Global monopole as a generator of a bulk-brane structure in Brans-Dicke bulk gravity

We investigate a braneworld model generated by a global monopole in the context of Brans-Dicke gravity. After solving the dynamical equations we study the relevant aspects of the resulting framework. Moving forward we apply the braneworld sum rules formalism to our case in trying to further generalize the setup. This study indicates the possibility of exclusively positive brane tensions in the model.


I. INTRODUCTION
In 1999 it was introduced a paradigmatic extra dimensional model of universe [1] whose geometrical/topological setup makes possible to approach the hierarchy problem. Since then a great amount of work was done in many contexts within the scope of extra dimensions. The model presented in Ref. [1] is built upon the R 4 × S 1 /Z 2 configuration and, as such, may be faced as a type of effective braneworld model of the Horava-Witten [2] investigation relating the eleven-dimensional orbifold R 10 × S 1 /Z 2 with ten-dimensional heterotic strings.
From the gravitational point of view, the branes presented in Ref. [1] are nothing but domain walls, one of them performing the large scale experienced Universe, with four non-compact dimensions. It became natural to wonder if other topological defects could be used to envisage a braneworld model with physical appealing characteristics. In this regard a branch of research was pursued by means of using different versions of cosmic strings to generate the bulk/brane structure. Global topological defects were used, leading to interesting braneworld models [3][4][5]. On the other hand, it is a well known fact that string theory encompasses a dilatonic scalar field in the low energy gravity [6]. The Brans-Dicke theory [7] appears in this context as a good laboratory to explore extra dimensional more involved scenarios. In Ref. [8] a local cosmic string generating the braneworld model was studied, while in Ref. [9] a braneworld was investigated with the aid of a global cosmic string. In both cases a six-dimensional bulk endowed with a five-dimensional brane arose and, therefore, the resulting models falls within the hybrid compactification scheme.
In the present work we pursue the idea of using another topological defect, the global monopole, engendering the bulk/brane structure. After solving analytically the Brans-Dicke dynamical equations in the Einstein's frame, we arrive at a hybrid compactification model with a seven-dimensional spacetime whose brane has four non-compact dimensions and two curled extra dimensions. We present a qualitative analysis contrasting the result obtained with its usefulness in serving as a braneworld model. The net result is that the model has an interesting setting, allowing an approach to the hierarchy problem while the size of the on-brane extra dimensions are in agreement with current experiments. The scalar field shape, however, is not free of ill defined behavior along the transverse extra dimension and this fact motivates our further study of braneworld models based on global monopole by means of the so-called braneworld sum rules. Applied to the case at hands we see that the sum rules point to the possibility of exclusively positive brane tensions constituting the model. While this last fact is shared by other models, and in this sense is not completely new, this is certainly a piece of good news, since a negative brane tension is a necessity in the paradigmatic model [1]. This paper is organized as follows. In the Section II we introduce the basic ingredients used to construct the desired framework, namely, a braneworld model based on a global monopole. This section of composed of two subsections: IIA, where we explicitly construct an analytic solution and IIB, devote to a discussion of the qualitative aspects of the obtained solution showing both the virtues and setbacks in engendering an alternative braneworld model. In the Section III we take one step further in our analysis by examining possible extensions via braneworld sum rules. Finally, in the Section IV we leave our concluding remarks.

II. TOWARDS THE MODEL
Let us assume a brane-bulk structure generated by a (p + 4)-dimensional global monopole within the Brans-Dicke gravity. The symmetry obeyed by such a configuration leads us to start from the spherically symmetric line of element below The corresponding field equations in the Einstein's frame (where the metric and the dilaton are decoupled) are and where G is the gravitational coupling constant and the parameterα is combination involving the Brans-Dicke one given byα 2 = 1/(2ω + 3). The conservation law of the energy-momentum tensor reads The energy-momentum tensor is defined in terms of the matter lagrangian as For the global monopole the corresponding lagrangian density is which exhibits the symmetry breaking of the SO(3) to U (1) groups. In (6) the parameter λ is a positive coupling constant and η is the symmetry breaking scale. The SO(3)-symmetric Higgs field φ a is an isotriplet of scalar fields which is assumed to have the form of the well-known hedgehog Ansatz, With the index a = 1, 2, 3 andx a = {sin θ cos φ, sin θ sin φ, cos θ}.
Using the expressions above we get the non-vanishing components of the energy-momentum tensor: as well as its trace The only non-trivial equation emerging from (4) is obtained for ν = 1: In order to better manage the equation above we can assume we are performing our study far from the monopole's core, where the Higgs field behaves as h ∼ 1. According to (13) this condition is achieved if e −2β → 0 outside the monopole's core. Besides, the components of the energy-momentum (9)-(11) outside the core assumes a much simpler form given by The notation used here, including particularly the last result, shall be useful to the next section. When investigating the braneworld sum rules we shall make use of another notation.

