Heavy Quark Energy Loss Far from Equilibrium in a Strongly Coupled Collision

Heavy quark energy loss far from equilibrium in a strongly coupled collision

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Citation Chesler, Paul M., Mindaugas Lekaveckas, and Krishna Rajagopal. “Heavy Quark Energy Loss Far from Equilibrium in a Strongly Coupled Collision.” J. High Energ. Phys. 2013, no. 10 (October 2013).

As Published http://dx.doi.org/10.1007/jhep10(2013)013

Publisher Springer-Verlag

Version Final published version

Citable link http://hdl.handle.net/1721.1/88553

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP10(2013)013 Springer June 26, 2013 : October 2, 2013 : September 4, 2013 : a,b Received 10.1007/JHEP10(2013)013 Published Accepted doi: Published for SISSA by [email protected] , and Krishna Rajagopal a,b [email protected] , Mindaugas Lekaveckas 1306.0564 a Holography and quark-gluon plasmas, Quark-Gluon Plasma, Gauge-gravity We compute and study the drag force acting on a heavy quark propagating direction as its velocity. And, generically, the force includes a component [email protected] same Center for Theoretical Physics, MassachusettsCambridge, Institute MA of 02139, Technology, U.S.A. Physics Department, Theory Unit,CH-1211 CERN, Genève23, Switzerland E-mail: b a Open Access provide cautionary lessons at higher rapidity. Keywords: correspondence ArXiv ePrint: quark, meaning that thein force the that the plasmaperpendicular exerts to the on direction of a motionapproach quark of the to moving quark. modeling through Our the it results dragin support acts heavy a on, straightforward ion and collisions, energy both loss before of, and heavy after quarks the quark-gluon with plasma modest hydrodynamizes, rapidity and instantaneous homogeneity and equilibrium provideresults a — semi-quantitative as description long as of thelarge our rapidity rapidity, of the the heavy gradients quark in isconsequences the not for too velocity the large. of ‘drag’ the For arequired force hydrodynamic heavy to fluid acting quark with result on drag in the the qualitative quark. quark through In the certain circumstances, plasma the can force point opposite to the velocity of the a homogeneous strongly coupledpressure plasma as with that the atresults. same the One instantaneous location energy interesting ofbefore exception density the the is or quark that heavy describes thereplasma quark many described is qualitative energy by a viscous features loss time hydrodynamics of has becomes delay our formed, after significant. expressions the based upon initial At assuming collision later times, once a liquid Abstract: through the matter produced ingauge the theory collision that of can twofrom sheets be equilibrium, of analyzed we energy holographically. find in a that Although strongly the this coupled equilibrium matter expression is for initially heavy far quark energy loss in strongly coupled collision Paul M. Chesler, Heavy quark energy loss far from equilibrium in a JHEP10(2013)013 14 ) limit, c N 12 26 33 4 31 20 ]. One challenge (not the only one) in using 6 – 1 – 1 – 14 spacetime [ 4 ]. 5 7 34 9 ] for homogeneous plasma in thermal equilibrium in strongly coupled 3 1 – 1 3.2 Heavy quark3.3 with zero transverse Heavy momentum quark with nonzero rapidity and transverse momentum 2.1 Gravitational description2.2 of colliding sheets String of dynamics 2.3 energy Extracting the drag force acting on the quark from3.1 the string profile Heavy quark with zero rapidity = 4 supersymmetric Yang-Mills (SYM) theory in the large number of colors ( plows through the expanding,this cooling, paper hydrodynamic we shall fluidto describe of meet this calculations strongly challenge. that coupled Forhave provide QGP. a yielded some In review qualitative qualitative insights of into many guidanceheavy properties other for ion of ways how collisions, in strongly see which coupled holographic ref. QGP calculations [ and dynamics in N where holography permits anamics semiclassical in description asymptotically of AdS energythese loss results in to terms glean ofsions qualitative string is insights dy- that into a heavy heavythrough quark quark the produced energy initially during loss the far-from-equilibrium in initial matter heavy collision ion produced event colli- must in first the propagate collision before it later strongly coupled non-Abelian plasmas.substantial For example, attention. heavy quark If energymuch one loss energy has shoots does received a itto heavy lose pull quark as a it through heavy propagates? abeen quark non-Abelian answered Equivalently, through what [ plasma, such is how a the plasma drag at force a required specified velocity? This question has 1 Introduction The discovery that stronglyativistic coupled heavy quark-gluon ion plasma (QGP) collisions is has produced prompted in much ultrarel- interest in the real-time dynamics of A Trailing string solution in anB equilibrium background Varying temperature 4 Conclusions and lessons for heavy ion physics 3 Results Contents 1 Introduction 2 Holographic description JHEP10(2013)013 ]) and 19 QCD hydro t α z hydro- ]. The 8 orthogonal an arbitrary µ x, y = ⊥ ~x ), with 2 π ] and analyzed there and 500 MeV. This is about a ]) is that the fluid 2 8 / 18 ∼ 2 c – N ] that the matter produced in T ( 11 3 21 µ ]. Because QCD is asymptotically , is the effective temperature (for ex- 21 20 when – corresponds to sheets with a thickness /c ] and from many other analyses of how 19 hydro of the Gaussian energy-density profile of 10 wµ T – 3 fm w = 0. They each have a Gaussian profile in 8 . 0 t – 2 – to hydrodynamize should not be seen as ‘rapid ∼ , where /c ). We shall describe this setup, and its holographic µ ] suggest that it would be interesting to repeat our hydro (2 10 year old result [ / , but we leave this for future work since here we shall only 10 /T , = 0 at time ∼ 9 1) = 1 wµ z − w 7 . Although there is no single right way to compare the widths . 2.1 corresponds to a time , the fluid can still have sufficiently large velocity gradients and pressure hydro ]. The incident sheets of energy move at the speed of light in the + hydro /T t 10 7 , i.e. comes to be described well by viscous hydrodynamics, after a time , . 9 0 ∼ direction and are translationally invariant in the two directions . At The principal lesson that has been learned to date from analyses of the collisions of We want a toy model in which we can reliably calculate how the energy loss rate of a . Their energy density per unit transverse area is z directions and collide at z z hydro hydro the colliding sheets of matterthe hydrodynamization are times strongly of coupled heavy fromis ion beginning that collisions to they per end se. teachRHIC as us The estimates collisions impact that for takes of the these atthermalization’ estimates most since this 0.6-1 fm timescale is comfortably longer than what we now know to expect from hydrodynamic modeling offree, RHIC the collisions dominant [ dynamicscollision at are the expected earliest to moments beevaluated of weakly at coupled, a the with sufficiently (short) the energetic distancein relevant heavy (weak) scale the coupling ion corresponding transverse being to plane the atlide. mean the spacing moment So, when between it gluons the would two be highly Lorentz-contracted inappropriate nuclei to col- take the estimates obtained in a context in which t anisotropies that the dissipativedrodynamic effects modeling of of viscosity heavy aret ion significant. collisions at In RHIC thefactor of (for context two a earlier of in recent time hy- example than the see upper bounds ref. on [ the hydrodynamization times inferred strongly coupled plasma formsstrongly from a coupled large initial number collisionsdynamizes of (for widely example, varied far-from-equilibrium seethat refs. is [ at mostample, around defined (0 from the fourth root of the energy density) at the hydrodynamization time recent investigations ofanalyses refs. for [ varying values of be seeking to draw qualitative lessons. strongly coupled sheets of energy as in refs. [ description, in section of these translationally invariant sheetsa of nucleus energy that with hasreasonable Gaussian estimates been profiles suggest to Lorentz-contracted that the by oursomewhere widths choice a between of of factor the thickness of of 107 the (RHIC) incident or nuclei 1470 at (LHC), RHIC and the LHC [ − the to scale with respect toare which working in all can dimensionfuleach be quantities sheet measured. in is chosen the The to conformal width be theory that we heavy quark moving through thecompares far-from-equilibrium to matter that present just inloss after strongly of a a coupled collision heavy plasma quarkof close moving energy through to the in equilibrium. debris strongly producedin We coupled by refs. study the SYM [ collision the theory of energy introduced planar sheets in ref. [ JHEP10(2013)013 ] ]. 6 22 with the case in which . increases. All these results 3.1 2 ] or, for that matter, in any 3.3 β 9 , − 8 , pulling the string through the and 1 ~ β -direction, meaning that it has zero p z / 3.2 1 ≡ , yields the energy loss rate of the heavy γ 2.3 – 3 – . The force needed to maintain the velocity of the 2.2 perpendicular to the ~ β ]. The addition of a heavy quark moving at constant velocity 8 and increases slowly as β at low spacetime [ 5 hydro ]. between the colliding sheets before the collision and calculating the force needed 2 ~ β , 1 /πT When we look at the drag force at late times, after the strongly coupled fluid has hy- The message from our results at early times is that there is no sign of any enhancement We describe our results in section III, beginning in section We compute heavy quark energy loss by inserting a heavy quark moving at constant amounts to including a classical string attached to the boundary of the geometry [ ~ analyses of far-from-equilibrium strongly coupled matter to date. drodynamized, we find differentor results large depending rapidity. on At small whethertitatively rapidity, the the by drag heavy assuming force quark that a has we small homogeneous calculate is plasma described in semiquan- equilibrium with an appropriate time- to what it would beat in in more an detail, equilibrium itquite plasma can different with be than the significantly at less weak same coupling, bycan energy where virtue density; arise instabilities of in when the and the looked initial far-from-equilibrium canThere delay matter result in are, its in however, rise. substantially no Thisof enhanced signs is the rates of sheets of of any heavy-quark strongly instabilities energy coupled in loss matter the [ that debris we produced analyze [ in the collisions are robust, in particularquark in moving with the some sense nonzero that rapidity we in see sections themin the again energy when loss we experiencedmoving consider by through a a being heavy far heavy quark from by equilibrium. virtue In of broad the terms, matter the that energy it loss finds is itself comparable the drag force inHowever, a we static find plasma with thattime the both relative same to the instantaneous what initial energy they risedensity density would be or or in in pressure. the pressure. a drag staticorder In plasma force 1 with appendix and the B its same instantaneous we peak energy provide are some delayed evidence in that this time delay is of rapidity. We compare the drag forceplasma that we in calculate thermal to what equilibrium it thatlongitudinal would be pressure has in as the a the homogeneous same matterWe energy that find density the or quark that transverse finds the pressure itselfproduced peak in or in at value the a of collision given the instant is in drag still time. force, far which from occurs equilibrium, at is comparable a to time the when peak the value matter of this dynamical background in section string endpoint, which we computequark in [ section the heavy quark is moving with to keep its velocity constantsheets throughout of the collision. energy in Viatotically holography, SYM the AdS theory colliding planar map intoβ colliding planar gravitationaland waves dragging in asymp- the stringcolliding gravitational endpoint wave geometry. at We constant show how velocity to compute the profile of the string in calculated the drag forcesheets on of energy a density, heavy wedrawn quark shall from seek that the similarly finds analyses qualitative itself lessons of to in the those collisions the that themselves. midst havevelocity been of the colliding if the physics of heavy ion collisions were strongly coupled from the start. After we have JHEP10(2013)013 c N (2.2) (2.1) , the λ direction z and large , c 2 N du + = 4 SYM theory with 2 − N dx , − h the gauge coupling constant, and + g 2 + 2Λ) = 0 dx − + 2 h u R ( 4 u , can be attributed to the presence of gradients – 4 – MN + G 3.3 2 ⊥ 1 2 d~x − and 6. In this limit, the back reaction of the string on the + − − MN 3.2 R dx + of the gauge theory with c dx ! We conclude in section IV with a look at the lessons on how N − z 2 g = ] ≡ 2 23 λ ds ]. In the dual gauge theory living on the boundary this geometry corresponds to 8 , black brane spacetime. For an infinitely massive quark, the endpoint of the string is We shall focus on the case in which the gauge theory is 5 -component of the force that is required to move the quark in the positive- in ref. [ colliding planar sheets of energy density.metric In Fefferman-Graham is coordinates given the pre-collision by [ of the stringbackground equations, given and by the then solution subsequently to Einstein’s determine equations. 2.1 the shape Gravitational of description theThe of geometry string colliding we sheets in choose of to a energy study is that of the colliding gravitational shockwaves studied evolution of the black brane geometry is governed by Einstein’s equations with cosmological constant Λgeometry = is negligible meaning that we can solve Einstein’s equations first, independently coincides with the trajectory of the quark. colors, although our results canwith immediately a be gravity generalized to dual any’t upon conformal Hooft making gauge coupling theory a suitablecorresponding modification quantities to in the the relationship between gravity the dual. In the limit of large quark moving through out-of-equilibriumcoupled matter plasma that consists is on ofAdS its a way string to moving becomingattached in to strongly some the four-dimensional non-equilibrium, boundary but ofis the asymptotically geometry. that The of geometry of Minkowski the space boundary and the trajectory of the string endpoint on the boundary from our results. 2 Holographic description In a strongly coupled conformal gauge theory with a dual gravitational description, a heavy the quark in orderthe to direction move of it motion alongIn of its certain the trajectory cases quark, includes we a az also component component find that perpendicular that, can to as bepoints a substantial toward consequence in negative of magnitude. gradientsbest to in model the the fluid drag on velocity, heavy the quarks produced in heavy ion collisions that can be drawn effects that, we show inin sections the fluid velocity. Wethe find energy that loss a of velocity the gradientclosely heavy in quark aligned the when with fluid the the direction hasare of velocity the motion larger greatest gradient. of effect at the As on larger heavy a quark rapidity. consequence, is most We effects find of generically velocity that gradients the force that must be exerted on dependent temperature. At large rapidity, however, this approximation misses qualitative JHEP10(2013)013 ], 8 (2.3) and position along the t ) that specifies the profile ± , x 2 ( h /w 2 ± = 0, the energy density is that of two direction at the speed of light. If x t 1 2 z ]. The metric that we have described − ± e 2 23 ) do not overlap in the distant past and / 1 − ]. In the boundary gauge theory the en- )[ − x ± ) ( 2 25 x h , ( h – 5 – πw 24 [ (2 3 ) and µν µ + ) for some function T x ± ( x ) = h ( ± h x ( ≡ h ± h . µ and z for the colliding sheets of energy. Before z ± ) is an exact pre-collision solution to Einstein’s equations. Following ref. [ t = 0 and leave debris in the forward light-cone that subsequently hydrodynamizes as t 2.2 = defines the energy scale, meaning that we shall measure all other dimensionful . A plot of the boundary energy density as a function of time ± x µ ) has compact support then For a given solution to Einstein’s equations, the near-boundary behavior of the metric ± = 0. The gravitational shockwaves move in the x ( where quantities in units of encodes the boundary stressergy tensor density of each shock is proportional u h the metric ( we choose Gaussian profiles it expands, becoming strongly coupled, liquid, plasma. where of the incident gravitationalboundary shockwaves theory. and hence The the boundary incident of sheets of the energy geometry in is the located at AdS radial coordinate Figure 1 ‘beam’ direction sheets with Gaussian profilesaround moving time toward each other at the speed of light. The sheets collide JHEP10(2013)013 , ) ) of /µ 2.4 2.2 5 (2.5) (2.4) µν . T , Σ and ]. = 0 8 B ]. In what w , (and spatial A t 10 , 8 ) ) in the form of ( du 2.2 − F dz ] and we shall not review it ( ) as initial data in the distant dt 10 , by demanding that the location 8 2.2 ξ + 2 but is an arbitrary function of the ) begin to overlap, the metric ( 2 ± , u ) x dz ( 1 . Lines of constant time B h t, z u 2 2 ( − u ξ 1. The numerical procedure for determining e + ]. Therefore, a natural choice is to excise the and + ) is invariant under the residual diffeomorphism – 6 – 1 u 2 ⊥ z 12 . One may fix , as well as the bulk metric functions , 2.4 z ) used for numerical evolution. Therefore, for initial data < u < t d~x → 8 ξ B 2.4 1 u e , and the incident sheets each have an energy per unit ]. We choose the width of each shock to be and 8 2 w t ]. = 1. With this choice of coordinates, the boundaries of ]. )[ 8 2 u + Σ 10 π , 2 8 (2 / 3 Adt µ − 2 c are functions of directions at the speed of light. The energy density profiles of the ) are infalling null geodesics. N . The geometry we study in this paper contains a black brane with = ) onto the metric ( z z u F 2 ± 2.2 ds and ⊥ Σ and ~x 3 than those in ref. [ / does not depend on the radial coordinate A, B, ξ An important practical matter when solving Einstein’s equations is fixing the compu- Near the collision time, when the functions To the best of our knowledge there does not exist a closed form coordinate transformation taking the 1 and solving Einstein’s equations is described in refs. [ , whose asymptotic near-boundary behavior determines the stress-energy tensor the colliding sheets of energy in the boundary gauge theory,pre-collision as metric described ( inwe ref. compute [ thenumerically. coordinate For transformation details see required refs. to [ put the initial metric ( boundary spacetime coordinates of the apparent horizonthe be computational domain at are static:ξ 0 here. Following this procedureF yields particularly simple. The metric ansatz ( where collision takes place ongeometry the inside boundary the [ horizon,ometry. as To this perform region thealways is lies excision causally inside the we disconnected event identify horizon) fromfurther and the the into choose location the to outside black stop of brane ge- integrating Einstein’s at an equations its apparent any location. horizon Our (which choice of coordinates makes this procedure where coordinates tational domain in planar topology. Moreover, the event horizon exists in the infinite past, even before the numerical evolution is that of infallingtakes Eddington-Finkelstein the coordinates form where the metric factor of 2 ceases to be apast, solution one to must Einstein’s thereforenumerical equations. compute scheme the Using for future ( solving evolutionfollows Einstein’s of we the simply equations state geometry can some numerically. be of Our found the salient in features. refs. A [ useful choice of coordinates for the extent, and translation-invariant, ineach the other transverse in dimensions) the thatsheets are are moving Gaussians with towards widths area that is given by meaning that we shall be probing the collision of sheets of energy that are thinner by a therefore corresponds in the boundary theory to two sheets of energy density (infinite in JHEP10(2013)013 6= 0 = 0 1: (2.7) (2.8) (2.6) . z z − 2 c is given = /N µ ) ). At any µ 2 u u π µ 2.6 u (2 ) to determine 00 ], to regulate this 2.7 T 8 = E = 0, the energy density = 0, during the collision t = 0, during the collision. = z z ε . ) = = . The fluid velocity is defined k t | we compare the pressures in t P z | , 2 ⊥ 0) and , P , 0 , = 0 to what those pressures would µ , , normalized such that which corresponds to an initial back- ⊥ . In the distant past the effect of the 0 ν µ P , z µ ε u T 00 ε, 2 2 c T π 085 2 = (1 N 2 2 c . π − N 2 diag( = 0 and leave debris in the forward light-cone. background – 7 – = 0 2 2 = c T t π ν N 2 E ≡ u µ rest frame ν µ = u T = background T µν rest frame being the proper energy density rescaled as in ( T µ lab frame ε u 0). In this local fluid rest frame, the stress-energy tensor at the ) diverge deep in the bulk. As described in ref. [ , 0 , 2.2 0 , = 0 the velocity of the fluid vanishes by symmetry, meaning that at = (1 z shows a plot of the rescaled energy density 1 and then boosting to a frame in which, at this spacetime point, the eigenvalue, ε ) we have simply − rest frame µ Before we turn to describing the dynamics of the string and then to calculating the After the collision, the fluid in the forward lightcone is expanding meaning that at Figure In the distant past the apparent horizon lies very deep in the bulk. This presents a 2.7 u µ lab frame in ( rate at which the heavyis quark formed loses in energy, we thethe should forward ‘beam’ justify light-cone referring direction as to and the a matter perpendicular fluid. that to In it figure at by spacetime point of interest is diagonal and can be written as Note that at with spacetime point, we canu find the local fluid rest frame by first using ( geometry during and after the collision. it has a velocity thatto points be in the the future-directed direction time-like of eigenvector increasing of We shall calculate the ratemoving of through energy the loss far-from-equilibrium of matterwhen (i.e. the right the energy near ‘drag density force’ ishow acting largest. it on) We loses a shall energy then heavy ingravitational follow quark the the terms, expanding, quark we cooling, forward wish fluid in to that time study forms and the in see the dynamics forward of lightcone. a In string moving in the black brane for our colliding sheets asis a that function of of two sheets position of andof energy time. with light. Gaussian Before profiles The moving sheets toward each collide other around at the time speed thereby control the sizeof of background the temperature metric is ground coefficient energy functions density 213 deep times inof smaller than energy the the and bulk. energy 450 density times at Our smaller the choice than center of the the energy sheets density at computational problem for solving Einstein’stions equations in numerically the as the metric coefficient ( problem func- we choose toture study background study plasma shocks of whichbackground temperature propagate temperature and is collide to in a push low the tempera- apparent horizon up towards the boundary and JHEP10(2013)013 ) 2 the 2.7 hydro t ) and in hydro k t P . We see that 2 from the proper , when the matter ⊥ P hydro t

