Contents

CQUeST-2010-0395

Zero Sound in Effective Holographic Theories

Bum-Hoon Lee, Da-Wei Pang and Chanyong Park

Department of Physics, Sogang University

Seoul 121-742, Korea

Center for Quantum Spacetime, Sogang University

Seoul 121-742, Korea

bhl@sogang.ac.kr, pangdw@sogang.ac.kr, cyong21@sogang.ac.kr

We investigate zero sound in -dimensional effective holographic theories, whose action is given by Einstein-Maxwell-Dilaton terms. The bulk spacetimes include both zero temperature backgrounds with anisotropic scaling symmetry and their near-extremal counterparts obtained in 1006.2124 [hep-th], while the massless charge carriers are described by probe D-branes. We discuss thermodynamics of the probe D-branes analytically. In particular, we clarify the conditions under which the specific heat is linear in the temperature, which is a characteristic feature of Fermi liquids. We also compute the retarded Green’s functions in the limit of low frequency and low momentum and find quasi-particle excitations in certain regime of the parameters. The retarded Green’s functions are plotted at specific values of parameters in , where the specific heat is linear in the temperature and the quasi-particle excitation exists. We also calculate the AC conductivity in -dimensions as a by-product.

## 1 Introduction

The AdS/CFT correspondence [1, 2] has revealed the deep relations between gauge theories and string theories and has provided powerful tools for understanding the dynamics of strongly coupled field theories in the dual gravity side. In recent years, this paradigm has been applied to investigate the properties of certain condensed matter systems [3]. The correspondence between gravity theories and condensed matter physics(sometimes is also named as AdS/CMT correspondence) has shed light on studying physics in the real world in the context of holography.

It is well known that in realistic condensed matter systems, the presence of a finite density of charge carriers is of great importance. According to the AdS/CFT correspondence, the dual bulk gravitational background should be charged black holes in asymptotically AdS spacetimes. The simplest example of such charged AdS black holes is Reissner-Nordström-AdS(RN-AdS) black hole, which has proven to be an efficient laboratory for studying the AdS/CMT correspondence. For instance, investigations of the fermionic two-point functions in this background indicated the existence of fermionic quasi-particles with non-Fermi liquid behavior [4, 5, 6], while the symmetry of the extremal RN-AdS black hole is crucial to the emergent scaling symmetry at zero temperature [7]. Moreover, adding a charged scalar in such background leads to superconductivity [8, 9, 10].

A further step towards a holographic model-building of strongly-coupled systems at finite charge density is to consider the leading relevant (scalar) operator in the field theory side, whose bulk gravity theory is an Einstein-Maxwell-Dilaton system with a scalar potential. Such theories at zero charge density were analyzed in detail in recent years as they mimic certain essential properties of QCD [11, 12, 13, 14, 15]. Solutions at finite charge density have been considered in [16, 17, 18, 19, 20, 21] in the context of AdS/CMT correspondence.

Recently a general framework for the discussion of the holographic dynamics of Einstein-Maxwell-Dilaton systems with a scalar potential was proposed in [22], which was a phenomenological approach based on the concept of Effective Holographic Theory (EHT). The minimal set of bulk fields contains the metric , the gauge field and the scalar (dual to the relevant operator). appears in two scalar functions that enter the effective action: the scalar potential and the non-minimal Maxwell coupling. They studied thermodynamics of certain exact solutions and computed the DC and AC conductivity. The main advantage of this EHT approach is that it permits a parametrization of large classes of IR dynamics and allows investigations on important observarables. However, it is not clear whether concrete EHTs can be embedded into string theories. For subsequent generalizations see [23, 24, 25, 26, 27, 28, 29, 30].

On the other hand, strongly coupled quantum liquids play an important role in condensed matter physics, where quantum liquids mean translationally invariant systems at zero (or low) temperature and at finite density. By now there are two successful phenomenological theories of quantum liquids: Landau’s Fermi-liquid theory and the theory of quantum Bose liquids, describing two different behaviors of a quantum liquid at low momenta and temperatures. In particular, the specific heat of a Bose liquid at low temperature is proportional to in spatial dimensions, while the specific heat of a Fermi liquid scales as at low , irrespective of the spatial dimensions.

