Probing Two Holographic Models of Strongly Coupled Anisotropic Plasma

Probing Two Holographic Models of Strongly Coupled Anisotropic Plasma

Abstract:

Quark-gluon plasma during its initial phase after its production in heavy-ion collisions is expected to have substantial pressure anisotropies. In order to model this situation by a strongly coupled super-Yang-Mills plasma with fixed anisotropy by means of AdS/CFT duality, two models have been discussed in the literature. Janik and Witaszczyk have considered a geometry involving a comparatively benign naked singularity, while more recently Mateos and Trancanelli have used a regular geometry involving a nontrivial axion field dual to a parity-odd deformation of the gauge theory by a spatially varying parameter. We study the (rather different) implications of these two models on the heavy-quark potential as well as jet quenching and compare their respective predictions with those of weakly coupled anisotropic plasmas.

Gauge-gravity correspondence, AdS-CFT Correspondence, Holography and quark-gluon plasmas

1 Introduction

In ultrarelativistic heavy-ion collisions, quark-gluon plasma is produced far from equilibrium with strong anisotropy caused by the fact that initially the system expands mainly along the collision axis. This complicates enormously any theoretical analysis and makes it difficult to decide whether the strong collectivity observed is indeed proving that the quark-gluon plasma behaves as a intrinsically strongly coupled, near-perfect fluid and that a description based on a perturbative plasma with collective effects from strong gluon fields can be ruled out [1].

At weak coupling, an anisotropic quark-gluon plasma leads to (non-Abelian) plasma instabilities [2, 3, 4, 5, 6, 7, 8, 9] leading to nonperturbatively large fields and turbulent behavior [10, 11, 12] which could be responsible for the strong collectivity. Using the framework of hard-loop effective theory [13], experimental signatures such as anisotropic photon and dilepton emission [14], momentum broadening of jets [15, 16] as well as heavy-quark potentials [17, 18] and quarkonium dissociation [19] have been studied with fixed anisotropy as an approximation to the actual dynamical situation.

At strong coupling, for which conventional lattice gauge theory is of no help for analyzing strongly nonequilibrium dynamics, holographic gauge/gravity duality [20, 21] offers the prospect of providing qualitative and semi-quantitative insights. In this framework, heavy-ion collisions and the subsequent anisotropic dynamics and eventual thermalization of a (super-)Yang-Mills plasma can be modeled e.g. by collisions of shock waves in anti-de Sitter space and horizon formation [22].

At least for certain observables, it may be meaningful to consider approximations with a temporarily fixed anisotropy in the gravity dual. One such attempt to model the effects of a system with anisotropic pressures was proposed by Janik and Witaszczyk in [23], where the gravity dual involves a comparatively benign naked singularity. In [24] we have studied electromagnetic signatures of this model which are qualitatively similar to weak-coupling results at high frequencies.

More recently, Mateos and Trancanelli [25, 26] have proposed a regular gravity dual of an anisotropic but equilibrium super-Yang-Mills plasma where a stationary anisotropy is introduced through a (parity-violating) deformation of the gauge theory with a parameter that depends linearly on one of the spatial coordinates. In contrast to the singular gravity dual of Ref. [23], this model has a hydrodynamical limit, and in [27] we have shown that it involves the remarkable feature that certain components of the viscosity tensor break the usual holographic bound of Einstein gravity duals. This model has been explored in [28, 29, 30, 31] with regard to the heavy quark potential, the drag force on quarks, and jet quenching. (Other holographic models of anisotropic fluids have been introduced and studied in Refs. [32, 33, 34].)

The aim of the present paper is to compare the effect of anisotropy on the heavy quark potential and jet quenching in the two holographic models by Janik and Witaszczyk (JW) and Mateos and Trancanelli (MT) based on a singular gravity dual and on axion-dilaton gravity, respectively. These results are moreover compared with perturbative results for a weakly coupled anisotropic plasma for both prolate and oblate anisotropies.

