全文

翻譯

開啟檢索術語導覽

http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-015-3597-0&domain=pdf

Web End = http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-015-3597-0&domain=pdf

Web End = Eur. Phys. J. C (2015) 75:368DOI 10.1140/epjc/s10052-015-3597-0

Regular Article - Theoretical Physics

http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-015-3597-0&domain=pdf

Web End = http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-015-3597-0&domain=pdf

Web End = The structure of f (R)-brane model

Zeng-Guang Xu1,a, Yuan Zhong1,2,b, Hao Yu1,c, Yu-Xiao Liu1,3,d

1 Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, Peoples Republic of China

2 IFAE, Universitat Autnoma de Barcelona, Bellaterra, 08193 Barcelona, Spain

3 Key Laboratory for Magnetism and Magnetic Materials of the MoE, Lanzhou University, Lanzhou 730000, China Received: 13 May 2015 / Accepted: 1 August 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract Recently, a family of interesting analytical brane solutions were found in f (R) gravity with f (R) = R + R2

in Bazeia et al. (Phys Lett B 729:127 2014). In these solutions, the inner brane structure can be turned on by tuning the value of the parameter . In this paper, we investigate how the parameter affects the localization and the quasilocalization of the tensorial gravitons around these solutions. It is found that, in a range of , despite the brane having an inner structure, there is no graviton resonance. However, in some other regions of the parameter space, although the brane has no internal structure, the effective potential for the graviton KaluzaKlein (KK) modes has a singular structure, and there exist a series of graviton resonant modes. The contribution of the massive graviton KK modes to Newtons law of gravity is discussed briey.

1 Introduction

The braneworld scenarios [15], have attracted more and more attention. They present new insights and solutions for many issues such as the hierarchy problem, the cosmological problem, the nature of dark matter and dark energy, and so on. In some braneworld scenarios, all the elds in the standard model are assumed to be trapped on a submanifold (called brane) in a higher-dimensional spacetime (called bulk), and only gravity transmits in the whole bulk. So a natural and interesting question is how to reproduce the effective four-dimensional Newtonian gravity from a bulk gravity.

In models with compact extra dimension, such as the ADD braneworld model [2,3], the effective gravity on the brane is transmitted by the massless graviton KaluzaKlein (KK)

a e-mail: mailto:xuzg12@lzu.edu.cn

Web End =xuzg12@lzu.edu.cn

b e-mail: mailto:zhongy2009@lzu.edu.cn

Web End =zhongy2009@lzu.edu.cn

c e-mail: mailto:yuh13@lzu.edu.cn

Web End =yuh13@lzu.edu.cn

d e-mail: mailto:liuyx@lzu.edu.cn

Web End =liuyx@lzu.edu.cn

mode (also called the graviton zero mode). Since the graviton zero mode is separated from the rst KK excitation by a mass gap, gravity is effectively four-dimensional at low energy. In models with innitely large extra dimensions, like the one proposed by Randall and Sundrum (the RS2 model) [5], although the graviton spectrum is gapless now, the effective gravity on the brane is shown to be the Newtonian gravity plus a subdominant correction, so the low energy effective gravity is also four-dimensional [69]. This is because the wave functions of all the massive graviton KK modes are suppressed on the brane, so the massive modes only contribute a small correction to the Newtonian gravity at large distance.

Another class of interesting models with innite extra dimensions is the so called thick branes. In these modes, the original singular thin brane in the RS2 model is replaced by some smooth thick domain walls generated by one or a few background scalars [1012]. One of the interesting features of the thick domain wall brane is that the massive graviton modes feel an effective volcano-like potential, which might support some resonances. This feature was rst noticed in Ref. [10].

Such resonant modes of the graviton can be interesting both phenomenologically and theoretically. Phenomenologically, the wave function of a graviton resonance peaks at the location of the brane and behaves as a plain wave in the innity of the extra dimension. As compared to the RS2 model, the massive modes of a thick brane might contribute a different correction to the Newtonian gravity. As for the theoretical aspect, the metastable massive graviton in thick brane models provides an alternative for massive gravity theory [13,14]. Unfortunately, the early proposal for a thick brane [1012] failed in nding a massive graviton resonance.

To construct a model that supports a graviton resonance, one has to tune the shape of the effective potential for the graviton. For typical thick brane models, where the gravity is taken as in general relativity, the only possible way is to

123

368 Page 2 of 10 Eur. Phys. J. C (2015) 75:368

tune the shape of the warp factor. Some successful models can be found in Refs. [1518]. There is another way, however, to tune the effective potential for the graviton if the gravity is described by a more general theory. For example, in f (R) gravity (see [1921] for comprehensive reviews on f (R) gravity and its applications in cosmology), the effective potential is determined by both the warp factor and the form of f (R) (see Ref. [22] for details). So, in principle, graviton resonances can be turned on by tuning gravity.