One may wonder if the newly developed techniques in AdS/CFT correspondence can help us understand the behavior of quantum liquids. In [31] the authors considered a class of gauge theories with fundamental fields whose holographic dual in the appropriate limit was given in terms of the Dirac-Born-Infeld (DBI) action in AdS space. They found that the specific heat in spatial dimensions at low temperature and the system supported a sound mode at zero temperature, which was called “zero-temperature sound”. One interesting feature was that the “holographic zero sound” mode was almost identical to the zero sound in Fermi liquids: the real part of the dispersion relation was linear in momentum () and the imaginary part had the same dependence predicted by Landau. The crucial difference was that the zero-temperature sound velocity coincided with the first-sound velocity, while generically the two velocities are not equal for a Fermi liquid. Such analysis was performed in the case of massive charge carriers in [32] and in the case of Sakai-Sugimoto model in [33]. The specific heat of general systems was calculated in [34] and the specific heat of Lifshitz black holes was discussed in [35] and [36], while the zero sound was also investigated in  [36].

In this paper we will study the low-temperature specific heat and the holographic zero sound in effective holographic theories. Here the bulk effective theory is -dimensional Einstein gravity coupled to a Maxwell term with non-minimal coupling and a scalar. It was found in  [37] that the theory admitted both extremal and near-extremal solutions with anisotropic scaling symmetry. We consider dynamics of probe D-branes in the above mentioned backgrounds and find that by appropriately fixing the parameters in the effective theory, the specific heat can be proportional to , resembling a Fermi liquid. We also compute the current-current retarded Green functions at low frequency and low momentum, and clarify the conditions when a quasi-particle excitation exists. Moreover, we also explore the possibility of observing the existence of Fermi surfaces in such a system by numerical methods. We find that although the system possesses some features of Fermi liquids, such as linear specific heat and zero sound excitation, we do not observe any characteristic structure in the wide range of . In addition, the AC conductivity is also obtained as a by-product.

The rest of the paper is organized as follows: the exact solutions of the effective bulk theory will be reviewed in section 2 and the thermodynamics of massless charge carriers will be discussed in section 3. We shall calculate the correlation functions in section 4 and identify the quasi-particle behavior, while the existence of Fermi surfaces will be explored in section 5 via numerics. We will calculate the AC conductivity in section 6, including the zero density limit. Finally we will give a summary and discuss future directions.

## 2 The solution

In this section we will review the solutions obtained in [37], which can be seen as generalizations of the four-dimensional near-extremal scaling solution discussed in [22]. In the beginning we consider the following action in -dimensions, without any reference to string theory or M/theory origin nor specifying the forms of the gauge coupling and the scalar potential explicitly,

 S=−116πGD∫dDx√−g[R+f(ϕ)FμνFμν+12(∂ϕ)2+V(ϕ)]. (2.1)

The resulting solutions are charged dilaton black holes, which have been investigated in the literature for a long period [38, 39, 40, 41]. Let us focus on solutions carrying electric charge only. The general configuration with planar symmetry can be written as follows

 ds2 = −U(r)dt2+dr2U(r)+V(r)D−2∑i=1dx2i, ϕ = ϕ(r),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak At=At(r),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak Ar=Ai=0. (2.2)

After plugging in the scaling ansatz

 U(r)∼rβ,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak V(r)∼rγ (2.3)

into the equations of motion, we can arrive at several constraints on the parameters and the scalar functions:

• We require that , so that the extremal solutions have smooth connections to the finite temperature solutions;

• The field equations indicate that and and when the scale invariance is restored.

• The equations of motion determine that the scalar field must take the form

 ϕ(r)=C2logr+ϕ0, (2.4)

and both and are constrained to be exponential in , power law in .

• Once we have fixed , the metric must have , where saturation occurs for vanishing flux, e.g. in with .

Subsequently, according to the constraints discussed above, we take the following forms of and .

 f(ϕ)=eαϕ,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak V(ϕ)=−V0eηϕ. (2.5)

Then we will consider charged dilaton black holes with a Liouville potential [42]. Now the scaling ansatz turns out to be

 ds2 = −C1rβdt2+dr2C1rβ+C3rγD−2∑i=1dx2i, ϕ(r) = C2logr+ϕ0,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak A′(r)=QrαC2+γD−22. (2.6)

Since and can be eliminated by rescaling and , we shall set and . The remaining parameters can be explicitly given in terms of ,

 β = 2−2(D−2)(α+η)(α+η)2+2(D−2)η,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak γ=2(α+η)2(α+η)2+2(D−2), C2 = −(D−2)α+ηγ,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak Q2=V022−η2−αη2+α2+αη, C1 = V0[(α+η)2+2(D−2)]2(D−2)(2+α2+αη)[2(D−2)+(D−1)α2−η2(D−3)+2αη]. (2.7)

It can be easily seen that there is no scale invariance in such backgrounds. We will restrict to and without loss of generality, which implies that must diverge to positive infinity at the horizon.

