AdS/CFT energy loss in time-dependent string configurations

Andrej Ficnar*

Department of Physics, Columbia University, New York, New York 10027, USA(Received 24 January 2012; published 28 August 2012)

We analyze spacetime momentum currents on a classical string world sheet, study their generic

connection via AdS/CFT correspondence to the instantaneous energy loss of the dual field theory degrees

of freedom and suggest a general formula for computing energy loss in a time-dependent string configu-

ration. Applying this formula to the case of falling strings, generally dual to light quarks, reveals that the

energy loss does not display a well-pronounced Bragg peak at late times, as previously believed. Finally, we

comment on the possible implications of this result to the jet quenching phenomena in heavy ion collisions.

DOI: 10.1103/PhysRevD.86.046010 PACS numbers: 11.25.Tq, 12.38.Mh

I. INTRODUCTION

Gauge/gravity duality has been a very insightful toolused to study many properties of strongly coupled non-Abelian plasmas, especially after many indications that thequark-gluon plasma created in heavy ion collisions at theRelativistic Heavy Ion Collider is a strongly coupled sys-tem [1]. The AdS/CFT correspondence [2–4] is a dualitybetween a (3þ 1)-dimensional N ¼ 4 SUðNcÞ super-Yang-Mills (SYM) gauge theory and type IIB string theoryon AdS5 � S5 spacetime. Using this conjecture and by tak-ing the limit Nc � � � 1, one can study this gauge theoryat strong coupling by studying classical, two-derivative(super)gravity. Due to phenomenological differences be-tween thermal N ¼ 4 SYM plasma and finite-temperatureQCD, it is important to consider gravity duals to nonconfor-mal gauge theories [5–12]. However, since the informationabout the medium (on the field theory side) is fully encodedin the spacetime metric (on the string side), in this work wewill keep our results as general as possible by considering ageneral metric G�� and only use the AdS5 metric for final

numerical evaluations.In recent years, an important application of the AdS/

CFT correspondence has been to study the phenomenon ofjet quenching [13–17] in strongly coupled systems, espe-cially after the pioneering work of Refs. [18,19], whostudied energy loss of heavy quarks in a strongly coupledN ¼ 4 SYM plasma. To study the plasma at a finitetemperature T, one introduces a black hole in the AdS5geometry with an event horizon at some radial coordinaterh, proportional to 1=T [3]. Then, one introduces degreesof freedom in the fundamental representation (‘‘quarks’’)in SYM by introducing a D7-brane in the AdS-BH (blackhole) geometry, which spans from r ¼ 0 (boundary) tosome r ¼ rm [20]; on the field theory side, this procedurecorresponds to the introduction of an N ¼ 2 hypermul-tiplet whose mass mQ is proportional to 1=rm. Dressedquarks are then dual to strings in the bulk with one orboth endpoints on the D7-brane and the physics of the

quark energy loss (on the field theory side) will be directlyrelated to the dynamics of these strings. In the large Nc andlarge � limit, one can neglect the backreaction of themetric due to the introduction of the string (the probeapproximation) and neglect the quantum corrections tothe motion of the strings. This means that we need to studythe motion of classical strings in the background given by aspacetime metric G��. For a review of AdS/CFT corre-

spondence and especially its applications to heavy ionphenomenology, the reader is referred to Refs. [21–23].Recently, in the light of Relativistic Heavy Ion Collider

results, as well as the new LHC results [24,25] on thesuppression of light hadrons in nucleus-nucleus (AA) colli-sions, a more consistent grasp on the energy loss of lightquarks in gauge/gravity duality has become necessary, inorder to be able to compute jet energy loss observables suchas the nuclear modification factor RAA and the elliptic flowparameter v2 and directly compare them to the experimentalresults [26]. If we wish to describe a strongly coupledmedium with light quarks, the D7-brane will fill the entireAdS-BH geometry. On this D7-branewe can then study openstrings whose endpoints source a D7 gauge field which inturn ‘‘induces’’ (in the sense of the field/operator correspon-dence) an image baryon density current in the field theory. Inother words, these open strings will represent dressed q �qpairs on the field theory side. Then the main idea, advocatedin Ref. [27], is that by studying the free motion of thesefalling strings, we can study the energy loss of light quarks.This application is just one example of the need to

understand the details of energy loss in (explicitly) time-dependent string configurations, such as the falling strings.The hope is that, by examining such configurations, we canmodel phenomena associated with the energy loss morerealistically, since in the quark gluon plasma formed inheavy ion collisions, quarks slow down and phenomenasuch as the instantaneous energy loss are expected todepend on the details of this nonstationary motion.Heavy quarks in such nonstationary situations have alreadybeen studied in Refs. [28,29].The authors of Ref. [27] have elegantly obtained one of

the first results for light quark energy loss in gauge/gravity*aficnar@phys.columbia.edu

