Drag and jet quenching of heavy quarks in a strongly coupled {\cal N}=2^{*} plasma

# Drag and jet quenching of heavy quarks in a strongly coupled N=2∗ plasma

Carlos Hoyos
Department of Physics,
University of Washington
Seattle, WA 98915-1560, USA
E-mail:
###### Abstract:

The drag of a heavy quark and the jet quenching parameter are studied in the strongly coupled plasma using the AdS/CFT correspondence. Both increase in units of the spatial string tension as the theory departs from conformal invariance. The description of heavy quark dynamics using a Langevin equation is also considered. It is found that the difference between the velocity dependent factors of the transverse and longitudinal momentum broadening of the quark admits an interpretation in terms of relativistic effects, so the distribution is spherical in the quark rest frame. When conformal invariance is broken there is a broadening of the longitudinal momentum distribution. This effect may be useful in understanding the jet distribution observed in experiments.

## 1 Motivation

Experiments of heavy ion collisions at RHIC show a quantitative deviation from perturbative QCD predictions of heavy quark jet production [1]. There are several indications that a deconfined state of matter that behaves as a strongly coupled plasma is formed at the center of the collisions. For instance, hydrodynamic simulations using a very low viscosity have been quite successful in describing the observed elliptic flow at low transverse momentum [2]. Another example is the disappearance of back-to-back jets, that could be understood as the effect of a highly dissipative medium. Further evidence comes from lattice theory [3], where it was observed that at the energy densities reached at RHIC the equation of state of QCD is still far from the free gas value.

Some of these observations agree with predictions of AdS/CFT duality [4] that provides a holographic description of some strongly coupled gauge theories. For instance, the pressure of the strongly coupled super Yang-Mills theory was predicted to be about 75% of the free gas [5], which is similar to the results found in the lattice theory in the range of energies of RHIC. Another prediction was that the plasma shows a hydrodynamic behavior with a very small viscosity [6]. In the context of heavy quarks, a classical string computation in AdS/CFT gives predictions for the drag force [7, 8, 9], and there are different estimates for the energy loss of a fast particle moving through the medium. The various approaches involve studying string configurations corresponding to massless quarks [10, 11], the jet quenching parameter from a dipole source [12] or the correlation functions of charged currents in the plasma [13]. The behavior of the drag and jet quenching in a non-conformal theory will be the main interest of this paper.

Both the drag force and the jet quenching have been studied in a large number of situations [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], including some non-conformal theories. As opposed to the shear viscosity, these quantities do not show an universal behavior. Although according to the lattice results the equation of state of QCD in the RHIC range is not very far from a conformal theory, it is necessary to quantify how the deviation from conformal invariance will affect the energy loss of the jets. For this, one needs to use a framework that can connect smoothly with the conformal case. There have been several recent attempts in the context of AdS/QCD that show an interesting behavior [38, 39, 40] , but it is certainly desirable to study a case where the deformation is fully understood. For this purpose, the theory [41, 42] is an ideal laboratory. Conformal invariance is explicitly broken by the introduction of a mass deformation in the theory. The ultraviolet fixed point is the theory, so the holographic correspondence is very well understood. The thermodynamic analysis [46, 47] shows that even when the equation of state starts deviating significantly from the conformal case there are no phase transitions. Therefore, one can tune the mass from zero to quite a large value and study the variation of the observables of interest.

Although the numerical results are given for the theory, the analytic results presented here do not depend on its particular properties, and should be general for a large class of models that describe scalar relevant deformations of a strongly coupled conformal theory in four dimensions. One should add that the deformation must have a dual description in terms of a scalar supergravity field, which it is not necessarily true for all of the possible relevant deformations in the field theory.

The paper is organized as follows. In section 2.1 there is a small review of the main features of the holographic dual. In section 2.2 some general properties of the black hole solutions and the thermodynamics of the theory are discussed. The choice of string configurations used to compute the different quantities is explained in section 2.3. Section 3 contains a derivation of the spatial string tension that will be used as reference scale. The next sections contain the computation of the drag coefficient and momentum broadening (section 4), and the jet quenching parameter (section 5). The numerical method used to calculate the values in the case is explained in the appendix A. Results are summarized and discussed in section 6.

