Azimuthal Asymmetries From Jets Quenched In Fluctuating Backgrounds
AAzimuthal Asymmetries From Jets Quenched InFluctuating Backgrounds
R Rodriguez , R J Fries Department of Mathematics and Physics, Ave Maria University, Ave Maria FL 34142, USA Cyclotron Institute and Department of Physics and Astronomy, Texas A&M University,College Station, TX 77843, USAE-mail: [email protected], [email protected]
Abstract.
High momentum jets and hadrons are important probes for the quark gluon plasma(QGP) formed in nuclear collisions at high energies. We investigate how fluctuations in thebackground density of the QGP and fluctuations in the spatial distribution of the hard processcreate azimuthal asymmetries of the high momentum hadron spectrum, described by the Fouriercoefficients v n , n > v in a simple energy loss model tunedto single inclusive hadron suppression. With the study of quark gluon plasma in collisions of relativistic heavy ions moving intoan increasingly accurate quantitative phase it has been found important to include fluctuationsin the space-time structure of the fireball into calculations of bulk quantities [1, 2]. Thesecan emerge from fluctuations in the initial energy density (cid:15) ( x, y ) in the plane transverse tothe beam axis and leave signature effects on bulk quantities like the azimuthal asymmetrycoefficients v n . For example fluctuations can lead to a sizeable triangular flow v which wouldbe vanishing in an averaged fireball due to the overall geometry of the nuclear overlap [2]. Atlarge transverse momentum P T fluctuations in the position of the hard process can also affectobservables accessible in current heavy ion experiments.Here we have explored the role of such fluctuations on the suppression of high- P T hadronsand their generalized azimuthal asymmetry coefficients v n . The v n are defined as a Fourierdecomposition of the azimuthal angle dependent spectrum dNP T dP T d Φ = dN πP T dP T (cid:34) (cid:88) n> v n ( P T ) cos( n Φ + δ n ) (cid:35) (1)where the angle Φ is measured with respect to the reaction plane defined below and the δ n are phases that encode a misalignment with the reaction plane. We note that for smooth,non-fluctuating fireballs we expect all odd coefficients v , v etc. at midrapidity to vanish forsymmetry reasons.However, in any given single event the initial energy density will typically exhibit a non-vanishing triangular eccentricity (cid:15) which could in turn lead to a non-vanishing v . The event-by-event fluctuations in initial energy density are driven by fluctuations of the positions of nucleonsin the initial nuclei and in the amount of energy deposited around midrapidity for every nucleon-nucleon collision [3]. We also expect that the triangular eccentricity is not correlated with the a r X i v : . [ nu c l - t h ] D ec (fm) -6 -4 -2 0 2 4 6 y (f m ) -6-4-20246 024681012141618202224 x(fm) -6 -4 -2 0 2 4 6 y (f m ) -6-4-20246 024681012141618202224 x(fm) -6 -4 -2 0 2 4 6 y (f m ) -6-4-20246 024681012141618202224 x(fm) -6 -4 -2 0 2 4 6 y (f m ) -6-4-20246 0510152025 Figure 1.
Engineered events with n = 3 , , , δ should appear random. Similar arguments can be made for other n > v n to acquire non-vanishing values with realistic fluctuations. While allof this has first been discussed for the bulk of the fireball [3] these statements easily transfer tohard probes. Fluctuations in the energy density lead to fluctuations in energy loss. In addition,the position of a hard process which creates a hard probe is subject to fluctuations.Systematic measurements of v n at large momentum could lead to further constraints on thetype of energy loss prevalent in QGP, and on the size of the transport coefficient ˆ q . It can alsogive an independent handle on the size and granularity of initial state fluctuations. Here wereport on a quantitative study of high momentum azimuthal coefficients using a simple energyloss model with realistic fluctuations.We calculate high momentum hadron spectra using our simulation package PPM [4, 5]. Itsamples initial momentum distributions of quark and gluon jets from a perturbative calculationand propagates leading partons through a given background fireball. Different energy loss modelscan be employed. Here we will show results from a simple LPM-inspired (sLPM) deterministicenergy loss model dE/dx ∼ ˆ qx where ˆ q scales with the 3/4th power of the local energy density [4].As a cross check we will sometimes also use the non-deterministic Armesto-Salgado-Wiedemann(ASW) model [6, 7]. Fitted to the same experimental data on single hadron suppression thesemodels cover a wide range of values for ˆ q . PPM will eventually fragment leading partons intohadrons and all results here will be shown for pions.We have used the Glauber Monte Carlo generator GLISSANDO [8] to produce an ensembleof Au+Au events at top RHIC energy using the three different centralities b = 3 . , . (cid:15) n [9] (cid:15) n = (cid:113) (cid:104) r n cos( nφ ) (cid:105) + (cid:104) r n sin( nφ ) (cid:105) (cid:104) r n (cid:105) (2)and the azimuthal asymmetry coefficients v m using “engineered” events with particular fixedeccentricities. These are created with the energy density modeled as simple Gaussians in thetransverse plane with a cos nφ undulation of the mean square radius, as shown in the examplesin Fig.1.First we scanned the space ( v n , (cid:15) m ), n, m = 1 , v n for a given (cid:15) m only if n = im where i > v n with the size of the eccentricity (cid:15) n . We expected a monotonically increasing function v n ( (cid:15) n )which can be seen confirmed in Fig. 2. Deviations from monotony only occur for unrealisticallylarge eccentricities. These basic results should hold if realistic fluctuations are considered. Fig. 3shows the correlation between v and (cid:15) for our ensemble of GLISSANDO events for all 3 impactparameters for Au+Au collisions at top RHIC energies. The basic linear correlation persists for n = 2 but is washed out. Correlations for n >
2, not shown here, are much more weakened to a ˛ v n=3 ˛ v n=4 ˛ v n=5 ˛ v n=6 Figure 2.