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duality. They have shown, by analyzing null geodesics in theAdS-BH spacetime and relating them to the energy of thefalling string, that the maximum stopping distance of light

quarks scales with energy as�xmax � E1=3. A similar resultwas obtained in Ref. [30] for energy loss of adjoint degreesof freedom (‘‘gluons’’) in N ¼ 4 SYM plasma. However,we emphasize that this maximum stopping distance is not atypical stopping distance of light quarks. It is, as such, arather crude quantity, that can be used as a phenomenologi-cal guideline, but cannot be used to extract the instantaneousenergy loss that enters in, for example, calculations ofRAA orv2 observables. To study the instantaneous energy loss, oneneeds to analyze the spacetime momentum currents on thestring world sheet, which, as demonstrated in Ref. [27], incase of falling strings become nontrivial, time-dependentquantities. In this paper we will extend the analysis of theworld sheet currents from Ref. [27] and suggest a perhapsmore appropriate definition of the energy loss (in the sense ofits identification with particular current components), inwhich the details of the geometry on theworld sheet becomeimportant. This small, but crucial detail will lead to poten-tially important phenomenological consequences: aswewillsee, in particular, the instantaneous energy loss will notexhibit a well-pronounced late-time Bragg-like peak, aspreviously believed.

II. DYNAMICS OF CLASSICAL STRINGS

Let us start by considering a classical string propagatingin a five-dimensional spacetime described by the metricG��. The dynamics of the string is described by the

Polyakov action:

SP ¼ � 1

4��0Z

d2�ffiffiffiffiffiffiffi�h

phabð@aX�Þð@bX�ÞG��; (2.1)

where �0 ¼ l2s , the squared fundamental string length;�a ¼ ð�; �Þ are the coordinates on the string world sheet;X�ð�; �Þ are the spacetime coordinates of the string (theembedding functions); and hab is the world sheet metric,which is considered as a dynamical variable in this action[here h � detðhabÞ]. If we vary this action with respect tohab, we get

�ab ¼ 1

2habðhcd�cdÞ; (2.2)

where �ab ¼ G��@aX�@bX

� is the induced world sheet

metric. Plugging this equation of motion in the Polyakovaction (2.1), we obtain the usual Nambu-Goto string action,which means that these two actions are classicallyequivalent.

The Polyakov action can thus be viewed as a classicalfield theory action of five scalar fields X� on a curved two-dimensional manifold described by the metric hab. This isthe reason why we will be using that action instead of themuch more common Nambu-Goto action, since in thederivation of the energy loss formula it will be necessaryto consider coordinate transformations on the world sheet

and see how the world sheet vectors and tensors changeunder them. In practice, this action will also be useful fornumerical evaluations, since a clever choice of the worldsheet metric will greatly improve the stability of the nu-merics [27], as we will see later.The equations of motion for the X� fields from the

Polyakov action are

@a½ffiffiffiffiffiffiffi�h

phabG��@bX

��¼1

2

ffiffiffiffiffiffiffi�hp

habð@�G�Þ@aX�@bX:

(2.3)

The expression in the brackets on the LHS are just thecanonical momentum densities:

�a� � 1ffiffiffiffiffiffiffi�h

p SPð@aX�Þ ¼ � 1

2��0 habð@bX�ÞG��: (2.4)

This definition can differ by a factor offfiffiffiffiffiffiffi�h

pfrom the usual

definition of these momenta found in the literature, whichis here just to ensure that the quantity in (2.4) is a properworld sheet vector. From now on, we will assume that thespacetime metric is diagonal and we will consider only�’ssuch that the metric does not depend on X�. In that case,equations of motion (2.3) are in fact just the covariantconservation law for the momentum densities:

