#### Radion tunneling in modified theories of gravity

Eur. Phys. J. C
Radion tunneling in modified theories of gravity
Tanmoy Paul 0
Soumitra SenGupta 0
0 Department of Theoretical Physics, Indian Association for the Cultivation of Science , 2A & 2B Raja S.C. Mullick Road, Kolkata 700 032 , India
We consider a five dimensional warped spacetime where the bulk geometry is governed by higher curvature F (R) gravity. In this model, we determine the modulus potential originating from the scalar degree of freedom of higher curvature gravity. In the presence of this potential, we investigate the possibility of modulus (radion) tunneling leading to an instability in the brane configuration. Our results reveal that the parametric regions where the tunneling probability is highly suppressed, corresponds to the parametric values required to resolve the gauge hierarchy problem.
1 Introduction
Over the last two decades models with extra spatial
dimensions [1–13] have been increasingly playing a central role
in search for physics beyond standard model of elementary
particle [14,15] and Cosmology [16,17]. Such higher
dimensional scenarios occur naturally in string theory and also are
viable candidates to resolve the well known gauge
hierarchy problem. Depending on different possible
compactification schemes for the extra dimensions, a large number of
models have been constructed. In all these models, our
visible universe is identified as a 3-brane embedded in a higher
dimensional spacetime and is described through a low energy
effective theory on the brane carrying the signatures of extra
dimensions [18–20].
Among various extra dimensional models proposed over
last several years, warped extra dimensional model pioneered
by Randall and Sundrum (RS) [6] earned a special attention
since it resolves the gauge hierarchy problem without
introducing any intermediate scale (between Planck and TeV) in
the theory. Subsequently different variants of warped
geometry model were extensively studied in [17,21–30]. A generic
feature of many of these models is that the bulk spacetime is
endowed with high curvature scale ∼ 4 dimensional Planck
scale.
It is well known that Einstein–Hilbert action can be
generalized by adding higher order curvature terms which naturally
arise from diffeomorphism property of the action. Such terms
also have their origin in String theory from quantum
corrections. In this context F (R) [31–43], Gauss-Bonnet (GB) [44–
46] or more generally Lanczos–Lovelock gravity are some
of the candidates in higher curvature gravitational theory.
In general the higher curvature terms are suppressed with
respect to Einstein–Hilbert term by Planck scale. Hence
in low curvature regime, their contributions are negligible.
However higher curvature terms become extremely relevant
in a region with large curvature. Thus for bulk geometry
where the curvature is of the order of Planck scale, the higher
curvature terms should play a crucial role. Motivated by this
idea, in the present work, we consider a generalized warped
geometry model by replacing Einstein–Hilbert bulk
gravity action with a higher curvature F (R) gravitational theory
[41,42,47–53].
One of the crucial aspects of higher dimensional two brane
models is to stabilize the interbrane separation (also known
as modulus or radion). For this purpose, one needs to generate
a suitable radion potential with a stable minimum [21–23].
The presence of such minimum guarantees the stability of the
modulus field. In Goldberger–Wise stabilization mechanism
[21,22], an external bulk scalar field was invoked to create
such a stable radion potential. However, when the bulk is
endowed with higher curvature F(R) gravity, then apart from
the metric there is an additional scalar degree of freedom
originating from higher derivative terms of the metric. It has
been shown that such a scalar degree of freedom can play
the role of the stabilizing field for appropriate choices of the
underlying F (R) model [26,27].
It is important to analyze the exact nature of the resulting
radion potential to explore whether there exists a metastable
minimum for the radion from which it can tunnel and leads
to an instability of the braneworld [54–59]. In this paper,
we aim to determine the radion tunneling in the presence of
higher curvature gravity in the bulk.
Our paper is organized as follows: following section is
devoted to brief review of the conformal relationship between
F (R) and scalar–tensor (ST) theory. In Sect. 3, we extend our
analysis of Sect. 2 for the specific F (R) model considered in
this work. Section 4 extensively describes the tunneling
probability for the dual ST model while Sect. 5 addresses these for
the original F (R) model. After discussing the equivalence,
the paper ends with some concluding remarks in Sect. 6.