A. Solving the field equations
From (2) and (3) we obtain the following set of dynamical equations As a first attempt let us adopt a radial coordinate such that we have a type of higher dimensional version of the "harmonic coordinate" introduced in [10], which brings an enormous simplification of gravitational problems involving scalar fields. In a scalar field model without a potential this choice is characterized by a radial coordinateũ satisfying ũ = 0 in which the condition γ − α + 2β + pσ = 0 holds. Besides, we assume the condition γ = σ, necessary to construct the braneworld model we are interested in 1 . Thus, we are left with the set of field equations below The set of equations above can be analytically integrated, as we shall now see in detail. The integration of (21) is quite straightforward, providing the expression where a and b are integration constants. Next, let us examine the equations (20) and (21) which shall give us a very useful relation. Notice that the equation (23) may be rewritten as This lead us to compare it with (20) and promptly set the interesting relation below By integrating twice this equation we are left with For convenience, we can fix to zero the first integration constant appearing above. The second one, ϕ 0 , shall be necessarily non-zero in order that the limit of the General Relativity (GR), namely ϕ → const. whenα → 0, may be properly recovered. Now let us define the constant factor multiplying β(r) as so that (27) will assume a simpler form Now we are ready to address the remaining field equation. The relation above allows us to express (22) as a purely differential equation for β(r). Using (24) and (29) in (22) and then collecting like terms we arrive at The next step, which shall make easier our job, is to adopt the change of variable u ≡ ar + b. So, by rewriting (31) in terms of the new radial coordinate u, we geẗ where the dot denotes derivative with respect to u. A new change of variable can be performed, namelẏ which leads (31) to the formḟ It is easy to see that (33) can be rewritten aṡ The equation above can be put in a nicer appearance through a further change of variable, in which we define andB Therefore, Eq. (34) turns out to be Rearranging this last expression we have which is now ready to be integrated 2 . After some algebraic manipulations we find 2 Let us recall that: Now we can move backwards along our multiple redefinition of variables. Using (32) in (39) we have Next, we use (32) in (31) which giveṡ Integrating Eq. (41) and resuming the usage of the original radial coordinate, r, we are left with the final form for β(r): Let us recall that the metric function α(r) relates to γ(r) and β(r) by means of the coordinate condition we adopted, γ − α + 2β + pσ = 0, which becomes when we take into account the previously assumed condition γ = σ. Therefore, the exact solutions both for the dilaton and the metric functions, considering a generic number of extra dimensions p read: We remark that such an exact result is an entirely new in the literature. Considering a realistic braneworld as modeling our universe we are led to fix p = 3. Besides, as it is expected that the scale of symmetry breaking is quite below the Planck scale it is reasonable to assume Gη 2 ≪ 1, which makes D 2 ≈ 0. Strictly speaking, there is no need for η to respect the usual four dimensional grand unification scale, but we shall keep this order of magnitude to fix ideas. These further assumptions impliesB ≈ 22 and thus