4 and 4 µ ). (Because we are making µ k / / 6 P ⊥ " 2.7 P P hydro 4 = 0 during the collision of the sheets of z . as dashed curves in figure /µ . It is clear from the figure that one could µ 8 . µ u /µ 2 t 8 . = 2 – 8 – that at the hydrodynamization time . The reason that with the strict criterion , determined via ( and µ 2 = 2 /µ ε u when we have an expanding, cooling, hydrodynamic onwards the flow of the fluid is described very well hydro t 0 /µ hydro t hydro t , although with the more strict operational criterion defined in = t /µ 2 − ), as a function of time at . We compare the pressures in the collision, shown as solid colored = 2 is so small that a 15% relative deviation corresponds to a very ⊥

t 1 k 0 1 P P 0.6 0.4 0.2 1.4 1.2 0.8 ], namely the time after which the actual pressures within the matter = 0, where the fluid is at rest in the lab frame, in this instance ( 8 ], we see from figure z 8 and the fluid velocity ε as in ref. [ . Pressure, both parallel to the direction of motion of the colliding sheets ( = 4 SYM plasma, to first order in a derivative expansion. In particular, the hydro t N We shall calculate the drag force on the heavy quark as it plows through the matter small absolute deviation. produced in the collision bothis far before from the equilibrium, hydrodynamization and time after liquid. As in ref. [ derived from the hydrodynamichydrodynamization occurs constitutive at relations.choose With a this reasonable but definition,hydrodynamization less in occurs strict well figure criterion before for 2 is hydrodynamization a with little respect later to is which that hydrodynamic constitutive relations to from a time thatby is hydrodynamics. clearly before We can 2 usetime the same operational definitionproduced of in the the hydrodynamization initially far-from-equilibrium collision are both within 15% of the values be if thethe matter were ahydrodynamic fluid constitutive described relations by allowenergy the density us equations to of determine this viscous comparison hydrodynamics at for is trivial as described above.) We plot the pressures determined by applying the first order dashed curves. We see thata the reasonable strongly degree coupled before fluidthe produced text, in the the hydrodynamization collision time hydrodynamizes is to Figure 2 the transverse directions ( energy illustrated in figure curves, to what they would be if the hydrodynamic constitutive relations were satisfied, shown as JHEP10(2013)013 is ab (2.9) ), we η (2.10) (2.13a) (2.12a) (2.13b) (2.12b) 2.4 yields dy- M X ) for the bulk space- , 2.4 N X b ∂ M , is a spatial worldsheet coordinate, 0, well before hydrodynamization, X a σ M ! 0 ∼ ∂ 1 = 0 (2.11) ≡ t − 2 B αX 0 MN σ ) 1 2 , . G X . . A − ab 1 0 u τ, σ ˙ ( = 0 = 0 X − M M ηη = α 2 2 – 9 – 0 − X X σ N AB − ˙ τ σ X √ X ∂ ∂

