Evolution of the Mean Jet Shape and Dijet Asymmetry Distribution of an Ensemble of Holographic Jets in Strongly Coupled Plasma
Jasmine Brewer, Krishna Rajagopal, Andrey Sadofyev, Wilke van der Schee
PP REPARED FOR SUBMISSION TO
JHEP
MIT-CTP-4943LA-UR-17-29843
Evolution of the Mean Jet Shape and DijetAsymmetry Distribution of an Ensemble ofHolographic Jets in Strongly Coupled Plasma
Jasmine Brewer, a Krishna Rajagopal, a Andrey Sadofyev, a,b
Wilke van der Schee a,c a Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b Theoretical Division, MS B283, Los Alamos National Laboratory, Los Alamos, NM 87545 c Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht Uni-versity, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands
E-mail: [email protected] , [email protected] , [email protected] , [email protected] A BSTRACT : Some of the most important experimentally accessible probes of the quark-gluonplasma (QGP) produced in heavy ion collisions come from the analysis of how the shape and energyof sprays of energetic particles produced within a cone with a specified opening angle (jets) in ahard scattering are modified by their passage through the strongly coupled, liquid, QGP. We modelan ensemble of back-to-back dijets for the purpose of gaining a qualitative understanding of howthe shapes of the individual jets and the asymmetry in the energy of the pairs of jets in the ensembleare modified by their passage through an expanding cooling droplet of strongly coupled plasma, inthe model in a holographic gauge theory that is dual to a 4+1-dimensional black-hole spacetime thatis asymptotically anti-de Sitter (AdS). We build our model by constructing an ensemble of stringsin the dual gravitational description of the gauge theory. We model QCD jets in vacuum usingstrings whose endpoints are moving “downward” into the gravitational bulk spacetime with somefixed small angle, an angle that represents the opening angle (ratio of jet mass to jet energy) thatthe QCD jet would have in vacuum. Such strings must be moving through the gravitational bulk at(close to) the speed of light; they must be (close to) null. This condition does not specify the energydistribution along the string, meaning that it does not specify the shape of the jet being modeled.We study the dynamics of strings that are initially not null and show that strings with a widerange of initial conditions rapidly accelerate and become null and, as they do, develop a similardistribution of their energy density. We use this distribution of the energy density along the string,choose an ensemble of strings whose opening angles and energies are distributed as in perturbativeQCD, and show that we can then fix one of the two model parameters such that the mean jet shapefor the jets in the ensemble that we have built matches that measured in proton-proton collisionsreasonably well. This is a novel way for hybridizing relevant inputs from perturbative QCD anda strongly coupled holographic gauge theory in the service of modeling jets in QGP. We send ourensemble of strings through an expanding cooling droplet of strongly coupled plasma, choosingthe second model parameter so as to get a reasonable value for R jet AA , the suppression in the numberof jets, and study how the mean jet shape and the dijet asymmetry are modified, comparing both tomeasurements from heavy ion collisions at the LHC. a r X i v : . [ nu c l - t h ] O c t ontents Ultrarelativistic heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the LargeHadron Collider (LHC) recreate droplets of the hot matter that filled the microseconds old universe,called quark-gluon plasma (QGP). Experiments at these facilities provide unique experimental ac-cess to the properties of QGP as well as to the dynamics via which droplets of QGP form, expandand cool. These experiments have demonstrated that in the experimentally accessible range of tem-peratures, up to several times hotter than the crossover temperature at which cooling QGP becomesordinary hadronic matter, droplets of QGP exhibit strong collective phenomena [1–7], with the dy-namics of the rapid expansion and cooling of the initially lumpy droplets produced in the collisionssuccessfully described by the equations of relativistic viscous hydrodynamics [8–21]. The ratioof the shear viscosity, η , to the entropy density, s , serves as a benchmark, because in a weaklycoupled plasma, η/s ∝ /g (with g the gauge coupling), meaning that this ratio is large, whereas η/s = 1 / π in the high temperature phase (conventionally called the plasma phase even thoughin reality it is a liquid) of any gauge theory that has a dual gravitational description in the limit ofstrong coupling and large number of colors [22–24]. Comparisons between hydrodynamic calcu-lations of, and experimental measurements of, anisotropic flow in heavy ion collisions indicate thatthe QGP in QCD has an η/s that is comparable to, and in particular not much larger than / π ,meaning that QGP itself is a strongly coupled liquid.The discovery that QGP is a strongly coupled liquid at length scales of order its inverse temper-ature and longer even though (because QCD is asymptotically free) it consists of weakly coupledquarks and gluons when probed with high resolution challenges us to find experimental means toprobe QGP at multiple length scales. The only probes that we have available are those producedin the same heavy ion collisions in which the droplets of QGP themselves are produced. Here weshall focus entirely on the use of high transverse momentum jets, produced at the moment of thecollision in initial hard scatterings, as probes. Jets are produced with some energy and virtuality,– 1 –he latter often also referred to as the jet mass. Assuming that the jet propagates in vacuum, bothare (almost) conserved during the development and branching of the partonic jet shower that occursafter the jet is produced in an initial hard scattering. (Only almost because the jet may exchangesoft momenta with the underlying event or with other jets.) The partonic shower develops within acone whose opening angle is proportional to the ratio of the jet mass to the jet energy. As a partonicjet shower propagates through the strongly coupled plasma created in a heavy ion collision, how-ever, the partons in the shower each lose energy and momentum as a consequence of their stronginteractions with the plasma, creating a wake in the plasma. These interactions lead to a reduc-tion in the jet energy (or quenching) and to modifications of the opening angle and shape of jetsproduced in heavy ion collisions relative to those of their counterparts produced in proton-protoncollisions, that propagate in vacuum. By pursuing a large suite of jet measurements, the differentLHC collaborations have observed strong modification of different jet observables in heavy ioncollisions [25–50], making jets promising QGP probes. The first experimental constraints on jetquenching came from hadronic measurements at RHIC [51–53]. Analyses of jets themselves andtheir modification are also being performed at RHIC [54–59] and are one of the principal scientificgoals of the planned sPHENIX detector [60].A complete theoretical description of the processes by which jets are modified via passagethrough QGP remains challenging for the same reason that it is interesting, namely because it isa multi-scale problem. The production of jets and the processes via which an initial hard partonfragments into a shower are weakly coupled hard processes. However, the dynamics of the dropletof QGP including the wake produced in it by the passing jets and, more generally, the interactionof the jets with the QGP are sensitive to strongly coupled physics at scales of order the temperatureof the QGP. One class of theoretical approaches is based upon assuming that suitably resummedweakly coupled analyses can be applied almost throughout. (See Refs. [61–67] for reviews. Basedon these approaches, Monte Carlo tools for analyzing jet observables are being developed [68–76]and many phenomenological studies of jets in medium have been confronted with LHC measure-ments of a variety of jet observables [73–75, 77–107].) However, since QGP is a strongly coupledliquid we know that physics at scales of order its temperature must be governed by strong cou-pling dynamics. This realization has opened the door to many connections between the physicsof the QCD plasma and gauge/gravity duality [108], which yields rigorous and quantitative accessto nonperturbative, strongly coupled, physics in a large family of non-abelian gauge theory plas-mas that have a dual holographic description in terms of a black hole spacetime in a gravitationaltheory with one higher dimension. The AdS/CFT correspondence has become very successful inrecent years for describing strongly-coupled dynamics in a variety of arenas. In its simplest form,AdS/CFT provides a duality between strongly coupled N = 4 supersymmetric Yang-Mills (SYM)theory in 3+1 dimensions and classical Einstein gravity in 4+1-dimensional AdS space, or a 4+1dimensional black hole that is asymptotically AdS in the case where the N = 4 SYM theory is ata nonzero temperature. Although this AdS/CFT duality has not been shown to apply to QCD, thestudy of the plasmas in gauge theories that do have a holographic description has led to many qual-itative insights into the properties and dynamics of QGP. (See Refs. [109–111] for reviews.) Withinthis context, there have been many interesting studies that address varied aspects of the interactionbetween high energy probes and strongly coupled plasma [112–146]. No holographic analysis can— by itself — treat the intrinsically weakly coupled processes of jet production and fragmentation,– 2 –ince in all examples that are currently accessible via gauge/gravity duality the gauge theory isstrongly coupled in the ultraviolet, rather than asymptotically free.There are now two quite different phenomenological approaches being developed with thegoal of addressing the multi-scale dynamics of QCD jets in strongly coupled plasma more fully,blending inputs from perturbative QCD calculations and holographic calculations where each maybe relevant. The authors of Refs. [147–150] have developed a hybrid strong/weak coupling modelin which perturbative QCD parton showers taken from P
YTHIA are modified, parton-by-parton,upon assuming that the interaction between each parton formed in the shower and the QGP followsthe rate of energy loss of an energetic quark in strongly coupled plasma obtained via the holo-graphic calculations in Refs. [141, 143]. They have confronted their hybrid model with varioussuites of experimental data and in so doing have obtained qualitative insights into the implicationsof measurements of jet suppression, jet shapes, jet fragmentation functions, and the suppression,energy asymmetry and angular distributions of dijets, gamma-jets and Z-jets for parton energy loss,transverse momentum broadening, the degree to which the wakes left in the plasma by passing jetshave time to equilibrate, and the resolving power of QGP.The second approach, which we shall further develop here, was introduced by three of us inRef. [145] and is more ambitious in its use of holography, as we model each jet in its entirety as anenergetic massless quark plowing through the plasma of N = 4 SYM theory. In holography, thedynamics of quarks in the fundamental representation is studied by adding spacefilling D7 branesto the bulk spacetime. Open strings can end anywhere within a D7 branes, and are dual to a quark-antiquark pair in the dual boundary CFT [151]. These open strings can be constructed in manydifferent kinds of configurations and have been used to model varied dynamical phenomena. Aswe shall discuss at greater length below, a pair of light quark jets in plasma is described by an openfundamental string whose endpoints shoot away from each other and the same time fall “down-wards” into the black hole in the additional dimension in the AdS spacetime, with the downwardangle of their motion representing (i.e. being proportional to) the opening angle of the jet in thegauge theory. One way of looking at the approach to modeling jets introduced in Ref. [145] is thatwe seek to use inputs from perturbative QCD that are in a sense minimal, namely only those inputsthat describe jet production. The way we do this is to construct an ensemble of holographic jetswith an initial probability distribution for their energy and opening angle taken from perturbativeQCD so as to reproduce this distribution as in proton-proton collisions. The qualitative insightobtained in Ref. [145] is that even though every jet in the ensemble widens as it propagates throughthe strongly coupled plasma, after passage through the plasma jets with a given energy in the en-semble can have a smaller mean opening angle than jets with that energy would have had if theywere in vacuum. This happens because there are far fewer jets with higher energies than with lowerenergies in the distribution (before quenching the distribution is ∼ E − ) and because those jets thatare initially wider lose more energy, meaning that the jets that remain with any specified energyare those narrow jets which suffered the least energy loss. This result highlights the importance ofanalyzing an ensemble of jets if one wishes to make comparisons, even qualitative comparisons,to jet phenomenology: because different jets with the same E jet that traverse the same plasma butthat start out with different initial opening angles lose very different amounts of energy, it is insuf-ficient and in fact quite misleading to attempt to draw phenomenological conclusions by lookingjust at single average jet with some given energy. This conclusion applies for very similar reasons– 3 –n perturbative [98], holographic [145], and hybrid [149] calculations.In the present study, which we reported on preliminarily in Ref. [146], we extend the modelof Ref. [145] in two important ways. First, we analyze the shape of the jets in the ensemble, ratherthan just their opening angle. This forces us to consider the initial distribution of energy along thestring more carefully. Our goal is to choose this distribution so as to reproduce the shape of jets invacuum, and then to study how this shape is modified by passage through the plasma. In Section 2we shall find a rather remarkable way of using quite nontrivial string dynamics in the holographicgauge theory to construct an ensemble of strings (in Section 3) whose mean jet shape does indeedreproduce the mean shape of QCD jets produced in proton-proton collisions. Second, we choosean ensemble of back-to-back dijets with the distribution of the energy asymmetry between the twojets in an event chosen to match that measured in proton-proton collisions and analyze how thisdijet asymmetry distribution is modified by passage through the plasma.With the goal of making this paper more self-contained, we shall spend the remainder of thisintroduction reviewing some aspects of various previous holographic calculations that provide thebasis and context for how we (and others) use selected strings in a holographic gauge theory asmodels for jets. Along the way we shall also set up key elements of our analysis that follows.Fig. 1, adapted from Ref. [143], provides a good starting point as it illustrates many key featuresof how to build a holographic model for jets in plasma. The depth of the black hole horizon inthe AdS direction is / ( πT ) ; it sets the inverse temperature of the strongly coupled plasma in thefield theory. If the Figure had been drawn in vacuum, there would be no horizon and the bluegeodesics would all be straight lines. The blue geodesics in the Figure as drawn curve downwardbecause of the presence of the horizon, which is to say because of the presence of the plasma.The essence of the holographic dictionary is that a depth z into the bulk corresponds to a length-scale z in the gauge theory. Hence, if a (red) bit of energy propagates along a (blue) trajectorythat is a straight line heading rightward and downward with some angle σ in the Figure, this bitof energy is holographically dual to energy in the gauge theory that expands in size linearly intime as it propagates rightward. This is to say it is dual to a flux of energy in the gauge theorythat fills a cone with an opening angle proportional to σ . In order to model a jet in vacuum, withsome unchanging value of its jet mass, we must therefore find a string whose endpoint travels withsome constant downward angle σ . That is, in vacuum the downward angle of the string endpoint, σ , in the gravitational description is proportional to the opening angle of the “jet” in the N = 4 SYM theory that we use to model a jet with that ratio of jet mass to jet energy in QCD. Withthis correspondence established, we can now read many qualitative features of jet quenching ina strongly coupled gauge theory directly from the Figure. The fact that string energy travellingalong blue geodesics falls into the black hole is the gravitational description is energy lost by thejet as the jet excites a wake in the plasma. The fact that the string endpoint trajectory, like anyof the blue trajectories, curves downward corresponds to the fact that the opening angle of the jetexpands as the jet propagates through the plasma, losing energy. The fact that trajectories whoseinitial downward angle is smaller go farther before falling into the horizon corresponds to the factthat jets whose initial opening angle, proportional to σ , is smaller lose energy more slowly, over alonger distance, and travel farther through the plasma before thermalizing in the plasma.Via the calculation that corresponds to the blue geodesics in Fig. 1 falling into the horizon, theauthors of Refs. [141, 143] obtained an analytic expression for the rate at which these jets, or better– 4 – T x -2 0 2 4 68 10 π T z Figure 1 : A null string (red) used to model a jet moving in the x -direction shown at severalcoordinate times t [143]. The string starts off at the point x = 0 on the boundary and expands atthe speed of light while falling downwards into the holographic z -direction, towards the horizonlocated at z = z h = 1 / ( πT ) . The blue curves represent the null geodesics that each (red) bit ofenergy that makes up the string follows; the black curve is the endpoint trajectory. Different bluecurves are parametrized by different values of σ , where σ is the initial downward angle in the ( x, z ) plane. The endpoint is the trajectory with σ = σ ; the figure is drawn with σ = 0 . . If thestring were in vacuum, there would be no horizon, the blue null geodesics would all be straightlines, and the red string would maintain its initially semi-circular shape forever [128]. The openingangle of the jet which such a vacuum string represents is proportional to σ . Due to the presence ofthe horizon, which is to say due to the presence of the strongly coupled plasma, a blue (or black)trajectory with a given σ curves downward: its angle in the ( x, z ) plane, which starts out equal to σ , steadily increases. Consequently, the opening angle of a jet increases as it propagates throughthe plasma [143]. The energy lost from the jet to the plasma corresponds to energy density alongthe string traveling along blue geodesics falling into the horizon. Clearly, geodesics with smaller σ ,i.e. with smaller initial angle, propagate the farther before reaching the horizon. The thermalizationlength of the jet, x therm , is the distance that the endpoint travels before it reaches the horizon. It isapparent from the figure that jets with a smaller σ , meaning a narrower initial opening angle, losetheir energy more slowly and have a longer x therm [143]. Figure adapted from Ref. [143].to say these models for jets constructed in strongly coupled N = 4 SYM theory, lose energy asthey propagate through the strongly coupled plasma: E init dE jet dx = − x πx (cid:113) x − x . (1.1)where E init is the initial energy of the jet represented by the null string in Fig. 1 and where x therm is the thermalization distance of the jet, namely the distance that the string endpoint travels beforefalling into the horizon. This distance is related to the initial downward angle of the string endpoint,– 5 – , by [143] T x therm = Γ (cid:0) (cid:1) π / √ σ , (1.2)which quantifies the fact that jets whose opening angle is initially smaller travel farther through theplasma. For a jet that travels a small distance and loses only a small fraction of its initial energy,we can expand and integrate (1.1), obtaining [143] dE jet dx = − E jet π / Γ (cid:0) (cid:1) T x σ / + O ( x ) . (1.3)When we construct our ensemble of strings with which we shall model an ensemble of jets inplasma in Section 3, we shall follow Ref. [145] in treating the proportionality constant between theinitial downward angle of the endpoint of a string, σ , and the opening angle of the jet that we wishto model with that string as a free parameter. This is the first of two free parameters in our model;we shall denote it by a and will define it precisely in Section 3.It is important to realize that not all string configurations that can be constructed in the gravi-tational dual of N = 4 SYM theory behave like the string in Fig. 1. Not all by any means. Stringshave a nonzero tension, but in Fig. 1 the string tension does not affect the dynamics of the redstring because each bit of string is following a null geodesic. One can (and in fact we will in thenext Section) instead construct string configurations with different initial conditions in which thestring worldsheet is, at least initially, not null and in which the string tension affects the dynamicsof the string to such a degree that the downward angle at which the endpoint of the string moveschanges substantially as it propagates even in vacuum.
