuthaldistribution of final-state particles in heavy-ion collisions.Denoting the final-state azimuthal distribution by P ( φ ),its Fourier decomposition reads P ( φ ) = 12 π + ∞ (cid:88) n = −∞ V n e − inφ , (1)and the quantity v n ≡ | V n | is the coefficient of anisotropicflow in the n th harmonic. In experiments, the num-ber of final-state particles is not large enough to al-low the computation of the Fourier series of Eq. (1) inevery event. Flow coefficients are computed from az-imuthal multi-particle correlations, which are averagedover many events. Since P ( φ ) is different in each colli-sion, anisotropic flow coefficients fluctuate on an event-by-event basis. Detailed information about the probabil-ity distribution of v n can be obtained by measuring itscumulants. A cumulant of order m involves m -particlecorrelations, as well as lower order correlations [21–23]: Itis constructed by an order-by-order subtraction of trivial a r X i v : . [ nu c l - t h ] M a y contributions coming from lower-order correlations. Cu-mulants are considered the best signature of the collectiveorigin of anisotropic flow in heavy-ion collisions. Nonzerovalues of higher-order cumulants have been measured ina wide range of collision systems, from Pb+Pb to p+pcollisions [24–26].The cumulants of the distribution of v n are combina-tions of moments. Explicit expressions up to order 8are [27] v n { } = (cid:104) v n (cid:105) ,v n { } = 2 (cid:104) v n (cid:105) − (cid:104) v n (cid:105) ,v n { } = 14 (cid:20) (cid:104) v n (cid:105) − (cid:104) v n (cid:105)(cid:104) v n (cid:105) + 12 (cid:104) v n (cid:105) (cid:21) ,v n { } = 133 (cid:20) (cid:104) v n (cid:105) − (cid:104) v n (cid:105) (cid:104) v n (cid:105) + 18 (cid:104) v n (cid:105) + 16 (cid:104) v n (cid:105)(cid:104) v n (cid:105) − (cid:104) v n (cid:105) (cid:21) , (2)where angular brackets denote an average over collisionevents in a given centrality class. Cumulants are definedin such a way that v n { k } = v n , if v n is the same for allevents.Any quantity which is linearly proportional to v n hasthe same cumulants as v n , up to a global factor. If thescaling between v n and ε n were exactly linear, then, forany even integers µ and ν [28], v n { µ } v n { ν } = ε n { µ } ε n { ν } . (3)Ratios of cumulants quantify the relative fluctuations of v n , which are equal to the relative fluctuations of ε n if thescaling is linear [8, 29]. In this work, we mainly focus onthe ratio v n { } /v n { } as a measure of the relative fluc-tuations of v n . This ratio depends on the event-by-eventfluctuations of v n . In particular, the larger the fluctu-ations of v n are, the smaller the ratio v n { } /v n { } is.Higher-order ratios of cumulants, such as v n { } /v n { } ,probe the non-Gaussianity of the fluctuations [27, 30].Ratios of cumulants are interesting because they areindependent of the hydrodynamic response (the propor-tionality coefficient between ε n and v n ), which is an im-portant source of uncertainty when trying to constrainmodels of initial conditions from experimental data [31].Equation (3) allows us to directly relate experimen-tal data (left-hand side) to models of initial conditions(right-hand side). The approximate linearity of the re-lation between v n and ε n in event-by-event hydrodynam-ics is typically measured using scatter plots [4] or thePearson correlation coefficient [3]. Nevertheless, theseapproaches do not give any information on ratios of cu-mulants, and on the accuracy of Eq. (3). More precisely, A similar analysis was recently carried out at Relativistic HeavyIon Collider (RHIC) energies within the AMPT model [28]. { } / v { } v hydro initial ATLAS centrality [%] { } / v { } v (b) FIG. 1. (color online) Comparison between v n { } /v n { } computed in hydrodynamics (full symbols) and ε n { } /ε n { } computed from the corresponding initial energy density pro-files (open symbols), for 2.76 TeV Pb+Pb collisions. Shadedbands: ATLAS data for v n { } /v n { } [24]. Symbols areshifted horizontally for readability. (a) Elliptic flow ( n = 2).(b) Triangular flow ( n = 3). if one models the deviation from linear scaling by a Gaus-sian noise, v n = κ n ε n + δ , where δ is a random fluctua-tion with a Gaussian distribution, this noise will typicallycontribute to the rms value of v n { } , not to higher-ordercumulants. Therefore, it is not at all trivial that ratios ofcumulants are preserved by the hydrodynamic evolution.In the next section, we analyze the validity of Eq. (3)more robustly, by testing this equation directly throughhydrodynamic calculations. III. Pb+Pb COLLISIONS
We first test the validity of Eq. (3) for v { } /v { } and v { } /v { } , by computing both sides of the equation inevent-by-event hydrodynamics. We run hydrodynamicsimulations of Pb+Pb collisions at √ s = 2 .
