# Holographic aspects of a higher curvature massive gravity

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## Abstract

We study the holographic dual of a massive gravity with Gauss–Bonnet and cubic quasi-topological higher curvature terms. Firstly, we find the energy–momentum two point function of the 4-dimensional boundary theory where the massive term breaks the conformal symmetry as expected. An *a*-theorem is introduced based on the null energy condition. Then we focus on a black brane solution in this background and derive the ratio of shear viscosity to entropy density for the dual theory. It is worth mentioning that the concept of viscosity as a transport coefficient is obscure in a nontranslational invariant theory as in our case. So although we use the Green–Kubo’s formula to derive it, we rather call it the rate of entropy production per the Planckian time due to a strain. Results smoothly cover the massless limit.

## 1 Introduction

For two decades, the AdS/CFT correspondence [1, 2, 3] has been at the center of attention in theoretical physics. It not only provides tools for performing calculations in strong coupling limit of field theories and condense matter phenomena, which otherwise undoubtedly was horrible if not impossible, but also opens new windows to understand different aspects of field theories and gravities as well. Much has been done for the Einstein gravity in the AdS bulk and investigated its CFT dual on the boundary. In the early of the AdS/CFT, the central charges of the boundary theory were found by the holography [4]. This was an important success which among others encouraged people to develop the duality for more complicated and realistic theories. In 4 dimensions, it is well known that CFT’s with Einstein gravity dual have two equal central charges *a* and *c* [4, 5]. This means that the dual Einstein gravity gives information about a very special class of CFT’s. To distinguish between central charges and explore more general conformal theories, one may add higher curvature (or higher derivative) gravities, which amongst them, the Gauss–Bonnet gravity serves as a simple model to study the duality. It has important features as its equation of motion includes only the second order derivatives, admits exact black hole solution and the corresponding dual theory would be a CFT with two distinguished central charges \(a\ne c\) [6]. It is possible to keep these advantages and add cubic curvature terms, the so called quasi-topological gravity which of course doesn’t generally admit a second order differential equation except for the AdS background which is in our interest [7, 8, 9]. The combination of Gauss–Bonnet and cubic quasi-topological gravity has not yet any stringy derivation, however as a toy model is rich enough to study different aspects of the dual conformal theory [9].

On the other hand, several studies have been performed for decades to generalize the graviton field to a massive one with different motivations from theoretical curiosity to phenomenological model buildings (for a recent review see [10]). Indeed, the problem of giving mass to the gravity is not an obvious one and was a challenge for several years. The first attempt was by Fierz and Pauli [11] proposing a linear massive model. Unfortunately, that doesn’t reduce to GR in the zero mass limit. Generalization to a nonlinear model in [12] was stopped by Boulware and Deser (BD) when they showed that this suffered from ghosts [13]. Finally in recent years, the BD ghost problem was resolved in [14, 15, 16] by a nonlinear massive gravity. This theory provides a fixed reference metric on which the massive gravity propagates. This breaks the general covariance with applications in holographic models with momentum dissipation [17, 18]. Then it was extended to include dynamics of this reference metric in the context of theories now known as bi-gravities [19, 20, 21] and higher dimensional massive graviton term is discussed in [22]. Many recent works include this massive gravity in the higher curvature gravities especially the Gauss–Bonnet (e.g., [23, 24, 25]) and various features mostly in the gravity side and some in the dual theories are derived.

Here our aim is to tackle the theory including Gauss–Bonnet cubic quasi-topological massive gravity. The important point about this combination is that while it has the rich structure of higher curvature theories, the presence of a mass scale breaks the conformal symmetry on the boundary. Indeed, the quantum field theory dual to the Gauss–Bonnet theory is not well-known and there are doubts if it even exists [26, 27]. This might be the case for a dual to the massive gravity. However, the Gauss–Bonnet gravity is widely discussed in holographic literature at least as a toy model or theoretical laboratory to study general aspects of quantum field theories or CFT’s. In this direction, one may consider the addition of massive gravity as a relevant perturbation of conformal symmetry. This may shed light on the boundary theory, if any, away from the fixed point. By the way, we emphasize that this set up should be considered as a toy model to study the holography.

