6.2 Quark energy loss in a thermal medium

Having learnt how to describe a massive quark in š’© = 4 SYM in terms of a string, this opens up the possibility of describing its energy loss as it propagates through a thermal medium. One can think of this process
  1. either from the bulk perspective, where the thermal medium gets replaced by a black hole and energy flows down the string towards its horizon,
  2. or from the gauge-theory perspective, where energy and momentum emanate from the quark and eventually thermalise.

In this section, I will take the bulk point of view originally discussed in [297, 268], with a related fluctuation analysis in [138]. The goal is to highlight the power of the techniques developed in Sections 4 and 5 rather than being self-contained. For a more thorough discussion, the reader should check the review on this particular topic [272Jump To The Next Citation Point].

The thermal medium is holographically described in terms of the AdS5-Schwarzschild black hole,

L2 ( dz2 ) ds2 = gmndxmdxn = --2 − h (z)dt2 + dāƒ—x2 +----- , (435 ) z h(z)
where h(z) = 1 − z44- zH determines the horizon size zH and the black-hole temperature T = -1-- πzH. The latter coincides with the gauge-theory temperature [498]. Notice z = 0 is the location of the conformal boundary and L is the radius of AdS5.

If one is interested in describing the dragging effect suffered by the quark due to the interactions with the thermal medium, one considers a non-static quark, whose trajectory in the boundary satisfies 1 X (t) = vt, assuming motion takes place only in the 1 x direction. One can parameterise the bulk trajectory as

1 X (t,z ) = vt + ξ(z), (436 )
where ξ(z) satisfies ξ → 0 as z → 0. To determine ξ(z), one must solve the classical equations of motion of the bosonic worldsheet action (16View Equation) in the background (435View Equation). These reduce to a set of conserved equations of the form
1 ∇ μπ μm = 0 , where πμm ≡ − ---′š’¢ μνgmn∂νXn (437 ) 2πα
is the worldsheet momentum current conjugate to the position Xm. Plugging the ansatz (436View Equation) into Eq. (437View Equation), one finds
ā”Œ ---------- dξ π ā”‚ā”‚ h − v2 ---= -ξāˆ˜ --4------, (438 ) dz h Lz4 h − π2ξ
where πξ is an integration constant. A priori, there are several allowed possibilities compatible with the reality of the trailing function ξ(z). These were analysed in [272Jump To The Next Citation Point] where it was concluded that the relevant physical solution is given by
2āˆ˜ ----- 2 ( ) πξ = − L-- h(z∗) = − √--v----L-- =⇒ ξ = − zHv- log 1-−-iy + ilog 1 +-y- , (439 ) z2∗ 1 − v2 z2H 4i 1 + iy 1 − y
where y is a rescaled depth variable y = z āˆ•zH.

To compute the rate at which quark momentum is being transferred to the bath, one can simply integrate the conserved current pμm over a line-segment and given the stready-state nature of the trailing string configuration, one infers [272Jump To The Next Citation Point]

dpm- √ --- z dt = − − gp m. (440 )
This allows us to define the drag force as
√ -- dp1 L2 v π λ v L4 Fdrag = ----= − ----2--′√------2 = − -----T2√------2 with λ = g2YMN = -′2. (441 ) dt 2πz Hα 1 − v 2 1 − v α
For a much more detailed discussion on the physics of this system see [272, 137Jump To The Next Citation Point]. The latter also includes a discussion of the same physical effect for a finite, but large, quark mass, and the possible implications of these results and techniques for quantum chromodynamics (QCD).

More recently, it was argued in [212] that one can compute the energy loss by radiation of an infinitely-massive half-BPS charged particle to all orders in 1āˆ•N using a similar construction to the one mentioned at the end of Section 6.1. This involved the use of classical D5-brane and D3-brane world volume reaching the AdS5 boundary to describe particles transforming in the antisymmetric and symmetric representations of the gauge group, respectively.

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