Warped Brane worlds in Critical Gravity

We investigate the brane models in arbitrary dimensional critical gravity presented in [Phys. Rev. Lett. 106, 181302 (2011)]. For the model of the thin branes with codimension one, the Gibbons-Hawking surface term and the junction conditions are derived, with which the analytical solutions for the flat, AdS, and dS branes are obtained at the critical point of the critical gravity. It is found that all these branes are embedded in an AdS$_{n}$ spacetime, but, in general, the effective cosmological constant $\Lambda$ of the AdS$_{n}$ spacetime is not equal to the naked one $\Lambda_0$ in the critical gravity, which can be positive, zero, and negative. Another interesting result is that the brane tension can also be positive, zero, or negative, depending on the symmetry of the thin brane and the values of the parameters of the theory, which is very different from the case in general relativity. It is shown that the mass hierarchy problem can be solved in the braneworld model in the higher-derivative critical gravity. We also study the thick brane model and find analytical and numerical solutions of the flat, AdS, and dS branes. It is find that some branes will have inner structure when some parameters of the theory are larger than their critical values, which may result in resonant KK modes for some bulk matter fields. The flat branes with positive energy density and AdS branes with negative energy density are embedded in an $n$-dimensional AdS spacetime, while the dS branes with positive energy density are embedded in an $n$-dimensional Minkowski one.


Introduction
The idea that the spacetime has more than four dimensions and our universe is a brane or domain wall embedded in higher dimensional spacetime [1,2,3,4,5,6] has been proposed for a long time and discussed extendedly. It is believed that the braneworld scenario can supply new insights for solving the gauge hierarchy problem and the cosmological constant problem [7,8,9,10,11,12,13,14,15,16,17].
There are many discussions about branes both in the frames of general gravity and modified gravities. In Refs. [18,19,20,21,22,23], non-minimal coupling branes in scalartensor gravity were discussed and the mass hierarchy problem can be solved in scalar-tensor thin branes model [18]. A brane model in the Recently presented EiBI gravity theory were constructed in Ref. [24] and it was found that the four-dimensional Einstein gravity can be recovered on the brane at low energy. Branes in spacetime with torsion were investigated in Ref [25,26] and it was shown that in the f (T ) gravity that the torsion of spacetime can effect the inner structure of branes [25]. Reference [27] investigated braneworld teleparallel gravity. For brane models in higher derivative gravity there are also many references, see for examples Refs. [28,29,30,31,32,33,34,35,36].
In this paper, we are interested in brane solutions in the framework of higher derivative gravities. As is known, general relativity is a non-renormalizable theory and it is suffered the singularity problem as well as other problems. In a quantum gravity theory or the low energy effective theory of string theory, higher-order curvature terms would be added to the Einstein-Hilbert action. The principal candidates for such corrections are contracted quadratic products of the Riemann curvature tensor. With this kind of corrections, the most general correction terms has the form of α ′ R 2 + β ′ R M N R M N + γ ′ R M N P Q R M N P Q , or αR 2 + βR M N R M N + γL GB , where is the Gauss-Bonnet term and it is topological invariant in four dimensions. Nevertheless, an action with quadratic curvature terms implies that the field equations contain the fourth derivations of the metric and thus would lead to massive ghost-like graviton. Recently, it is shown in Ref. [37] that the massive scalar mode can be eliminated and the massive ghost-like graviton becomes massless when the parameters of the quadratic curvature terms satisfies the critical condition. The corresponding theory is called critical gravity. It was generalized to higher dimensions in Ref. [38]. Therefore, it is interesting to reconsider brane scenario in critical gravity. Minkowski branes in five-dimensional critical gravity were investigated in Ref. [39]. It was found that at the critical point the equations of motion are of second order. The Gibbons-Hawking surface term was investigated and the junction conditions were obtained. With the junction conditions the analytic flat brane solutions were obtained. It as found that scalar perturbations for all these brane solutions are stable.
In this paper, we generalize the work of Ref. [39] and construct the flat and warped brane solutions in n-dimensional critical gravity. The organization of this paper is as follows. In Sec. II, we investigate the thin branes and construct analytic flat brane solutions with the junction conditions in critical gravity. In Sec. III, thick branes generated by a scalar field are investigated and the conditions of the splitting of the branes are obtained. Finally, our conclusion is given in Sec. IV.