## 2 Preliminaries

### 2.1 The supergravity dual of N=2∗

The contents of this section can be found in the references [42, 43, 44], but are included here for completeness. The field content of can be grouped in superfields as a vector multiplet and three chiral superfields with scalar components

 Φi=X2i−1+iX2i,  i=1,2,3. (1)

In the theory, two of the chiral fields and are grouped in a hypermultiplet and a mass is introduced in the Lagrangian. The remaining fields form an vector multiplet. The mass is introduced through the following relevant operators:

 O2=tr(−X21−X22−X23−X24+2X25+2X26),O3=tr(λ1λ1+λ2λ2)+(scalartrilinear)+h.c.. (2)

where are the fermionic components of the multiplets. This choice of operators correspond to the representations and , that map to two different scalar fields in the holographic dual, and . The last can couple to the scalar singlet mass operator,

 O1=tr(X21+X22+X23+X24+X25+X26), (3)

that is not BPS protected and has no associated supergravity field. The mass term comes from the combination

 O1−12O2=32tr(X21+X32+X33+X24). (4)

The dual theory is described by solutions of five-dimensional supergravity with the scalar fields turned on. This is a consistent truncation of type IIB supergravity in ten dimensions, and it is possible to uplift the five-dimensional solutions to ten dimensions, although this is far from trivial. In the zero temperature case the five-dimensional metric can be put in the form

 ds24,1=e2A(r)(−dt2+d→x2)+dr2. (5)

Thanks to supersymmetry, it is possible to obtain the function and the scalar fields , as solutions of a system of first order differential equations.

where

 W=−e−2α/√3−e4α/√32cosh(2χ). (7)

is the superpotential and is the five-dimensional gauge coupling. The value of sets the radius of curvature , related to the ’t Hooft coupling in the dual theory via  .

The zero temperature theory has an interesting structure that in the gauge theory can be formulated in terms of the Coulomb branch of the moduli space. On the gravity side the supergravity solution has a singularity at the origin of the space. This singularity is of the good kind [45] and from an analysis of D-brane and string probes in the geometry, it is possible to relate it to a particular configuration in the Coulomb branch of the theory where all dyonic states become massless on a ring (the enhançon) around the origin of moduli space [43, 44]. At finite temperature the Coulomb branch is lifted and there is no enhançon singularity in the geometry.

At non-zero temperature supersymmetry is broken and it is necessary to use the second order equations of motion [46, 47].

 □α=12∂P∂α,   □χ=12∂P∂χ. (8)

The potential for the scalar fields is

 P=g216[(∂W∂α)2+(∂W∂χ)2]−g23W2. (9)

The metric can be written as

 ds24,1=e2A(r)(−e2B(r)dt2+d→x2)+dr2, (10)

and the functions and can be obtained from the Einstein equations

 14Rμν=Tμν−13gμνTρρ (11)

where

 Tμν=∂μα∂να+∂μχ∂νχ−12gμν[∂ρα∂ρα+∂ρχ∂ρχ+P]. (12)

is the energy-momentum tensor of the scalar fields.

From the equations of motion one can deduce the asymptotic behavior of the scalar fields. As

 χ=ke−r/L(1+…),  ρ≡eα/√3=1−23k2rLe−2rL+…, (13)

where is the mass of the hypermultiplets. This radial dependence corresponds to scalar fields with masses

 m2χL2=−3,  m2αL2=−4. (14)

This matches the relation between the mass of a scalar field and the conformal dimension of the dual operator, , with for the operator and for the operator. At the horizon the scalar fields have a constant value

 χ=χ0+χ1r2+…,  ρ=ρ0+ρ1r2+…. (15)

As was mentioned before, it is possible to uplift the five-dimensional solutions to a full ten-dimensional geometry. In the Einstein frame it looks

 ds210=˜Ω2ds24,1+ds25, (16)

where

 ds25=L2˜Ω2ρ2(c−1dθ2+ρ6cos2θ(σ21cX2+σ22+σ23X1)+sin2θdϕ2X2) (17)

with the warp factor , and defined as

 ˜Ω2=(cX1X2)1/4ρ,  c=cosh(2χ),  ρ=eα/√3. (18)

The scalar functions are

 X1=cos2θ+ρ6csin2θ,X2=ccos2θ+ρ6sin2θ. (19)

The one-forms are the left-invariant forms satisfying .

