Universality of the diffusion wake in the gauge-string duality
PPUPT-2242LMU-ASC 63/07
Universality of the diffusion wake in thegauge-string duality
Steven S. Gubser , ∗ and Amos Yarom , † Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA Ludwig-Maximilians-Universit¨at, Department f¨ur Physik, Theresienstrasse 37,80333 M¨unchen, Germany
Abstract
As a particle moves through a fluid, it may generate a laminar wake behind it. Inthe gauge-string duality, we show that such a diffusion wake is created by a heavy quarkmoving through a thermal plasma and that it has a universal strength when compared tothe total drag force exerted on the quark by the plasma. The universality extends over allasymptotically anti-de Sitter supergravity constructions with arbitrary scalar matter. Wediscuss how these results relate to the linearized hydrodynamic approximation and how theybear on our understanding of di-hadron correlators in heavy ion collisions.September 2007 ∗ e-mail: [email protected] † e-mail: [email protected] a r X i v : . [ h e p - t h ] S e p ontents Consider a quark moving at constant velocity through an infinite, static, thermal medium.Assume that the temperature is above the confinement transition, so the medium is a plasmaof gluons and whatever dynamical matter there may be. The quark loses energy to theplasma, but where does this energy go? Sufficiently far from the quark, a complete accountof energy dissipation should be possible within the framework of linearized hydrodynamics.Hydrodynamics is usually applicable at length scales much larger than the mean free pathof thermal quasi-particles within the plasma, which is to say some multiple of 1 /T . (Weset (cid:126) = c = k B = 1 throughout.) The applicability of linearized hydrodynamics depends onbeing far enough from the quark for the perturbations to be very small. It is well understood(see for example [1]) that if one assumes axial symmetry of the fluid flow around the directionof motion of the quark, then the equations of linearized hydrodynamics partially decoupleinto a pair of equations describing compressive waves (sound) and a third equation describingshearing modes (the diffusion wake). Figure 1 shows a typical geometry for these two modes. In principle, there could be additional modes coming from conserved currents. In the scenarios weconsider, such currents do not appear.
1e denote the total energy of the moving quark as E . The total energy gained by theplasma is given by P t = − dE/dt , where in our conventions dE/dt < P t = P d + P s , (1)where P s and P d correspond respectively to the energy lost to sound and the energy lost todiffusion.We claim that in a fairly broad range of string theory constructions and using a sensiblebut non-unique way of separating diffusion modes from other modes, P d : P t = − v . (2)The relative minus sign means that the diffusion wake supplies energy to the quark. Weprove this in section 2 for any theory admitting a holographic dual asymptotic to AdS thatcan be described in terms of Einstein gravity coupled to scalars with no more than twoderivatives in the action. In our demonstration, the quark has to be infinitely massive, andit is represented as a string trailing into the five-dimensional dual geometry. The tension ofthe string is an arbitrary function of the scalars.Putting (1) and (2) together, we learn that P s : P d : P t = 1 + v : − v . (3)This means that energy flows into sound waves at a rate which is 1 + 1 /v times as fast asthe total outflow of energy from the quark. The diffusion wake supplies energy to the quarkin just the right amount to balance equation (1). In section 3 we show that we do not needto assume (1) in order to demonstrate (3), at least in the case of a single scalar. The result(3) holds for all v between 0 and 1, but the most interesting case is when v is greater thanthe speed of sound c s , in which case there is a sonic boom.The result (3) was demonstrated for N = 4 super-Yang-Mills with a large number ofcolors and a large ’t Hooft coupling in [2], which followed upon a number of works byourselves and other authors [3, 4, 5, 6, 7, 8, 9]; parallel lines of development have included Clearly an infinitely massive quark must also have divergent energy E . But P t = − dE/dt is finite andclosely related to the drag force as calculated for N = 4 super-Yang-Mills theory in [3, 4]; see also thediscussion at the end of section 2.1. uark at quark attime -t time 0 vvtconediffusionwake Mach flowenergyenergyflow s G t c t s xx p q M Figure 1: Top: A cartoon of the diffusion wake and the sonic boom. The Mach cone andthe diffusion wake experience comparable viscosity broadening, controlled by the parameterΓ s = 4 η/ T s . Bottom: The Poynting vector (cid:126)S produced by a heavy quark propagatingthrough a thermal plasma of N = 4 super-Yang-Mills theory, from [2]. The arrows indicatethe direction of (cid:126)S , and the color indicates its magnitude, with red meaning large | (cid:126)S | . Thedashed green line marks the Mach angle, and the gray line marks the curve where the farfield asymptotics of the wake falls to half its maximal value.310, 11, 12, 13, 14, 15, 16, 17]. We note in particular the close relation of [12] to the dragforce results of [3, 4].The backgrounds we consider can accommodate arbitrary speed of sound (limited, per-haps, by c s ≤ / √ The action we start with is S = 12 κ (cid:90) M d x √− G ( R + L φ ) − πα (cid:48) (cid:90) W d σ √− g q ( (cid:126)φ ) , (4)where L φ is the scalar Lagrangian L φ = − G µν Ω IJ ( (cid:126)φ ) ∂ µ φ I ∂ ν φ J − V ( (cid:126)φ ) . (5) G µν is the metric on the bulk spacetime M , and g αβ is the metric induced from it on thestring worldsheet W . We will consider backgrounds which are asymptotically anti-de Sitter,with (cid:126)φ → (cid:126)φ B near the conformal boundary of AdS. The functions V ( φ ), Ω( (cid:126)φ ) and q ( (cid:126)φ ) arearbitrary, except for the requirement that V ( φ ) should have a local (negative) maximumfor a finite value of the scalars, V ( (cid:126)φ B ) = − /L , in order for the desired anti-de Sitterasymptotics to arise near the conformal boundary.When the string is not present, the backgrounds of interest take the form ds = α ( r ) (cid:20) − h ( r ) dt + d(cid:126)x + dr h ( r ) (cid:21) (cid:126)φ = (cid:126)φ ( r ) . (6)This ansatz is the result of assuming translation invariance in the R , directions and rota-tional invariance in the R directions. It is assumed that h → r → α → L/r . There is an event horizon at r = r H > h ( r H ) = 0.The tt and ii components of the Einstein equations (where i runs over the three components4f (cid:126)x ) provide us with ddr (cid:0) h (cid:48) α (cid:1) = 0 (7a)32 h (cid:48) α (cid:48) hα + 3 α (cid:48)(cid:48) α = 12 α h L φ , (7b)where primes denote d/dr . The scalars satisfy ddr ∂L φ ∂φ I (cid:48) − ∂L φ ∂φ I = 0 . (7c)In addition, the rr component of the Einstein equation imposes a zero-energy constraint:32 h (cid:48) α (cid:48) hα + 6 α (cid:48) α = 12 α h (cid:0) L φ + Ω IJ φ I (cid:48) φ J (cid:48) (cid:1) . (8)The system of equations (7) and (8) has been extensively studied, and we briefly sum-marize aspects of the relevant literature. It was shown in [18] that a solution of the form (6)can exist in the case of a single scalar only if V ( φ H ) < V ( φ B ) with φ H ≡ φ ( r H ). Numericalconstruction of such black holes was first reported on in [18] for a specific potential. Furthernumerical solutions were exhibited more explicitly in [19] and later in [20, 21, 22, 23, 24],where analyses of thermodynamic quantities were also given. Following [25], some approxi-mate, analytical results for high temperature black holes were obtained in [26, 27, 28]. In [3, 4] the drag force acting on a moving quark at constant velocity v through a plasma hasbeen calculated using AdS/CFT. It is straightforward to generalize the drag force calculationof [3, 4] to the backgrounds described above. Various generalizations exist in the literature,for example [29, 30, 31]. The one we will describe here is quite a modest extension ofpreviously published results, which allows for an arbitrary dependence on the scalar field φ I coming from a Kaluza-Klein reduction of the ten dimensional Dilaton and metric. Theansatz for the trailing string is x = vt + ξ ( r ) , (9)and the string action takes the form S = (cid:90) dtdr L L = − q ( (cid:126)φ )2 πα (cid:48) α (cid:114) − v h + hξ (cid:48) . (10)5he conserved momentum conjugate to ξ is π ξ = ∂L∂ξ (cid:48) = − q ( (cid:126)φ )2 πα (cid:48) hα (cid:112) − v /h + hξ (cid:48) ξ (cid:48) . (11)The drag force exerted by the plasma on the quark is F = dp dt = − π ξ . (12)Note that π ξ >
0, so F is negative, meaning in a direction opposite the quark’s velocity.The total rate of energy loss from the quark to the plasma is P t = − dEdt = − v dEdx = − v dp dt = vπ ξ , (13)which is a positive quantity. The trailing string sources the metric and the scalar fields coupled to it. Treating κ as asmall parameter, we may evaluate the response of the various fields to the string to linearorder in κ .We consider linearized fluctuations around a background of the form (6). These fluctua-tions will be expressed as G µν → G µν + δG µν , (cid:126)φ → (cid:126)φ + δ (cid:126)φ, etc. (14)In axial gauge one sets δG rν = 0 for all ν . We further define δG mn = κ α H mn , (15)where the pre-factor κ α is for convenience.As in [32], we study the linearized fluctuations in Fourier space with a co-moving ansatz: H mn ( t, r, (cid:126)x ) = (cid:90) d k (2 π ) e ik ( x − vt )+ k x + k x H mn ( t, r, (cid:126)k ) . (16)Because of axial symmetry, the Fourier coefficients H mn can only depend on wave-number (cid:126)k through k and k ⊥ ≡ (cid:112) k + k . Let’s fix this symmetry by setting k = 0 and k = k ⊥ > H , H , and H ) are odd under reflection through the planeperpendicular to ˆ3. The trailing string is symmetric under this reflection, so it doesn’t sourcethe odd metric components. We may therefore set them to zero. The other components ( H , H , H , H , H , H , and H ) are even, and they are all sourced non-trivially by thetrailing string. A substantial simplification of the equations of motion arises from formingthe following linear combinations, which are essentially the same as the ones in [32]: A = 12 v (cid:32) − H + 2 k k ⊥ H − (cid:18) k k ⊥ (cid:19) H + (cid:18) kk ⊥ (cid:19) H (cid:33) (17) (cid:126)D = (cid:32) D D (cid:33) = v (cid:16) H − k k ⊥ H (cid:17) v (cid:16) − H + (cid:16) k k ⊥ − k ⊥ k (cid:17) H + H (cid:17) (18) (cid:126)E = E E E E = (cid:0) − h H + H + H + H (cid:1) v (cid:16) H + k ⊥ k H (cid:17) ( H + H + H ) (cid:16) H (cid:16) − (cid:0) k k (cid:1) (cid:17) + H (cid:16) − (cid:0) k k (cid:1) (cid:17) + 2 H − k k ⊥ k H (cid:17) . (19)The equation of motion for A decouples from all other metric components as well as fromthe scalars. The equations of motion for (cid:126)D mix with one another but decouple from A and (cid:126)E as well as the scalars. The equations of motion for (cid:126)E couple to the scalars. The definingrelations (17)-(19) are equivalent to H = 2 h − E + E ) H = 2 vk ( k ⊥ D + k E ) H = 2 vk k ⊥ k ( − D + E ) H = − v k ⊥ k A − v k k ⊥ k D + 23 E + − k + k ⊥ k E H = k k ⊥ k (cid:18) v k ⊥ k A + 2 v k − k ⊥ k D − E (cid:19) H = − v k k ⊥ k A + 4 v k k ⊥ D k + 23 E + k − k ⊥ k E H = v k ⊥ k A + 23 E + 13 E . (20)It will prove useful to have these inverse relations ready to hand when we want to extract7omponents of the gauge theory’s stress energy tensor.We shall restrict our attention to the vector modes since these are the only ones whichwill contribute to the diffusion wake. To see this consider the contribution of the movingquark to the Poynting vector (cid:126)S = (cid:16) S S S (cid:17) where S i = (cid:104) δT i (cid:105) = −(cid:104) δT i (cid:105) . (21)Recall that we employ mostly plus signature and that (cid:104) T i (cid:105) = 0 in the static background.We use (cid:126)S in preference to δ (cid:126)S for simplicity, keeping in mind that the non-zero Poyntingvector owes wholly to the presence of the moving quark. Using holographic renormalization(see appendix A) we can show that S i = − lim r → α L H i . (22)It is useful to decompose the Fourier components of the Poynting vector into a longitudinalpart and a transverse part (cid:126)S = S L ˆ k + (cid:126)S T , (23)where ˆ k = k ( k , k ⊥ ). The (cid:126)S T component is orthogonal to ˆ k and lies in the same plane as (cid:126)k and (cid:126)v due to the azimuthal symmetry around the direction of motion of the quark. It iswell understood (see for example [1]) that the diffusion wake is described in terms of (cid:126)S T .Referring to (20), we see that the contribution of E to (cid:126)S is in a direction parallel to thewave-number (cid:126)k , whereas the contribution to D is orthogonal to (cid:126)k . In short, E controls S L while D controls (cid:126)S T . Using (20) and (22) one finds (cid:126)S T = − vk ⊥ Lk (cid:16) − k k ⊥ (cid:17) lim r → α D . (24)The linearized Einstein equations in axial gauge imply the following second order equa-tions for the D modes, α − ddr (cid:0) D (cid:48) α (cid:1) − k h D + k v h D = π ξ h α ξ (cid:48) e − ik ξ α − h − ddr (cid:0) D (cid:48) hα (cid:1) − k h D + k v h = v − h ξ (cid:48) v h α ξ (cid:48) π ξ e − ik ξ . (25)In (25), the real-valued function ξ ( r ) is determined by solving (11). Additionally there is a8rst order constraint, D (cid:48) − hD (cid:48) = π ξ ik v α e − ik ξ . (26)Note that the source for the