Thin brane solutions
First we consider the thin brane model in the frame of a n-dimensional critical gravity. The action is [38] where and κ denotes the n-dimensional gravitational constant with κ 2 = 8π/M 4 * , where M * is the n-dimensional Planck mass scale. The parameters α and β satisfy the following critical condition [38] 4(n − 1)α + nβ = 0. (2. 3) The brane part S B of the above action is given by where V 0 is the brane tension and q µν is the induced metric on the brane, which is assumed located at the origin of the extra dimension x n = y. The capitals letters M, N, · · · = 0, 1, 2, 3, · · · , n − 2, n and the Greek letters µ, ν, · · · = 0, 1, 2, · · · , n − 2 denote the indices of the n-dimensional bulk and (n − 1)-dimensional brane, respectively. The equations of motion (EoMs) derived from the action (2.1) are read as In n-dimensional space, we have the following relation and C 2 := C M N P Q C M N P Q is the square of the n-dimensional Weyl tensor, Note that, under the critical condition (2.3), the last term R 2 in the right hand side of Eq. (2.10) vanishes. So, the Lagrangian density L G for the critical gravity can be reexpressed as where L EGB is the Einstein-Gauss-Bonnet (EGB) term: L GB , (2.14) In the following, we first generalize the result of the junction conditions in five dimensions in Ref. [39] to n dimensions for flat, AdS, and dS thin branes. Then we will use the generalized junction conditions to give the thin brane solutions.

Junction conditions
Following Ref. [39], we adopt the Gibbons-Hawking method to derive the junction conditions. The basic idea is as follows. The whole spacetime M is divided into two submanifolds by the thin brane, which is the boundary ∂M of the two submanifolds. The unit vector normal to the boundary ∂M is denoted by n Q and it is outward pointing. Then the induced metric on the brane is q M N = g M N − n M n N . The extrinsic curvature is defined as K M N = L n q M N /2. We denote [F ] ± := F (0+) − F (0−). In the following, we let n Q (0 + ) = n Q := (0, 0, 0, 0, −1) and n Q (0 − ) := (0, 0, 0, 0, +1) for the right and left sides, respectively. Due to the Z 2 symmetry of the extra dimension, we only need to calculate the right side. See e.g. Refs. [40,41,39] for the details.
We will deal with the C 2 term and EGB term, respectively. We first consider a general geometry instead of the special case of branes with ds 2 = e 2A(y)ĝ µν (x)dx µ dx ν + dy 2 . For the C 2 term, we have Here, the bulk term has been omitted and only the relevant boundary term is given explicitly. In order to have a well-posed variational principal, we introduce an auxiliary field ϕ M N P Q and replace C 2 with 2ϕ M N P Q C M N P Q − ϕ M N P Q ϕ M N P Q . Then from the EoM of the auxiliary field, ϕ M N P Q = C M N P Q , we can see that ϕ M N P Q has the same symmetry as the Weyl tensor and is also totally traceless. With the new field ϕ M N P Q , Eq. (2.15) becomes Then with the identity [39] Further, the first term in the above equation can be reduced to where ϕ M N := ϕ M P N Q n Q n P . So the surface term for the C 2 part is  [42] for the C 2 term So we have (considering the whole spacetime) Next, we come to the EGB term in Eq. (2.14), for which the Gibbons-Hawking surface term was given in Refs. [43,44,45]: withG µν =R µν − q µνR /2 the Einstein tensor of the induced metric q µν and J the trace of the following tensor: Then we have Thus, from Eqs. (2.25) and (2.29), for n-dimensional critical gravity theory, we finally get So, the junction conditions are Here T M N (brane) denotes the singular part of T M N . To avoid the δ-function in the junction conditions, we need the stronger condition [ϕ M N ] ± = 0.
For the special warped geometries of flat, AdS, and dS branes, whose metrics have the form the first condition (2.33) gives no more constraint for brane solutions becauseK M N ≡ 0; and C M P N Q is continuous and its contribution vanishes. So the above junction conditions for flat, AdS, and dS brane solutions in the critical gravity are simplified as where the nonvanishing components the brane energy-momentum tensor are T brane µν = −V 0ĝµν . The reduced metric is q µν =ĝ µν (x)e 2A(y) . With the constraint A(0) = 0 and the assumption of the Z 2 symmetry of the extra dimension (A(y) = A(−y)), we have Next, we mainly consider the branes with maximum symmetry, namely, flat (Minkowski), AdS, and dS branes. With the explicit junction conditions, we will give the thin brane solutions. The flat brane solutions in five-dimensional critical gravity has been found in Ref. [39].