B. Discussion
We shall present a qualitative analysis/description of the obtained result in the light of the braneworld model it may engender. First we call attention to the fact that, provided well behaved quantum fluctuations, the size of the compact dimensions on the brane is fixed by the dynamical equations [11]. This size, however, needs to respect observational constraints [12]. There is a well-known seesaw-like behavior more or less ubiquitous in higher dimensional braneworld models whose bulk/brane structure comes from a topological defect. This behavior is related to well defined extra dimensions on the brane and their impact on the hierarchy problem [13] in the hybrid compactification scheme. Usually when the compact on brane extra dimensions have a size compatible with experiments, the hierarchy problem is worsted and vice-versa.
For the case at hands, it can be readily verified, from Eqs. (50) and (49), that this seesaw behavior is reproduced here for a > 0. That is, supposing the brane surface at say r → r * , then σ(r → r * ) and β(r → r * ) are such that the seesaw is present. Notice, however, that this case cannot fulfill the requirement stressed right after Eq. (13), necessary to achieve h ∼ 1 outside the monopole's core. Nevertheless the case a < 0 (b < 0) shows that it is possible to envisage a brane surface 3 respecting |a|r * < (π/2 + |b|)/ √ 22, such that the size of the compact on brane dimensions may be keep small enough, along with a resulting warp factor which could serve, at least, to mitigate the hierarchy problem. Of course, the brane surface position must be fixed, for instance, by an internal solution and adequate junction conditions. In this case, however, at least part of the model inputs performed in Section II needs eventual reconsideration, resulting in a far from trivial problem. We shall postpone this approach to a future work.
The resulting line element for the solution found is clearly intricate and needs specific tools to make its physical content clear. In this regard we have computed all the Kretschmann scalars for the spacetime given by Eqs. (48-51) and the results indicate a spacetime completely free of singularities. This is a relevant output, since singularities appearing in gravitational solutions coming from global defects occurs more often than never. The functional form of the dilatonic field, however, indicates the necessity of a more involved setup. In fact, it may be readily verified that the dilatonic field is not everywhere well defined in its domain. It is conceivable wondering about a multiple brane scenario along with a nontrivial radial extra dimension topology in order to bypass this problem. The branes (or some of the branes) could, as usual, be placed at singular points of orbifolds, cutting of problematic radial regions. We shall leave this analysis for the next section, not by studying an specific example (whose complications may be easily underestimated), but otherwise by means of the so-called braneworld sum rules.

