# Regular string-like braneworlds

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## Abstract

In this work, we propose a new class of smooth thick string-like braneworld in six dimensions. The brane exhibits a varying brane-tension and an *AdS* asymptotic behavior. The brane-core geometry is parametrized by the Bulk cosmological constant, the brane width and by a geometrical deformation parameter. The source satisfies the dominant energy condition for the undeformed solution and has an exotic asymptotic regime for the deformed solution. This scenario provides a normalized massless Kaluza–Klein mode for the scalar, gravitational and gauge sectors. The near-brane geometry allows massive resonant modes at the brane for the *s* state and nearby the brane for \(l=1\).

## 1 Introduction

The braneworld paradigm brought new geometrical solutions for some of the most intriguing problems in physics, as the hierarchy problem [1, 2] and the origin of the dark energy [3] and the dark matter [4]. The warped geometry allows the bulk fields to propagate into an infinite extra dimension [5] and provides rich internal structure for the brane [6]. In five dimensions, the brane can be realized as a four dimensional domain wall, whose topological features guarantees the brane stability [7, 8, 9].

In the codimension-2, the vortex scenarios are a suitable source for an axisymmetric brane, known as a string-like braneworld [10, 11]. By adding one more extra dimension to the Bulk spacetime, the correction to the Newtonian gravitational potential turns out to be smaller than in 5D [12]. Moreover, in the thin string-like brane limit, no fine-tunning between the Bulk cosmological constant and the brane tension is needed [12]. The conical behaviour nearby the brane also provides a mechanism for the brane cosmological problem [13].

However, a global vortex geometry exhibits a mild singularity [14] and a regular local Abelian 3-brane which satisfies the dominant energy condition [15] and a baby skyrmion brane [16] which was only accomplished numerically. In the thin string-like brane limit, a vacuum \(AdS_6\) traps the massless modes of the bosonic and fermionic fields and provides a smaller correction to the gravitational potential [12, 17, 19, 20]. By considering a vanishing Bulk cosmological constant, a braneworld with a supersymmetric scalar fields and a cigar shape was found [21]. Some brane-core geometries were proposed considering more involved transverse spaces, such as the soliton cigar [22], the resolved conifold [23], the catenoid [24], an apple shapped manifold [25], the torus [26] and other.

In this article, we present a new class of smooth thick string-like model and explore some of its physical and geometrical features. The brane-tension varies inside the brane-core and attains the *AdS* regime asymptotically. The inclusion of a deformation parameter enables a continuous flow from the thin string into a thick brane with a core structure. The deformation parameter modifies the core properties, such as the behaviour of the stress-energy components and the variation of the curvature inside the core. Among the thick string-like solutions, we analyzed the properties of two models: the first solution has a bell-shaped source satisfying the dominant energy and the second, whose source exhibits an exotic asymptotic behaviour. For the dynamics of bulk bosonic fields in these thick brane scenarios, the Kaluza–Klein modes reveal interesting near and far brane characteristics.

This work is organized as follows. In Sect. 2 we present the smooth and deformed solutions and study their source and geometry properties. In Sect. 3, the Kaluza–Klein modes for the scalar, vector gauge and gravitational fields are studied and their features discussed. In Sect. 4, final remarks and perspectives are outlined.

## 2 Smooth string-like geometries

*r*. For a 3-brane at the origin, we impose the condition [10, 11, 12, 14, 15, 17, 18, 19, 20]

*A*. For \(A=B=-c=\sqrt{2\kappa _6(-\varLambda )/5}\), we obtain an \(AdS_6\) vacuum solution describing a thin string-like braneworld [12]. For \(\varLambda <0\), the most general vacuum solution is \(A(r)=c\tanh (\lambda r +c_1)\), where \(\lambda =\sqrt{-5\kappa _6 \varLambda /8}\) and \(c_1\) is an integration constant. Nonetheless, this solution does not vanish asymptotically and then, it does not provide a localized gravitational massless mode.

*A*(

*r*) extending the thin string-like solution in the form

*c*controls the asymptotic value of the warp function,

*p*modifies its variation inside the brane core and \(\lambda \) determines the brane width. Note that for \(p=0\) or for \(p\ne 0\) and far from the origin the warp function ansatz (9) reduces to the thin string-like solution. Therefore, the ansatz (9) represents a varying brane-tension solution. Integrating Eq. (9) we obtain the warp factor \(\sigma \) in terms of the hypergeometric function as

*A*has the angular pressure

*A*and

*B*. Let us consider the ansatz for

*B*in the form

Since we seek for a regular geometry converging asymptotically to the \(AdS_6\) spacetime and satisfying the regularity condition at the origin, let us adopt two models.