constraint equations (26) is of order O ( k − ) while that of thesecond order equations (25) is of order O (1). We are interested in the large distance asymptotics of the energy momentum tensor. In thisregime one would expect that linearized hydrodynamics will be a good approximation. Theequations for vector perturbations simplify drastically when only the leading order terms insmall k are retained: the equations of motion (25) become ddr ( α D (cid:48) ) = 0 ddr ( hα D (cid:48) ) = 0 , (27)and the first order constraint (26) becomes D (cid:48) − hD (cid:48) = π ξ iv k α . (28)The most general solution of the equations of motion (27) consistent with the boundarycondition D i → D ( r ) = d (cid:90) r d ˜ rα (˜ r ) D ( r ) = d (cid:90) r d ˜ rh (˜ r ) α (˜ r ) . (29)Because h → D has a logarithmic divergence there. Requiring that themetric fluctuations are finite at the horizon implies d = 0. This is equivalent to taking the k → d = 0 leads immediately to d = π ξ iv k . (30)The diffusion wake is a long, narrow, forward-directed stream of fluid along the negative x axis (see figure 1). Its narrowness in the x and x directions means that, in Fourierspace, we can take k and k large compared to k , effectively setting k ⊥ = k . In this limit,9e find a simple limiting form for (cid:126)S T : (cid:126)S T → − π ξ ivk (cid:16) (cid:17) ≡ (cid:126)S d . (31)To Fourier transform (cid:126)S d to real space, we need a prescription for passing the k = 0 pole.Passing it in the upper half plane results in a stream of energy along the negative x axis,while passing it in the lower half plane results in a stream of energy along the positive x axis—ahead of the quark. This latter possibility is unphysical in a steady-state solution. Wefind S d1 (0 , (cid:126)x ) = − π ξ iv (cid:90) d k (2 π ) e i(cid:126)k · (cid:126)x k + i(cid:15) = π ξ v θ ( − x ) δ ( x ) δ ( x ) , (32)where the integration is along real values, (cid:15) >
0, and θ ( x ) ≡ x >
00 for x <
0. (33)In [2, 8], it was observed that for the case of N = 4 super-Yang-Mills theory, effects of order O (1) lead to extra terms which correspond to replacing (cid:15) in (32) by η/svT k . This replacementbroadens the wake from a singular structure along the negative x axis to a stream inside aparabolic surface of revolution, as shown in figure 1.The energy carried by the narrow stream described in (32) per unit time is given by P d = (cid:73) (cid:126)S d · d(cid:126)a = (cid:90) ∆ (cid:126)S · d(cid:126)a = − π ξ v . (34)The first surface integral in (34) is over the boundary of a large volume enclosing the movingquark, and d(cid:126)a is an outward pointing area element. This integral can be evaluated im-mediately, using (32), to give the result − π ξ /v . In writing the second integral, we aim toreproduce the same result by integrating the full Poynting vector (cid:126)S over a carefully chosensurface ∆. From (32) we might expect that ∆ can be any sufficiently small surface inter-secting the negative real axis. But in the full Poynting vector, there is viscosity broadening,and there are also non-hydrodynamical effects close to the quark [6, 7, 33]. A good choice of∆ is shown in figure 2, where one takes (cid:96) → ∞ with (cid:96) (cid:29) (cid:96) p (cid:29) √ (cid:96) Γ s . The large (cid:96) limitinsures that linearized hydrodynamics is valid, and the specified range of (cid:96) p insures that theintegration over ∆ picks up the entire contribution from the diffusion wake and nothing else.10 ime 0 l quark at p D l p v wakediffusion energy x flow x Figure 2: The diffusion wake and the integration surface ∆ used in equation (34). Theinviscid limit of the diffusion wake is an infinitesmially narrow, forward directed stream ofenergy, as shown in lighter blue. After viscosity broadening, the diffusion wake thickens toa parabolic shape.The area element d(cid:126)a points toward negative x to agree with directionality of d(cid:126)a in the firstintegral in (34). Combining (13), (30), and (34), we have P d = − v P t , (35)This is the result we advertised in (2). In section 3.3 we will describe another way ofevaluating P d starting from the divergence of the stress tensor. To describe the sound modes, we need to know the first column of the stress energy tensorin the small momentum limit. For purposes of simplicity, we restrict attention to a singlescalar, so that L φ = − G µν ∂ µ φ∂ ν φ − V ( φ ) . (36)Referring to (20), we see that to determine the Poynting vector, it is enough to understandthe asymptotics of E . This field couples non-trivially both to the scalar and to E , E ,and E . However, in the small momentum approximation a simplification occurs: two of the Near the quark, the flow is not hydrodynamical. A near-field account can probably be given, along thelines of [6, 7, 33]. ddr (cid:0) E (cid:48) α (cid:1) = 0 (37) ddr (cid:0) E (cid:48) hα (cid:1) = 0 . (38)The other equations of motion, also at order O (1 /k ), are δφ (cid:48)(cid:48) + (cid:18) α (cid:48) α + h (cid:48) h (cid:19) δφ (cid:48) − α V (cid:48)(cid:48) ( φ ) h δφ + (cid:18) E (cid:48) + 23 E (cid:48) (cid:19) φ (cid:48) = 0 (39a) E (cid:48)(cid:48) + (cid:18) α (cid:48) α + 3 h (cid:48) h (cid:19) E (cid:48) + φ (cid:48) δφ (cid:48) + α V (cid:48) ( φ ) h δφ = 0 (39b) E (cid:48)(cid:48) + (cid:18) α (cid:48) α + h (cid:48) h (cid:19) E (cid:48) + 12 φ (cid:48) δφ (cid:48) + α V (cid:48) ( φ )2 h δφ = 0 . (39c)The first order constraints couple together all the E i ’s and the scalar: E (cid:48) + E (cid:48) − h (cid:48) h E − h (cid:48) h E + 12 φ (cid:48) δφ = iπ ξ k hα (40a)4 h α (cid:48) α E (cid:48) + (cid:18) h (cid:48) + 8 h α (cid:48) α (cid:19) E (cid:48) − φ (cid:48) δφ (cid:48) h + 2 α V (cid:48) δφ = 0 (40b) E (cid:48) + (cid:18) k v k h (cid:19) E (cid:48) + E (cid:48) + 12 h (cid:48) h E − k v h (cid:48) k h E − h (cid:48) h (cid:18) k v k h (cid:19) E + 32 (cid:18) k v k h (cid:19) φ (cid:48) δφ = − ik π ξ k α h (cid:18) v h (cid:19) (40c)where we have again written only the leading terms in a 1 /k expansion. Solving (37) and(38) together with the requirement that the metric fluctuations vanish at the asymptoticAdS boundary, we get E = e (cid:90) r d ˜ rα (˜ r ) = e h α ( h −