@a½ffiffiffiffiffiffiffi�h

p�a

�� ¼ffiffiffiffiffiffiffi�h

p ra�a� ¼ 0: (2.5)

In this case, the momentum densities are just the conservedNoether currents on the world sheet, associated with theinvariance of the action to the constant spacetime trans-lations X� ! X� þ ��. Due to this origin, these worldsheet currents describe the flow of the � component ofthe spacetime momentum of the string along the a direc-tion on the world sheet [31,32] and that is the reason whythey are important in the study of energy loss in the fieldtheory dual [33]. Because of that it is also necessary tounderstand how these currents transform under a genericchange of coordinates on the world sheet.A general coordinate transformation on the world sheet

ð�; �Þ ! ð~�; ~�Þ can be defined by the following matrix:

~Mab �

@~�a

@�b; Ma

b �@�a

@~�b¼ ð ~M�1Þab: (2.6)

The world sheet currents (2.4) and the world sheet metricthen transform as proper world sheet tensors:

~h ab ¼ McaM

dbhcd; (2.7)

~� a� ¼ ~Ma

b�b�: (2.8)

In practice, one solves the equations of motion (2.3) withconstraints (2.2) and an appropriate set of boundary con-ditions in some parametrization where the numerics arewell behaved (by choosing a convenient hab at the begin-ning of the calculation) and then, using formulas (2.7) and(2.8), transforms to some more ‘‘physical’’ coordinatesystem on the world sheet (for example � ¼ r, the radial

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AdS coordinate). From now on, we assume that, throughthis procedure, a specific parametrization ð�; tÞ has beenchosen, where t is the physical time (i.e., we are in thestatic gauge � ¼ t) and that � 2 ½0; �� for all t, i.e., the �coordinate parametrizes the string at some fixed time t.

III. WORLD SHEET CURRENTS

In general, whenever we have a covariant conservationlaw on a differentiable manifold, such as (2.5), one definesthe charge (whose flow is described by �a

�) that passes

through some hypersurface � as (following conventions inRef. [34])

p�� ¼ �

Z�?�� ¼ �

Z�d�na�

a�; (3.1)

where the second equation is written in terms of the one-form ��a ¼ hab�

b� and in the second equation, d� is the

induced volume element on the hypersurface� and na is theunit vector field normal to the hypersurface. In our case, wehave a two-dimensional manifold, and � represents an(open) curve on the world sheet and p�

� is the� componentof the spacetime momentum that flows through this curve.Here one should think of� as merely an index that denotesdifferent kinds of conserved currents on the world sheet.Therefore, in our case, we can simply write

p�� ¼ �

Z�dsnb�a

�hab; (3.2)

where ds is the line element of the curve �.In our choice of coordinates on the world sheet ð�; tÞ, we

can now ask what is the total � component of the momen-tum of the string at some fixed time t. This means that weneed to take the curve � to be the curve of constant t, whichmeans that its tangent vector ta is in the � direction, i.e.,ta ¼ ð1; 0Þ, where the first entry is the � coordinate and thesecond the t coordinate. The normal vector na ¼ ðn�; n�Þcan then be found by requiring its orthogonality to thetangent vector and the usual normalization condition:

t � n ¼ 0; n � n ¼ �1; (3.3)

where the dot products are taken with the world sheetmetric hab. Using these two equations, we can solve forthe components of the normal vector:

na ¼�� 1ffiffiffiffiffiffiffi�hp h��ffiffiffiffiffiffiffiffiffi

h��p ;

ffiffiffiffiffiffiffiffiffih��

pffiffiffiffiffiffiffi�h

p�: (3.4)

Finally, since for this particular curve we have d� ¼ 0, wecan express the line element ds as

ds2� ¼ habd�ad�b ¼ h��d�

2: (3.5)

Using (3.4) and (3.5) in (3.2), we have

p�ðtÞ ¼Z

d�ffiffiffiffiffiffiffi�h

p��

�ð�; tÞ: (3.6)

Note that this is true no matter what the parametrization ofthe string is. The only requirement here is that we are

dealing with a constant-t curve. Similarly, we can repeatthe same procedure for a constant-� curve, integrating oversome period of time:

p�ð�;�tÞ ¼Z�tdt

ffiffiffiffiffiffiffi�hp

���ð�; tÞ; (3.7)

which then gives the momentum that has flown down thestring (i.e., in the direction of increasing �) at position �during the time�t. Both (3.6) and (3.7) are the well-knownformulas that can be found in e.g., Refs. [18,19].Now consider an open string with free endpoint bound-

ary conditions:

���ð0; tÞ ¼ ��

�ð�; tÞ ¼ 0: (3.8)

Then, take a closed loop � on the string world sheet,composed of two constant-t curves at times t1 and t2, goingfrom� ¼ 0 to some chosen � ¼ ��, connected by the twocorresponding constant-� curves. Since the world sheetcurrents are conserved, we have, again following conven-tions in Ref. [34]:

I�?�� ¼ 0 ¼ �

Z t2

t1

dtffiffiffiffiffiffiffi�h

p��

�ð��; tÞ

þZ 0

��

d�ffiffiffiffiffiffiffi�h

p��

�ð�; t2Þ

�Z t1

t2

dtffiffiffiffiffiffiffi�h

p��

�ð0; tÞ

þZ ��

0d�

ffiffiffiffiffiffiffi�hp

���ð�; t1Þ: (3.9)

Due to the free endpoint boundary condition (3.8), the thirdterm on the RHS is zero, while the integrals over time,according to (3.6), represent the spacetime momentumof the part of the string between � ¼ 0 and � ¼ �� attimes t1 and t2:

p��� ðt2Þ � p��

� ðt1Þ ¼ �Z t2

t1

dtffiffiffiffiffiffiffi�h

p��

�ð��; tÞ: (3.10)

This equation clearly shows how the momentum of somepart of the string can change only if the ��

� component of

the world sheet current carries it away. The negative signon the RHS indicates that, for a positive ��

�, the momen-

tum of that part of the string will decrease, consistent withthe fact that this current component describes the flow ofthe momentum in the direction of increasing �, i.e., awayfrom the part of the string. Incidentally, if we take a stringconfiguration which is symmetric around �� ¼ �=2, then,due to this symmetry,��

�ð�=2; tÞmust vanish. In that case

p��� ðtÞ represents the momentum of half of the string and,

from (3.10), we see that this momentum for such a sym-metric string configuration does not change with time.

IV. ENERGY LOSS

To obtain the usual expression for the instantaneousenergy loss, we can let t1 ! t2 in (3.10):

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dp�

dtð�; tÞ ¼ � ffiffiffiffiffiffiffi�h

p��

�ð�; tÞ: (4.1)

This quantity gives the flow of the � component of themomentum along the string at a position � at time t. Wenote again that this is the well-known expression for energyloss from Refs. [18,19], but the previous analysis gave usinsight into its validity; namely, it pointed out that (4.1) isvalid only for constant-� curves.

Now, let us do the following coordinate transformationon the world sheet:

ð�; tÞ ! ð~�ð�; tÞ; tÞ; (4.2)

i.e., we stay in the static gauge and only change the stringparametrization using some well-defined function ~�ð�; tÞ.We can then repeat the analysis from the previous para-graph and see that in this coordinate system, for aconstant- ~� curve, we also have

d~p�

dtð~�; tÞ ¼ �

ffiffiffiffiffiffiffi�~h

p~���ð~�; tÞ: (4.3)

We can relate these to the corresponding quantities in theð�; tÞ coordinate system by using (2.7) and (2.8), which forthis particular transformation are given by

ffiffiffiffiffiffiffi�~h

p¼

ffiffiffiffiffiffiffi�hpj~�0j ; (4.4)

~� �� ¼ ~�0��

� þ _~��t�; (4.5)

where ~�0 � @~�=@� and _~� � @~�=@t. Plugging this in (4.3)we have

d~p�

dtð~�;tÞ¼ sgnð~�0Þ

�dp�

dt� ffiffiffiffiffiffiffi�hp _~�

~�0�t�

�ð�ð~�;tÞ;tÞ

: (4.6)

If we want to evaluate the energy loss at different times, wehave to make a choice of what points on the string(at different times) we are going to evaluate the currents in(4.6) on. We choose that these points on the string have aconstant ~� coordinate at all times (i.e., this is how we definethe, so far, arbitrary ~� parametrization), while in the �parametrization, these points are defined by a function��ðtÞ. The physical motivation behind such a choice is tosay that, at some time t, the jet is defined as the part of thestring between the endpoint � ¼ 0 and � ¼ ��ðtÞ. InRef. [27], for falling strings in AdS5, this choice was suchthat the spatial distance (i.e., the x coordinate in AdS5,assuming that the string is moving in the x-r plane) betweenthe string endpoint and those points was of the order�1=ð�TÞ. Now, since ~�ð��ðtÞ; tÞ is constant at all times,we have

d~�

dt¼ 0 ¼

�~�0 d��ðtÞ

dtþ _~�

��¼��ðtÞ

: (4.7)