### 2.1 Warped disk

### 2.2 Exotic string-brane

*p*. For \(p=1\), the curvature smoothly goes to an asymptotic \(AdS_6\) spacetime whereas for \(p=10\) there is a \(AdS_6\) plateau near the origin. In this scenario the bulk-brane relation mass is \(M_{4}^2=\frac{4\pi }{(3c\lambda -2)\lambda ^2}M_{6}^4,\) which increases as the source width \(\lambda \) decreases.

## 3 Bosonic fields

In this section we study the effects of the warped disk and exotic string-like models have upon the gravitational, scalar and vector gauge fields.

### 3.1 Gravity perturbations

*m*is the KK mass satisfying \(\Box _4\hat{h}_{\mu \nu }=-m^2\hat{h}_{\mu \nu }\)

**and**\(\beta =\gamma /\sigma \). For a factorizable Bulk, i.e., \(\sigma =1\) and \(\gamma =r^2\), the gravitational KK solutions are \(\chi _{g}=c_1 J_{l}(mr)+C_2 Y_{l}(mr)\). Thus, for a noncompact extra dimension the massive KK modes form a tower of non-normalized states. For a compact transverse space, the asymptotic behaviour of the massive spectrum is \(m_n \approx n\pi /R\). The warped scenarios bring a strong dependence of the cosmological constant and the brane source upon the KK modes, as we show in next sections.

### 3.2 Scalar field

### 3.3 Vector field

### 3.4 Massless modes

*AdS*regime similar to the thin string-like scenarios [12]. The divergence near the origin stems from the change to the conformal coordinate.

### 3.5 Massive modes

*l*at the origin. Then, for \(l=0\), \(y_{m,0,q}(r)\rightarrow A_{m,l,q}\) as \(r\rightarrow 0\), whereas for \(l\ne 0\), \(y_{m,l,q}(r)\rightarrow 0\). For \(r<< \sqrt{2/qc\lambda }\) the KK modes are described by \(y_{m,l,q}(r)=A_{m,l,q}J_{l}(mr)\). Hence, only the s-wave \(l=0\) state is allowed on the brane. Asymptotically, Eq. (28) becomes the thin string-like equation \(y_{m,l,q}^{''}-\frac{qc}{2}y_{m,l,q}^{'}+m^2 \frac{e^{cr}}{2^{c/\lambda }}y_{m,l,q}=0, \) whose solution is [12]:

The potential exhibits an infinite well at the origin for \(l=0\) and an infinite barrier for \(l\ne 0\). Then, only the \(l=0\) states are allowed at the brane, as previously discussed. For \(l=1\), it turns out that the potential has a finite well displayed from the origin, where resonant massive KK gravitons can be found.

## 4 Conclusions and perspectives

We proposed a new class of smooth thick string-like model with an \(AdS_6\) asymptotic regime. A localized and bell-shaped source satisfying the dominant energy condition was found, where the properties of the source and the geometry are dependent on the ratio between the cosmological constant and the brane width. A richer internal brane structure can be introduced by means of a varying brane-tension, which modifies how the curvature and the energy density vary inside the brane core. Amidst the thick string-like branes found, a bell-shaped source satisfying the dominant energy condition shares a great resemblance with the numerical Abelian vortex brane [15].

The scalar, gravitational and vector gauge sectors were also analysed. They showed similar features, as normalizable massless Kaluza–Klein modes and an attractive potential for the massive KK tower at the origin for the \(l=0\) states. For \(l\ne 0\), the infinite barrier at the origin avoids the detection of these modes at the brane. Nevertheless, for \(l=1\), we found a potential well besides the origin, where massive resonant states could be detected.

As perspectives, we point out the use of numerical analysis to deduce these geometrical solutions from a Lagrangian model, such as a deformed Abelian vortex [15]. For the KK spectrum and its phenomenological consequences, as the correction to the Newtonian and Coulomb potentials, numerical methods should also be carried out. We expect a strong influence of the brane width parameter and the bulk cosmological constant on the KK spectrum. The behavior of the massless mode and divergence the Schrödinger potential near the brane suggests the inclusion of an interaction term between the fields and the brane, as performed in the DGP models [29].

## Notes

### Acknowledgements

The authors would like to thank the Fundação Cearense de apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP), the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for financial support. R. V. Maluf and C. A. S. Almeida thank CNPq grant \(\hbox {n}^{\underline{\mathrm{o}}}\) 305678/2015-9 and 308638/2015-8 for supporting this project.

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