In this regards, we assume a boundary theory dual to our bulk model in Sect. 2. Firstly we derive the two-point function of the boundary energy–momentum tensor and show that it includes a massive operator on the boundary. We then look for an *a*-theorem which is originally based on renormalization group flows [28, 29, 30, 31] and indicates the truncation of degrees of freedom when going toward an IR fixed point. In the context of AdS/CFT correspondence, *a*-theorems are introduced by considering a generic background which asymptotes to AdS space [7, 32, 33, 34, 35, 36]. Then the *a*-function approaches the *a* charge (not the *c*) in the AdS limit. The function should be monotonically decreasing along the RG flow. This can be achieved by the null energy condition. In our case, we show that the same null energy condition as the massless theory gives the correct monotonic *a*-function.

In Sect. 3, we introduce an exact black brane solution in this background and derive its temperature and entropy then using the standard holographic methods to find the viscosity to entropy ratio. It is worth noting that in a theory where the translational symmetry is broken, e. g. by a mass term as in our case, the viscosity can not be interpreted as a hydrodynamic transport coefficient. Instead it can be considered as the rate of entropy production per the Planckian time due to a strain [37] and can be derived from the Kubo formula (see Eq. (49)).^{1} On the other hand, [40] introduces a shear viscosity from hydrodynamic constitutive relations. Both quantities approach the same viscosity in the massless limit. In our case we deal with the first one, but sometimes we may call it simply ‘viscosity’ rather than the rate of entropy production.

In the Einstein gravity with any matter content, the ratio \(\eta /s\) is found to be \(1/4\pi \) [41]. It was proposed by the KSS that this is a lower bound for relativistic quantum theories [42]. However, in the higher curvature gravities this bound is violated [43]. For the massive gravity, the ‘viscosity’ was calculated in [23] and [37] as a deformation of the metric component \(\delta g_{\mu \nu }\) and shown that the naive bounds on \(\eta /s\) are violated. However, [37] has different interpretation for the bound and argues that it is expected to exist due to a basic quantum mechanical uncertainty. Here we apply this interpretation and take \(\eta \) as the rate of entropy production and extend the calculation to include higher curvature theories with mass term. The result may lower the ‘viscosity’ to entropy ratio more than before.

In Sect. 4, we discuss on some bounds on parameters space. We consider the unitarity and causality bounds on the boundary theory.

## 2 Quasi-topological massive gravity and holography

*R*is the scalar curvature,

*L*the cosmological constant scale,

*f*a fixed rank-2 symmetric tensor known as reference metric and

*m*is the mass parameter. \(\mathcal {L}_{GB}\) and \(\mathcal {L}_{3}\) are respectively the Gauss–Bonnet and quasi-topology terms of gravity with \(\lambda \) and \(\mu \) their dimensionless couplings. A generalized version of the reference metric \( f_{\mu \nu } \) was proposed in [15] with the form \( f_{\mu \nu } = diag(0,0,c_0^2h_{ij})\) with \(h_{ij}=\delta _{ij}/L^2 \). In (1), \( c_i \)’s are constants and \( \mathcal {U}_i \) are symmetric polynomials of the eigenvalues of the \( 5\times 5 \) matrix \( \mathcal {K}^{\mu }_{\nu }=\sqrt{g^{\mu \alpha }f_{\alpha \nu }} \),

*AdS*solution as follows

*N*and

*f*functions are given in Sect. 3. These details are not important for our purposes in this section and we restrict ourselves to the near boundary behavior where \(\lim _{r \rightarrow \infty }f(r)=f_\infty \) and \(\lim _{r \rightarrow \infty }N(r)^2f(r)=1\) which corresponds to the following

*AdS*background,

### 2.1 Holographic picture on the boundary

*U*(

*a*,

*b*,

*x*) and \(_1F_1(a;b;x)\) are the Tricomi and the first kind confluent hypergeometric functions, respectively. Applying the boundary condition of \(\phi _p(r_\infty )=1\), we take \(b_2=0\) and