1) (41a) E = e (cid:90) r d ˜ rh (˜ r ) α (˜ r ) = e h α ln h , (41b)where we have used the equation of motion for h (7a) in the equalities on the right handside, and defined h ≡ h (cid:48) ( r H ). To avoid a logarithmic singularity at the event horizon, onemust set e = 0.The alert reader will have noted that (41b) is in precise analogy to (29). The horizonboundary condition e = 0 is likewise in analogy to d = 0. The justification of these12oundary conditions is actually somewhat subtle. At finite k , the possible leading near-horizon behaviors of E and D are h ± ivk /h . The minus sign corresponds to an outgoingwave and the plus sign corresponds to an infalling wave. Usually, the appropriate boundarycondition is to suppress the outgoing wave. But in a small k expansion, h ivk /h = 1 + ivk h log h . The relative factor of k means that the logarithmic term cannot be visible ina leading small k treatment. Another way to understand why one must forbid both theinfalling and outgoing waves is that the leading order small k analysis does not know aboutviscosity, so it also doesn’t know about the arrow of time; thus the boundary conditions, atthis order, should be symmetric between infalling and outgoing waves.Although it is unimportant for our subsequent analysis, we note for completeness thatthe A equation of motion completely decouples from the other modes and the scalar fieldsfor all values of k . At order 1 /k , it takes the simple form ddr (cid:0) A (cid:48) hα (cid:1) = 0 . (42)As with D and E , the horizon boundary condition forces A = O (1). The equations (37)-(40) are too complicated to solve in general as we did for the D set.Furthermore, equations (39) describe not only the response of the metric to the string, butalso static deformations of the black hole background. Although such deformations are notdirectly related to the response to the moving quark, it is helpful to describe them explicitlyon the dual gravity side as a warm-up to the treatment of sound modes. More precisely, wewant to consider perturbations that are invariant under translations in the R , directionsand rotations in the R dimensions. This allows us to set H = H = H and to set alloff-diagonal components of the metric to zero. So E = E = 0, while E and E may bedefined as in (19). It is straightforward to show that the equations of motion take the form(39) and that the zero-energy constraint (coming from the rr Einstein equation) takes theform (40b). The expectation is that there is only one allowed deformation, and it correspondsto uniformly changing the temperature in the gauge theory.These equations are still impossible to integrate explicitly for general V ( φ ). However,one may perform a near horizon analysis by switching the independent variable from r to h α = α + α h + α h + . . .r = r H + r h + r h + . . .φ = φ + φ h + φ h + . . . . (43)We find six linearly independent solutions:A: E = 1 E = 0 δφ = 0B: E = 0 E = 1 δφ = 0C: E = − r α V (cid:48) ( φ ) h E = − r α V (cid:48) ( φ ) h δφ = 1 + r α V (cid:48)(cid:48) ( φ ) h D: E = − r α V (cid:48) ( φ ) h E = − r α V (cid:48) ( φ ) h δφ = (cid:2) r α V (cid:48)(cid:48) ( φ ) h (cid:3) × log h × log h × log h E: E = 1 √ h E = 0 δφ = 23 r α V (cid:48) ( φ ) √ h F: E = 0 E = √ h δφ = − r α V (cid:48) ( φ ) h / . (44)Solutions A and B are exact, while the others, as expressed in (44), are accurate only tothe first two non-trivial orders in small h . Applying the constraint equation (40b) to thesolutions (44), one finds that the D solution is disallowed, and a relation is enforced betweenthe E and F solutions. The upshot is that for small h , E = e √ h + ˜ e − r α V (cid:48) ( φ ) δφ H h + . . .E = e − e r α V ( φ ) √ h − δφ H r α V (cid:48) ( φ ) h + . . .δφ = δφ H + 23 e r α V (cid:48) ( φ ) √ h + δφ H r α V (cid:48)(cid:48) ( φ ) h + . . . , (45)where e , ˜ e , e , and δφ H are integration constants. In principle, three of these constantscan be fixed in terms of the fourth by integrating E , E , and δφ out to the conformalboundary of AdS and imposing boundary conditions there corresponding to the absenceof any deformations of the lagrangian. This means that the boundary conditions on theasymptotic AdS boundary are such that E and E are O ( r ) near the conformal boundary,while δφ is O ( r ∆ ), where ∆ is the dimension of the operator dual to φ . Once these conditions There is a subtlety when ∆ is between 2 and 3: different QFT’s exist, which are conformal at least in alarge N limit, which allow ∆ to be exchanged with 4 − ∆ [34]. The simplest assumption is ∆ ≥
2, but theprecise form of the boundary condition on δφ does not matter in our analysis. For the
AdS -Schwarzschild background, the differential equations (39) can be integratedout to the conformal boundary of anti-de Sitter space, and the boundary conditions therecan be imposed explicitly. They are e = − ˜ e = − e , with no constraint on δφ H becausethe scalar and graviton perturbations decouple. To show all this, simply note that theterms shown explicitly in (45) provide an exact solution for AdS -Schwarzschild. For otherbackgrounds, one usually cannot explicitly perform the integration from the horizon to theconformal boundary. Generically, the scalar and graviton mix, and the three constraints atthe conformal boundary involve all of e , ˜ e , e , and δφ H .Since we can not integrate the equations, we proceed by expressing the various integrationconstants in terms of the entropy and temperature. The solutions (45) imply that the metricfluctuations are given by H = − e √ h + 23 ( e − ˜ e ) h + O ( h / ) H = 23 e + O ( h / ) (46)(recall that H = H = H .) Note that despite the 1 / √ h singular terms in (45), thefluctuations (46) are finite at the horizon. In (46) we have suppressed all terms beyond theones that affect the variation in the entropy and temperature. The entropy is straightforwardto compute: the perturbed metric to order κ takes the form ds = α (cid:20) − ( h − κ H ) dt + (1 + κ H ) d(cid:126)x + dr h (cid:21) . (47)So the entropy per coordinate volume of (cid:126)x is s = 2 πα κ (1 + κ H ) / (48)evaluated at the horizon. Thus δss = κ lim h → H = κ e . (49)Determining the variation of the temperature is a bit more subtle because G tt and G rr depend Intriguingly, we do not need to impose any boundary conditions at the horizon as we have in thediscussion following (41). It seems that at order 1 /k the constraint equation (40b) is enough to ensure that δφ and all the E i ’s are finite at the horizon. The
AdS -Schwarzschild case is special in that a constant shift of the dilaton is an exact deformation.So only two additional constraints arise from the conformal boundary. Usually there are three. h after the perturbation. One way to proceed is to change variables from r to ˜ r in such a way that the perturbed metric becomes ds = ˜ α (cid:20) − ˜ hdt + d(cid:126)x + d ˜ r ˜ h (cid:21) . (50)One easily finds ˜ α = α (cid:18) κ H (cid:19) ˜ h = h − κ H − κ hH d ˜ rdr = 1 − κ H h − κ H , (51)all up to O ( κ ) corrections which we ignore. The Hawking temperature of the unperturbedbackground (6) is easily seen to be 1 / πr . Suppose we expand ˜ r in a power series in ˜ h :˜ r = ˜ r H + ˜ r ˜ h + ˜ r ˜ h + . . . . (52)Then the Hawking temperature of the perturbed background is T = − π ˜ r . (53)Therefore δTT = 1 − ˜ r r = κ lim h → (cid:20) − dH dh + H h − h dH dh (cid:21) = κ ˜ e − e . (54)According to simple thermodynamic arguments summarized in [35], as long as there are nochemical potentials present, the speed of sound should be c s = d log Td log s = 13 (cid:18) ˜ e e − (cid:19) . (55)As a check, one may subsitute ˜ e = 2 e for AdS-Schwarzschild and recover c s = 1 /