For the computation of string solutions, one also has to take into account the dilaton

 e−φ=12((cX1X2)1/2+(cX1X2)−1/2). (20)

The axion and the three and five form fields are also present in the geometry, but they will not be important for the analysis presented here. This is the case because the components of the NS flux along the directions of the string probes studied here vanish. In other cases the situation can be different, an example of this would be the non-relativistic backgrounds of [48]. This would affect for instance the considerations of section 2.3.

### 2.2 Black hole properties and thermodynamics

Some general properties will be presented here. For concrete calculations the equations are solved numerically. Details are explained in the appendix A.

First recall the form of the five-dimensional black hole metric

 ds24,1=e2A(r)(−e2B(r)dt2+dx2)+dr2. (21)

For asymptotically spaces, the behavior as is

 A(r)→rL+A0, (22)

and

 B(r)→0. (23)

The constant is set to zero by a rescaling of the spacetime coordinates. It is always possible to choose a coordinate system where locally the metric component , so this form of the metric is completely general for a geometry that has spatial rotational invariance.

The horizon is at , where the blackening function vanishes. For a regular horizon the metric functions have the following expansions

 eB(r)=b0rL(1+b1r2L2+…), (24)
 eA(r)=a0(1+a1r2L2+…). (25)

The smoothness of the Euclidean solution and the area of the horizon determine the temperature and entropy density

 T=a0b02πL,  S=a304G5. (26)

( is the 5d Newton constant). Although in the uplift to the ten dimensional metric (16) the internal space and the conformal factor depend on the scalars, one can show that they do not affect the values of or 111The area of the horizon just gets a factor that is the volume of the undeformed ..

In ( theory), the values are

 a0=uHL,  b0=2, (27)

where is the horizon radius in Schwarzschild coordinates and is related to exponent in the blackening function , that in the space is fixed by the number of dimensions.

In the general case, the entropy can be cast in a more useful form in terms of the entropy. Using

 L34G5=N22π, (28)

the entropy density is

 S=4π2b30N2T3=8b30SN=4. (29)

So the ratio controls the deviation from the conformal case.

One of the Einstein equations can be integrated, giving a relation that will be useful for the computation of the drag coefficient. Notice that for scalar fields that depend only on the radial coordinate, the combination

 giiR00−g00Rii=0,   i=1,2or3, (30)

is independent of the scalar fields. This gives the equation

 B′′+(4A′+B′)B′=0  ⇒  lnB′+4A+B=const. (31)

This formula is true for any Lorentz-invariant deformation involving scalar fields, not only for the theory.

These formulas are quite general, but in order to describe the thermodynamic behavior of the plasma one has to solve the equations of motion and find the right geometries. This was done in [47, 46] and later on, hydrodynamic properties of the theory were also studied in [49, 50, 51, 52]. The free energy was computed numerically up to values . A good approximation is given by the fit

 FN=2∗≃FN=4e−m/(7T) (32)

For a value , the speed of sound was found to be , with . This implies that the equation of state of the plasma is quite close to the conformal case up to values of the mass . If the bulk viscosity over shear viscosity ratio approximately saturates the bound observed in [51]

 ζη≥2(13−v2s), (33)

then, at , is quite small, but starts to be comparable to the shear viscosity. In the numerical evaluation of quantities, the mass will be taken up to larger values , so the theory is in the region where deviations from conformality should be substantial.

### 2.3 Averaged Wilson loops

The analysis of the energy loss of heavy quarks in the plasma involves the computation of some Wilson loop configurations. In it is possible to use a classical string configuration in the holographic dual to evaluate the expectation value of a generalized version of a Wilson loop, involving the gauge and the scalar fields [53]

 ⟨W(C,θ)⟩=⟨ei∫Cdτ(Aμ˙xμ+θIXI√˙x2)⟩. (34)

where parameterizes the contour of the Wilson loop and the are components of a unit vector in , so they parameterize a five-sphere . In the holographic dual the is realized geometrically, and the generalized Wilson loops can be described using classical string configurations that sit at a point of the . In the theory the R-symmetry that acts on the is unbroken, so the values of the Wilson loops are independent of the value of .