It is not immediately apparent how such astring can serve as a model for a jet, since on the face of it it would seem to correspond to a jetwhose virtuality changes substantially after the jet has been created, something that does not hap-pen in QCD since a high energy jet once formed interacts at most softly with other jets or with theunderlying event, meaning that by momentum conservation the virtuality of a jet in QCD hardlychanges. In the gravitational description of strongly coupled N = 4 SYM theory, though, thereare (non-null) strings in vacuum in which an end of the string feels a force from the rest of thestring that changes its trajectory. This means that in strongly coupled N = 4 SYM theory there areconfigurations in which a flux of energy behaves completely differently from a jet in QCD. (Thisis unsurprising: most string configurations in the gravitational dual do not correspond to anythingthat looks like a jet. And, furthermore, in strongly coupled N = 4 SYM theory hard processes donot produce jets [124, 126].) It remains an open question whether a subset of strings whose endpoint trajectories change their downward angle can nevertheless be used directly as models for jets;investigating this would require computing the gauge theory energy flux and looking for instanceswhere virtuality and opening angle do not change, even when the string end point trajectory does.In the present work, we follow a more straightforward approach. We choose to model jets in QCDby choosing strings in the gravitational description of strongly coupled N = 4 SYM theory whoseendpoints follow a trajectory with some constant downward angle σ in vacuum, curving down-ward only because of the presence of the black hole horizon, modeling jets whose opening angleschange only because they are propagating through plasma. The simplest way that we know ofchoosing strings that constitute good models for jets in QCD is to choose null strings, as in Fig. 1,following an approach that goes back to Ref. [128].– 6 –ur discussion to this point has left the distribution of energy along the string unspecified.For null strings in vacuum, whatever distribution of energy along the string we choose initially(respecting open string boundary conditions) will simply propagate unchanged along the blue nullgeodesics (which are straight in vacuum). Since a null string like that in Fig. 1 propagates foran initial period of time (cid:28) / ( πT ) as if it were in vacuum, we have considerable freedom inchoosing the initial energy density along the string. After the string has propagated through theplasma for a distance that is (cid:29) / ( πT ) , its shape is no longer semicircular, as in vacuum. Infact, after propagation through the plasma it takes on the shape of a segment of the string thatdescribes an infinitely heavy quark being dragged through the plasma [143], a shape that wasfirst worked out in Refs. [112, 114, 115]. As the string propagates through the plasma over adistance (cid:29) / ( πT ) and blue trajectory after blue trajectory peels away and falls into the horizon,eventually the only aspect of the initial distribution of energy along the string that matters is theenergy that is initially very close to the endpoint of the string. For this reason, the authors ofRef. [143] chose an initial distribution of energy along the string that takes the near-endpoint form ∝ / ( σ √ σ − σ ) dictated by the open string boundary conditions everywhere along the string.Although operationally reasonable, the logic behind this choice is not fully satisfactory since it isbased upon using the form of the string energy density after the string has propagated for a longdistance through the plasma to choose the distribution of energy along the string initially, when thestring is still behaving as if it were in vacuum. It would be better to have an argument based uponthe physics of strings in vacuum for choosing the initial distribution of energy along the string. Weshall remedy this lacuna in Section 2.In Section 2 we study the dynamics of strings that are initially not null. As anticipated, invacuum their endpoints do not follow trajectories with a constant downward angle, meaning thatthey (initially) represent objects whose virtuality is not obviously conserved which makes it unclearhow they can be used to model jets in QCD. As an extreme example, extreme in the sense thatthey are the least apparently jet-like of any of the strings we analyze, we include strings similarto the ones considered in Ref. [139] in which the downward angle of the string endpoint changessuddenly. From our analysis of their dynamics, however, we find that a large class of strings that areinitially not null become null strings as they fall in the AdS vacuum: after a certain “nullificationtime” that we compute, every bit of string moves along a null geodesic, as anticipated in Ref. [141,143]. After nullification, the string endpoints follow trajectories with a constant downward anglemeaning that after nullification all these strings end up becoming jet-like. And, quite remarkably,we find that for a rather diverse set of initial conditions for the energy density along the string,as long as we don’t make the string null initially (in which case the energy distribution would notchange in vacuum) after the string nullifies the distribution of energy density along the string hasevolved such that it is approximated by a scaling form parametrized only by the downward angleof the string endpoint after nullification. Near the string endpoint this scaling form agrees with theexpression for the distribution of energy density along the string obtained from the near-endpointexpansion of Refs. [141, 143], as it must. We show that in plasma, which is to say when thegravitational description of the physics includes a horizon, the strings that we analyze nullify whilethey are still far above the horizon meaning that their nullification occurs as it would in vacuum.The results for the dynamics of strings that are initially not null, in Section 2, motivate ourconstruction, in Section 3, of an ensemble of null strings as a model for an ensemble of jets.– 7 –e choose the distribution of energy along a null string in this ensemble with a specified initialdownward angle σ according to the scaling form for this distribution, namely the scaling formattained by initially non-null strings. This means that we are using nontrivial string dynamics in(the dual of) strongly coupled N = 4 SYM theory, dynamics that does not itself appear to bejet-like, to determine how to distribute the energy density along the strings in the ensemble of nullstrings that we subsequently use to model an ensemble of jets. Another way of describing this isthat among all the possible null strings that can be constructed in the dual of N = 4 SYM theory,the subset that we choose to use in our ensemble of strings are those with a particular scaling formfor their energy distribution such that they can be formed either by starting with null strings fromthe beginning or by starting with strings that are initially not null and evolving them until theynullify.As in Ref. [145], we choose the distribution of the initial jet energies and opening angles forthe jets in our ensemble from perturbative QCD calculations as appropriate for QCD jets in proton-proton collisions. For each jet, we choose the initial distribution of energy along the string thatrepresents that jet in our model according to the scaling form obtained via our holographic analysisof the nullification of strings. This specifies the jet shape for each jet in our ensemble. Remarkablygiven that the strongly coupled dynamics by which the strings nullify has no apparent analogue inQCD, we find in Section 4 that, upon fitting the single parameter a , our model yields a very gooddescription of the mean jet shape in QCD, as measured in proton-proton collisions by the CMScollaboration [33].With our ensemble of strings fully specified, including via using the distribution of energyalong each string taken from the scaling form obtained via our holographic analysis of nullification,in Section 4 we send the ensemble through an expanding cooling droplet of hydrodynamic fluid(described in Section 3). In so doing, we introduce a second free parameter in the model, whichwe denote by b , which is the proportionality constant between the temperature of the QCD plasmathat we are modeling and the temperature of the N = 4 SYM plasma (with more degrees offreedom) that we are using as a model. We choose b such that the modification in the number ofjets with a given energy in the ensemble after it has passed through the droplet of plasma relativeto that in the initial ensemble is comparable to that seen in data. We then compute the modificationto the mean jet shape, and compare to experimental data [33]. We find a narrowing in the jetshape at small angles that is comparable to that seen in data, but because we are not including thecontribution to reconstructed jets coming from the wake in the plasma our model cannot describethe modification to the mean jet shape at larger angles. We then consider an ensemble of dijets, withthe dijet asymmetry distribution chosen to reproduce that measured in proton-proton collisions, andcompute the modification to this distribution caused by passage through the droplet of plasma. Hereagain we compare to experimental data from heavy ion collisions at the LHC [27]. We close witha look ahead at possible future improvements to the model. Our results for the modification of thedijet asymmetry distribution are promising but they are not in quantitative agreement with the data;this motivates a future analysis of an ensemble of trijets, since in reality (and unlike in our modeldijet ensemble) much of the dijet asymmetry seen in proton-proton collisions comes from eventsin which there is a third jet present. – 8 – String Dynamics
With the aim of studying the dynamics and evolution of an ensemble of null strings in N = 4 SYM plasma as a model for an ensemble of jets, and in particular for the purpose of choosing theshape of the distribution of energy density along the null strings, we begin with a study of stringsin 4+1-dimensional AdS space that are, initially, not null. In holography, a pair of light quarks isrepresented by an open fundamental string in AdS [151]. The 5-dimensional metric in AdS whichcorresponds to a constant-temperature plasma in the 4-dimensional N = 4 SYM theory on itsboundary is d s = L z (cid:18) − f ( z ) d t + d (cid:126)x ⊥ + d y + d z f ( z ) (cid:19) , (2.1)where z is the additional direction in AdS space, f ( z ) = 1 − z /z h , and the black hole is locatedat z = z h ≡ /πT . Here (cid:126)x ⊥ and y are field theory coordinates specifying the transverse planeand the beam direction, respectively. This metric is an exact solution to Einstein’s equations for aconstant-temperature plasma. We shall later (in Section 3) choose a temperature T that varies inspace and time so as to model an expanding cooling droplet of plasma but for a spatially-varyingtemperature profile this model neglects transverse flow, fluid viscosity, and gradients.The dynamics of strings in this geometry are most conveniently solved numerically using thePolyakov action (see Refs. [128, 129, 152]): S P = − T (cid:90) dτ ws dσ ws √− η η ab ∂ a X µ ∂ b X ν G µν , (2.2)with T = √ λ/ π the string tension with λ the ’t Hooft coupling, τ ws , σ ws the string worldsheetcoordinates, G µν the bulk AdS metric and η ab the string worldsheet metric, which determines thegauge choice in mapping τ ws and σ ws coordinates to target space coordinates X µ . The gaugedegree of freedom η ab can be solved for by varying the action, giving the constraint equation: γ ab = 12 η ab η cd γ cd , (2.3)where γ ab = ∂ a X · ∂ b X is the induced metric on the string worldsheet. Since the worldsheetmetric is a gauge choice, the functions X µ ( τ ws , σ ws ) can be chosen to make the numerics morestraightforward. Since we will typically solve the equations of motions in steps along τ ws , this forinstance requires that different parts of the string cover the part of spacetime in a similar pace inthe τ ws variable. This can be done by defining [129] η ab ≡ (cid:32) − Σ( X µ ) 00 1 / Σ( X µ ) (cid:33) , (2.4)with the stretching function Σ( X µ ) , which is commonly chosen to cancel singularities in theequations of motion. In all the string evolutions presented in this paper, we shall use Σ = (cid:18) − z − z (cid:19) α (cid:16) z z (cid:17) β (2.5)– 9 –ith α and β typically 1 or 2. From the action we can also obtain the target space energy-momentum density: π aµ ( τ ws , σ ws ) = 1 √− η δS P δ ( ∂ a X µ ( τ ws , σ ws )) = − T η ab ∂ b X ν G µν . (2.6)The energy-momentum density at some time t is then given by p µ ( t, σ ws ) = √ λ π √− η (cid:18) π τµ ( t, σ ws ) − π σµ ( t, σ ws ) ∂ σ ws t∂ τ ws t (cid:19) . (2.7)We shall analyze the string dynamics throughout in classical Einstein gravity and using classicalequations of motion, which means everything is done in the limit of strong coupling. In the strict λ → ∞ limit there are no quasi-particles and hence also no distinguishable quark-antiquark pairs.In AdS this is dual to the statement that in this strict limit creating a string requires an infinite energyof order O ( √ λ ) . Our limit hence has to interpreted as a large but finite coupling, where it doesmake sense to consider a quark-antiquark pair, dual to a string. Later, when we quote numericalresults we shall always take λ = 5 . , as in Ref. [153].We create each string at a single point ( z , t , x ) in the AdS spacetime. Without loss ofgenerality we can set x = 0 and t = 0 . One way of varying the initial conditions for our stringsis to vary z . We must also specify initial conditions for the velocity of the string in the AdSspacetime as a function of the string worldsheet parameter σ ws , subject to open string boundaryconditions. The aim of this Section is to introduce several classes of initial conditions for the stringand to show that when we choose the string to not be null initially its dynamics turn it into a nullstring (a string where each segment travels along an independent null geodesic) after a period oftime, which we compute. We find it striking that, although the nullification process occurs throughstrongly-coupled dynamics which may have no direct analog in the dynamics of jets in QCD, ityields an approximate scaling form for the distribution of energy density along the string after thestring nullifies. We shall use this scaling form in Sections 3 and 4 when we follow the evolution ofan ensemble of null strings, which serve as models for jets, as they pass through an expanding andcooling droplet of plasma.We shall analyze six classes of initial conditions for the strings, depicted in Fig. 2. The initialvelocity of the string in the x -direction is given as a function of σ ws by the six expressions: ∂ τ ws x ( σ ws ) = A cos( σ ws ) , (2.8) ∂ τ ws x ( σ ws ) = A (cid:18)
12 tanh (cid:16) (cid:16) σ ws − π (cid:17)(cid:17) − sech (2 π ) sin(2 σ ws ) (cid:19) (2.9) ∂ τ ws x ( σ ws ) = A cos( σ ws ) , (2.10) ∂ τ ws x ( σ ws ) = A cos ( σ ws ) , (2.11) ∂ τ ws x ( σ ws ) = A (cid:18)