3. It willalso prove interesting to note that the first law of thermodynamics predicts δ E = T δs = − α r H = − α r e , (56)where E is the energy density and we again assume that no chemical potential is present.16 .2 Small momentum asymptotics A striking feature of the full O (1 /k ) equations (37)-(40) is that the equations of motion for E , E , and δφ , together with the zero energy constraint (40b), are the same as for staticdeformations. The two extra non homogeneous constraints (40a) and (40c) will determine e and one other integration constant. Heuristically, this means that we can describe soundwaves as a modulation of the static deformation, plus some profile for E that incorporatesnon-zero Poynting vector. Technically, it means we can set˜ e = (1 + 3 c s ) e . (57)This is not really a constraint on ˜ e , instead we are simply trading ˜ e /e for c s ≡ d log T /d log s as a parameter for describing temperature-deformations. Only by fully integrating the dif-ferential equations (39) and (45) and imposing appropriate constraints at the conformalboundary could we determine c s . Plugging (45) and (41) into (40), and imposing e = 0as well as (57), one obtains a pair of linear equations in e and e . Solving these equationsleads to e = π ξ ik k + c s k v k − c s k e = 2 r k π ξ iα v v k − c s k . (58)Using (30) and (58) in (22) we obtain the Poynting vector to leading order in small k : S = − π ξ ivk + ic s k π ξ v v v k − c s k + O (1) S ⊥ = ic s k ⊥ π ξ v v v k − c s k + O (1) . (59)When c s = 1 / √
3, (59) reduces to the leading order results found in [2]. It is natural toidentify the contribution of sound modes as (cid:126)S s = (cid:16) S s1 S s ⊥ (cid:17) , (60)where S s1 = k k ⊥ S s ⊥ = ic s k π ξ v v v k − c s k . (61)Defining (cid:126)S d as in (31), we see that (cid:126)S = (cid:126)S s + (cid:126)S d .In order to completely characterize the energy dissipation, we need also to computethe fluctuations in the energy density. This is difficult to work out in a general settingbecause the Brown-York tensor probably does not correctly capture (cid:104) T (cid:105) owing to effects17rom holographic renormalization. Also one apparently has to solve for E and E out tothe boundary, which seems impossible. But there is a simple shortcut—though admittedlyheuristic: it is simply to use the thermodynamic relation δ E = T δs in the (cid:126)k -th Fourier mode.This should work at leading order in small k . The result, carried over directly from (56), is δ E = − α r e = ik π ξ v v k − c s k . (62)When c s = 1 / √
3, (62) reduces to the leading order results found in [2].
The results (59) and (62) are enough information to determine the first row of the stresstensor completely at small k . More precisely, consider the subtracted stress energy tensorexpectation value at time t = 0 (still in the rest frame of the plasma): δT mn ≡ (cid:104) T mn (0 , (cid:126)x ) (cid:105) − (cid:104) T mn (cid:105) bath , (63)where (cid:104) T mn (cid:105) bath is from the infinite, static, thermal medium. What we have computed, inFourier space, with k = 0 and k = k ⊥ >
0, is δT m = (cid:16) δ E S S ⊥ (cid:17) . (64)(Recall that we use (cid:126)S for the response of the Poynting vector to the string instead of δ (cid:126)S .)Rotational invariance around the ˆ1 axis determines δT m for other values of (cid:126)k . As discussedin [2, 32], the failure of the conservation law ∂ m δT mn = 0 is a good measure of energydissipation from the quark. The non-conservation law can be expressed in Fourier space as ik m δT m = P t , (65)where k m = (cid:16) − vk k k ⊥ (cid:17) (66)(setting k = − vk comes from the co-moving ansatz.) The right hand side of (65) wascomputed in (13). It is the total rate at which the quark deposits energy into the plasma:a positive quantity. A check on our computations so far is to plug (59), (62), and (64) into(65) and check that the right hand side agrees with (13). It does.18s in [2], we associate δ E entirely with dissipation into sound waves. This makes sensebecause, in a linearized hydrodynamical analysis, the diffusion wake shows up only in thePoynting vector. Thus we have a splitting δT m = δT m sound + δT m diffuse , (67)where δT m sound = (cid:16) δ E S s1 S s ⊥ (cid:17) δT m diffuse = (cid:16) S d1 (cid:17) . (68)There is an associated splitting of the non-conservation effect: ik m T m sound = P s ik m T m diffuse = P d . (69)We regard (69) as a definition of P s and P d . It is immediate from the preceding discussionthat P s : P d : P t = 1 + v : − v , (70)which is the result (3) that we advertised from the start.It is clear that in splitting up δT m as in (68) we are making choices rather than doingsomething inevitable. To justify this splitting we note that in linearized hydrodynamics,neglecting viscosity, the Fourier space form describing sound waves involves ω − c s k , whilethe analogous form for the diffusion wake is just ω (see for example [1]). This preciselycoincides with our choice of sound and diffusion modes in equations (31), (61) and (62). Thediffusion part of the stress tensor is exactly the long, narrow stream of plasma behind thequark, which we justified in a different way in the discussion leading up to (31).Furthermore, when v > c s , the split we use leads to exponential suppression of all com-ponents of δT m sound and δT m diffuse for x >
0, which is sensible because this region is causallyinaccessible to sound waves emanating from the quark. Another natural split is to associatecontributions to the stress tensor from the D set purely with the diffusion wake, and contribu-tions from the E set purely with sound modes. This is the same transverse/longitudinal splitdiscussed in [1] and it leads to power law tails of components of the stress tensor cancellingdelicately between diffusion and sound modes.It is interesting to note that when c s →
0, the total perturbation in the stress tensor19ecomes δT m ( (cid:126)k ) = iπ ξ vk (cid:16) v v (cid:17) , (71)with our usual convention that k = 0 and k = k ⊥ >