σ + Γ 2 ≡ ≡ = d N 0 is the string tension. We choose worldsheet coordinates ab M M ˙ Z 0 )) are the string embedding functions that describe where η λ X ˙ π 0 X 2 √ X 2 T τ, σ ( = − 0 , u = T ) ) by demanding P S τ, σ τ, σ ( ( are infalling null worldsheet geodesics. As we describe further below, α , ~x ) τ τ, σ ( is a temporal worldsheet coordinate, t are Christoffel symbols associated with the metric ( τ ) being an arbitrary function. Just as we did in the spacetime metric ( ≡ ) = ( N AB 1 τ, σ ( σ α τ, σ Varying the Polyakov action with respect to the embedding functions ( M The constraint equations are even simpler: where Γ time and namical equations of motion.worldsheet metric Likewise, yields varying the aEddington-Finkelstein Polyakov system bulk action and of with worldsheet constrainttion respect coordinates, equations. take to the the the simple With dynamical form equations our of choice mo- of infalling with have chosen worldsheet coordinatesLines that of are constant infalling Eddington-Finkelsteinwe fix coordinates. the function X the string is locatedthe within worldsheet the metric and spacetime metricsuch that that we the are worldsheet interested metric in takes probing, the form The dynamics of the string are governed by the Polyakov action where For a given solutionthe to dynamics Einstein’s of equations,hydrodynamic a flow for black of example the brane stronglyinfinitely the in coupled heavy one matter quark the in above moving bulk thethe through that boundary bulk and the describes theory, geometry boundary we the whose theory can endpoint collision, add matter follows an by the hydrodynamization, adding trajectory and a of the string quark in along the boundary. that it hydrodynamizes andthat becomes shown locally in isotropic the onlywhen figure. it at We is a shall first therefore much inand the later be then far-from-equilibrium time, studying matter later the beyond at in drag an force expanding on cooling a2.2 hydrodynamic heavy fluid quark that is String far dynamics from isotropic. parallel and transverse pressures are very different. The fluid is very anisotropic at the time JHEP10(2013)013 , ) ) σ M as ˙ X = 0, M t (2.17) (2.16) (2.14) (2.15) ˙ as the 2.13b 2.13b X M ˙ X ) imply it will be = 0 at time ) constitute a linear z , ) 2.11 at the boundary. The t ) and the constraint equa- γ ( 1 2.11 t 2 o ) is satisfied every where . ) and the boundary con- ~x ~ β − 2.11 = 2.16 2.13a ˙ u ). , given an initial string profile 0 τ → 2.17 lim σ — which we do since we have been . ) and ( ) ) imply that it will be satisfied at all t , M ) are first order differential equations 0 ( ) is a boundary constraint. If ( τ, ~ 2 β o 2 X 2.15 ~x . To solve these equations, in addition to 2.11 = 2.11 ~ βt, = 0. In doing so one can directly incorporate the and that finds itself at . We can solve these equations for ) = M ˙ ~x σ 2.13b ˙ σ ) = ~ β 0 X ) = τ, σ → t takes the form ( ( lim the dynamical equations ( σ – 10 – ( τ, σ t one must specify five boundary conditions at all near ( , the equations of motion ( o 0 M σ τ ~x ~x ˙ σ → , X 0 M lim σ → X lim 1 σ γ — and as long as we have boundary conditions for . The algorithm has two steps. First we need to obtain τ τ = 0 provided that ) into the series expansions. With the expansion known to order . We do this by observing that if we think of 1 + σ ) is a temporal constraint. If ( corresponds to the coordinate time τ as long as we know τ 2.16 . These boundary conditions correspond to choosing to study 1 2 τ ~ β = 2.13a ˙ t at some value of 0 as usual. ) and ( σ → 2 lim σ ~ β at the initial 2.15 at some initial σ − at the initial = 0. The boundary conditions ( . 1 for all σ M at the initial σ ) are satisfied at ˙ q u M X σ / ˙ 1 for all ~x, X ˙ ) with a power series expansion in t, ≡ with the independent variable being 2.13b M γ X M 2.13 = ˙ We can now see how to evolve the string forward in Two more boundary conditions are required. For convenience, we choose Given To see that this is true one can solve the string equations of motion ( X 2 M at the AdS boundary, where the string endpoint is located. Three boundary conditions ˙ functions of given tions ( boundary conditions ( one can easily see that the boundary limit of specified by X dynamical variables, the equationsfor of motion ( where so that the worldsheet time remaining boundary condition comes from demandingis that the satisfied boundary at constraint ( straint ( string, that is moving withmeaning constant that velocity this heavy quarkto finds probe. itself We right shall in present the results center for of several the different collision values of that we wish We choose the trajectory for some constant velocity a heavy quark, whose location after all coincides with the location of the endpoint of the system of ordinary differential equationsspecifying for the initial conditionsτ for are simply that the string endpoint moves on a given trajectory in space at onesubsequent time, times. the dynamical The equationsis constraint ( satisfied equation at ( allsatisfied times at at all one value of The constraint equation ( JHEP10(2013)013 M X τ . This ∂ ] moving 2 , ), rewritten at an initial 1 . With M background = 0 and obtain u T 2.12a X 2 ˙ σ − = α . To do this we need the M X τ ∂ ) starting from 2.11 , thus completing the evolution of the = 0 when the sheets collide on the boundary, τ t , when the centers of the incident sheets of /µ 3 , it is given simply by – 11 – u − 3 ] will lead to a small violation of the temporal constraint so large that at our initial time the heavy quark is 2 | = , 1 ~ at the new β , even in the distant past before the sheets of energy collide | σ t ). So, we solve ( the collision event is negligible. Because in an actual M . Such initial conditions satisfy the temporal constraint X one time-step later. We then repeat. A 2.17 τ . In the second step of the algorithm, we use ( , to compute the field velocities τ to a M 0 coordinate using pseudo-spectral methods. Specifically, we decom- τ apart. In the distant past, well before the sheets of energy collide, σ αX w 1 2 during and after + = 12 ) near the AdS boundary. M ˙ ) deep in the bulk. However, as time progresses the violation of the constraint decreases at the initial , given in appendix /µ X . ~ β dependence of all functions in terms of the first 20–35 Chebyshev polynomials. σ . With the gauge choice 2.13a A = α σ 2.13a M that we shall typically take to be t X = 0 — which we have in ( at all We discretize the Constructing initial string profiles that are equilibrium solutions to the pre-collision All that we still need to specify is our choice of initial conditions for τ Our choice of initial conditions is slightly complicated by our choice of infalling Eddington-Finkelstein σ ∂ 3 M ˙ of this, choosing theequation trailing ( string ofin refs. magnitude [ and thenthe as portion time of progress thehorizon further string of that towards the violates the blackcausally temporal disconnected brane. from constraint physics equation Becausein near is of appendix the rapidly this, boundary enveloped and the by hence the initial is event violation of of no concern. the We temporal discuss constraint this equation further is pose the coordinates. In infalling Eddington-FinkelsteinAt coordinates any the given bulk Eddington-Finkelsteinon is time the the boundary, causal future the of gravitational the shocks boundary. are colliding somewhere deep in the bulk. As a consequence of energy. This, together withthat the we insensitivity are of the interestedstudy results in during could to and be initial afterchanges the repeated conditions to collision and with our associated conclusions. other transients, choices means of that initial our conditions for the string without describe below, although perturbingtransient the before initial the collision string happens,the profile the heavy does change quark result that in resultsheavy an in ion the early-time collision rate a of heavyinterested energy quark in loss is any of aspect produced of during the the motion collision, of we our are heavy not quark particularly before the collision of our sheets equation ( geometry ensures that thechange future in non-trivial the evolution bulk of geometrywould associated the come with along string the as is artifacts collision entirely of event, due any not other due to choice to the of transients initial that conditions. However, as we shall within one of the incident sheets.shall make We sure shall that make at sure our notthat initial to are time do soon the this, going heavy which to quark is hitsuch has to it that not say to the yet that any felt significant we string degree. the profileat sheets We coincides velocity of therefore with energy choose initial the conditions trailing string solution of refs. [ time energy are 6 the near-boundary geometry between theis sheets located, of is energy, that which of isis an where so equilibrium the provided black heavy that brane quark we with do a not small choose temperature X as function known, we can thensystem determine from the initial at JHEP10(2013)013 . ν f (2.20) (2.19) (2.24) (2.21) (2.22) (2.23) (2.18) ]. The 26 , according /µ = +6 only has support at t . to EM ) S x ( /µ , , ) µν 3 em the variation in the Polyakov T M − N ~ βt µ M T − χ = x ∂ G t ~x + 3 ( − d , 3 , M δ √ ν Z µν X EM . EM M f , T S χ − ν → ) = + f N + = 0 x M ( P = µν ) = x n S µν X x tot T µν 4 ν ( – 12 – T d T ν = dt µ = dp µ F ∂ Z tot µν 0 tot x ∂ S 3 T → ) = d u x ( Z = lim µν P − T µ δS is thus given by ∂ ) = t ν ( f ν f can be identified as the flux of four momentum down the string [ does not transform as a four-vector under Lorentz transformations. It is ν ν f f ) only has support at the boundary and reads is the four momentum lost by the quark per unit time. Note that, as always ν the four momentum of the quark, even though in the setup we are analyzing the four 2.9 f ν p Under an infinitesimal diffeomorphism Just as the near-boundary behavior of the metric encodes the expectation value of to drag the quark (which is to say theaction string ( endpoint) at constant velocity. flux down the stringsystem, can be extracted by noting that the total actionmust for be the holographic diffeomorphism invariant.the boundary The and electromagnetic couples the action string endpoint to the classical background electric field used the boundary stress tensor,Specifically, the near boundary behavior of the string profile encodes is conserved Via the quark, energymedium. and The quantity momentum flow from the electric field into the surrounding to move the quark atthe constant medium speed surrounding and, the indragging quark; so the the doing, quark transfers quark is energy does a and not classical momentum slow background to electric down. field, If the the total external stress agent tensor for a force, conventional to write with momentum of the quark does not actually change. The external agent does work in order medium at a constantmomentum speed to by the some surrounding externalconserved: medium agent, meaning it must that be the transferring boundary energy stress and tensorwhere is not to the algorithm that wedifferential have equation described solver. above using a fourth order Runge-Kutta2.3 ordinary Extracting theBecause drag the force quark acting is on the a quark point-like from source the which string is profile being pushed through the surrounding We then time evolve the string profile, typically from JHEP10(2013)013 N = n that used M ), (2.28) (2.25) (2.26) (2.29) (2.30) (2.27) n ξ 4 ] are not 2.4 27 ). Simply 2.5 . This choice ξ M , ∂ ) where we present our 2 ). Upon demanding u X 3 − . + . . 2.18 i 5 ) shows that , Y N ( µ M ). In other words, α 5 δ 5 X F 5 δ b µ 2.29 X T ∂ 2.5 N a = α µ diffeomorphism ( xχ ab δ X 4 ξ ∂ M b η ). d α ∂ − n , ∂ defined in ( ) and ( µ α Z M µ a µ , one might naively conclude N MN µ 5 2.18 T π radial X T . As a consequence of this divergence − F 2 2 M a 2.25 G ξ η G δ u ∂ = α ]. The drag forces calculated in ref. [ /u − − , we conclude that = 0 but is not correct once we make the ∂ − ab µ = 2 27 √ √ with ξ EM ξ χ u 0 ν N µ η η µ µ M T ∂ n T + − F n ν 0 σ − 5 , transformed by ( – 13 – µ 5 − √ ∂ π → N µ σ = T u 5 2 = 2 δ u ) d xχ = lim ) is invariant under all infinitesimal diffeomorphisms, G ). We therefore conclude that depends on one’s choice of the gauge parameter 4 M = 0, and + µ EM d Z − P µ ν 5 σ µ u µ F X 2.5 2.27 a π √ 0 T G T Z δS ∂ h ν T − ( G 0 ∂ 0 = − in ( δ √ − = 0 the usual identification of force in terms of the canonical → → u u µ ξ √ ≡ EM F 0 = lim = lim ) = can easily be extracted via ( , a M δS → µ µ Y u π µ M f . However, this cannot be correct. Near the boundary some com- ( F can also be expressed in terms of the canonical worldsheet fluxes f µ n 5 µ ) that we needed to make in order to solve Einstein’s equations. With T f MN G transforms nontrivially under the 2.5 T − µ 5 √ T 0 is the covariant derivative with respect to the bulk spacetime metric ( G ] is reproduced. → u − 2 M , √ 1 D 0 would have been correct if we had used We note that The unique remedy to the above problem is to set Setting the normal to the boundary → = lim As we did, in a preliminary version of this study [ u 5 4 µ M We see that for thefluxes choice [ correct but, comparing the figuresresults, from the that differences preliminary are report not to large. those in section A straightforward exercise using the string stress ( is simply the near-boundaryδ limit of transformation ( the correct choice of including those arising from eq. ( ponents of the stringlim stress tensor diverge likeput, 1 the expression lim in solving Einstein’s equations. from which the drag F where in the last linethat we the used variation in the action vanish for all is the normal to the boundary at is the string stress tensor. Likewise, the variation in the electromagnetic action is where JHEP10(2013)013 , see E . We shall initially 1 = 0 direction, i.e. with zero z in the plane perpendicular to the direction -direction along which the sheets of energy ~ β z – 14 – -axis as perpendicular to the page, the quark is moving x )-plane of a quark moving in the . we consider a toy example in which the temperature increases x z, t β B = 0 path along which the heavy quark is moving is illustrated in z . Thinking of the 4 µ . The = 0 in a heavy ion collision with zero rapidity moving with some perpendic- ~ β t . The path in the ( ), in units of 2.6 rection, which is tocollide. say perpendicular Although to our the logical heavy perspective quark this was calculation presentproduced may at inform before how the we collision, thinkular from about velocity a a heavy phenomeno- quark that is the drag force, in appendix with time but theexpansion. plasma remains homogeneous and isotropic and3.1 does not undergo Heavy any quarkWe with begin zero with rapidity the case where the heavy quark is moving perpendicular to the ‘beam’ di- velocity of the quarkdirection. has We components shall both compareforce parallel that our we results to calculated to and tothe expectations perpendicular same what instantaneous based it to energy would upon density the have orthe comparing been parallel ‘beam’ position the pressure in of or drag a the transverse static quark. pressure homogeneous as To further that plasma explore at with the effect of a time-dependent background on In this section we presenton and a discuss heavy the quark results beingchoose that dragged the we quark through have to the obtained be collidingof moving for sheets with motion the of a of drag velocity figure the force colliding sheets. Later we shall consider the more general case where the Figure 3 rapidity, is shown as theeq. red ( line superimposed onout a of color-plot the of page the rescaled with energy velocity density 3 Results JHEP10(2013)013 95 . (3.2) (3.1) . The = 0 2 x β through the ~ β 5 and . = 0 x