12 tanh (cid:16) (cid:16) σ ws − π (cid:17)(cid:17) − sech (2 π ) sin(2 σ ws ) (cid:19) (2.12) ∂ τ ws x ( σ ws ) = A (cid:18) σ ws − e − π − σ ws ) + e − σ ws − π (cid:19) (2.13)with A a parameter that we specify as described below. The velocity in the holographic z -directionis given by ∂ τ ws z ( σ ws ) = A (cid:16) − cos(2 σ ws ) (cid:17) + Aσ s (2.14)– 10 – igure 2 : We show the shapes of the profiles that we use for the initial conditions on the velocitiesin AdS of the strings that we analyze, as given in Eqs. (2.8-2.13) and Eqs. (2.14-2.15). The solidcurves show the initial velocity of the string in the x -direction as a function of the worldsheetcoordinate σ ws . (The string endpoints are at σ ws = 0 and π ; the string midpoints are at σ ws = π/ .)The dashed curves give the initial velocity in the z -direction (“downward” into the bulk of AdS),enhanced by a factor of 10 for visibility. As discussed in the text, the small but nonzero z velocityis useful numerically to make ∂ τ ws t continuous at the midpoint of the string. The relationshipbetween the worldsheet time coordinate τ ws and the AdS time t is given in terms of these functionsin Eq. (2.16).for (2.8), (2.10), (2.11) and (2.13) and by ∂ τ ws z ( σ ws ) = A (cid:20) tanh (cid:18) (cid:16) σ ws − π (cid:17) + 12 (cid:19) + 1 (cid:21) (cid:20) tanh (cid:18) (cid:16) π − σ ws (cid:17) + 12 (cid:19) + 1 (cid:21) + Aσ s (2.15)for (2.9) and (2.12). This z -velocity is small but useful to ensure that ∂ τ ws t is continuous at σ ws = π/ . We shall set the parameter σ s = 0 in (2.8), (2.9), (2.11) and (2.13). This means that inthese four classes of initial conditions, the string initially has a nonzero velocity in the z -directiononly near σ ws = π/ . And, in these four classes of initial conditions the initial velocity of thestring endpoints (at σ ws = 0 and π ) are horizontal , corresponding initially to a collimated flow ofenergy with vanishing opening angle. If it were possible to create a jet with zero initial virtuality inQCD, its virtuality would remain zero; it would never fragment into a shower and would never filla cone. No production mechanism for a collimated object like this is known in QCD. We shall seebelow, however, that in N = 4 SYM the strongly coupled dynamics ensures that an object createdwith zero opening angle like this does not stay that way. The strongly coupled dynamics turns thisinitially collimated object into something that later becomes jet-like. Which is to say that the string– 11 –which is initially not null) nullifies. In initial conditions (2.10) and (2.12) we choose a nonzerovalue of the parameter σ s such that the initial downward angle of the string endpoints is . ◦ and . ◦ respectively.To complete the specification of the initial conditions that we shall analyze, we note that thevelocity in the time direction follows from the constraint equation η = 0 and from our assumptionthat the string starts at a single point. It is given by ∂ τ ws t = (cid:112) ( ∂ τ ws z ( σ ws )) + ( ∂ τ ws x ( σ ws )) . (2.16)Last, we describe how we choose the value of the parameter A . By conformal invariance we cankeep one parameter fixed and then vary other parameters without loss of generality. In this Section,we choose to fix A such that the energy (2.7) is E = 1000 . We performed several numericalchecks, verifying for all of our evolutions that the constraint equations are satisfied and that thetotal energy as obtained from (2.7) is conserved up to our numerical precision ( − or better). At the start of the evolution of the strings with initial conditions (2.8), (2.9), (2.11) and (2.13), inwhich the initial downward angle of the string endpoints vanishes, the string tension is a crucialingredient in determining the dynamics of the string. The string stretches initially, losing kineticenergy as it does so. This effect is especially strong near the boundary of the AdS spacetime,which is to say near the endpoint of the string, where the larger proper distance due to the largeAdS metric factor requires the string to have a large initial energy to off-set the potential energycost of the stretching. This suggests that there should be a sense in which strings of this type whichstart closer to the AdS boundary (smaller z ) have larger energy. We shall make this precise below.We shall see that what happens to these strings is that the string tension succeeds in pullingthe string endpoints away from the AdS boundary and giving them a nonzero downward angleeven though their initial downward angle vanishes. (This dynamics has no apparent analogue inthe physics of jets in QCD.) After some time of evolution, then, the strings have fallen into thebulk AdS space and the increasing kinetic energy becomes dominant over the potential energy. Atthis stage the string tension no longer has a significant effect on the dynamics of the string. Toa good approximation, each segment of the string travels on a null geodesic: the string nullifies.Focusing on the endpoint of the string, the string tension initially curves its trajectory downward,away from the AdS boundary, but after some time its downward angle stops increasing and ithenceforth follows a null geodesic with a constant downward angle. We shall call the downwardangle reached by the string endpoint as the string becomes null σ . We can now state the precisesense in which strings which start closer to the AdS boundary have larger energy: we shall showthat, for strings whose initial downward angle vanishes, the smaller the z we choose the larger theenergy we must choose if we wish to end up with a specified value of σ after nullification.If instead of starting off with a string whose endpoint initially moves horizontally we start offwith a string that is initially null, the discussion above changes completely. There is no stretchingeffect. The string tension never plays a significant role in the string dynamics. And, the initialdownward angle of the string endpoint keeps its initial nonzero value σ throughout. These are thestrings that we shall use as models of QCD jets in vacuum. However, in this case the dynamics– 12 –eaves the distribution of energy density along the string unchanged also, which gives us no clue asto what distribution to choose. The initial conditions (2.10) and (2.12) with initial downward anglesfor the string endpoints that are nonzero are not null, but they (in particular (2.12)) are much closerto being null than any of our initial conditions for strings whose endpoints are initially horizontal.We therefore expect that these strings should nullify more quickly, with less rearrangement of theenergy density along the string. And, we expect that for these strings it need not be the case thatreducing z means increasing the energy needed to achieve a specified σ .Note that since in vacuum null geodesics in AdS are just straight lines it is clear that a nullifiedstring is fully specified in vacuum by how much energy goes downward at what angle in AdS. Aswe illustrated in Fig. 1, it is also true in plasma that a nullified string is completely specified bythe initial downward angle of each bit of energy along the string. However, in this case the bluenull geodesics in Fig. 1 curve downward toward the black hole horizon meaning that the downwardangle of each bit of energy along the string increases.Figure 3 shows two examples of string evolution in vacuum, for a string with the initial con-dition (2.8) created at z = 0 . (above) and for a string with the initial condition (2.9) created at z = 0 . (below). We plot the time evolution of the downward angle of the point on the stringabove which a fraction (cid:15) of the string energy is found. That is, (cid:15) starts at 0 at the string endpoint and (cid:15) = 0 . corresponds to the string midpoint, which by symmetry moves straight downward into theAdS bulk. For both strings in the figure, the initial downward angle of the string endpoint is zero.As a consequence of the string tension, during the early time dynamics each segment of the stringchanges its downward angle σ . For each segment of the string, though, after some time passes itsdownward angle no longer changes. That is, the string nullifies. The constant nonzero downwardangle of the trajectory that the string endpoint follows after nullification, σ , is . ◦ in the upperpanel of Fig. 3 and . ◦ in the lower panel.The early time dynamics is particularly dramatic in the lower panel of Fig. 3, where the end-point of the string propagates almost horizontally for a time before relatively suddenly turningdownwards with a nonzero angle that soon becomes constant. These strings, with the initial con-dition (2.9), describe a flow of energy that is initially collimated, with zero opening angle, beforelater, suddenly, acquiring a substantial opening angle. We shall describe this dynamics further insubsection 2.2. First, though, we shall provide a further description of the nullification processin vacuum and shall then illustrate that it works quite similarly in plasma — because nullificationhappens while the strings are still far enough from the horizon that their dynamics is similar to thatin vacuum. We shall see that even though the strings with initial conditions (2.9) have dramaticdynamics early on, after they nullify and become jet-like they look rather similar to the jet-likestrings that form starting from the other initial conditions that we analyze. In this subsection, we shall compute the nullification timescales and the resulting distribution ofenergy along the string as a function of the downward angle of the trajectory followed by a bitof string after nullification, for strings whose initial conditions are given by Eqs. (2.8-2.13) thatnullify in vacuum. The nullification process depends on the initial velocity profile, the energy ofthe half-string E and the AdS depth z of the point at which we initialize the string. We haveanalyzed nullification for the six classes of initial conditions presented above with varying values– 13 – igure 3 : We depict the time evolution in vacuum of two strings whose endpoints initially movehorizontally, with zero downward angle. The upper panel is for the initial condition (2.8) with z = 0 . and the lower panel is for the initial condition (2.9) with z = 0 . . Each curve showsthe time evolution of the downward angle σ (not to be confused with the worldsheet coordinate σ ws )of the point on the string such that a fraction (cid:15) of the total energy of the string is found betweenthat point and the string endpoint. Hence, (cid:15) = 0 corresponds to the string endpoint and (cid:15) = 0 . corresponds to the string midpoint. The calculation is done entirely in vacuum. And, in the AdSvacuum null geodesics are trajectories with a constant downward angle σ . We see that each pointon the string “nullifies”: after some time, each bit of energy along the string moves along a nullgeodesic with some constant σ . After nullification, the string endpoints move along trajectorieswith nonzero downward angles. – 14 –f z . Without loss of generality, in vacuum we can choose units of energy such that E = 1000 .For each string that we evolve, we compute the opening angle σ with which the string endpoint isdescending into the bulk after the string has nullified as well as the time t null it takes for the endpointto reach this angle within accuracy of 10%. In Figures 4, 5 and 6 we show these observables forour strings, and more.In Figure 4 (bottom) we see that if we choose initial conditions from one of our classes ofinitial conditions in which the initial downward angle of the string endpoint is zero then, after theinitial phase of the string dynamics, when the string nullifies its endpoint is moving downward intothe AdS bulk at a constant angle σ that is well approximated by σ ∼ cE z , (2.17)for a profile-dependent constant c . Here, we have reinstated the E -dependence by dimensionalanalysis. (Strings that are initially closer to null, like those with initial conditions (2.10) and inparticular (2.12), do not satisfy this relationship.) In Ref. [143], it was shown analytically that ina limit in which σ → null strings as in Fig. 1 have an energy E ∝ σ − / . We now see that wecan reproduce this limit by choosing a sequence of strings with zero initial downward angle andincreasing E as long as we initialize the strings at a z that we choose to be ∝ E − / . The stringsin this sequence will nullify with a σ ∝ E − / .In Fig. 5 we plot the downward angle σ of every point on each of the strings that we havestudied, after nullification, rather than just focusing on the string endpoint. We find that for thestrings with initial conditions in which the initial downward angle of the string endpoint was zero,after nullification the downward angle of a bit of string, scaled by the downward angle σ of theendpoint of that string, takes on an approximate scaling form as a function of what fraction of theenergy is found above that bit of string. This is equivalent to saying that the energy distributionalong the string takes on an approximate scaling form after nullification. In Section 3 when we usean ensemble of null strings to model an ensemble of QCD jets, we shall choose to distribute theenergy density along these null strings according to the scaling form found in the lower panel ofFig. 5. Specifically, we shall take the form for σ/σ as a function of (cid:15) found after nullification forstrings that start out at z = 0 . with the initial condition (2.8). This scaling form is the principalresult of this Section; it is the result from this Section that we shall employ when we model anensemble of jets in Section 3.In the lower panel of Fig. 5 the scaling form that we find after nullification is compared tothe result from Ref. [141] (the black curve labelled 1511.07567), in which the near-endpoint ap-proximation for the energy density distribution as a function of σ , namely e ( σ ) ∝ σ √ σ − σ , wasemployed for the entire string. This provides a good approximation to the scaling form that wehave found for (cid:15) (cid:46) / , which is to say for the half of the energy of the string that is closer to itsendpoint. Farther away from the endpoint, the near-endpoint approximation does not describe thescaling form that we have found.Finally, in Figure 6 we show the nullification times for every point on each of the strings thatwe have studied. There it can be seen that for those strings which start out far from null, with theirendpoints moving horizontally, and subsequently nullify via the strongly coupled dynamics that wehave focused on in this Section do so after a nullification time that is around (2 − Ez , dependingsomewhat on the initial condition as well as on the position on the string.