0. In position space, δT m (0 , (cid:126)x ) = π ξ v θ ( − x ) δ ( x ) δ ( x ) (cid:16) v v (cid:17) , (72)which describes energy deposition and flow only in the path of the quark. There is ofcourse some viscosity broadening, presumably visible at higher orders in k , but no pressurebroadening—because there is no pressure. Sound waves don’t really exist in this limit. Butthe diffusion wake, as defined in (31), is completely insensitive to the speed of sound. Itsform and its strength appear to be a universal feature of string theory constructions startingfrom Einstein gravity coupled to strings and scalars. The stress energy tensor of the thermal plasma is conserved except at the location of thequark: ∂ m δT mn = J n δ ( x − vt ) δ ( x ) δ ( x ) . (73)In Fourier space, ik m δT mn = J n , (74)where k m = (cid:16) − vk k k ⊥ (cid:17) (75)as in (66). The n = 0 component of (74) is precisely (65). From (12) and (13) we read off J n = (cid:16) vπ ξ π ξ (cid:17) . (76)We emphasize that our derivation of (74), for n = 0, did not directly use a hydrodynamicapproximation: it came from the small k limit of an AdS/CFT computation. The purpose ofthis section is to understand how the AdS/CFT computation matches with linearized hydro,including (for N = 4) a sub-leading effect in small k .The constitutive relations for linearized hydro amount to δT hydro ,ij = c s (cid:15)δ ij −
34 Γ s (cid:18) ik i S j + ik j S i − δ ij ik l S l (cid:19) , (77)20here (cid:15) = δT S i = δT i . (78)(Recall that we use mostly plus signature.) The equations of motion for linearized hydrody-namics are ik m δT mn hydro = J n hydro . (79)But now there is no reason why the source term J n hydro should be pointlike. Instead, inposition space, J n hydro probably has width comparable to Γ s (the attenuation length) andmust be parametrized in some fashion. The authors of [1, 36] used an equivalent of J n hydro = (cid:16) e g + k g k ⊥ g (cid:17) . (80)In [1, 36] two scenarios where considered. In scenario 1, e = vg is a Gaussian in real spacewith width Γ s , while g = 0. Scenario 2 corresponds to the opposite extreme, e = g = 0and g a Gaussian with width Γ s .A natural though non-unique prescription in matching our results to linearized hydro isto set δT m = δT m (81)for m = 0 , , , δT mn hydro . If we follow this procedure to order O ( k − ), see (59) and (62), we find δT mn hydro = ik ( v ) − c s k + k v − ik v − ic s k (1+ v ) v ( c s k − k v ) − ic s k ⊥ (1+ v ) v ( c s k − k v ) − ik v − ic s k (1+ v ) v ( c s k − k v ) ik c s ( v ) − c s k + k v − ic s k ⊥ (1+ v ) v ( c s k − k v ) ik c s ( v ) − c s k + k v
00 0 0 ik c s ( v ) − c s k + k v π ξ + O (1) . (82)If we plug (82) into (79), we obtain an expression for J n hydro coinciding with (80) with e = vπ ξ + O ( k ) g = π ξ + O ( k ) g = O (1) . (83)Fourier transforming the leading terms shown in (83) leads to position results which aredelta-functions, as in (73). Because of the explicit factors of k in (80), non-vanishing e and One possible alternative prescription for determining δT m is to require that at order O ( k ), J i hydro hasonly longitudinal components. We’ve checked that the results we describe below remain unchanged undersuch a redefinition. tend to dominate over non-vanishing g at sufficiently long distances. More specifically,given e , g and g in Fourier space, one may define a dimensionless figure of merit, γ = 1 R (cid:12)(cid:12)(cid:12)(cid:12) g ( k = 1 /R ) e ( k = 1 /R ) (cid:12)(cid:12)(cid:12)(cid:12) , (84)such that γ (cid:28) γ (cid:29) R is a typical distanceat which one is interested in the strength of the flow induced by e and g as compared tothe flow induced by g . (We assume that e and g have similar magnitudes.) If J n hydro islocalized at a scale Γ s , then for a source such as the one we have described, where e and g are non-vanishing at leading order, one expects γ ∼ Γ s R . (85)Because one only trusts linearized hydro for R (cid:29) Γ s , our results correspond to scenario 1.To get scenario 2, one needs γ (cid:29) γ , as well as more information about thestructure of the source terms e and g , by calculating the stress tensor at the first subleadingorder in k . That is, we expand δT mn = δT (0) mn k + δT (1) mn + O ( k ) δT mn hydro = δT (0) mn hydro k + δT (1) mn hydro + O ( k ) (86)and impose (77) and (81) at order O (1) to get δT (1) mn hydro . These subleading contributionsto the energy momentum tensor are currently available only for the N = 4 theory [32].Adapting the notation in [32] to the current conventions, we find that δT (1) mn = δT (1) mn hydro + diagonal (cid:16) − (cid:17) vπ ξ πT . (87)This corresponds to J n = (cid:16) v − vk Γ s vk ⊥ Γ s (cid:17) π ξ + O ( k ) , (88) J. Casalderrey-Solana has suggested to us that γ could be changed by sending R → √ Γ s R in (84). Thisresults in a rough expectation γ ∼ (cid:112) Γ s /R in place of (85). s = 1 / πT . From (88) we read off e ( (cid:126)k ) = π ξ v + O ( k ) g ( (cid:126)k ) = π ξ (1 − vk Γ s ) + O ( k ) g ( (cid:126)k ) = π ξ v Γ s + O ( k ) . (89)These results agree with the expectation that J n hydro has structure on the scale Γ s , and itimplies that γ = Γ s R + O ( R − ) . (90)Taking Γ s = 0 .
07 fm and R = 7 fm to make a rough comparison with heavy ion collisions,one finds γ = 0 . γ will be be small for phenomenologically interesting values of R evenfor AdS/CFT constructions including scalars, unless perhaps the scalar lagrangian has somevery special or extreme form. Nevertheless, it would be interesting to see what form thesesubleading expressions take in, for example, the N = 2 (cid:63) theory [37, 28, 21], or the casacadinggauge theory [38, 26, 24, 23]. It seems to us more likely that γ could be altered significantlyby considering a different source from the heavy quark we have employed throughout. In this section we attempt to bridge between our results from AdS/CFT and the phenomeno-logical literature on heavy ion collisions. In doing so we should bear in mind all the usualcaveats to such comparisons. In particular: the AdS/CFT results apply in the limit of large N and large ’t Hooft coupling g Y M N ; the quark in our treatment is infinitely heavy and hasconstant velocity; and the holographic dual to QCD is not known.In heavy ion collisions, a well studied experimental probe of the interaction of a hardparton with the medium is the relative azimuthal angle ∆ φ between two energetic hadronsemitted from the interaction region. Histograms of hadron pairs always show a peak at∆ φ = 0, most likely meaning that the two hadrons were part of the same jet. In proton-proton collisions and sufficiently peripheral heavy ion collisions where the extent of themedium is small, another peak at ∆ φ = π arises, owing presumably to events where two hardpartons are created with back-to-back momenta—at least, back-to-back in the azimuthaldirection. See figure 3.Jet-broadening or jet-splitting refers to the change in shape of the peak at ∆ φ = π . Di-hadron histograms published by the PHENIX collaboration in [39] show a definite double-peaked structure: a minimum at ∆ φ = π and peaks at ∆ φ ≈ π ± .
