Energy deposited by a quark moving in an N � 4 super Yang-Mills plasma

Amos YaromLudwig-Maximilians-Universitat, Department fur Physik, Theresienstrasse 37, 80333 Munchen, Germany*

(Received 26 March 2007; published 31 May 2007)

We evaluate the energy-momentum tensor of a massive quark as it moves through an N � 4 SYMquark-gluon plasma at constant velocity. We find that in the near-quark region, where the dynamics isexpected to be dominated by dissipative behavior, the energy density may be quantitatively characterizedby a transient at velocities above the speed of sound of the plasma.

DOI: 10.1103/PhysRevD.75.105023 PACS numbers: 12.38.Mh, 11.25.Tq

I. INTRODUCTION AND SUMMARY

The Relativistic Heavy Ion Collider (RHIC) at theBrookhaven National Laboratory [1– 4] (see [5] for a re-view) probes the dynamics of an expanding quark-gluonplasma (QGP) at temperatures slightly above the QCDdeconfinement temperature TQCD � 170 MeV. One of theprominent features of the QGP at RHIC is the quenching(lack of energy) of hadronic jets emitted from the plasma.It is understood that the plasma acts as a dissipative me-dium for the partons moving through it. The exact mecha-nism for the energy loss of the partons is unclear partlybecause the coupling constant gYM is quite large andcertain features of the QGP are not accessible perturba-tively, and partly due to the difficulty in modeling theinteraction of a parton with the surrounding quark-gluonplasma.

One approach which overcomes these difficulties is toconsider, instead, a quark in a strongly coupled N � 4large N SU�N� SYM plasma. While there are severaldifferences between the former and a QCD plasma, oneof the lessons that has been learned over the past year isthat many features of an N � 4 SYM plasma roughlycoincide with those of the QGP at RHIC. A comparisonincludes the energy density, the viscosity to entropy densityratio [6,7], the friction coefficient for the moving quark[8,9] and the jet quenching parameter [10,11]. One mayextend the analysis to include theories which break con-formal invariance or N � 4 supersymmetry. For example[12,13] study the viscosity to entropy density ratio in anonconformal theory and a theory with massive quarks inthe fundamental representation. The diffusion coefficientfor various deformations of the superconformal N � 4theory have been calculated and analyzed in [14–22].Similar investigations of the jet quenching parameterhave been carried out in [23–30]. Some other features of

the quark-gluon plasma in N � 4 SYM and related theo-ries have been studied in [31–41].

This is not to say that N � 4 SYM theory is a goodapproximation to QCD. While there is qualitative agree-ment between the two, on the quantitative level there is theexpected mismatch (see [11,42] for some discussions).Here, we shall adopt a somewhat conservative approachand try to understand the characteristics of the energydeposition of a moving quark in an N � 4 SYM plasmaas a first step towards understanding quark-QGP dynamics.While being motivated by the comparisons between N �4 SYM and the results of the RHIC experiment listedabove, whether our analysis is actually related to QCD ornot remains somewhat speculative.

To study the energy associated with a quark we shallevaluate the near-quark energy-momentum tensor as thequark travels through a strongly coupled QGP at constantvelocity. This may be done by appealing to the AdS/CFTduality [43]. There, one may evaluate the energy-momentum tensor of a heavy quark in N � 4 SYM atfinite temperature by considering metric fluctuations due toa string hanging down from the boundary of an AdS-Schwarzschild (AdS-SS) background geometry [39,44,45].

We are interested in the temperature dependent contri-bution of the quark motion to the energy-momentum ten-sor. Apart from this subleading term, the energy-momentum tensor will also be influenced by the leadingzero temperature contribution of the moving quark,1 andby the QGP itself. Our main result is the real space value ofthe energy-momentum tensor hT��ijd (the ‘‘d‘‘ is a re-minder that the energy-momentum tensor of the conformalplasma and the leading contribution from the quark motionat zero temperature have been subtracted) which is valid inthe vicinity of the quark

*Electronic address: yarom@theorie.physik.uni-muenchen.de

1The value of the leading zero temperature contribution of themoving quark to the energy-momentum tensor may be obtainedup to an overall constant by imposing conformal and Poincaresymmetry [39]. We reproduce this result in Eq. (31).

PHYSICAL REVIEW D 75, 105023 (2007)

1550-7998=2007=75(10)=105023(10) 105023-1 © 2007 The American Physical Society

��������������1� v2p

�����pT2hTttijd �

v�r2��5� 13v2 � 8v4� � ��5� 11v2�x2��x�

72�r2�1� v2� � x2��

5=2; (1a)��������������

1� v2p

�����pT2hTtx�ijd � �

v2�2x2� � �1� v2�r2�x�

24�r2�1� v2� � x2��

5=2; (1b)��������������

1� v2p

�����pT2hTtrijd � �

�1� v2�v2�11x2� � 8r2�1� v2��r

72�r2�1� v2� � x2��

5=2; (1c)��������������

1� v2p

�����pT2hTx�x�ijd �

v�r2�8� 13v2 � 5v4� � �11� 5v2�x2��x�

72�r2�1� v2� � x2��

5=2; (1d)��������������

1� v2p

�����pT2hTx�rijd �

v�1� v2��8r2�1� v2� � 11x2��r

72�r2�1� v2� � x2��

5=2; (1e)��������������

1� v2p

�����pT2hTrrijd � �

v�1� v2��5r2�1� v2� � 8x2��x�

72�r2�1� v2� � x2��

5=2; (1f)��������������

1� v2p

�����pT2hT��ijd � �

v�1� v2�x�9�r2�1� v2� � x2

��3=2: (1g)

In Eqs. (1) O�T4� corrections to hT��ijd have been omittedand contributions with delta-function support wereignored.