In the theory the moduli space is partially lifted by the masses, so the in the holographic dual is deformed. The symmetry is broken to a subgroup. In this case one should be careful with the choice of Wilson loop, since the expectation values now depend on the values of . Other situations where this can happen include the R-charged black holes, as shown in [54]. A possible definition of a Wilson loop that is independent of the ’s is an averaged Wilson loop

 ⟨¯¯¯¯¯¯W(C)⟩=1V(S5)∫S5dΩ5(θ)⟨W(C,θ)⟩. (35)

Where the average is over the parameters that define the Wilson loop in the field theory. This corresponds to an object with zero R-charge, so it is closer to the properties of a Wilson loop involving only gauge fields. A similar kind of average was considered in [55] for Wilson lines in the theory with dependence on the internal directions, as a way to estimate the quark-antiquark potential for QCD.

Clearly in the theory the averaged Wilson loop has the same expectation value as (34). Although it is possible to find an explicit expression for , one can use a saddle point approximation instead. For this, one writes the Wilson loop as

 ⟨W(C,θ)⟩=eiS(C,θ)  or  ⟨W(C,θ)⟩=e−S(C,θ), (36)

depending on whether the Wilson loop is timelike or spacelike. The action computed in holographic models has a large factor, so in the strong coupling limit the integral over the volume will be dominated by minimal action configurations222Strictly speaking, for timelike Wilson loops the condition is that the action is extremized, but in the cases studied here the physically sensible saddle point corresponds to the position on the where the action is minimized.

 ⟨¯¯¯¯¯¯W(C)⟩≃⟨W(C,θmin)⟩. (37)

where is the value that minimizes the action. There is actually a full set of values related by the remaining symmetry, but as for the case, it is enough to take one representative. Notice that this formula would be valid only as long as the string configuration is at a fixed point in the compact space or moves along an isometric direction. That would not be the case if for instance an NS flux induces a force on the string, as was mentioned in section 2.1. The minimal action configurations at a fixed point will be the ones used to compute the string tension, drag and jet quenching.

A generalized Wilson loop (34) has a holographic description in the semiclassical limit as a classical string configuration in ten dimensions. The string is a two-dimensional surface with a boundary at that is the contour of the Wilson loop . The configuration is determined by the equations derived from the Nambu-Goto action in the Einstein frame,

 SNG=12πα′∫d2σL=12πα′∫d2σΩ2√g2,   Ω2≡eφ/2˜Ω2, (38)

where is the pullback of the five-dimensional metric (10), is the dilaton (20) and is the warp factor (18). From (13), the factor as . On the other hand, when the scalar fields tend to a constant (15) and the factor in the string action depends on the position of the string in the internal space.

String configurations with minimal action are localized at , that corresponds to the part of the geometry associated to the moduli space. In this case the factor is minimized for all values of ,

 Ω2=ρ2 ⎷2cosh2(2χ)1+cosh2(2χ). (39)

So there will be a dependence on the value of the scalar fields. In the limit, the factor becomes a constant:

 Ω20=ρ20 ⎷2cosh2(2χ0)1+cosh2(2χ0). (40)

## 3 Spatial string tension

Spatial Wilson loops in the deconfined phase show an area law behavior when the size of the Euclidean time is very small compared to the size of the loop. It is interesting to compute the spatial string tension using holography for two different reasons. In the first place, it is a physical quantity that one may compare directly with results from lattice computations [3, 56] (see [57] for a comparison with different methods). In a phenomenological application to the QCD quark-gluon plasma, one could fix the string tensions of lattice QCD and of the holographic model to be equal and extract the values of other parameters, in the spirit of previous works, as for example in [19, 8]. The second reason, and the one that motivates its calculation here, is that it has the same parametric dependence on the ’t Hooft coupling as the drag force coefficient and the jet quenching, so it is canceled out in the ratio. In principle one could accomplish the same by taking the ratio with the values of the conformal theory, but notice that this does not give useful information about intrinsic properties of the theory. In the next sections the values of the drag coefficient and jet quenching parameter will be expressed in terms of the string tension.

