Predictions of elliptic flow and nuclear modification factor from 200 GeV U + U collisions at RHIC

Predictions of elliptic flow ($v_2$) and nuclear modification factor ($R_{AA}$) are provided as a function of centrality in U + U collisions at $\sqrt{s_{_{NN}}}$ = 200 GeV. Since the $^{238}$U nucleus is naturally deformed, one could adjust the properties of the fireball, density and duration of the hot and dense system, for example, in high energy nuclear collisions by carefully selecting the colliding geometry. Within our Monte Carlo Glauber based approach, the $v_2$ with respect to the reaction plane $v_2^{RP}$ in U + U collisions is consistent with that in Au + Au collisions, while the $v_2$ with respect to the participant plane $v_2^{PP}$ increases $\sim$30-60% at top 10% centrality which is attributed to the larger participant eccentricity at most central U + U collisions. The suppression of $R_{AA}$ increases and reaches $\sim$0.1 at most central U + U collisions that is by a factor of 2 more suppression compared to the central Au + Au collisions due to large size and deformation of Uranium nucleus.


Introduction
Most striking findings at RHIC are the large elliptic flow v 2 [1] and the strong suppression of nuclear modification factor R AA [2]. The v 2 is defined by the second harmonic Fourier coefficient of azimuthal particle distribution with respect to the reaction plane, and the R AA is defined by the ratio of invariant yield in A + A collisions to that in p + p collisions scaled by number of collisions. Recent systematic measurements of v 2 [3] as well as developments of viscous hydrodynamical models [4,5,6,7] provide a conservative upper limit of the viscosity η to the entropy s ratio η/s ≤ 0.5. This corresponds to the 6 times larger value of an absolute lower bound η/s = 1/4π predicted by strongly coupled gauge filed theories based on the AdS/CFT correspondence [8,9]. It has been observed that the ratio v 2 /ε in different systems from AGS to RHIC scale like 1/SdN ch /dy [10] as it was predicted by a low density limit of v 2 [11,12], where ε is the initial geometrical anisotropy (eccentricity), S is the transverse area and dN ch /dy is the charged particle rapidity density. The saturation of v 2 /ε would indicate that the system is approaching the hydrodynamical limit and the collectivity no longer increases when the system size becomes larger. The measurements of transverse momentum spectra of charged hadrons showed that the yield at most central Au + Au collisions at √ s NN = 200 GeV is suppressed by a factor of ∼5 compared to the p + p reference scaled by number of binary collisions N coll [13,14] and the similar level of suppression persists for neutral pions up to p T = 20 GeV/c [15]. The integrated R AA above p T > 5 GeV/c and > 10 GeV/c decreased monotonically as a function N part and there were no sign of saturation [15]. Assuming the underlying dynamics remains the same, we ask what would happen to v 2 and R AA for a larger colliding system 238 U + 238 U collisions ? Comparing to the 197 Au nucleus, the 238 U has a much larger mass and, more importantly, it is largely deformed. The planned U + U collisions at RHIC will be important for us to understand how those observables behave at higher particle density. Monte Carlo Glauber simulations showed that the transverse number density 1/SdN ch /dy increases ∼ 35% at most central events in ideal tip-tip collisions (head-on collisions along the longest axes) [16]. The U + U collisions will become possible when the Beam Ion Source becoming operational in 2012 [17].
In this letter, we will report a geometrical approach based on the Monte Carlo Glauber model to predict the elliptic flow v 2 as well as the nuclear modification factor R AA in U + U collisions at top RHIC energy. In the Section 2, we will discuss our parameterization of Glauber model and define geometrical quantities which are used in this study. In the Section 3, the results of v 2 and R AA in U + U collisions will be presented and compared to the data in Au + Au collisions at √ s NN = 200 GeV.