76 TeV. Theinitial conditions from which initial anisotropies are com-puted are given by a Glauber Monte Carlo model [12, 32].Initial density profiles are evolved by means of the vis-cous relativistic hydrodynamical code
V-USPHYDRO [33–35]. We implement a shear viscosity over entropy ra-tio of η/s = 0 .
08 [36], and we compute flow coefficientsat freeze-out [37] for pions in the transverse momentumrange 0 . < p t < c . We compute v { } /v { } and v { } /v { } as function of centrality percentile. Be-tween 1000 and 5000 events are simulated in each cen-trality window, each event corresponding to a differentinitial geometry. Results are shown in Fig. 1, and arecompared to the measurements of the ATLAS Collabo-ration [24]. A first remark is that v { } /v { } is smallerthan v { } /v { } . This means that v fluctuations arelarger than v fluctuations, as expected since v is solelydue to fluctuations [38]. The smallness of v { } explainsthe large statistical error on the corresponding ratio. Wenow discuss, in turn, v { } /v { } and v { } /v { } . Inthe centrality intervals where Eq. (3) holds to a goodapproximation, we test initial condition models againstexperimental data. A. Elliptic flow fluctuations
We start with v [Fig. 1–(a)]. Equation (3) holdsapproximately up to 20 −
30% centrality, and gradu-ally breaks down as the centrality percentile increases.The difference between ε { } /ε { } and v { } /v { } canbe attributed to a cubic response term, proportional to( ε ) [39]. Once this nonlinear hydrodynamic response istaken into account, agreement with ATLAS data is excel-lent all the way up to 70% centrality. As we shall explainbelow, a similar nonlinear hydrodynamic response is alsoneeded for other models of initial conditions in order tomatch experimental data.Between 0% and 20% centrality, Eq. (3) holds to agood approximation. Therefore, in this centrality win-dow, the ratio ε { } /ε { } provided by initial conditionmodels can be tested directly against experimental datafor v { } /v { } . We test the sensitivity of this observ-able to initial conditions using TRENTo [42], a flexibleparametric Monte Carlo model which effectively encom-passes most of existing initial condition models [43]. Theinitial entropy density in TRENTo is expressed in termsof thickness functions, T A and T B , associated with eachof the colliding nuclei. Each thickness function is a sumof Gaussians, centered around the participant nucleons.The weight of each participant nucleon is a random vari-able, so that the contribution of a participant to the de-posited energy density may fluctuate. The strength ofthese fluctuations is regulated by a parameter, k (see theAppendix for details). Another parameter is the widthof the Gaussians, σ . The initial density profile is as-sumed to be a homogeneous function of degree 1 of thethickness functions T A and T B , and a third parameter p specifies this dependence. The values p = 1, p = 0, centrality [%] { } / v { } v ALICE CMS ATLAS TR p=1 TR p=0 TR p=-1 rcBK
FIG. 2. (color online) Test of initial condition models using v { } /v { } measured in Pb+Pb collisions at 2 .