– 15 – igure 4 : For each of the six classes of initial conditions in Eqs. (2.8-2.13), we plot the downwardangle σ reached by the endpoint of the string after it nullifies, which is to say after the downwardangle no longer changes. In the upper panel, we show how σ varies as we change the AdS depth z at which we initialize the string, while keeping the energy of the string fixed. We find that forstrings with the initial conditions (2.8), (2.9), (2.11) and (2.13) in which the downward angle of thestring endpoint is initially zero, to a good approximation the opening angle σ reached by the stringendpoint after nullification is given by σ ∼ E z , with a prefactor c that depends weakly on theinitial string profile. For the initial conditions (2.10) and (2.12) in which the downward angle of thestring endpoint is initially nonzero and in which the strings are closer to null from the beginning,this relationship is not satisfied. We shall now analyze the dynamics of strings with the initial conditions that we have introducedabove in plasma, rather than in vacuum. When the strings are initially far above the black hole– 16 – igure 5 : After nullification, the point on the string above which a fraction (cid:15) of the energy ofthe string is found moves along a null geodesic with some constant downward angle σ . In theupper panel, we plot this σ as a function of (cid:15) , after nullification, for strings with each of our sixclasses of initial conditions with varying values of z . σ is the value of σ at (cid:15) = 0 . In the lowerpanel, we rescale each of the many curves in the upper panel by its own σ . We find that for thestrings with the initial conditions (2.8), (2.9), (2.11) and (2.13) in which the initial downward angleof the string endpoint was zero, after nullification the rescaled curves in the lower panel all takeon a rather similar shape. This means that, for these initial conditions, the nullification dynamicsrearranges the way that energy density is distributed along the string as a function of σ/σ suchthat it reaches an approximate scaling form. The purple curves show that for initial conditions like(2.12) that are close to null from the beginning, because the energy density distribution along thestring hardly changes it need not reach the scaling form.– 17 – igure 6 : For a point on the string above which a fraction (cid:15) of the string energy is found, wedefine the nullification time t null as the time that the angle found in Fig. 5 is within 10% of its finalvalue. In the upper panel we plot t null as a function of (cid:15) for the strings that we have analyzed. Inthe lower panel, we plot t null / ( Ez ) , and find that the nullification time is roughly proportionalto z for the strings whose endpoints had an initial downward angle of zero. Those strings whichwe initialized with initial conditions (2.10) with their endpoints moving with a nonzero downwardangle are closer to null from the beginning, meaning that it is no surprise that they nullify faster.horizon, near the boundary, they are in a region of the spacetime where the metric is nearly thesame as in vacuum. This means that strings which nullify quickly compared to the time it takesthem to fall close to the black hole horizon will nullify via dynamics that is nearly the same as the– 18 – igure 7 : We compare the downward angles of the string endpoints after nullification σ definedas described in the text (top panel) and the nullification times (bottom panel) for strings that areproduced and then nullify in plasma (solid curves) to those that are produced and then nullify invacuum (dashed curves) as in the previous subsection. All strings were produced at z = 0 . except for those with initial conditions (2.12), which were produced at z = 0 . .dynamics in vacuum that we have analyzed above. We shall show in this subsection that, to goodaccuracy, this is indeed the case for strings with the initial conditions that we have chosen.In vacuum, null geodesics are straight lines. This simplifies many computational aspects ofstudying nullification in vacuum compared to in plasma, because the angles of null geodesics stayfixed and the deviation from null is given by the deviation of the trajectory from a straight lineand hence is easily assessed and quantified. As we discussed in Section 1, in the AdS black holespacetime dual to the plasma, null geodesics curve downward toward the horizon and a stringgets represented by a congruence of null geodesics that loses energy to a wake in the plasmaand ultimately thermalizes over a distance x therm , the distance that the string endpoint travelsbefore falling into the horizon. Because nullification happens relatively quickly compared to thethermalization time x therm , we expect that it happens near the boundary where the spacetime is– 19 –lose to vacuum AdS. However, to test this we must have a way of assessing whether the stringhas nullified that can be applied in the regime where null geodesics curve downwards, namely inthe regime that is not near the boundary. At any given time t we construct a null geodesic that istangent to the actual trajectory of the string endpoint (or for that matter to any chosen bit of string)and follow that null geodesic backwards/upwards all the way to the boundary. In the near-boundaryregion, this null geodesic is straight and has some downward angle. Once the string has nullified,its endpoint is following a null geodesic. This means that once the string has nullified, if we go toa later time and repeat the exercise of shooting a null geodesic that is tangent to the string endpointtrajectory back upwards we will find the same null geodesic with the same initial downward angleas we found at the earlier time. We define the angle σ as this initial downward angle, defined fromthe near-boundary slope of the null geodesic that the string endpoint follows at late times. And wedefine the nullification time t null as the time after which this initial downward angle changes by lessthan 10%. Defined in this way, both σ and t null can be compared directly to their values in vacuum,allowing for the quantitative comparison between the nullification dynamics in plasma to that invacuum depicted in Figure 7. We see that the nullification dynamics is similar indeed, confirmingthat nullification happens quickly enough that, in plasma, it happens in the near-boundary regimewhere the spacetime is very similar to vacuum AdS.We conclude that the strings that we investigate nullify far above the horizon, near the bound-ary. This means that the expression (1.2) provides a reasonable approximation to the relationshipbetween the distance that they travel between nullification and thermalization, x therm , and theirinitial downward angle after nullification σ , defined as described above. It also completes thejustification for how we shall model jets in Section 3: we shall use null strings as in Fig. 1 with aspecified initial downward angle σ and with energy density distributed along the string accordingto the scaling form that we have found via our analysis in subsection 2.1.1 of the dynamics of howstrings nullify in vacuum. Before turning to modeling jets, we close this section with a further look at the nullification ofstrings with the initial condition (2.9), noting that this is close to a particular string initial conditionfrom Ref. [139] in which all of the energy and momentum of the string is initially localized at itsendpoint, and hence travels initially in the same direction. We have seen that the dynamics of thesestrings before nullification is dramatic. However, these strings nullify like the others and have thebenefit that they allow a partial analytic treatment, one that will allow us to confirm (in this specialcase) some of the scaling expressions that we found above more generally, but numerically. It hasbeen shown in the literature that if we choose initial conditions in which all of the string energy islocalized at its endpoint with the string endpoint initially moving parallel to the boundary at some z (cid:28) / ( πT ) , with zero downward angle, then the endpoint loses energy according to [139, 140] dEdx = − √ λ π z . (2.18)– 20 –he endpoint will follow a straight line (i.e. a null geodesic) at constant z until all its energy hasbeen depleted, which happens after a distance: x snap = 2 π √ λ E z . (2.19)At that point the endpoint cannot continue further along its initial null geodesic and must changedirection (a “snapback”), as can clearly be seen in the lower panel of Fig. 3. (With E = 1000 , λ = 5 . and z = 0 . , the expression (2.19) yields x snap = 0 . , in agreement with the behaviorseen in that Figure.) This means that at this point in time the string is clearly different from a nullstring and hence we find that for this particular type of strings t null ≥ (2 π/ √ λ ) E z , consistentwith the approximate scaling that we found more generally, but numerically, above.After this snapback, nullification occurs: the string endpoint finds itself moving along a newnull geodesic with some nonzero downward angle σ that does not change further. In order for theendpoint to continue on a null geodesic without another snapback we find a condition on the initialendpoint energy E , its starting position z , and σ , the opening angle after nullification, that isgiven by E ≥ (cid:90) ∞ z dz dEdz = − (cid:90) ∞ z dz √ λ π z (cid:112) − f /R (2.20)where R = − f √ ˙ x + ˙ z / ˙ x = 1 / cos( σ ) , which is to say by E ≥ (cid:90) ∞ z dz z sin( σ ) = 1sin( σ ) z , (2.21)which is again consistent with the more general scaling that we found numerically above. We are now ready to construct the ensemble of null strings in the dual description of N = 4 SYMtheory that we shall use as a model for an ensemble of jets in heavy ion collisions. The first stepis to understand the relationship between the energy density distributed along an individual nullstring and the shape of the individual jet that this string represents. Here by shape we mean thedistribution of P out , the outward-directed flux of power at infinity, as a function of the angle r measured from the center of the jet. We use the result from Ref. [141]: dP out d cos r = 12 (cid:90) σ dσ e ( σ ) γ ( σ ) [1 − v ( σ ) cos r ] , (3.1)where as in Fig. 1 we have parametrized the null string worldsheet by σ , the initial downward angleof a blue null geodesic along which a bit of energy travels, where e ( σ ) is the energy density alongthe string as a function of σ with E final = (cid:82) σ dσe ( σ ) , where γ ( σ ) ≡ (1 − v ( σ ) ) − / , and where v ( σ ) = cos σ for a null geodesic. This formula is relevant for a null string propagating througha finite droplet of QGP that emerges from that droplet and then propagates onward to infinity invacuum. The domain of integration is over the angles σ that label those blue null geodesics thatdo not fall into the black hole. We will choose an ensemble of strings with differing values of theinitial downward angle of the string endpoint, σ . We shall specify the probability distribution for– 21 – and for the initial energy of the string below. For an individual string in the ensemble with someparticular value of σ , we choose e ( σ ) to be given by the approximate scaling form that we foundin Section 2.1.1 for the distribution of energy along the string after nullification for strings that wereinitially not null, whose endpoints initially had no downward angle. The approximate scaling formfound after nullification is illustrated in the lower panel of Fig. 5, where what is plotted is σ/σ asa function of (cid:15) , for points on the string above which a fraction (cid:15) of the total energy of the string.Specifically, we use the curve from Fig. 5 for strings that start out at z = 0 . with the initialcondition (2.8). For a string of total energy E whose downward endpoint angle at nullification isgiven by σ , we denote the curve in the lower panel of Fig. 5 by σ ( (cid:15) ) and express the jet shape P out ( r ) as a function of the angular coordinate r away from the jet axis as P out ( r ) = (cid:90) d(cid:15) E sin rγ ( σ ( (cid:15) )) [1 − v ( σ ( (cid:15) )) cos r ] . (3.2)This relation allows us to compute the shape of individual model jets in the N = 4 SYM gaugetheory corresponding to null strings with a specified initial downward angle σ .Next, we must specify the distribution of the initial energy E and downward angle σ of thestrings in the ensemble that we shall use to model an ensemble of jets. We shall then be able tocompute the mean jet shape in the ensemble in vacuum. We will then construct an ensemble of di-jets, using the probability distribution of the dijet asymmetry measured in proton-proton collisions.We will also need a model for the evolution of the plasma and a distribution for the starting pointsand directions of the jets in the transverse plane. After sending our jets through an expanding cool-ing droplet of strongly coupled N = 4 SYM plasma, we shall investigate how the mean jet shapeand the distribution of the dijet asymmetry are modified by passage through the plasma.As in Ref. [145], we shall utilize perturbative QCD calculations of the probability distributionfor a useful measure of the opening angle of a jet in QCD defined by C (1)1 ≡ (cid:88) i,j z i z j θ ij R , (3.3)where the sum is over all pairs of hadrons in the jet, θ ij is the angle between hadrons i and j ,and z i is the momentum fraction of hadron i . We shall consider jets recontructed with the anti- k t algorithm [154] with reconstruction parameter R = 0 . , as in the CMS data that we shall compareour results to below. We shall (quite arbitrarily) take the quark and gluon fractions each to be 0.5in the formulae for the C (1)1 distribution calculated in perturbative QCD in Ref. [155]. The openingangle of a holographic jet is proportional to the downward angle of the string endpoint σ [143],but we have no direct analogue of C (1)1 since the holographic calculation does not have hadronsmeaning that we cannot calculate Eq. (3.3) explicitly. Therefore, as in Ref. [145] we shall take C (1)1 = aσ , (3.4)introducing a free parameter a in our model. This allows us to translate the perturbative QCDcalculations for the distribution of C (1)1 into a probability distribution for the initial downward angle σ of the strings in our ensemble. Note that this probability distribution depends on the initial jetenergy E . We complete the specification of our ensemble of strings by choosing a distribution ofinitial jet energies which falls as E − . – 22 – igure 8 : Mean jet shape in vacuum as a function of r , the angle in ( η, φ ) -space from the centerof the jet, computed from the ensemble of null strings described in the text compared to CMSmeasurements of the mean jet shape for jets with energy above 100 GeV reconstructed with anti- k t reconstruction parameter R = 0 . in proton–proton collisions with √ s NN = 2 . TeV at theLHC from Ref [33], shown as black symbols. The pink band shows the results obtained from ourensemble of strings for a range in the free parameter given by a = 1 . − . , with a = 2 shown indark red. Although the effects of doing so are negligible here, the experimental data and the resultsof our calculations have been smeared in order to take into account the CMS jet energy resolution,as described in Ref. [33] and later in the text.The differential jet shape for an individual jet