# Langevin Diffusion in Holographic Backgrounds with Hyperscaling Violation

Authors: J. Sadeghi; F. Pourasadollah
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

We consider a relativistic heavy quark which moves in the quark-gluon plasmas. By using the holographic methods, we analyze the Langevin diffusion process of this relativistic heavy quark. This heavy quark is described by a trailing string attached to a flavor brane and moving at constant velocity. The fluctuations of this string are related to the thermal correlators and the correlation functions are precisely the kinds of objects that we compute in the gravity dual picture. We obtain the action of the trailing string in hyperscaling violation backgrounds and we then find the equations of motion. These equations lead us to constructing the Langevin correlator which helps us to obtain the Langevin constants. Using the Langevin correlators we derive the spectral densities and simple analytic expressions in the small- and large-frequency limits. We examine our works for planar andR-charged black holes with hyperscaling violation and find new constraints on θ in the presence of velocity v.


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## Body

## 1. Introduction

It is certainly important and interesting to understand the strongly coupled quark-gluon plasma (QGP) [1–4], since heavy-ion collisions experiments have provided a variety of evidences for creation of QGPs at RHIC. Over the recent years, there have been a lot of efforts to study the features of heavy-ion collisions and the QGP. In this context, the AdS/CFT correspondence [5–8] has provided a powerful tool to study strongly coupled field theory. It maps relativistic conformal field theories holographically to gravitational (or stringy) dynamics in a higher dimensional spacetime. This gauge/gravity duality provides the possibility of computing some properties of QGP.QGPs can be thought of as a soup of quarks and gluons. A heavy quark immersed in this fluid can be modeled in string theory (via the AdS/CFT correspondence) by an open string attached to the boundary of a bulk black hole. The end-point of this string receives the heavy quark on a boundary which is stretching in the UV part of bulk geometry. At a classical level, the straight string is a solution to the equation of motion and does not move in the absence of external force. In this case, the string extends from the boundary to the black-hole horizon (atr=rh). On the field theory side, a competition between the drag and the noise is balanced. And also the modes of string are in equilibrium at the Hawking temperature. The effect of thermal noise is not often considered in AdS/CFT. This seems to conflict with the fluctuation-dissipation theorem [9]. Clearly, one can expect that the Hawking radiation, which is emitted from the black brane, persuades the string to have a random motion. The fluctuations caused by the Hawking radiation are integrated within the stretched horizon rs=rh+ϵ. This gives a picture of the stochastic behavior of the string fluctuations as originating from the world-sheet horizon with the required noise at this horizon [10]. The fluctuations of the trailing string (quantum) provide the information about the heavy quark as it moves in the plasma. So, the dynamics of fields on the boundary can be dictated by the effective action at the stretched horizon [11–13].In this new scheme, in analogy with the dynamics of heavy quarks in heat bath giving rise to Brownian motion [14–20], one can consider the stochastic nature out of equilibrium systems. This involves a diffusive process that was first considered by using the Schwinger-Keldysh formalism adapted to AdS/CFT [21]. Another important improvement in this picture is related to relativistic Langevin evolution of the trailing string which is studied by [22–26]. The stochastic motion was formulated as a Langevin process [11–16] associated with the correlators of the fluctuations of the string.On the practical perspective, one can consider a fundamental string whose end-point lies in the UV region of a bulk black-hole background. The end-point of string is forced to move with velocityv. The string stretches in the bulk until the stretched horizon in this way becomes completely horizontal. When the quark is not moving (or moving with v→0) the stretched horizon approaches the black-hole horizon. The classical profile of the trailing string can be obtained by solving the Nambu-Goto equation of motion. By considering small fluctuations around the classical string profile at the quadratic level, second-order radial equations are obtained. These fluctuations are related to the thermal correlators with modified temperature Ts, through the second-order radial equations, and satisfy the fluctuation-dissipation relation associated with this temperature. Since in this case the system is out of equilibrium, the Hawking temperature of the induced string world-sheet metric Ts is in general different from the heat bath temperature T. A relativistic Langevin diffusion equation is associated with the correlators of the fluctuations of the string by the holographic prescription. The Langevin correlators obey the modified Einstein relation with modified temperature Ts and the relation between the diffusion constants is changed with this modified Einstein relation. [27, 28].There is already a huge amount of literature on the subject of holographic construction of a heavy quark immersed in the quark-gluon plasma. The motion of such a quark has been studied holographically in the classical and relativistical way in [16–20, 27, 28]. In various articles the entries have been assigned to the investigation of sting fluctuations in the gravity theories where the corresponding plasmas have different features (e.g., rotation, charge, and …) [18–20, 29–33]. The construction of some holographic setups in the literature is formed so that the boundary theory is not conformally invariant. The holographic techniques have been used in the study of the submerging quark in such plasmas in [27, 28, 33]. The purpose of the present paper is to investigate the relativistic Langevin evolution of a heavy quark in backgrounds with hyperscaling violation [33–40].Hyperscaling is a feature of the free energy based on naive dimension. For the theories with hyperscaling, the entropy behaves asS~Td/z, where T, d, and z are temperature, number of spatial dimensions, and dynamical exponent, respectively. Hyperscaling violation is first mentioned in context of holographic in [41]. In this context, the hyperscaling violation exponent θ is related to the transformation of the proper distance, and its noninvariance implies the violation of hyperscaling of the dual field theory. Then, the relation between the entropy and temperature has been modified as S~T(d-θ)/z. In general, theory with hyperscaling violation d-θ plays the role of an effective space dimensionality for the dual field theory. In theories with hyperscaling violation, the metric backgrounds are the sophisticated generalization of the AdS gravity. These metrics have a special characteristic so that they can be dual to the field theories which are not conformally invariant. The observations [34, 35, 42, 43] indicate that backgrounds whose asymptotic behavior coincides with these metrics may be of interest to condensed matter physics. So, it is natural to further explore gauge/gravity duality for these backgrounds.This paper is structured as follows. In Section2 we present the description of backgrounds with hyperscaling violation and the relevant classical trailing string solution in these backgrounds. In Section 3 we carry out the corresponding linear fluctuations; these fluctuations are utilized for the holographic computation of the Langevin correlators. Also, we obtain the Langevin coefficients and the spectral density associated with the Langevin correlators. In Sections 2 and 3 our computations are devoted to the planar black holes with hyperscaling violation. But, in Section 4, we investigate the above-mentioned computations for R-charged black hole with hyperscaling violation. In Section 5, we summarize our works and make some comments for future research.


## 2. Backgrounds with Hyperscaling Violation

In this section, we implement the backgrounds with hyperscaling violation as the bulk geometries. The string extends into the bulk and is a dual of a heavy external quark moving through the plasma on the boundary of bulk. In order to discuss the Langevin coefficients of a heavy quark with gauge/gravity duality techniques, we have to find the fluctuations of the trailing string. These fluctuations are related to the thermal correlators and the correlation functions are precisely the kinds of objects that we compute in the gravity dual picture. Therefore we obtain the action of the trailing string in the backgrounds with hyperscaling violation and then we find the equations of motion from that action. By solving these equations we are able to find the Langevin correlator which helps us obtain the Langevin constants.


### 2.1. Planar Black Holes with Hyperscaling Violation

As we indicated before, we want to utilize backgrounds with hyperscaling violation as the bulk geometries. These geometries arise generically as the solutions in appropriate Einstein-Maxwell-dilaton theories with the following action [40, 41, 44],
(1)S=-116πG∫‍dd+2x-g×[R-12(∂ϕ)2-f(ϕ)FμνFμν+V(ϕ)],
where ϕ is dilaton field and the symbols g and R are determinant of the metric and the scalar curvature, respectively. The gauge coupling g2=(f(ϕ))-1 and the potential V(ϕ) are both a function of the dilaton. The black-hole solution with hyperscaling violation from action (9) can be written as [34, 44]
(2)dsd+22=(Rr)2(rrF)2θ/d[-r-2(z-1)h(r)dt2+h(r)-1dr2+dxi2],h(r)=1-(rrh)d+z-θ.
We note here that the metric background includes a dynamical critical exponent z and a hyperscaling violation exponent θ; also d is the number of transverse dimensions and i=1,…,d, r=rh, is the location of horizon and r=rF is the boundary. This metric is not scale invariant under the following scaling:
(3)t⟶λzt,xi⟶λxi,r⟶λr
and transforms as
(4)ds⟶λθ/dds,
which is defining property of hyperscaling in holographic language. In order to understand the metric properties of this class of spacetimes, notice that (2) is conformally equivalent to a Lifshitz geometry [45, 46] as can be seen after a Weyl rescaling gμν→g~μν=Ω2gμν, with Ω=r-(θ/d). The scale-invariant limit is θ=0, which reduces to a Lifshitz solution.It is reasonable from the gravity side to demand that the null energy condition (NEC) [34, 40] be satisfied. For metric background (2), this imposes some constraints on θ and z as
(5)(d-θ)(d(z-1)-θ)≥0,(z-1)(d+z-θ)≥0.
These constraints have important consequences. First, in a Lorentz invariant theory, z=1 and then the first inequality implies that θ≤0 or θ≥d. On the other hand, for a scale-invariant theory (θ=0), we recover the known result z≥1. Notice that in theories with hyperscaling violation the NEC can be satisfied for z<1, while this range of dynamical exponents is forbidden if θ=0. In particular, {z<0;θ>d} gives a consistent solution to (5), as well as {0<z<1;θ≥d+z}. The NEC gives θ>d, but this range for θ leads to instabilities in the gravity side. So this choice of θ does not lead to the physically consistent theories.


### 2.2. Trailing String and Drag Force

Before going in detail we should indicate that we review the calculation of the unperturbed trailing solution that was discussed in [27, 28]. We consider an external heavy quark which moves in the quark-gluon plasma medium with a fixed velocity v on the boundary theory. It can be realized as the end-point of an open classical trailing string which is hanged from the boundary and moves at constant velocity v. The dynamics of this string is governed by the Nambu-Goto action,
(6)SNG=-12πα′∫‍dτdσ-det⁡gab,
where gab=Gμν∂aXμ∂bXν denotes the components of the bulk metric in the string frame. We choose the τ=t and σ=r to work in static gauge and take the following ansatz for trailing string:
(7)X1=vt+ξ(r),X2,X3,…,Xd=0.
By using the metric background (2), the induced metric on the world-sheet can be obtained as
(8)gab=(Rr)2(rrF)2θ/d(v2-r-2(z-1)h(r)vξ′(r)vξ′(r)h(r)-1+ξ′2).
So, the corresponding action becomes
(9)SNG=-12πα′∫‍dtdr(Rr)2(rrF)2θ/d×r-2(z-1)-v2h(r)+r-2(z-1)h(r)ξ′2(r).
The conjugate momentum πξ that flowed from the boundary to the bulk and interpreted as the total force experienced by the quark is written by
(10)ξ′2=(v2-r-2(z-1)h(r))C2[C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r)]r-2(z-1)h2(r),
through the relation C=2πα′πξ. The horizon for the induced world-sheet metric is obtained:
(11)rs-2(z-1)h(rs)=v2.
Since the numerator of the square root in (10) vanishes at the stretched horizon, reality requires that the denominator also vanishes. So, we have πξ2=(R/rs)4(rs/rF)4θ/drs-2(z-1)(h(rs)/4π2α′2). For z=1 the stretched horizon is given by
(12)rs=rh(1-v2)1/(d+1-θ).
In the special case θ=d+1 the stretched horizon in (12) tends to the infinity. It is an unacceptable case, since we expect that the stretched horizon is smaller than the horizon rh. Moreover, we eliminate θ>d+1 for the similar reason.The drag force on the quark can be obtained from the momentum conjugate which has the following form:(13)Fdrag=πξ=-vR2rs(2θ/d)-22πα′rF2(θ/d),Fdrag=-R2rh2((θ/d)-1)v(1-v2)2((θ/d)-1)/(d+1-θ)2πα′rF2(θ/d),R2rh2((θ/d)-1)v(1-v2)2((θ/d)-1)/(d+1-θ)2πα′rF2(θ/d)forz=1.
In the ultrarelativistic limit v→1 the drag force for the case z=1 vanishes, but one can check that this event does not happen for the range of z>1. The momentum friction coefficient η is responsible for the gradual loss in the momentum of a quark of mass M. It is related to the drag force via the Fdrag=-ηp, with p=Mγv [47]. So, one can obtain
(14)η=R2rh2((θ/d)-1)(1-v2)2((θ/d)-1)/(d+1-θ)2πα′MγrF2(θ/d),
where γ=1/1-v2 is the relativistic contraction factor. For the special case θ=d, the drag force and momentum friction coefficient reduce to the following expression:
(15)Fdrag=-R2v2πα′rF2,η=R22πα′MγrF2.
In this case, Fdrag and η are independent of the black-hole horizon.If we diagonalize the induced world-sheet metric by transforming the coordinate through the transformationdt→dt-(vξ′(r)/(v2-r-2(z-1)h(r)))dr, the resulting metric components are
(16)htt=(Rr)2(rrF)2θ/d(-r-2(z-1)h(r)+v2),hrr=r-2(z-1)(R/r)6(r/rF)6θ/dr-2(z-1)(R/r)4(r/rF)4θ/dh(r)-C2.
The effective Hawking temperature associated with the above black-hole metric can be found:
(17)Ts2=116π2[rs-2(z-1)h2(rs)(h′(rs)h(rs)+((4θ/d)-2(z+1))rs)×(h′(rs)h(rs)+-2(z-1)rs)].
For z=1, the above relation reduces to
(18)Ts=14π×([rh(1-v2)1/d+1-θ]-2([(4θd+d-3-θ)(1-v2)-(4θd-4)]=h14π×(1-v2)(d+1-θ)(4θd+d-3-θ))×[rh(1-v2)1/(d+1-θ)]-2)1/2.
In order to have correct value for Ts in the case of θ<d+1, we need to have the following condition:
(19)θ<d[1+d+(3-d)v2]d+(4-d)v2.
In the special case d=3, we have
(20)θ<123+v2.
The Hawking temperature related to the black-hole horizon in the presence of hyperscaling parameter and dynamical exponent z=1 is defined as
(21)T=(d+1-θ)4πrh.
From (18) and (21), one can easily find that
(22)Ts2=T2[1-((4θ/d)+d-3-θ)v2(d+1-θ)](1-v2)1-(2/(d+1-θ)).
For the special case θ=d, it is obvious from the above relation that the Hawking temperature and the modified temperature become equal. This equality is also confirmed for the range z>1. In the conformal limit, it means the hyperscaling parameter tends to zero and the background solution reduces to AdS-Schwarzschild. So, the stretched horizon position and temperature are given by
(23)Ts2=T2[1-d-3d+1v2](1-v2)1-(2/(d+1)),rs=d+14πT(1-v2)1/(d+1).
For d=3 we receive the expected relation [28]
(24)Ts=Tγ,rs=14πγT.