In the specific limit , the scalar potential becomes constant and the scale invariance of the solutions can be restored:

• When we set with vanishing flux, the scalar becomes constant and the resulting solution is ;

• When we set with flux through , becomes constant and the resulting solution is ;

• When we set arbitrary, with flux through , the scalar and the resulting solution is the modified Lifshitz solution, whose dynamical exponent 111The reason why such a solution is called “modified Lifshitz” is that the scalar field must be constant in Lifshitz background, which is required by scaling symmetry [43]. Properties of the modified Lifshitz solutions have been studied in [44] and [20].

Before coming to practical calculations we should determine the range of parameters. Firstly, by requiring that for small and the flux to be real, one can impose the bound on in terms of fixed ,

 −η<α<2η−η. (2.8)

Secondly, when the flux is zero,

 α=2η−η,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak β=γ=42+(D−2)η2≡γ′.

we should require to ensure a well-defined boundary in the sense of AdS/CFT 222For details see [37].,

 η2<2D−2.

Combining constraint derived in the general background in the beginning of this section, we can obtain the following complete restrictions

 1<β≤2,\leavevmode\nobreak \leavevmode\nobreak γ≤β≤2,\leavevmode\nobreak \leavevmode\nobreak −η<α≤2η−η,\leavevmode\nobreak \leavevmode\nobreak 0≤η<√2D−2. (2.9)

We will impose such constraints in the subsequent calculations.

The near extremal solution can be obtained in a similar way,

 ds2=−C1rβf(r)dt2+dr2C1rβf(r)+C3rγD−2∑i=1dx2i, (2.10)

where

 f(r)=1−(r+r)w,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak w=β−1+D−22γ, (2.11)

and the other parameters and fields remain the same as the extremal solutions. One can easily get the temperature

 T=14πC1wrβ−1+, (2.12)

and the entropy density

 s≡SBHVRD−2=14GDrD−22γ+. (2.13)

## 3 Thermodynamics of probe D-branes

In this section we will investigate thermodynamics of massless charge carriers in the backgrounds reviewed in previous section. According to AdS/CFT, probe D-branes correspond to fields in the fundamental representation of the gauge group in the probe limit  [45, 46]. An efficient method for evaluating the DC conductivity and DC Hall conductivity of probe D-branes was proposed in [47] and [48]. Moreover, a holographic model building approach to “strange metallic” phenomenology was initiated in [49], where the bulk spacetime was a Lifshitz black hole and the charge carriers were described by D-branes. Here we will consider probe D-branes as massless charge carriers and explore the thermodynamics in the near-extremal background.

The dynamics of probe D-branes is described by the Dirac-Born-Infeld (DBI) action

 SDBI = −NfTDVol(Σ)∫dtdrdqxe−ϕ√−det(gab+2πα′Fab), (3.1) = −τeff∫dtdrdqxe−ϕ√−det(gab+2πα′Fab),

where denotes the tension of D-branes, is the induced metric and is the field strength on the worldvolume. In the second line we set , where denotes volume of the internal space that the D-branes may be wrapping. Furthermore, we assume that the D-branes are extended along spatial dimensions of the black hole solution. If , the fundamental fields are propagating along certain -dimensional defect. We will introduce a nontrivial worldvolume gauge field and absorb the factor into . Since we are not studying realistic string theories, the Wess-Zumino terms will be omitted in the following discussions.

Before proceeding we should make sure that the backreaction of the probe branes onto the background can be neglected. Our discussion is along the line of [49]. Expanding the DBI action to quadric order of in the background, we can obtain

 SDBI=−τeff∫dtdrdqxe−ϕ√−g√1+gttgrrF2rt, (3.2)

To avoid backreaction of the probes on the background, the stress energy of the probes must be smaller than that generating the bulk spacetime. It can be easily seen that the stress energy of the original background, where denotes the Planck length in -dimensional spacetime and is the corresponding cosmological constant. Therefore by varying the quadric action of the probes with respect to , we can arrive at the following condition

 e−ϕ√1+gttgrrF2rt≪ℓD−2P|Λ|τeff. (3.3)

One can see that as long as the effective tension is sufficiently small, the backreaction can be neglected.