The organization of this paper is as follows. After briefly reviewing in Sect. 2 the two holographic models for an anisotropic super-Yang-Mills plasma of Refs. [23, 25, 26], in Sect. 3 we recall the effects of anisotropies on a weakly coupled plasma as described by hard anisotropic loop effective theory and we then consider a zero-coupling version of the MT model and its consequences. In Sect. 4 we consider the heavy-quark potential in the two holographic models as well as in the two weak-coupling models of an anisotropic plasma. In the case of the hard anisotropic loops we reproduce results for the real part of the heavy quark potential obtained previously in Refs. [17, 18]. In addition we note that in certain directions and at large distances the anisotropic heavy quark potential involves oscillatory behavior instead of an exponential tail, which is caused by electric plasma instabilities. Those are absent in the -deformed weakly coupled gauge theory, which however also has a nonmonotonic behavior at large distances in the anisotropy direction. The holographic models first of all show a completely different large-distance behavior since there is a finite distance where the string connecting the heavy quark breaks. Below the string-breaking distance, the heavy quark potential in the JW model agrees with the hard anisotropic loop results in that for oblate anisotropies the binding is stronger along the anisotropy direction than transverse to it and also in that this feature is reversed for prolate anisotropy. However, in the MT model as well as in its zero-coupling version the potential is always deeper in transverse directions, for both prolate and oblate anisotropy. In Sect. 5 we compare the two holographic models with regard to jet quenching and momentum broadening parameters. Again we find that in the JW model the effects of anisotropy are opposite for prolate and oblate anisotropies, while the MT model is more uniform in this respect. However, in the case of anisotropic jet quenching parameters, the weak coupling results of Ref. [15, 16] show the opposite trend than those of the JW model while they happen to agree qualitatively for an oblate anisotropy in the case of the MT model. Sect. 6 contains our conclusions.

2 Two holographic models of strongly coupled anisotropic plasma

A primary measure of the anisotropy of the boundary field theory is the pressure anisotropy

(1)

where () corresponds to an oblate (prolate) plasma. In the following we shall recapitulate the main features of the two gravity duals we want to consider and also discuss the behavior of the stress-energy tensor in either case.

2.1 JW model: singular gravity dual

The metric of the dual geometry given in Fefferman-Graham coordinates

(2)

can be related to the expectation value of the stress-energy tensor. Here is the holographic coordinate with defining the AdS boundary. The correspondence is such that near the metric should be given by

(3)

with

(4)

In [23] Janik and Witaszczyk fixed the (traceless) stress-energy tensor

(5)

and therefore the boundary conditions for the Einstein equations with a negative cosmological constant, without adding any further matter fields. The most general form of the metric respecting the symmetries of the stress-energy tensor is

(6)

Solving for the unknown functions , and one finds

(7)

where and are related to the energy density and the pressures by

(8)
(9)
(10)

is a dimensionful parameter which in the isotropic case () is related to temperature according to . Nonvanishing values of the dimensionless parameter characterize the anisotropy of the system, since and depend differently on . For negative the plasma is prolate, while it is oblate for positive . In the following we shall consider in particular the values , where (), and where (). In a plasma made of free particles, such values correspond to maximal anisotropies, but the above geometry permits also negative values of pressure components for larger (limited only by ).

The gravity dual we just described is pathological in the sense that a naked singularity appears whenever does not vanish. The singularity is however benign in the sense that one can still choose infalling boundary conditions at the singularity, such that the calculation of for example retarded current-current correlators is still possible [24]. It turns out that these current-current correlators show no hydrodynamical behavior and strongly deviate from the isotropic result even for arbitrarily small ’s for sufficiently small frequencies1. However, for larger frequencies the results are well behaved and smoothly approach the isotropic result for decreasing . In [24] we therefore assumed that this background can approximately describe the non-equilibrium physics on short enough time scales. Eventually the plasma will evolve towards equilibrium and therefore stationarity is not a valid assumption anymore. Asking questions about zero frequency limits, which strictly speaking probe an infinite time span are not meaningful in this model. In principle, this is also a problem for the potential between two heavy quarks in the plasma which we shall compute below, but we expect that the stationary approximation can still provide some qualitative insight.