However, the construction of a thick f (R)-brane model is not easy in practice (see Refs. [2333] for work on f (R)-branes). First of all, the dynamical equation in f (R)-gravity is of fourth order. So, traditional methods that help us to nd analytical brane solutions in second order systems, such as the superpotential method (also known as the rst order formalism [34,35]) do not work in a general f (R)-brane model. Usually one can only solve the system either by using numerical methods [27], or by imposing strict constraints on the model, for example, by assuming the scalar curvature to be constant [24].

The linearization of a f (R)-brane model is also a challenging task. Naively, the linear perturbations around an arbitrary f (R)-brane solution should also satisfy some fourth order differential equations. However, in Ref. [22], the authors found that the tensor perturbation equation can be nally rewritten as a second order Schrdinger-like equation. The results of Ref. [22] enable us to analyze the graviton mass spectrum of any thick f (R)-branes solutions. For example, in Ref. [28], the authors constructed an analytical thick brane solution in a model with f (R) = R + R2.

The solution is stable against the tensor perturbation. The localization of zero modes of both graviton and fermion is also shown to be possible. Unfortunately, no graviton resonance was found in the model of Ref. [28]. Recently, a more general analytical thick f (R)-brane solution was reported in Ref. [33]. The solution of Ref. [33] is a generalization of the one in [28]. The model in Ref. [33] has an interesting feature: an inner brane structure appears for a particular range of the parameter . Usually, the appearance of an inner brane structure is accompanied by graviton resonances [1518]. So it is interesting to see if it is possible to nd graviton resonances in the model of [33], and what the relation is between the inner brane structure and the graviton resonance. These two questions constitute the motivation for the present work.

In Sect. 2, we briey review the model and corresponding solutions in Refs. [28,33]. Then, in Sect. 3, we study the localization of the graviton zero mode and the condition for graviton resonances in the model of [33]. The correction from the massive graviton KK modes at small distance is discussed in order to compare with the constraints of breaking the Newton inverse square law from experiments given in Ref. [36]. The conclusion and discussions will be given in Sect. 4.

2 Review of the f (R)-brane model and solutions

We start with the ve-dimensional action of f (R) gravity minimally coupled with a canonical scalar eld,

S = [integraldisplay] d4xdyg [parenleftBigg]

225

f (R)

2MM V ()[parenrightBigg] ,

(1)

where f (R) is a function of the scalar curvature R, and 25 = 2M3 with M the fundamental ve-dimensional Planck

mass. In the following, we set M = 1. The signature of

the metric is taken as (, +, +, +, +), and the bulk coor

dinates are denoted by capital Latin indices, M, N, =

0, 1, 2, 3, 4, and the brane coordinates are denoted Greek indices, , , = 0, 1, 2, 3.

We are interested in the static Minkowski brane embedded in a ve-dimensional spacetime, so the line element is assumed as

ds2 = e2A(y)dxdx + dy2, (2) where e2A(y) is the warp factor, y = x4 stands for the extra

dimension, and is the induced metric on the brane.

For static brane solutions with the setup (2), the background scalar eld is only a function of the extra dimension, i.e., = (y). Therefore, the equations of motion for

the f (R)-brane system are

+ 4A = V, (3) f + 2 fR(4A 2 + A ) 6 f R A 2 f R = 25( 2 + 2V ), (4)

8 fR(A + A 2) + 8 f R A f = 25( 2 2V ), (5) where the prime denotes the derivative with respect to y, fR d f (R)/dR, and V dV ()/d.

In Ref. [28], a toy model with

f (R) = R + R2 (6) and the 4 potential

V () = (2 v2)2 + 5 (7) was considered, where > 0 is the self-coupling constant of the scalar eld, and v is the vacuum expectation value of the scalar eld. An analytical solution was found in Ref. [28]:

eA(y) = sech(ky), (8)

(y) = v tanh(ky), (9) where the parameters are given by

k = [radicalbigg]

3232 , =

3 784

, (10)

v = 7[radicalBigg]

3 2925

, 5 =

477 6728

1 25

. (11)

123

Eur. Phys. J. C (2015) 75:368 Page 3 of 10 368

The scalar eld satises (0) = 0 and () = v, and

the potential reaches the minimum (the vacuum) at = v.

The energy density = T00 = e2A(12 2 + V ()) peaks at

the location of the brane, y = 0, and it tends to vanish at the

boundary of the extra dimension. The brane is embedded in an anti-de Sitter spacetime.

The tensor perturbation of the above brane solution has been analyzed in Ref. [28]. It was shown that the solution is stable against the tensor perturbation and the gravity zero mode is localized on the brane. Furthermore, it was found that although there are fermion resonant KK modes on the brane, there are no graviton resonant modes [28].