In this background configuration, after performing the trivial integrations on and dividing out the infinite volume of , we can obtain the action density.

 S=−τeff∫drrm√1−A′2t, (3.4)

where

 m=12γq−C2=(α+η)[q(α+η)+2(D−2)(α+η)2+2(D−2). (3.5)

and the prime denotes derivative with respect to . Then the charge density is given by

 ρ≡δLδA′t=τeffrmA′t√1−A′2t. (3.6)

We can also solve for

 A′t=d√r2m+d2,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak d≡ρτeff. (3.7)

By plugging in the solution for , we can find the on-shell action density

 Son−shell=−τeff∫drr2m√r2m+d2. (3.8)

Following the methods used in [50], some interesting physical quantities like the chemical potential and the free energy, can be evaluated analytically. We will see that this is still the case for our background. The chemical potential is given by

 μ = ∫∞r+A′tdr, (3.9) = μ0−r+2F1(12m,12;1+12m,−r2m+d2),

where

 μ0=d1mB0(m),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak B0(m)=12B(1+12m,12−12m). (3.10)

Notice that in order to obtain the above results, we have made use of the following useful formulae for Beta function and incomplete Beta function

 B(a,b) = Γ(a)Γ(b)Γ(a+b)=∫∞0du(1+u)−(a+b)ua−1, B(x;a,b) = ∫x/(1−x)0du(1+u)−(a+b)ua−1.

as well as for Hypergeometric function

 B(x;a,b) = a−1xa2F1(a,1−b;a+1,x), 2F1(a,b;c,x) = (1−x)−a2F1(a,c−b;c,xx−1).

After choosing the grand-canonical ensemble, the free energy density is given by

 Ω = −Son−shell=τeff∫∞r+drr2m√r2m+d2, (3.11) = Ω0−τeff(2m+1)dr2m+1+2F1(1+12m,12;2+12m,−r2m+d2),

where

 Ω0=−τeff2(m+1)d1+1mB(1+12m,12−12m)=−τeff(m+1)B0(m)mμm+10. (3.12)

Moreover, other thermodynamic quantities can also be calculated from the thermodynamic relations. The charge density can be written as

 ρ=−∂Ω∂μ=τeff(μ0B0(m))m=τeffd, (3.13)

which is consistent with previous result. The entropy density is given by

 s=−∂Ω∂T=ρβ−1(4πC1w)1β−1T2−ββ−1+τeff2(β−1)d(4πC1w)2m+1β−1T2(m+1)−ββ−1, (3.14)

Notice that when , there exists a nontrivial contribution to the entropy density at like those observed in [35] and [36]. On the other hand, the entropy density is vanishing at extremality as long as . The specific heat is

 cV=T∂s∂T=ρ(2−β)(β−1)2(4πC1w)1β−1T2−ββ−1+τ2eff(2m+2−β)2(β−1)2ρ(4πC1w)2m+1β−1T2m+2−ββ−1. (3.15)

It is well known that for a gas of free bosons in spatial dimensions, the specific heat at low temperature is proportional to , while for a gas of fermions the low temperature specific heat is proportional to , irrespective of . When and is sufficiently small, the first term dominates. One can easily obtain when the specific heat is proportional to . Then the parameter can be expressed in terms of

 α1±=(2D−5)η±√2(D−2)[2(D−2)η2−1], (3.16)

Combining with (2.9), we can arrive at the following conclusions:

• When

 √12(D−2)≤η<√23(D−2),

both and are permitted solutions;

• When

 √23(D−2)≤η<√2D−2,

only is a permitted solution;

• When

 0<η<√12(D−2),

there is no solution, which means that we cannot realize in this regime.

When , the second term provides the only contribution. The linear dependence on fixes . Note that in this limit, . So

 α2±=−4(D−2)±√16(D−2)2+4(D−2)(2q−1)2(2q−1), (3.17)

By taking into account of (2.9), we can find that only is permitted.

