# A holographic study of the *a*-theorem and RG flow in general quadratic curvature gravity

## Abstract

We use the holographic language to show the existence of the *a*-theorem for even dimensional CFTs, dual to the AdS space in general quadratic curvature gravity. We find the Wess-Zumino action which is originated from the spontaneous breaking of the conformal symmetry in \(d\le 8\), by using a radial cut-off near the AdS boundary. We also study the RG flow and (average) null energy condition in the space of the couplings of theory. In a simple toy model, we find the regions where this holographic RG flow has a monotonic decreasing behavior.

## 1 Introduction

*c*-theorem [1, 2] states that the central charge monotonically decreases along the Renormalization Group (RG) flow. We can expect this from the Wilsonian approach in quantum field theory, in which, by integrating out the high energy modes, the number of degrees of freedom decreases. Komargodski and Schwimmer have proved a generalization of this theorem in [3, 4], after the conjecture of Cardy [5]. They prove an

*a*-theorem for four-dimensional unitary conformal field theories and show that for any RG flow between a UV and an IR fixed point \(a_{UV}\ge a_{IR}\). Here \(a_{UV}\) and \(a_{IR}\) are the coefficients of the four dimensional UV/IR conformal anomaly, which can be computed from the non-vanishing value of the trace of the energy-momentum tensor

*C*is the Weyl tensor and \(E_4\) is the Euler density (Gauss-Bonnet terms in four dimension). The computation of Weyl anomaly through the AdS/CFT correspondence first performed in [6, 7].

In the proof of [3], there is a Nambu-Goldstone boson \(\sigma \), corresponding to the spontaneously broken conformal symmetry and an effective action \(W[\sigma ]\), which is emerging by integrating out the degrees of freedom along the RG flows driven by adding the relevant operators.

In [8] the same idea is investigated in the context of the AdS/CFT. They holographically construct the \(W[\sigma ]\) and follow its changes along the RG flow. To find the effective action, they start from the gravity side by considering a bulk action together with the Gibbons-Hawking (GH) terms and counter-terms. The other ingredients are the AdS metric in the Poincare coordinate (flat boundary space) together with a radial cut-off near the AdS boundary. The later plays the role of the RG scale. By promoting *z*, the radial coordinate of the cut-off surface, to \(\sigma \) as a (spurion) field, *i.e. * \(z=e^{\sigma }\), and by computing the bulk and boundary actions, after a derivative expansion one will find a Wess-Zumino (WZ) action for the spurion field in the even dimensions. This effective action directly is related to the conformal anomaly in even dimensions as discussed in [8].

In this holographic approach, depending on which AdS throat we are dealing with, the coefficient of the effective dilaton action is equal to the value of \(a_{UV}\) or \(a_{IR}\). These AdS solutions correspond to the UV/IR fixed points of the RG flow. When one considers the contributions of both throats, the overall coefficient would be \(a_{UV}-a_{IR}\), which it has been proved in the quantum field theory side [9]. The study of the dilaton WZ effective action in even *d* dimensions and up to and including the 8-derivative terms has been performed in [10].

The generalization of the *c*-theorem to higher dimensions mostly has been done by using the gauge/gravity correspondence. For the early works including the holographic renormalization and RG flows through the relevant deformations of the Lagrangian see [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

Another direction for the generalization of the holographic *c*-theorem is the extension of the bulk Lagrangian to the higher curvature terms. The first attempts have been done in [26, 27] for quasi-topological gravities and in [28] for Lovelock and *f*(*R*) theories of gravity. Unlike the four dimensional holographic CFT dual to the AdS solution in the Einstein gravity, in quasi-topological theories \(a\ne c\). In these theories, it is possible to show that for a general RG flow there is a monotonically decreasing function *a*(*r*), assuming that the matter sector obeys the null energy condition. This function at the fixed points reproduces correct values for \(a_{UV}\) and \(a_{IR}\). With the same conditions, one cannot find a similar function for *c*(*r*), [26]. In this direction, an *a*-function for four dimensional general curvature square gravity has been found in [29]. The non-increasing behavior of this function is proved by using the Raychaudhuri equation. The non-increasing RG flow is restricted to a certain class of curvature square theories.

In [30] an a-function is introduced by using the Jacobson-Myers (JM) entropy functional. The non-increasing behavior of this function follows from the fact that the JM entropy functional satisfies the linearized second law of the causal horizon thermodynamics. This study includes the general curvature squared gravity and *f*(*R*) gravity. It also shows that in the absence of the null energy condition for certain theories which a scalar field is coupled to the gravity in AdS space, the second law would be enough condition for the monotonicity.