2 Transformation of a F(R) theory to scalar–tensor
theory
In this section, we briefly describe how a higher curvature
F(R) gravity model in five dimensional scenario can be recast
into Einstein gravity with a scalar field. The F(R) action is
expressed as,
1
S = 2κ2
d4x dφ√G F (R)
where x μ = (x 0, x 1, x 2, x 3) are usual four dimensional
coordinate and φ is the extra dimensional spatial angular
coordinate. R is the five dimensional Ricci curvature and G
is the determinant of the metric. Moreover 2κ12 is taken as
2M 3 where M is the five dimensional Planck scale.
Introducing an auxiliary field A(x , φ), the action (in Eq. (
1
)) can
be equivalently written as,
(
1
)
1
S = 2κ2
d4x dφ√G[F ( A)(R − A) + F ( A)].
(2)
d4x dφ√G
freedom. Clearly kinetic part of σ (x , φ) is non canonical.
In order to make the scalar field canonical, transform σ →
(x , φ) = √3 σ (xκ,φ) . In terms of (x , φ), the above action
takes the form,
S =
d4x dφ
2
+ √3κ
G˜ 2Rκ˜2 + 21 G˜ M N ∂M
∂N
G M N ∂M ∂N
˜
− U ( ) ,
1 A F(A)
where U ( ) = 2κ2 [ F (A)2/3 − F (A)5/3 ] is the scalar field
potential which depends on the form of F (R). Thus the action
of F (R) gravity in five dimension can be transformed into
the action of a scalar–tensor theory by a conformal
transformation of the metric.
3 Warped spacetime in F(R) model and corresponding
scalar–tensor theory
In the present paper, we consider a five dimensional
spacetime with two 3-brane scenario in F(R) model. The form of
F (R) is taken as F (R) = R + α Rn where n takes only
positive values, α is a constant and has the mass dimension
[2 − 2n]. Considering φ as the extra dimensional angular
coordinate, two branes are located at φ = 0 (hidden brane)
and at φ = π (visible brane) respectively while the latter one
is identified with the visible universe. Moreover the
spacetime is S1/Z2 orbifolded along the coordinate φ. The action
for this model is:
(4)
(5)
(6)
1
+ rc
where Vh , Vv are the brane tensions on hidden, visible brane
respectively. We also include Gibbons–Hawking boundary
terms on the branes, symbolized by Qh and Qv in the above
action (i.e Qh , Qv are the trace of extrinsic curvatures on
hidden, visible brane respectively).
This higher curvature F (R) model (in Eq. (5)) can be
transformed into a scalar–tensor theory by using the
technique discussed in the previous section. Performing a
conformal transformation of the metric as
G M N (x , φ) → G˜ M N = exp
1
√ κ (x , φ) G M N (x , φ)
3
By the variation of the auxiliary field A(x , φ), one easily
obtains A = R. Plugging back this solution A = R into
action (2), initial action (
1
) can be reproduced. At this stage,
one may perform a conformal transformation of the metric
as
G M N (x , φ) → G˜ M N = eσ (x,φ)G M N (x , φ).
M, N run form 0 to 5. σ (x , φ) is conformal factor and related
to the auxiliary field as σ = (2/3) ln F ( A). Using this
relation between σ (x , φ) and A(x , φ), one lands up with the
following scalar–tensor action
d4x dφ
G˜ R˜ + 3G˜ M N ∂M σ ∂N σ
+ 4G˜ M N ∂M ∂N σ −
A F ( A)
F ( A)2/3 − F ( A)5/3
1
S = 2κ2
where R˜ is the Ricci scalar formed by G˜ M N . σ (x , φ) is
the scalar field emerged from higher curvature degrees of
the above action (in Eq. (5)) can be expressed as a scalar–
tensor theory with the action given by:
S =
= 0.