2.2 MT model: axion-dilaton-gravity dual

In [25, 26] Mateos and Trancanelli presented a completely regular and well behaved gravity dual to an anisotropic but static plasma. Their model is based on the spatially anisotropic duals of Lifshitz-like fixed point of [36], but with AdS boundary conditions. This provides an anisotropic version of an super-Yang-Mills plasma where the anisotropy is kept fixed by a parity violating deformation of the gauge theory

(11)

where depends linearly on one spatial dimension. is a constant with dimensions of energy and can be interpreted as a density of branes. The complexified coupling constant of the SYM theory is related to the axion-dilaton of type IIB supergravity by

(12)

For the deformation in (11) the axion field is position dependent with written in terms of the ’t Hooft coupling . Since the axion is magnetically sourced by branes, the solution can be considered as a number of such branes dissolved in the geometry. It is important to note that since the branes do not extend in the holographic direction, they do not reach the AdS boundary and therefore do not introduce new degrees of freedom in the SYM theory. The brane setup is summarized in Table 1.

x x x x
x x x x

Table 1: Brane set up.

The solution for the ten-dimensional bulk geometry that we are eventually interested in is a direct product of a five-dimensional manifold with a negative cosmological constant and with radius given by . Therefore it suffices to consider the five dimensional axion-dilaton-gravity action

(13)

where and . The line element in the string frame is of the form2

(14)

In the following we will stop to write the dependence on the holographic coordinate explicitly. We note that reparametrization invariance is already used to fix the coefficient in front of and and that cannot be set to unity in general. If we would get an isotropic solution. is the blackening factor that must vanish at the position of the horizon . It turns out that all the functions and can be written in terms of the dilaton , which itself has to satisfy a third order nonlinear differential equation in .

We note that the temperature and the entropy density are well defined since the solution under consideration is static. The temperature can be found from the regularity condition on the metric after Euclidean continuation and is given by . The entropy density is given by a quarter of the horizon area over spatial volume.

The thermodynamics of this setup is discussed in detail in [26]. To summarize some of the most important points:

  • Holographic renormalization brings in a reference scale and therefore the stress-energy tensor of the boundary theory shows a conformal anomaly . The energy density and the pressures depend separately on and .

  • When we keep the temperature constant and increase the anisotropy parameter from the isotropic limit the pressure anisotropy first always becomes oblate. The maximal value of depends crucially on the temperature. After this initial oblate phase there always exists a special value for where the pressures in transverse and longitudinal direction coincide (without the bulk geometry becoming isotropic) and if is increased further the plasma becomes increasingly prolate. However regardless of the pressure anisotropy, the bulk geometry of the MT model is uniformly prolate, .

  • For small values of the plasma is unstable against filamentation along the -direction. It is thermodynamically favorable to have regions in (but infinitely extended in - and -direction) that are isotropic and regions with a larger value of . However, the interval of ’s for which these filamentation instabilities are present is smaller than the interval for the oblate plasma. In other words, the prolate phase is always stable, but there also exist oblate and stable phases.

1 0 0.318 0.159 0 1
201/200 1.32 0.318 0.163 0.08 1.01
3/50 107/100 6.43 0.318 0.201 1.69
48/625 7/5 50.51 0.318 0.383
Table 2: Choice of parameters and corresponding thermodynamic quantities for approximately constant temperature in the anisotropic axion-dilaton gravity dual.
1 0 0.318 0.159 0 1
41/40 1.13 0.314 0.159 0.18 0.97
9/250 6/5 4.23 0.289 0.159 1.79
2 27.37 0.231 0.159
Table 3: Choice of parameters and corresponding thermodynamic quantities for approximately constant entropy density in the anisotropic axion-dilaton gravity dual.

In the following sections we will compute the heavy quark potential and and the jet quenching parameter at constant temperature and at constant entropy density. In Table 2 and Table 3 we present our choice of parameters for these two situations and some related thermodynamic quantities (in units with )3. These parameters are chosen so as to be in the same ballpark as those considered in Ref. [25, 26]. With different parameters, also larger positive values of are possible, however the following results do not change qualitatively.