### 2.3. Fluctuations of the Trailing String

In order to study the stochastic motion of quark, we proceed to investigate the fluctuations around the classical trailing solution. We choose the static gauge, such that the string embedding becomesXμ(t,r)=(t,r,X1(t,r),X2(t,r),…,Xd(t,r)). So, we take the following ansatz for embedding:
(25)X1(t,r)=vt+ξ(r)+δX∥(t,r),X2(t,r)=δX2(t,r),…,Xd(t,r)=δXd(t,r).
By expanding the Nambu-Goto action in δXi(t,r) around the classical solution up to quadratic terms we have
(26)S2=-12πα′∫‍dtdrHab2×[N(r)∂aδX∥∂bδX∥+∑i=2dGii∂aδXi∂bδXi],
where
(27)N(r)=(r-2(z-1)h(r)(R/r))4(r/rF)4θ/d-C2(R/r)2(r/rF)2θ/d(r-2(z-1)h(r)-v2),Gii=(Rr)4(rrF)4θ/d,
and Hab=-hhab. The equations of motion can be found from the above action
(28)∂a(HabN(r)∂bδX∥)=0,∂a(HabGij∂bδX⊥)=0.
The definitions of δX⊥ and δX∥ are responsible for the longitudinal and the transverse fluctuations, respectively. By taking a harmonic ansatz as δXi(r,t)=eiωtδXi(r,ω), (28) become(29)∂r[(C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r))(v2-r-2(z-1)h(r))r-(z-1)∂rδX⊥]+ω2r-(z-1)(R/r)4(r/rF)4θ/d(C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r))(v2-r-2(z-1)h(r))δX⊥=0,∂r[(C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r))3/2r-(z-1)(v2-r-2(z-1)h(r))1/2∂rδX∥]+ω2r-(z-1)(C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r))1/2(v2-r-2(z-1)h(r))3/2δX∥=0.In what follows, we will construct solutions to (29) for the string fluctuations and obtain the diffusion constants and the spectral density from them. However, by using the method of the membrane paradigm, computation of diffusion constants can be done directly from the quadratic action (26). We will derive these constants through the two different methods in the next section.


## 2.1. Planar Black Holes with Hyperscaling Violation

As we indicated before, we want to utilize backgrounds with hyperscaling violation as the bulk geometries. These geometries arise generically as the solutions in appropriate Einstein-Maxwell-dilaton theories with the following action [40, 41, 44],
(1)S=-116πG∫‍dd+2x-g×[R-12(∂ϕ)2-f(ϕ)FμνFμν+V(ϕ)],
where ϕ is dilaton field and the symbols g and R are determinant of the metric and the scalar curvature, respectively. The gauge coupling g2=(f(ϕ))-1 and the potential V(ϕ) are both a function of the dilaton. The black-hole solution with hyperscaling violation from action (9) can be written as [34, 44]
(2)dsd+22=(Rr)2(rrF)2θ/d[-r-2(z-1)h(r)dt2+h(r)-1dr2+dxi2],h(r)=1-(rrh)d+z-θ.
We note here that the metric background includes a dynamical critical exponent z and a hyperscaling violation exponent θ; also d is the number of transverse dimensions and i=1,…,d, r=rh, is the location of horizon and r=rF is the boundary. This metric is not scale invariant under the following scaling:
(3)t⟶λzt,xi⟶λxi,r⟶λr
and transforms as
(4)ds⟶λθ/dds,
which is defining property of hyperscaling in holographic language. In order to understand the metric properties of this class of spacetimes, notice that (2) is conformally equivalent to a Lifshitz geometry [45, 46] as can be seen after a Weyl rescaling gμν→g~μν=Ω2gμν, with Ω=r-(θ/d). The scale-invariant limit is θ=0, which reduces to a Lifshitz solution.It is reasonable from the gravity side to demand that the null energy condition (NEC) [34, 40] be satisfied. For metric background (2), this imposes some constraints on θ and z as
(5)(d-θ)(d(z-1)-θ)≥0,(z-1)(d+z-θ)≥0.
These constraints have important consequences. First, in a Lorentz invariant theory, z=1 and then the first inequality implies that θ≤0 or θ≥d. On the other hand, for a scale-invariant theory (θ=0), we recover the known result z≥1. Notice that in theories with hyperscaling violation the NEC can be satisfied for z<1, while this range of dynamical exponents is forbidden if θ=0. In particular, {z<0;θ>d} gives a consistent solution to (5), as well as {0<z<1;θ≥d+z}. The NEC gives θ>d, but this range for θ leads to instabilities in the gravity side. So this choice of θ does not lead to the physically consistent theories.


## 2.2. Trailing String and Drag Force

Before going in detail we should indicate that we review the calculation of the unperturbed trailing solution that was discussed in [27, 28]. We consider an external heavy quark which moves in the quark-gluon plasma medium with a fixed velocity v on the boundary theory. It can be realized as the end-point of an open classical trailing string which is hanged from the boundary and moves at constant velocity v. The dynamics of this string is governed by the Nambu-Goto action,
(6)SNG=-12πα′∫‍dτdσ-det⁡gab,
where gab=Gμν∂aXμ∂bXν denotes the components of the bulk metric in the string frame. We choose the τ=t and σ=r to work in static gauge and take the following ansatz for trailing string:
(7)X1=vt+ξ(r),X2,X3,…,Xd=0.
By using the metric background (2), the induced metric on the world-sheet can be obtained as
(8)gab=(Rr)2(rrF)2θ/d(v2-r-2(z-1)h(r)vξ′(r)vξ′(r)h(r)-1+ξ′2).
So, the corresponding action becomes
(9)SNG=-12πα′∫‍dtdr(Rr)2(rrF)2θ/d×r-2(z-1)-v2h(r)+r-2(z-1)h(r)ξ′2(r).
The conjugate momentum πξ that flowed from the boundary to the bulk and interpreted as the total force experienced by the quark is written by
(10)ξ′2=(v2-r-2(z-1)h(r))C2[C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r)]r-2(z-1)h2(r),
through the relation C=2πα′πξ. The horizon for the induced world-sheet metric is obtained:
(11)rs-2(z-1)h(rs)=v2.
Since the numerator of the square root in (10) vanishes at the stretched horizon, reality requires that the denominator also vanishes. So, we have πξ2=(R/rs)4(rs/rF)4θ/drs-2(z-1)(h(rs)/4π2α′2). For z=1 the stretched horizon is given by
(12)rs=rh(1-v2)1/(d+1-θ).
In the special case θ=d+1 the stretched horizon in (12) tends to the infinity. It is an unacceptable case, since we expect that the stretched horizon is smaller than the horizon rh. Moreover, we eliminate θ>d+1 for the similar reason.The drag force on the quark can be obtained from the momentum conjugate which has the following form:(13)Fdrag=πξ=-vR2rs(2θ/d)-22πα′rF2(θ/d),Fdrag=-R2rh2((θ/d)-1)v(1-v2)2((θ/d)-1)/(d+1-θ)2πα′rF2(θ/d),R2rh2((θ/d)-1)v(1-v2)2((θ/d)-1)/(d+1-θ)2πα′rF2(θ/d)forz=1.
In the ultrarelativistic limit v→1 the drag force for the case z=1 vanishes, but one can check that this event does not happen for the range of z>1. The momentum friction coefficient η is responsible for the gradual loss in the momentum of a quark of mass M. It is related to the drag force via the Fdrag=-ηp, with p=Mγv [47]. So, one can obtain
(14)η=R2rh2((θ/d)-1)(1-v2)2((θ/d)-1)/(d+1-θ)2πα′MγrF2(θ/d),
where γ=1/1-v2 is the relativistic contraction factor. For the special case θ=d, the drag force and momentum friction coefficient reduce to the following expression:
(15)Fdrag=-R2v2πα′rF2,η=R22πα′MγrF2.
In this case, Fdrag and η are independent of the black-hole horizon.If we diagonalize the induced world-sheet metric by transforming the coordinate through the transformationdt→dt-(vξ′(r)/(v2-r-2(z-1)h(r)))dr, the resulting metric components are
(16)htt=(Rr)2(rrF)2θ/d(-r-2(z-1)h(r)+v2),hrr=r-2(z-1)(R/r)6(r/rF)6θ/dr-2(z-1)(R/r)4(r/rF)4θ/dh(r)-C2.
The effective Hawking temperature associated with the above black-hole metric can be found:
(17)Ts2=116π2[rs-2(z-1)h2(rs)(h′(rs)h(rs)+((4θ/d)-2(z+1))rs)×(h′(rs)h(rs)+-2(z-1)rs)].
For z=1, the above relation reduces to
(18)Ts=14π×([rh(1-v2)1/d+1-θ]-2([(4θd+d-3-θ)(1-v2)-(4θd-4)]=h14π×(1-v2)(d+1-θ)(4θd+d-3-θ))×[rh(1-v2)1/(d+1-θ)]-2)1/2.
In order to have correct value for Ts in the case of θ<d+1, we need to have the following condition:
(19)θ<d[1+d+(3-d)v2]d+(4-d)v2.
In the special case d=3, we have
(20)θ<123+v2.
The Hawking temperature related to the black-hole horizon in the presence of hyperscaling parameter and dynamical exponent z=1 is defined as
(21)T=(d+1-θ)4πrh.
From (18) and (21), one can easily find that
(22)Ts2=T2[1-((4θ/d)+d-3-θ)v2(d+1-θ)](1-v2)1-(2/(d+1-θ)).
For the special case θ=d, it is obvious from the above relation that the Hawking temperature and the modified temperature become equal. This equality is also confirmed for the range z>1. In the conformal limit, it means the hyperscaling parameter tends to zero and the background solution reduces to AdS-Schwarzschild. So, the stretched horizon position and temperature are given by
(23)Ts2=T2[1-d-3d+1v2](1-v2)1-(2/(d+1)),rs=d+14πT(1-v2)1/(d+1).
For d=3 we receive the expected relation [28]
(24)Ts=Tγ,rs=14πγT.


## 2.3. Fluctuations of the Trailing String

In order to study the stochastic motion of quark, we proceed to investigate the fluctuations around the classical trailing solution. We choose the static gauge, such that the string embedding becomesXμ(t,r)=(t,r,X1(t,r),X2(t,r),…,Xd(t,r)). So, we take the following ansatz for embedding:
(25)X1(t,r)=vt+ξ(r)+δX∥(t,r),X2(t,r)=δX2(t,r),…,Xd(t,r)=δXd(t,r).
By expanding the Nambu-Goto action in δXi(t,r) around the classical solution up to quadratic terms we have
(26)S2=-12πα′∫‍dtdrHab2×[N(r)∂aδX∥∂bδX∥+∑i=2dGii∂aδXi∂bδXi],
where
(27)N(r)=(r-2(z-1)h(r)(R/r))4(r/rF)4θ/d-C2(R/r)2(r/rF)2θ/d(r-2(z-1)h(r)-v2),Gii=(Rr)4(rrF)4θ/d,
and Hab=-hhab. The equations of motion can be found from the above action
(28)∂a(HabN(r)∂bδX∥)=0,∂a(HabGij∂bδX⊥)=0.
The definitions of δX⊥ and δX∥ are responsible for the longitudinal and the transverse fluctuations, respectively. By taking a harmonic ansatz as δXi(r,t)=eiωtδXi(r,ω), (28) become(29)∂r[(C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r))(v2-r-2(z-1)h(r))r-(z-1)∂rδX⊥]+ω2r-(z-1)(R/r)4(r/rF)4θ/d(C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r))(v2-r-2(z-1)h(r))δX⊥=0,∂r[(C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r))3/2r-(z-1)(v2-r-2(z-1)h(r))1/2∂rδX∥]+ω2r-(z-1)(C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r))1/2(v2-r-2(z-1)h(r))3/2δX∥=0.In what follows, we will construct solutions to (29) for the string fluctuations and obtain the diffusion constants and the spectral density from them. However, by using the method of the membrane paradigm, computation of diffusion constants can be done directly from the quadratic action (26). We will derive these constants through the two different methods in the next section.