Flat brane
The line-element of a flat brane with the most general (n − 1)-dimensional Poincaréinvariant is where e 2A(y) is the warp factor. Such a compactification is known as a warped compactification. Considering the Z 2 symmetry of the brane model, we have A(y) = A(−y). Furthermore, we can set e 2A(0) = 1 in order to get q µν = η µν on the brane. The bulk energy-momentum tensor reads from which the brane energy-momentum tensor is given by For arbitrary α and β, the field equations (2.5) are forth-order differential ones. However, at the critical point α = − n 4(n−1) β [38], the bulk field equations turn out to be where ζ is given by Eq. (2.11), and the prime and double prime stands for the first-order and second-order derivations with respect to y, respectively. Throughout this paper we will use the critical condition (2.3). The junction conditions read due to the Z 2 symmetry of the extra dimension. The solution for the warp factor A(y) is where k is a positive parameter since we are interested in the exponentially decreasing warp factor, which could solve the hierarchy problem if we consider the two-brane model with an S 1 /Z 2 extra dimension [7]. Then, from Eq. (2.42), the naked cosmological constant is given by The brane tension is determined by the junction condition (2.44): It is clearly that the result is consistent with the one in general relativity when ζ = 0. The naked cosmological constant and brane tension are respectively negative and positive when ζ = 0 (in this paper, we assume that n ≥ 5), and can be positive, zero, and negative when ζ < 0, depending on the magnitude of ζ compared with k −2 . If we require that the higher order terms in (2.2) are small compared with the R term, which implies ζk 2 ≪ 1, then we will have negative Λ 0 and positive brane tension for any such ζ. If we rewrite the Einstein equations (2.5) as G M N = κ 2 T eff M N , namely, and identify (E M N − γ 2 H M N )/κ 2 + T M N as an effective energy-momentum tensor, then we will always get an effective positive brane tension.
The flat thin brane is embedded in an AdS n spacetime, with the effective cosmological constant Λ given by Λ = −(n − 1)k 2 . Therefore, the higher order terms only effect the naked cosmological constant and brane tension.
For the case n = 5, the result reads which is the thin brane solution found in Ref. [39].

AdS brane
The metric describing an AdS brane embedded in an AdS n spacetime is assumed as The corresponding Einstein equations beyond the thin brane turn out to be The junction conditions read Then the relation between these two cosmological constants is from which one sees that Λ 0 = Λ only for vanishing ζ, and the naked cosmological constant can be positive, zero, and negative, depending on the value of the combine of the parameters β and γ (i.e., ζ = −4γ(n − 3) + β(n − 2)). The junction conditions (2.53) give the brane tension: which can be positive, negative, or zero.

dS brane
The metric describing a dS brane has the following form: The EoMs at y = 0 are The junction condition is similar with the case of AdS brane: The solution is The naked cosmological constant and other parameters are related by The cosmological constant of the AdS n is also Λ = −(n − 1)k 2 and the relation between Λ and Λ 0 is (2.57).
The junction condition gives the relation between the brane tension and other parameters: Just as the case of AdS brane, the brane tension here can also be positive, negative, or zero.

Effective action and mass hierarchy
To derive the effective action of gravity on the brane, we follow the procedure in Ref. [7]. The n-th dimension coordinate ranges from −y b to y b with the topological of S 1 /Z 2 , brane I locates at y = 0, and brane II locates at y = y b . We consider the massless gravitational fluctuations of the background metric (2.38):