76 TeV up to20% centrality. Stars: CMS data [40]. Full circles: ALICEdata [41]. Shaded band: ATLAS data [7]. Open symbols:Values of ε { } /ε { } given by the TRENTo model with p = − p = 0 (circles) and p = 1 (squares). Fulldiamonds: ε { } /ε { } from the Monte Carlo rcBK model. and p = − T A + T B ) /
2, a geometric mean, √ T A T B , and aharmonic mean, T A T B / ( T A + T B ). The case p = 1 cor-responds to the Glauber Monte Carlo model, where theenergy density is proportional to the number of woundednucleons [10]. The case p = 0 gives results close toQCD-inspired models such as IP-Glasma [18, 42] andEKRT [20, 43], while p = − ε { } /ε { } and ε { } /ε { } , in Pb+Pb collisions, depend little on theparameters k and σ . Therefore, we fix these parame-ters to the values suggested by the authors of TRENTo[42], which allow for a good description of the multiplicitydistributions [14, 42]. On the other hand, ratios of cu-mulants strongly depend on the third parameter, p . Re-sults for ε { } /ε { } are shown in Fig. 2, where they arecompared to available experimental data on v { } /v { } .The case p = 1, corresponding to wounded nucleon scal-ing, is in poor agreement with data. In particular, theratio ε { } /ε { } is below data. This means that the rel-ative fluctuations of ε are too large, causing ε { } to falltoo steeply in central collisions [29]. The other values of p , p = 0, and p = −
1, corresponding to saturation mod-els, are in fair agreement with data. Note that, in central A comparison of the behaviors of v { } and ε { } in the 0 − collisions, ε { } is essentially equal to the mean eccentric-ity in the reaction plane [27]. Saturation-inspired modelsare known to predict a larger mean eccentricity in the re-action plane than the Glauber model [44, 45]. The largermean eccentricity implies that relative fluctuations of ε are smaller. Therefore, the ratio ε { } /ε { } is larger.Figure 2 also displays, for comparison, results obtainedusing the Monte Carlo rcBK [16] initial state model. ThisQCD-inspired model predicts a mean eccentricity in thereaction plane comparable to the MC-KLN model [31],which explains why results are similar to TRENTo with p = − v { } /v { } , much as inFig. 1 (a). Therefore, for mid-central and peripheral col-lisions, all parameterizations of initial conditions requirea nonlinear hydrodynamic response, breaking Eq. (3), inorder to be compatible with data. B. Triangular flow fluctuations
We now test the validity of Eq. (3) in the case oftriangular flow fluctuations. Hydrodynamic results inFig. 1 (b) show that, as in the case of elliptic flow, ε { } /ε { } is systematically larger than v { } /v { } above 40% centrality. This can again be attributed to anonlinear hydrodynamic response, whose effect is, how-ever, smaller for v than for v . A possible explanationto this nonlinear effect could be a coupling between v and v [5]. In general, one expects any nonlinear effectto be associated with the large magnitude of v , whichis by far the largest Fourier harmonic [48]. Therefore,even though the large error bars in Fig. 1 (b) preventany definite conclusion, we expect the nonlinear responsebetween ε and v to be small in central collisions.By virtue of this conclusion, we compare v { } /v { } from experimental data to ε { } /ε { } from initial statemodels, across the full centrality range. We implementthe same models as in Fig. 2, and we also show resultsobtained using the IP-Glasma [18] model, for sake of com-parison. Results are displayed in Fig. 3, where the 0 − ε fluctua-tions, on the other hand, grow from central to peripheralcollisions in all the tested models. This centrality depen-dence has a simple explanation: Since the system sizedecreases as a function of the centrality percentile, therelative fluctuations of ε become larger [49]. In general,the nonlinear hydrodynamic response seen in Fig. 1–(b) centrality range also shows that the MC-KLN model is in betteragreement with data than the Glauber model [41]. A similar conclusion was drawn from simulations within the IPGlasma model [46]. would help in decreasing v { } /v { } above 40% central-ity and reducing the centrality dependence, which is seenin models and not in data. However, all configurations ofTRENTo in Fig. 3–(b) are compatible with ATLAS dataabove 40% centrality, and some points would fall belowdata if a nonlinear response were included.Figure 3–(a) presents results in the 20% most cen-tral collisions, where we use a finer centrality binningfor initial-state models. In this centrality range, we donot foresee any significant nonlinear hydrodynamic re-sponse, and initial state calculations should match data.Data points (in particular the measurements of the AL-ICE Collaboration) are, however, above the predictionsof all models. As observed for elliptic flow, the woundednucleon prescription (p=1) gives the worst results. Weconclude that initial state models overestimate