To derive the above equation of motion, is taken as
function of extra dimensional coordinate only. Considering the
variation of (φ) is small in the bulk [21,22], Eq. (12) turns
out to be,
+
1 e− 2 √53 κ
+ rc
Qv δ(φ − π ) ,
Vv + κ2
+ √23κ G˜ M N ∂M ∂N
where is chosen to be negative and the quantities in tilde
are reserved for ST theory. R˜ is the Ricci curvature formed
by the transformed metric G˜ M N . (x , φ) is the scalar field
corresponds to higher curvature degree of freedom and U ( )
is the scalar potential which for this specific form of F (R)
has the form,
U ( ) =
a ebκ ,
+ κ2
where a (mass dimension [2]) and b (dimensionless) are
related to the parameters α and n by the following
expressions:
α =
4b + 10
5 + 2√3b
n = 2 + 2√3b .
2√3a2κ8/3
Considering that the scalar field depends on extra
dimensional coordinate only (see Eq. (14)), the total derivative term
can be integrated once leading to the final form of the action
as follows:
In order to generate radion potential in ST theory, here we
adopt the GW mechanism [22] which requires a scalar field
in the bulk. For the case of ST theory presented in Eq. (10),
can act as a bulk scalar field. Considering a negligible
backreaction of the scalar field ( ) on the background spacetime,
the solution of metric G˜ M N is exactly same as well known
RS model i.e.
ds˜2 = e−2krc|φ|ημν d x μd x ν − rc2dφ2,
(11)
where k = 2−4M3 . Using these metric and explicit form of
U ( ), we obtain the scalar field equation of motion in the
bulk as follows,
(7)
(9)
− 4 rkc ∂∂φ + aκb ebκ
= 0.
With a non zero value of
has the following solution:
(8)
ebκ (φ)
4k 1
= ab2 y0 − rcφ
(12)
(
13
)
(14)
(15)
(16)
ebκvh
(18)
(19)
where y0 = a4bk2 e−bκvh and vh is the value of the bulk scalar
field on the hidden brane (φ = 0).
Using the solution of metric (see Eq. (11)), we obtain the
extrinsic curvature of φ = constant hypersurface as follows:
Qμν = ke−2krcφ ημν
and
Q = Qμν e2krcφ ημν = 4k.
The above expression of Q (trace of the extrinsic curvature)
leads to the boundary term of the action as,
Sb =
d4x √−gh e− 2 √53 κvh
+
√−gv e− 2 √53 κvv
4k
Vv + κ2
4k
Vh + κ2
ab
− 2√3kκ2
ab ebκvv
− 2√3kκ2
=
d4x √−gh Vhe f f +
√−gv Vve f f ,
(17)
(φ) (see Eq. (14))
where we use the explicit solution of
with vh = (0), vv = (π ). Further
Vhe f f = e− 2 √53 κvh Vh + κ4k2
ab
− 2√3kκ2
ebκvh
and
Vve f f = e− 2 √53 κvv Vv + κ4k2
ab
− 2√3kκ2
ebκvv ,
with gh , gv are the determinants of the induced metric on
hidden, visible brane respectively. It may be observed that the
boundary terms emerging from the total derivative of and
the Gibbons–Hawking terms modify the brane tensions of
the respective branes to produce the effective brane tensions
as Vhe f f and Vve f f .
Plugging back the solution of (φ) (Eq. (14)) into scalar
field action and integrating over φ yields an effective modulus
potential having the form as,
1 e−4krcπ 1
Ve f f (rc) = b2κ2 πrc − y0 + y0
− 2k
1 − e−4krcπ
where ’Ei’ denotes the exponential integral function.
It may be observed that the scalar field degrees of freedom
is related to the curvature as,
From the above expression, we can relate the boundary values
of the scalar field (i.e (0) = vh ) with the Ricci scalar as,
2
vh = √3κ
ln 1 + nα Rn−1(0) ,
where R(0) is the value of the curvature on Planck brane.
Thus the parameters that are used in the scalar–tensor theory
are actually related to the parameters of the original F (R)
theory.
Furthermore the various components of stress tensor of
the scalar field can be written as,
1 1 1
Tμν ( ) = − 2 ημν e−2krc|φ| | | − 2κ2b2 (rcφ − y0)2
(20)
(21)
(22)
and
and
Tφφ ( ) = 21 rc2 | | − 2κ32b2 (rcφ −1 y0)2 ,
where we use the solution of (φ) obtained in Eq. (14).