A spatial Wilson loop can be described holographically by a classical string configuration ending on the appropriate contour at the boundary. The contour of the Wilson loop is chosen to be rectangular in the and directions. The length in the direction is taken to be very large, so it is a good approximation to neglect any dependence on the coordinate. The choice of worldsheet coordinates is and , and the profile of the string is determined by a function . The string worldsheet Lagrangian is

 L=Ω2eA√1+e2Ax′2. (41)

From this, the following equation of motion is obtained

 x′=ce−A√Ω4e4A−c2. (42)

Choosing the integration constant implies that the string profile ends at . The length in the direction and the regularized action of the Wilson loop333Obtained by subtracting the straight configuration . are

 Lx2=∫∞r0drce−A√Ω4e4A−c2,
 S2=Ly2πα′(∫∞r0drΩ2eA[Ω2e2A√Ω4e4A−c2−1]−∫r00drΩ2eA). (43)

The length grows as . In that regime, the contributions to the integrals from are small, since the integrands decay exponentially and the integrand has a square root divergence at . One can use the approximation . Defining and ,

 Lx2≃1a0∫∞r0→0dr1√~Ω4e4~A(r)−1,
 S2≃Ω20a0Ly2πα′∫∞r0→0dr1√~Ω4e4~A(r)−1. (44)

Remember that , as defined in (25). At large distances the Wilson loop follows an area law

 S∼Ω20a202πα′LxLy=(2b0)2Ω20π√λT22LxLy≡σsLxLy. (45)

In terms of the theory result, the string tension is

 σsσN=4=(2b0)2Ω20. (46)

In figure 1 this approximate formula in the theory is compared with a direct evaluation using the string solutions for several values of . The agreement is good only for small masses, although it captures the right tendency of the string tension to decrease as the mass increases.

## 4 Drag force and Langevin equation

In the formation of the quark-gluon plasma, soft light quarks and gluons thermalize very fast but heavy quarks remain out of equilibrium for a relatively much longer time if their mass is much larger than the temperature . At weak coupling the quarks lose energy through bremsstrahlung and collision processes, that can be described within the framework of kinetic theory. The momentum change of the heavy quark may be described by a Langevin equation. A quark with spatial momentum moving through the plasma experiences a force [58]

 dpidt=−ηDpi+ζLi(t)+ζTi(t). (47)

This is a phenomenological description where the effect of the medium has been separated in two parts, the drag and a stochastic contribution , . The drag force opposes to the movement of the quark , where is the drag coefficient. The term coming from random collisions with the components of the plasma and broadens the momentum distribution in the longitudinal and transverse directions. Usually the random components are taken as Gaussian white noise, but in general the momentum transfer coefficients and can depend on the momentum

 ⟨ζLi(t)ζLj(t′)⟩=pipjp2κL(p)δ(t−t′),   ⟨ζTi(t)ζTj(t′)⟩=(δij−pipjp2)κT(p)δ(t−t′). (48)

In principle, the drag coefficient can also depend on the momentum . The fluctuation-dissipation theorem relates the zero-momentum values of the drag coefficient and the momentum transfer , giving the Einstein relation

 ηD(0)=κ2MT. (49)

The momentum transfer also determines the diffusion of the quark through the plasma. Consider a quark at rest at and . The mean squared position of the quark at later times will be

 ⟨xi(t)xj(t)⟩=2Dtδij,   D=2T2κ, (50)

where is the diffusion coefficient.

At strong coupling there is no well-defined kinetic description of the thermal medium, so it is not clear that a description in terms of the Langevin equation should work. Remarkably, in the holographic description of a heavy quark in the plasma, a drag force of the form appearing in (47) with a constant drag coefficient was found [7, 8, 9]. Later a computation of the momentum transfer showed that it depends on the momentum and that the Einstein relation (49) holds in the limit of low velocities [60, 59].

### 4.1 Holographic computation of the drag

A heavy quark moving through the plasma is described by a trailing string configuration in the black hole background [7, 8, 9]. The profile moves at constant velocity in the direction and it is extended from the boundary to the horizon in the radial direction and from to in the spatial direction. Identifying the time and with the worldsheet coordinates, the profile is described by a function

 x(r,t)=vt+ξ(r), (51)

and by the position in the internal space in the ten-dimensional geometry. As was argued in section 2.3, a reasonable choice is to pick a point that maps into the moduli space of the theory. Introducing (51) in the Nambu-Goto action, the Lagrangian for is

 L=−√−g=−Ω2eA(e2B−v2+e2A+2Bξ′2)1/2. (52)

The drag force in the direction is given by the momentum flow along the string. Labeling the momentum in the direction, the components of the background metric and the components of the induced metric on the worldsheet, the momentum flow is

 dpxdt=12πα′√−gGxxgrr∂rx(r,t)=12πα′√−gΩ4e4A+2Bξ′=−πξ2πα′. (53)