β e " × ) to frame expectations T T T ] . 3.1 ) 2 , 4 k = 0 plane is always moving 1 z 5. The dashed curves show the ,T . , 4 ⊥ 4 exact eq. with eq. with eq. with 2 = 0 ,T β 4 ⊥ x − ~ β = 0 the ﬂuid is at rest, meaning that β ,T 1 , respectively. 4 z e p T 5 2 2 ) µ t and πT ( diag(3 4 – 15 – 2 λ c π 0), meaning that in this case the local ﬂuid rest ) on a quark being pulled through the collision in the 2 , √ N 8 0 π 0 2 , (2 π = 0 , λ/ eq =

√ 2 dt d~p µ = (1 2 µ − ) which we now rewrite as u µν

rest frame 2.8

1 0 T

0.6 0.4 0.2 1.2 0.8

" µ dt ⊥

⊥

2 dp ⊥ = 0, i.e. at zero rapidity, with velocity z plasma in strongly coupled SYM theory, namely [ at any point in spacetime, as follows. At . Drag force in units of is the temperature of the equilibrium plasma. Out of equilibrium, the matter does . Our results for quarks moving at zero rapidity with speed , as is the energy density through which it moves. The parallel and perpendicular 2 T 3 We are interested in comparing our results to expectations based upon the classic result d~p/dt -direction at for the lab frame in whichthe we fluid are takes working is the the form local ( fluid rest frame and the stress tensor for where not have a well-defined temperature. We can nevertheless use ( to the direction of motionsection of the quark. Weare have shown calculated as the the drag solid force red as curves described in in figures for the drag forceequilibrium required to move a heavy quark with constant velocity case of a quark movingturn at to zero next rapidity for isthrough two simpler fluid reasons: than at the (i) more rest, aframe general with is quark case the moving that same along as we the the is will lab directed frame; antiparallel (ii) to the the drag velocity force vector, acting with on no the component quark of with the zero force rapidity perpendicular energy density, parallel pressure or perpendiculardashed curves pressure as are that described at further the in location the of text. the quark.figure The pressures of the material in which it finds itself are those that we plotted in figure Figure 4 x drag force that the quark would experience in an equilibrium plasma with the same instantaneous JHEP10(2013)013 , ) ) τ 5 ( 3.1 T 95. . ≡ and k the drag = 0 T 4 . In the x T = β ), obtaining ].) At leading ⊥ = T 3.1 13 , k = 11 T e background T T =

6 = ⊥ e " × T T T T k T = = e T ⊥ ] of the drag force on a heavy quark exact eq. with eq. with eq. with T 4 30 – = 28 e T ) for a static plasma with temperature 3.1 ) we have defined three different “effective 2 µ t 3.2 . Because the stress tensor is traceless, any one – 16 – 5 0 the proper time. (See, for example, refs. [ and ] includes an investigation of the next order corrections. τ 4 30 , with 3 ]. Ref. [ / 2 but for a quark moving at zero rapidity with a velocity 2 − 4

/τ

3 2 1 0 7 6 5 4

" µ

dt ⊥

2 28

⊥ )[ ]. In this approximation, the drag force on the heavy quark is indeed given by ( 2 dp ⊥ τ ( 31 [ T = 0. The dashed curves are devices through which we use what we know 3 z / 1 /τ 1 . Next, we see that the increase in the drag force due to the dramatic change . Same as figure ∝ ) τ replaced by ( 5 T T Comparing the drag force on the heavy quark, the solid red curves in figures Some support for this strategy is provided by analyses [ 5 and position background Figure 5 moving through a fluidinvariant that in is the undergoing otheran boost two dimensions. expansion invariant expansion in In in powersorder, this one of which setting, dimension 1 is the and to gradientwith is say expansion at translation of late hydrodynamics times, becomeswith the fluid can be treated as ideal meaning that to the threeright, dashed we curves first yields seeforce many that is observations. indeed at given very ReadingT by early the the times equilibrium figures where result from ( left to pressure or parallel pressure ast that present in theabout far-from-equilibrium the conditions drag at force time in ancase. equilibrium plasma to frame expectations for the nonequilibrium of the dashed curves canalso be explains why obtained wherever from two thethat of other none the two; of dashed only the curves twoforce cross three are experienced the dashed independent. by third curves the must This should heavy crosscollision be quark also. in seen of the Note as the far-from-equilibrium a conditions sheetsforce “prediction” created would of by for be the energy. the in actual a Rather, static drag plasma the with three the same dashed instantaneous curves energy density tell or us transverse what the drag If the fluidnonequilibrium were setting at of interest, rest throughtemperatures”, in ( no equilibrium, one of wethen which would use is each have a of true thesethe temperature three three since “temperatures” dashed none in curves the can in equilibrium be figures expression defined. ( We can JHEP10(2013)013 6 4 for a long ) with an effective ) with the effective background 3.1 T 3.1 is the same as that in 6 ) by solving the temporal constraint equation τ, σ ( t – 17 – 5 our result is a little below ( . = 0 β 95 our result is a little above this benchmark. What seems . for a discussion. A = 0 we have found that our results of interest, namely our results β 6 while for ⊥ T , while the blue, purple and green dashed curves correspond to initial string profiles 4 that the drag force is best approximated by the value that it would have in a static ) numerically; see appendix We are using initial conditions that correspond to a heavy quark which has been 5 In obtaining these results, we have determined 6 2.13a These perturbations to thetransient initial effects shape at of early theduring times string, and but which after are their the in effects collision no on are way the negligible. small, drag do force have felt by the( heavy quark as we illustrate infor figure the drag forceto on the the choice heavy of quarkfigure initial during and conditions. after The thechosen solid collision, such that red are the curve quite initial in insensitive drag figure force is zero, four or sixteen times that for the red curve. and after the collisionwell are before to the the collision. choiceschoice we If of are our initial making results conditions for duringto this the and the would initial after question be shape the of problematic, of collision whatbe since the the were since there initial sensitive in string actual conditions to can heavy for our be ion a no collisions heavy there right quark are answer before no the heavy quarks collision present should then. Fortunately, dragged through an equilibriumtime before plasma the with collision. the This,in low of a heavy course, temperature ion is collision, not where reminiscentimportant the at to heavy all quark check of is how what created sensitive during actually our the happens collision. results for It is the therefore drag force on the heavy quark during expanding, cooling, anisotropic hydrodynamicspanned by fluid the lies drag withinor force the parallel in band a pressure static of orno plasma expectations transverse qualitative with pressure. deviation the relative For same toconsequences a instantaneous these of energy heavy expectations, the density quark meaning presence that with of we zero gradients see of rapidity, no we the qualitative fluid see velocity. and plasma with the sameis instantaneous coincidental transverse as pressure. we seetemperature To that some for degree thisrobust agreement is the fact that at late times the actual force required to drag the quark through the with the same instantaneous energythat density. a These reasonable second first-cut andion approach third collision to observations would suggest modelling beafter the the to drag collision on turn and a thetemperature from heavy then determined drag quark on by force the use in onthe instantaneous the a expansion energy equilibrium roughly heavy density. of expression one ( the Fourth, “sheet at fluid late thickness” is times in described when well time by viscous hydrodynamics we see in figures delayed. And, the delay inheavy the quark increase velocity. of We the shallthe drag return drag force to force seems this is to below.that comparable increase the to Third, with increasing the we peak see expectations valuecollision that provided of the by is the peak the drag not dashed value force dramatically of curves, in smaller meaning the or far-from-equilibrium larger matter than produced what in it the would be in a static plasma in the stress tensor of the fluid corresponding to the collision of the sheets of energy is JHEP10(2013)013 . , . 8 5 βγ and 4 . The other three 4 = 0.

6 z β is certainly the increase ) and ask to what degree bkg bkg β bkg T T 3.1 T 5 and = =0 =2 =4 . -dependent effect that remains in in in in 4 β T T T T = 0 x β with with with with -dependence of the time delay in figure of the low temperature plasma present before 2 β µ t – 18 – background 0 . The interesting T βγ , the biggest effect of increasing 2 − 5 . We investigate the

1 0

and

we investigate this time delay further by watching how the drag

0.4 0.2 0.8 0.6

" µ dt ⊥

2 4

⊥ B

2 dp ⊥ is the time delay in the onset and peaking of the drag force. This time delay . Illustration of how the drag force depends on the initial string shape. The four curves 7 provides the answer. Almost all of the change in the overall magnitude of the drag . In appendix 7 γ -dependence in our results is described by assuming that the drag force scales with Let us now return to the question of how the drag force changes with quark velocity β . Looking at figures the time when (around thesame same with time as) the the same dragat instantaneous force energy higher in density an velocities (transverse equilibriumwith its plasma pressure) with would peak the peak is while force delayed responds by in a a background time in that which the increases temperature approximately of the linearly plasma is homogeneous in force, e.g. the changecan in be the understood height at asin which figure scaling it with peaksincreases seen with by increasing comparing figures From this figure wewhich see the that drag force a peaks reasonable is characterization that of at our low velocity results it for peaks the about time one at sheet thickness after in the overall magnitudedramatically of different from the the drag equilibriumthese force. expectations figures, provided it Since by is the we naturalthe dashed to have curves look seen in at that theFigure our equilibrium results formula ( are not are. In all cases, the heavy quark is dragged with velocity β zero to four times thethe actual collision. temperature Thecurves solid show red that curve by isinsensitive the to the the time same choice the of as initialvaries collision the conditions. between occurs solid zero Note the and red that the sixteen drag curveof initial times force in the drag that force string felt figure for felt illustrated by the by here the the red heavy are curve; heavy quark not the quark perturbations small, is to but quite the their initial residual shape effects during and after the collision Figure 6 show the drag forceheavy on quark a that quark is whose being string dragged initially through has an the equilibrium shape plasma with of temperatures the ranging string from trailing behind a JHEP10(2013)013 05 . = 0 can be β 5 -dependence β and -direction as in . 4 x , and the time at ) we see that, once /µ 7 53 , but the dependence 3.1 . γ

e 4.5 T , all in the 5 8 95 973 . . . . β . The remaining 4 , which is 0 =0 =0 =0 =0 0, is similar to the delay between 4 βγ 5 ⊥ ⊥ ⊥ ⊥ eq. with for velocities ranging from → γ β 3.5 and 4 vs. 2 p t µ 3 t 1, i.e. at ⊥ → -independent. γ – 19 – 2.5 β 0 -dependence of our results in ﬁgures at β p t 2 clearly has a component that is linear in peaks. We plot ∆ 2 −