## 3. Holographic Computation of Langevin Correlators and Diffusion Constants

### 3.1. Momentum Correlators from the Trailing String

The Langevin correlators can be computed holographically from the classical solutions for the fluctuations of the trailing string. Two types of independent retarded correlatorsgR∥ and gR⊥ for the longitudinal and transverse fluctuations [22] are reasonably expected from the structure of action (26) for the fluctuations
(30)g∥ab=N2πα′Hab,g⊥ab=Gii2πα′Hab.
In the holographic version for the retarded correlator of diagonal metric (16) we have
(31)GR(ω)=-[ψ*(r,ω)grr∂rψ(r,ω)]boundary,
where ψ is related to the fluctuations δX∥ and δX⊥. The expression in (31) must be evaluated at the boundary of the trailing string world-sheet. In theories with hyperscaling violation, existence of a dimensionful scale that does not decouple in the infrared requires the proper powers of this scale, which is denoted by rF. Also we note here, by following the effective holographic approach [48] in which the dual theory lives on a finite r slice, the metric background (2) provides a good description of the dual field theory only for a certain range of r, presumably for r≥rF anticipating the applications at the low energy regions. In the case of an infinitely massive quark, the string is attached at the AdS boundary at r=0; however, in our case with hyperscaling violation, the string is connected to the boundary at r=rF, where rF→0. In the case of finite mass quark, the trailing string is attached to a point rb and its mass is given through the following relation:
(32)M=12πα′∫0rb‍(RrFθ/d)2r(2θ/d)-2-z+1=12πα′(RrFθ/d)2rb(2θ/d)-z(2θ/d)-z.
For θ~(z/2)d, the mass of quark tends to the infinity. Moreover, we expect an infinitely massive quark for rb→0. Thus, θ<(z/2)d is the acceptable region for θ to make the expected mass for quark in the limit rb→0 [49].The solutions of fluctuation equations (29) have the same behavior for the transverse and longitudinal components at the world-sheet horizon and at the boundary. At the r→rs limit both equations take following form,
(33)∂r2ψ+1|r-rs|∂rψ+(ω4πTs|r-rs|)2ψ=0,
so, the solution in near world-sheet horizon is given by
(34)ψ(r,ω)~(rs-r)±i(ω/4πTs),
where Ts is the modified temperature which is given by (17) and (18) for special case (z=1). In the above relation the outgoing waves are brought in + sign while the incoming waves are in − sign.Near the boundaryr→rF, for (29), we have to discuss the range of θ and z. Let us consider θ<d+z, where h(r) vanishes in the limit of r→rF→0. In the special case z=1 both equations reduce to
(35)∂r2ψ+(2θ/d)-2r∂rψ+ω2γ2ψ=0;
therefore the solution of above equation is given by
(36)ψ=c1r(3d-2θ)/2dJ-(3d-2θ)/2d(γωr)+c2r(3d-2θ)/2dY-(3d-2θ)/2d(γωr),
where for small ω two independent solutions with normalizable and nonnormalizable modes become
(37)ψ=cs+cvr(3d-2θ)/d.
Near the boundary for the range of z>0, we have
(38)U=C2-(Rr)4(rrF)4θ/dr-2(z-1)h(r)⟶-R4rF2(z+1),V=v2-r-2(z-1)h(r)⟶v2forz<1,⟶-rF-2(z-1)forz>1.
Then, fluctuation equations (29) for z>1 near the boundary for both cases become
(39)∂r2ψ+(2θ/d)-1-zr∂rψ+ω2r2(z-1)ψ=0,
which has the following solutions:
(40)ψ=c1r((z+2)d-2θ)/2dJ-((z+2)d-2θ)/2zd(γωzrz)+c2r((z+2)d-2θ)/2dY-((z+2)d-2θ)/2zd(γωzrz).
Also, in the region ωr≪1 we receive the following equation:
(41)ψ=cs+cvr(2d+zd-2θ)/d.
For z<1 these equations reduce to
(42)∂r2ψ+(2θ/d)-2r∂rψ-γ2v2ψ=0,
and the solution of above equation will be
(43)ψ=c1r(3d-2θ)/2dI-((3d-2θ)/2d)(ωvr)+c2r(3d-2θ)/2dK-((3d-2θ)/2d)(ωvr),
and for the small ω we receive relation (37).The appropriate boundary conditions for the wave-functions in the expression (34) for the retarded correlator are in-falling behavior at the world-sheet horizon with the condition ψ(r)=1 [50]:
(44)ψ(rb,ω)=1rb={0M⟶∞,rqMfinite,(45)ψ(r,ω)~ψh(rs-r)-i(ω/4πTs)r~rs.
By utilizing the wave-functions, we can construct the propagator from relation (31). We consider the properties of real and imaginary parts of retarded Green’s functions separately.Real Part of Retarded Correlator. In the real part of retarded correlator (31) there are some ambiguities related to the UV divergencies in the on-shell action. To avoid these divergencies and receive finite results, one has to investigate the action on a regularized spacetime with boundary at r=rF. Then, after identifying the divergencies in the limit rF→0, the counterterms can be added to prevent infinite results.To evaluate the real part of (31), the wave-functions close to the boundary can be implemented for different regions of z. For z=1, we can expand the solution (36) for γωr≪1 near the boundary r=rF as
(46)ψUV(r)=c1(γω2)(-3d+2θ)/2d×[1Γ((-1/2)+(θ/d))-(γω)2rF24Γ((1/2)+(θ/d))]-c2π(γω2)(3d-2θ)/2drF(3d-2θ)/d×[Γ(-32+θd)+Γ((-5/2)+(θ/d))(γω)2rF24]+⋯.
According to relation (44) the value of c1=cs is fixed at r=rF:
(47)c1=cs=(γω2)(3d-2θ)/2dΓ(-12+θd).
To obtain the real part of (31) we must evaluate relation (30) at the boundary; then we have
(48)Hrr~1γ,Htt~-γ,N~γ2R2rF2.
Eventually, we find the following divergent term from the expression (46) for the transverse and longitudinal components:
(49)ReGR⊥≃γ-2ReGR∥≃γω22πα′(1-(2θ/d))rF×[1+γ2ω2rF22(1-(2θ/d))+O(rF4)].
Notice that in derivation of the above relation we neglect the effect of the second term in (46), which is proportional to c2=cv and starts at O(rF3-(2θ/d)ω(3d-2θ)/2d) (considering the region of θ<d+1). The correlators have UV divergences (in the ω2 term) which arise from the scheme dependence in their calculation.We now address the analysis of on-shell action to obtain the transverse and longitudinal Green’s functions and compare them with the results of relation (49). We study the divergence structure of action (6), expanded to quadratic order in the fluctuations defined in (26), around the classical trailing string solution [28]. So we write
(50)SNG=S0+S1+S2+⋯.
For each term in the above relation we derive the divergency around r=rF separately. The zeroth order term reads simply:
(51)S0=-12πα′∫‍dtdr-g=-12πα′∫‍dt∫rFrs‍drR2γrF2θ/dr(2θ/d)-2.
Around r=rF, the above integral shows a divergency of the order 1/rF:
(52)S0div⁡=R22πα′(1-(2θ/d))γrF∫‍dt.
For the second term in (30) the quadratic order action (26) is implemented. By inserting the solution (37) close to the boundary and using relation (48), we obtain the divergent part of (26) as
(53)S2div⁡=R22πα′(1-(2θ/d))γrF12×∫‍dωω2(γ2|cs⊥(ω)|2+γ4|cs∥(ω)|2)=R22πα′(1-(2θ/d))γrF12∫‍dtγ2(δX˙⊥)2+γ4(δX˙∥)2.
One can easily check that there is no divergency coming from the first-order action. Consequently, from the above action the divergent parts of the transverse and longitudinal Green’s functions are
(54)(ReGR⊥)=R22πα′(1-(2θ/d))rFγω2,(ReGR∥)=R22πα′(1-(2θ/d))rFγ3ω2.
The resorption of both (52) and (53) divergencies can be done by adding a single covariant boundary counterterm,
(55)Scount=ΔM(rF)∫‍dtX˙μX˙μ,
which is responsible for the renormalization of the quark mass. By expanding the above relation to the second order in X→=v→t+δX→, we find
(56)Scount=ΔM(rF)γ{∫‍dt+∫‍12∫‍dtγ2(δX˙⊥)2+γ4(δX˙∥)2}.
It is clear from (52) and (53) that the following choice for ΔM eliminates the total divergencies:
(57)ΔM(rF)=-R22πα′(1-(2θ/d))rF.
If we repeat the above progress for the case of z>1 for the zeroth order we have
(58)S0div⁡=R22πα′(z-(2θ/d))rFz∫‍dt.
The divergency coming from the second-order action is given by
(59)S2div⁡=R22πα′(2-z-(2θ/d))rF2-z12×∫‍dωω2(|cs⊥(ω)|2+|cs∥(ω)|2)=R22πα′(2-z-(2θ/d))rF2-z12∫‍dt(δX˙⊥)2+(δX˙∥)2.
So the divergent parts of the transverse and longitudinal Green’s functions are identical as
(60)(ReGR⊥)=(ReGR∥)=R22πα′(2-z-(2θ/d))rF2-zω2,
where this result can also explicitly be derived from the explicit expansions of the wave-functions (40) close to the boundary. We note that, to get the result of (59), we use the following relation:
(61)Hrr~rF-(z-1),Htt~-rFz-1,N~γ2R2rF2.
By comparing two divergencies in the zeroth and the second order, we find that the zeroth order divergency dominates over the second order since z>1. For the second order, the divergencies appear only in the range of 1<z<2. So we may consider the following term for renormalizing the quark mass and fixing the coefficient of the counterterm action:
(62)ΔM(rF)=-R2γ2πα′(z-(2θ/d))rFz.
Unfortunately, this choice for changes in the quark mass does not completely cancel divergencies coming from the on-shell action. It seems that the other counterterms should be added before cancelling these divergencies. However, we showed that the real parts of retarded correlator are equal to divergencies which come from the unrenormalized on-shell action. The discussion on removing the divergencies remains as a open problem we may investigate in the future.In the region ofz<1 all computations are similar to the case of z=1 except that in this region we must replace γ with 1/-v2. This means that in the renormalization of quark mass we have to add a virtual mass to the quark to receive the finite results and also we encountered an imaginary value in the computation of the real part of retarded correlator. This seems somewhat complicated, but as we know this range of dynamical exponents is forbidden if θ=0 [51, 52]. In particular, as discussed before the range of 0<z<1 with θ≥d+z gives a consistent solution to the null energy condition and to z<0, θ>0. So one can conclude that the range of 0<z<1 is not in agreement with θ<d+z (the range of θ that we assumed first at our work). In continuous paper, we do not consider this region for z.Imaginary Part of Retarded Correlator. In the imaginary part of retarded correlator we do not encounter the divergencies, since it is proportional to the conserved quantity (current) as
(63)Im⁡GR(ω)=-12igrrψR*∂rψR≡-Jr
and compute at the horizon. From definitions (27) and (30), we find, in the near-horizon limit,
(64)g⊥rr≃4πTs(rs-r)Gii(rs),g∥rr≃4πTs(rs-r)N(rs),g⊥rr≃4πTs(rs-r)Gii(rs)g⊥rr≃4πTs(rs-r)Giiir⟶rs,whereN(rs)=16π2R2Ts2rF2θ/dh2(rs)((h′(rs)/h(rs))+(-2(z-1)/rs))2N(rs)×rs(2θ/d)+2z-4.
By substituting the above relation in (63) and utilizing the solution (45) for ψ one can write
(65)Im⁡GR⊥=-Gii(rs)|ψh⊥|22πα′ω,Im⁡GR∥=-N(rs)|ψh∥|22πα′ω.
From [11, 12, 53, 54] we get the imaginary part of the retarded correlator which is given by
(66)G(ω)=cot(ω(2Ts))Im⁡GR(ω),
where GR(ω) is symmetrized correlator. The spectral density associated with Langevin dynamics are defined as
(67)ρR(ω)=-1πIm⁡GR(ω),ρ(ω)=-1πIm⁡G(ω)=cot(ω(2Ts))ρR(ω).
The above equations help us to investigate the spectral density at the large frequency. In order to study the behavior of Langevin correlators at the high frequency we have to use the WKB method.