2. On the other hand,23 y x li n e b ea m
12 3 m i d - r a p i d it y p l a n e Df Figure 3: A cartoon of a hard process in a heavy ion collision. The blue and green crescentsare the parts of the nuclei that didn’t interact. The gold region is the thermalizing medium.The red dot is the vertex of a hard process that occurs early in the collision, producingtwo partons with back-to-back momenta in the azimuthal direction. For simplicity we showall particles in the x - y plane meaning at mid-rapidity. If particles 1 and 2 escape to formhadrons, it would correspond to ∆ φ ≈ π . If instead particle 2 stops in the medium but emitsa high-angle secondary which is then observed as 3, then ∆ φ is significantly different from π . 24he di-hadron histograms published by the STAR collaboration in [40] do not exhibit apronounced double-peaked structure and are discussed in that paper in terms of softening in p T , broadening in ∆ φ and rapidity, and thermalization. Momentum cuts were significantlydifferent in these two studies:STAR : 0 .
15 GeV < p assoc T < < p trig T < < p assoc T < . < p trig T < . (91)STAR’s pseudo-rapidity acceptance, | η | <
1, is also significantly greater than PHENIX’s, | η | < .
35. Subsequent experimental analyses, including [41, 42, 43, 44, 45], support theconclusion that at least for intermediate momentum hadrons, similar to the PHENIX cutsshown in (91), splitting of the away side peak does occur.A candidate explanation for jet-splitting is that a hard parton traveling through a quark-gluon plasma (QGP) loses energy in large part by producing a sonic boom [1, 46]. The sonicboom would then propagate through the plasma until freeze-out converts it into an excessof hadrons emitted in roughly the direction of the Mach angle, relative to the direction ofthe original parton. There are theoretical alternatives (see for example [47, 48, 49, 50]), butour string theory results bear less directly upon them. In [1], two scenarios were presentedfor the distribution of energy into sound modes and diffusion modes. In section 4 we haveshown that in scenario 1 the relative strength of the sonic boom and the diffusion wake isthe same as in our string theory analysis. After Cooper-Frye hadronization on a surface offixed time, it shows no jet-splitting due to the relatively strong diffusion wake. In scenario 2,only longitudinal modes are sourced by the energy momentum tensor so the diffusion wakeis entirely suppressed and a jet-splitting effect is recovered after Cooper-Frye hadronization.Further work [36] showed that for scenario 2 to match to data, the energy loss per distanceneeds to be chosen quite large—perhaps unrealistically so. Another study [51] includedexpansion of the plasma and used source terms similar to scenario 1. It also concludedthat a minimum at δφ = π is hard to achieve. In [52, 53], a model including Mach conepropagation through the QGP has been argued to fit experimental data provided the fractionof energy going into the sonic boom is 0 .
75 or more. But it is not yet clear to us how thisfraction is related to the relative strength of the sonic boom and the diffusion wake as wecompute it.The robustness of the diffusion wake in the string theory construction we have exploredfavors scenario 1. It is therefore a concern that this scenario seems less likely to agree withdata. It is natural for us to hope that a diffusion wake really is generated when a hard parton25raverses a quark-gluon plasma, but that it is erased or disguised in the data. More realistictheoretical treatments should include expansion of the plasma, as in [51, 52, 53] and perhapsalso a more sophisticated treatment of hadronization; as compared to the sonic boom, thediffusion wake is deeper inside the expanding medium, so it has more chance to broaden,soften, and thermalize before freeze-out. Some hints of this were observed in [51]. Therealso issues of detector acceptance which may be relevant, see for example the discussion in[53]. Finally, we should keep firmly in mind our starting assumption that the quark is verymassive. If the trigger particle could be tagged as originating from a heavy quark, then thedistribution of associated hadrons would be more directly related to our analysis. Studiescited thus far do not include flavor tagging; see however [54].In summary: We have argued that the strength of the diffusion wake relative to thedrag force is universal in AdS/CFT constructions based on a trailing string in a backgroundsupported by arbitrary scalar matter. This universality is in some ways comparable to theuniversality of the η/s ratio [55, 56, 57] (see also [58]): it arises because of a decouplingof certain metric perturbations from the scalar dynamics, and it may suffer corrections ininverse powers of N and g Y M N . Also, it is worth remembering that our slightly weakerresult on the universality of dissipation through sound modes does not specify how wellthis dissipation is focused at the Mach angle. Phenomenologically, it may be challenging toaccommodate as strong a diffusion wake as we predict. Acknowledgments
The work of S. Gubser was supported in part by the Department of Energy under Grant No.DE-FG02-91ER40671, and by the Sloan Foundation. A. Yarom is supported in part by theGerman Science Foundation and by the Minerva foundation. S.S.G. thanks T. Hemmick,S. Pratt, and B. Jacak for useful discussions. A.Y. thanks J. Casalderrey-Solana, M. Haack,E. Shuryak and D. Teaney for useful discussions. A. Y. is grateful to the 5th Simons workshopat YITP and the Princeton University Physics Department for hospitality.
A Holographic renormalization
The prescription for calculating the expectation value of a gauge theory operator (cid:104)O Φ (cid:105) dualto a field Φ in the supergravity limit of string theory is to identify the generating functionof (cid:104)O Φ (cid:105) with the supergravity partition function [59, 60]. Sources for the operator (cid:104)O Φ (cid:105)
26n the gauge theory are identified with boundary values of Φ (modulo some issues with lowdimension operators [34].) In the saddle point approximation, calculating one point functionsreduces to varying the supergravity action with respect to the boundary values of the fieldsΦ near the AdS boundary. (cid:104)O Φ (cid:105) = lim r → (cid:112) g (0) ∂∂φ (0) S SUGRA . (92)Here g is defined through ds = L r ( dr + g mn dx m dx n ), and g (0) is the leading O ( r ) term ina series expansion of g . Similarly, φ (0) is the leading O ( r − ∆ ) term (with ∆ the dimensionof O Φ ) in a series expansion of Φ.At this point the prescription (92) is not well defined since generically (92) will give di-vergent results. Simply ignoring these divergences may lead to wrong Ward identities. Oneprescription for systematically dealing with these singularities is holographic renormalization[61, 62] (see [63] for a review). This prescription seems to work well even for non asymp-totically AdS backgrounds [64]. It comprises of adding a boundary action S ct to S SUGRA . S ct may be composed only of the boundary values of the field Φ, and is constructed so as toprecisely cancel the r → (cid:104) T mn (cid:105) = lim r → √ g ∂∂g mn ( S SUGRA + S ct ) (93)= T BYmn + X mn (94)where T BYmn is the Brown York stress tensor, which is what we would obtain by varying theon-shell Gibbons Hawking term in the action and simply discarding the singularities. Moreprecisely, we are interested in the contribution of the trailing string, or moving quark, to theenergy momentum tensor to linear order in κ , i.e. (cid:104) δT mn (cid:105) = δT BYmn + δX mn . (95)The purpose of this section is to show that δX mn is diagonal and to evaluate the off diagonalparts of (cid:104) δT mn (cid:105) in terms of the near boundary value of the metric fluctuations.Expanding the various scalar fields in a power series in r , we find φ I ( r ) = r − ∆ I ( φ I (0) + r φ I (2) + . . . ) + r ∆ I ( ¯ φ I (0) + . . . ) (96)27here ∆ I is the dimension of the operator dual to φ I . If ∆ I is an integer then additional log-arithmic terms will appear in the series expansion for φ I . Fields dual to irrelevant operatorshave ∆ I >