p 7

t 1.5 0 1 between the time at which the drag force on a heavy quark moving

⊥ 0.2 1.6 1.4 1.2 0.8 0.6 0.4

p × µ

dt ⊥ ⊥

2 t

⊥ ⊥

1 dp 2

" ⊥

" 1 2 1

0.9 0.8 0.7 0.6 0.5 0.4 1.3 1.2 1.1 , the dashed curve is p t µ ⊥ βγ . The dashed curve shows the rescaled drag force that the quark would experience at zero rapidity peaks, i.e. the peak in the solid curve in figure . We see that the largest β β 5 . Rescaled drag force for four different quark velocities . The time delay ∆ 973. The increase in ∆ and . 4 = 0 β which the dashed curveto in figure is not completely linear.the The peaks value in of the ∆ purple and blue dashed curves in figures Figure 8 with velocity seen here illustrates theincreases fact with that the delay inin an the equilibrium onset plasma of, with andwe the have subsequent same rescaled peak instantaneous by in, energy the density. drag From ( force Figure 7 figures understood by assuming that the drag force scales roughly with JHEP10(2013)013 and e /T the length γ 7 . γ ) later than the time when the energy ). These considerations are somewhat can be understood qualitatively in terms γ hydro hydro πT ( πT ( / – 20 – γ/ is the temperature defined from the fourth root of the energy , suggesting that the time delay for a fast quark should include in the ‘beam’ direction parallel to the direction of motion of the ) corresponds to the time it takes information from deep inside γ z hydro β T ). Furthermore, for a quark that is moving with a large hydro πT ( hydro / πT may be generic while the fact that (for the colliding sheets of energy above) ( / γ 1 ], in which it has also been attributed to the time it takes for the gluon fields around the moving ∼ 33 , 32 A time delay between a change in the stress tensor of the plasma and the resulting The analysis in appendix B supports a gravitationally intuitive picture in which a time A time delay between some change in the environment in which a heavy quark finds itself and the 7 refs. [ quark to rearrange. 3.2 Heavy quarkIn with this zero section transverse and momentum some the nonzero next velocity we shall consider more general casesresulting in change in which the the drag quark force has on the heavy quark has been seen in other contexts, see for example to grow back. Whetherthat or the not drag such force speculationsonly on are after the correct quark the in responds detail, fieldsthat our to carried takes results changes along some in indicate time. with thethe the If conditions time the moving around delay time quark the that this rearrange quark we takes evaluate themselves were would in constant be a in proportional the way to rest frame of the quark, plasma whose stress tensor isit similar — to only the if instantaneousas stress the tensor the fields of that energy the dress density fluidmoving the of around quark moving the must quark plasma be are increasessmaller. configured stripped the appropriately. off Perhaps longer Perhaps leaving when wavelength only fields the those that energy on dress density length the decreases scales those of longer order wavelength 1 fields need change in the drag forceof that the is boundary proportional to gaugeinstantaneous theory stress as tensor follows. of Clearly,of the the the plasma drag quark in force but which dependsthe also the not drag on quark only force the finds takes on on history the itself its of and “correct” the the value quark. — velocity i.e. In the particular, value it our would results have in suggest an that equilibrium region is proportional to a contribution that isheuristic, proportional however, since to thebehind changes the in heavy quark the responds bulk do geometry not to occur which only the in the string near-horizon dragging region. that the geometry near thethe horizon speed has of changed light, cannotnear propagate meaning the in that boundary the the — bulkat time which faster least it is than what takes determines forof the such the drag a string force change that on to stretches the affect heavy from the quark the string — quark is at the boundary down to the near-horizon density peaks, where density of the fluidanalyzing, at this the corresponds time to when a time the delay fluid of hydrodynamizes. arounddelay one For of sheet the order thickness. collision 1 the we are bulk, near the black hole horizon, to propagate to the boundary. The information investigation indicates that theincreasing fact that thewe time find delay that increases atpeaks roughly low is linearly velocity likely with the agenerically drag peak coincidence. force at peaks a It time close also that to indicates is when about that the 1 at transverse low pressure velocity the drag force may space but over some narrow range of time increases from a low value to a high value. This JHEP10(2013)013 at = 0. (3.3) to be /µ of the x 3 z β ~ β β − = t and 4 and . x β = 0 z β . We shall not consider values ~ β )-plane of two quarks with nonzero z, t 0 z = 0. We shall allow both u u x ≡ β – 21 – fluid z , we first need to ask what the velocity of the fluid β 10 shows the force on a quark with = 0 with some velocity t 10 4. The trajectories are superimposed on the plot of the rescaled = . z . We shall consider cases in which the quark has zero or nonzero = 0 3 z = 0, meaning that the quark passes through the spacetime point β , we shall only consider trajectories in which the velocity z 9 . In all cases that we shall consider, the quark is pulled with constant velocity 2 and . x β as in figure = 0 E z but has no transverse velocity, β z β is the fluid velocity four-vector obtained from the stress tensor in the lab frame . The red arrows show the trajectories in the ( µ u that are greater than 0.4 because we want to ensure that at the time -direction and its magnitude is given by The solid red curve in figure In order to obtain the greatest contrast with the case in which the quark has zero = 0 the quark is at z z t β through the energy density produced by the colliding sheets. ~ depend on its velocitythe relative to that of the moving fluid. The fluid velocity is always in where a nonzero nonzero in the next section. To interpret the force shown inis figure at the location of the quark as a function of time, since the drag on the quark should a heavy quark is producedof at which we choose our initialis about conditions to the catch quark up has with not it yet in felt anyrapidity, the analyzed significant sheet away. above, of in energy this that section we shall consider the case in which the quark has As illustrated in figure quark is constant and inat which the initial positionat of which the the quark sheets hastime, of been we energy chosen are collide. such of that course Although interested in in our gaining qualitative setup insights the into quark circumstances in has which existed for all energy density transverse velocity β colliding sheets, which is to say that we shall allow the quark to have nonzero rapidity. Figure 9 rapidity, with JHEP10(2013)013 at which we choose

/µ 6 e " × 3 T T T − 4, meaning that the fluid is . 6 0 = = 0. The force is shown in the t > x β 4 exact eq. with eq. with eq. with fluid . Reading this figure from left to β 4 4 and 11 . =0.4 2 z = 0 ⊥ µ z , t µ β 2 =0.4 t =0 z in figure x ⊥ – 22 – ⊥ 0 fluid β 0 , defined in the text, at the location of a quark moving along the 4. If the hydrodynamic fluid at late times were boost invariant, 2 : (i) We see that at the time . − 4. Note that at late times . 9 fluid 2 β = 0

− = 0 z 0 1 β

). We show

0.2 0.6 0.4 0.2 0.8 z

− β

µ 0 dt ⊥

2 2.7

0.4 0.3 0.2 0.1 0.9 0.8 0.7 0.6 0.5

⊥

uid fl 2 dp " with 9 . The ‘drag’ force (solid red curve) acting on a quark moving through the matter . The fluid velocity would be given by fluid moving faster than the quark. as described by eq.right while ( referring to figure our initial conditions the quark is already feeling some effects of the fast-approaching sheet Figure 11 trajectory in figure β given by the draginstantaneous force energy that density, the parallel quark pressurerest would frame or experience at perpendicular the in pressure location an as of equilibrium that the plasma in quark. with the The the local dashed same fluid curves are described further in the text. Figure 10 produced in the collisionlab along frame. a The trajectory dashed with curves show what the force would be if in the local fluid rest frame it were JHEP10(2013)013 so 11 can fluid k (3.4) (3.5) β T is near fluid β and fluid ⊥ 8 β T , to the local , -direction with e z ). 11 T 3.5 , = 0 at the location we must first boost ). This gives us the is moving is at rest t      3.1 fluid 10 β RF RF , , ) describes the force on a z 0 eq eq f f 3.1 should not really be thought is smaller or larger than the z z u u ) one obtains ( RF RF , , + + fluid fluid x y eq eq β β f f 2.30 RF RF ), plotted in figure , , t ( 0 z eq eq fluid f f fluid and compare the drag force there with 0 0 β β fluid u u z β ) in the expression ( 10 β − = 0, meaning that      z − z ), and use the temperatures 3.4 β . The expression ( 1 RF – 23 – z, 3.2 and evaluating ( 4 = β z ) on a heavy quark that is being dragged through 1 to confirm that upon boosting the bulk black brane metric u from ( fluid RF β 3.1 z, + 2.3 β RF 0 z, ]. At these early times u 6= 0 can be attributed to the Gaussian tail of the sheet of 9 β = fluid β = 0 at all times at rises toward the speed of light. Note that the quantity lab frame 4 at late times. fluid fluid . , through an equilibrium plasma with the same instantaneous energy β β µ eq 4. Note that if the hydrodynamic expansion at late times were boost . f = 0 = 0. (iv) Finally, at late times after the fluid has hydrodynamized, RF t z, = 0 β fluid z β of the quark in the lab frame. Next, we compute the stress tensor in the local β z β With an understanding of the velocity of the matter through which the heavy quark It is a worthwhile check of the formalism for extracting the drag force from the bulk gravitational 8 quantites that we haveand set the out trailing in string section profile by a velocity density, perpendicular pressure orFinally, parallel we pressure boost as the thatappropriate three Lorentz of forces transformation the computed for matter in forces, this at given way in its back the location. to present context the by lab frame, using the that can be positivevelocity or negative depending onfluid whether rest frame, where itobtained takes as the well form as ( thedrag velocity force that the quarkwith would velocity experience if, in the local fluid rest frame, it were moving at that time.fluid That rest is, frame. we In boosta the velocity by local a fluid velocity rest frame, the quark is moving in the is moving now in hand,expectations we based can upon return the toplasma force figure in ( equilibrium, as wequark did moving in through figure matter atto rest, a so frame to in use which it the at matter any through given which time the quark in figure invariant, the velocity of thethe fluid one at that late the timesother on quark words, a is if trajectory the following expanding with wouldwould fluid constant be at have rapidity given late like times by were the boost velocity invariant, the of curve the in figure quark itself. In as its energy density isof positive as [ the velocity ofthe a left-moving fluid, sheet although of for energy conveniencezero. we slams shall into By refer the symmetry, to right-moving sheet itof as and, the such. briefly, quark (iii) at Next, is close to meaning that the fact that energy which is approachingthe with velocity left 1. catches (ii) upbackground Then, plasma to the the sheet quark ofbe and energy computed incident as as from the we have energy described of even the when sheet the overwhelms matter that is far of from the equilibrium, as long of energy. The quark is initially immersed in the background plasma, which is at rest, JHEP10(2013)013 , k z T and , we (3.6) (3.7) than z , with 4 β z 10 or the -direction, ⊥ z T or the , e T for a quark moving in , z 4 u µ . If we apply this algorithm − u 2 arise at late times. First, we z ) 10 + β 4 x x µ q β β ( u z z meaning that there is an extended ν q and u u u ν /µ − + u 10 0 . x z β β 1 γ 0 0 ) can be either the that the maximum value of the drag force 2 = 5 u u ) t 3.7 10 γ γ πT 2 2 ( ). We plot the three lab-frame forces computed ) ) – 24 – λ π ) or ( 2.7 2 πT πT √ , meaning that their relative velocity is small. In ( ( − 3.6 λ λ π π ) and have allowed for a generic choice of coordinate 11 2 2 √ √ = ~ β , = = (1 , to the expectations for an equilibrium plasma moving with γ , we see in figure 4 ≡ 10 lab frame should be a force pulling backward on it, toward negative , µ q z lab frame lab frame µ eq u β , , f should be negative, as is indeed the case for the dashed curves in x z eq eq f f the fluid velocity and the quark velocity are equal — the dashed curves dp/dt /µ with the same instantaneous energy density or pressure, shown as the dashed 8 . used in the expressions ( therefore cross zero there. Interestingly, we see that the solid curve stays 2 T . This reflects the fact that at late times the quark and the fluid are moving fluid ' . (iii) Instead, the quark is still being dragged forward, toward positive 10 4 β the fluid four-velocity from eq. ( t µ positive. This means that in the local fluid rest frame the force that the external 10 u -direction. As in figure The most interesting differences between figures We can now compare the actual drag force on a quark moving in the x nonzero, the result is x we expect that figure dp/dt agent must exert inquark order to is move moving the towards quark the towards left the in left this acts frame toward the but right. the The force exerted on it by the liquid in figure positive for quite a longperiod time afterwards of until time when:than the (i) quark, The which fluidit would has moving suggest hydrodynamized. at that constant pulling the (ii) against ‘drag’ The the force fluid push on is from the moving faster quark the needed fluid to that keep is moving faster than the quark. That is, see that the magnitudein of figure the dragat force at comparable late velocities, times seefact, is at figure much smaller in figure for the maximum dragsee force spanned a by time the delayforce three in first dashed goes the curves. negative rise as And,in in the as the heavy the in same quark magnitude is figure direction hit and ofthe from only first. the the then left drag goes by force. positive the sheet as Here, of the energy second though, going sheet the of drag energy slams into the solid red curve invelocity figure curves. In many respects, thisthe comparison is as wein found the in far-from-equilibrium figure matter produced during the collision is within the expectations where we have defined axes. The obtained from the stress tensor in the local fluid rest frame. which can be written as as we have just described asin the the three more dashed curves general inβ case figure in which in the lab frame the quark is moving with both with JHEP10(2013)013 12 , the /µ 8 . = 2 RF t

e " × T T T 5 exact eq. with eq. with eq. with 4 µ RF these gradients are aligned parallel to the t =0.4. Fluid rest frame 3 10 z , the ‘drag’ force that must be exerted to maintain × – 25 – , /µ 0 . =0 x both because the eﬀects of the gradient on the ‘drag’ × = 5 acting toward the left, pushing it in the direction of its 4 2 , in ﬁgure RF t 4

, plotted here in the local ﬂuid rest frame and focusing on times show the gradient of the ﬂuid velocity in the lab frame and

also

⊥

0 ⊥ 10 and 13

0.1 ⊥ 0.2

0.05 0.15 0.05

⊥ −

/µ RF

" µ

⊥ 8

2 .