Further study including the Ricci polynomials in the bulk Lagrangian is presented in [31]. They show the existence of an *a*-theorem for the Ricci cubic theory by restricting the couplings of the theory. These constraints are inconsistent with the ghost-free condition of the theory, but at the level of the Riemann cubic theories, the constraints for non-increasing RG flow coincide with the ghost-free conditions. For further studies of the cubic gravities see also [32] and [33].^{1}

In this paper, we are going to study the holographic *a*-theorem for general quadratic curvature (GQC) gravity following the reference [8]. In Sect. 2, we use the perturbative method for maximally symmetric solutions to find an effective action for the GH terms. We also use the known algorithm for finding the counter-terms. These terms are sufficient to cancel the divergences of quadratic curvature gravity with dimension less than ten. By finding these terms, we are able to compute the related WZ actions in even dimensions. We also read the corresponding coefficients to find the value of the *a*-charge.

In Sect. 3 we study the holographic RG flow between the UV and IR fixed points. We use a kink solution, which we suppose it to satisfy the equations of motion in the presence of the matter field. This solution reduces to the AdS solution at both UV/IR limits. We use it to study the behavior (monotonicity) of the RG flow. We suppose a proper ansatz for the holographic RG flow with general coefficients and find the possible regions in the space of couplings where the value of this RG flow monotonically decreases. We also check the regions where the (average) null energy condition holds. In the last section, we summarize our computations and discuss the results.

## 2 Dilaton action in GQC gravity

*a*-theorem for the dual gauge theories. To construct the effective dilaton action, we begin from the following total action

*i.e.*

*L*by the following equation

### 2.1 Gibbons–Hawking surface terms

*h*is the determinant of the induced boundary metric and

*K*is the trace of the extrinsic curvature of the boundary surface, \(\partial \mathcal {M}\). The extrinsic curvature is defined by \(K_{\mu \nu }=2\nabla _{(\mu } n_{\nu )}\) and \(n_\mu \) is the space-like unit vector normal to the boundary. Moreover, there is a generalized GH action for the GB gravity [37]

In general, to find the related GH terms for the remaining Ricci curvature terms, \(\mathcal {L}^{R^2}= a _1 R^2 + a _2 R_{\mu \nu }^2\), the usual method does not work, *i.e. *by variation of the Lagrangian with respect to the metric one cannot find the suitable terms to have a well-defined variational principle. However, according to the perturbative method in [38], for a maximally symmetric solution, we can find an effective GH term.

*i.e.*\( R_{\mu \nu } = -\frac{d}{L^2}\, g_{\mu \nu }\) into the above equation. Finally, the related GH term is the GH term of the EH gravity with an effective coefficient,

*i.e.*

### 2.2 Counter-terms

In the following we are going to use the suggested algorithm in [41] to find all the proper counter-terms of the Ricci square terms of the bulk action. In our case, this approach is sufficient to remove the polynomial divergences of the on-shell action. However, one can find a systematic Hamiltonian approach to holographic renormalization in [42] which gives the complete counter-terms that cancel out all the divergences, including the logarithmic ones, see also [43].

*i.e.*\(\tilde{\Pi }^{(0)}_{ab}= \alpha h_{ab}\). Inserting this into the Eq. (2.14), one finds

### 2.3 a-theorem in GQC gravity

*a*-theorem for GQC gravity holographically. We consider the Euclidean AdS metric in the Poincare coordinate as follow

*a*-theorem for the GQC theory, then the final scalar action is a WZ action with an overall coefficient proportional to the conformal anomaly in even dimensions.

To accomplish this, we need to find the intrinsic and extrinsic curvatures constructed from the boundary metric of (2.24). All the computations related to this subsection are presented in Appendix A. The results of our calculations for various even dimensions are listed below:

#### 2.3.1 \(\mathbf d = 2 \)

#### 2.3.2 \(\mathbf d = 4 \)

#### 2.3.3 \(\mathbf d = 6 \)

#### 2.3.4 \(\mathbf d\,=\,8 \)

^{2}There are nine Weyl invariant terms in 8 dimensions, in other words the dilaton action can be written as [10]

*a*-anomaly coefficients reduce to the GB results in [8] when \(a_1=a_2=0\). Our results for \(a^*_d\) confirm the general

