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arXiv:1202.0129v1 [hep-th] 1 Feb 2012

Thick Brane Split Caused by Spacetime Torsion

Jie Yang∗, Yun-Liang Li†, Yuan Zhong‡, Yang Li§

Institute of Theoretical Physics,Lanzhou University,

Lanzhou 730000, People’s Republic of China

Abstract

In this paper we apply the five-dimensional f(T) gravity with f(T) = T +kTnto brane scenario

to explore the solutions under a given warp factor, and we find that the analytic domain wall

solution will be a double-kink solution when the geometric effect of spacetime torsion is strongly

enhanced. We also investigate the localization of fermion fields on the split branes corresponding

to the double-kink solution.

PACS numbers: 04.50.-h, 11.27.+d

∗yangjiev@lzu.edu.cn

†liyunl09@lzu.edu.cn

‡zhongy2009@lzu.edu.cn

§liyang09@lzu.edu.cn

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I.INTRODUCTION

The presence of extra dimensions is playing a fundamental role in solving the hierarchy

problem, explaining physical interactions based on common principles and other problems

in high energy physics [1–9]. Under the condition of no undesirable physical consequences

obtained, as we know so far, any realistic candidate for a grand unified theory should be

multidimensional. Because of the absence of observational and experimental data, preference

makes no difference in discriminating various kinds of multidimensional models of gravity.

Actually, all sorts of models have been studied in extra dimension gravity.

The concept of brane scenario was introduced in 1983 by Robakov and Shaposhnikov,

who pointed out that we live in a topological defect embedded in 5-dimensional spacetime,

i.e., domain wall, or thick brane in modern terminology which was used as a new approach

to solve the problem of the unobservability of the extra dimensions [10]. According to the

idea, particles corresponding to electromagnetic, weak and strong interactions are confined

on some hypersurface called a brane. Only gravitation and some exotic matter could prop-

agate in the extra dimension. And in [10] the authors found that particles with spin 0 and

1/2 can be trapped on the domain wall described by a scalar field without gravity. During

the 80s and the early 90s, one of the most striking facts which activated the studies on

brane models was the development in superstring theory and M-theory since the mid of 90s,

especially the discovery of D-brane solutions [11, 12]. In 1999, Randall and Sundrum (RS)

proved that gravitation also can be localized on the brane if one takes the gravity into con-

sideration [1]. This is the famous RS brane model which attracts much concentration from

physicists because of its theoretic value, observable effect and solving the long-standing hier-

archy problem and cosmological constant problem. And graviton resonances were previously

considered in thick brane scenario in [5, 6].

So far, various thick brane or domain wall solutions have been investigated (a review in

Ref. [13]) and the trapping of all kinds of matter fields on the single-brane or multi-branes

are also discussed for both thin or thick branes [14–31]. All of these works only considered

the contribution of spacetime curvature without torsion. In this paper we would like to

investigate thick brane solutions caused by the spacetime torsion. A applicable theory is

the teleparallel equivalent of General Relativity (TEGR)[32–37] which instead of using the

curvature defined via the Levi- Civita connection, it uses the Weitzenb¨ ock connection that

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has no curvature but only torsion. This theory allows us to interpret general relativity as a

gauge theory for a translation group. And in this context, gravity is not due to curvature,

but to torsion, and torsion accounts for gravitation not by geometrizing the interaction, but

by acting as a force.

A question that will be asked is that what is the role of torsion or the difference between

torsion and curvature [32, 38]. Although the equations of motion in teleparallel gravity

are dynamically equivalent to those in general relativity and relate to the same degrees of

freedom of gravity (more general relativity theories, like Einstein-Cartan and gauge theo-

ries for the Poincar´ e and the affine groups, consider curvature and torsion as representing

independent degrees of freedom), the teleparallel gravity describes a different geometry, the

Weitzenb¨ ock spacetime. The spacetime metric gµνplays no dynamical role in the teleparallel

description of gravitation.

If we want to investigate the influence of spacetime torsion, we should modify the telepar-

allel gravity. Following the spirit of f(R) gravity (see [39] for a review, [40–44] for applica-

tions in braneword), a generalization of teleparallel gravity is f(T) gravity which was first

proposed by Bengochea and Ferraro to explain the observed acceleration of the universe [45].

And models based on modified teleparallel gravity were also found to provide an alternative

to inflation without inflaton [46, 47]. It therefore has attracted some attention recently.

More recently, Linder [48] proposed two new f(T) models to explain the accelerating expan-

sion and found that the f(T) theory can unify a number of interesting extensions of gravity

beyond general relativity.

The fact that we should note is that f(T) gravity could be a phenomenological extension

of the teleparallel gravity, inspired by the f(R) generalization of the general relativity.