Our notation is such that x� � x� vt is the comovingcoordinate of the quark, T is the temperature of the quark-gluon plasma, v the velocity of the quark (in the x direc-tion), r is the direction transverse to the quark motion, � isthe polar angle and � is the large N ’t Hooft couplingconstant.

An analysis of hT��i was initiated in [39] where thelarge distance behavior of the energy-momentum tensorwas discussed. There it was shown that if the quark movesfaster than the speed of sound, v2 � 1=3, a Mach cone willbe created far from the moving quark. This fits nicely witha hydrodynamic model [46] (valid at large distances aswell) for the sound waves carried by the plasma as a resultof the quark motion. As argued in [46] when one getscloser to the moving quark he or she will reach a regimewhere hydrodynamics becomes nonlinear and turbulenteffects are expected. In the immediate vicinity of the quarkthere is a region where the hydrodynamic approximationbreaks down completely and strong dissipative effectsdominate the dynamics. This is precisely the regiondescribed by Eqs. (1). It is nice to see that the AdS/CFTduality provides for a method of probing the near-quark surroundings which is not accessible viastandard constructions due to strong dissipative behavior(see [46]) and the lack of a model for the relevantinteractions.

Focusing on the energy density hTtt�x�; r�ijd we find thatit exhibits some velocity dependent directional features(see Fig. 1) which do not appear for the other componentsof the energy-momentum tensor. As long as the velocity isrelatively low, 0< v2 < 5=13, the energy density has avanishing gradient in the x� direction along

tan 2!1 �

�rx�

�2

�5� v2 �

������15p ���������������������������������������

15� 50v2 � 47v4p

�20� 44v2 �1� v2�:

In this range of velocities no exceptional behavior is ob-served even when the quark passes the speed of soundv2 � 1=3. As mentioned earlier, this should be contrastedwith the Mach cone which is formed far from the quark asit passes the speed of sound of the conformal plasma[39,46].

At velocities squared above 5=13, the subleading dis-sipative contribution to the energy density begins develop-ing lobe-like features specified by a nontrivial, vanishinggradient in the r direction at an angle of

tan 2!2 �

�rx�

�2�

5� 13v2 � 8v4

�5� 13v2 :

As the velocity squared of the quark is increased to 5=11one finds that there are two contributions to hTttijd, theprevious contribution now deformed to a lobe of highenergy, and a new low energy contribution right behindthe quark.2 This new contribution is directed at an angle of

tan 2!3 �

�rx�

�2

�5� v2 �

������15p ���������������������������������������

15� 50v2 � 47v4p

�20� 44v2 �1� v2�:

2The terms high and low energy are, of course, relative to theleading zero temperature contribution of the quark and thecontribution of the plasma which have been subtracted. Wewould like to thank the authors of [47] for clearing up a signmixup in this context.

AMOS YAROM PHYSICAL REVIEW D 75, 105023 (2007)

105023-2

As the velocity is increased further, the energy deficitbehind the quark dominates and the contribution of thelobe becomes smaller until it vanishes at a velocity squaredof 5=8. We encourage the reader to have a look at [48]where this effect can be observed as a continuous functionof the velocity.

This behavior may suggest that there are two mecha-nisms that come into play. One at relatively low velocitiesv2 < 5=13, responsible for a region of high energy, and oneat high velocities v2 > 5=8, responsible for a region ofenergy depletion, with some overlap. It is interesting tonote that the overlap region is above the Mach velocity forthe quark, and that there is no remnant of the Mach conediscussed in [39].3 A different possibility is that in Fig. 1we are seeing an interference pattern of some sort. Thoughthen it is not clear what sort of waves are propagating in the

plasma since sound waves are not expected at such highmomentum scales and the gluon field strength does notexhibit lobe-like behavior at short distances [41]. It isperhaps worth mentioning that a region of energy defi-ciency behind the quark is characteristic of a Landau-Pomeranchuk-Migdal (LPM) effect [49,50] where the par-ton motion in the plasma is modeled by inelastic collisionswith fixed scatterers. The literature contains many moreinteresting discussions regarding the mechanism for en-ergy loss in a QGP (see for example [46,51–53]) though wecould not find quantitative correlations between the de-scriptions there and the behavior described above.

While contemplating over these issues, the authors of[47] brought to my attention that their work has someoverlap with this paper. I thank them for sharing theirresults with me prior to publication.