⊥

2 dp ⊥ = 2 but not in figure RF t Unlike in figure 12 9 to better illustrate the behavior after hydrodynamization. After . Forces as in figure and /µ 5 . 1 10 The red curves in figure > We have also neglected spatial gradients in the energy density of the fluid. In the following, we will show 9 RF that after hydrodynamization thewell effects with of the spatial behavior gradientsthe of on energy the the density gradients are drag of small the force at fluid that these velocity. we late We compute times, have correlate checked as that are the their spatial effects gradients on of the force that we compute. force are larger in absolutequark magnitude and when because the gradient insmall is this since aligned case the with the relative the motion velocity drag of of force the the in quark and the the absencein fluid of the is gradients local so would fluid small. be rest frame, very in each case projected onto the quark velocity in that frame, fluid velocity. direction in which thethe quark gradients in is the moving; fluidbe the velocity can exerted lesson have on we qualitative the effects learnfigures on quark. is the that ‘drag’ Qualitative force in effects that this of must the circumstance gradient in the fluid velocity arise in through which it is movingmotion is rather than draggingby it plotting in the the forcea opposite exerted result on direction! as the surprising We quarkvia as illustrate in the this this the dashed is in local curves, possible figure we fluid is are rest that completely frame. in neglecting formulating The the our reason effects expectations, that of as spatial shown gradients in the than the quark) and thethe dashed quark leftward curves in accordingly the leadsee local us that fluid to between rest expect frame thatthe should the be leftward force motion a needed force acts to acting toward drag toward the the right! left. Instead, we Figure 12 t quark is moving to the left in the local fluid rest frame (in the lab frame, the fluid is moving faster JHEP10(2013)013 0,

> (3.8) (3.9) 01 07 2 7 . . . . x 6 , which β z =0 =0 =0 =0 =0 β x x x x x 4 and add . ⊥ ⊥ ⊥ ⊥ ⊥ , signiﬁcantly 5 = 0 and but in the lab RF 2. Because the z z, . 4 x β = 0 and β β 2, we should ﬁnd . z µ = 0 β 4 . In all curves in both RF x z t = 0 β β 4, Fluid Rest Frame x . β =0 3 z RF ⊥ z, β is substantially reduced while in RF 4 and RF . z, 2 β β RF ˆ z β = 0

β β ·

RF 0

but now with ﬂips, which is to say that in the lab frame

⊥ 0.2 0.4 0.6 0.4 0.2

− −

RF β RF ) u (

⊥ RF ˆ z ) ∂z RF 11 ∂z ∂u z ∂u z, u β ≡ 4. The diﬀerent curves are for trajectories with shows that if we start with . ≡ ∇ and in the lab frame but that is large compared to . We shall conﬁrm this expectation in the next ˆ β

– 26 – 7 · = 0. For trajectories with z 13 RF = 0 ) 10 β 10 07 2 7 ˆ x β . . . z z · β u β =0 =0 =0 =0 in the lab frame, but is comparable to ∇ x x x x RF z 6 ( ⊥ ⊥ ⊥ ⊥ ) β , the sign of z u 4 and /µ . 8 ∇ . ( 5 2, the quantity ( = 0 . = 2 4 as in ﬁgures z . is not much changed. This suggests that if we analyze the drag µ β RF = 0 t t ˆ β Laboratory frame 4 x = 0 · . . 4 β . ) z z . At β RF u =0 x ˆ β z β · 3 ∇ ⊥ that is small compared to RF ) x z β u . Component of the gradient of the ﬂuid velocity in the direction of motion of the quark 2 ∇ , for example

⊥ that is quite small compared to

0.3 0.2 0.1 0.6 0.5 0.4

u ·

⊥ RF ˆ x z z, β β through the colliding sheetsis of to energy say along witha a both trajectory trajectory transverse momentum with with and nonzero velocity rapidity of nonzero. the quark We now startlocation has by a of considering component the perpendicular quark to (this the was velocity not of the the case fluid in at the both previous sections) we now expect and results that in the localframe fluid are rest more frame similar are tosection, more those but similar in in to figure those so in doing figure we shall3.3 discover a second surprise. Heavy quarkIn with this nonzero section rapidity we and analyze transverse the momentum force that must be exerted in order to move a heavy quark both quantities vanish identically. Figure a nonzero | the lab frame ( force on a quark that follows a trajectory with and for the trajectory with the velocity of thea fluid exceeds the velocityreduces ( of the quark. We see in the rightnamely panel that adding Figure 13 at the location ofpanel). the The quark differences inpanels, between the the the lab two trajectory frame panelsvarying of (left arises values panel) the of because and quark in has the local fluid rest frame (right JHEP10(2013)013 4, . /dt 13 ⊥ = 0 (3.10) dp z β , the dashed 2 and . 3.2 = 0 and, below, x and β

6 was zero, though, with e × ⊥ /dt 3.1 k T T T x β dp 4 exact eq. with eq. with eq. with 2 does not substantially reduce the . =0.4 , = 0 z . (Note that 2 ~ ⊥ β x , µ · β 14 β t ~ f =0.2 ≡ x – 27 – ⊥ k 0 dt dp 4, turning on . 2 − = 0

z 1 0

0.2 0.8 0.6 0.4 0.2

−

× µ dt "

are deﬁned via the pressures in the ﬂuid that act in the directions

⊥ -direction.) We see that, as in both sections

⊥ in the sense that the drag force in the direction of motion of the quark

z 2 " dp ⊥ T 3.2 and k . The force (solid red line) in the lab frame that must be exerted in the direction of T 2 at least the sign of the force is the same for the solid and dashed curves. The . At late times, after the fluid has hydrodynamized, we see behavior that is more similar Let us first look at the drag force parallel to the direction of motion the quark in the = 0 x β dashed curves nevertheless failafter to the give fluid a haswhich qualitative shows hydrodynamized. description that, of when Thismagnitude the of is actual the consistent drag component of with force the the gradient of left the panel fluid velocity of in figure the direction of motion curves provide a reasonable guidepeaks to later the than peak the value dashed of curves the do, drag a force timeto delay that but in that the section is actual by force is now affected familiar. by the gradientsexpectations in spanned the by fluid the velocity three to dashed such curves. a degree Unlike that when it is well outside the which is shown asrefer the solid to red the curve forcecontrast, in parallel figure and perpendicularparallel to to the and perpendicular directionof to of the the motion fluid, direction of i.e. of the the motion quark; of in the colliding sheets and hence has a component perpendicular to the direction of motionlab of frame, the quark. described in the previousquark section, show were what being this dragged forcedicular through would pressure a be or plasma if parallel with in pressure the the as local same that fluid instantaneous at rest energy the frame location density,find perpen- the of that the the quark. force that must be exerted in order to move the quark along this trajectory Figure 14 motion of the quarka (in trajectory the lab with frame) nonzero in transverse order momentum to and drag rapidity. it with The a dashed velocity curves, computed as JHEP10(2013)013 that (3.11) 12 2, we see . = 0 x β , when we turn z β |

e × ⊥ RF T T T z, . The dashed curves show β 5 | 16 exact eq. with eq. with eq. with . z 4 β β x µ f we see that the story is diﬀerent in the RF − t x =0.4. Fluid rest frame 15 β β z 3 z × – 28 – , f ≡ are nonzero. In the lab frame, the existence of a =0.2 the dashed curves show that even if there were no 2, we ﬁnd that the drag force in this frame does lie ⊥ . x z dt × dp β 16 2 = 0 , where we saw that, because

plotted here in the local ﬂuid rest frame and focusing on times

⊥ β and

⊥ 14 0

⊥ x

0.2 0.1

⊥ β

0.15 0.05 0.25

" µ dt "

⊥ ⊥

2 ⊥ dp = 0 the dashed curves that show what the drag force in the local fluid rest x β 4 and to better illustrate the behavior after hydrodynamization. We saw in figure . 2 we do substantially reduce the component of the gradient of the fluid velocity . Forces as in figure . /µ = 0 5 . z 1 = 0 β x The force perpendicular to the direction of motion of the quark is a new development, Turning now to the force acting perpendicular to the direction of motion of the quark, > β RF present only whenperpendicular both force isthrough no a surprise fluid since whosereason, in velocity in this includes the a frame left component of panel perpendicular reference of to the figure its quark own. is For moving this It is plotted aswhat the the solid force red perpendicular curveframe in to if, the the in the left direction local panel of fluidwith of rest motion the frame, figure same of the instantaneous the quark energy were quarkthat density, moving perpendicular through would at an pressure, be the equilibrium or location in plasma parallel of the pressure the as lab quarks. in the direction of motion of the quark —we in fix the our local sign fluid conventions by rest defining frame. this as of the quark —local in fluid the rest frame. labwithin frame. the With range In ofwith figure expectations the spanned right by panel of theon figure three dashed curves. This is consistent frame would be intransverse a spatially pressure homogeneous or plasma parallelthey with neglected pressure the the same failed effects instantaneous to ofthat energy gradients describe after density in or the hydrodynamization the fluid the actualspanned velocity actual force on by force the on the on force. dashed the the Here, curves. quark, quark with because falls within the range of expectations Figure 15 t when JHEP10(2013)013 . 14 4 are nonzero. A that the answer 2 z µ β 16 RF t =0.4. Fluid rest frame z and × 0 , x β =0.2 x ×

−

⊥

⊥ 0

⊥ 0.2 0.1 0.2 0.1 0.5 0.4 0.3

⊥ − −

" µ dt "

⊥ ⊥

2 dp ⊥

6 – 29 – e ⊥ × T T T , but here we plot the force that must be exerted perpendic- 4 exact eq. with eq. with eq. with 14 =0.4 z 2 ⊥ , µ t =0.2 ]. Here we see this phenomenon arising as a robust consequence of x ⊥ 0 36 – 34 2 − thus represents a stark consequence of the presence of spatial gradients in the

. Left panel: as in ﬁgure 0 0.1 0.2 16

0.25 0.05 0.15 0.05 − −

− − −

× µ dt "

⊥ ⊥

2 dp " drag force that includes athrough component perpendicular a to medium the arises directionbeen of in introduced motion other [ of contexts the quark ingradients which in some the anisotropy fluidcollisions. in velocity — the a medium form has of anisotropy that must be present in heavy ion matter produced in theis collision. far from The equilibrium.of effect the At is force these largest perpendicularvalue early at to of times, early the the the times, motion drag effectnonvanishing of when force at is the the that late large quark acts matter times, indeed: isresult parallel after about to to the the half the be peak expanding as velocity generic, value fluid large of arising has as the for hydrodynamized. the quark. any peak We trajectory The find in effect this which is both also in which the quark isalong moving its through trajectory a through this fluidof fluid at motion includes rest, of a the the component forcebe quark! perpendicular required no to In to the component this drag direction of frame, theof the quark in figure force the perpendicular absence of to gradients the in motion the of fluid the there quark. could The right panel velocity gradients in thedirection fluid of the motion force of wouldto the include the quark local a in fluid component the restdirection perpendicular lab frame of frame. to and motion ask the This of whether posesis the in no: quark. an that obvious even frame We in question: the show a drag in boost reference force the frame is right in parallel panel which to of at the figure each point in time we have boosted to a frame location of the quark)The the dashed force curves in therefore thisin all frame fact vanish could quite in only substantial the act atquark right parallel its includes panel. peak: to a The the component even actual direction in perpendicular force of the to local does the its fluid quark. not direction rest vanish of frame, and motion. the is force needed to drag the Figure 16 ular to the directionthis of component motion of of the theRight quark, force panel: in is the the significantly force labin smaller that frame. the than must local Note be the fluid the exerted component rest(e.g. scale perpendicular frame. the of on to equilibrium the If, the the plasma in force vertical with direction this in axis: the of frame, same figure motion the instantaneous of quark energy the were density moving quark or through pressure a as homogeneous that fluid at the JHEP10(2013)013 x 4. β .

6 e ⊥ × T T T = 0 z β 4 4 exact eq. with eq. with eq. with , but here for a 7 and . 4 after boosting . =0.4 16 2 . The important z 2 ⊥ µ = 0 , µ = 0 t RF 16 x t z β =0.4. Fluid rest frame =0.7 that by increasing β z x × ⊥ 0 0 , and 13 =0.7 14 7 and . x × 2

2 −

= 0 −

⊥ x

⊥ β 0 0

0.1 0.2 0.3 ⊥

0.1 0.4 0.3 0.2 0.1

0.15 0.25 0.35 ⊥ 0.05 0.05 − − −

− RF

" µ

− − − − "

× µ

dt "

⊥

2 dp and the left panel of ﬁgure ⊥ 2 dp " 14

– 30 – e × ⊥ e × ⊥ T T T T T T , but here in the local fluid rest frame. that we saw in figures 17 x 4 β 4 exact eq. with eq. with eq. with exact eq. with eq. with eq. with we see that at late times, after hydrodynamization, the 4. . =0.4 2 17 z 2 µ ⊥ , = 0 µ RF t z t . As in figure =0.4. Fluid rest frame β =0.7 z x × ⊥ , 0 0 7 and =0.7 . x , we show our results for the case with × Figure 18 = 0

18 2

2 x

we ﬁrst show our results in the lab frame, ﬁnding results that are similar in −

− ⊥

. Drag force parallel to (left panel) and perpendicular to (right panel) the direction of ⊥

⊥ 0 3 2 1 17

, here there are no qualitative eﬀects of the gradients in the ﬂuid velocity apparent.