4. In that case the power series expansion in r is divergent at the boundary andthe sources φ I (0) should be taken to be infinitesimal, so that they vanish on-shell [60, 61].For clarity, we shall omit the index I , and reinsert it only when required. The coefficients φ ( n ) in (96) can be related to φ (0) by solving the equation of motion for φ order by order in r . The coefficients ¯ φ ( n ) are related to ¯ φ (0) in a similar manner. ¯ φ (0) and φ (0) are determinedthrough the boundary conditions. For the case at hand the response of the field to the stringis such that the leading near boundary contributions vanish. Using the same notation as in(14) φ → φ + δφ (97)we have δφ (0) = 0. Also, in our construction the background value of the fields φ dependonly on the radial coordinate, meaning that in the notation of (97) φ (0) is a constant. Asimilar analysis follows for a near boundary expansion of the metric g mn = g (0) mn + g (2) mn r + ¯ g (0) mn r + . . . . (98)Consider now the boundary counterterm action S ct . As discussed earlier it must be com-posed of the near boundary metric γ mn ( (cid:15) ) = L (cid:15) g mn ( (cid:15) ) and fields φ ( (cid:15) ), where (cid:15) is the radialcoordinate which will be taken to zero at the end of the calculation. Writing the metricdependence explicitly, we find S ct = 12 κ (cid:90) √ γL ct d x = 12 κ (cid:90) √ γ (cid:0) L Λ + RL R + γ ij L ij + O ( (cid:15) ) (cid:1) d x, (99)where the integrals are over a surface which is a distance r = (cid:15) from the asymptotically AdSboundary. In (99) we have used γ mn = (cid:15) − √ γ ∼ (cid:15) − γ mn ∼ (cid:15) R mnkl ∼ (cid:15) (100) R mn ∼ (cid:15) R ∼ (cid:15) (cid:3) ∼ (cid:15) (101)and we’ve also rewritten any (cid:3) φ I L I expressions as γ mn ∂ n φ I ∂ m L I .28e wish to evaluate the finite terms in2 (cid:112) g (0) ∂∂g (0) mn ( S ct + S b ) = lim (cid:15) → L κ (cid:15) √ γ ∂∂γ mn ( S ct + S b ) (102)= lim (cid:15) → L κ (cid:15) (cid:18) − γ mn ( L ct + L b ) + 2 ∂L ct ∂γ mn (cid:19) . (103)where S b = (cid:82) √ γL b d x is the on-shell boundary action which gets contributions from thevarious scalar fields.We start from the last expression in the parenthesis in (103). Using (99), the leadingcontributions to ∂L ct ∂γ mn which come from the trailing string will look like (cid:15) − ( δ ( R mn L R ) + δL mn + . . . ) . (104)The second term in (104) vanishes since it must involve two derivatives in a direction tangentto the radial AdS direction. One of these will always act on a background field φ (0) whichdepends only on the radial direction, giving ∂ i φ (0) = 0.Consider the first term in (104). L R and δL R are polynomials of the scalar fields with L R , δL R ∼ (cid:15) n with n ≥
0. (The inequality for n follows by considering (96). If n < L R , δL R ∼ (cid:15) n would imply a non vanishing source term for a non-normalizable operator.)So we can expand L R = (cid:80) s =0 L ( s ) R (cid:15) s and δL R = (cid:80) s =0 δL ( s ) R (cid:15) s . Similarly R ij = 0 for i (cid:54) = j because the background metric g is independent of the x i directions and δR mn = (cid:80) s =4 R ( s ) mn (cid:15) s since the boundary conditions on the response of the metric to the string are that the metricvanishes close to the boundary. This brings us to the conclusion that δ ( R ij L R ) ∼ (cid:15) so thefirst term in (104) will not contribute to the stress energy tensor through (99) and (103).Finally we consider contributions coming from γ mn ( L b + L ct ). Expanding γ mn = L (cid:15) (cid:0) g (0) mn + g (2) mn (cid:15) + ¯ g (0) mn (cid:15) + . . . (cid:1) and L ct + L b = L (0)Λ + L (0) b + O ( (cid:15) ), we see that any finite non diagonal contributions of (cid:15) γ ij ( L ct + L b ) to (cid:104) T mn (cid:105) can only come from ¯ g (4) mn (cid:16) L (0)Λ + L (0) b (cid:17) . But such expressions willalso contribute to a divergent O ( (cid:15) − ) term in √ γ ( L Λ + L b ) which will not be compensated by In [32] it was shown that even when the boundary conditions imply that the metric fluctuations vanishat the AdS boundary, the stringy source induces an O ( r ) k independent term in the series expansion forthe metric fluctuations. This O ( r ) contribution is apparently related to the infinite mass of the quark andcorresponds to a delta function at the location of the quark when going to real space. Regardless, it doesnot appear in the 1 /k expansion that we are considering here. N µ . (Outward pointingmeans pointing toward the conformal boundary.) The definition is T BYmn = K mn − (cid:18) K + 3 L (cid:19) g Σ mn , (105)where g Σ mn is the induced metric on Σ and K mn is its extrinsic curvature. This expressioncomes from varying the Gibbons Hawking term in terms of the boundary metric γ Σ mn . Theindices m and n run from 0 to 3 (along Σ) while µ runs over all five dimensions of theasymptotically AdS spacetime, whose radius of curvature near the boundary is L . In general, g Σ mn = g mn − N m N n K mn = − g mk ∇ k N n , (106)but in axial gauge, when Σ is a surface at fixed r , one has the simpler expressions g Σ mn = g mn K mn = 12 √ G rr ∂ r g mn . (107)The metric of the boundary theory and the holographic stress tensor must be computed ina coordinated way, because both are affected by conformal transformations. A consistentprescription is g (0) mn = lim r → α g mn (cid:104) T mn (cid:105) = 1 κ lim r → α T BYmn . (108)For the ansatz (6), one finds g (0) = η = diag {− , , , } and (cid:104) δT mn (cid:105) = lim r → α L ( H mn − η lk H lk η mn ) , (109)where we have used (15). 30 eferences [1] J. Casalderrey-Solana, E. V. Shuryak, and D. Teaney, “Conical flow induced byquenched QCD jets,” J. Phys. Conf. Ser. (2005) 22–31, hep-ph/0411315 .[2] S. S. Gubser, S. S. Pufu, and A. Yarom, “Sonic booms and diffusion wakes generatedby a heavy quark in thermal AdS/CFT,” arXiv:0706.4307 [hep-th] .[3] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, and L. G. Yaffe, “Energy loss of aheavy quark moving through N = 4 supersymmetric Yang-Mills plasma,” hep-th/0605158 .[4] S. S. Gubser, “Drag force in AdS/CFT,” hep-th/0605182 .[5] J. J. Friess, S. S. Gubser, and G. Michalogiorgakis, “Dissipation from a heavy quarkmoving through N = 4 super- Yang-Mills plasma,” JHEP (2006) 072, hep-th/0605292 .[6] A. Yarom, “The high momentum behavior of a quark wake,” Phys. Rev.
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