Plugging this in (4.6) we arrive at

d~p�

dtð~�;tÞ¼ sgnð~�0Þ

�dp�

dtþ ffiffiffiffiffiffiffi�hp

�t�

d��

dt

�ð��ðtÞ;tÞ

: (4.8)

This is the central result of this paper. This formula gives theappropriate expression for energy loss in terms of quantitiesexpressed in any parametrization ð�; tÞ in which the function��ðtÞ is known. Here we were making use of the simpleexpression for the energy loss in the special ~� parametriza-tion (in which the coordinate of the points on which weevaluate the currents is constant), but in using this formulaone does not need to know what that parametrization reallyis, since the RHS of (4.8) is given only in terms of quantitiesin ð�; tÞ parametrization.Now, the argument for calling some quantity dE=dt the

energy loss comes from the idea that, when integrated oversome period of time �t, this integral should give theamount of energy that the jet (that is, some predefinedpart of the string) has lost over some period of time �t:

�Eð�tÞ ¼Z�tdt

dE

dt: (4.9)

By identifying dE=dt with �dp0=dt in the ð� ¼ r; tÞparametrization (essentially just the �r

t component of theworld sheet current), as implied in Ref. [27], means thatthis amount of energy lost should be given by

�Eappð�tÞ ¼ �Z�tdt

dp0

dtðr�ðtÞ; tÞ; (4.10)

where the subscript app stands for ‘‘apparent’’ and wherer�ðtÞ corresponds to the points at a fixed spatial distance�1=ð�TÞ from the string endpoint at all times. However,this formula (i.e., that the energy loss is given only by the�component of the world sheet current), as we showedbefore, is valid only if one uses a constant-� curve, whichis not the case in this parametrization. Then, in order to beable to use that simple expression, we need to find aparametrization ~� in which the coordinates of the pointsgiven by r�ðtÞ are constant. In this case, the energy lostwould indeed be

�Eð�tÞ ¼ �Z�tdt

d~p0

dtð~�; tÞ: (4.11)

The difference between this and the apparent energy loss isthen explicitly given by (4.8):

�Eð�tÞ¼�Eappð�tÞ�Z�tdt

� ffiffiffiffiffiffiffi�hp

�t0

dr�dt

�ðr�ðtÞ;tÞ

: (4.12)

In the following section we will numerically examine theeffect of this correction in the case of falling strings inAdS5 spacetime, dual to N ¼ 4 SYM.

V. NUMERICAL EVALUATION

In this section we will largely follow the proceduredescribed in Ref. [27], but for consistency we will review

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it here. We will work in the AdS5-BH geometry with theconformal boundary located at r ¼ 0:

ds2¼G��dx�dx�¼L2

r2

��fðrÞdt2þdx2þ dr2

fðrÞ�; (5.1)

where fðrÞ ¼ 1� r4=r4h, with rh being the radial position

of the horizon of the black hole. We choose the world sheetmetric hab to have the following form:

hab ¼ diagð�sð�; �Þ|fflfflfflffl{zfflfflfflffl}��

; 1=sð�; �ÞÞ|fflfflfflfflfflffl{zfflfflfflfflfflffl}��

: (5.2)

Here sð�; �Þ is the ‘‘stretching function,’’ which is chosenin such a way that the numerical computation is wellbehaved. Choosing the world sheet metric in this wayrepresents merely a choice of parametrization on the worldsheet (i.e., a ‘‘choice of gauge’’) and the constraint equa-tions (2.2) are there to ensure that the embedding functionschange accordingly. Explicitly, the constraint equations are

_X � X0 ¼ 0; (5.3)