These above expressions of TM N ( ) lead to the ratio of
corresponding component of stress tensor between bulk scalar
field and bulk cosmological constant as,
Tμν ( ) a2b2
Tμν ( ) max = 1 − 32κ2k2| |
Tφφ ( ) 3a2b2
Tφφ ( ) max = 1 − 32κ2k2| |
e2bκvh
e2bκvh ,
where Tφφ ( ) and Tμν ( ) are different components of stress
tensor for the bulk cosmological constant. It may be observed
that for ebκvh < κk√| | , the stress tensor for the scalar field
ab
( ) is less than that for the bulk cosmological constant ( ) for
entire range of extra dimensional coordinate (i.e 0 < φ < π ).
This condition allows us to neglect the back-reaction of the
scalar field in comparison to bulk cosmological constant.
To introduce the radion field we replace rc → T˜ (t ) [22],
where T˜ (t ) is the fluctuation of the modulus around its vev
and is known as radion field. Here, for simplicity we assume
[22] that this new field depends only on t . The corresponding
metric ansatz is,
ds˜2 = e−2kT˜ (t)|φ|ημν d x μd x ν − T˜ 2(t )2dφ2.
(23)
Recall that the quantities in tilde are reserved for ST
theory. As mentioned earlier, the bulk scalar field fulfills the
requirement for generating the radion potential.
With the metric in Eq. (23), the five dimensional Einstein–
Hilbert part of the action yields the kinetic part of the radion
field in the four dimensional effective action as [22],
Skin [T˜ ] =
12M 3
k
d4x ∂μ(e−kπ T˜ (t))∂μ(e−kπ T˜ (t)).
As we see that T˜ (t ) is not canonical and thus we redefine the
field by the following transformation,
T˜ (t ) −→ T˜can(t ) =
24M 3 e−kπ T˜ (t)
k
Correspondingly the radion potential is obtained from Eq.
(20) by replacing rc by T˜ (t ) i.e.
1
VST (T˜ ) = b2κ2
e−4kπ T˜ (t) 1
π T˜ (t ) − y0 + y0
− 2k
1 − e−4kπ T˜
4k
− b2κ2 e−4ky0 Ei [4k(y0 − π T˜ )] − Ei [4ky0] .
In terms of T˜can, the Lagrangian of radion field becomes
L[T˜can] =
21 T˜˙c2an − VST (T˜can) ,
which is the same Lagrangian for a particle moving in a
potential VST .
The potential VST has a minimum at
2k
< π T˜ >+ = < πrc >+= y0 − b2κ2
1 −
b2κ2
8k2
and a maxima at
< π T˜ >− = < πrc >−
2k
= y0 − b2κ2
1 −
b2κ2
8k2
− 1 .
respectively. Recall y0 = a4bk2 e−bκvh , a and b are given by
Eq. (9). Moreover, Eq. (25) clearly indicates that VST (T˜ )
becomes zero at T˜ = 0 and reaches a constant value =
b2κ12 y0 − 2k + b42κk2 e−4ky0 Ei [4ky0] asymptotically at large
T˜ . In Fig. 1, we give the plot between VST (T˜ ) versus T˜ (t ).
(24)
(25)
+ 1
(26)
(27)
VST
rc t
√
Fig. 1 VST vs rc(t) = T˜ (t) for a = 1, b = 32 , k = M = 1,
−1, κvh = 0.01
=
Consequently, using the form of radion potential in Eq.
(25) with the transformation given in Eq. (24), one arrives at
the following mass squared of radion field in scalar–tensor
theory given as,
m˜ r2ad (ST ) = e−2kπ<rc>+
2 2 2
b κ
12M 3k2
× ⎣
⎡
1 +
According to Goldberger–Wise (GW) stabilization
mechanism [21,22], the modulus is stabilized at that separation
for which the effective radion potential becomes minimum.