3 Two weak coupling models of a stationary anisotropic plasma

3.1 Hard anisotropic loop effective theory

0.3162 0.1678 0.681 0.640
0 0 1 1 1 1
10 6.442 3.317 4.075 1.491 1.421
100 55.72 10.05 12.74 2.158 1.889
Table 4: Anisotropy parameters in the hard anisotropic loop effective theory

In a weakly coupled (nearly collisionless) ultrarelativistic gauge theory plasma there is a hierarchy of scales, with hard scales defined as typical energies and momenta of plasma constituents, and soft scales , with coupling constant , pertaining to leading-order collective phenomena such as Debye screening and plasmon masses. In thermal equilibrium, the effective theory of soft scales is provided by the “hard thermal loop” effective action [37]. With anisotropic distribution functions for hard particles, the corresponding “hard anisotropic loop” effective theory [13] involves a rich spectrum of stable and unstable modes at momentum scales , which have been worked out completely for axisymmetric deformations of distribution functions of the form [3]

(15)

with anisotropy direction and some normalization factor with for vanishing anisotropy parameter . A prolate momentum distribution is obtained for , whereas parametrizes oblate momentum distributions.

While the pressure anisotropy is directly determined by (see Table 4), a comparison of quantities at different anisotropy is rather ambiguous [38]. This could be done, e.g., by keeping the number density or the energy density fixed, but in both cases it also depends on whether this is done by adjusting the normalization or the parameter . Keeping number densities of hard particles fixed by adjusting , as done in Ref. [18], leads to , whereas constant energy density in hard particles requires with

(16)

Alternatively, one could compare isotropic and anisotropic plasmas by fixing and adjusting . Keeping the number density constant requires

(17)

whereas for constant energy density one has

(18)

with . Following Refs. [19, 38], we shall mainly consider the option of rescaling in Sect. 4.2.

At leading order, an anisotropic distribution function of the form (15) gives rise to a polarization tensor of the form with four independent dimensionless structure functions and the isotropic Debye mass.

For any nonzero , the (chromo-)magnetostatic propagator turns out to involve spacelike poles and unstable modes corresponding to a filamentation (or Weibel) instability. Below we shall be interested also in the (chromo-)electrostatic potential given by the Fourier transform of the electrostatic propagator

(19)

where are elementary but rather unwieldy functions of and (for explicit expressions see [17]). This electrostatic propagator has poles at purely imaginary wave vectors, corresponding to Debye screening, but additionally poles at a range of real wave vectors, which correspond to electric plasma instabilities. The latter arise for wave vectors within (outside) a cone about the direction of anisotropy for the oblate (prolate) case.

3.2 Anisotropic Chern-Simons deformation of weakly coupled gauge theories

As an alternative model of a plasma with fixed anisotropy we consider the zero-coupling limit of a gauge theory with the deformation (11) present in the MT model, i.e. a gauge theory with Lagrangian

(20)

This is similar to axion electrodynamics with constant spacelike axion gradient [39, 40]4 but in the spirit of the MT model we interpret , which has dimension of inverse length, thermodynamically as a density of some conserved charge distributed along the spatial direction.

The Lagrangian (20) implies an anisotropic dispersion law for gauge boson modes with two gauge-invariant branches [40, 42]

(21)

Since , there are no tachyonic modes, in contrast to the case of a timelike axion gradient [41] and also in contrast to the hard anisotropic loop effective theory.

Introducing a finite temperature, the free energy per gauge boson turns out to be given by

(22)

in the limit [42]. In the high-temperature limit we can ignore renormalization ambiguities, which appear in the terms in the present case of zero coupling (a more complete discussion will be given in [42]). With the interpretation of as a density of a conserved charge (as in [26]) which can be increased by compressing the volume such that scales inversely to its longitudinal extent, we obtain the pressure anisotropy

(23)

which indicates a prolate pressure anisotropy at sufficiently large . This is to be contrasted with the holographic infinite-coupling result of the MT model [26]

(24)

which corresponds to an oblate anisotropy when .5

In the following we shall compare the various models in particular with regard to anisotropies in the heavy quark potential. In the weak-coupling model given by (20), this is obtained from the anisotropic electrostatic propagator which has the simple form

(25)

This corresponds to anisotropic Debye screening, without the complication of electric instabilities which are present in the electrostatic propagator of the hard anisotropic loop effective theory. (In fact, there are also no magnetic instabilities in the -deformed magnetostatic propagator, although as will be discussed in [42] the phase diagram in the --plane has regions of metastability and absolute instability against filamentation towards inhomogeneous densities along the direction of anisotropy.)

