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
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
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 . 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  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 . 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. ) 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
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  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 .
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  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
 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
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
where ηab = diag(−1,1,1,1) is the Minkowski metric for the tangent space. From the
relation (1), it follows that
Instead of using the Levi-Civita connection Γρ
µν, we would like to apply the Weitzenb¨ ock
and the torsion
to establish the teleparallel gravity. The difference between the Levi-Civita connection and
Weitzenb¨ ock connection is the well-known contortion tensor 
µ ν+ T
ν µ− Tρ
By defining a tensor Sµν
one can write the Lagrangian of the teleparallel gravity as [32–36]
LT = −
16πGT = −
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
where we have taken
4πG5= 1 for convenience. The corresponding field equations read
) + fTTS
∂QT − t
where f ≡ f(T),fT ≡ ∂f(T)/∂T,fTT ≡ ∂2f(T)/∂T2and tM
is the energy-momentum tensor of the matter field. Capital Latin indices M,N ... =
0,1,2,3,5. Here the field equations are expressed in purely spacetime form, not containing
coordinates of the tangent space.
In our work we consider the static flat braneworld scenario with the metric
ds2= e2A(y)ηµνdxµdxν+ dy2,(10)
where ηµν = diag(−1,1,1,1) is the four-dimensional Minkowski metric, and e2A(y)is the
warped factor. Then the tetrad fields are ha
µ= diag(eA,eA,eA,eA,1), T = −12A′2. From
now on, the prime always denotes the derivative with respect to y, unless specified.
In our model, we take f(T) = T + kTn, and LM= h(−1
φ ≡ φ(y) depends only on the extra dimension y. And then the field equations are given as
2∂Mφ ∂Mφ − V (φ)) , where
A′′+ 4A′φ′=dV (φ)
?12A′2+ (−1)n−112nk(2n − 1)A′2n?= −V +1
2A′′+ (−1)n−122n−33nk(2n − 1)A′2n−2(2A′2+ nA′′) = −V −1
Note that there are only two independent equations in the above equations. Therefore, we
need only to consider eq. (12) and the following one:
φ′2= −[12A′2+ (−1)n−112nkn(2n − 1)]A′′
which is obtained by the combining of eqs. (12) and (13).
III. SOLUTIONS FOR f(T) BRANE
Although the model is a second-order derivative theory, it is hard to give an analytic
solution for general cases. For simplicity, let us take
e2A(y)= cosh−2b(αy),(b > 0),(15)
and consider the following cases.
For n = 1/2, eqs. (12) and (13) reduce to
3A′2= −(V −1
2A′′= −(V +1
They are the same equations as those in Refs. [5, 30], where the gravity is described by
general relativity. As a consequence, the solutions of this case are equivalent to those in the
case f(T) = T. A domain wall solution has been obtained in [5, 30] by using a superpotential
V (φ) =3bα2
(1 + 4b)cos2(2φ
√6b) − 4b
Obviously, α is a parameter which fixes the thickness of the wall. As stated in Ref. , as
y → ±∞, A(y) → −bc|y|. Thus, the spacetime described by the metric (10) and (15) is
B. Other positive integers n
With the substitution of (15) into (14), we should have the following condition
∆ ≡ (−1)n12n−1n(2n − 1)kb2n−2α2n−2≤ 1, (20)
to make the right side of (14) non-minus. Only when (20) is satisfied, we can obtain a real
function solution. Note that the existence of (−1)nconstraints the values of n and k.
For n = 2, we yield an analytic domain wall solution:
i√2?E(iαy;1 − 72kb2α2) − F(iαy;1 − 72kb2α2)?
1 + 72kb2α2+ (1 − 72kb2α2)cosh(2αy)tanh(αy)
where F(iαy;1−72kb2α2),E(iαy;1−72kb2α2) are the first and second kind elliptic integrals,
respectively. One can prove that φ(y) is real provided that 1 − 72kb2α2≥ 0 as required by
(20). Specially, when 1 − 72kb2α2= 0,
which is a kink solution. However, for large enough 1 − 72kb2α2, the solution (21) turns to
be a double-kink, as shown in Fig.1.