II. EVALUATION OF hT��i

We wish to evaluate the expectation value of the energy-momentum tensor in response to the motion of a quark in a

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 514

H

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 513

H

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 25

H

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 512

H

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 1023

H

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 511

H

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 1021

H

L-14 -12 -10 -8 -6 -4 -2 0

0

5

10

15

20

x 1 T

r 1 T v2 12

H

L

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 59

HL

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 1017

L

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 58

L

-14 -12 -10 -8 -6 -4 -2 00

5

10

15

20

x 1 T

r 1 T v2 57

L

FIG. 1 (color online). Contour plots of the leading dissipative contribution to the energy density 1����pT4 hTtt�x�; r�ijd due to the motion

of a quark moving at constant velocity v. The horizontal axis corresponds to the comoving coordinate of the quark, x� � x� vt, whilethe vertical one corresponds to the direction transverse to the quark motion, r. Large positive values are marked with an ‘‘H‘‘ whilelower values are marked ‘‘L‘‘. See [48] for an animation.

3Note however, that when the Mach cone is generated there isa buildup of energy in its vicinity (see Fig. 2, and Eq. (80) of[39]) which may extend all the way down to the near-quarkregion at the intermediate velocity range 5=11< v2 < 5=8.

ENERGY DEPOSITED BY A QUARK MOVING IN AN . . . PHYSICAL REVIEW D 75, 105023 (2007)

105023-3

thermal plasma in N � 4 SYM. Using the AdS/CFTdictionary this is given by the response of the AdS-SSmetric to the motion of a string with one endpoint locatedon the AdS boundary, tracing the worldline of the quark.

Ignoring for the moment the effect of the string on thespacetime geometry, one can find the string’s profile byminimizing the Nambu Goto action

SNG � �1

2��0Z ���������wp

d�d�

in an AdS-SS background. Hence, w is the induced metricw� � G�0���@�X

�@X� with G�0� the metric of an AdS-SS

black hole; G�0� may be read off of the line element

ds2 �L2

z2

��g�z�dt2 � dx2 �

Xi�2;3

dx2i �

dz2

g�z�

�;

with g�z� � 1� � zz0�4. The coordinate z runs from 0 at the

asymptotically AdS boundary to z0 at the black hole hori-zon. We have set the value of the dilaton to its trivialbackground value since it decouples from the equation ofmotion at the linearized order that we are considering.

The equations of motion for the embedding functions ofthe string X���; �� are supplemented by boundary condi-tions which require that the string endpoint moves atconstant velocity v, and that no energy is fed to the stringfrom the horizon. Using a static gauge and an ansatzX���; �� � ��; ��� � vt; 0; 0; �� one finds [8,9]

�z� �vz0

4

�ln

1� zz0

1� zz0

� 2 arctanzz0

�: (2)

The differential equation describing the combined dy-namics of the string and the target space metric (within thesupergravity approximation) can be found by minimizingthe action

S � SEH � SNG; (3)

where

SEH �1

16�G5

Z ���������Gp �

R�12

L2

�d5x: (4)

Since we will be looking for the expectation value of theenergy-momentum tensor, it will be enough to consider thelinearized equations of motion for the metric fluctuationsh, defined through G � G�0� � h, in the presence of thestring whose profile is determined by Eq. (2).

The linearized equations for h resulting from minimiz-ing (3) may be written in condensed form as

D ����h�� � T��; (5)

with D���� a second order linear differential operator. Theboundary conditions for the metric fluctuations are thatthey vanish at the AdS boundary (implying that there isno source term for the metric) and that there are no out-

going modes coming from the horizon. Once a solution, h,to Eq. (5) is obtained the energy-momentum tensor in theSYM theory hTi � hTplasmai � hTquarki may be determinedthrough [39]

hTquarki �1

�

���������������

1� v2

sQ; (6)

where Q�� is obtained from h � . . .�Qz4 � . . . througha small z expansion. hTplasmai ��2

8 N2T4diagonal�3; 1; 1; 1� �O�N0� is the contribution of

the thermal plasma to the energy-momentum tensor [54].Since we are interested in the dissipative effects of thequark motion, we subtract this background value and so,for our purposes it may be ignored.

In Eq. (6) one should use the relations

N2 ��L3

2G5;

1

�T� z0 and

�����p�L2

�0: (7)

Apart from that, one should also make sure that all lowerorder terms in the expansion of h in small values of zcorrespond to contact terms (and therefore may be renor-malized away).4)

In [39] it was shown that by choosing a gauge such thath5� � 0 and working in cylindrical coordinates, thecoupled equations of motion for the various componentsof the metric tensor decouple into three sets of secondorder equations and two sets of first order constraints.Explicitly, we Fourier transform the metric fluctuations,

h�k; z� �Zh���x; z�e�i�k�x��k2x2�k3x3�d3k;

and define

h ���k�; k?; z� �4G5L

�0��������������1� v2p

1

z2 H���k�; k?; z�; (8)

where we have also rotated h to a cylindrical coordinatesystem. By considering the combinations

A �1

2v2

��H11 � 2

k�k?H12 �

�k�k?

�2H22 �

�kk?