4 Heavy Quark Potential

4.1 Holographic calculations

We begin by discussing the heavy quark potential obtained from the Wilson-Polyakov loop which is dual to a fundamental string with spacelike separated endpoints at the AdS boundary [43, 44, 45] for the generic form of a metric describing stationary but spatially anisotropic geometries

(26)

The action for the hanging string is

(27)

with being the induced metric on the worldsheet. Here the indices are either or and . Due to the symmetry in the transverse plane we can always choose a coordinate system such that the -coordinate vanishes. Parametrizing the string worldsheet by and and making a stationary ansatz for and , we obtain

(28)

where primes denote derivatives with respect to the holographic coordinate and is a constant coming from the time integration. We need to find the string profile and therefore evaluate the equations of motion for and , which are of the form

(29)
(30)

and being constants of motion. Disentangling the above equations we end up with

(31)
(32)

For a hanging string that connects two spatially separated points at the boundary we expect and to become negative for . Since the numerator is manifestly positive (note that in our conventions is negative for Lorentzian signature) the denominator has to vanish at some point and then becomes negative for increasing values of . This is also in line with the requirement that at the turning point . Evaluating the zero in the common factor of the denominators eventually leads to a equation that can be written as

(33)

This is the defining equation of an ellipse and therefore

(34)
(35)

and are completely determined by and the angle . To obtain the functions and we can make one further choice, namely that both and vanish at the turning point . It is then easy to find the distance between the two string endpoints

(36)

and the energy of the configuration

(37)

To calculate the action above we integrate from to the turning point in (4.1), which covers only half of the string and therefore we have to multiply by two in order to obtain the full result. The last term above is the energy of two straight strings hanging from the boundary to the horizon at and is necessary to regularize the amount of energy of the hanging string. This also means that the connected configuration is energetically favored as long as . It can be checked easily that for an isotropic geometry with we recover the already well known expression for the heavy quark static potential. If in the anisotropic case we restrict to the simpler cases where the string endpoints are either separated exactly along the or direction the above equations simplify and we reproduce the same solutions as given previously in [28]. Our expressions above however are valid for any separation of the string endpoints in the -plane. The generic situation allows us to probe the geometry by letting the string hang down in the bulk and study how it deforms as a function of .

Figure 1: Potential energy of heavy quarks in the JW model for plasmas with different anisotropies but constant energy density. The isotropic case corresponds to (black lines), oblate anisotropy with to (blue) and prolate anisotropy with to (red). Full (dashed) lines refer to a separation of the quarks along (perpendicular to) the direction of the anisotropy.
Figure 2: Difference in the distance between two connected quarks at a given potential with its isotropic value for a plasma with oblate (, , blue) and prolate (, , red) anisotropy at the same energy density. Full (dashed) lines correspond to a separation of the quarks along (perpendicular to) the direction of the anisotropy.

We start by discussing the results for the JW model, the singular anisotropic gravity dual. In Fig. 2 we plot the potential between the two heavy quarks, where we have adjusted the parameter of the model such that the energy density is kept constant for different anisotropies. Full (dashed) lines correspond to quarks separated along (transverse to) the direction of anisotropy.

We note that in the oblate phase quarks separated along a transverse direction have a slightly shallower potential and consequently a smaller dissociation distance. (By dissociation distance we are referring to the maximal distance between two quarks, for which it is still energetically favorable to be connected by a hanging string in the bulk.6) For prolate plasmas the heavy quark potential is instead shallower for longitudinally separated quarks than for transverse separations. Evidently the anisotropy only mildly influences the heavy quark potential even though we are considering extreme anisotropic plasmas with () and (). In Fig. 2 we have made these small effects more conspicuous by plotting the difference in the separation of two quarks at a given potential energy compared to the isotropic case.