### 3.2. The WKB Approximation at Large Frequency

In this section we are going to derive the large-frequency limit of the spectral density, so in order to do this process we need to apply the WKB method. So, in order to arrange the equation as Schrödinger-like form (29) equations, we have to rescale the corresponding wave-function. The large ω solution can be obtained by an adaptation of the WKB method [28, 55]. The Schrödinger-like form of this equations is
(68)-ϕ′′+Vs(r)ϕ=0,Vs(r)=-ω2Gii2L2+12(log⁡L)′′+14(log⁡L)′2,
where ϕ=Lψ with
(69)ψ={δX⊥δX∥,L={LLNGii,L=(C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r))(v2-r-2(z-1)h(r))r-(z-1).
In order to solve Schrödinger equation with some approximations, we divide the range rF<r<rH in three regions r≪rH, r≃rH, and rtp≪r<rH.Near Boundary
(
r
≪
r
s
). In this region for z=1 and z>1 we have different limits as
(70)L={R2r(2θ/d)-2rF2θ/dγR2r(2θ/d)-1-zrF2θ/d,N={R2r(2θ/d)-2rF2θ/dz=1,R2r(2θ/d)-2rF2θ/dz>1.
By using these relations the Schrödinger potential is given by
(71)Vs≃{-γ2ω2+((θ/d)-1)((θ/d)-2)r2z=1,-ω2r-2(z-1)+((θ/d)-(1/2)(z+1))((θ/d)-(1/2)(z+3))r2z>1.
We replace these potentials to relation (68); then the solution of Schrödinger equation will be
(72)ϕ=A1rJ(3d-2θ)/2d(rγω)+A2rY(3d-2θ)/2d(rγω),z=1,ϕ=A1rJ((z+2)d-2θ)/2zd(rzωz)+A2rY((z+2)d-2θ)/2zd(rzωz),z>1.Near Horizon (r≃rs). In this region for both z=1 and z>1 case we obtain
(73)L=(4πTs)R2rs(2θ/d)-2rF2θ/d(rs-r).
If we implement the above relation and (64) for N(rs) in the Schrödinger potential one can arrive at
(74)Vs≃-(ω~2+14)1(r-rs)2,r⟶rs,
where ω~=ω/4πTs. The solution of Schrödinger equation (68) after substituting this potential in-falling boundary condition at the horizon is given by
(75)ϕh≃Ch(rs-r)-iω~+(1/2).WKB Region
(
r
t
p
<
r
≪
r
s
). This region is allowed classically and it covers almost all ranges as rF<r<rs. For large ω’s, the first term of (68) dominates and the potential becomes
(76)Vs≃-ω2Gii2R2,rtp<r≪rs.
For a small region close to the boundary, including turning point rtp, the approximation (76) breaks down. The turning point for large ω’s is found by solving the equation, Vs(r)=0,
(77)rtp={((θ/d)-1)((θ/d)-2)γωz=1,[((θ/d)-(1/2)(z+1))((θ/d)-(1/2)(z+3))ω2]1/2zz>1.
The crucial fact is that, for large ω, rtp≪rs, the regions 1 and 3 overlap. On the other hand, also regions 2 and 3 overlap and are close to r≃rs. Therefore, the solution in WKB region can be used to connect the near-boundary and near-horizon. By inserting the expression (76) in (68) two independent solutions to -ϕ′′+Vϕ=0 in the region V≪0 are written as
(78)ϕ1=1pcos⁡∫rp,ϕ2=1psin∫rp,p(r)=-Vs(r).
Explicitly, the general solution has the following form:
(79)ϕwkb=C1LGiicos⁡∫rω2GiiL+C2LGiisin∫rω2GiiL,ϕwb=C1LGiicos⁡∫rω2GiiL+C2LGiisinrtp<r≪rs.
We should note that the solutions in three regions are applied for both transversal and longitudinal equations.In the next step, we consider the cases that three regions overlap. As we mentioned before, regions 2 and 3 overlap close to the horizon. By expanding the solutions (79) for large ω’s near the horizon we have the following equation,
(80)ϕwkb≃(4πTs)1/2(rs-r)1/2×{C1cos⁡[υ-ω~log⁡(rs-r)]+C2sin[υ-ω~log⁡⁡(rs-r)]},[υ-ω~log⁡(rs-r)]+1HC2sin[υ-ω~log⁡⁡(rs-rr)]r⟶rs,
where υ=∫0r1‍(ω2Gii/L). By comparing relations (75) and (80) we find that
(81)C1=-iC2=Ch(4πTs)1/2e-iυ.
Next we consider the near-boundary region r≪rs. We know that, for the large ω’s, the UV region overlaps with the WKB region. For matching the UV solutions (72) for large ω’s, we need the following expansion for Bessel functions:
(82)Jν(x)≃2πx[cos⁡(x-νπ2-π4)+⋯],|x|⟶∞,Yν(x)≃2πx[sin(x-νπ2-π4)+⋯],|x|⟶∞.
So, the large ω’s expansion for solutions (72) becomes
(83)ϕuv=A12πγω[cos⁡(γωr-π2(2d-θd))]+A22πγω[sin(γωr-π2(2d-θd))],2πγω[sin(γiiωr-π2(2d-θd))]z=1,(84)ϕuv=A12zπωrz-1[cos⁡(ωrzz-π2((z+1)d-θzd))]+A22zπωrz-1[sin(ωrzz-π2((z+1)d-θzd))],2zπωrz-1[sin(ωrzz-π2i((z+1)d-θzd))]z>1.
On the other hand for WKB solutions in the r≪rs we have
(85)ϕwkb≃C1γcos⁡(γωr)+C2γsin(γωr),r≪rsforz=1,ϕwkb≃C1rz-1cos⁡(ωrzz)+C2rz-1sin(ωrzz),r≪rsforz>1.
Comparing between (83), (85) for z=1 and (84), (85) for z>1 gives us the following equation:
(86)C1=A1e-i(π/2)((2d-θ)/d)2πω,A2=iA1forz=1,C1=A1e-i(π/2)(((z+1)d-θ)/zd)2zπω,A2=iA1forz>1.
Finally, all coefficients depend on determination of A2. By imposing unit normalization of the function ψ=(1/L)ϕ at r=rb, that is, the point where the string is attached, one can find this coefficient.Infinite Quark Mass. In this case the end-point of string is attached to the boundary rb=rF→0 and we normalize the wave-functions on this location. Then by imposing the ψ(rF→0,ω)=1 we have
(87)A2=-πR(ω/2)(3d-2θ)/2dγ1-(θ/d)rFθ/dΓ[(3d-2θ)/2d],z=1,A2=-πR(ω/2z)((z+2)d-2θ)/2zdrFθ/dΓ[((z+2)d-2θ)/2zd],z>1.
Consequently, by using (86) and (87) with (81) we get the following result:
(88)Ch=π(ωγ/2)1-(θ/d)(4πTs)1/2RrFθ/dΓ[(3d-2θ)/2d]ei(υ-(π/2)((d-θ)/d)),z=1,Ch=π(ω/2z)(d-θ)/zd(4πTs)1/2RrFθ/dΓ[((z+2)d-2θ)/2zd]ei(υ-(π/2)((d-θ)/zd)),z>1.
Eventually from the above expressions we derive the coefficient ψh as
(89)ψh=π(ωγ/2)1-(θ/d)rs(θ/d)-1Γ[(3d-2θ)/2d]ei(υ-(π/2)((d-θ)/d))×{1⊥Gii(rs)N(rs)∥,z=1,(90)ψh=π(ω/2z)(d-θ)/zdrs(θ/d)-1Γ[((z+2)d-2θ)/2zd]ei(υ-(π/2)((d-θ)/zd))×{1⊥Gii(rs)N(rs)∥,z>1.
By inserting these expressions in (65) we get
(91)Im⁡GR⊥=R2ω2α′(ωγ/2)2-(2θ/d)rF2(θ/d)[Γ((3d-2θ)/2d)]2,z=1,Im⁡GR⊥=R2ω2α′(ω/2z)2(d-θ)/zdrF2(θ/d)[Γ(((z+2)d-2θ)/2zd)]2,z>1.
The longitudinal component of retarded correlator can be found easily by relations (64) and (65). As mentioned before the imaginary part of retarded correlator is proportional to the conserved current. By using the above expression for imaginary part of retarded correlator and (67), we determine the spectral densities associated with the Langevin dynamics in the limit ω≫1/rs:
(92)ρ⊥(ω)≃γ-2ρ∥(ω),z=1,ρ⊥(ω)≃ρ∥(ω),z>1.
So this result is interesting only for z=1 for the transversal and the longitudinal components of spectral density. In the region z>1, this situation approximately converts to the equality for components of spectral densities.Finite Quark Mass. Here we are going to study the finite mass quark which is a similar work for an infinitely case. Now, also we use relations (72) but with the normalization condition at the cutoff rb=rq≠0. In this case one can obtain
(93)A2=L(rq)rq×[-iJ(3d-2θ)/2d(rqγω)+Y(3d-2θ)/2d(rqγω)],ijjJ(3d-2θ)/2d(rqγω)+Y(3d-2θ)/2di(rqγω)z=1,A2=L(rq)rq×[-iJ((z+2)d-2θ)/2zd(rqzωz)+Y((z+2)d-2θ)/2zd(rqzωz)],iJ((z+2)d-2θ)/2zd(rqzωz)+Y((z+2)d-2θ)/2zdhh(rqzωz)z>1.
The other coefficients can be derived through the above relation as with the previous way for the infinite mass case.


### 3.3. Langevin Diffusion Constants via the Retarded Correlator

So far, we found the correlators and spectral densities which are required to establish the generalized Langevin equation. Now we want to find the diffusion coefficients from the information of last section for bothz=1 and z>1 case. We consider a long-time limit which makes the generalized Langevin equation. This limit is expressed in the zero-frequency limit of Green’s functions [27, 28, 53, 54]. Therefore, we investigate the zero-frequency limit of Green’s functions which allow us to give the analytic results for the diffusion constants. The diffusion constant is defined in terms of the symmetric correlator Gsym as
(94)κ=lim⁡ω→0Gsym=-2Tslim⁡ω→0GR(ω)ω.
By going back to the definition of correlator (63), it seems that the evaluation of wave-function in the zero-frequency limit is necessary. For this purpose, we write the small-frequency limit for the horizon asymptotic of the ψR in (45):
(95)ψR(r,ω)=ψh(rs-r)-i(ω/4πTs)≃(1--iω4πTslog⁡|rs-r|+⋯).
This solution reduces to the ψR=ψh in the strict limit ω=0. It matches with the boundary solution results ψh=1 for both transverse and longitudinal modes. This condition is applied consistently for finite and infinite massive quarks, since the radius value for boundary does not have effect in (63) for low frequency limit. Therefore, by using the explicit expressions (65) in (94) and ψh=1, we receive the following results:
(96)κ⊥=R2rs(2θ/d)-2πα′rF2θ/dTs,κ∥=16πR2rs(2θ/d)+2z-4Ts3α′rF2θ/dh2(rs)((h′(rs)/h(rs))+(-2(z-1)/rs))2.
From the above expression for the diffusion constants, it is obvious that there is not any dependence on dynamical exponent z for transversal component but for longitudinal component. The ratio between transversal and longitudinal component can be written as
(97)κ∥κ⊥=[4πrszTs2(1-z)+(2+d-z-θ)(v2-rs-2(z-1))]2.
For the special case z=1 and using the definition of Ts in (18), we get
(98)κ∥κ⊥=1+4(d-θ)d(d+1-θ)[v21-v2].
For particular gauge/gravity dualities the inequality κ∥>κ⊥ has been noticed to hold [56, 57]. In the absence of hyperscaling parameter θ one can check that this inequality is maintained, but, in the presence of θ, it seems that we need the θ<d condition. We note that at condition NEC; the range θ>d is allowed. But, in range of θ we have some instabilities in the gravity side. For the range of z>1 the ratio between the transversal and longitudinal components is given by
(99)κ∥κ⊥=1+4(d-θ)v2d[(d+z-θ)rs-2(z-1)-(2+d-z-θ)v2].
It is obvious that the range of θ<d is also necessary for the case z>1. In this case there is also another condition, rs2(z-1)<(d+z-θ)/(v2(d+2-z-θ)), with rs defined in (11). In the special case θ=d the universal inequality κ∥>κ⊥ converts to the equality κ∥=κ⊥ for both z=1 and z>1. For this special θ in the last section, we found that the Hawking temperature and modified temperature are identical.The jet-quenching parameters can be defined in terms of the diffusion constants as [27, 28]
(100)q^⊥=2κ⊥v,q^∥=κ∥v.
Therefore we obtain
(101)q^⊥=2R2rs(2θ/d)-2vπα′rF2θ/dTs,q^∥=κ⊥v[4πrszTs2(1-z)+(2+d-z-θ)(v2-rs-2(z-1))]2.


