A Note on Asymmetric Thick Branes

In this work we study asymmetric thick braneworld scenarios, generated after adding a constant to the superpotential associated to the scalar field. We study in particular models with odd and even polynomial superpotentials, and we show that asymmetric brane can be generated irrespective of the potential being symmetric or asymmetric. We study in addition the nonpolynomial sine-Gordon-like model, also constructed with the inclusion of a constant in the standard superpotential, and we investigate gravitational stability of the asymmetric brane. The results suggest robustness of the new braneworld scenarios and add further possibilities for the construction of asymmetric branes.


I. INTRODUCTION
The concept of braneworld with a single extra dimension of infinite extent [1] plays a fundamental role in high energy physics and cosmology. In such braneworld scenario, the particles of the standard model are confined to the brane, while the graviton can propagate in the whole space [1][2][3][4]; see also [5] for further details.
Most braneworld scenarios assume a Z 2 -symmetric brane, as motivated by the Horava-Witten model [6]. Notwithstanding, more general models that are not directly derived from M-theory, are obtained by relaxing the mirror symmetry across the brane [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Asymmetric branes are usually considered in the literature as a framework to models where the Z 2 symmetry is not required. The parameters in the bulk, such as the gravitational and cosmological constants, can differ on either sides of the brane. The term coined "asymmetric" branes means also that the gravitational parameters of the theory can differ on either side of the brane. In some cases, the Planck scale on the brane differs on either side of the domain wall [25,26]. One of the prominent features regarding asymmetric braneworld models is that they present some self-accelerating solutions without the necessity to consider an induced gravity term in the brane action [19].
Braneworld models without mirror symmetry have been investigated in further aspects. For instance, different black hole masses were obtained on the two sides on the brane [7], as well as different cosmological constants on the two sides [9][10][11][12][13][14]. In addition, one side of the brane can be unstable, and the other one stable [21,22]. If the Z 2 symmetry is taken apart, it is also possible to get in-frared modifications of gravity, comprehensively considered in the asymmetric models in [25] and in the hybrid asymmetric DGP model in Ref. [16] as well. In both models, the Planck mass and the cosmological constant are different at each side of the brane. In the asymmetric model, there is a hierarchy between the Planck masses and the AdS curvature scales on each side of the brane [20].
Asymmetric braneworlds can be further described by thick domain walls in a geometry that asymptotically is led to different cosmological constants, being de Sitter (dS) in one side and anti de Sitter (AdS) in the other one, along the perpendicular direction to the wall. The asymmetry regarding the braneworld arises as the scalar field interpolates between two non-degenerate minima of a scalar potential without Z 2 symmetry. Asymmetric static double-braneworlds with two different walls were also considered in this context, embedded in a AdS 5 bulk. Furthermore, an independent derivation of the Friedmann equation was presented in the simplest for an AdS 5 bulk, with different cosmological constants on the two sides of the brane [24]. Finally, a braneworld that acts as a domain wall between two different fivedimensional bulks was considered, as a solution to Gauss-Bonnet gravity [9,19]. Models with moving branes have been further considered [27] in order to break the reflection symmetry of the brane model [7,10,12,13,[28][29][30].
In the current work, we shall further study asymmetric branes, focusing on general features, which we believe can be used to better qualify the braneworld scenario as symmetric or asymmetric. The key issue is to add a constant to the superpotential that describe the scalar field, which affects all the braneworld results. In particular, one notes that it explicitly alters the quantum potential, that responds for stability, evincing and unraveling prominent physical features, as the asymmetric localization of the graviton zero mode. We analyze all the above mentioned features of asymmetric branes in the context of superpotentials containing odd and even power in the scalar field, up to the third and fourth order power, respectively, and the sine-Gordon model.
The investigation starts with a brief revision of the equations that govern the braneworld scenario under investigation, getting to the first-order framework, where first-order differential equations solve the Einstein and field equations if the potential has a very specific form. We deal with asymmetric branes, in the scenario where the brane is generated from a kink of two distinct and well-known models, with the superpotentials being odd and even, respectively. Subsequently, the sine-Gordon model is also studied and the associated linear stability is investigated. The quantum potential and the graviton asymmetric zero mode are explicitly constructed.