*d*dimensional relation suggested by [27]. For the GQC action in the Euclidean background the

*a*-anomaly can be simply evaluated by computing the value of the bulk action on the AdS space

## 3 Holographic RG flow in GQC gravity

In the previous section, we established a holographic *a*-theorem for the GQC gravity by finding the WZ action. In this section, we are going to study the holographic renormalization group (RG) flow of this theory in the presence of a matter field. This RG flow is a function of the radial coordinate (RG scale) and the couplings of the theory ,*i.e. * \(a=a(r; a_1,a_2,a_3)\). We are interested in those functions, which are decreasing monotonically as we decrease the RG scale and are stationary at the UV/IR fixed points. The values of this function at these fixed points are given by \(a^*_d\), the coefficients of the WZ action that we found in the previous section for even *d* dimensions. The *a*-theorem ensures that for any RG flow which connects the UV fixed point to the IR fixed point, \(a_{UV}\ge a_{IR}\).

*A*(

*r*), varies from \(a_{UV}\) to \(a_{IR}\). By using the above geometry, the equations of motion in the presence of the matter field can be written as

### 3.1 An ansatz for RG flow

*a*-anomaly is proportional to the value of the bulk Lagrangian computed on the AdS space-time [27]. Away from the fixed points, we expect that the value of the RG flow as a function of the energy, or holographically, the value of the

*a*-anomaly as a function of the radial coordinate

*r*, is given by a function \(a(r)=a(A(r),A'(r),A''(r),\ldots )\). We define the following function as an ansatz for the holographic RG flow

^{3}

*a*(

*r*) and \(a'(r)\) consequently. If we restrict ourselves to the above RG flow (ignoring the other possible terms in ...), after the differentiation with respect to

*r*, the \(a'(r)\) has the same order of the derivatives as the equations of motion (3.3a) and (3.3b). As a result, the monotonicity of

*a*(

*r*) depends on the behavior of \(A'(r)\), \(A''(r)\) and the energy-momentum components of the matter field, since we can get rid of \(A^{(3)}(r)\) and \(A^{(4)}(r)\) by using the equations of motion.

*i.e.*\(A(r)=r/\tilde{L}\), into the (3.5), we must achieve the values of \(a^*\) in the UV/IR regions in Eq. (2.39)

*UV*and

*IR*fixed points. Since \( {{a}^{*}_{UV}}\ge \ {{a}^{*}_{IR}}\), by a simple algebraic analysis one may show the following restrictions on the value of \(\kappa _2\) These conditions only depend on the asymptotic behavior of the solutions of the equations of motion.

### 3.2 The NEC and ANEC

*t*,

*r*) direction, then the null energy condition can be written as

*A*(

*r*).

*A*(

*r*) or \(A''(r)\) but this dependence appears as a positive coefficient in (3.10) and therefore, ANEC has a universal behavior for all the possible solutions of the equations of motion.

### 3.3 A toy model

*A*(

*r*) and matter fields. Here in the presence of the higher curvature terms, we consider such a super-potential exists (or even simpler, we can just consider a massless scalar field with a kinetic term) and the equations of motion (3.3a) and (3.3b) support the above solution.

*B*, so we suppose that \(B<0\). The ansatz of (3.11) asymptotically admits the AdS solution \(A(r)= r/\tilde{L}_\infty \) so that

In the following subsections, we will need to know the behavior of the *a*(*r*). Specifically, we are interested in its monotonic behavior. As a result, one should examine the behavior of the \(a'(r)\) under the various values of \(\zeta _i\) coefficients in (3.5). This provides various paths in the RG flow depending on the choice of \(\zeta _i\) coefficients, although all paths asymptotically have the same behavior at \(r\rightarrow \pm \infty \) where \(a'(r)\rightarrow 0\).

#### 3.3.1 Regions of NEC and ANEC

*A*, therefore the expression in the parenthesis above, which is a quadratic polynomial of

*x*, must be positive/negative everywhere in the interval of \(0<x<+\infty \) when \(A>0\) or \(A<0\).

By a simple numerical analysis, we can find the regions in \((\kappa _1,\kappa _2)\) space where the null energy condition holds, see Fig. 2. In this figure the upper wedge corresponds to the \(A>0\) and the lower wedge to the \(A<0\). The size of wedges depends on the values of *A*, *B* and \(\tilde{L}\). We have also considered (3.7a) and (3.7b) conditions in the drawing of the Fig. 2.