is the power P out ( r ) as a function of the angle r from the jet axis, as given in Eq. (3.2). The (normalized) meanx jet shape is the average of theindividual jet shapes over the ensemble. We plot our results for the mean jet shape in Fig. 8, binningthem in bins of width ∆ r = 0 . for consistency with the CMS data, which we also show in theFigure. We find that upon making a suitable choice for the free parameter in the model a we obtaina rather good description of the mean jet shape measured in proton-proton collisions! The resultshown in Figure 8 has the best fit value a = 2 shown in red, which is in reasonable agreementwith the crude estimate of a ∼ . given in [145] for smooth jets, as well as a band of predictionscorresponding to varying a from 1.8 to 2.5. We find it pleasing, and perhaps even remarkable,that even though we picked the initial energy distribution along the null strings that we are usingto model jets from a strongly coupled calculation of the dynamics of strings as they nullify thatis quite different from the dynamics of jets in QCD, the mean jet shape that we obtain agrees sonicely with measurements made in proton-proton collisions.To compute the modification to the mean jet shape in our ensemble caused by the passage ofthe jets through the strongly coupled plasma, we need a model for the dynamics of the droplet ofplasma. We assume boost invariant longitudinal expansion and initialize the droplet of plasma at aproper time τ = 1 fm/c after the collision. As in Ref [145], we make the overly simplified assump-– 23 –ion that our null strings are produced at the same time that the hydrodynamic plasma is initialized,thus completely neglecting the possibility of energy loss before the plasma hydrodynamizes. Againas in Ref. [145], we also make the overly simplified assumption that all quenching stops and thestrings propagate as in vacuum after the droplet of plasma has cooled below T = 175 MeV. Thedroplet of N = 4 SYM plasma and its evolution are encoded, via the AdS-CFT correspondence,in changes to the 5-dimensional metric in AdS space. An expanding and cooling droplet of plasmain the field theory corresponds to a black hole in 5-dimensional AdS space whose horizon is ex-panding in the spatial directions while shrinking “downward”, away from the AdS boundary, inthe z -direction. As in Ref. [145], we take a simple blast-wave profile to model the temperatureevolution in the transverse plane and assume boost-invariant longitudinal expansion, choosing T ( τ, (cid:126)x ⊥ ) = b (cid:20) dN ch dy N part ρ part ( (cid:126)x ⊥ /r bl ( τ )) τ r bl ( τ ) (cid:21) / . (3.5)Here, τ ≡ √ t − z is the proper time, ρ part ( (cid:126)x ⊥ ) is the participant density in the transverse plane asgiven by an optical Glauber model, and r bl ( τ ) ≡ (cid:112) v T τ /R ) with v T = 0 . and R = 6 . fm.We consider 2.76 TeV Pb-Pb collisions at mid-rapidity and 0-10% centrality at the LHC, and basedupon averaging the results for 0-5% and 5-10% centrality from Ref. [156] we take the number ofparticipants to be N part (cid:39) and based upon summing the results for pions, kaons and protons inRef. [157] we take dN ch /dy (cid:39) . b is the second free parameter in our model; we shall use itto parameterize differences in the number of degrees of freedom between N = 4 SYM and QCD,meaning that we should model a QCD plasma with temperature T by an N = 4 SYM plasma withsome lower temperature. The constant b is a measure of the multiplicity per entropy, and for a QCDplasma is b ≈ . [145]. Since we are modeling this plasma with an N = 4 SYM plasma, in ourmodel we must scale the temperature that we use down by choosing a value of b that is substantiallysmaller than this [145].In this work, we consider an ensemble of ≈ , jets which sample distributions in jetopening angle, energy, and the starting position and direction of the jets within the droplet ofplasma. We take the initial position of the quark-antiquark pair in the transverse plane to be dis-tributed according to a binary scaling distribution proportional to ρ part ( (cid:126)x ⊥ ) with their directionsrandomly distributed in the transverse plane. We have described our choice for the initial distri-bution of the energy and opening angle, and hence downward angle for the null string endpoint,above. For the analysis of the dijet asymmetry, we additionally sample the initial dijet asymmetrydistribution, which increases the size of our ensemble by a factor of roughly 30.After we send our ensemble of strings through the expanding cooling droplet of plasma, werecompute the energies and opening angles (given by a times the downward angle of the stringendpoint after it emerges from the droplet of plasma) of each string in the ensemble. In the nextSection, we shall compute the modification to various observables from the ensemble after passagethrough the plasma. The simplest observable to calculate is the ratio of the number of jets with aspecified E in the ensemble after quenching to that number in the ensemble before quenching, aratio that is the analogue in our model of R jetAA . In Fig. 9 we show that if we choose b = 0 . weobtain a value for this ratio that is comparable to the measured value of R jetAA [31]. This value of b is qualitatively consistent with what has been found in other models. For example, in the hybridstrong/weak coupling model for jet quenching, fitting the single model parameter in that model to– 24 – igure 9 : Jet suppression R jet AA , the ratio of the number of jets with a given energy in our ensembleafter the jets have been quenched via passage through the expanding cooling droplet of plasma tothe number of jets with the same energy in the initial ensemble, before quenching. As in Fig. 8, thepink band shows the results obtained from our ensemble of strings for a range in the parameter a given by a = 1 . − . , with a = 2 shown in dark red. We have chosen the value of b , the secondfree parameter in the model, defined in the text, to be b = 0 . so as to obtain a jet suppression thatis comparable to experimental measurements of jet R AA , for example those in Refs. [31, 38, 46].experimental measurements of R jetAA yields the conclusion that the thermalization length x therm forjets is about 3-4 times longer in QCD plasma with temperature T than in strongly coupled N = 4 SYM plasma with the same temperature [147, 148].The blast wave temperature evolution is of course much simpler than the true hydrodynamicevolution of the strongly-coupled plasma in heavy ion collisions. In addition, the use of the metric(2.1) with a T that varies in space and time has the downside that it neglects flow, viscosity, andgradients in the plasma: (2.1) itself is the metric corresponding to a constant-temperature plasmaand when we insert a T that varies in space and time into it what we obtain is not a solution toEinstein’s equations for a plasma whose temperature varies in space and time, meaning that it doesnot describe a solution to hydrodynamics: as noted, it neglects flow, viscosity and gradients. Wehave done brief and preliminary investigations where we have chosen more realistic hydrodynamicbackgrounds. First, we have tried taking the temperature evolution from the viscous hydrodynamicsimulation of a heavy ion collision in Ref. [158] and used the metric (2.1) for this T that variesin space and time. Second, we have tried taking both the temperature and fluid velocity fromRef. [158] and implementing higher order gradient corrections to the metric to include the effects offlow and viscosity, as well as gradients in the fluid as in Ref. [159]. From this brief study, it appearsthat the alternative temperature profiles have only small quantitative effects on the energy loss,while the presence of flow and gradients may have somewhat larger effects but effects that are stillonly quantitative, not qualitative. Because of the computational complexity of such calculations, we– 25 –ostpone the inclusion of a full hydrodynamic background including fluid velocity, viscosity andgradients in the analysis of the full ensemble of jets to future works. In our limited study, includingflow in the plasma profile appears to decrease the energy loss. Improving upon our oversimplifiedtreatment (aka neglect) of energy loss before hydrodynamization and after hadronization wouldincrease the energy loss; we leave these investigations to future work also.Before we turn to our results, we note that we shall be comparing our results for how variousobservables are modified by passage through the plasma to CMS measurements of jets in PbPbcollisions from which the effects of the jet energy resolution of the detector have not been un-folded. This means that, as described in Refs. [27, 33], the appropriate baseline against which tocompare these measurements is not data from proton-proton collisions per se , because the jet en-ergy resolution of the CMS detector differs in PbPb and proton-proton collisions. What we needto do, therefore, is to take as inputs to our calculations distributions appropriate for unsmearedproton-proton collisions (we shall use P YTHIA simulations thereof), calculate the modifications toobservables after we run our ensemble of jets through a droplet of plasma, and then before compar-ing to measured data we must smear the jet energy in our initial (proton-proton) ensemble and inour final ensemble after quenching, in both cases using the Gaussian smearing functions for 0-10%centrality PbPb collisions provided by the CMS collaboration in Ref. [28]. In Figs. 8 and 9 and inthe following Section, we can then compare to CMS measurements in PbPb collisions that have notbeen unfolded, and to the pp baseline against which CMS compares these measurements, namelysimulated proton-proton jets smeared to take into account the difference in the jet energy resolutionof the CMS detector in PbPb and proton-proton collisions.
In previous Sections, we have described our construction of an ensemble of null strings in the dualgravitational description of N = 4 SYM theory that we shall use as a model for an ensemble ofjets. Using null strings means that we are automatically describing flows of energy whose openingangles do not change, in vacuum, making them natural as models for high energy jets in QCDwhose jet mass and jet energy do not change, in vacuum. Building an ensemble in such a waythat it can serve as a model for an ensemble of jets produced in proton-proton collisions requiredus to use key further inputs from several different directions. First, we chose the probability dis-tribution for the energy and opening angle of the jets represented by the strings in our ensemblefrom perturbative QCD calculations of these quantities in proton-proton collisions. This alone isnot enough, however, as we must specify the distribution of energy density along our null strings.We have done so in a way that incorporates a striking regularity of the dynamics of strings in AdSthat we identified in Section 2. We found that a large class of strings that are initially not null, andin particular that have a vanishing initial opening angle, evolve to become null, and as they do theenergy density distributed along them takes on a particular scaling form, when scaled relative to thedownward angle of the string endpoint, which is proportional to the opening angle of the jet thatthe string models. In Section 3 we have seen that if we choose the probability distribution for theopening angles of the jets in our ensemble according to the results of a perturbative QCD calcula-tion, and then distribute the energy density along the string that represents each jet according to thescaling form that we identified from our analysis of string dynamics, we obtain an ensemble of jets– 26 –ith a mean jet shape that is in excellent agreement with that measured in proton-proton collisions,see Fig. 8, upon fixing a , the first of two free parameters in our model.One way of thinking about the way that we have chosen our ensemble of strings is that wehave only included a subset of all possible null strings, the subset whose energy is distributedalong them such that they can be obtained by starting with non-null strings whose initial openingangle vanishes initially and letting these non-null strings evolve. The string dynamics turns theseinitially non-null strings into jet-like, null, strings with a particular form for their energy densitydistribution. Like any null string, these null strings can be created by initializing them that way.But, this subset of null strings can also be created by starting with non-null initial conditions andletting the string evolve and nullify. Although the property of the string dynamics that we areemploying is striking, we have no first-principles argument for why we should choose this subsetof null strings. Similarly, we do not know whether the only strings that can reasonably be used asmodels for jets are strings that are null from the beginning. We leave to future work constructing thebulk-to-boundary propagator for non-null strings which nullify, computing the gauge theory stress-energy tensor at early times, and investigating whether there are some initially non-null stringswhose opening angle is reasonably constant at all times, including before nullification. With such acomputation of the gauge theory stress-energy tensor at early times in hand, further investigationswould also be possible, including trying to design initially non-null strings that have a reasonablyconstant opening angle at early times and that at the same time have (fractal) substructure, as QCDjets do. For the present, we have an ensemble of strings that model an ensemble of jets in vacuumwith a mean jet shape as shown in Figure 8.Next, we send this ensemble of strings through the simplified model for the expanding, coolingdroplet of plasma that we have described in Section 3. There are many ways in which our treatmentof the medium that our ensemble of strings sees could be improved in future work. As noted in Sec-tion 3, we have neglected any interactions between the strings and the medium before τ = 1 fm /c and after the time when the plasma cools below T = 175 MeV. Both these oversimplifications canbe revisited in future work. Again as described in Section 3, we have used a blast wave modelfor the dynamics of the expansion and cooling of the droplet of plasma; this can be revisited too.At the same time in future when a full relativistic viscous hydrodynamic treatment of the plasmais used instead of a blast wave model, the AdS black hole metric should be augmented to includethe effects of shear viscosity and of fluid gradients. For the present, we have the blast wave modelfor the expanding, cooling droplet of plasma, as described in Section 3. There, we have fixed theparameter b (which is the proportionality constant between the temperature of the QCD plasma andthe temperature of the N = 4 SYM plasma — with its greater number of degrees of freedom —that we are using to model the QCD plasma). The value of b that we find corresponds to choosingthe N = 4 SYM plasma to be between 3 and 4 times cooler than the QCD