Therefore, in the present context, the stable value of
interbrane separation is given by rc =< rc >+, which is
determined in Eq. (26). But due to quantum fluctuation, the radion
field has a non zero probability to tunnel from rc =< rc >+
to rc = 0, which in turn makes the brane configuration
unstable. So it is worthwhile to calculate the quantum tunneling
for radion field from rc =< rc >+ to rc = 0. In order to
do so, the radion potential is approximately considered as a
rectangle barrier having width (w) =< rc >+ and height (h)
= Ve f f (< rc >−) respectively. For such a potential barrier,
the tunneling probability ( PST ) is given by,
where m˜ rad is the mass of radion field, determined in Eq. (28)
and VST = VST (< rc >−) − VST (< rc >+), which can
be easily calculated from the expression of radion potential.
Obviously, PST depends on the parameters a and b. For
a = 1 (in Planckian unit), we give the plot between PST
versus b (see Fig. 2):
(28)
∗
(29)
Figure 2 clearly depicts that the tunneling probability
increases with the parameter b and asymptotically reaches
to unity at large b. It is expected, because with increasing
value of b, the height as well as the width both are ∝ b12
of the potential barrier decreases and as a consequence, PST
increases. Moreover, PST becomes zero as b tends to zero,
because for b → 0, the height of the potential barrier goes to
infinite and as a result, PST = 0. This character of global
minimum (as b tends to zero) actually overlaps with the
Goldberger–Wise result [21,22].
However, resolution of the gauge hierarchy problem
requires kπ < rc >+= 36, which in turn makes b 3 √13
(for a = 1). With these values of a and b, PST becomes
drastically suppressed and comes as ∼ 10−32. This small value
of tunneling probability guarantees the stability of the
interbrane separation (and hence of the radion field) at < rc >+.
Thus it can be argued that the smallness of tunneling
probability is intimately connected with the requirement of resolving
the gauge hierarchy problem. Further it may be mentioned
that these values of a and b are consistent with the condition
ebκvh < κk√| | , necessary for neglecting the backreaction of
ab
the scalar field in the background spacetime (as mentioned
earlier).
Now we turn our focus on radion potential as well as on
radion tunneling probability for the original F (R) model (Eq.
(5)).
5 Radion potential and tunneling probability in F(R)
model
Recall that the original higher curvature F (R) model is
described by the action given in Eq. (5). Solutions of metric
(G M N ) for this F (R) model can be extracted from the
solutions of corresponding scalar–tensor theory (Eqs. (11) and
(14)) with the help of Eq. (6). Thus the line element (in the
bulk) in F (R) model turns out to be
−T (t )2dφ2].
(T (t )) and for simplicity, this new field is assumed to be the
function of t . The metric takes the following form,
ds2 = e− √κ3 (t,φ)[e−2kT (t)|φ|ημν d x μd x ν
From the angle of four dimensional effective theory, T (t ) is
known as radion field. Moreover (t, φ) is obtained from
Eq. (14) by replacing rc to T (t ). Plugging back the solution
(see Eq. (35)) into five dimensional action and integrating
over φ generates a kinetic as well as a potential part for the
radion field T (t ). Kinetic part comes as
In the present context, we take the form of F (R) as
F (R) = R + α Rn and thus the above field equation is
simplified to the form:
Skin[T ] = 21 f 2
d4x ∂μ(e−kπ T (x))∂μ(e−kπ T (x))
where the factor f has the following form:
(32)
f =
24M3
k
1 +
ds2 = e− √κ3 (φ) e−2krc|φ|ημν d x μd x ν − rc2dφ2 ,
(30)
where (φ) is given in Eq. (14).
At this point, we need to verify whether the above solution
of G M N (in Eq. (30)) satisfies the field equations of the
original F (R) theory. The five dimensional gravitational field
equation for F (R) theory is given by,
21 G M N R − RM N + α2 G M N Rn − nα Rn−1 RM N
−nαG M N
Rn−1 + nα∇M ∇N Rn−1. = 0
It may be shown that the solution of G M N in Eq. (30)
satisfies the above field equation to the leading order of κvh .
It may be recalled that the equivalence of the chosen F (R)
model was transformed to the potential of the scalar–tensor
model in the leading order of κvh . Thus it guarantees the
validity of the solution of spacetime metric (i.e. G M N ) in the
original F (R) theory.