III. SUM RULES ANALYSIS
The problem approached in this paper pointed, in the previous section, to the necessity of a more involved setup, in order to avoid completely the ill defined regions to the dilatonic field along the radial extra dimension. However, a direct approach, putting two (or several) branes in the spacetime within a radial nontrivial topology, aiming to solve the field equations, seems to be out of order, given the complexity of this endeavor. Therefore we shall circumvent this issue by investigating consistency conditions to be respected by a braneworld model with (at least two) branes and orbifold-like radial extra dimension.
The braneworld sum rules where first proposed in Ref. [14] applied to the five-dimensional paradigmatic braneworld [1] and short after generalized to a plethora of possible scenarios [15]. The basic idea is to make use of the compact internal space (the radial extra dimension in our case) along with an ingenious manipulation of the bulk/brane geometric quantities, in order to extract physical information about a given model without properly solving the field equations. The sum rules method was previously applied in the context of Brans-Dicke gravity in Ref. [16], where it was shown that the dilatonic gravity allows for smooth branes in five dimensions and do not needs negative brane tensions as well. All the results obtained so far, in the scalar-tensor formalism, formulate the sum rules in the Jordan's frame. Our model, however, was built in the Einstein's frame and since the very existence of the branes lead to a conformal invariance breaking, there is no general argument pointing to the possibility of using here the results obtained in the Jordan's frame a priori. Therefore we shall recompute the braneworld sum rules in the Einstein's frame.
Since here we shall deal with bulk, brane and internal space in a systematic manner, the notation used in this section must differ from the used so far. Therefore, we shall fix the notation as follows: before to particularize the formulation to our case, we denote by D = p+ 4 the number of bulk dimensions, byp+ 1 the number brane dimensions and by D −p − 1 the number of dimensions of the internal space. The indices for the whole bulk shall be denoted by capital Latin letters, while for the brane dimensions we reserve Greek letters. Latin small letters stands for the internal space dimensions. Besides, while the bulk scalar of curvature is denoted by R,R denotes the brane scalar of curvature, whileR stands for its internal space counterpart. From this general perspective the line element may be denoted by Regarding the notation here used, notice that the brane coordinates may also encompass compact extra dimensions (this is indeed our case of interest) while the transverse dimensions to the brane are reserved to the internal space. A relatively simple manipulation of geometric quantities (see Ref. [15] for details) allows one to relate the scalar of curvatures and partial traces (see below) with a total derivative in the internal space as ∂ · (W χ (r)∂W (r)) = W χ+1 p(p + 1) where χ is a free parameter and the partial traces are defined as R µ µ = W −2 (r)g µν R µν and R m m = g mn R mn , such as R = R µ µ + R m m . When the specific model/gravitational theory is implemented via partial traces and an integral over the internal compact space is performed, a one-parameter (χ) family of consistency conditions arise. These conditions are the sum rules.
In the new notation Eq. (3) is given by from which the partial traces can be read. Taking into account that the Brans-Dicke field has no dynamics on the brane and recalling that g µν g µν =p + 1 we have with T µ µ (T m m ) defined in a manner akin to R µ µ (R m m ). Inserting Eqs. (55) and (56) into Eq. (53) we have Working out the right hand side of Eq. (57), by taking into account that T = T µ µ + T m m , we arrive at Now we are in position to apply the formalism to a braneworld scenario motivated by the investigations performed in the previous section. We shall first particularize the dimensions and in a second moment entering the global monopole stress tensor. The case investigated in the previous section pointed to a bulk with D = 7 (and hence p = 3) andp = 5. Therefore the brane is six-dimensional and the internal space (D −p − 1) has 1 dimension. Notice that for the codimension one caseR = 0. Therefore, denoting ∂ r ϕ = ϕ ′ , we arrive at The stress tensor partial traces occurring in Eq. (59) are given by where T (i) 5 stands for the i th −brane tension. Notice that these partial traces comes from a generic bulk stress tensor, without cosmological constant, allowing for several branes placed at r 1 , r 2 , · · · . The monopole stress tensor, denoted here by τ MN , is the usual one Hence the partial traces read where the first two terms of the partial trace survives due to the monopole hedgehog configuration. Inserting back Eqs. (62) into (60) and finally into (59) we have There are some important remarks in order to carry the calculations a little further. To achieve a more concrete braneworld scenario we can think of two branes placed at the singular points of a Z 2 orbifold. Within this case we certainly may avoid the radial problematic regions present in the last section. Moreover, this context is the simplest realization of a compact internal space for which ∂ · (W χ (r)∂W (r)) = 0, leading to the one parameter family of consistency conditions. Hence, integrating over the internal space we have Eq. (64) may be recast in a slightly more informative fashion by means of a simplification. In trying to reproduce our Universe in large scales it is pretty safe to setR ∼ 0 and, therefore, Eq. (64) reads Now the choice χ = −1 is particularly illustrative since it eliminates most warp factor contributions and the contribution coming from the potential as well. Within this choice it is fairly simple to see that the monopole Higgs field dynamics may counterbalance the Brans-Dicke sign contribution in a way that no negative brane tension is needed. This aspect is in fact relevant, since in codimension one braneworld models, the brane tension is directly proportional to the effective four-dimensional Newtonian constant [17].

IV. CONCLUSION
In this paper we provide some contributions to the study of alternative braneworld models in a twofold approach. We construct a framework where a global monopole is used to generate the bulk-brane setup, within a Brans-Dicke gravity. For such a system we derive and solve exactly the dynamical equations, obtaining a model in which a seven dimensional bulk accommodates a brane endowed with four non-compact dimensions and two (on-brane) compactified ones, obeying the hybrid compactification scheme. Next, is analyzed the viability of the obtained braneworld scenario and how useful could it be in addressing the so-called hierarchy problem. In this vein, we discussed at which extent the solution can be used in order to avoid the seesaw-like effect, a typical behavior characterizing braneworld models based on topological defects.
The full model, whose brane surface position is stabilized by the solution inside the monopole's core along with appropriate junction conditions at the brane surface (or eventually by another stabilization mechanism), is currently under investigation, but the usual approach has been shown quite non-trivial so far.
Motivated by the problems concerning the dilatonic field behavior along the radial direction we performed an analysis of possible model extensions via the braneworld sum rules. This approach reveals us that it is possible to envisage a more involved model containing, for instance, two positive tension branes placed at the singular points of a radial orbifold.