# A smoothed string-like braneworld in six dimensions

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## Abstract

We propose a static and axisymmetric braneworld in six dimensions as a string-like model extension. For a subtle warp function, this scenario provides near brane corrections. By varying the bulk cosmological constant, we obtain a source which passes through different phases. The solution is defined both for the interior and for the exterior of the string and satisfies all the energy conditions. A smoothed gravitational massless mode is localized on the brane, of which the core is displaced from the origin. In contrast to the thin-string model, the massive solutions have a high amplitude near the brane. Furthermore, by means of an analog quantum potential analysis, we show that only s-wave gravitational Kaluza–Klein modes are permissible.

## Keywords

Massive Mode Warp Factor Warp Function Massive Graviton Massless Mode## 1 Introduction

In the last decade, the extra dimension models turned out to be a cornerstone of high energy physics [1, 2]. In particular, the Randall–Sundrum (RS) model brought the incredible idea of infinity extra dimensions through a warped geometry [2]. Some authors enhanced the RS model results to six dimensions. Since the two-dimensional transverse manifold has its own geometry, it leads to some nonexistent features in the RS models. Indeed, for an axisymmetric brane, the so-called string-like defect, the brane tension is related with the conical deficit angle of the transverse space [3, 4, 5, 6, 7]. Further, the Kaluza–Klein (KK) modes produce a smaller correction to the Newtonian potential [4]. The gauge bosons can be trapped to the brane by means of only the gravitational interaction [8]. Moreover, it is possible to localize the vector and spinor fields in the same geometry [9].

In spite of the aforementioned results, the string-like models exhibit some issues about their sources. In fact, these branes can be realized as a stable solution of a gauge and scalar fields [6, 7]. Nonetheless, the solution for the Einstein field equations for these models are still lacking. Amongst the string-like models, one which the width of the string vanishes and it is possible to concern with only the vacuum solution is the Gherghetta–Shaposhnikov (GS) model [4]. However, the thin-string-like models do not satisfy the dominant energy condition [10]. In order to overcome it, some authors solved numerically the equations for an Abelian vortex in six dimensions which exhibits a smooth geometry and satisfies all the energy conditions [11]. In the supersymmetric approach, a realistic and smooth cigar solution was also found by numerical means [12].

Another issue of the thin-string-like models, is that all the regularity conditions are not satisfied at the origin [10]. However, some authors added a conical behavior near the origin and studied its consequences [13, 14, 15, 16], whereas others tried to smooth this conical behavior [16, 17, 18, 19, 20, 21, 22, 23, 24].

In this article we explore some features of a smooth extension of the GS model built analytically. Besides, using a smoothed warp function, the metric satisfies all the regularity conditions. We analyze the geometrical and physical properties of this model which yields an interior and exterior string-like solution, allowing near brane corrections to the GS model. The brane core is displaced from the origin. By studying the gravitational modes, we find a smoothed localized zero mode peaked around the shifted brane core. By means of a numerical analysis, we find a smooth near brane correction to the massive KK modes. Further, the quantum analog potential possesses an infinite well around the origin which barrier is strongly dependent on the bulk cosmological constant value.

This work is organized as follows: in Sect. 2, we review the main characteristics of the string-like branes and we present our extension. In Sect. 3, we study the properties of the source from the Einstein equations. Section 4 is devoted to the study of the massless and massive gravitational modes upon the geometry. Finally, in Sect. 5, some final remarks and perspectives are outlined.

## 2 Smoothed string-like geometry

In this section, we review the main properties of the string-like branes and propose another complete solution.

In the Poincaré patch, the asymptotic point, also known as the \(AdS\) horizon, is located at \(w=0\). The decreasing behavior of the warp factor yields an asymptotic conical behavior. Ponton and Poppitz [5] proposed that it is possible to smooth out this horizon feature by embedding the GS model in the AdS–CFT correspondence. However, it is worthwhile to notice that, since the Ricci scalar is everywhere constant, the singularity at the horizon is not an essential singularity.

The metric factors (10) can be regarded as an exterior solution of the string-like brane of width \(\epsilon \). For \(\epsilon \rightarrow 0\), the brane is infinitely thin. An awkward property of the thin-string-like branes is that they do not satisfy the dominant energy condition, which turns this model a fairly exotic scenario [10]. In order to overcome these issues, some authors derived numerically the geometry from an Abelian vortex [11] and in a supersymmetric approach [12]. Here, we propose an extension of the thin-string-like solution and we investigate the modifications on the geometrical and physical properties of this scenario.

## 3 Sources properties

In this section, we study the components of the energy-momentum tensor as well as the value of the cosmological constant.