*A*the ANEC exists above the line of \(d \lambda ^2 \kappa _1 +4 A (\lambda ^2+3\lambda +3)\kappa _2-6A(d-1)\tilde{L}^2=0\) in \((\kappa _1,\kappa _2)\) space and for the negative

*A*, below that line (see Fig. 2 for a specific choice of parameters).

#### 3.3.2 A monotonically decreasing RG flow

As it was mentioned in the introduction, we expect that the RG flow monotonically decreases because in the Wilsonian approach by integrating out the high energy modes, the number of degrees of freedom decreases. This means that we must look for the RG flows that \(a'(r)\ge 0\). Generally, it is hard to find a set of specific values for \(\zeta _i\) coefficients in (3.5) or a condition on the matter field, such as the (A)NEC, to prove \(a'(r)\ge 0\). To investigate the behavior of the RG flow we study two examples by fixing the free parameters \(\zeta _i\) in the ansatz (3.5). After that, we discuss about a more general case.

#### 3.3.3 Example 1

*i.e.*\(\zeta _0=\zeta _1=\zeta _2=\zeta _3=0\) then

*A*, the overall coefficient is positive/negative therefore the expressions inside the parenthesis must be positive/negative everywhere in \(x>0\). A simple analysis of the quadratic polynomials shows that, to have a monotonically decreasing RG flow for every value of \(\kappa _1\), it requires the \(\kappa _2\) takes the following values

*a*(

*r*) and \(a'(r)\) is sketched in Fig. 3.

#### 3.3.4 Example 2

*a*-function, one possible choice is \(\kappa _1=0\), together with a matter field that satisfies the NEC, \(\Delta \mathcal {T}=\kappa ^2({T^{r}}_r-{T^{t}}_t)>0\), (this coincides with the example 1 for \(\kappa _1=0\)). The pure gravitational part of the theory restricts to a specific type of the quadratic curvature Lagrangian

*A*, the coefficient of \(\mathcal {F}(x)\) is a positive/negative function. Moreover this coefficient asymptotically goes to zero on both UV \((x\rightarrow +\infty )\) and IR \((x\rightarrow 0)\) sides and it has just one extremum point. The \(\mathcal {F}(x)\) itself is a fourth order polynomial of

*x*, therefore if we demand a monotonically decreasing function of

*a*(

*r*), we must find conditions which \(\mathcal {F}(x)\) is negative/positive for all the values of \(x>0\). It means that this function should not have any root in this interval.

*a*,

*b*,

*c*and

*d*regions in Fig. 4

#### 3.3.5 A general analysis

*a*(

*r*) in Eq. (3.5). By a differentiation with respect to

*r*we have

*x*. For simplicity let’s suppose that the \(\zeta _i\) coefficients are linear combinations of the \(\kappa _1\) and \(\kappa _2\),

*i.e.*\(\zeta _i=a_i \kappa _1+b_i \kappa _2\) for \(i=0, 1, 2, 3\). To follow the effect of each term in (3.5) individually, we separate each \(\zeta _i\) by setting all the other \(\zeta \)’s equal to zero, see the Fig. 5a–h

In all the left hand side Fig. 5a, c, e, g we have fixed all \(b_i=0\). The lower wedges belong to \(A<0\) and the upper ones are for \(A>0\). On the other hand in all the right hand side Fig. 5b, d, f, h we have fixed all \(a_i=0\). These strips (half plane) belong to \(A<0\), in fact, \(A>0\) has not any allowed region. For all cases, we have drawn the regions of validity of ANEC. Again these behaviors are general in every dimension and independent of the values of *A*, *B* and \(\tilde{L}\).

If all \(a_i=0\) then the allowed region would be a strip (half plane) in \((\kappa _1,\kappa _2)\) plane. The width of this strip depends on the choice of the coefficients as well as the values of

*A*,*B*and \(\tilde{L}\). This case is similar to example 1.If at least one of the \(a_i\ne 0\) then there will be two wedge-like regions in the \((\kappa _1,\kappa _2)\) plane similar to the example 2.

## 4 Summary and discussion

This paper is divided into two main parts, we first holographically show the existence of the *a*-theorem for even dimensional conformal field theories which are dual to the AdS space in general quadratic curvature gravity. In the second part, we discuss on the holographic RG flow between two CFT’s at the UV and IR fixed points.

In Sect. 2 we generalize the method in reference [8], which in the context of the gauge/gravity correspondence we find the effective dilaton action corresponding to the spontaneously broken conformal symmetry in even dimensions. At the first step, we need the GH terms and counter-terms corresponding to the bulk action of (2.3).