plasma that we aremodeling, consistent with other estimates made in quite different ways [147, 148].After sending our ensemble of jets through the droplet of expanding, cooling plasma, wecalculate the new mean jet shape in the ensemble, after quenching. Note that, as in any experi-mental analysis, we impose a cut on the jet energy. In order to compare our results for the mod-ification of the mean jet shape to the experimental measurements in Ref [33], we impose the cut p jet T > GeV, meaning that any jet whose transverse momentum drops below 100 GeV uponpropagation through the plasma is removed from the ensemble. This means that even though every– 27 – igure 10 : The ratio of the mean shape of the jets in our ensemble after they have been quenchedby their passage through the expanding cooling droplet of plasma, as described in Section 3, to theirmean shape before quenching, shown in Fig. 8. As in Figs. 8 and 9, the pink band shows our resultsfor a = 1 . − . with a = 2 shown in dark red. The CMS measurement [33] of the modificationof the mean jet shape, namely the ratio of its measured value for jets with energy above 100 GeVreconstructed with anti- k t R = 0 . in 0-10% central PbPb collisions with √ s NN =2.76 TeV to thatin proton-proton collisions, as in Fig. 8, is shown as the blue symbols. The discrepancy betweenour calculation and the experimental measurements at large angles r relative to the center of the jetis expected, since we do not include any analogue of the particles originating from the wake thatthe jet leaves behind in the plasma, some of which must necessarily be reconstructed as a part ofthe jet in any experimental analysis even after background subtraction [149].jet in the ensemble gets wider as it propagates through the plasma, because jets that are initiallywider lose more energy and hence are more likely to drop below 100 GeV the mean opening angleof the jets in the ensemble can decrease [145]. Indeed, when we plot the ratio of the mean jet shapeof the jets in our ensemble after quenching to that in our ensemble before quenching in Fig. 10we see that quenching makes the mean jet shape get narrower in our calculation. In Fig. 10 wecompare the results of our calculation to the CMS data of Ref. [33], finding qualitative agreementat small r , namely close to the core of the jet, where we see that the ratio plotted in the figure dropsbelow one in our calculation and in data. This confirms that even though every jet in the ensemblegets wider as it propagates through the plasma, the mean jet shape of the jets in the ensemble with p T > GeV gets narrower. At larger r , our model does not include the soft particles comingfrom the wake in the plasma — which carry the momentum lost by the jet and which thereforemust contribute to the reconstructed jet [149]. Including these effects increases the number of softparticles at all angles in the jet cone, which pushes the ratio plotted in Fig. 10 significantly upwardsat larger r [149].A by now classic signature of the modification of jets in heavy ion collisions due to their– 28 – igure 11 : The dijet asymmetry distribution in our ensemble of holographic jets, before quenching(black curve) and after propagation through the strongly-coupled plasma of Section 3 (red curve).The red curve is drawn for a = 2 , with the pink band indicating a = 1 . − . . The CMSmeasurement of the dijet asymmetry distribution in the –
10 % most central Pb–Pb collisions with √ s NN = 2 . TeV from Ref. [27] is shown in blue symbols. In order to make a comparison to thisdata, as we described at the end of Section 3 we smeared both the unquenched input distribution thatwe obtained from P
YTHIA and the distribution after quenching that is the output of our calculation,doing the smearing as described in Refs. [27, 33],passage through the strongly coupled plasma is a significant enhancement in the dijet asymmetry.In events in which at least two jets are reconstructed, the dijet asymmetry is defined as A J ≡ ( p T, − p T, ) / ( p T, + p T, ) , where p T, and p T, are the transverse momenta of the jets with thelargest and second-to-largest transverse momenta. The A J distribution is reasonably broad alreadyin proton-proton collisions, see the black curve in Fig. 11. The two jets in a dijet need not beback-to-back and need not have the same energy first of all because there may be three or morejets in the event and second of all because of the interplay between the substructure of jets andthe algorithms via which jets are found and reconstructed. In heavy ion collisions, jet quenchingintroduces a significant new source of dijet asymmetry, since one jet in the dijet will always losemore energy as it traverses the plasma than the other. A further broadening of the A J distributionis thus a signature of jet quenching.The A J distribution in the absence of quenching cannot be captured fully in our model, sincewe have no analogue of jet finding or reconstruction and since we have no events with more thantwo jets. What we have done is to construct an ensemble of back-to-back dijets whose A J distri-bution is as in proton-proton collisions, taking that input distribution from P YTHIA , and using ahalf of one of our strings to represent each jet in a dijet pair. We have constructed an ensembleof roughly five million dijet events with a distribution of asymmetries, in addition to the distri-– 29 –utions of opening angles, energies, starting positions, and directions within the plasma as in thecomputation of the jet shape modification. The A J distribution from this ensemble, with the jetenergies suitably smeared to take account of the jet energy resolution of the CMS detector, as wedescribed at the end of Section 3, is shown as the black curve in Fig. 11. Note that although inthe analysis reported in Fig. 11 we have only used back-to-back dijets, we have also constructeda (smaller) ensemble in which we have not assumed that the jets are back-to-back, instead takingthe distribution of angles between the jets in dijet events from that measured in Ref. [27]. Thisdistribution is peaked, favoring jets which are back-to-back, and we have checked that our resultsfor the modification of the distribution of the dijet asymmetry A J are not significantly different inthis case than in our (larger) ensemble that is the basis for Fig. 11, in which the dijets are alwaysformed from a back-to-back pair. What is important is the choice of initial A J distribution.Next, we send each dijet in the ensemble through its droplet of strongly coupled plasma.Following Ref. [27], we then smear the jet energies as described at the end of Section 3, selectthose events in which p T, >
120 GeV / c and p T, >
30 GeV / c after quenching, and compute the A J distribution for this ensemble of dijets that have been quenched via their propagation throughthe plasma. Our results are shown in red in Figure 11, compared with CMS data from the –
10 % most central PbPb collisions.We find qualitative agreement between the modification to the distribution of the dijet asym-metry A J computed in this simple holographic model and heavy ion collision data from CMS. Weanticipate that the largest systematic effect not represented in the pink band in Figure 11 arisesfrom the absence of three-jet events in our calculation, since these are in fact the origin of muchof the dijet asymmetry in proton-proton collisions. We leave the construction of an ensemble ofholographic three-jet events to future work. Acknowledgments
We thank Francesca Bellini, Jorge Casalderrey-Solana, Paul Chesler, David Chinellato, Yang-TingChien, Andrej Ficnar, Alexander Kalweit, Yen-Jie Lee, Simone Marzani, Chris McGinn, Guil-herme Milhano, Daniel Pablos and Jesse Thaler for useful discussions. We are especially gratefulto Simone Marzani for providing the formulas of Ref. [155]. KR acknowledges the hospitality ofthe CERN Theory Group. AS is partially supported through the LANL/LDRD Program. This workis supported by the U.S. Department of Energy under grant Contract Number DE-SC0011090.
References [1] PHENIX Collaboration, K. Adcox et al., “Formation of dense partonic matter inrelativistic nucleus-nucleus collisions at RHIC: Experimental evaluation by the PHENIXcollaboration” , Nucl. Phys.
A757 , 184 (2005), nucl-ex/0410003 .[2] BRAHMS Collaboration, I. Arsene et al., “Quark gluon plasma and color glasscondensate at RHIC? The Perspective from the BRAHMS experiment” , Nucl. Phys.
A757 , 1(2005), nucl-ex/0410020 .[3] B. B. Back et al., “The PHOBOS perspective on discoveries at RHIC” , Nucl. Phys.
A757 ,28 (2005), nucl-ex/0410022 . – 30 –4] STAR Collaboration, J. Adams et al., “Experimental and theoretical challenges in thesearch for the quark gluon plasma: The STAR Collaboration’s critical assessment of theevidence from RHIC collisions” , Nucl. Phys.
A757 , 102 (2005), nucl-ex/0501009 .[5] ALICE Collaboration, K. Aamodt et al., “Elliptic flow of charged particles in Pb-Pbcollisions at √ s NN = 2 . TeV” , Phys. Rev. Lett. , 252302 (2010), arXiv:1011.3914 .[6] ATLAS Collaboration, G. Aad et al., “Measurement of the pseudorapidity and transversemomentum dependence of the elliptic flow of charged particles in lead-lead collisions at √ s NN = 2 . TeV with the ATLAS detector” , Phys. Lett.
B707 , 330 (2012), arXiv:1108.6018 .[7] CMS Collaboration, S. Chatrchyan et al., “Measurement of the elliptic anisotropy ofcharged particles produced in PbPb collisions at √ s NN =2.76 TeV” , Phys. Rev. C87 ,014902 (2013), arXiv:1204.1409 .[8] D. Teaney, J. Lauret & E. V. Shuryak, “Flow at the SPS and RHIC as a quark gluon plasmasignature” , Phys. Rev. Lett. , 4783 (2001), nucl-th/0011058 .[9] P. Huovinen, P. F. Kolb, U. W. Heinz, P. V. Ruuskanen & S. A. Voloshin, “Radial andelliptic flow at RHIC: Further predictions” , Phys. Lett. B503 , 58 (2001), hep-ph/0101136 .[10] D. Teaney, J. Lauret & E. V. Shuryak, “A Hydrodynamic description of heavy ion collisionsat the SPS and RHIC” , nucl-th/0110037 .[11] T. Hirano, U. W. Heinz, D. Kharzeev, R. Lacey & Y. Nara, “Hadronic dissipative effects onelliptic flow in ultrarelativistic heavy-ion collisions” , Phys. Lett. B636 , 299 (2006), nucl-th/0511046 .[12] P. Romatschke & U. Romatschke, “Viscosity Information from Relativistic NuclearCollisions: How Perfect is the Fluid Observed at RHIC?” , Phys. Rev. Lett. , 172301(2007), arXiv:0706.1522 .[13] M. Luzum & P. Romatschke, “Conformal Relativistic Viscous Hydrodynamics:Applications to RHIC results at √ s NN = 200 GeV” , Phys. Rev.
C78 , 034915 (2008), arXiv:0804.4015 , [Erratum: Phys. Rev.
C79 , 039903 (2009)].[14] B. Schenke, S. Jeon & C. Gale, “Elliptic and triangular flow in event-by-event (3+1)Dviscous hydrodynamics” , Phys. Rev. Lett. , 042301 (2011), arXiv:1009.3244 .[15] T. Hirano, P. Huovinen & Y. Nara, “Elliptic flow in Pb+Pb collisions at √ s NN = 2 . TeV: hybrid model assessment of the first data” , Phys. Rev.
C84 , 011901(2011), arXiv:1012.3955 .[16] C. Gale, S. Jeon, B. Schenke, P. Tribedy & R. Venugopalan, “Event-by-event anisotropicflow in heavy-ion collisions from combined Yang-Mills and viscous fluid dynamics” ,Phys. Rev. Lett. , 012302 (2013), arXiv:1209.6330 .[17] C. Shen, Z. Qiu, H. Song, J. Bernhard, S. Bass & U. Heinz, “The iEBE-VISHNU code – 31 – ackage for relativistic heavy-ion collisions” , Comput. Phys. Commun. , 61 (2016), arXiv:1409.8164 .[18] C. Shen, J.-F. Paquet, U. Heinz & C. Gale, “Photon Emission from a MomentumAnisotropic Quark-Gluon Plasma” , Phys. Rev.
C91 , 014908 (2015), arXiv:1410.3404 .[19] S. Ryu, J. F. Paquet, C. Shen, G. S. Denicol, B. Schenke, S. Jeon & C. Gale, “Importanceof the Bulk Viscosity of QCD in Ultrarelativistic Heavy-Ion Collisions” , Phys. Rev. Lett. , 132301 (2015), arXiv:1502.01675 .[20] J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu & U. Heinz, “Applying Bayesianparameter estimation to relativistic heavy-ion collisions: simultaneous characterization ofthe initial state and quark-gluon plasma medium” , Phys. Rev.
C94 , 024907 (2016), arXiv:1605.03954 .[21] S. A. Bass, J. E. Bernhard & J. S. Moreland, “Determination of Quark-Gluon-PlasmaParameters from a Global Bayesian Analysis” , arXiv:1704.07671 .[22] G. Policastro, D. Son & A. Starinets, “The Shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma” , Phys. Rev. Lett. , 081601 (2001), hep-th/0104066 .[23] A. Buchel & J. T. Liu, “Universality of the shear viscosity in supergravity” ,Phys. Rev. Lett. , 090602 (2004), hep-th/0311175 .[24] P. Kovtun, D. T. Son & A. O. Starinets, “Viscosity in strongly interacting quantum fieldtheories from black hole physics” , Phys. Rev. Lett. , 111601 (2005), hep-th/0405231 .[25] ATLAS Collaboration, G. Aad et al., “Observation of a Centrality-Dependent DijetAsymmetry in Lead-Lead Collisions at √ s NN = 2 . TeV with the ATLAS Detector at theLHC” , Phys. Rev. Lett. , 252303 (2010), arXiv:1011.6182 .[26] CMS Collaboration, S. Chatrchyan et al., “Observation and studies of jet quenching inPbPb collisions at √ s NN = 2 . TeV” , Phys. Rev.
C84 , 024906 (2011), arXiv:1102.1957 .[27] CMS Collaboration, S. Chatrchyan et al., “Jet momentum dependence of jet quenching inPbPb collisions at √ s NN = 2 . TeV” , Phys. Lett.
B712 , 176 (2012), arXiv:1202.5022 .[28] CMS Collaboration, S. Chatrchyan et al., “Studies of jet quenching usingisolated-photon+jet correlations in PbPb and pp collisions at √ s NN = 2 . TeV” ,Phys. Lett.
B718 , 773 (2013), arXiv:1205.0206 .[29] CMS Collaboration, S. Chatrchyan et al., “Measurement of jet fragmentation into chargedparticles in pp and PbPb collisions at √ s NN = 2 . TeV” , JHEP , 087 (2012), arXiv:1205.5872 .[30] ATLAS Collaboration, G. Aad et al., “Measurement of the jet radius and transverse – 32 – omentum dependence of inclusive jet suppression in lead-lead collisions at √ s NN = 2.76TeV with the ATLAS detector” , Phys. Lett. B719 , 220 (2013), arXiv:1208.1967 .[31] CMS Collaboration, S. Chatrchyan et al., “Nuclear modification factor of high transversemomentum jets in PbPb collisions at √ s NN = 2 . TeV” , CMS-PAS-HIN-12-004 .[32] ATLAS Collaboration, G. Aad et al., “Measurement of the Azimuthal Angle Dependence ofInclusive Jet Yields in Pb+Pb Collisions at √ s NN = ,Phys. Rev. Lett. , 152301 (2013), arXiv:1306.6469 .[33] CMS Collaboration, S. Chatrchyan et al., “Modification of jet shapes in PbPb collisions at √ s NN = 2 . TeV” , Phys. Lett.
B730 , 243 (2014), arXiv:1310.0878 .[34] ALICE Collaboration, B. Abelev et al., “Measurement of charged jet suppression in Pb-Pbcollisions at √ s NN = 2.76 TeV” , JHEP , 013 (2014), arXiv:1311.0633 .[35] CMS Collaboration, S. Chatrchyan et al., “Evidence of b-Jet Quenching in PbPb Collisionsat √ s NN = 2 . TeV” , Phys. Rev. Lett. , 132301 (2014), arXiv:1312.4198 ,[Erratum: Phys. Rev. Lett. , 029903 (2015)].[36] CMS Collaboration, S. Chatrchyan et al., “Measurement of jet fragmentation in PbPb andpp collisions at √ s NN = 2 . TeV” , Phys. Rev.
C90 , 024908 (2014), arXiv:1406.0932 .[37] ATLAS Collaboration, G. Aad et al., “Measurement of inclusive jet charged-particlefragmentation functions in Pb+Pb collisions at √ s NN = 2 . TeV with the ATLASdetector” , Phys. Lett.