the rela-tive fluctuations of ε in central Pb+Pb collisions. C. Predictions for v { } We now use Eq. (3) to make predictions for v { } in Pb+Pb collisions. The number of events in our hy-drodynamic calculations is not large enough to test di-rectly the validity of Eq. (3) for v { } /v { } . How-ever, we have noted that the nonlinear hydrodynamicresponse is smaller for v than for v . In addition, a pre-vious study [27] has shown that, even for v , the ratio v { } /v { } is little affected by the nonlinear response,so that Eq. (3) applies, to a good approximation, upto very peripheral collisions. Therefore, we assume thatEq. (3) yields a reasonable estimate of v { } /v { } , andwe make predictions on this basis using our TRENToconfigurations and the rcBK model.It has been argued that the probability distribution of ε [51], which is solely due to fluctuations, is well de-scribed by the power distribution [50], which has a singlefree parameter characterizing the rms value of ε . If thedistribution of ε follows the power distribution, then,the ratio ε { } /ε { } is a simple function of the ratio ε { } /ε { } , which is displayed as a dashed line in Fig. 4.By running Monte Carlo simulations of the initial state,we can test whether the results fall on this line. To thispurpose, we simulate a large number of initial conditionsfor Pb+Pb collisions, and we compute ε { } /ε { } in the20 −
80% centrality range.Results are shown as symbols in Fig. 4. The centralitypercentile corresponding to each symbol can be inferredfrom Fig. 3 (b). For a given model, ε { } /ε { } increaseswith the centrality percentile. The rcBK model agreeswith the prediction of the power distribution, while thevarious parametrizations of the Trento model give in gen-eral values of ε { } /ε { } which fall below the expectedcurve. The fact that the power distribution can be a poorapproximation for large systems such as Pb+Pb colli-sions, even if the anisotropy is solely due to fluctuations,has already been pointed out in Ref. [52]. Even thoughprecise figures depend on the particular model used, we { } / v { } v (a) ALICE CMS ATLAS centrality [%]
20 30 40 50 60 70 80 (b) TR p=1 TR p=0 TR p=-1 rcBK IP Glasma
FIG. 3. (color online) Test of initial condition models using v { } /v { } measured in 2.76 TeV Pb+Pb collisions: (a) up to 20%centrality; (b) between 20% and 80% centrality. Stars: CMS data [47]. Full circles: ALICE data [41]. Shaded band: ATLASdata [24]. ALICE and CMS data are not shown in panel (b) for the sake of readability, but are compatible with ATLAS data.Remaining symbols correspond to values of ε { } /ε { } from several models of initial conditions. Open symbols: TRENTo,with p = − p = 0 (circles), p = 1 (squares). Full crosses: IP-Glasma [18]. Full diamonds: Monte Carlo rcBK [16]. predict on the basis of our Monte Carlo calculations, andof Eq. (3), that v { } /v { } should lie between 0.75 and0.85 in the 30 −
50% centrality range.
IV. HIGH-MULTIPLICITY p+Pb COLLISIONS
In this Section, we study relative flow fluctuations inhigh-multiplicity p+Pb collisions at √ s = 5 .
02 TeV, andwe make quantitative predictions for higher-order cumu-lants of v and v . Nonzero elliptic and triangular flowvalues have been measured in p+Pb systems [25, 53–55].In particular, a positive v { } has been reported by allcollaborations, suggesting that the measured azimuthalcorrelations originate from a collective effect. Hydrody-namic simulations have also been carried out [56–61],using either IP-Glasma or Glauber Monte Carlo initialconditions. Satisfactory agreement with data was found,which supports the hydrodynamic picture as a valid de-scription of the p+Pb system [62]. Since elliptic flow issignificantly smaller in p+Pb collisions than in Pb+Pbcollisions [63], one does not expect a significant nonlin-ear hydrodynamic response, and we assume that Eq. (3)always holds. Event-by-event hydrodynamic simulationsconfirm that v and v scale linearly with the correspond-ing initial anisotropies, ε and ε [56].We first select a model of initial conditions by requiringthat it reproduces the first nontrivial ratio of cumulants, v { } /v { } , which has been measured by the CMS Col-laboration [54], as a function of centrality percentile. Asin the previous section, we employ the TRENTo model.However, the sets of parameters that give a reasonabledescription of Pb+Pb data fail to describe p+Pb data.Specifically, the values p = − p = 0, which providea good description of experimental data in Fig. 2, yielda negative ε { } in p+Pb collisions (i.e., an undefined ε { } ), and values of ε which are much smaller thanneeded in order to explain the magnitude of the mea-sured v . This is due to the fact that, with these param-eters, the initial density profile is always included in thetransverse area spanned by the proton, which is circular.For the same reason, the IP-Glasma model underpredicts v by a large factor, unless one allows the proton to be“eccentric” [59]. On the other hand, previous hydrody-namic calculations have shown that the implementationof Glauber Monte Carlo initial conditions yields results ingood agreement with p+Pb data. We therefore choosethe value p = 1, corresponding to the Glauber model,even though it does give a bad description of flow fluctua-tions in Pb+Pb data. We fix the parameter governing themultiplicity fluctuations to the value k = 0 . σ of the source associated with each nucleon tovary. Previous calculations implement σ = 0 . {2} / v{4} v { } / v { } v
