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Symmetric and asymmetric thick brane structures

This work deals with the presence of thick branes in a model with two source scalar fields that interact with one another in a very specific way. The model is new, capable of generating kinklike configurations that engender important modifications in the energy density of the system, inducing the presence of symmetric and asymmetric structures inside the brane.

1. Introduction. The braneworld scenario first proposed in [1] in the form of a thin brane, was soon generalized to describe thick branes in the presence of scalar fields; see, e.g., Refs. [2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein. In the thick brane scenario the models deal with a fivedimensional AdS warped geometry with a single extra dimension of infinite extent in the presence of real scalar fields as source fields to generate the brane. For a review see Ref. [15].
In the thick brane scenario with a single extra dimension, the presence of source scalar fields that support kinklike configurations capable of generating robust braneworld solutions has increased the interest in the study of localized structures in the real line. In this sense, in the present work we consider a model described by two real scalar fields which are coupled in a very specific way, as recently suggested in [16]. The model is different from the Bloch brane model investigated before in Ref. [10], where the scalar fields were described under the presence of standard kinematics. In the present context, in the work [16] the coupling between the two fields modified the kinematics of one of the two fields, and this allowed the presence of novel kinklike configurations, engendering interesting internal structure. In the present work, we want to understand how the new solutions may contribute to change the braneworld configuration.
To make the investigation pedagogical, we first deal in Section 2 with the braneworld scenario on general grounds, considering the two source scalar fields to be driven by the Lagrange density suggested in [16]. We then move on and in Section 3 we investigate four distinct models, in particular two models that support kinklike solutions which unveil the presence of branes with multiple internal structures, in which the warp factor and energy density depend on the extra dimension symmetrically or asymmetrically. In the asymmetric case, the brane may connect asymptotic AdS 5 and Minkowski geometries, or two distinct AdS 5 geometries. Interestingly, these possibilities only depend on a single real parameter that controls the asymmetry of the model. We end the work in Section 4, where we summarise the main results and add comments capable of fostering new investigation into the subject.
2. New braneworld scenario. Let us now investigate scalar fields in the braneworld scenario with a single extra dimension of infinite extent, described by the line element where e 2A is the warp factor, A = A(y) is the warp function, y is the extra dimension and η µν = diag(+, −, −, −) describes the four-dimensional Minkowski metric tensor, with µ, ν = 0, 1, 2, 3. The five-dimensional metric In the present study, we deal with the action in the form where we are using natural units and 4πG 5 = 1, for simplicity. The Lagrangian density describes two real scalar fields and is written as Here, φ and χ denote the scalar fields and f (χ) is in principle an arbitrary function. Notice that the kinematical term related to φ presents a factor controlled by the χ field. A similar Lagrangian density was recently considered in Ref. [16] in the study of novel kinklike configurations, that arise in (1, 1) spacetime dimensions, where the χ field simulates, through the function f (χ), geometrical constrictions that modifies the usual kink profile. The variation of the previous action with respect to the scalar fields and to the metric tensor leads us to the equations where V φ = ∂V /∂φ, V χ = ∂V /∂χ and f χ = ∂f /∂χ. In Eq. (4c), G ab is the Einstein tensor and T ab represents the energy-momentum tensor. We follow the usual route and consider that the scalar fields are all static, depending only on the extra dimension. In this case, Eqs. (4a) and (4b) become where the prime represents derivative with respect to y, i.e., φ = dφ/dy and χ = dχ/dy. Also, the non-vanishing components of Einstein's equations (4c) are where A = dA/dy and A = d 2 A/dy 2 .
As it is now standard in the study of braneworlds, here it is of great interest to find first order equations that solve the above Eqs. (5) and (6). We follow the lines of Refs. [3,17] and implement a first order formalism, which arise for potentials with the form where W = W (φ, χ) is in principle an arbitrary function of scalar fields. In this case, we obtain the following first order differential equations and which are compatible with the scalar field equations of motion (5) and the Einstein's equations (6). Notice that, in principle, since W presents terms that mix φ and χ, the equations (8) must be solved together as a system. However, a special case arises when W (φ, χ) = g(φ) + h(χ), such that the first order equations in (8) become φ = g φ /f (χ) and χ = h χ . In this situation, the latter equation can be solved independently and the χ field acts as a source for the field φ since it feeds the function f (χ). It was shown in Ref. [16], in models with two scalar fields in (1, 1) dimensions, that a similar mechanism may be able to simulate geometrical constrictions on kinks, which may give rise to kinks with multiple internal structures, and here we want to investigate this possibility in the braneworld context, in the case of warped five-dimensional geometries with a single extra dimension of infinite extent. In this model, the energy density can be calculated standardly; it is given by With the use of the first order equations, it can be rewritten in the simple form The energy of this system is found by integration of the above expression with respect to the extra dimension, so we get that the energy of the brane associated with the scalar fields is zero, for models described by W (φ, χ) that behave adequately asymptotically, since the warp factor exp(2A) has to vanish asymptotically, to ensure finiteness of the energy of the braneworld configuration.