_X 2 þ s2ðX0Þ2 ¼ 0; (5.4)

where _X� � @�X�, ðX�Þ0 �@�X

� and, in general, A � B �G��A

�B�. We assume that the string is moving in the

x� r plane and choose the ‘‘pointlike’’ initial conditions,where the string is initially a point at some radial coordi-nate rc:

tð�; 0Þ ¼ 0; xð�; 0Þ ¼ 0; rð�; 0Þ ¼ rc: (5.5)

Then ðX�Þ0ð�; 0Þ ¼ 0, which automatically satisfies thefirst constraint equation (5.3), and now we have to choosean initial velocity profile [i.e., functions _X�ð�; 0Þ] suchthat the second constraint equation _X2 ¼ 0 and the freestring endpoint boundary conditions (3.8) are satisfied.Following Ref. [27], we choose

_xð�; 0Þ ¼ Arc cosð�Þ; (5.6)

_rð�; 0Þ ¼ rc

ffiffiffiffiffiffiffiffiffiffiffifðrcÞ

qð1� cosð2�ÞÞ; (5.7)

where A is a constant determining the ‘‘amplitude’’ of thevelocity profile. Then _tð�; 0Þ is determined by the con-straint equation (5.4):

_tð�; 0Þ ¼ rcffiffiffiffiffiffiffiffiffiffiffifðrcÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2cos2ð�Þ þ ð1� cosð2�ÞÞ2

q: (5.8)

For this set of initial conditions we choose the followingstretching function:

sð�; �Þ ¼ sðrÞ ¼ 1� r=rh1� rc=rh

�rcr

�2: (5.9)

In particular, its most important feature is that it matchesthe singularity of the Gtt metric component near the hori-zon rh, so that the embedding functions can remain wellbehaved as parts of the string approach the horizon.

Initial conditions (5.5), (5.6), and (5.7) have been chosenas in Ref. [27] (and in Ref. [19]), in order to be able toexactly compare the effect of the correction on the resultsfrom that work. As discussed in Refs. [19,27], the physicalmotivation behind the choice of initial conditions (5.5) isthat they should resemble a quark-antiquark pair producedby some local current, with the quarks having enoughenergy to move away in the opposite directions. The ve-locity profiles (5.6) and (5.7) are one of the simplestprofiles that satisfy the open string endpoint boundaryconditions and uniformly evolve the string towards theblack hole (which should resemble the process of thermal-ization of the interaction energy described by the body ofthe string) with the endpoints moving away in the oppositedirections. The energy density profile in the boundarytheory dual to such a string evolution will have two peaksconcentrated around the endpoints, and the smoothU-shaped profile between the peaks will slowly decreasein magnitude.With this choice of initial and boundary conditions, we

can solve the equations of motion (2.3) numerically, obtainthe embedding functions X�ð�; �Þ and then evaluate theenergy loss in the radial � ¼ r parametrization withand without the correction in (4.8). To obtain the actualenergy loss, we simply use the formula (4.8) with theworld sheet fluxes expressed in the static gauge ð�; tÞ using(2.7) and (2.8):

dE

dt¼ L2

2��0

�f

r21

j _tj�st0 � d��

dt

�sðt0Þ2 � _t2

s

���ð��ðtÞ;tÞ

:

(5.10)

The apparent energy loss is given by (4.1) in the ðr; tÞparametrization, so we need to use formulas (2.7) and(2.8) again, giving

�dE

dt

�app

¼ L2

2��0

�f

r2sr0t0 � 1

s_r _t

jr0 _t� _rt0j�ð��ðtÞ;tÞ

: (5.11)

The results are shown in Fig. 1. One can clearly see that thecorrection, derived in (4.8), becomes especially importantat late times, when dr�=dt grows, as the relevant parts ofthe string start falling towards the black hole faster andfaster.We should note that the energy of the part of the string

between � ¼ 0 and � ¼ ��ðtthÞ at the thermalization timetth (when the string endpoint stops moving in the x direc-tion) is generally nonzero. This means that the area underthe solid blue curve in Fig. 1 is always less than 1, andrepresents the relative amount of energy lost from the partof the string defined by ��ðtÞ, as evident from (3.10). Onthe other hand, the area under the dashed red curve is notknown a priori, and could be <1 or >1, depending on themagnitude of the correction in (4.8). Specifically, if wedecrease the spatial distance from the endpoint at which

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we evaluate the energy loss (keeping the same initialconditions), the area under the red curve increases andeventually becomes >1 (in the figure it is already slightlyhigher than 1).