Figure 3: String profiles in the JW model for boundary forces acting on the quarks pointing in a fixed direction (as indicated by ) and projected onto the boundary. The left panel shows profiles for strings at the point of string breaking () for (black, dotted), (blue) and (red) with . The right panel shows strings with but different turning points for the hanging string. For () the colors going from red to yellow (blue to green) correspond to , , and with the value at string breaking. Here dashed lines indicate unstable string configurations.

Let us finally study the profile of the hanging string in the singular anisotropic geometry of the JW model in more detail. Due to the deformation of the spacetime as we go away from the boundary, the string projected onto the boundary will not be a straight line. The direction of the force acting on the string endpoint at the boundary can be defined by an angle

(38)

If then the force acts along the -axis. One could now think of the following experiment. We act with forces pointing in a specified direction in the -plane on two heavy quarks that are initially close together. We choose the strength of the forces such that the heavy quarks slowly start to separate more until they dissociate. When we keep the direction of the forces fixed the whole time the quarks will however not follow a straight line along the force due to the deformation of the space in the holographic coordinate. Instead we observe the behavior shown in Fig. 3. We note that for the JW background the strings bend differently depending on the sign of the parameter. In the right panel of Fig. 3 we consider a string endpoint with a force acting in direction and vary the depth of the turning point of the hanging string. Therefore we can probe the geometry up to a certain value of the holographic coordinate. As increases the deformation of the string gets stronger and stronger. For strings hanging almost down to the singularity we notice that the strings get deformed in such a way that they smoothly fit in the remaining space. When we take a look at the line element of the singular gravity dual we note that for the -direction disappears while for the transverse directions vanish and the space degenerates into an infinite line as we go to the singularity. Therefore in the JW model the pressure anisotropy is encoded very directly in the geometry which is probed by the hanging string.

Figure 4: Potential for heavy quarks in the MT model. The left panel compares varying anisotropies at constant temperature: (isotropic; black line), (oblate; blue lines), (prolate; green) and (prolate; red); the right panel at constant entropy density for different anisotropies (oblate; blue), (prolate; green) and (prolate; red). Full (dashed) lines correspond to a separation of the quarks along (perpendicular to) the direction of the anisotropy.

The anisotropic plasma of the MT model dual to axion-dilaton gravity is actually in thermal equilibrium and therefore we can compare the heavy quark static potential at constant temperature and at constant entropy density. The difference can be clearly seen in Fig. 4. At fixed temperature the dissociation length gets smaller for any separation in the -plane as we increase the anisotropy parameter . At constant entropy density the difference of the dissociation length compared to an isotropic plasma depends on whether we separate the quarks along a transverse direction (string breaking occurs at a larger distance) or along the longitudinal direction (string breaking happens at a smaller distance compared to the isotropic result).

However, regardless of the sign of the pressure anisotropy we find that in the MT model the heavy quark potential is always deeper for transverse separation of the quarks. (Note that the blue lines in the figures correspond to oblate configurations, while the green and red lines are for an increasingly prolate plasma, see Tables 2 and 3). This is a striking difference to the situation in the JW model where oblate and prolate anisotropies lead to opposite deformations of the heavy quark potential. The situation in the MT model is instead always similar to that in the JW model for prolate anisotropy. This appears to be a direct consequence of the fact that in the MT model for any whereas in the singular geometry of the JT model is larger (smaller) than unity for prolate (oblate) pressure anisotropy.

In the remaining plots we will only show the results for constant entropy density keeping in mind that for constant temperature the distance at which the string breaks becomes smaller and smaller as we increase .

Figure 5: String profiles in the MT model for boundary forces acting on the quarks pointing in the same direction (as indicated by ) and projected onto the boundary. The left panel shows the profiles for strings at the point of string breaking () for (black, dotted), (blue), (green) and (red) with . The right panel shows the profiles for and but at different turning points for the hanging string: at string breaking (green), (cyan) and (purple). Here dashed lines indicate unstable string configurations.