### 3.4. The Diffusion Constants via the Membrane Paradigm

This method allows us to achieve the diffusion constants directly from action (26); here we do not need to derive the wave-function as a pervious section, which is described in detail by [58]. By using this method for the following metric background,
(102)ds2=a(r)2dt2+b(r)2dr2+ci2(r)dxi2,
we obtain the transversal and longitudinal diffusion constant as
(103)κ⊥=ci2(rs)πα′Ts,κ∥=±16πα′|a2|b2ci2(a2/ci2)′|(a2/ci2)′||r=rsTs3.
By inserting the component of metric background (2), we receive expected result (96) for diffusion constants. In the next section we will study R-charged black hole with hyperscaling violation.


## 3.1. Momentum Correlators from the Trailing String

The Langevin correlators can be computed holographically from the classical solutions for the fluctuations of the trailing string. Two types of independent retarded correlatorsgR∥ and gR⊥ for the longitudinal and transverse fluctuations [22] are reasonably expected from the structure of action (26) for the fluctuations
(30)g∥ab=N2πα′Hab,g⊥ab=Gii2πα′Hab.
In the holographic version for the retarded correlator of diagonal metric (16) we have
(31)GR(ω)=-[ψ*(r,ω)grr∂rψ(r,ω)]boundary,
where ψ is related to the fluctuations δX∥ and δX⊥. The expression in (31) must be evaluated at the boundary of the trailing string world-sheet. In theories with hyperscaling violation, existence of a dimensionful scale that does not decouple in the infrared requires the proper powers of this scale, which is denoted by rF. Also we note here, by following the effective holographic approach [48] in which the dual theory lives on a finite r slice, the metric background (2) provides a good description of the dual field theory only for a certain range of r, presumably for r≥rF anticipating the applications at the low energy regions. In the case of an infinitely massive quark, the string is attached at the AdS boundary at r=0; however, in our case with hyperscaling violation, the string is connected to the boundary at r=rF, where rF→0. In the case of finite mass quark, the trailing string is attached to a point rb and its mass is given through the following relation:
(32)M=12πα′∫0rb‍(RrFθ/d)2r(2θ/d)-2-z+1=12πα′(RrFθ/d)2rb(2θ/d)-z(2θ/d)-z.
For θ~(z/2)d, the mass of quark tends to the infinity. Moreover, we expect an infinitely massive quark for rb→0. Thus, θ<(z/2)d is the acceptable region for θ to make the expected mass for quark in the limit rb→0 [49].The solutions of fluctuation equations (29) have the same behavior for the transverse and longitudinal components at the world-sheet horizon and at the boundary. At the r→rs limit both equations take following form,
(33)∂r2ψ+1|r-rs|∂rψ+(ω4πTs|r-rs|)2ψ=0,
so, the solution in near world-sheet horizon is given by
(34)ψ(r,ω)~(rs-r)±i(ω/4πTs),
where Ts is the modified temperature which is given by (17) and (18) for special case (z=1). In the above relation the outgoing waves are brought in + sign while the incoming waves are in − sign.Near the boundaryr→rF, for (29), we have to discuss the range of θ and z. Let us consider θ<d+z, where h(r) vanishes in the limit of r→rF→0. In the special case z=1 both equations reduce to
(35)∂r2ψ+(2θ/d)-2r∂rψ+ω2γ2ψ=0;
therefore the solution of above equation is given by
(36)ψ=c1r(3d-2θ)/2dJ-(3d-2θ)/2d(γωr)+c2r(3d-2θ)/2dY-(3d-2θ)/2d(γωr),
where for small ω two independent solutions with normalizable and nonnormalizable modes become
(37)ψ=cs+cvr(3d-2θ)/d.
Near the boundary for the range of z>0, we have
(38)U=C2-(Rr)4(rrF)4θ/dr-2(z-1)h(r)⟶-R4rF2(z+1),V=v2-r-2(z-1)h(r)⟶v2forz<1,⟶-rF-2(z-1)forz>1.
Then, fluctuation equations (29) for z>1 near the boundary for both cases become
(39)∂r2ψ+(2θ/d)-1-zr∂rψ+ω2r2(z-1)ψ=0,
which has the following solutions:
(40)ψ=c1r((z+2)d-2θ)/2dJ-((z+2)d-2θ)/2zd(γωzrz)+c2r((z+2)d-2θ)/2dY-((z+2)d-2θ)/2zd(γωzrz).
Also, in the region ωr≪1 we receive the following equation:
(41)ψ=cs+cvr(2d+zd-2θ)/d.
For z<1 these equations reduce to
(42)∂r2ψ+(2θ/d)-2r∂rψ-γ2v2ψ=0,
and the solution of above equation will be
(43)ψ=c1r(3d-2θ)/2dI-((3d-2θ)/2d)(ωvr)+c2r(3d-2θ)/2dK-((3d-2θ)/2d)(ωvr),
and for the small ω we receive relation (37).The appropriate boundary conditions for the wave-functions in the expression (34) for the retarded correlator are in-falling behavior at the world-sheet horizon with the condition ψ(r)=1 [50]:
(44)ψ(rb,ω)=1rb={0M⟶∞,rqMfinite,(45)ψ(r,ω)~ψh(rs-r)-i(ω/4πTs)r~rs.
By utilizing the wave-functions, we can construct the propagator from relation (31). We consider the properties of real and imaginary parts of retarded Green’s functions separately.Real Part of Retarded Correlator. In the real part of retarded correlator (31) there are some ambiguities related to the UV divergencies in the on-shell action. To avoid these divergencies and receive finite results, one has to investigate the action on a regularized spacetime with boundary at r=rF. Then, after identifying the divergencies in the limit rF→0, the counterterms can be added to prevent infinite results.To evaluate the real part of (31), the wave-functions close to the boundary can be implemented for different regions of z. For z=1, we can expand the solution (36) for γωr≪1 near the boundary r=rF as
(46)ψUV(r)=c1(γω2)(-3d+2θ)/2d×[1Γ((-1/2)+(θ/d))-(γω)2rF24Γ((1/2)+(θ/d))]-c2π(γω2)(3d-2θ)/2drF(3d-2θ)/d×[Γ(-32+θd)+Γ((-5/2)+(θ/d))(γω)2rF24]+⋯.
According to relation (44) the value of c1=cs is fixed at r=rF:
(47)c1=cs=(γω2)(3d-2θ)/2dΓ(-12+θd).
To obtain the real part of (31) we must evaluate relation (30) at the boundary; then we have
(48)Hrr~1γ,Htt~-γ,N~γ2R2rF2.
Eventually, we find the following divergent term from the expression (46) for the transverse and longitudinal components:
(49)ReGR⊥≃γ-2ReGR∥≃γω22πα′(1-(2θ/d))rF×[1+γ2ω2rF22(1-(2θ/d))+O(rF4)].
Notice that in derivation of the above relation we neglect the effect of the second term in (46), which is proportional to c2=cv and starts at O(rF3-(2θ/d)ω(3d-2θ)/2d) (considering the region of θ<d+1). The correlators have UV divergences (in the ω2 term) which arise from the scheme dependence in their calculation.We now address the analysis of on-shell action to obtain the transverse and longitudinal Green’s functions and compare them with the results of relation (49). We study the divergence structure of action (6), expanded to quadratic order in the fluctuations defined in (26), around the classical trailing string solution [28]. So we write
(50)SNG=S0+S1+S2+⋯.
For each term in the above relation we derive the divergency around r=rF separately. The zeroth order term reads simply:
(51)S0=-12πα′∫‍dtdr-g=-12πα′∫‍dt∫rFrs‍drR2γrF2θ/dr(2θ/d)-2.
Around r=rF, the above integral shows a divergency of the order 1/rF:
(52)S0div⁡=R22πα′(1-(2θ/d))γrF∫‍dt.
For the second term in (30) the quadratic order action (26) is implemented. By inserting the solution (37) close to the boundary and using relation (48), we obtain the divergent part of (26) as
(53)S2div⁡=R22πα′(1-(2θ/d))γrF12×∫‍dωω2(γ2|cs⊥(ω)|2+γ4|cs∥(ω)|2)=R22πα′(1-(2θ/d))γrF12∫‍dtγ2(δX˙⊥)2+γ4(δX˙∥)2.
One can easily check that there is no divergency coming from the first-order action. Consequently, from the above action the divergent parts of the transverse and longitudinal Green’s functions are
(54)(ReGR⊥)=R22πα′(1-(2θ/d))rFγω2,(ReGR∥)=R22πα′(1-(2θ/d))rFγ3ω2.
The resorption of both (52) and (53) divergencies can be done by adding a single covariant boundary counterterm,
(55)Scount=ΔM(rF)∫‍dtX˙μX˙μ,
which is responsible for the renormalization of the quark mass. By expanding the above relation to the second order in X→=v→t+δX→, we find
(56)Scount=ΔM(rF)γ{∫‍dt+∫‍12∫‍dtγ2(δX˙⊥)2+γ4(δX˙∥)2}.
It is clear from (52) and (53) that the following choice for ΔM eliminates the total divergencies:
(57)ΔM(rF)=-R22πα′(1-(2θ/d))rF.
If we repeat the above progress for the case of z>1 for the zeroth order we have
(58)S0div⁡=R22πα′(z-(2θ/d))rFz∫‍dt.
The divergency coming from the second-order action is given by
(59)S2div⁡=R22πα′(2-z-(2θ/d))rF2-z12×∫‍dωω2(|cs⊥(ω)|2+|cs∥(ω)|2)=R22πα′(2-z-(2θ/d))rF2-z12∫‍dt(δX˙⊥)2+(δX˙∥)2.
So the divergent parts of the transverse and longitudinal Green’s functions are identical as
(60)(ReGR⊥)=(ReGR∥)=R22πα′(2-z-(2θ/d))rF2-zω2,
where this result can also explicitly be derived from the explicit expansions of the wave-functions (40) close to the boundary. We note that, to get the result of (59), we use the following relation:
(61)Hrr~rF-(z-1),Htt~-rFz-1,N~γ2R2rF2.
By comparing two divergencies in the zeroth and the second order, we find that the zeroth order divergency dominates over the second order since z>1. For the second order, the divergencies appear only in the range of 1<z<2. So we may consider the following term for renormalizing the quark mass and fixing the coefficient of the counterterm action:
(62)ΔM(rF)=-R2γ2πα′(z-(2θ/d))rFz.
Unfortunately, this choice for changes in the quark mass does not completely cancel divergencies coming from the on-shell action. It seems that the other counterterms should be added before cancelling these divergencies. However, we showed that the real parts of retarded correlator are equal to divergencies which come from the unrenormalized on-shell action. The discussion on removing the divergencies remains as a open problem we may investigate in the future.In the region ofz<1 all computations are similar to the case of z=1 except that in this region we must replace γ with 1/-v2. This means that in the renormalization of quark mass we have to add a virtual mass to the quark to receive the finite results and also we encountered an imaginary value in the computation of the real part of retarded correlator. This seems somewhat complicated, but as we know this range of dynamical exponents is forbidden if θ=0 [51, 52]. In particular, as discussed before the range of 0<z<1 with θ≥d+z gives a consistent solution to the null energy condition and to z<0, θ>0. So one can conclude that the range of 0<z<1 is not in agreement with θ<d+z (the range of θ that we assumed first at our work). In continuous paper, we do not consider this region for z.Imaginary Part of Retarded Correlator. In the imaginary part of retarded correlator we do not encounter the divergencies, since it is proportional to the conserved quantity (current) as
(63)Im⁡GR(ω)=-12igrrψR*∂rψR≡-Jr
and compute at the horizon. From definitions (27) and (30), we find, in the near-horizon limit,
(64)g⊥rr≃4πTs(rs-r)Gii(rs),g∥rr≃4πTs(rs-r)N(rs),g⊥rr≃4πTs(rs-r)Gii(rs)g⊥rr≃4πTs(rs-r)Giiir⟶rs,whereN(rs)=16π2R2Ts2rF2θ/dh2(rs)((h′(rs)/h(rs))+(-2(z-1)/rs))2N(rs)×rs(2θ/d)+2z-4.
By substituting the above relation in (63) and utilizing the solution (45) for ψ one can write
(65)Im⁡GR⊥=-Gii(rs)|ψh⊥|22πα′ω,Im⁡GR∥=-N(rs)|ψh∥|22πα′ω.
From [11, 12, 53, 54] we get the imaginary part of the retarded correlator which is given by
(66)G(ω)=cot(ω(2Ts))Im⁡GR(ω),
where GR(ω) is symmetrized correlator. The spectral density associated with Langevin dynamics are defined as
(67)ρR(ω)=-1πIm⁡GR(ω),ρ(ω)=-1πIm⁡G(ω)=cot(ω(2Ts))ρR(ω).
The above equations help us to investigate the spectral density at the large frequency. In order to study the behavior of Langevin correlators at the high frequency we have to use the WKB method.