�2H33

�;

(9)

~D �D1

D2

� ��

12v �H01 �

k�k?H02�

12v2 ��H11 � �

k�k?� k?

k��H12 �H22�

0@ 1A(10)

and

4A holographic renormalization prescription was presented in[55,56], see also [57] for some recent applications. If one iscareful when taking the limit where the AdS radial coordinatereaches the asymptotically AdS boundary then a subtraction ofdivergent contact terms will give the same results. See forexample [58–61].

AMOS YAROM PHYSICAL REVIEW D 75, 105023 (2007)

105023-4

~E �

E1

E2

E3

E4

0BBB@1CCCA �

12 ��

3gH00 �H11 �H22 �H33�

12v �H01 �

k?k�H02�

12 �H11 �H22 �H33�

14 �H11�2� 6�k�k �

2� �H22��4� 6�k�k �2� � 2H33 � 12 k�k?

k2 H12�

0BBBB@1CCCCA; (11)

one finds that (5) reduces to

�I@2

z �

��

3

zI �

g0

gKX

�@z � VX�z�

�~X �

zge�ik��z� ~SX�z�;

(12)

� ~KX�z�@z � ~VX�z�@z� ~X � ze�ik��z� ~~SX�z�; (13)

with ~X � ~E; ~D or A and g0 � @zg�z�. I is the identitymatrix in appropriate dimensions. The matricesKX; VX; ~KX and ~VX and the vectors SX and ~~SX can be foundin [39] and have been reproduced in Appendix A forcompleteness. An important feature of Eq. (12) that wewill use is that in the z0 ! 1 limit,5 g0

g KX ! 0.Because of the linear relations between ~E, ~D, A and H,

Eqs. (9) to (11), the fourth order contribution to a small zexpansion of h�� can be inferred from the fourth ordercoefficient in an expansion of the fields X. Defining X �NXz�OXz2 � 1

3PXz3 �QXz4 �O�z5� (recall that h�0� �

0), we find that NX � OX � 0 and that the first orderconstraints (13) imply that

PA � PD1� PD2

� PE2� �PE3

� 1; (14)

PE1� 1� v2; PE4

� v2

�3k2�

k2 � 1�: (15)

Inverting the relations between the A, ~D, ~E variables andthe H variables, Eqs. (9) to (11), one may obtain the thirdorder term P�� in the small z expansion of h, h�� �� 1

3P��z3 �Q��z

4 �O�z5�,

P �

4� 2v2 �6v 0 0�6v 2� 4v2 0 0

0 0 2� 2v2 00 0 0 2� 2v2

0BBB@1CCCA:

Since P is independent of the momenta, any divergences

that it may cause will be contact terms. This allows us touse Eq. (6).

The constraint Eqs. (13) also relate the values of QX toone another. One finds

QD2� QD1

�1

4ik�vz20

; (16a)

QE2�

1

2QE1�

v

4ik�z20

; QE3� �

1

2QE1

; (16b)

QE4�

1

2

��1� 3v2 k

2�

k2

�QE1�

3ik�v�1� v2�

4k2z20

: (16c)

Therefore, we need only findQE1,QD1

andQA to obtain allthe fourth order terms in an expansion of X. Then, asdiscussed in the previous paragraph, one may invertEqs. (9) to (11) to obtain the fourth order term Q�� in anexpansion of h��. Plugging it into Eq. (6) we will obtainthe Fourier transformed energy-momentum tensor we arelooking for. To make our notation slightly more precise, weshould have used Q�� for the fourth order term in theexpansion of h instead of Q�� which was defined belowEq. (6) to be its real space counterpart. However, since inthe rest of this section we will be working in Fourier space,we choose an unhatted notation which is less cumbersome.

To proceed, we note that the equations of motion (12)have the property that in the z0 ! 1 limit, the kineticterms become diagonal and correspond to the kinetic termsobtained for a scalar field in an AdS background. Thisimplies that in the z0 ! 1 limit it is more useful to changeto a basis where the potential is diagonal as well. Ourapproach to solve (12) and (13) is to first redefine thevariables ~D and ~E so that in the z0 ! 1 limit we shallget the AdS equations of motion. To do this, we multiplythe matrix Eq. (12) for X � ~D from the left by

dD �

���2p��2 � 1� �

���2p�2

���2 � 1������������������������4

��2�1�2� 1

q��2 � 1�

�����������������������4

��2�1�2� 1

q0@ 1A (17)

and for X � ~E by

5One should use the dimensionless quantity ~kz0 instead of z0 (with ~k some scale with units of inverse length) when considering anexpansion at ‘‘large’’ values of z0. At the moment, we shall refrain from writing this out explicitly to avoid cluttering the notation.

ENERGY DEPOSITED BY A QUARK MOVING IN AN . . . PHYSICAL REVIEW D 75, 105023 (2007)

105023-5

dE �

29 ��

2 � 1��2�2 � 1� � 43�

2�2�2 � 1� � 49 ��

2 � 1��2�2 � 1� 19 �1� 2�2�2

0 1 1 0� 2

3 ��2 � 1� 4�4�4�2�3

�2�143 ��

2 � 1� 13 �1� 2�2�

23 ��

2 � 1� �4�2 23 �1� 2�2� 2

3 ��2 � 1�

0BBB@1CCCA: (18)

In Eqs. (17) and (18) we have defined � � v k�~k

with ~k2 � k2? � �1� v

2�k2�.