1 0 ⊥ 2 3

0.5 3.5 2.5 1.5 RF

" µ

1.5 0.5 " 2.5

2 × µ

dt "

2 14

⊥

⊥ In ﬁgure To get a sense of how generic our results are, we close this section by illustrating them

⊥ dp 2 " dp from 0.2 to 0.7velocity we in have the substantially direction reduced of the motion component of of the quark theto in gradient the the of local lab the frame. fluid fluid moving rest through a frame. fluid that Again, is we instantaneously at find rest that the even force in required to a drag frame the in quark which the quark is In figure most respects to thosedistinction at is smaller that indrag force figure that we calculatefigure is well described by theWe dashed can curves understand meaning this that, unlike by in noting from the left panel of figure for the case where the heavy quark is dragged along a trajectory with Figure 17 motion of the quarkquark in with the lab frame, as in figure JHEP10(2013)013 ]. 42 ) for – 37 3.1 ) to learn about through the matter 3.1 is the temperature of β 2. This reflects the fact . hydro T = 0 x β increases. This would correspond 2 ) (where β that appears in it through the energy , where − T hydro 1 16 p πT ( / / in a heavy ion collision at RHIC. This delay – 31 – /c the component of the gradient of the fluid velocity in the z β ) turned into a parameter to be fit to data; (iv) Use the resulting spacetime- 3.1 increases at fixed x β 7 than it was in the right panel of figure . It also seems to be a generic feature of our results that there is some time delay after Our results indicate that a straightforward approach along the lines above can reason- = 0 x simpler setting suggests that itthe is fluid of when order it hydrodynamizes) 1 is for not a heavy relativistic, quark andto whose increases velocity a slowly delay assuggests of that 1 something it takes like a little 0.1-0.2 fm time after the heavy quark is enveloped by matter with a high most robust lesson for heavy ion physics that wethe can collision draw before from the ourthough rate results. it of is energy during loss thishas of very the the earliest very heavy time highest quark energy that risesthe density. the to case Although matter its of we in peak have the which not value, the collision characterized even quark the of is delay sheets immersed time of in energy quantitatively, our analysis of this delay in a The message of ourheavy quark calculation energy seems loss to incollision, be heavy even ion that before collisions, if it equilibration,density. we is defining reasonable The want the to to error apply that usein it one ( throughout this would the way make is by likely treating to be the smaller far-from-equilibrium than energy other loss uncertainties. We anticipate that this is the though the peak value ofsheets the of energy energy loss that inthe the we matter have matter analyzed produced produced in occurs in the beforetheless the collision reasonably hydrodynamization, collision of well at is the reproduced a still two bysign time far the of when from straightforward any equilibrium, approach. significant this Certainly “extra” peak there energy value is loss is no arising never- by virtue of being far from equilibrium. prefactor in ( dependent drag force, and consequent energymodel loss the rate, dynamics in of a heavy Langevin quarks equation in employed heavy to ion collisions,ably as be for applied example even in at refs. very [ early times, before hydrodynamics applies. In particular, even described via viscous hydrodynamics)and into an time; effective (ii) temperature Use asand perturbative a momenta QCD of function heavy to of quarks space calculateheavy produced ion the via collision; distribution hard scattering (iii) of atthe Use the the drag the initial earliest force effective positions moments in temperature of a the from homogeneous (i) plasma in in the thermal expression equilibrium, ( perhaps with the overall 4 Conclusions and lessonsThe for straightforward approach heavy to ion modelling the physics in rate of a energy loss heavy of ion heavyturn quarks collision the produced proper proceeds energy as density follows: as a (i) function of Use space the and time equilibrium in equation the of collision (for state example to this effect of gradientsmuch smaller in relative the to the fluidβ drag velocity force is in the somewhatthat direction smaller of as motion in of absolute thedirection terms, quark, of here and motion where of the quark decreases. along its trajectory includes a component perpendicular to the trajectory. We see that JHEP10(2013)013 – 32 – ) for a homogeneous fluid and therefore neglects all 3.1 From a more theoretical perspective, next steps that our results motivate include re- We have found qualitative consequences for the drag force on heavy quarks with larger The straightforward approach to modelling the drag force on a heavy quark in a heavy We would like to thank JorgeStanislas Casalderrey-Solana, Mrowczynski, Michal Juan Heller, Hong Pedraza,mann Liu, Silviu for David Pufu, Mateos, helpful Wilke discussions. vanunder This der work cooperative was Schee research supported and agreement by DE-FG0205ER41360. Urs the Wiede- U.S. Department of Energy analytical understanding of the effectson of heavy gradients quark in energy the loss. fluid velocity (and temperature) Acknowledgments effects. That beingthe for straightforward the approach to future,should modelling at be heavy present used quark the with dynamics caution fourth for in lesson heavy heavy from quarks ion at our collisions peating high results our rapidity. analysis is for the that collisions of sheets of energy with varying widths and seeking an maximum drag force acting parallel tononzero the at velocity late of times the too, quark.well when The the by perpendicular quark viscous force is is propagating hydrodynamics.presence through of a gradients This liquid in that perpendicular the isof fluid described force heavy velocity. can quark Here also too, dynamics it be in would heavy attributed be ion interesting to to collisions the use to a investigate model the consequences of these the same direction ascally, the we velocity find of that the theperpendicular quark, force to rather exerted the than by velocity dragging theperpendicular of fluid on the force on quark, it. can a even be And, as heavydynamization, generi- large; seen quark its in we will peak the found have value local a instances in fluid component where the rest at frame. local early This fluid times, rest before frame hydro- was about half as large as the rapidity, moving closer topresence parallel of a to velocity the gradient,at which gradient all is in in to the the say qualitative straightforwardfound fluid phenomena approach that velocity, that that because are we arising have not of from sketchedquark present the above. the that gradients For has in example, a the we small have fluid velocity the relative force to exerted the by fluid the at its fluid location on can a sometimes heavy point in ion collision is built uponeffects the result of ( gradients inour the results fluid is velocity. that A thiswhose third neglect velocities qualitative works are lesson reasonably close that well to for we perpendicular heavy can to quarks infer with the from small gradient rapidity, in the fluid velocity. the matter, with thestrongly drag coupled calculation and cannot energy provide amoments loss complete of characterization rising a of only heavy the ion very afterIt earliest collision, would this it be response. is interesting possible toinvestigate Although that whether use this our a a qualitative time model lesson delay of may along carry heavy these over. quark lines dynamics has in observable heavy consequences. ion collisions to energy density before the gluon fields around the heavy quark respond to the presence of JHEP10(2013)013 ) . σ and A.4 (A.4) (A.1) (A.2) (A.3) z and that, in ) as our background τ u T ) and ( A.4 ) we choose and A.3 2.5 ) the temporal τ . ) and ( 2.10 h u ) A.3 ). With this choice of γ uξ √ ), the trailing string pro- , as functions of u γ ~x h 2.5 √ u (1 + u 1 h 1 and background , − u − t as a function of ) πT ) meaning that the event horizon , t ( tan u uξ at which we are initializing the string tan h u / u u γ t 2.5 γ 1 ). Because we are initializing at a time √ √ − h = 1 (1 + h u u 1 h τ, σ , at any given early Eddington-Finkelstein ( tan − u + t + h τ τ 2.2 u tan – 33 – = = h + u t h h u u . This together with the expressions ( u + , upon making the transformation ( ) , which is to say σ ~ 1 t β uξ − M u = = 2.2 ~ X β u ~x tan = (1 + h u 1 ~x through an equilibrium plasma with temperature − + ~ β t , even deep in the bulk, even though deep in the bulk where the tan σ . After we make the transformation ( h 2 ) does not satisfy the temporal constraint equations. The other option u ~ β + − A.4 t ) becomes a differential equation for 1 ) by a numerically determined q ) and then to solve the temporal constraint differential equation numerically, / 2.13a A.4 , has the solution A.3 = 1 ) is given by γ ] , even well before the sheets of energy collide on the boundary, the gravitational t 2 A.1 If all we were interested in doing was dragging a heavy quark through static plasma, , directions. As we mentioned in section 1 z background -direction with velocity initial conditions at all gravitational shocks are already colliding atprofile the the early choice ( is to use ( replacing ( time shocks are already colliding somewhere deepon in the the bulk. worldsheet Eddington-Finkelstein are coordinates the advantageous evolution from equations the but point fromfeature of the view is point of a of making view complication.have it of followed possible specifying in There to the obtaining are initial solve all two conditions this the possible results ways that to we proceed. show is The to route use that ( we there would be nothing to add.heavy quark The is background initially of in interest afuture to region the us, of quark however, spacetime is will filled one be with where− slammed static the by plasma two but sheets in of which energy, incident in upon the it from the + As we have describedworldsheet coordinates in in which section give us our initial conditions for file ( and for the case of a static plasma the solution to the temporal constraint equation is now T where where we have notof the yet black made brane thestring in transformation profile, the ( when bulkconstraint we is ( choose located worldsheet at the coordinates case of of a the time-independent form metric ( corresponding to a static plasma with temperature The trailing string solution that describes an infinitelyis heavy [ quark being dragged in the A Trailing string solution in an equilibrium background ~x JHEP10(2013)013 (B.2) (B.1) ) where there setup is not a A.3 ad hoc ) with 2.4 , 4 ! ) t ∂t ∂ξ ( 2 h ) at all times. , u ) change with time. This of course t − t ( )) ) ( ) itself solves the temporal constraint t T = 0 ( u T u, t = F uξ A.4 ( k f = T 2 (1 + , and in particular which always has zero fluid = ) t – 34 – T ⊥ ( ,B ξ T ) − t + ( = ξ e 1 u T + ) = 1 1 u

u, v = ( that when the energy density and pressures of the matter f Σ = A 3.1 0) at all times. Also, , 0 , 0 , The metric that describes a spatially homogeneous plasma with a time-dependent = (1 when the quark at the boundary is in a region of static plasma, these two options yield µ with u temperature is given in Eddington-Finkelstein coordinates by ( bulk metric that providessolution the to dual Einstein’s gravitational equations. descriptionanything; of The it this setup is in simplythe this a drag appendix device force is with on thereforefinds which a not itself. to heavy a Note evaluate quark model that the to in for time a this setup delay change there in in are the the no conditions spatial response gradients in of of which the the fluid heavy velocity quark since delay in a time-dependentenergy background that are that our focus is throughoutthat, much the at simpler rest of all than this times, paper. theis is We colliding in a shall thermal consider sheets spatially a equilibrium homogeneous of background velocity at plasma temperature throughout, whose properties but are byviolates exactly hand energy as we conservation if shall in it make the boundary theory. It is therefore no surprise that the We discovered in section through which the heavyresponse quark of is the moving dragthe change force drag there experienced force seems would by to changeinstantaneous the be if energy heavy it a quark, density were time relative given or delay to by pressure. in its the value the way In in in a this which static appendix plasma with we the shall same quantify this time boundary. Since the dragasymptotics force of on the the string profile, heavy nothing quark in is this computed paragraphB from affects the it. near-boundary Varying temperature the distinction between them is irrelevantthat is well that before it the sheets arises collide onlythese at deep considerations the enough has boundary within been the the enveloped region by bulk ofthis, the the the event string horizon initial that of violation is the of affected blackoption) by the brane. and temporal Because the constraint of initial equation arbitrariness (assumingis associated we no pursue reason with to the choosing do first the so profile (in either ( the first or second option) are causally disconnected from the identical results close toequation. the They boundary, are inequivalent where deepfor in ( how the to bulk, initialize and the indeed string thereWe profile is have deep no checked, in one however, the “right that bulk answer” these inon Eddington-Finkelstein two the coordinates. options heavy yield quark, identical meaning results for that the for drag our force purposes they are equivalent. The reason that t JHEP10(2013)013 ) ). ), t ( kµ 3.1 ( T (B.3) (B.4) / 1 at early = 0. In µ ∼ t = πT at early times and µ . We see that the 2 )

= ) and shows what the /µ moving through a spatially ) 3.1 πT t β ( 5 9 95 . . . πT =0 =0 =0 . = 0 the apparent horizon is at equilibrium 2 ⊥ ⊥ ⊥ ) into ( ξ )) t ) that increases from ( t ( T , T k t µ 1 ). The diﬀerent solid curves show the scaled -dependent even when rescaled as in the − 1 µ -dependent, time delay. The temperature )) becomes a step function at β t ) t B.4 t β ( ( 1 h T (1 + erf( u = 1 corresponds to choosing by demanding that the apparent horizon be at 1 4 – 35 – and is a plot of ( u ξ − ) describes the residual diﬀeomorphism introduced ) = β 0 t t ( ( ξ ξ ) = 1 , for a heavy quark with velocity t ( h βγ u ) = 0. 1

−

4 3 2 1

, t ) (and therefore

⊥ t ) there is only a single dashed curve.