FIG. 1: The shape of the scalar φ(y) plotted with n = 2,b = 1,α = 1. With the decrease of k,
double-kink solutions will be more notable.
For other values n, an analytical solution like (21) is hard to obtain, but when ∆ = 1,
(14) reduces to
2bα2sech2(αy)tanh2n−2(αy) = φ′2. (23)
For n = 2, it gives (22). For n = 3,
2)ΞΠ(3 + 2√2;iarcsinh(Z),17 + 12√2) + sech(αy)tanh(αy))
2)ΞF(iarcsinh(Z);17 + 12√2)
3 + 2√2
Ξ ≡ sech(2αy)
(3 + 2√2)(1 + Z2)
1 + (3 + 2√2)2Z2,
and Π(3 + 2√2;iarcsinh(Z),17 + 12√2) is the third kind elliptical integral. For n = 4,
4) + F(2iαy;3
b(5 + 3cosh(αy))tanh3(αy).(25)
Generally, for different values n, via numerical approach, we find that the solution is a
kink when 0 < ∆ < 1 and a double kink as ∆ is less than some negative value. From
eq. (14), we can see that the domain wall solution turns out to be a double-kink solution
when the contribution from the second term of right side exceeds the first term. So ∆ shows
the strength of the geometrical effect of torsion.
Commonly, the appearance of a double-kink solution means that the domain wall at y = 0
symmetrically splits into two branes. This can be seen from the distribution of the energy
2bα2sech2(αy) − 3b2α2tanh2(αy)
−(−3)n22n−3kb2n−1α2n(2n − 1)(ncsch2(αy) − 2b)tanh2n(αy), (26)
as shown in fig. 2. Locations of those two peaks are where two sub-branes inhabit. At the
boundary of the spacetime
ρ(±∞) = −3b2α2+ (−3)n22n−2kb2nα2n(2n − 1) (27)
is a minus constant if eq. (20) is satisfied.
FIG. 2: The density ρ(y) of the scalar field φ(y) with n = 2,α = 1, b = 1 (left) and b = 0.01
(right). A trend of brane-splitting can be seen from the transition.
C.The split of brane
In Ref. , the authors investigated the split of thick brane, which is generated by a
complex scalar field coupled to gravity. They showed that the split of the brane is due
to a first-order transition when the temperature approaches the critical value and a new
disordered phase would appear between these two sub-branes.
At zero temperature, the split of thick brane was realized by using a real scalar field .
In this model, the engendered internal structure depends on a real parameter, which changes
the self-interaction of the scalar field. The split of brane was also investigated by using two
real scalar fields [68, 77–80].
In our work, the internal structure is different, because the energy density of the scalar
field is non-vanished at y = 0, i.e., ρ(0) =3
2bα2?= 0. Such a structure indicates that the split
of the brane is incomplete, and there is a connection between the two sub-branes. While for
the case with ρ(0) = 0, the original brane is completely split, and the newly generated branes
are independent. The energy density dwelling on the split branes becomes more notable with
the increase of the contribution from torsion to which the phase transition is due. It indicates
that the geometric effect will influence the distribution of the energy density. It should be
note that the distance between the two split branes is mainly determined by α and b, which
also determine the thickness of the domain wall.
A probable explanation to the changeable k is that k might relates to the evolution
of universe. Note that the temperature of the cosmological background is a characteristic
parameter relevant to the evolution, so we can recognize k as a function of temperature.