Finally we also present the results for strings where the forces acting on the endpoints point in certain directions specified by the angle . In Fig. 5 we also note that the situation is qualitatively the same as in the prolate case for the singular gravity dual. Also indicated in the plot are the trajectories the endpoint of the string follows as we increase the anisotropy and keep fixed. Here again we see once more that the geometry does change monotonically with increasing irrespectively of the behavior of the pressure anisotropy in the boundary theory. In the right panel of Fig. 5 we also probe the geometry by varying the location of the turning point of the hanging string.

Figure 6: Static potential for a weakly coupled anisotropic plasma in the hard anisotropic loop formalism as a function of the quark separation (units set by the isotropic Debye mass). The blue and red lines correspond to longitudinal (full) and transverse (dashed) orientation for and , respectively. The isotropic result is shown in black. In the upper two plots, the particle number density for the anisotropic and the isotropic plasma are the same, in the lower two plots the energy density is kept fixed.
Figure 7: Static potential in free anisotropic -QCD for . Full lines correspond to the -direction and dashed lines to the -direction.

4.2 Comparison with weak-coupling calculations

At weak coupling, the real part of the heavy quark potential is given by the Fourier transform of the electrostatic propagator. In an axisymmetric situation integration over the azimuth angle leads to

(39)

where and with our choice of the anisotropy direction along . In the case of the hard anisotropic loops, the propagator given by (19) involves poles at real corresponding to electric plasma instabilities which are integrated over with a principal value prescription, while in the zero-coupling version of the MT model only has poles at imaginary .

In Fig. 6 we have evaluated (39) with the hard anisotropic loop propagator for strongly oblate () and prolate () anisotropy (cf. Table 4), keeping alternatively the hard particle density and the energy density fixed for different anisotropies. The details of the deviation from the isotropic result slightly depend on whether or is kept constant, and in either case we find that for oblate anisotropies the heavy quark potential is slightly deeper along the anisotropy direction than transverse to it, while for prolate anisotropies this situation is reversed7. In order to make these effects more visible, we also plot divided by the modulus of the vacuum (Coulomb) potential, .

Comparing with the results of the JW model, we find a remarkable qualitative agreement in the dependence on the sign of the anisotropy and the direction of the quark separation. Moreover, the absolute deviation from the isotropic result is rather small both at weak coupling and in the JW model.

On the other hand, as we have seen above, the MT model has a qualitatively different dependence on the direction of the quark separation in the case of oblate anisotropies (which are usually considered in the context of heavy ion collisions).

Turning to the zero-coupling version of the MT model introduced in Sect. 3.2, the heavy quark potential is given by the Fourier transform of (25) which is plotted in Fig. 7. As we have discussed in Sect. 3.2, the high-temperature limit of this weak coupling model corresponds to a prolate anisotropy (in contrast to the holographic MT model), whereas for general both prolate and oblate anisotropies are possible, depending on the renormalization scale. Curiously enough, the potential shown in Fig. 7 (which does not depend on UV renormalization) has qualitatively similar dependence on the direction of quark separation as the holographic MT model (and the hard anisotropic loop potential in the prolate case).

We finally also consider the behavior of the quark potentials at large distances. In the two holographic models, there is a finite separation beyond which the string connecting the heavy quarks becomes unstable because strings entering the horizon or the naked singularity are energetically favored, and at a somewhat larger distance even no unstable connecting solution can be found.

To leading order at weak coupling, the isotropic quark potential is simply given by a Yukawa potential with exponential decay at large distance. The anisotropic weak coupling results show curious deviations. In the anisotropic -deformed zero-coupling case Fig. 7 shows a nonmonotonic behavior of the potential along the anisotropy direction such that beyond (where the potential is actually already extremely small) there is even a repulsive behavior.