Stability
Before going on to investigate specific models, the us now focus on the stability of the brane associated to the model described by Eqs. (2) and (3). We deal with the stability of the gravity sector, considering small perturbations in the metric associated to the line element in Eq. (1), in the form with as a very small parameter and h µν = h µν (x µ , y). We use transverse traceless gauge, h µν →h µν , to show that the metric fluctuation decouples from the fields [3], leading to where = η µν ∂ µ ∂ ν and ∂ y = ∂/∂y. Introducing the z coordinate to make the metric become conformally flat, with the choice dz = e −A(y) dy, and defininḡ h µν = e ip·x e −3A(z)/2 H µν (z), the above equation becomes where U (z) represents the stability potential, given by Here A z and A zz correspond to the first and second derivative of the warp function with respect to the z variable. We remark that the stability equation (14) is a Schrödinger-like equation, and the stability is ensured when Eq. (14) only admits non negative eigenvalues, i.e., p 2 ≥ 0. One can show that Eq. (14) can be written in the form with the operators S and S † given by The above factorization shows that the stability equation cannot support states with negative eigenvalues. Hence, the brane is stable under small fluctuations of the metric.
3. Specific Models. To see how the general procedure works for building thick brane solutions, let us now illustrate it with several models, which we describe below.

Model 1
The first model is described by the function and the choice We notice that the contribution of the χ field in the above W (φ, χ) is the very same of the well-known χ 4 potential in the flat spacetime scenario. On the other hand, the φ terms in the above W , combined with the f (χ), brings novel configurations to light. The potential is given by Eq. (7); we do not write it here because the expression is awkward. For the above function, the first order equations for the scalar fields (8) become One first solves the equation in the right and then substitute the solution in the equation on the left. By doing so, one finds that the solutions are φ(y) = 3 tanh(y) and χ(y) = tanh(y), which can be seen in the two top panels in Fig. 1. We notice that the φ solution presents infinite derivative at the origin, which is a consequence of the f (χ) function fed by the χ solution. Thus, our method allows us to modify the derivative of the φ solution that plays an important role in the energy density.
In order to calculate the warp function, we use Eq. (9), which reads From the above equation, we can see that A (0) = 0 and A (0) → −∞. This makes the warp function to develop a sharp peak at the origin, a phenomenon which is similar to the behavior of the thin brane scenario proposed in Ref. [1]. In this sense, the above model somehow simulates a thin brane, even though we are considering topological structures that appear from the two field model used to generate the brane. We were able to solve the above equation and calculate the warp function analytically with the condition A(0) = 0. However, since the full expression is very long, we omit it here. In the two bottom panels of Fig. 1 we depict the warp factor and the energy density of the model. Notice that, as we stated before, at y = 0, the warp factor shows a sharp peak, as in a thin brane, and the smooth behavior of a thick brane outside the origin. This hybrid behavior is the opposite of the one found in Ref. [18], where the brane behaves as a thick brane inside a compact space around the origin and as a thin brane outside the compact space. Moreover, the total energy vanishes, as expected.

Model 2
The next model we investigate is described by where α is a non-negative real parameter. This model is motivated by the one found Ref. [16], which simulates the geometrical constrictions that give rise to double kink profiles in the magnetization of specific magnetic materials [19]. For the function given above, the first order equations (8) become Notice that the parameter α controls the slope of the χfield solution. These equations support kinklike solutions given by φ(y) = tanh(ξ(y)) and χ(y) = tanh(αy), with ξ(y) = y − tanh(αy)/α. The profile of each solution is depicted in the two top panels of Fig. 2. Notice that, near the origin, the φ kink shows a plateau whose width is controlled by α. This occurs because, for y ≈ 0, we have φ ∝ α 2 y 3 ; thus, as α decreases, the plateaux gets wider.
The warp function is driven by Eq. (9), which now reads We have not been able to find the warp function analytically for this model. So, we perform the calculations numerically for some values of α. As before, we take A(0) = 0. In the two bottom panels of Fig. 2 we display the warp factor and the energy density of the model. Notice that the warp factor gets wider as α decreases. In the energy density, we see that α gives rise to an internal structure, which appears for α ≤ α crit ≈ 1/2. We recall that the presence of internal structure in branes was studied previously in Ref. [11], but here we included a parameter that can be used to control the internal structure of the brane. As we can see, as α gets smaller, the internal structure gets larger.