20 - 80 %
TR p=1 TR p=0 TR p=-1 rcBK
FIG. 4. (color online) Predictions for v { } /v { } in 2.76 TeVPb+Pb collisions, from several models of initial conditions, inthe 20 −
80% centrality range. Empty symbols: Predictionsof TRENTo with p = 1 (squares), p = 0 (circles), and p = − we shall see, results depend somewhat on the value of σ .Figure 5–(a) displays the comparison between ε { } /ε { } from the TRENTo model, and v { } /v { } measured by the CMS Collaboration [54]. The centralitypercentile in our TRENTo configuration is defined fromthe multiplicity of produced particles, thus mimickingthe experimental situation. For σ = 0 . etal. [60], who find a v { } which matches data, and aslightly underpredicted v { } . Agreement with experi-mental data mildly improves if the participant nucleonswidths are lowered to σ = 0 . σ yieldmore spiky initial density profiles, and are known to in-crease the magnitude of ε and ε in small systems [14].In central p+Pb collisions, we find that the rms ε in-creases by 8% when σ is lowered from 0 . . ε increases by 12%). Larger values of ε n areknown to yield larger values of ε n { } /ε n { } [50]. Evenwhen σ = 0 . v { } /v { } . Note, how-ever, that the experimental measurements of v { } and v { } differ in the implementation, and the comparisonwith our results may not be consistent: v { } is mea-sured with a large pseudorapidity ( η ) gap to suppressnonflow effects, but no η gap is implemented in the mea- -
10 1 10 { } / v { } v (a) = 0.3 fm s = 0.4 fm s CMS centrality [%] -
10 1 10 { } / v { } v (b) FIG. 5. (color online) v { } /v { } (a) and v { } /v { } (b) asfunctions of centrality percentile in 5.02 TeV p+Pb collisions.Full circles: TRENTo parametrization with σ = 0 . σ = 0 . surement of v { } . Therefore, measurements of v { } may be affected by nonflow, short-range (near side) cor-relations. In addition, the η gap typically reduces v { } ,because of pseudorapidity dependent event-plane fluctu-ations [64]. Recently, a novel method to measure multiparticle cumulants in small systems was proposed [65].It implements pseudorapidity gaps for the measurementsof four-particle correlations. The results reported by theauthors of this method suggest that, in proton+protoncollisions, the measured four-particle correlations ( v { } and v { } ) may originate entirely from nonflow contri-butions. We expect agreement between our model andexperimental data to be improved if v { } and v { } aremeasured using the same sample of detected particles.We now make predictions for the ratio v { } /v { } ,which has not yet been measured in p+Pb collisions. v { } in p+Pb collisions has been computed in event-by-event hydrodynamics [60]. Nevertheless, the ratio { } / v { } v (a) {2} / v{4} v { } / v { } v (b) CMS = 0.3 fm s = 0.4 fm s FIG. 6. (color online) Eccentricity-driven predictions for v { } /v { } and v { } /v { } as function of v { } /v { } in 5.02 TeV p+Pb collisions. Full symbols: TRENToparametrization with σ = 0 . σ = 0 . v { } /v { } is a more robust quantity, in the sense thatdepends little on model parameters (such as viscosity, orfreeze-out temperature) and kinematic cuts ( p t ) . Ourresults, from the TRENTo configuration with p = 1, areshown in Fig. 5–(b). We find v { } /v { } to be slightlysmaller than v { } /v { } in Fig. 5–(a). The sensitivityto the value of σ is somewhat stronger for v than for v .The CMS Collaboration has also measured v { } /v { } and v { } /v { } [25] in p+Pb colli-sions. Our TRENTo results for these ratios are shownin Fig. 6. As in Fig. 4, we plot them as a function of The fact that the ratios of cumulants are not sensitive to thevalue of η/s is clearly inferable from the results of [60]. There,the authors show explicitly that both v { } and v { } increase(decrease) by the same amount when the value of η/s is raised(lowered). {2} / v{4} v { } / v { } v = 0.3 fm s = 0.4 fm s FIG. 7. (color online) Prediction for v { } /v { } as func-tion of v { } /v { } in central 5.02 TeV p+Pb collisions, fromdifferent TRENTo parametrizations. Circles: σ = 0 . σ = 0 . the lowest-order ratio, v { } /v { } . We observe thatour Monte Carlo results are in perfect agreement withthe prediction of the power distribution (dashed line inFig. 6). This confirms that the power distribution isa good description of eccentricity fluctuations in smallsystems, irrespective of the details of the simulatedconfigurations [52]. Existing CMS data exhibit as wellgood agreement with this theoretical prediction. Futuremeasurements with smaller error bars will provide acrucial test of the eccentricity-driven nature of v inproton+nucleus collisions.Eventually, we make a prediction for v { } /v { } asfunction of v { } /v { } in central p+Pb collisions. Re-sults are displayed in Fig. 7, for both σ = 0 . σ = 0 . σ = 0 . V. DISCUSSION AND OUTLOOK