Even more curious behavior can be found in the hard anisotropic loop potential at large distances. In this case there is nonmonotonic behavior along (transverse to) the anisotropy direction for prolate (oblate) anisotropy, and here the nonmonotonic behavior is moreover oscillatory. This behavior, which has not been noted in the previous studies of the hard anisotropic loop potential [17, 18], is shown in Fig. 8, where the potential is plotted at large distances in the plane (enhanced by dividing by the modulus of the vacuum (Coulomb) potential). This oscillatory behavior, which is reminiscent of Friedel oscillations at finite chemical potential (for a recent discussion see [47]), has its origin in the presence of poles in the electrostatic propagator at real wave vector corresponding to electric plasma instabilities. It is however rather clear that this curious behavior is devoid of physical implications even at weak coupling, because the plasma instabilities imply that a stationary anisotropy is only a justifiable approximation at sufficiently small time scales and correspondingly small length scales.

Figure 8: Ratio of static potential to vacuum potential for and at same energy density as in the isotropic plasma.

5 Jet Quenching

5.1 Holographic calculations

The computation of the jet quenching parameter for an anisotropic plasma with an axion-dilaton-gravity dual, the MT model, has been presented in [28, 30]. Here we will reproduce the result for the most general case with an ultrarelativistic quark moving in an arbitrary direction [30] and compare the results with those of the singular geometry of the JW model.

According to the prescription of [48, 49] we calculate the string worldsheet with endpoints moving in the same direction at the speed of light and separated a small distance along a direction perpendicular to their motion. The jet quenching parameter can then be obtained from

(40)

In the following we consider a quark endpoint moving in the -plane. The direction is parametrized by an angle such that for the quark moves along the -axis. We therefore start by two subsequent coordinate transformations. First we define

(41)
(42)
(43)

and then we introduce light-cone coordinates

(44)

The metric then takes the form

(45)

Writing the new metric coefficients in terms of our original ones we find8

(46)
(47)
(48)
(49)
(50)

We choose the worldsheet coordinates and let , and depend on the holographic coordinate in the following. It is interesting that we must allow for a non-constant embedding of the string in to find a solution in most general case. The Nambu-Goto action of the string is then given by

(51)

The expression under the square root is actually negative which leads to an imaginary action. The reason is that we consider a spacelike string worldsheet. However this is expected because it is exactly what we need to obtain a jet quenching parameter that is real.

Since the Lagrangian does not depend on , or explicitly we can find three constants of motion , and . In the limit where these constants are small9 we obtain

(52)
(53)
(54)

In [30] the coefficients are given explicitly for the metric of the axion-dilaton-gravity dual. Since we are interested in comparing the results of two different gravity duals we express these coefficients in terms of the general form of the metric given in (26).

(55)
(56)
(57)
(58)

This agrees with [30] if we insert the precise form of the metric in the axion-dilaton-gravity case. However it is now also straightforward to consider any background whose metric is of the form (26).

The string endpoints at the boundary are not separated along the direction and integrating (52) gives

(59)

Along the -axis the separation of the endpoints is while in -direction it is . The constants of motion are then

(60)
(61)

If we insert the expressions (52)-(54) into the action (51) and expand to second order in ’s we obtain

(62)

The action is imaginary because we considered a spacelike worldsheet and is the length of the Wilson line in -direction. The first, independent term is divergent, however, the jet quenching parameter is proportional to and therefore just contained in the second finite term. Upon inserting (59)-(61) into the action and considering the defining relation for the jet quenching parameter (40) we eventually obtain

(63)

with

(64)
(65)
Figure 9: Jet quenching parameter normalized to the isotropic result in the JW model. Left panel for an oblate plasma with , right panel for a prolate plasma with , with .
Figure 10: Jet quenching parameter normalized to the isotropic result in the MT model. Left panel for a plasma at the same entropy density for and which corresponds to oblate pressure anisotropy; right panel for prolate pressure anisotropy with .

The average

(66)

is the total jet quenching parameter for a quark moving with angle with respect to the anisotropy direction, while contains the information about momentum broadening in directions transverse to the motion of the quark, with being perpendicular to both the direction of the jet and the anisotropy direction. When , i.e. the quark moving along the direction of anisotropy, is independent of . In the context of heavy-ion collisions, one is of course mostly interested in jets at larger . For one can define transverse and longitudinal jet quenching parameters