(i) \(c_1=0\):

*N*is a constant number. On the other hand, the Fourier transform of the two-point function (8), based on a CFT, read as [44]:

*p*. Comparison with (17) shows the replacement of \(p^2\rightarrow p^2+\alpha _m^2\) which indicates the deviation from conformal symmetry by the dimensionful parameter \(m^2\). In terms of the inverse Fourier transform of (17), considering only the logarithmic term, we find

^{2}

(ii) \(c_1\ne 0\):

### 2.2 The *a*-theorem

*a*-theorem in the context of the massive gravity. For a moment consider \(m=0\) and start from a conformal field theory where in 4-dimensions, it includes two central charges,

*c*and

*a*which can be derived by putting the conformal field theory on a curved background. Then the trace read as,

*Ln*subscript means the logarithmic divergent part. Now simply extremize the action with respect to \(\alpha \) and \(\beta \) and plug them back to the action, one can easily read the central charges as coefficients of \(I_4\) and \(E_4\) where they appear as

*c*or

*a*-theorem of [34] and we are going to consider it in the context of massive gravity.

*a*-theorem, the task is to find a monotonically decreasing function of the scale which matches with the central charge at the fixed point. In holography, most of

*a*-theorems are based on the null energy condition in the bulk theory [35, 36, 47]. Let us consider it for the massive gravity. We start with the following metric,

*r*limit, we assume \(A(r)=r/\tilde{L}\) and the metric becomes asymptotically

*AdS*. Suppose a theory consisted of the action (1) as the gravity part with some matter source not included there, then the generalized Einstein equation has the following form,

*G*is the Einstein tensor and

*H*,

*Z*and

*M*are respectively variations of Gauss–Bonnet, quasi-topological and massive terms.

*a*-function through,

*AdS*background where \(A(r) \sim r/\tilde{L}\), then the

*a*central charge previously introduced in (31) will be recovered which is exactly the Euler density coefficient in the trace anomaly (22). Notice that this analysis is in the presence of the mass term in the action and gives the same functional form as in the literature for massless case [34]. This follows from the fact that the null energy condition used in (35) was based on \(\xi _{(1)}^{\mu }\xi _{(1)}^{\nu }T_{\mu \nu }\ge 0\) with \(\xi _{(1)}=(e^{-A(r)},0,0,0,1)\) which has no component in non-vanishing directions of the reference metric.

## 3 Quasi-topological black brane in Gauss–Bonnet massive gravity

In this section, we study a black brane solution to the action (1). We derive its metric, temperature and the entropy, then compute the rate of entropy production due to a strain which in the massless limit corresponds to viscosity of the hydrodynamic limit of the dual theory on the boundary.

### 3.1 The black brane solution

*N*(

*r*) we have [8],

*f*is given by solution of the following equation,

*b*is a constant of motion and can be determined as a function of the radius of horizon at which \(f(r_0)=0\):

*N*(

*r*) is constant by variation of

*f*(

*r*) from (39). The speed of light in the boundary CFT is simply \(c=1\), thus we have \(\lim _{r \rightarrow \infty }{N^2 f(r)}=1\) so we take \(N=1/\sqrt{f_\infty }\).

### 3.2 The rate of entropy production

In our previous work in [23], we calculated the viscosity in the massive Gauss–Bonnet gravity by the direct application of Green–Kubo formula, of course in the special case of \(m_1=m_2=0\) while \(m_3\ne 0\). We are now going to generalize the formalism to include more general massive higher curvature theories with two derivatives equation of motions. We take the advantage of pole method of [52] which can be generalized for a massive field as we discuss shortly.