## 3.2. The WKB Approximation at Large Frequency

In this section we are going to derive the large-frequency limit of the spectral density, so in order to do this process we need to apply the WKB method. So, in order to arrange the equation as Schrödinger-like form (29) equations, we have to rescale the corresponding wave-function. The large ω solution can be obtained by an adaptation of the WKB method [28, 55]. The Schrödinger-like form of this equations is
(68)-ϕ′′+Vs(r)ϕ=0,Vs(r)=-ω2Gii2L2+12(log⁡L)′′+14(log⁡L)′2,
where ϕ=Lψ with
(69)ψ={δX⊥δX∥,L={LLNGii,L=(C2-(R/r)4(r/rF)4θ/dr-2(z-1)h(r))(v2-r-2(z-1)h(r))r-(z-1).
In order to solve Schrödinger equation with some approximations, we divide the range rF<r<rH in three regions r≪rH, r≃rH, and rtp≪r<rH.Near Boundary
(
r
≪
r
s
). In this region for z=1 and z>1 we have different limits as
(70)L={R2r(2θ/d)-2rF2θ/dγR2r(2θ/d)-1-zrF2θ/d,N={R2r(2θ/d)-2rF2θ/dz=1,R2r(2θ/d)-2rF2θ/dz>1.
By using these relations the Schrödinger potential is given by
(71)Vs≃{-γ2ω2+((θ/d)-1)((θ/d)-2)r2z=1,-ω2r-2(z-1)+((θ/d)-(1/2)(z+1))((θ/d)-(1/2)(z+3))r2z>1.
We replace these potentials to relation (68); then the solution of Schrödinger equation will be
(72)ϕ=A1rJ(3d-2θ)/2d(rγω)+A2rY(3d-2θ)/2d(rγω),z=1,ϕ=A1rJ((z+2)d-2θ)/2zd(rzωz)+A2rY((z+2)d-2θ)/2zd(rzωz),z>1.Near Horizon (r≃rs). In this region for both z=1 and z>1 case we obtain
(73)L=(4πTs)R2rs(2θ/d)-2rF2θ/d(rs-r).
If we implement the above relation and (64) for N(rs) in the Schrödinger potential one can arrive at
(74)Vs≃-(ω~2+14)1(r-rs)2,r⟶rs,
where ω~=ω/4πTs. The solution of Schrödinger equation (68) after substituting this potential in-falling boundary condition at the horizon is given by
(75)ϕh≃Ch(rs-r)-iω~+(1/2).WKB Region
(
r
t
p
<
r
≪
r
s
). This region is allowed classically and it covers almost all ranges as rF<r<rs. For large ω’s, the first term of (68) dominates and the potential becomes
(76)Vs≃-ω2Gii2R2,rtp<r≪rs.
For a small region close to the boundary, including turning point rtp, the approximation (76) breaks down. The turning point for large ω’s is found by solving the equation, Vs(r)=0,
(77)rtp={((θ/d)-1)((θ/d)-2)γωz=1,[((θ/d)-(1/2)(z+1))((θ/d)-(1/2)(z+3))ω2]1/2zz>1.
The crucial fact is that, for large ω, rtp≪rs, the regions 1 and 3 overlap. On the other hand, also regions 2 and 3 overlap and are close to r≃rs. Therefore, the solution in WKB region can be used to connect the near-boundary and near-horizon. By inserting the expression (76) in (68) two independent solutions to -ϕ′′+Vϕ=0 in the region V≪0 are written as
(78)ϕ1=1pcos⁡∫rp,ϕ2=1psin∫rp,p(r)=-Vs(r).
Explicitly, the general solution has the following form:
(79)ϕwkb=C1LGiicos⁡∫rω2GiiL+C2LGiisin∫rω2GiiL,ϕwb=C1LGiicos⁡∫rω2GiiL+C2LGiisinrtp<r≪rs.
We should note that the solutions in three regions are applied for both transversal and longitudinal equations.In the next step, we consider the cases that three regions overlap. As we mentioned before, regions 2 and 3 overlap close to the horizon. By expanding the solutions (79) for large ω’s near the horizon we have the following equation,
(80)ϕwkb≃(4πTs)1/2(rs-r)1/2×{C1cos⁡[υ-ω~log⁡(rs-r)]+C2sin[υ-ω~log⁡⁡(rs-r)]},[υ-ω~log⁡(rs-r)]+1HC2sin[υ-ω~log⁡⁡(rs-rr)]r⟶rs,
where υ=∫0r1‍(ω2Gii/L). By comparing relations (75) and (80) we find that
(81)C1=-iC2=Ch(4πTs)1/2e-iυ.
Next we consider the near-boundary region r≪rs. We know that, for the large ω’s, the UV region overlaps with the WKB region. For matching the UV solutions (72) for large ω’s, we need the following expansion for Bessel functions:
(82)Jν(x)≃2πx[cos⁡(x-νπ2-π4)+⋯],|x|⟶∞,Yν(x)≃2πx[sin(x-νπ2-π4)+⋯],|x|⟶∞.
So, the large ω’s expansion for solutions (72) becomes
(83)ϕuv=A12πγω[cos⁡(γωr-π2(2d-θd))]+A22πγω[sin(γωr-π2(2d-θd))],2πγω[sin(γiiωr-π2(2d-θd))]z=1,(84)ϕuv=A12zπωrz-1[cos⁡(ωrzz-π2((z+1)d-θzd))]+A22zπωrz-1[sin(ωrzz-π2((z+1)d-θzd))],2zπωrz-1[sin(ωrzz-π2i((z+1)d-θzd))]z>1.
On the other hand for WKB solutions in the r≪rs we have
(85)ϕwkb≃C1γcos⁡(γωr)+C2γsin(γωr),r≪rsforz=1,ϕwkb≃C1rz-1cos⁡(ωrzz)+C2rz-1sin(ωrzz),r≪rsforz>1.
Comparing between (83), (85) for z=1 and (84), (85) for z>1 gives us the following equation:
(86)C1=A1e-i(π/2)((2d-θ)/d)2πω,A2=iA1forz=1,C1=A1e-i(π/2)(((z+1)d-θ)/zd)2zπω,A2=iA1forz>1.
Finally, all coefficients depend on determination of A2. By imposing unit normalization of the function ψ=(1/L)ϕ at r=rb, that is, the point where the string is attached, one can find this coefficient.Infinite Quark Mass. In this case the end-point of string is attached to the boundary rb=rF→0 and we normalize the wave-functions on this location. Then by imposing the ψ(rF→0,ω)=1 we have
(87)A2=-πR(ω/2)(3d-2θ)/2dγ1-(θ/d)rFθ/dΓ[(3d-2θ)/2d],z=1,A2=-πR(ω/2z)((z+2)d-2θ)/2zdrFθ/dΓ[((z+2)d-2θ)/2zd],z>1.
Consequently, by using (86) and (87) with (81) we get the following result:
(88)Ch=π(ωγ/2)1-(θ/d)(4πTs)1/2RrFθ/dΓ[(3d-2θ)/2d]ei(υ-(π/2)((d-θ)/d)),z=1,Ch=π(ω/2z)(d-θ)/zd(4πTs)1/2RrFθ/dΓ[((z+2)d-2θ)/2zd]ei(υ-(π/2)((d-θ)/zd)),z>1.
Eventually from the above expressions we derive the coefficient ψh as
(89)ψh=π(ωγ/2)1-(θ/d)rs(θ/d)-1Γ[(3d-2θ)/2d]ei(υ-(π/2)((d-θ)/d))×{1⊥Gii(rs)N(rs)∥,z=1,(90)ψh=π(ω/2z)(d-θ)/zdrs(θ/d)-1Γ[((z+2)d-2θ)/2zd]ei(υ-(π/2)((d-θ)/zd))×{1⊥Gii(rs)N(rs)∥,z>1.
By inserting these expressions in (65) we get
(91)Im⁡GR⊥=R2ω2α′(ωγ/2)2-(2θ/d)rF2(θ/d)[Γ((3d-2θ)/2d)]2,z=1,Im⁡GR⊥=R2ω2α′(ω/2z)2(d-θ)/zdrF2(θ/d)[Γ(((z+2)d-2θ)/2zd)]2,z>1.
The longitudinal component of retarded correlator can be found easily by relations (64) and (65). As mentioned before the imaginary part of retarded correlator is proportional to the conserved current. By using the above expression for imaginary part of retarded correlator and (67), we determine the spectral densities associated with the Langevin dynamics in the limit ω≫1/rs:
(92)ρ⊥(ω)≃γ-2ρ∥(ω),z=1,ρ⊥(ω)≃ρ∥(ω),z>1.
So this result is interesting only for z=1 for the transversal and the longitudinal components of spectral density. In the region z>1, this situation approximately converts to the equality for components of spectral densities.Finite Quark Mass. Here we are going to study the finite mass quark which is a similar work for an infinitely case. Now, also we use relations (72) but with the normalization condition at the cutoff rb=rq≠0. In this case one can obtain
(93)A2=L(rq)rq×[-iJ(3d-2θ)/2d(rqγω)+Y(3d-2θ)/2d(rqγω)],ijjJ(3d-2θ)/2d(rqγω)+Y(3d-2θ)/2di(rqγω)z=1,A2=L(rq)rq×[-iJ((z+2)d-2θ)/2zd(rqzωz)+Y((z+2)d-2θ)/2zd(rqzωz)],iJ((z+2)d-2θ)/2zd(rqzωz)+Y((z+2)d-2θ)/2zdhh(rqzωz)z>1.
The other coefficients can be derived through the above relation as with the previous way for the infinite mass case.


## 3.3. Langevin Diffusion Constants via the Retarded Correlator

So far, we found the correlators and spectral densities which are required to establish the generalized Langevin equation. Now we want to find the diffusion coefficients from the information of last section for bothz=1 and z>1 case. We consider a long-time limit which makes the generalized Langevin equation. This limit is expressed in the zero-frequency limit of Green’s functions [27, 28, 53, 54]. Therefore, we investigate the zero-frequency limit of Green’s functions which allow us to give the analytic results for the diffusion constants. The diffusion constant is defined in terms of the symmetric correlator Gsym as
(94)κ=lim⁡ω→0Gsym=-2Tslim⁡ω→0GR(ω)ω.
By going back to the definition of correlator (63), it seems that the evaluation of wave-function in the zero-frequency limit is necessary. For this purpose, we write the small-frequency limit for the horizon asymptotic of the ψR in (45):
(95)ψR(r,ω)=ψh(rs-r)-i(ω/4πTs)≃(1--iω4πTslog⁡|rs-r|+⋯).
This solution reduces to the ψR=ψh in the strict limit ω=0. It matches with the boundary solution results ψh=1 for both transverse and longitudinal modes. This condition is applied consistently for finite and infinite massive quarks, since the radius value for boundary does not have effect in (63) for low frequency limit. Therefore, by using the explicit expressions (65) in (94) and ψh=1, we receive the following results:
(96)κ⊥=R2rs(2θ/d)-2πα′rF2θ/dTs,κ∥=16πR2rs(2θ/d)+2z-4Ts3α′rF2θ/dh2(rs)((h′(rs)/h(rs))+(-2(z-1)/rs))2.
From the above expression for the diffusion constants, it is obvious that there is not any dependence on dynamical exponent z for transversal component but for longitudinal component. The ratio between transversal and longitudinal component can be written as
(97)κ∥κ⊥=[4πrszTs2(1-z)+(2+d-z-θ)(v2-rs-2(z-1))]2.
For the special case z=1 and using the definition of Ts in (18), we get
(98)κ∥κ⊥=1+4(d-θ)d(d+1-θ)[v21-v2].
For particular gauge/gravity dualities the inequality κ∥>κ⊥ has been noticed to hold [56, 57]. In the absence of hyperscaling parameter θ one can check that this inequality is maintained, but, in the presence of θ, it seems that we need the θ<d condition. We note that at condition NEC; the range θ>d is allowed. But, in range of θ we have some instabilities in the gravity side. For the range of z>1 the ratio between the transversal and longitudinal components is given by
(99)κ∥κ⊥=1+4(d-θ)v2d[(d+z-θ)rs-2(z-1)-(2+d-z-θ)v2].
It is obvious that the range of θ<d is also necessary for the case z>1. In this case there is also another condition, rs2(z-1)<(d+z-θ)/(v2(d+2-z-θ)), with rs defined in (11). In the special case θ=d the universal inequality κ∥>κ⊥ converts to the equality κ∥=κ⊥ for both z=1 and z>1. For this special θ in the last section, we found that the Hawking temperature and modified temperature are identical.The jet-quenching parameters can be defined in terms of the diffusion constants as [27, 28]
(100)q^⊥=2κ⊥v,q^∥=κ∥v.
Therefore we obtain
(101)q^⊥=2R2rs(2θ/d)-2vπα′rF2θ/dTs,q^∥=κ⊥v[4πrszTs2(1-z)+(2+d-z-θ)(v2-rs-2(z-1))]2.