However this solution of G M N immediately leads to the
separation between hidden (φ = 0) and visible (φ = π )
branes along the path of constant x μ as follows:
π d = rc
0
π
dφe− 2√κ3 (φ),
where d is the interbrane separation in F (R) model. Using the
explicit functional form of (φ) (Eq. (14)), above equation
can be integrated and simplified to the following one,
κ ab2
kπ d = kπrc 1 − 2√3 vh + 16√3k πrc ,
where the sub-leading terms of κ are neglected. rc is the
modulus in the corresponding ST theory and has a stable
value at < rc >+, which is shown in the previous section (see
Eq. (26)). So, it can be argued that due to the stabilization of
ST theory, the modulus d in F (R) model has also a stable
value at,
(33)
kπ < d >+ = kπ < rc >+
κ √1 e−bκvh
1 − 2 3
√ vh + 4 3
a
− 8√3κ2
1 +
1 −
b2κ2
8k2
. (34)
A fluctuation of branes around the stable configuration
is now considered. This fluctuation can be taken as a field
(35)
(36)
.
(37)
(38)
(39)
It may be observed that VF(R)(Tcan) goes to zero as the
parameter a tends to zero. This is expected because for
a → 0, the action contains only the Einstein part ((recall
from Eq. (9) that the higher curvature parameter α is
proportional to a)) which does not produce any potential term
for the radion field [22]. Thus for five dimensional warped
geometric model, the radion potential is generated from the
higher order curvature term α Rn. Again the Lagrangian for
the canonical radion takes the following form:
Due to the appearance of f, T (x ) is not canonical and in
order to make it canonical, we redefine the field as
T (x ) −→ Tcan(x ) = f e−kπ T (x).
For a → 0, the action contains only the linear term in Ricci
scalar and the factor f goes to 24M 3/ k which agrees with
[22].
Finally the potential part of radion field is as follows,
20
VF(R)(Tcan) = √
3
f 4k 51++2√√33bb
M6
4
× 1 + √3
5+2√3b
κvh 4+4√3b
L[Tcan] =
21 T˙c2an − VF(R)(Tcan) ,
which matches with the Lagrangian for a point particle
moving under a potential VF(R).
The radion potential in F (R) model has also a minima and a
maxima at < Tcan >+ and at < Tcan >− respectively, where
< Tcan >+= f e−kπ<d>+
and
< Tcan >−= f e−kπ<d>−
with < d >+, < d >− have the following expressions:
< d >± = < rc >±
where < rc >± are determined in Eq. (26) and in Eq.
(27) respectively. We emphasize that due to the presence of
conformal factor connecting the two theories, the value of
< d >± (in F (R) model) is different from < rc >± (in ST
model). Finally the squared mass of radion field is as follows,
mr2ad = e−2kπ<d>+
√3aκ8/3 2+23√3b k 71++4√√33bb
It is evident that mass of the radion field also goes to zero as
a → 0 (higher curvature parameter α is proportional to a).
Using the form of VF(R)(Tcan) along with the transformation
Eq. (38), now we give the plot between radion potential and
T (t ) (see Fig. 3).
Figure 3 clearly depicts that the radion potential goes to
zero at T (x ) = 0 and reaches a constant value asymptotically
at large value of T (x ). Comparing Figs. 1 and 3, it is clear that
the nature of radion potential does not change in comparison
to that in ST theory. However, due to the conformal factor, the
extremas of the potential are shifted in F (R) model, which
is clear from Eq. (33).
As per GW mechanism, the radion field is stabilized at
< d >+. But as mentioned earlier, due to quantum
mechanical tunneling effect, there exists a non zero tunneling
probability of the radion field from d =< d >+ to d = 0. Again
considering the radion potential as a rectangle barrier having
width (w =< d >+) and height (h = VF(R)(< d− >)), we
(40)
(41)
(42)
(43)
0.8
0.6
0.4
0.2
Fig. 4 PF(R) vs b for a = 1, k = M = 1,
calculate the tunneling probability ( PF(R)) from d =< d >+
to d = 0 and is given by,
1
PF(R)
= 1 +
VF(R)(< d >−)
VF(R)
where mrad is given in Eq.(43) and VF(R) = VF(R)(<
d− >) − VF(R)(< d+ >). From Eq. (44), it is clear that
PF(R) depends on both the parameters a and b. Here we take
a = 1 (in Planckian unit) and give the plot demonstrating the
variation of PF(R) with respect to b (see Fig. (4)).