Although the f(R) gravity is probably not the low-energy limit of some fundamental theory,

it does include models that can be motivated by effective field theory. In contrast, f(T)

gravity seems at this stage to be just an ad hoc generalization. Recently it was pointed out

that f(T) gravity violates the local Lorentz invariance [49, 50]. Nevertheless, it still attracts

an increasing interest in the literature because of its advantage over f(R) gravity, namely,

its field equations are at most second-order instead of fourth-order. The validity of f(T)

gravity as an alternative also has been investigated by analyzing the large-scale structure

[51] and the observational constraints on model parameters [52, 53]. Apart from obtaining

acceleration, one can reconstruct a variety of cosmological evolutions [54–57], can consider

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the possibility of the phantom divide crossing [58–60], and can investigate the vacuum and

matter perturbations [61–63] beyond the background evolution.

Considering the increasing interest in f(T) gravity and the possibility as an alternative

to general relativity, in this paper we investigate the impact of torsion instead of curvature

on the structure of thick branes and the localization of fermions on the thick branes. The

spacetime torsion can result in the splitting of the thick brane with an internal structure

which is called double-kink defect since it seems to be composed of two standard kinks. In

the works of Bazeia and his collaborators [64–68] a class of defect structures were obtained

by a φ4potential. The appearance of the structure will result in a split in the matter energy

density in the center of the brane. And the resonances of gravitons and fermions in the such

structure scenario have been considered in recent works [69–74]. In our work we use the

resonance detecting method to analyze the KK modes of fermion fields and how the internal

structure due to geometric effect influence the resonances of fermion in the splitting brane.

The paper is organized as follows: In Sec. II, we first give a brief review of the teleparallel

gravity and then give the field equations for the five-dimensional f(T) brane. In Sec. III,

because their equations are still second order, we obtain some exact analytic domain wall

solutions for a given warped factor. In Sec. IV, we study the localization of fermion fields

on the thick branes by presenting the potential of the Schr¨ odinger equations.

II.SET UPS AND DYNAMICAL EQUATIONS

Before we set up our model, let us briefly give a review of the teleparallel gravity. In

teleparallel gravity, it is the vierbein or tetrad fields, ha(xµ) (rather than the metric) that

work as the dynamical variables. At each point of the manifold, the tetrad fields form an

orthonormal basis for the corresponding tangent space of the point. In four-dimensional

teleparallel gravity, Latin indices a,b,... and Greek indices µ,ν,... both run from 0 to

3, label coordinates of the tangent space and the spacetime, respectively. For a specified

spacetime coordinate basis the components of ha(xµ) are hµ

a. Clearly, hµ

aare both spacetime

vectors and Lorentz vectors.

The relation between the tetrad fields and the metric is given by

gµν= ηabha

µhb

ν, (1)

where ηab = diag(−1,1,1,1) is the Minkowski metric for the tangent space. From the

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relation (1), it follows that

hµ

aha

ν= δµ

ν,hµ

ahb

µ= δb

a. (2)

Instead of using the Levi-Civita connection Γρ

µν, we would like to apply the Weitzenb¨ ock

tensor

˜Γρ

µν= hρ

a∂νha

µ, (3)

and the torsion

Tρ

µν=˜Γρ

νµ−˜Γρ

µν, (4)

to establish the teleparallel gravity. The difference between the Levi-Civita connection and

Weitzenb¨ ock connection is the well-known contortion tensor [75]

Kρ

µν≡˜Γρ

µν− Γρ

µν=1

2[T

ρ

µ ν+ T

ρ

ν µ− Tρ

µν]. (5)

By defining a tensor Sµν

ρ

:

Sµν

ρ

=1

2[Kµνρ− δν

ρTθµθ+ δµ

ρTθνθ], (6)

one can write the Lagrangian of the teleparallel gravity as [32–36]

LT = −

= −c4h

c4h

16πGT = −

?1

c4h

16πGS

µν

ρ

Tρ

µν

16πG4Tρ

µνT

µν

ρ

+1

2Tρ

µνTνµ

ρ− T

ρ

ρµTνµ

ν

?

,(7)

where h = det(ha

µ) =√−g, with g the determinant of the metric gµν. It is well known that

the teleparallel gravity is equivalent to general relativity. Therefore, in order to discuss the

effects of the torsion, we have to generalize the gravity.

As to the f(T) gravity, we need only to replace the T in Lagrangian (7) by an arbitrary

differentiable function of T, and then the action in five-dimensional gravity is

S = −1

4

?

d5xhf(T) +

?

d5xLM, (8)

where we have taken

c4

4πG5= 1 for convenience. The corresponding field equations read

h−1fT∂Q(hS

MQ

N

) + fTTS

MQ

N

∂QT − t

M

N

= −T

M

N

, (9)

where f ≡ f(T),fT ≡ ∂f(T)/∂T,fTT ≡ ∂2f(T)/∂T2and tM

TM

N

is the energy-momentum tensor of the matter field. Capital Latin indices M,N ... =

N

= fT˜ΓR

SNSMS

R

−1

4δM

Nf,

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