## 3.4. The Diffusion Constants via the Membrane Paradigm

This method allows us to achieve the diffusion constants directly from action (26); here we do not need to derive the wave-function as a pervious section, which is described in detail by [58]. By using this method for the following metric background,
(102)ds2=a(r)2dt2+b(r)2dr2+ci2(r)dxi2,
we obtain the transversal and longitudinal diffusion constant as
(103)κ⊥=ci2(rs)πα′Ts,κ∥=±16πα′|a2|b2ci2(a2/ci2)′|(a2/ci2)′||r=rsTs3.
By inserting the component of metric background (2), we receive expected result (96) for diffusion constants. In the next section we will study R-charged black hole with hyperscaling violation.


## 4.R-Charged Black Holes with Hyperscaling Violation

Generally, theR-charged black holes have three independent charges and are static solutions of N=2 supergravity. The bosonic part of the effective gauged supersymmetric N=2 Lagrangian describes the coupling of vector multiples in supergravity [59] which is given by the following expression,
(104)e-1L=(g2V+e-148ϵμνρσλCIJKFμνIFρσJAλKR2-12gxy∂μϕx∂μϕy-14aIJFμνIFμνJ-g2V+e-148ϵμνρσλCIJKFμνIFρσJAλK),
where e=-g is the determinant of vielbein, R is the Ricci scalar, gxy is a metric on the scalar manifold, aIJ is a kinetic gauge coupling of the field strength, g is a constant gauge coupling, and gxy, aIJ, and V are functions of scalar field ϕx. The variation of Lagrangian (104) with respect to gxy, ϕx, and FμνI gives the field equations of motion. In [36] we found the deformed R-charged black-hole metric background with hyperscaling violation. In this section we use the form of R-charged black hole with hyperscaling violation in the flat space k=0. The metric background of this black hole is given by
(105)ds2=(rrF)2(θ/3)R2r2[H1/3(-hdt2+dxi2)+H-2/3h-1dr2],
with
(106)H=(rrF)θ+Q1r2+Q2rFθr4-θ+Q3rF2θr6-2θ,h=1-(rR)2(rr0)2-θH-1,
where Q1=(q1+q2+q3)/R2, Q2=(q1q2+q2q3+q1q3)/R4, and Q3=q1q2q3/R6. Notice that, in the definition of metric background (105), we use the radial coordinate transformation r→(R2/r). We introduce parameter rh as the location of horizon in the r coordinate such that it is the largest root of h=0. Now, by using the above information, we are ready to proceed as pervious section.


### 4.1. Trailing String and Drag Force

In order to study the stochastic motion of the quark in a plasma on the boundary ofR-charged black hole with hyperscaling violation, we repeat the process of Section 2 in this section. The trailing string corresponding to a quark moving on the boundary of R-charged black hole with a constant velocity v—through parameterization (6)—is characterized by the following induced world-sheet metric:
(107)gab=(Rr)2(rrF)2θ/d×H(r)1/3(v2-h(r)vξ′(r)vξ′(r)(h(r)H(r))-1+ξ′2).
Constructing the Nambu-Goto action we find the momentum πξ flowing from the bulk to the horizon which is equal to the drag force:
(108)Fdrag=πξ=vR2H2/3(rs)2πα′rF2.
The stretched horizon rs is defined through the relation h(rs)=v2 and this expression reduces to solve the following equation,
(109)(1-v2)rsθ(1+ax+bx2+cx3)=0,withx=rs2-θ,
where a=Q1rFθ,  b=Q2rF2θ+(rFθ/R2r02-θ(1-v2)), and c=Q3rF3θ. The stretched horizon for θ≠2 in the terms of a, b, and c is given by
(110)x=13[B2c-2(3ac-b2)cB-bc]withB=[1234a3c-a2b2-18abc+4b3+27c2cwithB=+36abc-8b3-108c24a3c-a2b2-18abc+4b3+27c2]1/3.
For the special case θ=2, in order to receive rs≠0, there must be the condition 1+a+b+c=0. If we diagonalize world-sheet induced metric (107), then the modified temperature is obtained as
(111)Ts2=h′(rs)h(rs)H(rs)16π2×[4((θ/3)-1)rs+h′(rs)h(rs)+23H′(rs)H(rs)].
By calculating h′(rs) in detail one can obtain that for θ=3-1=2 this expression becomes zero and hence the temperature. For this value of θ, we obtained in [36] that the temperature and total particle number got zero. Therefore for θ=2 the Hawking temperature and the modified temperature both become equal to zero.


### 4.2. Fluctuations of Trailing String

In order to investigate the fluctuations of trailing string we utilize quadratic Nambu-Goto action (26) with
(112)N(r)=h(r)(R/r)4(r/rF)4θ/dH(r)2/3-C2(R/r)2(r/rF)2θ/dH(r)1/3(h(r)-v2),Gii=(Rr)4(rrF)4θ/dH(r)1/3.
Therefore, by using the above expressions equations of motion (28) become(113)∂r[H(r)(C2-(Rr)4(rrF)4θ/dH(r)2/3h(r))(v2-h(r))∂rδX⊥]+ω2(R/r)4(r/rF)4θ/dH(r)(C2-(R/r)4(r/rF)4θ/dH(r)2/3h(r))(v2-h(r))δX⊥=0,∂r[(C2-(R/r)4(r/rF)4θ/dh(r))H(r)3/2(v2-h(r))1/2H(r)1/6∂rδX∥]+ω2(C2-(R/r)4(r/rF)4θ/dH(r)2/3h(r))1/2(v2-h(r))3/2H(r)1/2δX∥=0.At the world-sheet horizon r→rs both of the above equations reduce to (33) with Ts defined in (111). Therefore, the solution to this equation is similar to what is obtained in relation (34). Near the boundary limit, r→rF, of (113), we receive relation (35) with d=3, since in this limit h(r) and H(r) have the following behavior:
(114)lim⁡r→rF→0H(r)=1,lim⁡r→rF→0h(r)=1.
So the solutions to the equations of motion near the boundary are similar as before, that is, relation (36) with d=3.


### 4.3. Momentum Correlator of Trailing String

The classical solutions that we obtained for a trailing string in black-hole background (105) are implemented in computing the Langevin correlators. As we discussed before, these correlators involve two parts: real and imaginary.


#### 4.3.1. Real Part of Retarded Correlator

The real part of the correlators from fluctuation modes (36) is obtained through relation (31) as
(115)ReGR⊥≃γ-2ReGR∥≃γω22πα′(1-(2θ/3))rF×[1+γ2ω2rF22(1-(2θ/3))+O(rF4)].
As this relation demonstrates, in the limit of rF→0, there are some divergencies of the order of (1/rF). These divergencies can be canceled in a similar way to Section 2 by the identification of ΔM(rF)=-(R2/(2πα′(1-(2θ/3))rF)) in boundary counterterm (55).


#### 4.3.2. Imaginary Part of Retarded Correlator

For computing the imaginary part of retarded correlator, we require relation (63). The expressions g⊥rr and g∥rr in relation (30) reduce to the following for metric background (105):
(116)g⊥rr≃4πTs(rs-r)Gii(rs),g∥rr≃4πTs(rs-r)N(rs),g⊥rr≃4πTs(rs-r)Gii(rs)g⊥rr≃4πTs(rs-r)Giihr⟶rs,withN(rs)=16π2R2Ts2rF2θ/dH2/3(rs)h′2(rs)rs(2θ/d)-2.
By inserting the above expressions in (63) and using solution (44) for ψ we write
(117)Im⁡GR⊥=-Gii(rs)|ψh⊥|22πα′ω,Im⁡GR∥=-N(rs)|ψh∥|22πα′ω.
The spectral densities forming these correlators can be found via relation (67). To obtain the high-frequency behavior of the Langevin correlators, we will implement the WKB method in the next subsection.


### 4.4. The WKB Approximation at Large Frequency

In this section we implement the WKB method to obtain the high-frequency behavior of spectral densities. Everything is the same as Section2, except that the function L has the following definition in Schrödinger potential (68):
(118)L=H(r)(C2-(Rr)4(rrF)4θ/dh(r)H2/3(r))(v2-h(r)).
As before, we divide the range of rF<r<rH in three regions and derive the solutions to the Schrödinger equations.Near Boundary
(
r
≪
r
s
). In this region with
(119)L=R2r(2θ/3)-2rF2θ/3γ,N=R2r(2θ/3)-2rF2θ/3
and so, the Schrödinger potential,
(120)Vs≃-γ2ω2+((θ/3)-1)((θ/3)-2)r2
we receive the following solutions:
(121)ϕ=A1rJ(3/2)-(θ/3)(rγω)+A2rY(3/2)-(θ/3)(rγω).Near Horizon
(
r
≃
r
s
). In this region, one can obtain the following for L function:
(122)L=(4πTs)R2rs(2θ/2)-2rF2θ/3H1/3(rs)(rs-r).
If we implement the above relation and relation (117) for N(rs) in the Schrödinger potential; then we get
(123)Vs≃-(ω~2+14)1(r-rs)2r⟶rs,
where ω~=ω/4πTs. The solution to Schrödinger equation (68) after substituting this potential is identical to what is found in solution (75).WKB Region
(
r
t
p
<
r
≪
r
s
). In this region, for large ω’s, the first term in (68) dominates and the Schrödinger potential becomes
(124)Vs≃-ω2Gii2R2,rtp<r≪rs.
The turning point for large ω is found as
(125)rtp=((θ/3)-1)((θ/3)-2)γω.
We should note that the solutions in three regions are applied for both transversal and longitudinal equations.Now, with the above information, we are able to find the coefficients by considering the limits that three regions overlap. Eventually, by repeating the process followed in Section2, we obtain the following expression for ψh for the infinite massive quark case:
(126)ψh=π(ωγ/2)1-(θ/3)rs(θ/3)-1Γ[(3/2)-(θ/3)]ei(υ-(π/2)((3-θ)/3))×{1⊥Gii(rs)N(rs)∥
and then by inserting the above expressions in (63) we attain
(127)Im⁡GR⊥=R2ω2α′(ωγ/2)2-(2θ/3)rF2(θ/3)[Γ((3/2)-(θ/3))]2.
The longitudinal component can be found in the same way and by using relation (65). The finite mass case is obtained through the way described in Section 2.


### 4.5. Langevin Diffusion Constants

The diffusion coefficients can be found either from the direct evaluation of the correlators or from using the membrane paradigm. Both of these methods give rise to the following identical results for diffusion constants:(128)κ⊥=R2H1/3(rs)rs(2θ/3)-2πα′rF2θ/3Ts,κ∥=16πR2rs(2θ/3)-2Ts3α′rF2θ/3H2/3(rs)h′2(rs).
There is a problem in computing the longitudinal component for the special case θ=2. In this case both H(rs) and h′(rs) tend to zero, so the longitudinal diffusion constant tends to the infinity. It seems that this value for θ is unacceptable. By studying [36], we perceive that the null energy condition imposes some condition on θ as
(129)θ≤0.365,θ≥1.315.
In a similar way to the previous work and because we see some instabilities in the range of θ≥1.315, we choose the range of θ≤1.315. This case is in analogy with the case θ<d in the previous section.The ratio between longitudinal and transversal components of diffusion constants is(130)κ∥κ⊥=1+2h(rs)h′(rs)[(2θ/3)-2rs+H′(rs)(rs)].
We expect the universal inequality κ∥>κ⊥ to be held. Therefore, in order to keep the inequality for the following relation,
(131)κ∥κ⊥=1+2v2(1-v2)×[(rs/rF)θ+(2Q1rs2/3)+(Q2rFθrs4-θ/3)2(rs/rF)θ+Q1rs2-Q3rF2θrs6-2θ],
with Qi>0, the condition 2+ax>cx3 must be satisfied. For Qi<0, both of the numerator and denominator on the right-hand side of relation (131) can be negative or positive, where each of these situations puts some conditions on Qi and rs. Since, due to relation (110), the rs is related to Qi, rF, and v2, to hold the inequality, there should be some conditions on Qi, rF, and v2. The jet-quenching parameters can be obtained easily from relation (100) as in the previous section.