For large \(\rho \), the components of the energy-momentum tensor vanish and the vacuum solution of the Einstein equation yields the relation between \(c\) and \(\Lambda \) given by Eq. (13). This equation determines the bulk to be asymptotically an \(AdS_{6}\) spacetime. Besides, it also means that, by varying \(c\), we can study the changes in the source and fields for different bulk cosmological constant values.

It is important to note that the near brane corrections, provided by the ansatz (19) and (20), lead to a displacement of the core of the source from the origin. Similar results have been obtained by Giovannini et al. [11] numerically for higher winding number Abelian vortices and for the string-cigar model [36]. Moreover, the brane width defined as \(\epsilon \approx \bar{\rho } - \rho _{\text {max}}\), where \(\bar{\rho }\) and \(\rho _{\text {max}}\) are the positions of the half-maximum and the maximum of \(t_0\), respectively, tends to zero when \(c\rightarrow \infty \). Therefore, the GS model can be seen as a \(c\rightarrow \infty \)-limit of this smoothed version.

## 4 Gravity localization

### 4.1 Massless mode

Analysis of the massless mode graph, shown in Fig. 5, reveals new results. Differently from thin-strings models, the maximum of \(\psi _0\) is displaced from origin. It agrees with the fact that the brane core is not at \(\rho = 0\). As discussed in Sect. 3, the brane core approaches the origin when \(c\) increases. This behavior directly reflects on the zero-mode solution. This effect is due to the smooth correction near the origin induced by the \(\beta \) factor. This mode goes along with the energy-momentum tensor. As \(c\) increases, the maximum of the zero mode and the energy density tend to coincide.

### 4.2 Massive modes

Solutions of Eq. (37) for \(M^2(\rho ) \ne 0\), called massive modes, are difficult to obtain analytically. Before embarking on a numerical analysis two different regimes are studied.

In order to obtain a complete domain solution, we integrated Eq. (37) numerically. For this purpose, we used the matrix method [37] with second order truncation error for the domain \(\rho \in [0,6]\). We have plotted in Fig. 6 numerical solutions for \(c = 1.0\), comparing with the analytical GS model solution (44). Near the origin, the solutions behave as a Bessel function, whereas far from the brane, they behave as the GS massive modes solutions. The main advantage of the numerical approach is that we obtain the full domain solution, so we may construe *inside brane* massive gravitons. In contrast to the thin-string model, the massive solutions are non-zero with a high amplitude near the brane.

### 4.3 Analog quantum potential

It is important to note from the massless mode solution that the Neumann boundary conditions are satisfied around the brane core position. Moreover, for \(\rho \rightarrow 0\), \(\psi _0^{\prime } \rightarrow \infty \), which agrees with the infinite potential well (see Fig. 7a).

## 5 Conclusions and perspectives

We have studied the geometrical and physical properties of a six-dimensional thin-string braneworld extension considering a square dependence on radial extra coordinate on the angular metric component. We proposed a subtle warp function that agrees with the thin-string-like model far from the brane, yielding near brane corrections. The geometry possesses an axial symmetry about the origin and the curvature is well behaved. Although we have not been concerned with a specific physical model, the source of this geometry satisfies all the energy conditions. The energy density has a maximum displaced from the origin and a non-null thickness.

We also performed the gravity localization in this scenario. The massless mode is localized and, differently from thin-string models, also has a maximum displaced from the origin, which for great values of the cosmological constant tends to coincide with the energy density one. This shift of the core is due to the near brane smooth correction. Moreover, for large values of the cosmological constant, the model is reduced to a thin-string model slightly apart from the origin. More attention was given to massive modes. We firstly analyzed the graviton massive modes equation for its asymptotic regimes and showed that near brane solutions are expressed in terms of Bessel functions of the first kind for angular momentum values \(l \ne 1\). Far from the brane, we recover the Gherghetta–Shaposhnikov thin-string model without degenerate states. Numerical solutions of the equation for the massive modes revealed that, in contrast to the thin-string model, the massive solutions have a considerable amplitude near the brane, indicating the possibility of gravitational massive states interacting with the defect. This result was reinforced from a Schödinger approach, where we have studied the analog quantum potential. From this formalism we proved the massless mode, which must satisfy a Schödinger-like equation for \(m = 0\). We also concluded that massive states may interact with the brane only for \(l = 0\) which refers to four-dimensional gravitons.

As a future perspective, the KK spectrum can be achieved from the analog quantum potential by means of suitable numerical methods to predict corrections to the Newtonian potential.

## Notes

### Acknowledgments

The authors are grateful to the Brazilian agencies CNPq, CAPES, and FUNCAP for financial support.

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