This will diagonalize the equations of motion in the z0 ! 1 or g! 1, g0 ! 0 limit. Actually the equation of motion for~E cannot be diagonalized. Instead we have brought it into Jordan form. The precise equations of motion one obtains are

rather messy, and we shall not write them out explicitly. What is important, is that in our new basis we are assured that the‘‘off diagonal’’ terms are, by construction, at least of order z�4

0 . Therefore, we may first solve the diagonal equations andthen use perturbation theory to find corrections. These will be at least of order z�4

0 and will be neglected. Changing to themore convenient variables Z � ~kz, Z0 � ~kz0, we can concentrate on the following three equations

�@2Z �

��

3

Z�

4Z3

Z40 � Z

4

�@Z �

Z40�Z

40 � Z

4��2 � 1��

�Z4 � Z40�

2

�A0 �

Ze�ik��Z=~k�

g�Z�~k; (19)

�@2Z �

��

3

Z� 4

Z3�2

Z40 � Z

4

�@Z �

Z40�Z

40 � Z

4��2 � 1��

�Z4 � Z40�

2

�Da �

���2pZe�ik��Z=~k�

g�Z�~k; (20)

�@2Z �

��

3

Z�

4Z3�4�2��2 � 1� � 3�

3�Z4 � Z40�

�@Z �

�Z04�Z04� Z4��2� 1��

�Z4 � Z40�

2

��Ea

�Ze�ik��Z=~k�

g�Z�~k

��2�2 � 1����2 � 1�v2 � �2 � 2�

3��2 � 1��

2v2Z4��2 � 1��2�2 � 1�

3�Z4 � Z40�

�; (21)

where

~k�2Da ����2p�D1��2 � 1� �D2�2�; (22)

~k�2Ea2�2 � 1

� �4

3E2�

2 �1

9E4�2�

2 � 1�

�2

9E1��2 � 1� �

4

9E3��2 � 1�; (23)

~k�2A0 � A: (24)

The Da and Ea equations are the first diagonal componentsof (12) after multiplying it from the left by dD, Eq. (17), orby dE, Eq. (18), respectively.

The A0 equation of motion is identical to the equation ofmotion for the dilaton in an AdS-SS black hole backgroundas it is sourced by the trailing string [38,39,41,62]. In [41]the large momentum asymptotics of Eq. (19) were foundand the near-quark value of the field strength,�TrF2, wasevaluated. The WKB method was used to solve the homo-geneous equation at large momenta; the solution to thenonhomogeneous equation then follows by constructingthe Green’s function from the homogeneous solutions.

Alternatively, to obtain the large Z0 asymptotics of A0

one may expand the equation of motion (19) and thefunction A0�Z� in an inverse power series in Z0. The uniquenormalizable solution to order Z�2

0 is given by

A0�Z� ��Z

2~k�Z�I2�Z� � L0�Z�� � 2L1�Z�� �

i�

3Z20

Z4;

(25)

where I2 is a modified Bessel function of the first kind andLi are modified Struve functions of order i. A caveat in thisapproach is that the boundary conditions are imposed at Z0

which is the perturbative parameter we are expanding in.To properly impose the boundary conditions on genericsolutions one should resum the whole power series.Nonetheless, the solution obtained in this somewhathand-waving manner coincides with the one obtained us-ing the WKB method of [41]. In both cases, one finds thatfor

A0 � . . .�QA0Z4 � . . . ; (26)

QA0 is given by

~kQA0 �1

16��

i�

3Z20

�O�Z�40 �: (27a)

In order to find the O�Z4� coefficients QX in the small Zexpansion of X � Da and X � Ea (in analogy withEq. (26)), one needs to solve the equations of motion forDa and Ea, (20) and (21). Following the somewhat heu-ristic approach used to obtain (27a), one finds that anexpansion of Da, Ea and their equations of motion in aninverse power series in Z0 leads to equations of motionwhich are identical to Eq. (19) for A0 up to a different

AMOS YAROM PHYSICAL REVIEW D 75, 105023 (2007)

105023-6

normalization of the source terms. Using Eqs. (25) and(27a), one may easily read off the coefficients

QDa�

���2pQA0 ; (27b)

QEa �

��2�2 � 1���2�v2 � 1� � �v2 � 2��

3��2 � 1�

�QA0 ; (27c)

where terms of order O�Z�40 � have been neglected. A more

careful analysis following [41] which leads to the sameresult has been left to Appendix B. There we also find thatthe regime of validity of Eqs. (27) is given by large Z0 and�< Z2=3

0 .Putting together (16), (22), (23), and (27) we find

QA �1

16�~k�

i�

3~kz20

; (28)

QD1� QA �

i�

4~kz20

; (29)