69)
M pl is the (n − 1)-dimensional Plank scale on the brane, M is the n-dimensional Plank scale satisfying 2M n−2 = 1 2κ 2 , and terms likeĝ andR are constructed byĝ µν (x). Note that in the action (2.2) the terms like R 2 and R M N R M N et al. are considered as higherorder terms comparing with the R term. This is equivalent to αR 2 , βR M N R M N , γL GB ≪R, which imply βk 2 , γk 2 ≪1. So we have |1 − a| ≪ 1. The action of the n-dimensional gravity is reduced to where the (n − 1)-dimensional effective action is From this, we can see that the relationship between the fundamental scale and the effective one reduces to the case in RS1 model because a . = 1. Hence, the n-dimensional critical gravity reduces to the (n − 1)-dimensional critical gravity on the brane. Substituting the metric (2.67) into the junction condition (2.37), we obtain the brane tensions of the two branes: Let us consider a Higgs field on the brane II with the action (for the case n = 5) where v 0 is the vacuum expectation value of the Higgs field. Redefining the fieldĤ = e −ky b H, we obtain the canonical normalized action of the Higgs fieldĤ: From the above expression, we see that the redshift of the the vacuum expectation value of the Higgs field and the mass of the particles are the same with the RS1 model in Ref. [7]. So, the mass hierarchy problem is also solved in the higher-order braneworld model in the critical gravity.

Thick branes generated by a scalar field
In this section we study thick branes generated by a scalar field. The brane part of the action (2.1) is where the scalar field is assumed as φ = φ(y) for flat, AdS, and dS branes considered below.

Flat brane
The line-element of a flat brane generated by a scalar field is also assumed as (2.38). The EoMs (2.5) reduce to the following second-order coupled equations: where V φ ≡ dV /dφ. Note that the above three equations are not independent. To solve these equations, we introduce the superpotential function W (φ), which is defined as: , we obtain Ref. [39]: where

The case ζ = 0
For the case ζ = 0, we have c 1 = c 2 = 0. In order to support a kink solution for the scalar filed, we first use the superpotential W = kv 0 φ − φ 3 3v 2 0 , for which the potential is The naked cosmological constant Λ 0 = − 4(n−1)k 2 v 4 0 κ 4 9(n−2) 2 is negative. Substituting the superpotential into Eqs. Also we can take another superpotential W = kφ 2 0 sin(φ/φ 0 ). The solution is Note that the potential here is the Sine-Gordon potential and the scalar has a single kinklike configuration. The naked cosmological constant Λ 0 is also negative.

The case ζ = 0
In the last subsection we cannot get a physical brane solution for a usual φ 4 potential for vanishing ζ with the superpotential method. However, for the case ζ = 0, we can consider the usual φ 4 potential by setting W = aφ. The potential turns to and the corresponding cosmological constant is Here, we consider the case of n > 4 and require v 2 0 > 0 and b > 0, this leads to ζ < 0 and Λ 0 < 0. The solutions of the scalar field and the warped factor are φ(y) = v 0 tanh(ky), (3.18) If a trigonometric superpotential is used, the scalar field can be a single kink, double kink, or even multi-kink, and there can be various kinds of structure of the brane. The warped factor A(y) and the scalar field are related with the extra dimension y by For the case W (φ) = qφ 0 sin φ φ 0 , we find that the parameters ζ and q can affect the structure of the brane. To see this, we first plot the scalar potential V (φ) in Figs. 1(a) and  1(b), which show the influence of the parameters ζ and q, respectively. We can see that as ζ and q get larger, a fake vacuum of the scalar potential will emerge, which is different from the case in general relativity, i.e. ζ = 0. So we can expect that the scalar field has a double kink solution, which is shown in Figs. 2(a) and 3(a); and the brane is a double brane which can be seen from the energy density ρ(y) = T M N U M U N = −T 0 0 = 1 2 φ ′2 + V in Figs. 2(b) and 3(b).
The corresponding cosmological constant is The condition that the single brane split into a double brane is i.e., (3.25) From Fig. 3(b), it can be seen that with the increase of the parameter ζ, the brane become fatter. When ζ reaches the critical value ζ c1 , there will be a wide platform around the brane location. When ζ > ζ c1 , there will be a minimum for the energy density at the center of the brane and two sub-branes appear. Such brane with inner structure may support resonant KK modes for various bulk matter fields.

AdS thick Brane
Now we consider the AdS thick brane, for which the line-element is also assumed as (2.50) and the EoMs read as

The case ζ = 0
For the case ζ = 0, we consider the Sine-Gordon potential Then we get the following solution e 2A(y) = cosh 2 (ky), (3.30) The cosmology constant is which is negative. Therefore, the AdS thick brane is embedded in an asymptotic AdS spacetime. It is interesting to note that the single kink scalar connects the adjacent locations of the extrema of the scalar potential, and the energy density is negative. This is very different from the case of flat branes.