Model 3
The third model is described by the same W (φ, χ) of the previous model, in Eq. (23a), but here we choose another f (χ), of the form We notice that the above function is periodic, since the cosine vanishes when its argument is equal to kπ/2, with k being an odd integer. As shown in Ref. [16], this feature creates several plateaux in the kinklike solution, controlled by n. Here, we investigate its effect on the brane. For this model, the first order equations for the fields are given as in Eq. (8), which leads us to φ = (1 − φ 2 ) cos 2 (nπχ) and χ = α(1 − χ 2 ). (28) As in the previous model, the first order equation for χ is solved by χ(y) = tanh(αy), which can be seen in the second top panel in Fig. 3 for some values of α. By using this solution to feed the above first order equation for φ, we obtain the solution φ(y) = tanh (η(y)), (29a) where Ci(x) is the cosine integral function and ξ ± n (y) = 2nπ(1 ± tanh(αy)). The above solution is depicted for n = 2 and α = 1/2 and 1/4 in the top panel of Fig. 3. Notice that we now have a kinklike structure with 2n plateaux.
To calculate the warp function, one must use Eq. (9), which has the form Unfortunately, we have not been able to find the analytical solution for A(y) in this model. We then perform the calculations numerically with the condition A(0) = 0. In the two bottom panels of Fig. 3 we depict the warp factor and the energy density, respectively, for n = 2 and α = 1/2 and 1/4. Notice that, even though the warp function is not significantly modified by the plateaux in the solutions, one can see the energy density now gets interesting internal structures. Furthermore, there is an odd number of maxima in the energy density for finite y; they are controlled by n, counted as 2n − 1.

Model 4
Let us now consider the model with the same f (χ) defined in Eq. (27), but now changing Eq. (23a) to the new form where k is a real parameter. The addition of k is inspired in the recent work [20], which develops the possibility of inducing asymmetric behavior to the brane; see, e.g., Refs. [21][22][23] for other related issues. In the present work, we want to investigate how the parameter k works in the new model. We first notice that the constant k does not change the equations of motion for both φ and χ, so they behave as in the previous model. However, the warp function A = A(y) feels the presence of k through the equation tanh(η(y)) − tanh 3 (η(y)) 3 We solve this equation numerically: for n = 2 and α = 1/2, and for k = 0.1 and 0.2 the results are shown in Fig. 4. They clearly show that k induces an asymmetric behavior to the brane, with the asymmetry being controlled by k. The bigger the k, the larger the asymmetry. Another issue related to the addition of k concerns the presence of asymmetric five-dimensional cosmological constants, as explored before in [20]. In the present case, asymptotically we can define where + and − are used to identify the asymptotic behavior as the extra dimension y tends to ±∞. We consider the case α = 1/2 to write with k constrained to vary in the interval [-1,1] to avoid breaking the brane model robustness [20]. In the case of k = ±1, the brane lives inside asymptotically AdS 5 and Minkowski geometries. Also, for |k| ∈ (0, 1), the brane connects two distinct AdS 5 geometries.

Conclusion.
In this work we have studied the modelling of branes in warped five-dimensional geometries by two scalar fields coupled as in the Lagrange density in Eq. (3). We have calculated the equations of motion that describe our system and, also, we have shown that the energy of the brane is null. Since the equations of motion are of second order, to simplify the problem, we have developed a first order formalism. In this framework, the potential is required to have a specific form, in which the coupling between the scalar fields appear. The first order formalism shows that the brane is stable against small fluctuations of the metric in the gravity sector of the brane.
In this context, we have explored a special case of the first order equations, in which one field may be used as a source for the first order equation that describes the other field. We have then studied four distinct models, the first one engendering a surprising feature: the warp factor presents a sharp peak at the origin, which remind us of the thin brane behavior, and a smooth behavior outside the origin, as a usual thick brane. This induces a divergence in the energy density, at y = 0, but this divergence is integrable and the total energy is still zero. In the second model, we have taken specific scalar field configurations that engender a parameter which is useful to tune an internal structure to the brane. In the third model, we have used the novel kinklike configuration described before in Ref. [16] to create a brane with multiple internal structures, in which a parameter adjusts their distances to the center of the brane. Finally, in the fourth model we added a constant, which introduced an asymmetry to the brane, in addition to the multiple internal structure already shown in the third model. The asymmetry also changed the cosmological constant, giving rise to different scenarios, with distinct asymptotic geometries.
An interesting perspective could be the addition of fermions and gauge fields, to see how the models are able to localize those fields inside the brane; see, e.g., Refs. [24][25][26][27] and references therein. Another issue concerns the use of the models studied in the present work to describe specific issues, such as black hole formation [28] and modified models in the presence of Lagrange multiplier [29]. Another line of research related to modifications of the gravity sector of the braneworld model, concerns changing R to f (R), or to the Born-Infeld or Gauss-Bonnet or other generalized possibilities, as the ones described for instance in Refs. [30][31][32][33][34]. We are now investigating some possibilities, hoping to report on them in the near future.