Glauber Model
The nucleon density distribution is parameterized by a deformed Woods-Saxon profile [18] where ρ 0 is the normal nuclear density, R and a denote the radius of nucleus and the surface diffuseness parameter, respectively. We have used R = 6.38 fm and a = 0.535 fm for 197 Au nucleus, and R = 6.81 fm and a = 0.55 fm for 238 U nucleus. The Y m l (θ) denotes the spherical harmonics and θ is the polar angle with the symmetry axis of the nucleus. Deformation parameters are β 2 = 0.28 [19] and β 4 = 0.093 [20] for Uranium. The presence of β 4 modifies the shape of Uranium compared to that only with β 2 , which was implemented in several different models [16,19]. The radius increases ∼6% (3%) at θ = 0 (θ = π/2), while it decreases ∼3% around θ = π/4. We have assumed that Au nucleus is spherical (β 2 = β 4 = 0), thus Eq. (1) reduces the spherical Woods-Saxon profile. Recent calculation [21] shows that the ground-state deformation of 197 Au affects the eccentricity of initial geometry overlap only at most central collisions from both optical and Monte Carlo Glauber simulations. The positions of nucleons are sampled by 4πr 2 sin (θ)ρ(r) dθdφ, where the absolute normalization of ρ(r) is irrelevant.
Both projectile and target U nuclei are randomly rotated along the polar and azimuthal directions event-by-event with the probability distribution sin Θ and uniform distribution for Θ and Φ, respectively. The sin Θ weight needs to be implemented to simulate unpolarized nucleus-nucleus collisions. The results are averaged over all possible orientations unless otherwise specified.
A binary nucleon-nucleon collision take places if where d is the distance between nucleons in the transverse direction orthogonal to the beam axis, and σ N N = 42 mb is the inelastic nucleon-nucleon cross section at √ s = 200 GeV. For each event, the total number of binary collisions N coll is calculated by the sum of individual number of collision and the total number of participant nucleons N part is the number of nucleons that interacts at least once. Charged particle pseudorapidity density is obtained by a two component model [22] where n pp = 2.29 and x = 0.145 are fixed to reproduce the PHOBOS results [23]. Event-by-event multiplicity fluctuations have been taken into account by convoluting Negative Binomial Distribution for a given N part and N coll where µ = n pp is the mean of the distribution and 1/k = 0.5 corresponds to deviation from a Poisson distribution. In this study, we have generated 1 million events for U + U collisions by randomly selecting an impact parameter b according to the dσ/db = 2πb. Figure 1 shows the comparison of dN ch /dη distributions in Au + Au and U + U collisions at √ s NN = 200 GeV from our Monte Carlo Glauber model. The maximum dN ch /dη in U + U collisions increases ∼15% compared to that in Au + Au collisions. We have defined the event centrality bins by the fraction of events in dN ch /dη. The centrality bins are summarized in Table 1.
and L for each centrality bin in U + U collisions at √ s N N = 200 GeV. Errors denote systematic uncertainties, see texts for more details of systematic error evaluations.  Since the positions of nucleons fluctuate event-by-event, the principal axes of the participant nucleons in the transverse plane are tilted and rotated with respect to the original coordinate system. We define the participant plane (PP) which is the relevant plane to take into account the event-by-event position fluctuations of participant nucleons. The transverse area and eccentricity with respect to the reaction plane (RP) and participant plane are defined as where σ 2 The curly brackets {...} denote the average over all participants for a given event. We have also calculated the averaged transverse path length L from the RMS width which could be a relevant geometrical quantity for the R AA . The path length is very close to ρL defined in [24], while Eq. (10) takes into account the event-by-event center of mass shift of the nuclei within the transverse plane. Average quantities, N part , N coll , S , ε and L have been calculated for each centrality bin where ... describe the average over all events. Systematic uncertainties on the average quantities have been estimated (i) by varying input parameters R, a, n pp , x as well as the total cross section within ±5% and (ii) by using different density profiles for nucleons in the Monte Carlo Glauber simulations. The dominant source of systematic uncertainty is the total cross section. Total systematic uncertainty has been evaluated by the quadratic sum of individual systematic uncertainty. Table 1 summarizes the centrality bins, average quantities and their systematic uncertainties obtained in the Monte Carlo Glauber simulation. Figure 2 compares the N coll , transverse number density 1/ S RP dN ch /dη, reaction plane eccentricity ε RP , second order cumulant of participant eccentricity ε PP {2} ≡ ε 2 PP , and path length L as a function of N part together with their systematic uncertainties in Au + Au and U + U collisions at √ s NN = 200 GeV. We found that the all geometrical quantities essentially scale with N part . The N coll , S , L increase and ε RP decreases at most central U + U collisions compared to those in Au + Au collisions because of the larger size of Uranium. One can see that the ε PP {2} in U + U collisions starts deviating the N part scaling around N part = 200, and increases ∼60% at top 5% central U + U collisions. The higher values of ε PP {2} in U + U collisions for large N part is purely from the ground-state deformation of Uranium. We have confirmed that the ε PP {2} becomes the same if we assume the Uranium is spherical. The relevance of the ε PP {2} will be discussed in the next section.