QE1�

2

3��2� v2� � �1� v2��2�QA

�i�~kz2

0

�1

6�1� 5v2� �

1

3�2�1� v2�

�: (30)

As discussed earlier, we may now find the z4 contribu-tion Q�� to the metric fluctuations, h��, due to the stringby inverting the relations (9)–(11). Plugging the resultingQ�� into (6) one finds that the contribution of the quarkmotion to the Fourier transform of the energy-momentum

tensor���������1�v2

�

qhTquark��i �O��Tk�

4� is given by

~k24

��2� v2 � �1� v2� v2k2�

~k2� 3v� �1�v

2�vk2�

~k2 � �1�v2�vk?k�~k2 0

3v� �1�v2�vk2

�~k2 ��1� 2v2� � �1�v

2�k2�

~k2�1�v2�k?k�

~k20

� �1�v2�vk?k�~k2

�1�v2�k?k�~k2 � �1�v

2�2k2�

~k2 0

0 0 0 ��1� v2�

0BBBBB@

1CCCCCA

� iv�k�

�T~k

�2

� 13v�

19 �1� v

2�v k2�

~k2k?v2k�� k�k?�1�v2�v

9~k2 013v�

19 �1� v

2�v k2�

~k2 pk� � k?2k�� 1

9 �1� v2� k�k?

k2 0k?v2k�� k�k?�1�v2�v

9~k2 � k?2k�� 1

9 �1� v2� k�k?k2 pk? 0

0 0 0 p�

0BBBBB@

1CCCCCA;(31)

where

� �1

18�5� 11v2� �

1

9�1� v2�v2 k

2�

~k2; (32)

pk� �1

18�5v2 � 11� �

1

9�1� v2�

k2�

~k2; (33)

pk? �1

2�1� v2� �

1

9�1� v2�2

k2�

~k2; (34)

p� �7

18�1� v2�: (35)

It is satisfying to see that the leading T � 0 contributiondue to the quark motion is exactly what was predicted in[39] based on conformal invariance and Poincare symme-try. Fourier transforming the leading dissipative contribu-tion to position space, we obtain Eq. (1).

ACKNOWLEDGMENTS

I would like to thank O. Aharony, R. Brustein,A. Buchel, J. Friess, S. Gubser, G. Michalogiorgakis,S. Pufu and U. Wiedemann for useful correspondence

and comments and M. Haack for many interesting discus-sions. I am supported in part by the German ScienceFoundation and by the Minerva Foundation.

APPENDIX A: THE EQUATIONS OF MOTION FORTHE ENERGY-MOMENTUM TENSOR

In order to decouple the components of Eq. (5), we havedefined in Eqs. (9) through (11) the variables A, ~D and ~E.The resulting equations of motion were given in (12) and(13). The matrices KX, VX, ~KX and ~VX and the vectors ~SXand ~~SX in these equations are given by

KA � 1; VA �k2

g2

�v2

�k�k

�2� g

�; SA � 1;

(A1)

KD �0 00 1

� �; VD �

k2

g2

�g gv2�k1

k �2

�1 v2�k1

k �2

!;

~SD �11

� �;

(A2)

ENERGY DEPOSITED BY A QUARK MOVING IN AN . . . PHYSICAL REVIEW D 75, 105023 (2007)

105023-7

KE �

32 0 0 0

0 0 0 0

0 0 12 0

0 0 0 1

0BBBBB@

1CCCCCA; ~SE �

1� v2

g

1

�1� v2 � v2

g

v2��1� 3�k�k �2�

0BBBBBB@

1CCCCCCA; (A3)

VE �k2

3g2

�2g 12v2�k�k �2 6v2�k�k �

2 � 2g 00 0 2g g0 0 �2g �g

2g �12v2�k�k �2 0 3v2�k�k �

2 � g

0BBB@

1CCCA; (A4)

~K D �0 01 �g

� �; ~~SD �

0z2

ivz20k�

!; ~VD � 0; (A5)

~K E �0 1 1 0�g 0 �3v2�k�k �

2 � g �gg 0 2 0

0@ 1A; ~~SE �

ivz2

k�z20g

� 3ivk�z2

k2z20�v

2

g � 1�

z�1� v2

g �

0BBB@1CCCA; (A6)

~V E �1

6g

0 �6g0 �3g0 0

�3gg0 18v2 k2�

k2 g0 3�3v2 k2�

k2 � g�g0 0

2k2gz �12k2�v2z �2�3v2k2

� � gk2�z 2k2gz

0B@1CA: (A7)

APPENDIX B: WKB APPROXIMATION

In this section we would like to solve the equations ofmotion (20) and (21) with boundary conditions such thatDa�0� � Ea�0� � 0 and that Da and Ea have only ingoingmodes at the horizon. In effect, we are not interested in thefull solutions to the equations of motion, but rather in thefourth order terms in a series expansion of Da and Ea insmall Z.

This problem has been addressed in [41] for the relatedEq. (19). As discussed in the text, the latter equation ofmotion has been solved by first obtaining the homogeneoussolutions and then using them to construct the appropriateGreen’s function. Here we shall follow the same method,using some of the arguments made in [41].