Therefore, there might exists a critical temperature Tc, at which, the brane splits into two
IV.LOCALIZATION OF SPIN-1
Whether various bulk fields could be confined to the brane by a natural mechanism is
an interesting and important issue to build up the standard model. It has been known
that massless scalar fields  and gravitons [1, 5, 6] can be localized on branes of different
types. Abelian vector fields can be localized on the RS brane in some higher-dimensional
cases  or on thick dS branes and Weyl branes [25, 26]. The localization of fermion fields
is also interesting. In order to localized fermions, the coupling between the fermion fields
and the background scalars should be introduced. With different scalar-fermion couplings,
a single bound state and a continuous gapless spectrum of massive fermion KK states can
be obtained, see for example [21–24]. In some other models, there exist finite discrete KK
states (mass gap) and a continuous gapless spectrum starting at a positive m2[26–28, 69, 70]
or even only exist bound KK modes.
From the results in Refs. [26, 28] we note that the effective potentials of the KK modes
of scalar and vector fields are free of gravity model and only dependent on A(y). It can be
easily verified that the zero mode of these fields can be localized on the brane we obtianed
here. However, the effective potentials of fermion fields couple to the background scalar, so
the localization is model-dependent.
In this section we will investigate how the spacetime torsion influences the localization of
fermion fields on the brane. Via performing the conformal transformation dz = e−A(y)dy ,
we can rerepresent the metric in conformal coordinates. Taking the simply Yukawa coupling,
the 5-dimensional Dirac action of a massless spin 1/2 fermion coupled to the background
scalar φ is
d5xh(¯ΨΓM(∂M+ ωM)Ψ − η¯ΨφΨ), (28)
The non-vanishing components of the spin connection ωMfor the background metric are
2A′γµγ5+ ˆ ωµ, (29)
where prime denotes the derivation with respect to conformal coordinate z from now on,
and ˆ ωµis the spin connection on the brane and vanishes here. Then the equation of motion
is given by
[γµ∂µ+ γ5(∂z+ 2A′) − ηeAφ]Ψ = 0. (30)
The sign of the coupling η of the spinor Ψ to the scalar φ is arbitrary, and without loss of
generality, we assume η > 0.
According to (30) Ψ can be expanded by
[ΨL,n(x)fL,n(z) + ΨR,n(x)fR,n(z)]e−2A
with ΨL= −γ5ΨLand ΨR= γ5ΨRbeing the left-handed and right-handed components of
a 4D Dirac field respectively. By demanding ΨL,Rsatisfy the 4D massive Dirac equations
γµ∂µΨL,R= mΨL,R, we yield the following coupled equations
[∂z+ ηeAφ]fL(z) = mfR(z), (32)
[∂z+ ηeAφ]fR(z) = mfL(z). (33)
These equations can be reduced to the Schr¨ odinger-like equations for the KK modes of left
and right chiral fermions
z+ VL(z)]fL(z) = m2fL(z), (34)
z+ VR(z)]fR(z) = m2fR(z), (35)
where the effective potentials are given by
VL(z) = (ηeAφ)2− η∂z(eAφ),
VR(z) = VL(z)|η→−η.
The index n is dropped for convenience.
Since the Yukawa coupling is an odd function of the extra dimension z, the effective
potential VL,R(z) of left- and right-chiral fermions are invariant under the reflection symmetry
z → −z. Here we discuss the case n = 2, and get the effective potential in proper coordinate
y from eqs. (15) and (21):
ζ2sinh2(αy) + i√2ζξ sinh(2αy) − 2ξ2cosh2(αy)
2bζ sinh2(αy) + i√2bξ sinh(2αy) − 2ζ
ξ= E(iαy|1 − 72kb2α2) − F(iαy|1 − 72kb2α2),
ζ 1 + 72kb2α2+ (1 − 72kb2α2)cosh(2αy).
For simplicity, we take b = 1, and obtain the analytical transformation y =
Further, we can reexpress VL(y) as the function of the conformal coordinate z, VL(z) and
VL(z) are plotted in Fig. 3 for different values of k.