We can also formally evaluate the “speed of sound”. In grand-canonical ensemble, the pressure is given by

 P=−Ω0=τeff(m+1)B0(m)mμm+10, (3.18)

while the energy density is

 ε=Ω0+μ0ρ=mτeff(m+1)B0(m)mμm+10. (3.19)

Therefore,

 ε=mP,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak ⇒\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak c2s=∂P∂ε=1m. (3.20)

However, it was emphasized in [36] that this quantity is only the speed of normal/first sound in the relativistic case . Actually the speed of normal sound is dimensionful in a system with . We will calculate the holographic zero sound in the next section.

## 4 The holographic zero sound

In this section we will calculate the holographic zero sound in the anisotropic background at extremality. The basic strategy is to consider fluctuations of the worldvolume gauge field on the probe D-branes in the background with nontrivial . Such analysis was performed for background in [31] and for Lifshitz background in [36]. We will calculate the holographic zero sound in a similar way and classify the behavior of the zero sound in different parameter ranges.

### 4.1 The retarded Green’s functions

Zero sound should appear as a pole in the density-density retarded two-point function at extremality [31]. In [36] the authors provided a general framework for evaluating the corresponding retarded Green’s functions with background metric

 ds2=gttdt2+grrdr2+gxxD−2∑i=1dx2i. (4.1)

Here we will take the nontrivial dilaton into account. The symmetries in the spatial directions allow us to consider fluctuations of the gauge fields with the following form

 Aμ(r)\leavevmode\nobreak →\leavevmode\nobreak Aμ(r)+aμ(t,r,x),

where denotes one of the spatial directions. The quadratic action for the fluctuations is given by

 Sa2=τeff2∫dtdrdqxe−ϕgq/2xx[grrf2tx−|gtt|a′2xgxx(|gtt|grr−A′2t)1/2+|gtt|grra′2t(|gtt|grr−A′2t)3/2], (4.2)

where . Note that we are working in the gauge of . After performing the Fourier transform

 aμ(t,r,x)=∫dωdk(2π)2e−iωt+ikxaμ(ω,r,k),

the linearized equations of motion can be written as

 ∂r[e−ϕgq/2xx|gtt|grra′t(|gtt|grr−A′2t)3/2]−e−ϕgq/2−1xxgrr√|gtt|grr−A′2t(k2at+ωkax)=0, (4.3)
 ∂r[e−ϕgq/2−1xx|gtt|a′x(|gtt|grr−A′2t)1/2]+e−ϕgq/2−1xxgrr√|gtt|grr−A′2t(ω2ax+ωkat)=0. (4.4)

In addition, the following constraint can be obtained by writing ’s equation of motion in gauge

 grrgxxωa′t+(|gtt|grr−A′2t)ka′x=0. (4.5)

The above equations are not independent, as we can obtain the equation of motion for by combining the constraint equation and the equation of motion for . Therefore it is sufficient to solve the constraint and the equation for only. By introducing the gauge-invariant electric field

 E(r,ω,k)=ωax+kat,

we can obtain the equation of motion for

 E′′+[∂rln(e−ϕgq−32xx|gtt|g−1/2rru(k2u2−ω2))]E′−grr|gtt|(u2k2−ω2)E=0, (4.6)

where

 u(r)=√|gtt|grr−A′2tgrrgxx. (4.7)

Moreover, the quadratic action can also be expressed in terms of ,

 Sa2=τeff2∫drdωdke−ϕgq−32g1/2rru[E2+|gtt|grr(u2k2−ω2)E′2], (4.8)

For our specific background, the metric can be rewritten in terms of the new radial coordinate as follows

 ds2=−C1zβdt2+zβ−4C1dz2+1zγD−2∑i=1dx2i. (4.9)

In the new coordinate system, the solution of the worldvolume gauge field and the function are given by

 ˙A2t=d2z2mz4(1+d2z2m),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak u2=C1zβ−γ(1+d2z2m), (4.10)

where dot denotes derivative with respect to . Integrating the quadratic action by parts

 Sa2=τeff2∫dωdke−ϕgq−32xxg−1/2rr|gtt|u(u2k2−ω2)˙EE, (4.11)

introducing a cutoff at and taking the limit , the quadratic action turns out to be

 Sa2=−τeff2∫dωdkϵ2−mk2˙EE. (4.12)

After imposing the incoming boundary condition at the “horizon” and plugging in the solutions of , the retarded correlation function reads [51]

 GRtt(ω,k)=δ2δat(ϵ)2Sa2=(δE(ϵ)δat(ϵ))2δ2δE(ϵ)2Sa2, (4.13)