First, we shall find the homogeneous solutions toEqs. (20) and (21) by bringing them to Schrodinger form,

00 � V � 0. This may be done by defining X ����������bX�Z�Z3

qX with X � Ed, Db, and

bDa�Z� � g�Z��

2; (B1a)

bEa�Z� � g�Z��1��4=3���1��2�; (B1b)

and dividing the resulting equation by���������Z3

bX�Z�

qg�Z�. The

Schrodinger equations obtained this way are characterizedby the potentials

VDb�Z� � V0 �

4��2 � 1�2� ZZ0�6

Z20g�Z�

2 ; (B2a)

VEd�Z� � V0 �64�4��2 � 1�2� ZZ0

�6

9Z20g�Z�

2 ; (B2b)

where

V0 � �15

4Z2 �1� �1� �2�� ZZ0

�4

g�Z�2�

4� ZZ0�6

Z20g�Z�

2 : (B2c)

In Eq. (B2c), V0 is the Schrodinger potential correspondingto the equation of motion for a massless scalar field in anAdS-SS background.

In the notation of Eqs. (B2), it is clear that the VX’s differfrom V0 only in their near horizon asymptotics. In [41] itwas shown that as long as the Schrodinger potential de-scribing the asymptotically AdS system has a parametri-cally long flat region for which V�Z� ’ �1 followed by aV�Z�> 0 region near the horizon, then the O�Z4� coeffi-cient QX in a small Z expansion of X is given by the Z

AMOS YAROM PHYSICAL REVIEW D 75, 105023 (2007)

105023-8

independent terms in

1

16�~k�1 �

1

8

Z Z2=30

Z

JX�x�������������bX�x�

px

K2�x�dx; (B3)

where exponentially suppressed terms, e�Z2=30 , should be

consistently neglected. J�x� is the nonhomogeneous termin the equation of motion for X. In our case (Eqs. (20) and(21)) we find

JDa�Z� �

���2pJ0�Z�; (B4)

JEa�Z� ���2�2 � 1����2 � 1�v2 � �2 � 2�

3��2 � 1�

�2v2Z4��2 � 1��2�2 � 1�

3�Z4 � Z40�

�J0�Z�; (B5)

where

J0�Z� �Z~ke�ik��Z=~k�g�Z� (B6)

is the source term for the equation of motion for the A0

component of the metric (which coincides with the equa-tion of motion of the dilaton). b�x� is given in Eqs. (B1) andK2�x� is a modified Bessel function of the second kind.

Clearly, as long as �< Z2=30 the potentials (B2a) and

(B2b) satisfy the flatness and positivity requirements dis-cussed above. Therefore, one may use formula (B3) toobtain the fourth order coefficients of Da, Ea and A0. TheA0 case has been discussed in [41], where it was shown thatas long as Z0 <�2 one may expand the integral in (B3) in apower series in Z0 owing to the exponential decay of themodified Bessel function K2�x�. One finds ~kQA0 �

116��

i�3Z2

0�O�Z�4

0 �. To obtain the other two solutions, we note

that up to order O�Z�40 �, the expressions JDa

��������bDa

pand

JEa�������bEa

pdiffer from JA0

�������bA0p

� J0

����������g�Z�

pby a constant

multiplicative term. This leads us to Eqs. (27).

[1] I. Arsene et al. (BRAHMS), Nucl. Phys. A757, 1(2005).

[2] K. Adcox et al. (PHENIX), Nucl. Phys. A757, 184 (2005).[3] B. B. Back et al., Nucl. Phys. A757, 28 (2005).[4] J. Adams et al. (STAR), Nucl. Phys. A757, 102 (2005).[5] B. Muller and J. L. Nagle, Annu. Rev. Nucl. Part. Sci. 56,

93 (2006).[6] G. Policastro, D. T. Son, and A. O. Starinets, Phys. Rev.

Lett. 87, 081601 (2001).[7] P. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. Lett.

94, 111601 (2005).[8] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, and L. G.

Yaffe, J. High Energy Phys. 07 (2006) 013.[9] S. S. Gubser, Phys. Rev. D 74, 126005 (2006).

[10] H. Liu, K. Rajagopal, and U. A. Wiedemann, Phys. Rev.Lett. 97, 182301 (2006).

[11] H. Liu, K. Rajagopal, and A. Wiedemann, J. High EnergyPhys. 03 (2007) 066.

[12] A. Buchel, Phys. Rev. D 72, 106002 (2005).[13] D. Mateos, R. C. Myers, and R. M. Thomson, Phys. Rev.

Lett. 98, 101601 (2007).[14] J. Casalderrey-Solana and D. Teaney, Phys. Rev. D 74,

085012 (2006).[15] C. P. Herzog, J. High Energy Phys. 09 (2006) 032.[16] E. Caceres and A. Guijosa, J. High Energy Phys. 11 (2006)

077.[17] E. Caceres and A. Guijosa, J. High Energy Phys. 12 (2006)

068.[18] M. Chernicoff and A. Guijosa, J. High Energy Phys. 02

(2007) 084.[19] M. Chernicoff, J. A. Garcia, and A. Guijosa, J. High

Energy Phys. 09 (2006) 068.