## 4.1. Trailing String and Drag Force

In order to study the stochastic motion of the quark in a plasma on the boundary ofR-charged black hole with hyperscaling violation, we repeat the process of Section 2 in this section. The trailing string corresponding to a quark moving on the boundary of R-charged black hole with a constant velocity v—through parameterization (6)—is characterized by the following induced world-sheet metric:
(107)gab=(Rr)2(rrF)2θ/d×H(r)1/3(v2-h(r)vξ′(r)vξ′(r)(h(r)H(r))-1+ξ′2).
Constructing the Nambu-Goto action we find the momentum πξ flowing from the bulk to the horizon which is equal to the drag force:
(108)Fdrag=πξ=vR2H2/3(rs)2πα′rF2.
The stretched horizon rs is defined through the relation h(rs)=v2 and this expression reduces to solve the following equation,
(109)(1-v2)rsθ(1+ax+bx2+cx3)=0,withx=rs2-θ,
where a=Q1rFθ,  b=Q2rF2θ+(rFθ/R2r02-θ(1-v2)), and c=Q3rF3θ. The stretched horizon for θ≠2 in the terms of a, b, and c is given by
(110)x=13[B2c-2(3ac-b2)cB-bc]withB=[1234a3c-a2b2-18abc+4b3+27c2cwithB=+36abc-8b3-108c24a3c-a2b2-18abc+4b3+27c2]1/3.
For the special case θ=2, in order to receive rs≠0, there must be the condition 1+a+b+c=0. If we diagonalize world-sheet induced metric (107), then the modified temperature is obtained as
(111)Ts2=h′(rs)h(rs)H(rs)16π2×[4((θ/3)-1)rs+h′(rs)h(rs)+23H′(rs)H(rs)].
By calculating h′(rs) in detail one can obtain that for θ=3-1=2 this expression becomes zero and hence the temperature. For this value of θ, we obtained in [36] that the temperature and total particle number got zero. Therefore for θ=2 the Hawking temperature and the modified temperature both become equal to zero.


## 4.2. Fluctuations of Trailing String

In order to investigate the fluctuations of trailing string we utilize quadratic Nambu-Goto action (26) with
(112)N(r)=h(r)(R/r)4(r/rF)4θ/dH(r)2/3-C2(R/r)2(r/rF)2θ/dH(r)1/3(h(r)-v2),Gii=(Rr)4(rrF)4θ/dH(r)1/3.
Therefore, by using the above expressions equations of motion (28) become(113)∂r[H(r)(C2-(Rr)4(rrF)4θ/dH(r)2/3h(r))(v2-h(r))∂rδX⊥]+ω2(R/r)4(r/rF)4θ/dH(r)(C2-(R/r)4(r/rF)4θ/dH(r)2/3h(r))(v2-h(r))δX⊥=0,∂r[(C2-(R/r)4(r/rF)4θ/dh(r))H(r)3/2(v2-h(r))1/2H(r)1/6∂rδX∥]+ω2(C2-(R/r)4(r/rF)4θ/dH(r)2/3h(r))1/2(v2-h(r))3/2H(r)1/2δX∥=0.At the world-sheet horizon r→rs both of the above equations reduce to (33) with Ts defined in (111). Therefore, the solution to this equation is similar to what is obtained in relation (34). Near the boundary limit, r→rF, of (113), we receive relation (35) with d=3, since in this limit h(r) and H(r) have the following behavior:
(114)lim⁡r→rF→0H(r)=1,lim⁡r→rF→0h(r)=1.
So the solutions to the equations of motion near the boundary are similar as before, that is, relation (36) with d=3.


## 4.3. Momentum Correlator of Trailing String

The classical solutions that we obtained for a trailing string in black-hole background (105) are implemented in computing the Langevin correlators. As we discussed before, these correlators involve two parts: real and imaginary.


### 4.3.1. Real Part of Retarded Correlator

The real part of the correlators from fluctuation modes (36) is obtained through relation (31) as
(115)ReGR⊥≃γ-2ReGR∥≃γω22πα′(1-(2θ/3))rF×[1+γ2ω2rF22(1-(2θ/3))+O(rF4)].
As this relation demonstrates, in the limit of rF→0, there are some divergencies of the order of (1/rF). These divergencies can be canceled in a similar way to Section 2 by the identification of ΔM(rF)=-(R2/(2πα′(1-(2θ/3))rF)) in boundary counterterm (55).


### 4.3.2. Imaginary Part of Retarded Correlator

For computing the imaginary part of retarded correlator, we require relation (63). The expressions g⊥rr and g∥rr in relation (30) reduce to the following for metric background (105):
(116)g⊥rr≃4πTs(rs-r)Gii(rs),g∥rr≃4πTs(rs-r)N(rs),g⊥rr≃4πTs(rs-r)Gii(rs)g⊥rr≃4πTs(rs-r)Giihr⟶rs,withN(rs)=16π2R2Ts2rF2θ/dH2/3(rs)h′2(rs)rs(2θ/d)-2.
By inserting the above expressions in (63) and using solution (44) for ψ we write
(117)Im⁡GR⊥=-Gii(rs)|ψh⊥|22πα′ω,Im⁡GR∥=-N(rs)|ψh∥|22πα′ω.
The spectral densities forming these correlators can be found via relation (67). To obtain the high-frequency behavior of the Langevin correlators, we will implement the WKB method in the next subsection.


## 4.3.1. Real Part of Retarded Correlator

The real part of the correlators from fluctuation modes (36) is obtained through relation (31) as
(115)ReGR⊥≃γ-2ReGR∥≃γω22πα′(1-(2θ/3))rF×[1+γ2ω2rF22(1-(2θ/3))+O(rF4)].
As this relation demonstrates, in the limit of rF→0, there are some divergencies of the order of (1/rF). These divergencies can be canceled in a similar way to Section 2 by the identification of ΔM(rF)=-(R2/(2πα′(1-(2θ/3))rF)) in boundary counterterm (55).


## 4.3.2. Imaginary Part of Retarded Correlator

For computing the imaginary part of retarded correlator, we require relation (63). The expressions g⊥rr and g∥rr in relation (30) reduce to the following for metric background (105):
(116)g⊥rr≃4πTs(rs-r)Gii(rs),g∥rr≃4πTs(rs-r)N(rs),g⊥rr≃4πTs(rs-r)Gii(rs)g⊥rr≃4πTs(rs-r)Giihr⟶rs,withN(rs)=16π2R2Ts2rF2θ/dH2/3(rs)h′2(rs)rs(2θ/d)-2.
By inserting the above expressions in (63) and using solution (44) for ψ we write
(117)Im⁡GR⊥=-Gii(rs)|ψh⊥|22πα′ω,Im⁡GR∥=-N(rs)|ψh∥|22πα′ω.
The spectral densities forming these correlators can be found via relation (67). To obtain the high-frequency behavior of the Langevin correlators, we will implement the WKB method in the next subsection.


## 4.4. The WKB Approximation at Large Frequency

In this section we implement the WKB method to obtain the high-frequency behavior of spectral densities. Everything is the same as Section2, except that the function L has the following definition in Schrödinger potential (68):
(118)L=H(r)(C2-(Rr)4(rrF)4θ/dh(r)H2/3(r))(v2-h(r)).
As before, we divide the range of rF<r<rH in three regions and derive the solutions to the Schrödinger equations.Near Boundary
(
r
≪
r
s
). In this region with
(119)L=R2r(2θ/3)-2rF2θ/3γ,N=R2r(2θ/3)-2rF2θ/3
and so, the Schrödinger potential,
(120)Vs≃-γ2ω2+((θ/3)-1)((θ/3)-2)r2
we receive the following solutions:
(121)ϕ=A1rJ(3/2)-(θ/3)(rγω)+A2rY(3/2)-(θ/3)(rγω).Near Horizon
(
r
≃
r
s
). In this region, one can obtain the following for L function:
(122)L=(4πTs)R2rs(2θ/2)-2rF2θ/3H1/3(rs)(rs-r).
If we implement the above relation and relation (117) for N(rs) in the Schrödinger potential; then we get
(123)Vs≃-(ω~2+14)1(r-rs)2r⟶rs,
where ω~=ω/4πTs. The solution to Schrödinger equation (68) after substituting this potential is identical to what is found in solution (75).WKB Region
(
r
t
p
<
r
≪
r
s
). In this region, for large ω’s, the first term in (68) dominates and the Schrödinger potential becomes
(124)Vs≃-ω2Gii2R2,rtp<r≪rs.
The turning point for large ω is found as
(125)rtp=((θ/3)-1)((θ/3)-2)γω.
We should note that the solutions in three regions are applied for both transversal and longitudinal equations.Now, with the above information, we are able to find the coefficients by considering the limits that three regions overlap. Eventually, by repeating the process followed in Section2, we obtain the following expression for ψh for the infinite massive quark case:
(126)ψh=π(ωγ/2)1-(θ/3)rs(θ/3)-1Γ[(3/2)-(θ/3)]ei(υ-(π/2)((3-θ)/3))×{1⊥Gii(rs)N(rs)∥
and then by inserting the above expressions in (63) we attain
(127)Im⁡GR⊥=R2ω2α′(ωγ/2)2-(2θ/3)rF2(θ/3)[Γ((3/2)-(θ/3))]2.
The longitudinal component can be found in the same way and by using relation (65). The finite mass case is obtained through the way described in Section 2.


## 4.5. Langevin Diffusion Constants

The diffusion coefficients can be found either from the direct evaluation of the correlators or from using the membrane paradigm. Both of these methods give rise to the following identical results for diffusion constants:(128)κ⊥=R2H1/3(rs)rs(2θ/3)-2πα′rF2θ/3Ts,κ∥=16πR2rs(2θ/3)-2Ts3α′rF2θ/3H2/3(rs)h′2(rs).
There is a problem in computing the longitudinal component for the special case θ=2. In this case both H(rs) and h′(rs) tend to zero, so the longitudinal diffusion constant tends to the infinity. It seems that this value for θ is unacceptable. By studying [36], we perceive that the null energy condition imposes some condition on θ as
(129)θ≤0.365,θ≥1.315.
In a similar way to the previous work and because we see some instabilities in the range of θ≥1.315, we choose the range of θ≤1.315. This case is in analogy with the case θ<d in the previous section.The ratio between longitudinal and transversal components of diffusion constants is(130)κ∥κ⊥=1+2h(rs)h′(rs)[(2θ/3)-2rs+H′(rs)(rs)].
We expect the universal inequality κ∥>κ⊥ to be held. Therefore, in order to keep the inequality for the following relation,
(131)κ∥κ⊥=1+2v2(1-v2)×[(rs/rF)θ+(2Q1rs2/3)+(Q2rFθrs4-θ/3)2(rs/rF)θ+Q1rs2-Q3rF2θrs6-2θ],
with Qi>0, the condition 2+ax>cx3 must be satisfied. For Qi<0, both of the numerator and denominator on the right-hand side of relation (131) can be negative or positive, where each of these situations puts some conditions on Qi and rs. Since, due to relation (110), the rs is related to Qi, rF, and v2, to hold the inequality, there should be some conditions on Qi, rF, and v2. The jet-quenching parameters can be obtained easily from relation (100) as in the previous section.


## 5. Summary

In this paper, we used AdS/CFT correspondence and studied the stochastic motion of an external quark in a plasma. It corresponds to fundamental string whose end-point lies in the UV region of a bulk black-hole background and is forced to move with velocityv. By using the Nambu-Goto action, we obtained the equations of motion for this string in the planar black holes with hyperscaling violation background. The solution of the corresponding equations of motion leads us to finding the classical profile of the trailing string. Next, we considered small fluctuations around the classical string profile. These fluctuations satisfied the second-order radial equations and are related to the associated thermal correlators. We achieved the modified temperature Ts; it was felt by string fluctuations. In order to have a positive temperature, we got a constraint on θ for the case of z=1. We derived the drag force on the quark in the presence of dynamical exponents z=1 and z>1. Here, we found that in the ultrarelativistic limit this force becomes zero for the case z=1, while for the range of z>1 it is not so. We computed holographically the full Langevin correlators and the associated spectral densities, including real and imaginary parts of correlators for large- and small-frequency limits. In the large-frequency regime, the spectral densities are obtained via the modified WKB method. We have shown that only for the case z=1 the ratio between transversal and longitudinal components of spectral density depends on the velocity of quarks. For the range of z>1 this relation tends to a constant with the value of 1. We investigated the large-time limit of the fluctuations and for the constant diffusion we need small-frequency modes to obtain the diffusion. In order to hold inequality κ∥≥κ⊥, we have shown that the range of θ≤d is an acceptable region for θ. This region is in agreement with what is found in literature. We note that the region θ>d leads to some instabilities on the gravity side, so for this region we cannot have the physically consistent theories. For the special case θ=d the inequality κ∥≥κ⊥ converts to the equality κ∥=κ⊥ for both z=1 and z>1 cases. In this case we found that the Hawking temperature and the modified temperature become equal; however the world-sheet horizon and the black-hole horizon are different.We repeated all of the above procedures for theR-charged black hole with hyperscaling violation. For this black hole we found that the range of θ≤1.315 can provide a plausible region for θ which is in agreement with our results in [36]. We also realized that the universal inequality of diffusion constants can be confirmed by some constraints on the world-sheet horizon and charges.As we indicated before, we obtained the real part and imaginary part of retarded correlator for both planer andR-charged black-hole with hyperscaling violation. For real part, we encountered some divergencies in the limit r→rF. In that case we introduced a boundary counterterm action for the case z=1 in planar black hole and R-charged black hole with hyperscaling violation. For the case of z>1 in planar black hole with hyperscaling violation, the boundary counterterm with the last definition cannot help us to remove the divergencies. It seems that a new definition of boundary counterterm is needed to overcome this problem. We would like to investigate this problem in future works. We are also interested to study different black holes with hyperscaling violation and investigate the Langevin diffusion process.



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*Source: 670598-2014-10-20.xml*