B739 , 320 (2014), arXiv:1406.2979 .[38] ATLAS Collaboration, G. Aad et al., “Measurements of the Nuclear Modification Factorfor Jets in Pb+Pb Collisions at √ s NN = 2 . TeV with the ATLAS Detector” ,Phys. Rev. Lett. , 072302 (2015), arXiv:1411.2357 .[39] ALICE Collaboration, J. Adam et al., “Measurement of jet suppression in central Pb-Pbcollisions at √ s NN = 2.76 TeV” , Phys. Lett. B746 , 1 (2015), arXiv:1502.01689 .[40] ALICE Collaboration, J. Adam et al., “Measurement of jet quenching with semi-inclusivehadron-jet distributions in central Pb-Pb collisions at √ s NN = 2 . TeV” , JHEP , 170(2015), arXiv:1506.03984 .[41] ATLAS Collaboration, G. Aad et al., “Measurement of the production of neighbouring jetsin lead-lead collisions at √ s NN = 2 . TeV with the ATLAS detector” , Phys. Lett.
B751 ,376 (2015), arXiv:1506.08656 .[42] CMS Collaboration Collaboration, S. Chatrchyan et al., “Study of Isolated-Photon + JetCorrelations in PbPb collisions at √ s NN = 5 . TeV” , CMS-PAS-HIN-16-002 .[43] CMS Collaboration, V. Khachatryan et al., “Measurement of transverse momentum relativeto dijet systems in PbPb and pp collisions at √ s NN = 2 . TeV” , JHEP , 006 (2016), arXiv:1509.09029 .[44] CMS Collaboration, V. Khachatryan et al., “Correlations between jets and charged – 33 – articles in PbPb and pp collisions at √ s NN = 2 . TeV” , JHEP , 156 (2016), arXiv:1601.00079 .[45] CMS Collaboration, V. Khachatryan et al., “Decomposing transverse momentum balancecontributions for quenched jets in PbPb collisions at √ s NN = 2 . TeV” , JHEP , 055(2016), arXiv:1609.02466 .[46] CMS Collaboration, V. Khachatryan et al., “Measurement of inclusive jet cross sections in pp and PbPb collisions at √ s NN = , Phys. Rev. C96 , 015202 (2017), arXiv:1609.05383 .[47] ATLAS Collaboration, M. Aaboud et al., “Measurement of jet fragmentation in Pb+Pb and pp collisions at √ s NN = 2 . TeV with the ATLAS detector at the LHC” , Eur. Phys. J.
C77 , 379 (2017), arXiv:1702.00674 .[48] ALICE Collaboration, S. Acharya et al., “First measurement of jet mass in Pb-Pb and p-Pbcollisions at the LHC” , arXiv:1702.00804 .[49] CMS Collaboration, A. M. Sirunyan et al., “Study of Jet Quenching with Z + jetCorrelations in Pb-Pb and pp Collisions at √ s NN = 5 .
02 TeV ” , Phys. Rev. Lett. ,082301 (2017), arXiv:1702.01060 .[50] ATLAS Collaboration, M. Aaboud et al., “Measurement of jet p T correlations in Pb+Pband pp collisions at √ s NN = 2 . TeV with the ATLAS detector” , arXiv:1706.09363 .[51] PHENIX Collaboration, K. Adcox et al., “Suppression of hadrons with large transversemomentum in central Au+Au collisions at √ s NN = 130 GeV” , Phys. Rev. Lett. , 022301(2002), nucl-ex/0109003 .[52] STAR Collaboration, C. Adler et al., “Centrality dependence of high p T hadronsuppression in Au+Au collisions at √ s NN = 130 GeV” , Phys. Rev. Lett. , 202301(2002), nucl-ex/0206011 .[53] STAR Collaboration, C. Adler et al., “Disappearance of back-to-back high p T hadroncorrelations in central Au+Au collisions at √ s NN = 200 GeV” , Phys. Rev. Lett. ,082302 (2003), nucl-ex/0210033 .[54] STAR Collaboration, M. Ploskon, “Inclusive cross section and correlations of fullyreconstructed jets in √ s NN = 200 GeV Au+Au and p+p collisions” , Nucl. Phys.
A830 ,255C (2009), arXiv:0908.1799 , in “Proceedings, 21st International Conference onUltra-Relativistic nucleus nucleus collisions (Quark matter 2009): Knoxville, USA, March30-April 4, 2009” , p. 255C-258C.[55] PHENIX Collaboration, D. V. Perepelitsa, “Reconstructed Jet Results in p + p, d + Au andCu + Cu collisions at 200 GeV from PHENIX” , Nucl. Phys.
A910-911 , 425 (2013), in “Proceedings, 5th International Conference on Hard and Electromagnetic Probes ofHigh-Energy Nuclear Collisions (Hard Probes 2012): Cagliari, Italy, May 27-June 1,2012” , p. 425-428.[56] STAR Collaboration, L. Adamczyk et al., “Jet-Hadron Correlations in √ s NN = 200 GeV – 34 – + p and Central Au + Au Collisions” , Phys. Rev. Lett. , 122301 (2014), arXiv:1302.6184 .[57] STAR Collaboration, P. M. Jacobs & A. Schmah, “Measurements of jet quenching withsemi-inclusive charged jet distributions in Au + Au collisions at √ s NN =200 GeV” ,Nucl. Phys. A956 , 641 (2016), arXiv:1512.08784 , in “Proceedings, 25thInternational Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (Quark Matter2015): Kobe, Japan, September 27-October 3, 2015” , p. 641-644.[58] STAR Collaboration, L. Adamczyk et al., “Dijet imbalance measurements in Au + Au and pp collisions at √ s NN = 200 GeV at STAR” , Phys. Rev. Lett. , 062301 (2017), arXiv:1609.03878 .[59] STAR Collaboration, L. Adamczyk et al., “Measurements of jet quenching withsemi-inclusive hadron+jet distributions in Au+Au collisions at √ s NN = 200 GeV” ,Phys. Rev. C96 , 024905 (2017), arXiv:1702.01108 .[60] A. Adare et al., “An Upgrade Proposal from the PHENIX Collaboration” , arXiv:1501.06197 .[61] P. Jacobs & X.-N. Wang, “Matter in extremis: Ultrarelativistic nuclear collisions atRHIC” , Prog. Part. Nucl. Phys. , 443 (2005), hep-ph/0405125 .[62] J. Casalderrey-Solana & C. A. Salgado, “Introductory lectures on jet quenching in heavyion collisions” , Acta Phys. Polon. B38 , 3731 (2007), arXiv:0712.3443 , in “Theoretical physics. Proceedings, 47th Cracow School, Zakopane, Poland, June 14-22,2007” , p. 3731-3794.[63] A. Majumder & M. Van Leeuwen, “The Theory and Phenomenology of Perturbative QCDBased Jet Quenching” , Prog. Part. Nucl. Phys. , 41 (2011), arXiv:1002.2206 .[64] Y. Mehtar-Tani, J. G. Milhano & K. Tywoniuk, “Jet physics in heavy-ion collisions” ,Int. J. Mod. Phys. A28 , 1340013 (2013), arXiv:1302.2579 .[65] J. Ghiglieri & D. Teaney, “Parton energy loss and momentum broadening at NLO in hightemperature QCD plasmas” , Int. J. Mod. Phys.
E24 , 1530013 (2015).[66] J.-P. Blaizot & Y. Mehtar-Tani, “Jet Structure in Heavy Ion Collisions” , Int. J. Mod. Phys.
E24 , 1530012 (2015), arXiv:1503.05958 .[67] G.-Y. Qin & X.-N. Wang, “Jet quenching in high-energy heavy-ion collisions” ,Int. J. Mod. Phys.
E24 , 1530014 (2015), arXiv:1511.00790 .[68] K. Zapp, J. Stachel & U. A. Wiedemann, “A Local Monte Carlo implementation of thenon-abelian Landau-Pomerantschuk-Migdal effect” , Phys. Rev. Lett. , 152302 (2009), arXiv:0812.3888 .[69] K. Zapp, G. Ingelman, J. Rathsman, J. Stachel & U. A. Wiedemann, “A Monte CarloModel for ’Jet Quenching’” , Eur. Phys. J.
C60 , 617 (2009), arXiv:0804.3568 .[70] N. Armesto, L. Cunqueiro & C. A. Salgado, “Q-PYTHIA: A Medium-modified – 35 – mplementation of final state radiation” , Eur. Phys. J.
C63 , 679 (2009), arXiv:0907.1014 .[71] B. Schenke, C. Gale & S. Jeon, “MARTINI: An Event generator for relativistic heavy-ioncollisions” , Phys. Rev.
C80 , 054913 (2009), arXiv:0909.2037 .[72] I. P. Lokhtin, A. V. Belyaev & A. M. Snigirev, “Jet quenching pattern at LHC in PYQUENmodel” , Eur. Phys. J.
C71 , 1650 (2011), arXiv:1103.1853 .[73] K. C. Zapp, F. Krauss & U. A. Wiedemann, “A perturbative framework for jet quenching” ,JHEP , 080 (2013), arXiv:1212.1599 .[74] K. C. Zapp, “JEWEL 2.0.0: directions for use” , Eur. Phys. J.
C74 , 2762 (2014), arXiv:1311.0048 .[75] K. C. Zapp, “Geometrical aspects of jet quenching in JEWEL” , Phys. Lett.
B735 , 157(2014), arXiv:1312.5536 .[76] JETSCAPE Collaboration, S. Cao et al., “Multistage Monte-Carlo simulation of jetmodification in a static medium” , Phys. Rev.
C96 , 024909 (2017), arXiv:1705.00050 .[77] I. Vitev & B.-W. Zhang, “Jet tomography of high-energy nucleus-nucleus collisions atnext-to-leading order” , Phys. Rev. Lett. , 132001 (2010), arXiv:0910.1090 .[78] J. Casalderrey-Solana, J. G. Milhano & U. A. Wiedemann, “Jet Quenching via JetCollimation” , J. Phys.
G38 , 035006 (2011), arXiv:1012.0745 .[79] G.-Y. Qin & B. Muller, “Explanation of Di-jet asymmetry in Pb+Pb collisions at the LargeHadron Collider” , Phys. Rev. Lett. , 162302 (2011), arXiv:1012.5280 , [Erratum:Phys. Rev. Lett.108,189904(2012)].[80] C. Young, B. Schenke, S. Jeon & C. Gale, “Dijet asymmetry at the energies available at theCERN Large Hadron Collider” , Phys. Rev.
C84 , 024907 (2011), arXiv:1103.5769 .[81] Y. He, I. Vitev & B.-W. Zhang, “ O ( α s ) Analysis of Inclusive Jet and di-Jet Production inHeavy Ion Reactions at the Large Hadron Collider” , Phys. Lett.
B713 , 224 (2012), arXiv:1105.2566 .[82] J. Casalderrey-Solana, J. G. Milhano & U. Wiedemann, “Jet quenching via jetcollimation” , J. Phys.
G38 , 124086 (2011), arXiv:1107.1964 , in “Quark matter.Proceedings, 22nd International Conference on Ultra-Relativistic Nucleus-NucleusCollisions, Quark Matter 2011, Annecy, France, May 23-28, 2011” , p. 124086.[83] T. Renk, “On the sensitivity of the dijet asymmetry to the physics of jet quenching” ,Phys. Rev.
C85 , 064908 (2012), arXiv:1202.4579 .[84] R. B. Neufeld & I. Vitev, “The Z -tagged jet event asymmetry in heavy-ion collisions at theCERN Large Hadron Collider” , Phys. Rev. Lett. , 242001 (2012), arXiv:1202.5556 .[85] T. Renk, “Energy dependence of the dijet imbalance in Pb-Pb collisions at 2.76 ATeV” ,Phys. Rev. C86 , 061901 (2012), arXiv:1204.5572 .– 36 –86] W. Dai, I. Vitev & B.-W. Zhang, “Momentum imbalance of isolated photon-tagged jetproduction at RHIC and LHC” , Phys. Rev. Lett. , 142001 (2013), arXiv:1207.5177 .[87] L. Apolinario, N. Armesto & L. Cunqueiro, “An analysis of the influence of backgroundsubtraction and quenching on jet observables in heavy-ion collisions” , JHEP , 022(2013), arXiv:1211.1161 .[88] X.-N. Wang & Y. Zhu, “Medium Modification of γ -jets in High-energy Heavy-ionCollisions” , Phys. Rev. Lett. , 062301 (2013), arXiv:1302.5874 .[89] G.-L. Ma, “Dijet asymmetry in Pb+Pb collisions at √ s NN =2.76 TeV within a multiphasetransport model” , Phys. Rev. C87 , 064901 (2013), arXiv:1304.2841 .[90] J. Huang, Z.-B. Kang & I. Vitev, “Inclusive b-jet production in heavy ion collisions at theLHC” , Phys. Lett.
B726 , 251 (2013), arXiv:1306.0909 .[91] F. Senzel, O. Fochler, J. Uphoff, Z. Xu & C. Greiner, “Influence of multiple in-mediumscattering processes on the momentum imbalance of reconstructed di-jets” , J. Phys.
G42 ,115104 (2015), arXiv:1309.1657 .[92] R. Perez-Ramos & T. Renk, “In-medium jet shape from energy collimation in partonshowers: Comparison with CMS PbPb data at 2.76 TeV” , Phys. Rev.
D90 , 014018 (2014), arXiv:1401.5283 .[93] T. Renk, “A study of the constraining power of high p T observables in heavy-ioncollisions” , arXiv:1408.6684 .[94] R. Perez-Ramos & T. Renk, “A Monte Carlo study of jet fragmentation functions in PbPband pp collisions at √ s = 2 . TeV” , arXiv:1411.1983 .[95] Y.-T. Chien, A. Emerman, Z.-B. Kang, G. Ovanesyan & I. Vitev, “Jet Quenching fromQCD Evolution” , Phys. Rev. D93 , 074030 (2016), arXiv:1509.02936 .[96] J. Huang, Z.-B. Kang, I. Vitev & H. Xing, “Photon-tagged and B-meson-tagged b-jetproduction at the LHC” , Phys. Lett.