[20] P. C. Argyres, M. Edalati, and J. F. Vazquez-Poritz, J. HighEnergy Phys. 01 (2007) 105.

[21] J. F. Vazquez-Poritz, arXiv:hep-th/0605296.[22] G. Bertoldi, F. Bigazzi, A. L. Cotrone, and J. D. Edelstein,

arXiv:hep-th/0702225.[23] A. Buchel, Phys. Rev. D 74, 046006 (2006).[24] F.-L. Lin and T. Matsuo, Phys. Lett. B 641, 45 (2006).[25] S. D. Avramis and K. Sfetsos, J. High Energy Phys. 01

(2007) 065.[26] N. Armesto, J. D. Edelstein, and J. Mas, J. High Energy

Phys. 09 (2006) 039.[27] S. D. Avramis, K. Sfetsos, and D. Zoakos, Phys. Rev. D 75,

025009 (2007).[28] Y.-h. Gao, W.-s. Xu, and D.-f. Zeng, arXiv:hep-th/

0611217.[29] S. S. Gubser, arXiv:hep-th/0612143.[30] P. C. Argyres, M. Edalati, and J. F. Vazquez-Poritz, J. High

Energy Phys. 04 (2007) 049.[31] K. Peeters, J. Sonnenschein, and M. Zamaklar, Phys. Rev.

D 74, 106008 (2006).[32] H. Liu, K. Rajagopal, and U. A. Wiedemann, Phys. Rev.

Lett. 98, 182301 (2007).[33] E. Caceres, M. Natsuume, and T. Okamura, J. High

Energy Phys. 10 (2006) 011.[34] A. Buchel, S. Deakin, P. Kerner, and J. T. Liu, arXiv:hep-

th/0701142.[35] D. Mateos, R. C. Myers, and R. M. Thomson, arXiv:hep-

th/0701132.[36] J. Casalderrey-Solana and D. Teaney, arXiv:hep-th/

0701123.[37] O. Aharony, S. Minwalla, and T. Wiseman, Classical

Quantum Gravity 23, 2171 (2006).

ENERGY DEPOSITED BY A QUARK MOVING IN AN . . . PHYSICAL REVIEW D 75, 105023 (2007)

105023-9

[38] J. J. Friess, S. S. Gubser, and G. Michalogiorgakis, J. HighEnergy Phys. 09, 072 (2006).

[39] J. J. Friess, S. S. Gubser, G. Michalogiorgakis, and S. S.Pufu, Phys. Rev. D 75, 106003 (2007).

[40] Y.-h. Gao, W.-s. Xu, and D.-f. Zeng, arXiv:hep-th/0606266.

[41] A. Yarom, arXiv:hep-th/0702164.[42] S. S. Gubser, arXiv:hep-th/0611272.[43] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).[44] V. Balasubramanian and P. Kraus, Commun. Math. Phys.

208, 413 (1999).[45] J. M. Maldacena, Phys. Rev. Lett. 80, 4859 (1998).[46] J. Casalderrey-Solana, E. V. Shuryak, and D. Teaney,

arXiv:hep-ph/0602183.[47] S. S. Gubser and S. S. Pufu, arXiv:hep-th/0703090.[48] See EPAPS Document No. E-PRVDAQ-75-041712 for an

animation. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.

[49] X.-N. Wang, M. Gyulassy, and M. Plumer, Phys. Rev. D51, 3436 (1995).

[50] M. Gyulassy, I. Vitev, X.-N. Wang, and B.-W. Zhang, inQuark-Gluon Plasma 3, edited by R. C. Hwa and X.-N.Wang (World Scientific, Singapore, 2004).

[51] A. Kovner and U. A. Wiedemann, arXiv:hep-ph/0304151.[52] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and

D. Schiff, Nucl. Phys. B483, 291 (1997).[53] A. Majumder and X.-N. Wang, Phys. Rev. C 73, 051901

(2006).[54] S. S. Gubser, I. R. Klebanov, and A. W. Peet, Phys. Rev. D

54, 3915 (1996).[55] M. Bianchi, D. Z. Freedman, and K. Skenderis, Nucl.

Phys. B631, 159 (2002).[56] M. Bianchi, D. Z. Freedman, and K. Skenderis, J. High

Energy Phys. 08 (2001) 041.[57] O. Aharony, A. Buchel, and A. Yarom, J. High Energy

Phys. 11 (2006) 069.[58] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys.

Lett. B 428, 105 (1998).[59] D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli,

Nucl. Phys. B546, 96 (1999).[60] I. R. Klebanov and E. Witten, Nucl. Phys. B556, 89

(1999).[61] O. Aharony, A. Buchel, and A. Yarom, Phys. Rev. D 72,

066003 (2005).[62] U. H. Danielsson, E. Keski-Vakkuri, and M. Kruczenski,

J. High Energy Phys. 01 (1999) 002.

AMOS YAROM PHYSICAL REVIEW D 75, 105023 (2007)

105023-10