Where is the canonical momentum

 πξ=−∂L∂ξ′=Ω2e3A+2Bξ′(e2B−v2+e2A+2Bξ′2)1/2. (54)

From this expression one finds the following equation for the profile

 ξ′=πξe−A−B ⎷e2B−v2Ω4e4A+2B−π2ξ. (55)

In order to have a profile that can go all the way to the horizon, it is necessary that numerator and denominator flip sign at the same point. This imposes a condition on the canonical momentum

 π2ξ=Ω4e4A+2B∣∣v2=e2B=v2Ω4e4A∣∣v2=e2B. (56)

Using (56) in (53), one can deduce the value of the drag force. For , ,

 dpxdt=−πξ2πα′=−vΩ2e2A2πα′∣∣∣v2=e2B. (57)

The expression (57) leads to the known result in the theory,

 dpxdt=−π√λ2T2v√1−v2=−π√λ2T2pxM≡−μN=4pxM, (58)

where a factor of the mass is extracted from the drag coefficient for convenience. Notice that is independent of the velocity. In the general case this is not true, and it is interesting to extract the velocity dependence.

In terms of the functions appearing in the metric, the drag coefficient becomes

 μ=−Mpxdpxdt=12πα′(1−e2B)1/2Ω2e2A∣∣v2=e2B. (59)

At large velocities , the value of the drag coefficient depends on the asymptotic behavior of the solution . The scalar fields do not affect to the result, since . The leading terms in the metric are

 e2B=1−~μ21e−4r/L+…,   eA=er/L(1+…). (60)

where the constant corresponds precisely to the value of the drag coefficient up to factors. One can extract the value of using the relation (31). Comparing the (60) and (25) limits, this gives

 ln(2~μ21)=lnb0+4lna0, (61)

so the drag coefficient for ultrarelativistic quarks is

 μ2v→1=b0a402(2πα′)2. (62)

Contrary to previous claims the drag coefficient is sensitive to the infrared physics even at large velocities. Technically, the high-velocity drag appears in the expansion of the metric (60) as a normalizable term so it has a similar status to an expectation value, it should be possible to relate it to transport coefficients of the plasma.

At small velocities , the drag coefficient is determined by the expansions at (25)

 μ2v→0=Ω40a40(2πα′)2. (63)

The ratio depends on and the value of the scalar fields at the horizon

 (μv→0μv→1)2=2b0Ω40. (64)

The drag coefficient can be written in terms of the coefficient:

 μ2v→1=(2πLT)42(2πα′)2b30=(2b0)3μ2N=4, (65)

hence

 μ2v→0=(2b0)4Ω40μ2N=4. (66)

Using (46), the drag coefficient would be

 μv→0=σs,  μv→1=√b021Ω20σs. (67)

For , this implies that , as was pointed out in [65]. However, this relation does not seem to hold in the non-conformal cases, because of the deviation of the string tension from the analytic result. The numerical value of the drag coefficient in string tension units as a function of the velocity for several mass deformations has been plotted in figure 2. Formula (64) is also compared with the numerical results, with very good agreement. The main effect of breaking conformal invariance is that the drag coefficient depends on the velocity, in the theory it is larger at higher velocities. Similar behavior was observed in other non-conformal theories [15].

### 4.2 Holographic computation of momentum broadening

In the field theory side, the stochastic forces , in the Langevin equation (47) correspond to operators built with heavy quark fields, so it is natural to identify them with small fluctuations of the string due to thermal radiation coming from the black hole [59, 60]. Notice that a force has to be applied to the quark to make it move at constant velocity, so the quark is not in thermal equilibrium with the plasma. However, the state described by the trailing string is a steady state with a constant momentum flow.

In the derivation of the momentum transfer, the worldsheet black hole is proposed to be a holographic description of the Wilson line spanned by the heavy quark, so the steady state is described effectively as a thermal state of a one-dimensional theory with temperature . The worldsheet temperature depends on the velocity of the quark and it is in general different from the black hole temperature. At zero velocity the quark is at equilibrium with the plasma, so . Using a path integral approach, the degrees of freedom beyond the worldsheet horizon can be integrated out. The effect is to introduce a random force for the string endpoint at the worldsheet horizon that propagates to the boundary and introduces the stochastic forces of the Langevin description. This is equivalent to consider the Hawking radiation in the worldsheet theory. This introduces the picture of an ensemble of strings with fluctuating velocities. The large time average configuration is the classical string solution and the mean deviation is related to the magnitude of the random forces [61, 62, 63, 64].