Azimuthal asymmetry v n vs eccentricity (cid:15) n in engineered events for n = 3, 4, 5, 6. epsilon20 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 v2 b=3.2 fm epsilon20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 v2 b=74 fm epsilon20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 v2 b=11 fm Figure 3.
Correlation between v and (cid:15) for an ensemble of Au+Au collisions at top RHICenergy in GLISSANDO for three impact parameters 3.2, 7.4 and 11 fm.point that makes it hard to predict v n for a known (cid:15) n . All results shown are for sLPM energyloss but the corresponding result for ASW show no noticeable difference, making the conclusionsrather robust against large variations in the microscopic origin of energy loss.We determined the phases δ n for high momentum pions in our ensemble of GLISSANDOevents for three different impact parameters. In Fig. 4 we plot those phases relative to thereaction plane determined by the eccentricity (cid:15) . In other words our definition of a reactionplane is given by the fundamental initial ellipticity, which in general deviates from the planedefined by beam axis and impact vector. We observe that δ and δ are randomly distributed,so there is no correlation between the reaction plane and the fluctuations which create (cid:15) or (cid:15) .This had also been found for the bulk azimuthal asymmetries before [9]. δ and δ on the otherhand show a correlation with the reaction plane which is consistent with v and v receivingcontributions from (cid:15) .The transverse momentum dependence of the coefficients v n for pions for our ensemble ofGLISSANDO Au+Au events is shown in Fig. 5. We observe two hierarchies of coefficients, onebeing v > v > v and the second one being v > v > v . v is always the largest coefficient,even in the most central events. Interestingly v is non-zero and the second largest coefficient,beating v and v by more than a factor 2. Momentum conservation dictates a sum rule for v integrated over P T . The recoil of the medium in which energy is lost would lead to a negative v at lower momentum. Such a back reaction is not included in this calculation. v at intermediateand large P T could be sensitive to the mechanism of medium recoil. Generally we point outthat our results start to become unreliable below 4 to 6 GeV since hydrodynamic expansionwas not included in the calculation. Fig. 5 shows the results for both sLPM and ASW energyloss. We have to conclude that azimuthal asymmetry coefficients are not particularly useful to d -1 -0.5 0 0.5 1 C oun t s d -0.6 -0.4 -0.2 0 0.2 0.4 0.6 C oun t s d -0.6 -0.4 -0.2 0 0.2 0.4 0.6 C oun t s d -0.4 -0.2 0 0.2 0.4 C oun t s Figure 4.
Distribution of the phases δ n for n = 3, 4, 5, 6 in our ensemble of Au+Au events. (GeV) T p n v v v v v v v (a) sLPM, b = 3 . (GeV) T p n v v v v v v v (b) ASW, b = 3 . (GeV) T p n v v v v v v v (c) sLPM, b = 7 . (GeV) T p n v v v v v v v (d) ASW, b = 7 . Figure 5.
Pion v n vs P T for two impact parameters for Au+Au collisions at top RHIC energy,calculated with either sLPM or ASW energy loss.discriminate between energy loss models. We also find that the P T dependence of the coefficientsbecomes rather weak at large momenta.To summarize, in this study we have explored higher order azimuthal asymmetry coefficientsat large momentum in heavy ion collisions. We find that in general v n rises with (cid:15) n but thiscorrelation weakens for larger n . We also find that there are only a few cross correlationsbetween v n and (cid:15) m for n (cid:54) = m . We have also classified the preferred angular orientation of theazimuthal asymmetries as given by the phases δ n with respect to the reaction plane. We find adecorrelation with the reaction plane for all odd n . Finally we have made predictions for v n asa function of P T in two different energy loss models. We find mostly consistent results betweenthose two models with v being the largest coefficient followed by v . In general the v n carrygeometrical information and measurements at large momentum can be complementary to thosefor bulk observables, but they seem less useful to distinguish different energy loss models.This work was supported by NSF CAREER Award PHY-0847538 and by the JETCollaboration and DOE grant DE-FG02-10ER41682. References [1] B. Alver et al.
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