3.5 2.5 1.5

× µ

⊥ t ⊥ (

2 (

⊥

⊥ = 1

= 3. We have plotted the drag force rescaled in such a way that

T 1 dp 2 u ⊥ " " )) and where 2 t u , ( k = ( k f k T π T at late times, rising smoothly during a window in time that is ( at late times and is described by ( = / µ µ ⊥ a parameter. We choose T = 2 = 2 k ) = 1 = . Drag force, scaled by t e ( , we choose ) whereas ensuring that it is at πT T πT h t ( ). As we discussed there, we fix u h 19 u We shall choose a time-dependent temperature that starts at 2.5 = 1. In this simplified setup, in coordinates in which = drag force would beis in independent an of equilibriumactual the plasma drag with quark force, the shown velocity instantaneousfigure as temperature and the shows solid a curves,of significant, is the and somewhat plasma has risen quite significantly before the drag force begins to rise; once the For large values of figure the dashed curve, which is obtained by substituting which ensures that ends at wide, with in ( u u times to drag force on quarks with differentBecause velocities. The dashed curve is the equilibrium expectation ( where Figure 19 homogeneous plasma with a time dependent temperature JHEP10(2013)013 is the final ) and then T we show how initial time dependent, 20 πT u ( ) where / as the delay between t .

6 final γ πT ( as described in the text, as a / we see that at low velocities , the parameter that controls k 19 k 5 moves the drag force only learns of h u 4 = we see that the time delay increases with u ⊥ k when they cross 2.5. In figure – 36 – that controls the rapidity with which the temperature 3 19 k 2 ⊥ . We shall define the time delay ∆ 5, which we note is around 1 / . 2 19 =1 =3 , defined from results as in figure k k t ) changes. For both values of t ( ). This interpretation should not be taken too literally, however, T

ﬁnal

0.6 0.5 0.7

0.65 0.55 t µ ⊥ time dependent makes the entire bulk metric at all πT ( This article is distributed under the terms of the Creative Commons h / u in a way that is close to, but not exactly, linear with for two values of the parameter β γ . The delay time ∆ From a gravitational perspective, the fact that there is a time delay can be described There are various ways in which we could choose an operational definition of the time depends on the quark velocity for two different values of t not just the metric near the horizon. Open Access. Attribution License which permits anyprovided use, the distribution original and author(s) reproduction and in source any are medium, credited. horizon travelling through the bulkthe to heavy reach quark the is near-boundarythe region, encoded. setup where of the If this dragdrop taken on appendix to literally, the around this 1 time interpretationbecause delay would making should suggest start that out in around 1 final temperature. And,increasing for both values of in qualitative terms asthis if when at the a horizon time at that is delayed by of order the light-travel-time for information from the delay from the results inthe figure times when theforce dashed and the and final drag solid force,∆ curves i.e. cross in figure the midpointthe between rapidity the with initial which there drag is a time delay of around 0 temperature reaches its final plateaufinally, the the drag drag force force over-shoots is before only approaching about its half new equilibrium way up value from its above. rise; and, Figure 20 function of increases. JHEP10(2013)013 ] ] ] ] ] D 82 ]. ] Adv. , ]. = 4 SPIRE (2013) N Gauge/String IN [ SPIRE Phys. Rev. (1998) 253 , IN D 87 hep-th/0205236 2 ]. arXiv:0812.2053 [ arXiv:1202.0981 ][ ]. hep-th/9711200 [ hep-th/0605182 ]. [ From full stopping to ][ [ (2006) 013 arXiv:1011.3562 [ SPIRE 07 SPIRE IN SPIRE [ IN Phys. Rev. IN Strong Coupling Isotropization (2002) 043 (2012) 201602 ][ , ][ arXiv:1305.4919 JHEP 06 , (1999) 1113 Energy loss of a heavy quark , (2009) 211601 (2012) 191601 108 (2006) 126005 38 arXiv:0910.0748 ]. [ (2011) 021601 102 108 JHEP , D 74 106 Adv. Theor. Math. Phys. SPIRE , arXiv:1101.0618 IN hep-ph/0605199 [ hep-th/0512162 The characteristics of thermalization of , [ ]. [ supersymmetric Yang-Mills theory – 37 – Phys. Rev. Lett. Black brane entropy and hydrodynamics: The (2009) 126013 , Phys. Rev. Heavy quark diffusion in strongly coupled SPIRE , ]. = 4 Phys. Rev. Lett. Phys. Rev. Lett. IN , , Int. J. Theor. Phys. N Asymptotic perfect fluid dynamics as a consequence of ][ D 80 Phys. Rev. Lett. limit of superconformal field theories and supergravity , SPIRE (2006) 085012 Horizon formation and far-from-equilibrium isotropization in Boost invariant flow, black hole formation and Holography and colliding gravitational shock waves in Numerical solution of gravitational dynamics in asymptotically (2006) 045013 ]. N IN ]. ]. ][ arXiv:1309.1439 , D 74 Adding flavor to AdS/CFT supersymmetric Yang-Mills plasma D 73 SPIRE (1998) 231 [ Phys. Rev. SPIRE SPIRE spacetime IN , 2 IN IN 5 = 4 Holographic thermalization with radial flow ][ ][ ][ The Large- N arXiv:0906.4426 [ Drag force in AdS/CFT Phys. Rev. Anti-de Sitter space and holography Phys. Rev. , , ]. ]. ]. ]. ]. ]. arXiv:1211.2218 [ SPIRE SPIRE SPIRE SPIRE SPIRE SPIRE IN arXiv:1103.3452 IN IN hep-th/9802150 IN IN hep-th/0605158 IN of Non-Abelian Plasmas Simplified [ 061901 Boost-invariant case boost-invariant plasma from holography [ supersymmetric Yang-Mills plasma [ far-from-equilibrium dynamics in (2010) 026006 transparency in a holographic model of heavy ion collisions anti-de Sitter spacetimes AdS/CFT Duality, Hot QCD and Heavy Ion Collisions asymptotically AdS [ [ [ Yang-Mills Theor. Math. Phys. [ moving through [ [ P.M. Chesler and L.G. Yaffe, J. Casalderrey-Solana, M.P. Heller, D. Mateos and W. van der Schee, E. Witten, A. Karch and E. Katz, J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U.A. Wiedemann, J. Casalderrey-Solana and D. Teaney, J.M. Maldacena, C. Herzog, A. Karch, P. Kovtun, C. Kozcaz and L. Yaffe, S.S. Gubser, M.P. Heller, D. Mateos, W. van der Schee and D. Trancanelli, W. van der Schee, I. Booth, M.P. Heller and M. Spalinski, M.P. Heller, R.A. Janik and P. Witaszczyk, P.M. Chesler and L.G. Yaffe, P.M. Chesler and L.G. Yaffe, P.M. Chesler and L.G. Yaffe, R.A. Janik and R.B. Peschanski, [8] [9] [5] [6] [7] [3] [4] [1] [2] [16] [17] [14] [15] [12] [13] [10] [11] References JHEP10(2013)013 , ] , ]. 09 , , , = 200 08 Int. J. (2012) , SPIRE NN ]. s JHEP IN , √ C 86 JHEP ][ , (2001) 595 5 SPIRE ]. ]. IN (2009) 002 217 AdS ][ nucl-th/0407067 (2012) 065 [ 06 ]. SPIRE SPIRE 06 Phys. Rev. IN IN , ]. Kerr-AdS Black Holes ][ ][ SPIRE D JHEP 4 IN , JHEP nucl-th/0305084 ]. (2005) 163 Drag force in a strongly coupled SPIRE , ][ IN Holographic isotropization ]. ][ 739 hep-ph/0509185 SPIRE [ IN ][ SPIRE ]. ]. ]. Commun. Math. Phys. IN , arXiv:1211.2186 Parton Energy Loss in the Extremly Prolate arXiv:1301.4563 Holographic reconstruction of space-time and [ ][ [ Systematic parameter study of hadron spectra Far-from-equilibrium heavy quark energy loss at (2006) 011 SPIRE SPIRE SPIRE Retardation effect for collisional energy loss of arXiv:1202.3696 IN IN IN [ 04 – 38 – ][ ][ ][ ]. AIP Conf. Proc. ]. arXiv:1010.1856 ]. , [ ]. Energy loss in a strongly coupled anisotropic plasma (2013) 861c On the collision of two shock waves in JHEP SPIRE Anisotropic Drag Force from arXiv:1304.5172 SPIRE , IN [ SPIRE IN (2012) 100 hep-th/0010138 SPIRE Early-Time Energy Loss in a Strongly-Coupled SYM Plasma ][ IN ]. ]. [ ][ A904 IN [ Hydrodynamic description of ultrarelativistic heavy ion collisions ]. Moving Quark in a Viscous Fluid 08 ][ Holographic Jets in an Expanding Plasma (2010) 054904 , R.C. Hwa et al. eds. (2003), pg. 634 [ SPIRE SPIRE arXiv:1102.4893 arXiv:1012.3800 arXiv:1206.2271 SPIRE (2013) 026 PoS(Confinement X)175 [ [ [ IN IN JHEP , IN , ][ ][ (2001) 740 09 (1983) 140 ][ C 82 Nucl. Phys. , Asymptotically Anti-de Sitter space-times and their stress energy tensor Energy Loss of Heavy Quarks from Asymptotically AdS Geometries Thermalization at RHIC Heavy quark in an expanding plasma in AdS/CFT A 16 hep-th/0605191 arXiv:0803.3226 Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region D 27 JHEP [ [ (2011) 108 (2011) 070 (2012) 085 , ]. arXiv:1110.2943 05 04 10 [ Phys. Rev. , Quark gluon plasma SPIRE arXiv:1202.2737 arXiv:0904.1874 hep-th/0002230 IN anisotropic plasma JHEP JHEP hard partons produced in a QGP JHEP [ Phys. Rev. strong coupling [ 054905 Mod. Phys. (2006) 032 (2008) 027 renormalization in the AdS/CFT[ correspondence in [ quark-gluon Plasma GeV linearized and elliptic flow from viscous hydrodynamic simulations of Au+Au collisions at A. Nata Atmaja and K. Schalm, K.B. Fadafan and H. Soltanpanahi, S. Peigne, P.-B. Gossiaux and T. Gousset, A. Guijosa and J.F. Pedraza, M. Chernicoff, D. Fernandez, D. Mateos and D. Trancanelli, N. Abbasi and A. Davody, J. Bjorken, G. Giecold, A. Stoffers and I. Zahed, K. Skenderis, C.P. Herzog, P. Chesler, M. Lekaveckas and K. Rajagopal, D. Grumiller and P. Romatschke, S. de Haro, S.N. Solodukhin and K. Skenderis, U.W. Heinz, M.E. Carrington, K. Deja and S. Mrowczynski, P.F. Kolb and U.W. Heinz, C. Shen, U. Heinz, P. Huovinen and H. Song, M.P. Heller, D. Mateos, W. van der Schee and M. Triana, [35] [36] [32] [33] [34] [30] [31] [28] [29] [25] [26] [27] [23] [24] [21] [22] [20] [19] [18] JHEP10(2013)013 , , ]. ]. ] SPIRE IN SPIRE ][ IN ][ Quark Gluon Plasma Heavy-flavour spectra arXiv:0809.1499 , in [ arXiv:0903.1096 (2011) 1666 arXiv:0709.2884 ]. [ ]. C 71 SPIRE (2009) 054907 SPIRE IN Nonperturbative heavy-quark diffusion in IN ][ ][ C 79 (2008) 192301 Heavy Quark Diffusion with Relativistic Langevin Eur. Phys. J. 100 – 39 – , Heavy-quark probes of the quark-gluon plasma at RHIC Phys. Rev. , hep-ph/0412346 nucl-th/0508055 [ [ ]. How much do heavy quarks thermalize in a heavy ion collision? Heavy Quarks in the quark-gluon Plasma Phys. Rev. Lett. , SPIRE IN ][ (2005) 064904 (2006) 034913 C 71 C 73 ]. SPIRE , R.C. Hwa, X.-N. Wang eds., World Scientific (2010), pg. 111 [ arXiv:1101.6008 IN in high energy nucleus-nucleus[ collisions Dynamics in the quark-gluon[ Fluid 4 Phys. Rev. Phys. Rev. the quark-gluon plasma W. Alberico, A. Beraudo, A. De Pace, A. Molinari, M. Monteno et al., Y. Akamatsu, T. Hatsuda and T. Hirano, R. Rapp and H. van Hees, H. van Hees, V. Greco and R. Rapp, H. van Hees, M. Mannarelli, V. Greco and R. Rapp, G.D. Moore and D. Teaney, [42] [40] [41] [38] [39] [37]