Using this approach, the value of the momentum transfer coefficient can be found from a Kubo formula involving the retarded Green’s function of fluctuations of the string profile

 κa=−limω→02TstωImG(a)R(ω),  a=L,T. (68)

Consider small perturbations around the trailing string

 x1=vt+ξ(r)+δx1(t,r)   x2=δx2(t,r),   x3=δx3(t,r). (69)

Expanding to quadratic order, the Lagrangian for the fluctuations is444There are also first order terms but those are total derivatives, so they do not affect to the equations of motion and will be ignored here.

 L(2)=−GαβT∂αδx1∂βδx1−∑i=1,2GαβL∂αδxi∂βδxi, (70)

where

 Gαβa=12fa√−hhαβ, (71)

and

 fT=Ω2e2A, (72)
 fL=Ω4e4A+2B−π2ξΩ2e2A(e2B−v2). (73)

The ratio for large values of becomes the right one for the zero temperature theory, . The velocity dependent factor corresponds to the anisotropy between the transverse and longitudinal fluctuations at short times or large frequencies. However, the small frequency behavior should be related to values of close to the worldsheet horizon , determined by . Expanding close to the horizon and using (56), one finds

 fL(rH)=fT(rH)(1+2A′(rH)B′(rH)+2Ω′(rH)Ω(rH)B′(rH)). (74)

In the theory this gives

 fL(rH)=1(1−v2)3/2,  fT(rH)=1(1−v2)1/2, (75)

so the zero temperature anisotropy persists at the worldsheet horizon.

The metric can be diagonalized555This is done by changing the time variable to , with . After this change of variables, the new components of the metric are , , . In the text the new variables are relabeled to , . with

 α=[(e2B−v2)(Ω4e4A+2B−π2ξ)]1/2Ω2eA+B. (76)

Close to the horizon, the function vanishes

 α≃α0(r−rH)+⋯=
 =2eA(rH)+B(rH)[B′(rH)(2Ω′(rH)Ω(rH)+2A′(rH)+B′(rH))]1/2(r−rH)+⋯. (77)

Requiring that the Euclidean solution is regular at the horizon implies that the temperature of the worldsheet black hole is . The temperature at coincides with the black hole temperature . In the theory, the worldsheet temperature has a simple dependence with the velocity . The deviation from the conformal case has been represented in figure 3.

It is convenient to use a Fourier decomposition of the fluctuations. A Fourier mode obeys the following equation ()

 φ′′a+(log(αfa))′φ′a+ω2α2φa=0. (78)

In order to compute a retarded correlator, the solution must be ingoing close to the horizon

 φa=(r−rH)−iω/α0Fa(r)   Fa(rH)=const.. (79)

One can use a low frequency expansion

 F=F(0)a(r)+ωF(1)a(r)+⋯. (80)

The leading term is just a constant and can be fixed to . The equation for the next term can be simplified by choosing

 W′a+(log(αfa))′Wa+1α0(r−rH)(1r−rH−(log(αfa))′)=0. (81)

One can rewrite this as

 [αfaWa]′−1α0[αfar−rH]′=0. (82)

Then, imposing regularity at the horizon , the solution is

 Wa=−fHaαfa+1α0(r−rH), (83)

where .

In order to compute the Green’s function a cutoff is introduced and the normalization of the solutions is fixed at the cutoff , so

 Ya(ω,r)=e−iωtφa(r)φa(rΛ). (84)

The retarded Green’s function is proportional to the boundary action as the cutoff is taken to infinity. To leading order in , the result is

 G(a)R(ω)=12πα′limrΛ→∞fa√−hhrrY∗a(ω,rΛ)∂rYa(ω,rΛ)=−iωfHa2πα′+O(ω2). (85)

Using (74) and (85) in (68), the value of the transverse momentum transfer is

 κT=2γTstμ. (86)

The longitudinal momentum transfer can be simplified using that and

 ∂~μ∂v≡(∂~μ/∂r)(∂eB/∂r)