We have shown that ratios of cumulants are a powerfultool to test models of initial conditions directly againstexperimental data. The Glauber Monte Carlo model,which is by far the most employed model in both ex-perimental and theoretical analyses, is excluded by ex-perimental data on elliptic flow fluctuations in centralPb+Pb collisions. On the other hand, saturation models(mimicked by the TRENTo parametrizations with p = 0or p = −
1) provide a good description of the experimen-tal results. However, even if these models predict the cor-rect fluctuations of v , they overpredict the fluctuationsof v in central Pb+Pb collisions. A possible explanationis that they overestimate both the fluctuations and themean eccentricity, ε , in the reaction plane. In this way,the error cancels in the ratio v { } /v { } , but not in thecorresponding ratio for v , which is solely due to fluctua-tions. It will be of crucial importance to reduce the errorbars on experimental data on v { } in central Pb+Pb col-lisions, in order to check whether the ratio v { } /v { } isindependent of centrality, as suggested by ALICE data.Indeed, this observation does not seem compatible withexisting models of initial conditions.The parametrizations of the initial state that are suit-able for describing central Pb+Pb collisions, can not beemployed in central p+Pb collisions, and vice versa. In-deed, the Glauber model, which is excluded by Pb+Pbdata, provides the only reasonable description of p+Pbcollisions. We do not consider this as a contradiction,because we are merely trying to identify the parametriza-tion which captures the initial geometry in a given sys-tem, and we do not aim at a unified description of all sys-tems. We predict that the ratio v { } /v { } is very closeto v { } /v { } in high-multiplicity p+Pb collisions, andboth the distributions of v and v to follow the powerdistribution. These results imply that, up to small cor-rections, the same non-Gaussianities drive the fluctua-tions of ε and ε . Our explicit test of the power behav-ior up to higher-order cumulants, in particular, suggeststhat the main non-Gaussianity driving the fluctuationsis the fact that the distributions are bounded by unity.However, nonflow effects differ for v and v (back-to-back correlations typically increase v , and decrease v )and must be carefully removed in the analysis.As a final remark, we stress that the conclusions drawnin our p+Pb analysis should hold in any small systemmodel where ε and ε originate solely from fluctuations.It would be rather natural, then, to extend this analysisto the case of high-multiplicity proton+proton collisions,where the observed azimuthal multi particle correlationshint at the onset of collective effects [26, 66]. Thesenew data have triggered novel models of initial condi-tions [13, 67], which can be tested against experimentaldata using ratios of cumulants, as done in this work forp+Pb collisions. ACKNOWLEDGEMENTS
J.N.H. acknowledges the use of the Maxwell Clusterand the advanced support from the Center of AdvancedComputing and Data Systems at the University of Hous-ton and was supported by the National Science Foun-dation under Grant No. PHY-1513864. We thank MattLuzum for useful discussions. G.G. wishes to thank ScottMoreland for kind assistance with the use of TRENTo.
Appendix A: The TRENTo model
TRENTo is a flexible parametric Monte Carlo modelfor the initial conditions of heavy-ion collisions, which en-compasses several other models of initial conditions [42].Consider the case of a nucleon A colliding with a nu-cleon B. Each participant nucleon deposits entropy in thetransverse plane according to a Gaussian distribution ofwidth σ , which reads S A,B = w A,B πσ exp (cid:20) ( x − x A,B ) + ( y − y A,B ) σ (cid:21) . (A1)The normalization, w , is a random number which is as-signed to each participant nucleon. Its probability dis-tribution is a Γ distribution, whose mean value is equalto unity, and whose width is regulated by a parameter, k . The total initial entropy profile is computed througha generalized average of Gaussian sources, S ( p ; S A , S B ) = (cid:18) S p A + S p B (cid:19) p , (A2)where p is an arbitrary real parameter. The previousformula can be generalized to the case of a nucleus Acolliding with a nucleus B [42]. Note that, for p = 1,nuclear density profiles are superimposed ( S ∝ S A + S B ).If p = 0 or p = −
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