The GH terms for Einstein–Hilbert and Gauss–Bonnet terms are known but with the standard method of variation, one cannot find a proper GH term for general quadratic curvature terms. We do this by computing the effective GH term on a maximally symmetric AdS space [38]. The final result (2.8) is a GH term for EH action but with an effective coefficient. The total GH surface terms are the sum of the GH terms in (2.6) and (2.8).

The counter-terms for general quadratic curvature gravity already are computed by various approaches up to the quadratic boundary curvatures [38]. We use the algorithm in [41] to compute these counter-terms up to the cubic curvature terms which are needed to study the conformal field theories with dimensions \(d\le 8\). The total counter-terms are the sum of (2.9), (2.19) and (2.20a)–(2.20c).

After finding all the necessary Lagrangians, following to [8], we introduce a radial cut-off as a scalar function of the boundary variables. By using the induced metric on this cut-off surface and computing the bulk and surface terms we find the WZ action of the dilaton field in \(d=2, 4, 6, 8\). In all of these dimensions, the coefficient of the WZ action is the value of *a*-anomaly and agrees with the known relation of (2.39) for \(a^*_d\) in term of the bulk action computed on the AdS background. Moreover, in \(d=8\) as well as the WZ terms we can find some non-vanishing Weyl invariant terms which already are introduced in [10].

The existence of this WZ action holographically shows that the *a*-theorem exists for conformal field theory dual to the AdS space in GQC gravity in even *d* dimensions.

In Sect. 3, we study the holographic renormalization group flow in GQC gravity in the presence of a matter field. We try to find those RG flows that monotonically are decreasing as we decrease the RG scale and are stationary at the UV/IR fixed points. The value of the RG flow at these fixed points is given by \(a^*_d\) that we found in Sect. 2. The *a*-theorem ensures that for any RG flow which connects the UV fixed point to the IR fixed point, \(a_{UV}\ge a_{IR}\).

In this section we use the ansatz of (3.5) for RG flow which is constructed from the warped factor of the kink solution (3.2). This kink solution is interpolating between the two AdS solutions in the UV/IR fixed points. The *a*-theorem makes restrictions (3.7a) and (3.7b) on the value of the couplings.

To study the RG flow we need to know the exact form of the kink solution from equations of motion. In the presence of a matter field, this is not a simple job, instead, we use the toy model of (3.11) which has all the properties we need. Meanwhile, since we are studying the gravity in the presence of a matter field, it is important to check the regions of the validity of the (average) null energy condition. We have presented a numerical sample of our results in Fig. 2. Our numerical analysis shows that the NEC in this toy model allows not all the possible values of the couplings. On the other hand, the ANEC as a weaker condition provides a wider region. We expect that by imposing the ANEC the dual quantum field theories do not suffer from the negative energy fluctuations as proved by [45].

Finally, we have studied the RG flow (3.5) in two examples by fixing the free parameters in (3.5). We observe a general behavior for the allowed region where the monotonically decreasing RG flow exists. The numerical results are summarized in Figs. 4 and 5. We show that the regions of monotonically decreasing RG flow may or may not have overlap with the regions where the null energy condition holds, for example, see Fig. 4.

The analysis of Fig. 5 suggests that, if we demand the ANEC together with a monotonically decreasing RG flow then the unknown \(\zeta _i\) coefficients in the RG flow of (3.5) must be just a function of \(\kappa _1\) and not \(\kappa _2\). Therefore we believe that example 2 is a good description for the RG flow and changing the numerical coefficients of \(\zeta _i\) coefficients does not alter the whole picture.

## Footnotes

- 1.
- 2.
To evaluate the 8-dimensional total action we would need the proper GB counter-terms, which in principle its calculation is possible due to earlier proposed methods,

*e.g.*[39]. - 3.
Such functionality comes into the mind by thinking to the Wilsonian approach in quantum field theory. It is possible that the RG flow, similar to its fixed points, is proportional or related to the effective Lagrangian, which is computed on the background solution. This may contain higher curvature counter-terms for the gravitational field or the counter-terms for the matter field.

## Notes

### Acknowledgements

A. G. would like to thanks A. Sinha for reading the manuscript and drawing our attention to ANEC subject. We would also like to thanks M. M. Sheikh-Jabbari for his valuable comments on the draft. This work is supported by Ferdowsi University of Mashhad under the Grant 3/47298 (1397/05/16).

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