II. THE FRAMEWORK
We start with a five-dimensional action in which gravity is coupled to a scalar field in the form where and a = 0, 1, ..., 4. We are using 4πG = 1, with field and space and time variables dimensionless, for simplicity. By denoting V φ = dV /dφ, the Einstein equations and the Euler-Lagrange equation are G ab = T ab and ∇ a ∇ a φ + V φ = 0, respectively. We write the metric as where A = A(y) describes the warp function and only depends on the extra dimension y. Taking the scalar field with same dependence, we obtain where prime stands for derivative with respect to the extra dimension. As one knows, solutions to the firstorder equations also solve the equations (4), if the potential has the following form where W = W (φ) is a function of the scalar field φ. This result shows that if one adds a constant to W , it will modify the potential, and consequently, all the results that follow from the model. To better explore this idea, we study three distinct models, which generalize previous investigations.
A. The case of an odd superpotential Let us introduce the function where c is a real constant. This W is an odd fucntion for c = 0. The potential given by Eq. (6) has now the form It has the Z 2 symmetry only when c = 0; the symmetry is now φ → −φ and c → −c. There are minima at The maxima depend on c and are solutions of the algebraic equation 4φ 3 max − 39φ max − 2c = 0. See Fig. 1. We focus attention on the scalar field. There is a topological sector connecting the minima. In this sector, there are two solutions (kink and antikink), which can be mapped to one another when φ → −φ or y → −y. Thus, we study solely the kinklike solution. The first order equation for the field does not depend on the parameter c, and is provided by φ ′ = 1 − φ 2 . The solution for this equation is with φ(±∞) = φ ± = ±1. It obeys φ(y) = −φ(−y), and it connects symmetric minima. The warp function can be obtained by using Eq. (5b) A(y) = − 1 9 tanh 2 (y) + 4 9 ln (sech (y)) − c 3 y , with A(0) = 0. We note that the behavior of this function outside the brane can be written as and asymptotically the five-dimensional cosmological constant is  The energy density is given by and it also depends on c ρ(y) = e 2A c 2 − 16 81 + 4 27 sech 4 (y) + 31 81 sech 6 (y) − 4c 27 tanh(y)(2 + sech 2 (y)).
It is symmetric only for c = 0. When c = 0, there exist contributions to the asymmetry from both the warp factor e 2A and the Lagrange density L as well. In Fig. 3, we depict the profile for Brane I (solid curve), II (dashed curve) and III (dot-dashed curve). Therefore, for c = 0, the warp factor and the energy density are asymmetric. This model shows that although the kinklike solution connects symmetric minima, the potential is asymmetric and gives rise to braneworld scenario which is also asymmetric, unless c = 0.

B. The case of an even superpotential
Let us now investigate a different model, which presents kinklike solution that connects asymmetric minima. It is defined by This W is an even function of φ. The potential given by Eq. (6) is now It is depicted in Fig. 4, and it has the Z 2 symmetry for any value of the parameter c. The minima are φ − = −1, φ 0 = 0 and φ + = 1, with There are two topological sectors. The first connects the minima φ − and φ 0 , while the second connects the minima φ 0 and φ + . Note that these sectors can be mapped by taking the transformation φ → −φ. In each sector, there are two solutions (kink and antikink); they can be mapped into each other with the coordinate transformation y → −y. Thus, we only study one of these solutions. Using the first-order equation, φ ′ = φ 1 − φ 2 , we find the solution which connects asymmetric minima.
Here the warp function is given by and presents the asymptotic values The bulk cosmological constant is provided by For c < −1/2 and c > 0, the warp factor diverges. We obtain the asymmetric branes, separating two bulk spaces In Figs. 5 and 6 we depict the warp factor and the energy density, for some values of c. This model is different from the previous one. The potential is always symmetric, and the kinklike solution connects asymmetric minima. The model gives rise to warp factor that is always asymmetric, although at c = −1/4 it connects same AdS 5 bulk spaces.

III. STABILITY
An issue of interest concerns stability of the gravity sector of the braneworld model. For the models studied in the previous section, such investigations can be implemented numerically [31], an issue to be described in the longer work under preparation [32]. Here, however, to gain confidence on the stability of the suggested braneworld scenarios, we develop the analytical procedure: we get inspiration from the first work in Ref. [5], where it is shown that the sine-Gordon model is good model for analytical investigation. Thus, we consider the sine-Gordon-type model with with |c| ≤ √ 6. We note that for φ small, the model is similar to the case of an odd superpotential studied previously; thus, the current investigation engenders results that are also valid for the model investigated in Sec. II A. The point here is that we want to study stability analytically, so the sine-Gordon model is the appropriate model.
To study stability, we follow Ref. [31]. We work in the tranverse traceless gauge, to decouple gravity form the scalar field. Also, we have to change from y to the conformal coordinate z = z(y) to get to a Schroedingerlike equation with a quantum mechanical potential. This investigation cannot be done analytically anymore, unless we take c = 0. Thus, we resort on the approximation, taking c very small and expanding the results up to first-order in c. In this case we can write the conformal coordinate as We note that f (y) is an even function, while the c-term contribution is odd. Therefore z(y) is an asymmetric function. The inverse is After some algebraic calculations, we could find the associated potential where U c (z) is the contribution up to first-order in c. The sine-Gordon-type model is nice, and we could find the explicit results: the conformal coordinate z = e −A(y) dy in (24a) is computed as z(y) = sinh(y) + c 3 (y sinh(y) − cosh(y)) .
Since c is small, the above expression can be inverted to give It follows from Eq.(26) that the quantum potential is written as It is depicted in Fig. 7. The maxima of the potential U (z) are provided by  In the limit z → ∞, the asymptotic value of the potential reads