B750 , 287 (2015), arXiv:1505.03517 .[97] Y.-T. Chien & I. Vitev, “Towards the understanding of jet shapes and cross sections inheavy ion collisions using soft-collinear effective theory” , JHEP , 023 (2016), arXiv:1509.07257 .[98] J. G. Milhano & K. C. Zapp, “Origins of the di-jet asymmetry in heavy ion collisions” ,Eur. Phys. J.
C76 , 288 (2016), arXiv:1512.08107 .[99] X. Zhang, L. Apolinario, J. G. Milhano & M. Ploskon, “Sub-jet structure as adiscriminating quenching probe” , Nucl. Phys.
A956 , 597 (2016), arXiv:1512.09255 ,in “Proceedings, 25th International Conference on Ultra-Relativistic Nucleus-NucleusCollisions (Quark Matter 2015): Kobe, Japan, September 27-October 3, 2015” ,p. 597-600.[100] N.-B. Chang & G.-Y. Qin, “Full jet evolution in quark-gluon plasma and nuclear – 37 – odification of jet production and jet shape in Pb+Pb collisions at 2.76ATeV at the CERNLarge Hadron Collider” , Phys. Rev.
C94 , 024902 (2016), arXiv:1603.01920 .[101] A. H. Mueller, B. Wu, B.-W. Xiao & F. Yuan, “Probing Transverse Momentum Broadeningin Heavy Ion Collisions” , Phys. Lett.
B763 , 208 (2016), arXiv:1604.04250 .[102] L. Chen, G.-Y. Qin, S.-Y. Wei, B.-W. Xiao & H.-Z. Zhang, “Probing TransverseMomentum Broadening via Dihadron and Hadron-jet Angular Correlations in RelativisticHeavy-ion Collisions” , Phys. Lett.
B773 , 672 (2017), arXiv:1607.01932 .[103] Y. Mehtar-Tani & K. Tywoniuk, “Groomed jets in heavy-ion collisions: sensitivity tomedium-induced bremsstrahlung” , JHEP , 125 (2017), arXiv:1610.08930 .[104] Y. Tachibana, N.-B. Chang & G.-Y. Qin, “Full jet in quark-gluon plasma withhydrodynamic medium response” , Phys. Rev.
C95 , 044909 (2017), arXiv:1701.07951 .[105] R. Kunnawalkam Elayavalli & K. C. Zapp, “Medium response in JEWEL and its impact onjet shape observables in heavy ion collisions” , JHEP , 141 (2017), arXiv:1707.01539 .[106] J. G. Milhano, U. A. Wiedemann & K. C. Zapp, “Sensitivity of jet substructure tojet-induced medium response” , arXiv:1707.04142 .[107] Y. Mehtar-Tani & K. Tywoniuk, “Quenching of high- p T jet spectra” , arXiv:1707.07361 .[108] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity” ,Int. J. Theor. Phys. , 1113 (1999), hep-th/9711200 , [Adv. Theor. Math. Phys. , 231(1998)].[109] J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal & U. A. Wiedemann, “Gauge/String Duality, Hot QCD and Heavy Ion Collisions” , Cambridge University Press(2014), see also 1101.0618.[110] O. DeWolfe, S. S. Gubser, C. Rosen & D. Teaney, “Heavy ions and string theory” ,Prog. Part. Nucl. Phys. , 86 (2014), arXiv:1304.7794 .[111] P. M. Chesler & W. van der Schee, “Early thermalization, hydrodynamics and energy lossin AdS/CFT” , Int. J. Mod. Phys. E24 , 1530011 (2015), arXiv:1501.04952 .[112] C. Herzog, A. Karch, P. Kovtun, C. Kozcaz & L. Yaffe, “Energy loss of a heavy quarkmoving through N = 4 supersymmetric Yang-Mills plasma” , JHEP , 013 (2006), hep-th/0605158 .[113] H. Liu, K. Rajagopal & U. A. Wiedemann, “Calculating the jet quenching parameter fromAdS/CFT” , Phys. Rev. Lett. , 182301 (2006), hep-ph/0605178 .[114] J. Casalderrey-Solana & D. Teaney, “Heavy quark diffusion in strongly coupled N = 4 Yang-Mills” , Phys. Rev.
D74 , 085012 (2006), hep-ph/0605199 .[115] S. S. Gubser, “Drag force in AdS/CFT” , Phys. Rev.
D74 , 126005 (2006), hep-th/0605182 . – 38 –116] H. Liu, K. Rajagopal & U. A. Wiedemann, “An AdS/CFT Calculation of Screening in aHot Wind” , Phys. Rev. Lett. , 182301 (2007), hep-ph/0607062 .[117] H. Liu, K. Rajagopal & U. A. Wiedemann, “Wilson loops in heavy ion collisions and theircalculation in AdS/CFT” , JHEP , 066 (2007), hep-ph/0612168 .[118] S. S. Gubser, “Momentum fluctuations of heavy quarks in the gauge-string duality” ,Nucl. Phys. B790 , 175 (2008), hep-th/0612143 .[119] M. Chernicoff, J. A. Garcia & A. Guijosa, “The Energy of a Moving Quark-Antiquark Pairin an N = 4 SYM Plasma” , JHEP , 068 (2006), hep-th/0607089 .[120] J. Casalderrey-Solana & D. Teaney, “Transverse Momentum Broadening of a Fast Quark ina N = 4 Yang Mills Plasma” , JHEP , 039 (2007), hep-th/0701123 .[121] P. M. Chesler & L. G. Yaffe, “The Wake of a quark moving through a strongly-coupledplasma” , Phys. Rev. Lett. , 152001 (2007), arXiv:0706.0368 .[122] S. S. Gubser, S. S. Pufu & A. Yarom, “Sonic booms and diffusion wakes generated by aheavy quark in thermal AdS/CFT” , Phys. Rev. Lett. , 012301 (2008), arXiv:0706.4307 .[123] P. M. Chesler & L. G. Yaffe, “The Stress-energy tensor of a quark moving through astrongly-coupled N = 4 supersymmetric Yang-Mills plasma: Comparing hydrodynamicsand AdS/CFT” , Phys. Rev. D78 , 045013 (2008), arXiv:0712.0050 .[124] D. M. Hofman & J. Maldacena, “Conformal collider physics: Energy and chargecorrelations” , JHEP , 012 (2008), arXiv:0803.1467 .[125] S. S. Gubser, D. R. Gulotta, S. S. Pufu & F. D. Rocha, “Gluon energy loss in thegauge-string duality” , JHEP , 052 (2008), arXiv:0803.1470 .[126] Y. Hatta, E. Iancu & A. H. Mueller, “Jet evolution in the N = 4 SYM plasma at strongcoupling” , JHEP , 037 (2008), arXiv:0803.2481 .[127] F. Dominguez, C. Marquet, A. H. Mueller, B. Wu & B.-W. Xiao, “Comparing energy lossand p ⊥ - broadening in perturbative QCD with strong coupling N = 4 SYM theory” ,Nucl. Phys.
A811 , 197 (2008), arXiv:0803.3234 .[128] P. M. Chesler, K. Jensen & A. Karch, “Jets in strongly-coupled N = 4 super Yang-Millstheory” , Phys. Rev. D79 , 025021 (2009), arXiv:0804.3110 .[129] P. M. Chesler, K. Jensen, A. Karch & L. G. Yaffe, “Light quark energy loss instrongly-coupled N = 4 supersymmetric Yang-Mills plasma” , Phys. Rev. D79 , 125015(2009), arXiv:0810.1985 .[130] F. D’Eramo, H. Liu & K. Rajagopal, “Transverse Momentum Broadening and the JetQuenching Parameter, Redux” , Phys. Rev.
D84 , 065015 (2011), arXiv:1006.1367 .[131] P. Arnold & D. Vaman, “Jet quenching in hot strongly coupled gauge theories revisited:3-point correlators with gauge-gravity duality” , JHEP , 099 (2010), arXiv:1008.4023 . – 39 –132] P. Arnold & D. Vaman, “Jet quenching in hot strongly coupled gauge theories simplified” ,JHEP , 027 (2011), arXiv:1101.2689 .[133] P. Arnold & D. Vaman, “Some new results for ’jet’ stopping in AdS/CFT: long version” ,J. Phys.
G38 , 124175 (2011), arXiv:1106.1680 , in “Quark matter. Proceedings, 22ndInternational Conference on Ultra-Relativistic Nucleus-Nucleus Collisions, Quark Matter2011, Annecy, France, May 23-28, 2011” , p. 124175.[134] M. Chernicoff, J. A. Garcia, A. Guijosa & J. F. Pedraza, “Holographic Lessons for QuarkDynamics” , J. Phys.
G39 , 054002 (2012), arXiv:1111.0872 .[135] P. M. Chesler, Y.-Y. Ho & K. Rajagopal, “Shining a Gluon Beam Through Quark-GluonPlasma” , Phys. Rev.
D85 , 126006 (2012), arXiv:1111.1691 .[136] P. Arnold, P. Szepietowski & D. Vaman, “Coupling dependence of jet quenching in hotstrongly-coupled gauge theories” , JHEP , 024 (2012), arXiv:1203.6658 .[137] P. Arnold, P. Szepietowski, D. Vaman & G. Wong, “Tidal stretching of gravitons intoclassical strings: application to jet quenching with AdS/CFT” , JHEP , 130 (2013), arXiv:1212.3321 .[138] P. M. Chesler, M. Lekaveckas & K. Rajagopal, “Heavy quark energy loss far fromequilibrium in a strongly coupled collision” , JHEP , 013 (2013), arXiv:1306.0564 .[139] A. Ficnar & S. S. Gubser, “Finite momentum at string endpoints” , Phys. Rev.
D89 , 026002(2014), arXiv:1306.6648 .[140] A. Ficnar, S. S. Gubser & M. Gyulassy, “Shooting String Holography of Jet Quenching atRHIC and LHC” , Phys. Lett.
B738 , 464 (2014), arXiv:1311.6160 .[141] P. M. Chesler & K. Rajagopal, “Jet quenching in strongly coupled plasma” , Phys. Rev.
D90 , 025033 (2014), arXiv:1402.6756 .[142] R. Rougemont, A. Ficnar, S. Finazzo & J. Noronha, “Energy loss, equilibration, andthermodynamics of a baryon rich strongly coupled quark-gluon plasma” , JHEP , 102(2016), arXiv:1507.06556 .[143] P. M. Chesler & K. Rajagopal, “On the Evolution of Jet Energy and Opening Angle inStrongly Coupled Plasma” , JHEP , 098 (2016), arXiv:1511.07567 .[144] J. Casalderrey-Solana & A. Ficnar, “Holographic Three-Jet Events in Strongly Coupled N = 4 Yang-Mills Plasma” , Nucl. Part. Phys. Proc. , 115 (2016), arXiv:1512.00371 .[145] K. Rajagopal, A. V. Sadofyev & W. van der Schee, “Evolution of the jet opening angledistribution in holographic plasma” , Phys. Rev. Lett. , 211603 (2016), arXiv:1602.04187 .[146] J. Brewer, K. Rajagopal, A. Sadofyev & W. van der Schee, “Holographic Jet Shapes andtheir Evolution in Strongly Coupled Plasma” , Nucl. Phys.
A967 , 508 (2017), arXiv:1704.05455 . – 40 –147] J. Casalderrey-Solana, D. C. Gulhan, J. G. Milhano, D. Pablos & K. Rajagopal, “A HybridStrong/Weak Coupling Approach to Jet Quenching” , JHEP , 19 (2014), arXiv:1405.3864 , [Erratum: JHEP , 175 (2015)].[148] J. Casalderrey-Solana, D. C. Gulhan, J. G. Milhano, D. Pablos & K. Rajagopal, “Predictions for Boson-Jet Observables and Fragmentation Function Ratios from a HybridStrong/Weak Coupling Model for Jet Quenching” , JHEP , 053 (2016), arXiv:1508.00815 .[149] J. Casalderrey-Solana, D. Gulhan, G. Milhano, D. Pablos & K. Rajagopal, “AngularStructure of Jet Quenching Within a Hybrid Strong/Weak Coupling Model” , JHEP ,135 (2017), arXiv:1609.05842 .[150] Z. Hulcher, D. Pablos & K. Rajagopal, “Resolution Effects in the Hybrid Strong/WeakCoupling Model” , arXiv:1707.05245 .[151] A. Karch & E. Katz, “Adding flavor to AdS / CFT” , JHEP , 043 (2002), hep-th/0205236 .[152] R. Morad & W. A. Horowitz, “Strong-coupling Jet Energy Loss from AdS/CFT” , JHEP , 017 (2014), arXiv:1409.7545 .[153] S. S. Gubser, “Comparing the drag force on heavy quarks in N = 4 super-Yang-Millstheory and QCD” , Phys. Rev. D76 , 126003 (2007), hep-th/0611272 .[154] M. Cacciari, G. P. Salam & G. Soyez, “The Anti- k t jet clustering algorithm” , JHEP ,063 (2008), arXiv:0802.1189 .[155] A. J. Larkoski, S. Marzani, G. Soyez & J. Thaler, “Soft Drop” , JHEP , 146 (2014), arXiv:1402.2657 .[156] ALICE Collaboration, E. Abbas et al., “Centrality dependence of the pseudorapiditydensity distribution for charged particles in Pb-Pb collisions at √ s NN = 2.76 TeV” ,Phys. Lett. B726 , 610 (2013), arXiv:1304.0347 .[157] ALICE Collaboration, B. Abelev et al., “Centrality dependence of π , K, p production inPb-Pb collisions at √ s NN = 2.76 TeV” , Phys. Rev. C88 , 044910 (2013), arXiv:1303.0737 .[158] M. Habich, J. L. Nagle & P. Romatschke, “Particle spectra and HBT radii for simulatedcentral nuclear collisions of C + C, Al + Al, Cu + Cu, Au + Au, and Pb + Pb from √ s = 62 . - GeV” , Eur. Phys. J.
C75 , 15 (2015), arXiv:1409.0040 .[159] K. Rajagopal & A. V. Sadofyev, “Chiral drag force” , JHEP , 018 (2015), arXiv:1505.07379arXiv:1505.07379