Elliptic flow v 2
It has been found that the elliptic flow v 2 divided by initial anisotropy ε in coordinate space scaled like 1/SdN ch /dy among different energies and collision systems from AGS to RHIC [10]. A simple formula that has been proposed in [25] describes very well the variation of v 2 with 1/SdN/dy where dN/dy is the rapidity density of total particles, h is the v 2 /ε in the ideal hydrodynamical limit when 1/S dN/dy → ∞, and B contains informations about the equation of state and the partonic cross section [25]. The Eq. (11) reduces v 2 /ε ∼ (h/B)1/S dN/dy when 1/S dN/dy → 0 for leading order in 1/S dN/dy, thus the above equation satisfies both low density and ideal hydrodynamical limit of v 2 . The integrated v 2 for unidentified charged hadrons from the PHOBOS collaboration can be well described by the Eq. (11) [26]. Assuming no change in the collision dynamics, we will study the v 2 /ǫ distributions versus the collision centrality in U + U collisions.  [27,28] and that from standard event plane method [29], respectively. The dN/dy is obtained by multiplying 3/2 to the measured dN ch /dy at STAR [30] to take into account the neutral particles. The simultaneous fit has been performed for (a) four particle cumulant v 2 {4}, six particle cumulant and q-distribution method and for (b) standard event plane v 2 {EP}, scalar product method, η subevent, random subevent, and two particle cumulant. The results of v 2 are taken from [31]. The two different groups of v 2 are categorized based on the multi-particle methods for (a) and two particle methods for (b). As long as the distribution of eccentricity is 2D gaussian in the transverse plane, the effect of fluctuation on the v 2 {4} is negligible and thus the ε RP can be used to scale the v 2 {4} [32]. This assumption holds except for the peripheral 60-80% centrality, where the distribution of eccentricity becomes non-Gaussian. The v 2 from two particle methods are expressed as (v α 2 ) 1/α where v 2 is the true v 2 value and α varies from 1 to 2 depending on the event plane resolution [33,34]. In this study, ε part {2} = ε 2 part was used by assuming α = 2. We confirmed that the resulting v 2 values unchanged by using ε part (i.e. α = 1). Because the v 2 values were extrapolated from (v 2 /ε) multiplied by ε, most of the difference between ε part {2} and ε part is canceled out and thus the resulting v 2 is the same regardless of the choice of eccentricity. Figure 4 shows the extracted v 2 in U + U collisions compared to those in Au + Au collisions as a function of centrality at √ s NN = 200 GeV. The v RP 2 (v PP 2 ) denotes the v 2 measured with respect to the reaction (participant) plane. The v RP 2 and v PP 2 have been calculated by multiplying the ε RP and ε PP {2} to the fitting results of v 2 /ε shown in Fig. 3 for each centrality bin. Since we have calculated the dN ch /dη in the Monte Carlo Glauber simulation, it is necessary to convert the dN ch /dη to dN/dy for calculating the v 2 for each centrality bin. We assume that dN ch /dy ≈ 1.15dN ch /dη to extrapolate the v 2 /ε for each centrality [35]. The v RP 2 in U + U collisions is consistent with that in Au + Au collisions for centrality 0-80%. The v PP 2 is also consistent with each other in U + U and Au + Au collisions for centrality 20-80%, whereas the v 2 in U + U collisions at top 0-10% centrality is 30-60% larger than that in Au + Au   collisions. The larger v 2 is attributed to the larger participant eccentricity due to the ground-state deformation in top 0-10% centrality in U + U collisions compared to that in Au + Au collisions. The extracted v 2 in Au + Au collisions are slightly smaller than the data at peripheral collisions. Since the dN ch /dη has been tuned to reproduce the PHOBOS results and is smaller than the STAR dN ch /dy at peripheral 60-80%, the resulting v 2 /ε (and hence the v 2 ) become smaller than the STAR v 2 .

