六维平滑弦状 braneworld: 弦论扩展与能量条件满足的解决方案

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本文主要探讨了"六个维度上的平滑弦状braneworld"这一新颖的理论模型，它是弦理论的一种扩展，旨在理解宇宙在额外维度中的行为。作者们，J.C. Araújo、J.E.G. Silva、D.F.S. Veras 和 C.A.S. Almeida，来自巴西费尔南多·卡埃罗联邦大学物理系，他们在2015年的《欧洲物理杂志C》（Eur.Phys.J.C）上发表了这项研究。研究的核心内容是构建了一个静态且轴对称的六维空间结构，它类似于弦世界模型，但具备更精细的几何特性。通过设计一个微妙的翘曲函数，这个模型能够提供接近于弦本身的修正，从而更好地处理高维度下可能存在的微扰效应。这有助于我们更好地理解引力和其他基本力在这些额外维度中的表现。作者们特别关注了宇宙学常数在模型中的作用，通过调整它，他们能够实现光源的不同相位传递，这是一种动态宇宙学现象。这种模型不仅适用于弦的内部结构，还适用于外部环境，确保了所有能量条件的满足，符合物理定律的稳定性。一个关键的发现是，在这个平滑的braneworld中，存在一个无质量的重力模式，它被精确地定位在六维空间的"米糠"（brane）上，但核心位置并非原点，这与传统的细线模型有所区别。有趣的是，相比于细线模型，这里的大型质量解在靠近braneworld的地方有较高的振幅，这可能暗示着在某些物理条件下，重力场的表现形式有所不同。此外，作者们通过模拟量子电势分析，进一步揭示了在这个六维体系中，只有s波的重力Kaluza-Klein模式是稳定的，这对于理解量子效应在高维度空间中的行为具有重要意义。这可能是对量子引力理论的一个重要贡献，因为它提供了一种在更高维度背景下检验量子力学与广义相对论兼容性的新视角。总结来说，这篇论文深入探讨了六维平滑弦状braneworld模型，它为理解弦理论的扩展以及额外维度的物理效应提供了一个独特的框架，同时也对量子引力的计算和宇宙学常数调控等问题带来了新的见解。这一研究对于未来的理论物理学和宇宙学探索具有潜在的重要价值。

Eur. Phys. J. C (2015) 75:127

DOI 10.1140/epjc/s10052-015-3350-8

Regular Article - Theoretical Physics

A smoothed string-like braneworld in six dimensions

J. C. B. Araújo

,J.E.G.Silva

, D. F. S. Veras

,C.A.S.Almeida

Departamento de Física, Universidade Federal do Ceará, C.P. 6030, Fortaleza, Ceará 60455-760, Brazil

Received: 12 October 2014 / Accepted: 9 March 2015 / Published online: 17 March 2015

Abstract We propose a static and axisymmetric braneworld

in six dimensions as a string-like model extension. For a sub-

tle warp function, this scenario provides near brane correc-

tions. 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 gravita-

tional 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 poten-

tial analysis, we show that only s-wave gravitational Kaluza–

Klein modes are permissible.

1 Introduction

In the last decade, the extra dimension models turned out

to be a cornerstone of high energy physics [1,2]. In particu-

lar, the Randall–Sundrum (RS) model brought the incredible

idea of inﬁnity extra dimensions through a warped geome-

try [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 [3–7]. 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 inter-

action [8]. Moreover, it is possible to localize the vector and

spinor ﬁelds in the same geometry [9].

e-mail: julio@ﬁsica.ufc.br

e-mail: euclides@ﬁsica.ufc.br

e-mail: franklin@ﬁsica.ufc.br

e-mail: carlos@ﬁsica.ufc.br

In spite of the aforementioned results, the string-like mod-

els 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 dimen-

sions 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 numer-

ical 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 [13–16], whereas others

tried to smooth this conical behavior [16–24].

In this article we explore some features of a smooth exten-

sion of the GS model built analytically. Besides, using a

smoothed warp function, the metric satisﬁes all the regu-

larity 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 local-

ized 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,wereviewthe

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

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