Figure 4 reveals that just as in ST theory, PF(R) (tunneling
probability in F(R) model) increases with increasing value of
b and acquires the maximum value (= 1) asymptotically at
large b. For b → ∞, the higher curvature parameter (α)
goes to zero (see Eq. (9)) and the action reduces to Einstein–
Hilbert action. This in turn makes the brane configuration
unstable [22] and as a consequence the tunneling probability
becomes unity. As the parameter b decreases, the effect of
higher curvature term starts to contribute and as a result, the
modulus is stabilized at a certain separation and hence the
probability for tunneling becomes less than one. Furthermore
for b → 0, higher curvature parameter α → ∞, which in
turn makes the height of the radion potential barrier infinity
(height ∝ b12 ) and thus the potential acquires a global
minimum. As a consequence, the tunneling probability tends to
zero, which is shown in Fig. 4). The character of global
minimum actually mimics the result of Goldberger and Wise
[21]. It is expected because for b → 0, the bulk scalar
potential in the present context (U ( )) becomes quadratic (all the
other terms are proportional to higher power of b and can be
neglected) as same as the potential considered in [21].
Finally we examine whether the solution of gauge
hierarchy problem in F(R) model leads to a small value of the
tunneling probability or not. We find that the resolution of
gauge hierarchy pro√blem requires kπ < d >+= 36, which
in turn makes b = 32 . For this value of b, PF(R) is highly
suppressed and takes the value of ∼ 10−32. Therefore, in
original F (R) theory, the requirement for solving the gauge
hierarchy problem is correlated with the smallness of radion
tunneling probability (a similar analysis is also obtained in
ST theory as discussed in Sect. 4).
6 Conclusion
In this work, we consider a five dimensional compactified
warped geometry model with two 3-branes embedded within
the spacetime. Due to large curvature (∼ Planck scale), the
bulk spacetime is governed by a higher curvature theory like
F (R) = R + α Rn. In this scenario, we determine the radion
potential from the scalar degrees of freedom of higher
curvature gravity and investigate the possibility of tunneling for
the radion field. Our findings and implications are as follows:
• Due to the presence of higher curvature gravity in the
bulk, a potential term for the radion field is generated, as
shown in Eq. (39). This is in sharp contrast to a model with
only Einstein term in the bulk where the modulus
potential can not be generated without incorporating any
external degrees of freedom such as a scalar field. However for
the higher curvature gravity model, this additional degree
of freedom originates naturally from the higher curvature
term. It may also be noted that the radion potential goes
to zero as the higher curvature parameter α → 0.
• The radion potential (VF(R)) has a minimum (< d >+)
and a maximum (< d >−) respectively where the
height between minimum and maximum of the potential
depends on both the parameters α and n. Moreover, VF(R)
becomes zero at T (x ) = 0 (T (x ) is the radion field) and
reaches a constant value asymptotically at large T (x ), as
depicted in Fig. 3.
• According to GW mechanism, the modulus is stabilized
at < d >+. But due to quantum mechanical effect, there
exists a possibility of tunneling for the radion field from
d =< d >+ to d = 0, which in turn makes the
aforementioned brane configuration unstable. We calculate
this tunneling probability ( PF(R)) which depends on the
parameters a and b (a and b can be written in terms of
α and n, see Eq. (9)). For a certain choice of a, PF(R)
increases with increasing value of b, as demonstrated in
Fig. 4. It may be observed that this behaviour of PF(R)
with the parameter b is expected, because the height of
the potential barrier decreases as b increases and as a
result, PF(R) increases. Finally we find that the solution
of gauge hierarchy problem requires kπ < d >+= 36,
which in turn highly suppresses the tunneling
probability and as a consequence, PF(R) comes as ∼ 10−32. This
small value of the tunneling probability guarantees the
stability of interbrane separation at < d >+. Therefore,
it can be argued that the smallness of tunneling
probability is interrelated with the requirement of solving the
gauge hierarchy problem.
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