By defining

 Π(ω,k)≡δ2δE(ϵ)2Sa2, (4.14)

the retarded correlation functions can be written in terms of

 GRtt(ω,k)=k2Π(ω,k),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak GRtx(ω,k)=ωkΠ(ω,k),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak GRxx(ω,k)=ω2Π(ω,k). (4.15)

### 4.2 Matching the solutions

In order to evaluate the retarded correlation functions, we should try to solve (4.6), whose analytic solutions are always difficult to find. We will leave the numerical work to section 5, while here we will obtain the low-frequency behavior of by solving (4.6) in different limits and matching the two solutions in an overlapped regime, following the spirit of [31] and [36]. To be concrete, we will solve (4.6) in the limit of large and then expand the solution in the small frequency and momentum limit. Next we will take the small frequency and momentum limit first and then perform the large expansion. The integration constants can be fixed by matching the two solutions.

First let us take , which leads to the following equation for

 ¨E+2−β+γz˙E+ω2z2β−4C21E=0. (4.16)

The solution can be given in terms of a Hankel function of the first kind,

 E=D0(x2)νH(1)ν(x),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak x=ωC1(β−1)zβ−1,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak ν=12−γ2(β−1), (4.17)

In the limit of small frequency with , the asymptotic expansion reads

 E = D0Γ(12+γ2(β−1))−1(1−itanπγ2(β−1)) (4.18) −iD0πΓ(γ2(β−1)−12)(ω2C1(β−1))1−γβ−1zβ−γ−1,

It should be pointed out that the case of must be treated separately. In this case the corresponding parameter is given by

 α3+=−(D−1)η+√(D−2)2η2+2(D−2). (4.19)

Now the expansion contains a logarithmic term

 E≃D0+2iπD0(log(ωzβ−1)−log(2C1(β−1))+γE), (4.20)

where is the Euler constant.

Next we take with being fixed. Then the last term in (4.6) can be neglected and the equation of becomes

 ¨E+[∂zln(e−ϕgq−32xx|gtt|g−1/2rru(k2u2−ω2))]˙E=0. (4.21)

When and , the solution is given by

 E = D1+D2[C1k2zm−1m(1√1+d2z2m+1m−12F1(12,12−12m;32−12m,−d2z2m)) (4.22) −ω2zm+nm+n2F1(12,12+n2m;32+n2m,−d2z2m)],

where . For either or , the powers of do not match, which will be displayed in Appendix A. We will make use of the following useful formulae for the asymptotic expansion

 2F1(a,b;c,x) = (1−x)−aΓ(c)Γ(b−a)Γ(b)Γ(c−a)2F1(a,c−b;a−b+1,11−x) +(1−x)−bΓ(c)Γ(a−b)Γ(a)Γ(c−b)2F1(b,c−a;b−a+1,11−x),
 2F1(12,12;32,−x2)=x−1log(x+√1+x2).

Therefore the large limit is given as follows when ,

 E≃D1+D2[C1μ0k2md−ω2znnd−ω2d−nm2mdB(12+n2m,−n2m)]. (4.23)

When the expansion also contains a logarithmic term

 E ≃ D1+D2[C1μ0k2md−ω2mdlog(2dzm)] (4.24) ≃ D1+D2[C1μ0k2md−ω2mdlog2d+ω2(β−1)dlogω−ω2(β−1)dlog(ωzβ−1)].

To evaluate the retarded correlation functions, we need small expansion of the solution. It can be seen that the second term in the expansion always tends to zero more rapidly, being irrespective of or not. Therefore we have

 E≃D1+D2C1k2m−1zm−1, (4.25)

Assuming that 333the can be dealt with in a similar fashion, see footnote 7 of [36]., the leading order behavior of reads , so the quadratic action turns out to be

 Sa2 = −τeff2∫dωdkϵ2−mk2E˙E (4.26) = −τeffC12∫dωdkD1D2,

Thus

 Π(ω,k)=limϵ→0δ2δE(ϵ)2Sa2=δ2δD21Sa2∣∣ϵ→0, (4.27)

The relation between the integration constants and can be obtained by matching the expansions of the solutions in different limits and eliminating the other integration constant .

Finally we summarize our result for

 Π(ω,k)∝τeffC1δ1k2−δ2ω2−δ3G0(ω). (4.28)

When the parameters are given by

 δ1 = −C1μ0nπm(2C1(β−1))nβ−1Γ(γ2(β−1)−12)Γ(γ2(β−</