Since b = 1, we can get a simple expression of the warped factor in conformal coordinate,
i.e., A(z) = −ln√1 + z2α2. Note that at z = 0, A(0) = A′(0) = φ(0) = 0, so VL(0) =
−ηeA(0)φ′(0) = −
2ηα. For positive η, VL(0) < 0. From fig. 3, it can be seen that VL(±∞)
FIG. 3: The effective potential of left-chiral fermions with n = 2,b = 1,α = 1 in different coordinate
vanishes at infinity, therefore there is only one bound massless mode for left-chiral fermions
followed by a continuous gapless spectrum of KK states with η > 0.
From fig. 3, we find that, with the decrease of k, the height of the potential well will
increase, then there will be two minima in the potential well, namely double well. Although
there is only one bound massless mode, but some resonances may appear which can tunnel
from the brane to the bulk. In the case shown in fig. 3, only the zero mode exists. But
for k = 0.5 there exists an extra resonance with m2= 3.2369, probability 0.472493 and
odd wavefuction. For k = −1.5, there are two resonances with m2= 5.2925,10.3974. For
k = −4, the resonances increase to three, m2= 7.84658,17.899,25.032. So the resonant
states will increase with the contribution from torsion. For the right-chiral KK modes, there
have no bound modes but continuous and gapless spectra which are the same with the
left-chiral KK modes.
Note that the height of the potential well will also increase with b and α, but the width
becomes narrower, then there will be a double well with a smaller VL(0). The coupling
constant η can also affect the width and the height of the well, but there will be no transition
from one minimum in the potential well to two minima. Similarly the well has a smaller
Next we discuss the condition of the localization. The zero mode for the left-chiral
fermions reads [69, 70, 79]
fL0(z) ∝ exp
In order to check whether the zero mode can be localized on the brane, we should check
whether the normalization condition for the zero mode is satisfied, namely, whether the
is finite. Since eA(ω)φ(ω) → 0 when ω → ∞, so it is clear that the integral (40) is finite for
positive η, namely, the zero mode for left-chiral fermions can be localized on the brane for
Since b = 1, we can get a simple expression of the warped factor in conformal coordinate,
i.e., A(z) = −ln√1 + z2α2. At the infinity, eA→
fL0(z → ±∞) → |z|−ηφ∞
α , (41)
φ(z → ∞) →
+i(1 − 72kα2)E(
1−72kα2) + i72kα2K(
√1 − 72kα2
− iE(1 − 72kα2) + K(72kα2)
If the normalization condition is satisfied, we can get the following equivalent condition,
α dz < ∞. (43)
Only when η > η0=
2φ∞, the above integral is convergent, which means that the left-chiral
zero mode can be localized on the brane under this condition.
From eq. 20, we can find that the consequences here can also be obtained for the even
integer n. For the odd integer n and with k bigger than some positive value, we also obtain
the similar consequences here. Note that k represents the strength of the contribution from
torsion, so the results are applicable only when the torsion have a significant effect.
In this paper, we investigate the geometric effect of torsion on thick branes in gauge
theory, and find some analytic domain wall solutions for some specific values of n. We
also find that the geometric effect determines whether the domain wall solution is a kink Download full-text
or double-kink. With the increase of the contribution of torsion, the configuration of the
solution changes from a kink to double kink. The more significant the effect is, the more
energy dwells on the sub-branes. We also study the localization of fermion fields on the brane
described by the domain wall solution. It is shown that there is only one bound massless
mode on the brane, but when the spacetime torsion has a significant effect, the potential
well for the fermion KK modes will become more and more deeper and more resonant states
of left-chiral fermions with short lifetime will appear. With the coupling parameter k being
a function of temperature, the evolution of universe can influence not only the split behavior
of the thick brane via changing the contribution from the spacetime torsion, but also the
number of fermion resonate states.
This work was supported by the National Natural Science Foundation of China (No.
11075065), the Fundamental Research Funds for the Central Universities (No. lzujbky-2012-
k30), and the Natural Science Foundation of Gansu Province, China (No. 096RJZA055).
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