The case ζ = 0
For the case ζ < 0, we find a solution for a φ 4 model: The energy density ρ(y) of the system is From the above expression of the energy density we can see that the brane tension of the AdS brane is negative, which is different from the cases of flat and dS branes. The solution of the scalar field is a double kink. At the boundaries of the extra dimension, the scalar field φ → ±φ 0 , which are locations of the extrema of the scalar potential, but not the locations of the minima. This is the reason why the energy density is negative. The shape of the scalar field, the potential and the energy density are shown in Figs. 4(a)-4(c). The cosmology constant is which yields ρ ≤ 0, ρ ′′ | y=0 > 0. (3.45) Therefore, the brane is a single brane with negative tension. It is interesting to note that, the scalar is a double kink when ζ = − 2 (n−4)H 2 . This double-kind structure could contribute to the resonant structure of bulk fermions. The cosmology constant is

dS thick brane
In order to simplify the EoMs in the case of dS brane, we introduce the conformal coordinates. The line-element is assumed as Then the EoMs are turned out to be where the prime denotes the derivative with respect to z.

The case ζ = 0
For the case ζ = 0, the EoMs reduce to the ones in general relativity. The solution for n = 5 has been given in Ref. [46,47,48] in general relativity. For arbitrary n, we consider the following potential where the parameter p satisfies 0 < p < 1. The solution of the warped factor and scalar field is The corresponding naked cosmology constant is Since R M N (z → ∞) → 0, the effective cosmological constant is also zero. So the dS thick brane is embedded in an n-dimensional Minkowski spacetime. The energy density is given by  We can see that as H and ζ get larger, the scalar field turns to a double kink (see Figs. 6(a) and 7(a)); and the brane splits into two sub-branes, which can be seen from the energy density ρ in Figs. 6(b) and 7(b). This is different from the case of ζ = 0.
The condition that the single brane splits into the double brane is (3.58)

Conclusion
In this paper, we generalized the Minkowski brane models in five-dimensional critical gravity in Ref. [39] to warped ones in n dimensions. For thin brane models in arbitrary dimensional critical gravity theory, the Gibbons-Hawking surface term and the junction conditions were derived. It was found that for the special case of flat, AdS, and dS thin branes the C 2 term in the action has no contribution to these junction conditions. The solutions for both thin and thick branes were obtained at the critical point α = − n 4(n−1) β. We found that the combination of the parameters β and γ in the action (2.2), i.e., ζ ≡ β(n − 2) − 4γ(n − 3), has nontrivial effect on the brane solutions. All the flat, AdS, and dS thin branes are embedded in an AdS n spacetime, and the effective cosmological  constant Λ of the AdS n spacetime equals the naked one Λ 0 only when the combine coefficient ζ = 0. The naked cosmological constant and brane tension can be positive, zero, and negative, depending on the value of ζ. Following the procedure in Ref. [7], we reduce the n-dimensional critical gravity to the (n − 1)-dimensional critical gravity on the brane, and the mass hierarchy problem was also solved in the higher-order braneworld model in the critical gravity. For the thick flat branes, when ζ = 0, we got two analytical solutions, both of which describe a single brane generated by a kink-like scalar. When ζ = 0, the analytical and numerical solutions were obtained. It was found that the brane will split into a double brane when one of the parameters ζ and k is larger than its critical value. Such brane with inner structure may support resonant KK modes for various bulk matter fields. All these flat branes are embedded in an n-dimensional AdS spacetime.
For the thick AdS branes, the scalar connects the adjacent locations of the extrema of the scalar potential, and the energy density is negative. This is very different from the cases of flat and dS branes. The scalar can have single or double kink configuration, but the brane has no inner structure. These AdS branes are also embedded in an n-dimensional AdS spacetime.
For the thick dS branes, when ζ = 0 and the scalar potential is taken as the Sine-Gordon one, the brane has positive energy density but has no inner structure. When ζ = 0, the inner structure of the dS brane will appear when the parameter ζ or H is larger than its critical value. The energy density of the brane system is positive for any ζ. These dS branes are embedded in an n-dimensional Minkowski spacetime.