*r*and \(E_2\) is the summation of a term in order of \(\omega ^2\) and a mass term. Here we provide

*q*(

*r*) as an effective

*r*-dependent coupling which encodes the higher curvature nature of the theory. This is in the spirit of [49]. Following the procedure of [51], the final result for the viscosity is as follows,

*f*at horizon. The above equation is the same as [9] up to \(\phi _0(0)^2\) factor. This is natural, since as stated above it includes the effective coupling and does not explicitly depend on mass parameter which is implicit in

*f*’s derivatives. In [9] the \(\phi _0(0)^2\) factor is one due to translational invariance in the massless theory.

*f*derivatives in (63) from (45) after rewritten in

*z*-coordinate,

Checking this result for Einstein massive gravity we set \(\lambda =\mu =0\) in (68) and find \(\eta /s=\phi _0(0)^2/(4\pi )\). The result is the same as [37].^{3}

**Finding** \(\phi _0(0)^2\):

*z*and

Let us firstly consider a special case where \(m_1=m_2=0\) while \(m_3\ne 0\). This gives \(K_2=0\) and the only regular solution is the constant one \(\phi _0(0)=1\). This matches exactly with our previous work [23] for massive Gauss–Bonnet gravity in which we considered \(m_1=m_2=0\).

Figure 1 displays the profile of \(\phi _0(z)\) from the horizon to boundary for different values of \(m_1\) where by (46) and setting \(m_2=m_3=0\), it indeed shows different temperatures with \(m_1=-4\) corresponding to zero temperature.

## 4 Physical constraints

*c*-central charge should be positive, then

^{4}It reduces to the following equation where only terms proportional to \(q^2\), \(\omega ^2\) and their higher degrees are survived in the short wavelength limit,

*f*derivatives can be found by (43) rewritten in \(\rho =r_0/r\) coordinate. Then

## 5 Conclusion

The higher curvature gravities are important in the holographic study of conformal field theories. In contrast to Einstein gravity which duals to four dimensional CFT’s with equal central charges, higher curvature theories provide dual CFT’s with two distinguished charges. A fundamental higher derivative gravity is expected to be derived from a string theory calculation, however the Gauss–Bonnet and cubic quasi-topological higher curvature gravities may be considered as toy models with rich structure to investigate the dual quantum field theory on the boundary [7, 8, 9]. Of course, the existence of a dual theory to the Gauss–Bonnet gravity is under question [26, 27], however, we merely considered it as a toy model.

Here we studied a higher curvature massive gravity. The latter is important as generalization of the Einstein gravity and has phenomenological applications and theoretical consequences. It is known that when one considers massive gravity as a bulk theory in the context of AdS/CFT correspondence, the boundary theory violates the Lorentz invariance [17, 18]. This is related to introducing the reference metric. Here we investigated the conformal structure of the boundary theory, if any. It was shown that adding a mass term to the bulk is equivalent to existence of a massive operator on the boundary. It appears like a short range Yukawa potential in the energy–momentum two-point function. The constant coefficient of this two-point function is proportional to the *c* central charge and remains intact in the massive theory. Based on the null energy condition, we introduced a monotonically *a*-function which is a candidate for an *a*-theorem.

In the second part of this article, we worked out an exact black brane solution. We derived the temperature to be mass dependent. Then we considered the hydrodynamic limit in the dual theory and calculated the shear viscosity to the entropy density, \(\eta /s\) which is better to be interpreted as the rate of logarithm of the entropy production due to a strain. We found in (61) that it includes two factors, an effective coupling due to higher curvature terms and an extra factor \(\phi _0(r_0)^2\) from the massive gravity which we found it numerically. This later factor is less than one in the physical range of parameters which indicates that generally the graviton mass effect is reduction of the viscosity.

At the final stage, we investigated physical constraints as unitarity, ghost free and causality. The latter set condition on mass parameters \(c_1\), \(c_2\) and \(c_3\). It was found that \(c_1\le 0\), if \(c_1=0\) then \(c_2\le 0\) and so on. Of course, they are bounded from below due to non-negativity of temperature (89).

## Footnotes

## Notes

### Acknowledgements

Authors would like to thank A. Imaanpur and M. M. Sheikh-Jabbari for useful discussions.

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