Rajeev S. Bhalerao

Elliptic flow and incomplete equilibration at RHIC

Abstract We argue that RHIC data, in particular those on the anisotropic flow coefficients v 2 and v 4 , suggest that the matter produced in the early stages of nucleus–nucleus collisions is incompletely thermalized. We interpret the parameter ( 1 / S ) ( d N / d y ) , where S is the transverse area of the collision zone and d N / d y the multiplicity density, as an indicator of the number of collisions per particle at the time when elliptic flow is established, and hence as a measure of the degree of equilibration. This number serves as a control parameter which can be varied experimentally by changing the system size, the centrality or the beam energy. We provide predictions for Cu–Cu collisions at RHIC as well as for Pb–Pb collisions at the LHC.

Determining initial-state fluctuations from flow measurements in heavy-ion collisions

We present a number of independent flow observables that can be measured using multiparticle azimuthal correlations in heavy-ion collisions. Some of these observables are already well known, such as v{sub 2}(2) and v{sub 2}(4), but most are new--in particular, joint correlations among v{sub 1}, v{sub 2}, and v{sub 3}. Taken together, these measurements will allow for a more precise determination of the medium properties than is currently possible. In particular, by taking ratios of these observables, we construct quantities that are less sensitive to the hydrodynamic response of the medium and thus more directly characterize the initial-state fluctuations of the event shape, which may constrain models for early-time, nonequilibrium QCD dynamics. We present predictions for these ratios using two Monte Carlo models and compare them to available data.

Eccentricity fluctuations and elliptic flow at RHIC

Abstract Fluctuations in nucleon positions can affect the spatial eccentricity of the overlap zone in nucleus–nucleus collisions. We show that elliptic flow should be scaled by different eccentricities depending on which method is used for the flow analysis. These eccentricities are estimated semi-analytically. When v 2 is analyzed from 4-particle cumulants, or using the event plane from directed flow in a zero-degree calorimeter, the result is shown to be insensitive to eccentricity fluctuations.

Understanding anisotropy generated by fluctuations in heavy-ion collisions

Event-by-event fluctuations are central to the current understanding of ultrarelativistic heavy-ion collisions. In particular, fluctuations in the geometry of the early-time collision system are responsible for new phenomena such as triangular flow, which have solved important puzzles in existing data. We propose a simple model where initial fluctuations stem from independent flux tubes randomly distributed in the transverse plane. We calculate analytically the moments of the initial anisotropies (dipole asymmetry, eccentricity, triangularity), which are the sources of anisotropic flow, and their mutual correlations. Our analytic results are in good agreement with calculations from commonly-used Monte-Carlo codes, providing a simple understanding of the fluctuations contained in these models. Any deviation from these results in future experimental data would thus indicate the presence of non-trivial correlations between the initial flux tubes and/or extra sources of fluctuations that are not present in current models.

Event-plane correlators

Introduction.−High-energy heavy-ion collision experiments at the Relativistic Heavy-Ion Collider (RHIC), BNL and the Large Hadron Collider (LHC), CERN have firmly established the formation of strongly interacting matter which exhibits a strong collective flow [1–3]. This not only suggests that the matter formed is close to local thermal equilibrium but also provides a window to the initial state of the fireball immediately after the collision. Collective flow in the plane transverse to the beam axis is typically measured in terms of two-particle angular correlations [4–7] and its small anisotropies captured in its Fourier harmonics [8–11]. Recently, a new tool, namely correlations among event planes corresponding to different harmonics [12–14], is emerging with a promise to throw additional light on the initial-state phenomena. Pair correlations (i.e., the single-particle anisotropic flow vn extracted from two-particle correlations) are reasonably well understood [8]. Event-plane correlators represent higher-order correlations, involving at least three particles. Such higher-order correlations thus open a new direction in heavy-ion physics, much in the same way as studies of non-Gaussian fluctuations [15] in the early Universe. They bring in a large number of new observables which provide new, detailed insight into the hydrodynamic response and on the initial stage, and will significantly improve our understanding of heavy-ion collisions. The ATLAS collaboration [14] has recently released preliminary measurements of eight two-plane correlators (e.g., the correlation between the second and fourth harmonics) and six three-plane correlators (e.g., the mixed correlation between second, third and fifth harmonics) in Pb-Pb collisions at √ sNN = 2.76 TeV, as a function of the centrality of the collision. These correlators are qualitatively understood within event-by-event hydrodynamics [16] and provide new insight [17] into the interplay between the linear and nonlinear [18] hydrodynamic response [19, 20] to the initial density profile. The ATLAS analysis is very demanding in terms of detector acceptance: each event plane is determined in a separate pseudorapidity window; windows must be pairwise separated by gaps in order to suppress nonflow correlations; finally, each window must be wide enough to achieve a significant resolution in every Fourier harmonic up to order six. In this article, we show that the analysis can be done more simply, and that even three- and four-plane correlators can be safely analyzed with just two symmetric pseudorapidity windows, in the same way as two-plane correlators. The analysis can thus be performed by other experiments with smaller detector acceptance. We also explain in detail how to generalize the scalar-product method used earlier for two-particle flow analysis [21], to mixed correlations, so as to eliminate the ambiguity brought about by the event-plane method [22]. We carry out realistic simulations within a multiphase transport (AMPT) model [23]. Results are for the first time in quantitative agreement with the ATLAS measurements. In addition, we present predictions for several new correlators, in particular four-plane correlators. Method.−The azimuthal distribution of outgoing particles is decomposed in Fourier harmonics in each collision event. The flow vector in harmonic n is defined as [24],

Statistical model for the nucleon structure functions

Abstract A phenomenological model for the nucleon structure functions is presented. Visualising the nucleon as a cavity filled with parton gas in equilibrium and parametrizing the effects due to the finiteness of the nucleon volume, we obtain a good fit to the data on the structure function F 2 p . The model then successfully predicts other unpolarized structure function data.

Equilibration of the quark-gluon plasma produced in relativistic heavy ion collisions

Abstract Formation and equilibrium of the quark-antiquark plasma are studied in the framework of the relativistic covariant kinetic theory. The Boltzmann equation is solved with the collision term, the source term, and the chromoelectric field term all present simultaneously. We find that the collision term strongly damps the oscillations of the background color field and of the particle energy density, and causes a large enhancement of the number density. The field term, on the other hand, makes the field decay faster, and thus increases the rate of evolution of the plasma. We also present our results for the particle distribution functions.

Relativistic viscous hydrodynamics for heavy-ion collisions: A comparison between the Chapman-Enskog and Grad methods

Derivations of relativistic second-order dissipative hydrodynamic equations have relied almost exclusively on the use of Grads 14-moment approximation to write

Characterizing flow fluctuations with moments

f(x,p)

Anisotropic flow from Lee-Yang zeros: a practical guide

, the nonequilibrium distribution function in the phase space. Here we consider an alternative Chapman-Enskog-like method, which, unlike Grads, involves a small expansion parameter. We derive an expression for