Regular string-like braneworlds

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 $l=1$.


I. 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 the codimension-2, the vortex scenarios are a suitable source for an axisymmetric brane, known as a string-like braneworld [7,8]. However, a global vortex geometry exhibits a mild singularity [9] and a regular local Abelian 3-brane which satisfies the dominant energy condition was only accomplished numerically [10]. 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 [11,[13][14][15]. Some brane-core geometries were proposed by considering more involved transverse spaces, such as the cigar [16], the Hamilton cigar [17], the resolved conifold [18], the catenoid [19] and others [20,21].
In this article, we present a new class of smooth string-like model and explore some of its physical and geometrical features. 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 behavior 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 behavior.
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 Sec. II we present the smooth and deformed solutions and study their source and geometry properties. In Sec. III, the Kaluza-Klein modes for the scalar, vector gauge and gravitational fields are studied and their features discussed. In Sec. IV, final remarks and perspectives are outlined.
The equations (6), (7) and (8) form a complex system of coupled equations. For a global vortex source, the geometry exhibits a mild singularity [9]. For a local vortex, only numerical solutions are known [10]. The angular Einstein equation (8) is a decoupled nonlinear nonhomogeneous Riccati equation. For A = B = −c = 2κ 6 (−Λ)/5, we obtain a AdS 6 vacuum solution describing a thin string-like braneworld [14]. Since we are looking for smooth and localized geometries, let us assume an ansatz for the warp function A(r) extending the thin string-like solution in the form where the parameter c controls the asymptotic value of the warp function, p modifies its variation inside the brane core and λ determines the brane width. Note that for p = 0 or for p = 0 and far from the origin the warp function ansatz (9) reduces to the thin string-like solution. Integrating Eq. (9) we obtain the warp factor σ in terms of the hypergeometric function as We plotted the warp factor (10) in Fig. 1. For p = 1 we obtain the bell-shaped σ whereas for p = 3 and p = 6, the warp factor exhibits a plateau around the origin. The deformed p = 2 warp factor reduces to the string-cigar model [17]. For p ∈ N, the source for the warp function A has the angular pressure The Fig. 2 shows the behaviour of the angular pressure for p = 1 (thick line), p = 3 (thin line) and p = 6 (dotted line). For p = 1 the angular pressure has a bell-shape t θ = (c(5c + 4λ)/2) sech 2 (λr) localized around the origin. For p > 1, the angular pressure exhibits two modified patterns inside the brane core. Therefore, we recognize the solutions for p > 1 as deformed string-like branes.
The radial Einstein equation (7) provides a constraint between A and B. Let us consider the ansatz for B in the form For f = 0 and c = 2λ, the bulk metric has the form ds 2 = sech 2 (λr)η µν dx µ dx ν + dr 2 + R 2 0 sech 2 (λr)dθ 2 , a smoothed string-like solution whose transverse space is a catenoid [19]. Likewise the thin string-like model, this metric does not satisfy the condition expressed in Eq. (4), and then, at r = 0, we have a 4-brane.
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.

A. Warped disk
Consider f 1 (r) := m λr . For this choice the metric has the form In order to satisfy the regularity conditions, we have to set m = 2λ. Then, the metric (14) represents a warped product of the 3-brane and the two dimensional flat disk. The angular metric component is shown in Fig. 3 for p = 1 where the high of the function increases with the ratio c/λ. Fig. 4 shows the Ricci scalar which for p = 1 has a bell-shape and for p = 10 exhibits a plateau near the origin. The components of the stress-energy tensor for p = 1 are t 0 (r) = κ 6 (5c + 4λ) 2 sech 2 (λr) + 5c 2 tanh(λr) r , t r (r) = κ 6 5c 2 2 sech 2 (λr) + 2c tanh(λr) r .
We plotted these components for c = 2λ in Fig. 5. Note that the source of this geometry is localized around the origin and satisfies the dominant energy condition. For p = 6, the core shows an internal structure where the peak of the components are shifted from the origin and the radial pressure is almost constant. The relationship between the bulk and brane Planck masses is given by For c/λ = 4/3, we obtain M 2 4 = 2π ln 2 λ 3 M 4 6 . Then, the ratio M 4 /M 6 increases as the brane width decreases.

B. Exotic string-brane
For f (r) = m tanh(λr) , the bulk metric has the form The regularities conditions are satisfied for m = 2λ. The components of the stress-energy tensor are whose are sketched in Fig. 7 for c/λ = 2. The source satisfies the dominant energy condition inside the core and exhibits an exotic radial pressure asymptotically. For c = 2λ 5 , t 0 = t θ and the source is dominated by the radial pressure. Fig. 8 shows the behaviour of the Ricci scalar as we change the deformation parameter 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.

III. 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.

B. Massive modes
For m = 0, the KK radial equation in the warped disk model becomes y m,l,q + 1 r − qc 2 tanh(λr) y m,l,q + cosh(λr) c/λ m 2 − l 2 r 2 y m,l,q = 0, where q = 5 for the graviton and q = 3 for the photon. Near the brane, by expanding Eq.(24) up to first order, we obtain whose solutions are where A m,l,q is an integration constant. The KK modes have a zero of order l at the origin.
Then, for l = 0, y m,0,q (r) → A m,l,q as r → 0, whereas for l = 0, y m,l,q (r) → 0. For r << 2/qcλ 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.(24) becomes the thin string-like equation y m,l,q − qc 2 y m,l,q + m 2 e cr 2 c/λ y m,l,q = 0, whose solution is [14]: For the exotic string model, the KK tower satisfies y m,l,q + λ coth(λr) − qc 2 tanh(λr) y m,l,q + cosh(λr) c/λ m 2 − (lλ) 2 sinh 2 (λr) y m,l,q = 0, (28) which has the same near brane behavior of the warped disk model. However, far from the brane the KK eigenfunctions are given by The presence of the width brane parameter λ shows that the exotic source modifies the KK tower even outside the brane core.
Unlike the massless modes, the massive states forms a tower of non-normalizable states.
We plotted the graviton Schrödinger potential for the warped disk in Fig. 11 and for the exotic string in Fig. 12. Since asymptotically both potentials vanish, the massive modes are gappless free states. The scalar and vector gauge potentials have similar behavior. The potential exhibits an infinite well at the origin for l = 0 and an infinite barrier for l = 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.

IV. 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 geometrical deformation parameter, 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 [10].
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 = 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 [10]. 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 [24].