Nuclear modification factor R AA
The integrated R AA over a certain p T range in Au + Au collisions at √ s NN = 200 GeV has been described by R AA = (1 − S 0 N part a ) n−2 , where n = 8.1 is the power-law exponent of p T distribution, and S 0 = (9.0 ±6.1) ×10 −3 and a = 0.57 ±0.14 for N part > 20 and p T > 5 GeV/c [15]. We have assumed that the path length L determines the R AA in both Au + Au and U + U collisions. The R AA in U + U collisions has been extrapolated by fitting the R AA (L) in Au + Au collisions with an ansatz from above equation where n = 8.1, S ′ 0 and b are free parameters that have been evaluated by fitting the data. Figure 5 shows integrated R AA for p T > 5 GeV/c from [15] as a function of L in Au + Au collisions at √ s NN = 200 GeV. We have assumed that the definition of our centrality bins is the same as that of the PHENIX in order to plot the R AA as a function of L for each centrality. Result of the fit with Eq. (12) is shown by the solid line in Fig. 5 and holds quite well over entire range of L since the L scales like N part 1/3 . Figure 6 shows integrated R AA as a function of N part extrapolated for U + U collisions at √ s NN = 200 GeV. The R AA in U + U collisions has been evaluated for a given L in each centrality. The calculated R AA in Au + Au collisions (open boxes) is consistent with the data within the systematic error as it should. We found that the R AA reaches ∼0.1 at most central U + U collisions, which is by a factor of 2 more suppression compared to the central Au + Au collisions due to the larger size and the deformation of Uranium. Heinz and Kuhlman pointed out in [19] that the radiative energy loss ∆E of a fast parton moving through the medium is almost independent of the orientations of nuclei for the out-of-plane direction in the full overlap U + U collisions. Whereas the ∆E in the in-plane direction decreases by about 35% towards the ideal body-body collisions (head-on collisions along the shortest axes). For the body-body collisions, they found that the difference of ∆E between out-of-plane and in-plane directions is more than twice in U + U collisions that achieved in Au + Au collisions. They also found that the total energy loss is larger by up to a factor of 2. More differential study, such as selecting the orientations of Uranium and directions with respect to the reaction plane, will be needed to see whether the R AA would have such dependences or not. Since the L in U + U collisions is slightly larger (∼3% in central, and ∼5% in peripheral collisions) than that in Au + Au collisions, the R AA in U + U collisions would be even more suppressed for a given N part . Due to the large errors on the extrapolated R AA , we have not observed any difference of the R AA between Au + Au and U + U collisions for a given N part .

Summary
In summary, we have predicted the v 2 and R AA in U + U collisions at  with that in Au + Au collisions over all centrality range, whereas the v PP 2 increase by 30-60% at most central 0-10% collisions due to the larger ε PP {2} in U + U collisions. The R AA at top 5% central U + U collisions further suppressed and reaches ∼0.1, which is by a factor of 2 more suppression compared to the most central Au + Au collisions. It is clear that the larger mass and deformation form the U nucleus will allow us to study the matter at higher density. By selecting the relative orientation of the colliding Uranium nuclei, the discussed effects may be further enhanced. We will report the method in a separate paper.