Also note that the ‘‘jet definition’’ we are using hereis taken from Ref. [27], where the jet is defined as thepart of the string within a certain �x distance from theendpoint. As discussed in Ref. [27], the physical motiva-tion behind this is that the baryon density in the boundarytheory should be well localized on scales of order�x� 1=�T. There are also other physically well-motivated jet definitions [35], exploration of which is leftfor future work.

VI. DISCUSSION

We have derived, by analyzing transformations of space-time momentum fluxes on the classical string world sheet,a general expression (4.8) for calculating the instantaneousenergy loss for time-dependent string configurationsvalid in any choice of world sheet parametrization. Thisformula shows that the energy loss in time-dependentstring configurations receives a correction to the simple��

� expression. This correction comes from the fact that

the points on the string at which we want to evaluate theenergy loss at different times do not necessarily haveconstant coordinates in the chosen world sheet parametri-zation. The importance of the correction depends on howfast the coordinates of these points change in time in that

parametrization, i.e., on the magnitude of the d��ðtÞ=dtfunction. In the example of falling strings, we have seenthat this correction becomes especially important at latetimes and substantially decreases the magnitude of theBragg-like peak (Fig. 1).One should point out that this correction does not affect

the results of Ref. [27] for the maximum stopping distance

ð�xÞmax � E1=3, since this expression was derived frompurely kinematical considerations, by analyzing the equa-tions of motion and relating the total energy of the stringto the approximate endpoint motion described by thenull geodesics. In other words, the world sheet currents(actually, their identification with the energy loss) were notused in that derivation.We should also point out that this correction does not

affect the well-established drag force results ofRefs. [18,19], since the trailing string is a stationary stringconfiguration where d��=dt ¼ 0.One is tempted to speculate about the implications of the

results shown in Fig. 1 to the jet quenching phenomena.Our preliminary numerical studies suggest that, althoughthe early time behavior of the energy loss is susceptible tothe initial conditions (as noted in Ref. [27]), the linearity ofit seems to be a remarkably robust feature. Of course, amore thorough numerical analysis is needed to confirmsuch a claim (and is left for future work), but if this indeedremains to be true and dE=dt scales like �t1, then taking

into account that ð�xÞmax � E1=3, it can be shown that thisis similar to the typical qualitative behavior of energy lossof light quarks in perturbative QCD in the strong Landau-Pomeranchuk-Migdal regime [26]. This suggests a tempt-ing idea that the phenomenon of light quark jet quenchingmay have a roughly universal qualitative character, regard-less of whether we are dealing with a strongly or a weaklycoupled medium.However, as emphasized before, a more thorough

quantitative analysis and estimate of the relative magnitudeof energy loss and stopping distances are necessary. Itwould be also interesting to more thoroughly examinethe effects of varying the string initial conditions, as wellas choosing a different ��ðtÞ function (i.e., a different jetdefinition), on the shape of the instantaneous energy loss.By choosing some set of these, one can then compute theRAA in a dynamical, expanding medium with a realisticset of nuclear initial conditions, and, finally, inspect therobustness of that result by varying the string initial con-ditions and the ��ðtÞ choice and seeing how much RAA

would be affected.

ACKNOWLEDGMENTS

We especially thank J. Noronha and M. Gyulassy forvaluable input and help. We also thank M. Mia, S. Endlich,A. Buzzatti, C. Herzog and S. Gubser for helpful discus-sions. We acknowledge support by U.S. DOE NuclearScience Grant No. DE-FG02-93ER40764.

1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

1.2

Tt

dEdt

TE

0

FIG. 1 (color online). Comparison of the (normalized) instan-taneous energy loss as a function of time with and without thecorrection in (4.8). The dashed red curve shows the apparentenergy loss ðdE=dtÞapp in the radial � ¼ r parametrization

[Eq. (5.11)], while the solid blue curve is the actual energyloss dE=dt, as given by (5.10). The energy loss was evaluated atpoints at a fixed spatial distance from the string endpoint, chosenin such a way that the correction in (4.8) appears clearly. Thenormalization constant E0 is the energy of half of the string andT ¼ 1=ð�rhÞ is the temperature. The numerical parameters usedare A ¼ 50 and rc ¼ 0:1rh.

ANDREJ FICNAR PHYSICAL REVIEW D 86, 046010 (2012)

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