Abstract
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.
Highlights
Theory in the large number of colors 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 anti–deSitter (AdS) spacetime, with the heavy quark represented by a string moving through this spacetime [6,7,8,9,10,11]
The earliest work on heavy quark dynamics in the equilibrium plasma of strongly coupled N = 4 SYM theory [9,10,11] 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 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 been investigated in the equilibrium plasmas of many gauge theories with gravitational duals [15,16,17,18,19,20,21,22,23,24,25,26,27,28]
Summary
The stress-energy tensor for the conformal fluid of N = 4 SYM theory flowing hydrodynamically in 3 + 1-dimensions with a temperature T and 4-velocity uμ that vary as functions of space and time is given to first order in gradients by. The metric (2.7) is the zeroth approximation to the gravitational dual of the moving fluid; it would be a complete description if gradients made no contribution to the fluid stress-energy tensor, which is to say if the fluid were an ideal fluid with zero shear viscosity. The second term in the metric (2.6) is the dual gravitational description of the contribution of first order gradients in uμ and b to the fluid stress-energy tensor. It is given by [33].
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