Effects of fluid velocity gradients on heavy quark energy loss

Mindaugas Lekaveckas 0 Krishna Rajagopal 0 0 Center for Theoretical Physics, Massachusetts Institute of Technology , Cambridge, MA 02139 U.S.A We use holographic duality to analyze the drag force on, and consequent energy loss of, a heavy quark moving through a strongly coupled conformal fluid with non-vanishing gradients in its velocity and temperature. We derive the general expression for the drag force to first order in the fluid gradients. Using this general expression, we show that a quark that is instantaneously at rest, relative to the fluid, in a fluid whose velocity is changing with time feels a nonzero force. And, we show that for a quark that is moving ultra-relativistically, the first order gradient corrections become larger than the zeroth order drag force, suggesting that the gradient expansion may be unreliable in this regime. We illustrate the importance of the fluid gradients for heavy quark energy loss by considering a fluid with one-dimensional boost invariant Bjorken expansion as well as the strongly coupled plasma created by colliding sheets of energy. 1 Introduction and summary 2 3 4 Hydrodynamic fluid and a heavy quark moving through it 2.1 Gravitational description of a moving fluid 2.2 Gravitational description of a moving heavy quark Computing the drag force on the heavy quark 3.1 Drag force in the instantaneous fluid rest frame 3.2 Generalizing to a frame in which the fluid is moving 3.3 General fluid motion Applications 4.1 A quark at rest in a fluid that is, instantaneously, at rest 4.2 Bjorken flow 4.3 Colliding sheets of energy 5 Future directions 1 Introduction and summary The analysis of how a heavy quark moving through the strongly coupled liquid quark-gluon plasma produced in ultrarelativistic heavy ion collisions loses energy and, subsequently, diffuses in the flowing plasma is of considerable theoretical interest because experimentalists are developing the detectors and techniques needed to use heavy quarks as tracers or probes of the strongly coupled liquid. If one assumes that the interactions between the heavy quark and the quark-gluon plasma are weak then perturbative techniques originally formulated for energetic light quarks [14] can be employed to analyze heavy quark energy loss [5]. The discovery that the plasma produced in heavy ion collisions is a strongly coupled liquid has prompted much interest in the real-time dynamics of strongly coupled non-Abelian plasmas and in the dynamics of heavy quarks therein. Although it remains to be seen to what degree treating all aspects of the dynamics of heavy quarks as strongly coupled is a good approximation, this approach is certainly of value as a benchmark: thorough understanding of the physics in this tractable setting can provide valuable qualitative insights. What makes these calculations tractable is holographic duality, which maps questions of interest onto calculations done via a dual gravitational description of the strongly coupled plasma and the heavy quark probe. The simplest theory in which these holographic calculations can be done is strongly coupled N = 4 supersymmetric Yang-Mills (SYM) theory in the large number of colors (large Nc) limit, whose plasma with temperature T is dual to classical gravity in a 4+1-dimensional spacetime that contains a 3 + 1-dimensional horizon with Hawking temperature T and that is asymptotically antideSitter (AdS) spacetime, with the heavy quark represented by a string moving through this spacetime [611]. The earliest work on heavy quark dynamics in the equilibrium plasma of strongly coupled N = 4 SYM theory [911] yielded determinations of the drag force felt by a heavy quark moving through the static plasma and the diffusion constant that governs the subsequent diffusion of the heavy quark once its initial motion relative to the static fluid has been lost due to drag. The basic picture of heavy quark dynamics that emerges, with all but the initially most energetic heavy quarks being rapidly slowed by drag and then becoming tracers diffusing within the (moving) fluid, is qualitatively consistent with early experimental investigations [12]. For a review, see ref. [13]. Subsequently, the holographic calculational techniques were generalized to any static plasmas whose gravitational dual has a 4+1-dimensional metric that depends only on the holographic (i.e. radial) coordinate in ref. [14] and heavy quark energy loss and diffusion has by now been investigated in the equilibrium plasmas of many gauge theories with gravitational duals [1528]. More recently, in ref. [29] we and a coauthor have calculated how the drag force and energy loss rate of a heavy quark moving through the far-from-equilibrium matter present just after a collision compares to that in strongly coupled plasma close to equilibrium. We studied the energy loss of a heavy quark moving through the debris produced by the collision of planar sheets of energy in strongly coupled SYM theory introduced in ref. [30] and analyzed there and in refs. [31, 32]. The matter produced in these collisions is initially far from equilibrium but then rapidly hydrodynamizes: its expansion and cooling is described well by viscous hydrodynamics after a time thydro that is at most around (0.7 1)/Thydro, where Thydro is the effective temperature defined from the fourth root of the energy density at the hydrodynamization time thydro. In ref. [29] we computed the drag force on a heavy quark moving through the initially far-from-equilibrium matter and the subsequent hydrodynamic fluid. We compared our results to what the drag force would have been in an equilibrium fluid with the same instantaneous energy density, and found that there is no dramatic extra energy loss in the far-from-equilibrium matter. However, even at late times when the expansion of the fluid is well-described by viscous hydrodynamics we found deviations between the actual drag force and what the drag force would have been in a spatially homogeneous equilibrium fluid with the same energy density. That is, we found that the gradients in the actual fluid do affect the drag force felt by the heavy quark moving through the fluid. Our goal in the present paper is a thorough investigation of the effects of gradients in the temperature and velocity of the fluid, up to first order, on the drag force. We begin by computing the drag force on a heavy quark moving through a fluid whose own motion is only in one direction, which we shall take to be the z-direction. If we denote the fluid 4velocity by u then at this stage the only gradients that we are considering are zuz and tuz as well as zT and tT . Throughout this paper, we shall only work to first order in spatial gradients and time derivatives of the fluid temperature and velocity. The gravitational dual for a slowly changing fluid, including the effects of first order derivatives but neglecting higher derivatives, was first obtained in ref. [33], where Einsteins equations (...truncated)