Recently, a general solution was constructed in Ref. [33] with the same f (R) as given in Eq. (6). The warp factor is assumed as the general form of (8) with two positive parameters B and k:

eA(y) = sechB(ky). (12)

The corresponding Ricci tensor at the boundary of the extra dimension is given by

RM N (y ) 4B2k2gM N effgM N , (13) from which we can see that the background spacetime is asymptotical AdS with the effective cosmological constant eff = 4B2k2. The curvature is

R(y) = 4Bk2[5B 5(B + 2)sech2(ky)]. (14)

Then from Eq. (4) we can give the derivative of the scalar eld:

2 = Bk2sech2(ky)[braceleftbigg]

32 4k2[bracketleftBig]5

B2 + 16B + 8

(5B2 + 32B + 12)sech2(ky)[bracketrightbigg][bracerightbigg]. (15)

Note that the here corresponds to in Ref. [33]. Thus,

2 0 implies 1

3 32(1 + 4B)k

2.0

1.5

1.0

0.5

3 2 1

1 2 3 y

0.5

(a)

2 1

1 2 y

(b)

Fig. 1 The energy density (y) of the brane system. The parameters are set to B = 1 (a) and B = 4 (b), k = 1, and = 1 (solid thin

purple line), = s (dotteddashing red line), = 0 (dashed blue

line), and 2 (thick black line).

Solving d

dy2

[vextendsingle][vextendsingle]

y=0 = 0 results in

= s

3 + 9B

8k2(16 + 60B + 49B2)

, (18)

where 1 < s < 2 for nite B. So y = 0 is an inection

point of when = s, and the brane will have an internal

structure when s. The energy density (y) of the brane

system is shown in Fig. 1.

For an arbitrary B and a xed , the analytical solutions for the scalar eld and scalar potential were found in Ref. [33]: when = 1, the solution is(y) = v1[1 sech(ky)]sign(y), (19)

V () = c2[parenleftbig]||

8(8 + 16B + 5B

2)k2 2.

(16)

2 (|| v1)2 c1[bracketrightbig] c0; (20)

when = 0, we have

(y) = 6B arctan [bracketleftbigg]tanh [parenleftbigg]

The energy-momentum tensor density is

= e2A(y) [parenleftbigg]

[parenrightbigg][bracketrightbigg] , (21)

V () = 3B2k2 + 3Bk2 [parenleftbigg]B +

2MM + V ()[parenrightbigg]

= Bk2sech2B(ky)[braceleftbigg] 3B + 3 [parenleftbigg]B +

[parenleftBigg][radicalbigg]

3B [parenrightBigg] ; (22)

when = 2, the result is just the one found in Ref. [28]: (y) = v2 tanh(ky), (23)

V () = 1(2 v22)2 0. (24)

1 4

cos2

sech2(ky)

+ 4k2[bracketleftbig]5B3

(10B3 + 37B2 + 32B + 8)sech2(ky)

+ (5B3 + 37B2 + 44B + 12)sech4(ky)[bracketrightbig][bracerightbigg]

. (17)

123

368 Page 4 of 10 Eur. Phys. J. C (2015) 75:368

Here ci, vi, and i are positive parameters determined by B and k. We only list the expressions of vi:

v1 = [radicalbigg]

3B(6 + B)(2 + 5B)

8 + 32B

(z) of the tensor perturbations, which is a Schrdinger-like equation [22]:

[2z + W(z)](z) = m2(z), (33) where the effective potential W(z) is

W(z) =

, (25)

v2 = [radicalBigg]

3B(6 + B)(2 + 5B)

16 + 2B(16 + 5B)

. (26)

Now, it is clear that, when B = 1 and = 2 =

3232k2 , the

exact solution described by Eqs. (12), (23), (24), and (26) is just the one given in Eqs. (6)(11), for which the brane has no internal structure.

3 Localization and resonant KK modes of the tensor uctuation

In this section, we investigate the stability of the solution against tensor uctuations of the metric as well as the localization of gravity on the brane. Especially, we will nd that the effective potential for the KK modes of the tensor uctuation has a rich structure and it will support some resonant KK modes when B is large enough and > 0.

We start with the transverse-traceless (TT) tensor perturbations of the background spacetime:

ds2 = e2A(y)( + h)dxdx + dy2, (27) where h = h(x, y) depends on all the spacetime coor

dinates and satises the TT condition

h = h = 0. (28) It can be shown that the TT tensor perturbations h is decoupled from the scalar and vector perturbations of the metric as well as the perturbation of the scalar eld =

(za)2

2za

a +

a2 +

3 2

zaz fR

a fR

(z fR)2

2z fR

1 4

fR . (34)

Equation (33) can be factorized as

KK(z) = m2(z), (35) with

K = +z + [parenleftbigg]

3 2

f 2R +

a +

z fR

, (36)

K = z + [parenleftbigg]

3 2

a +

z fR

, (37)

which indicates that there is no graviton mode with m2 < 0.

For Eq. (33), the solution of the zero mode with m = 0 is (0)(z) = N0a3/2(z) f 1/2R(z). (38)

It is easy to show that (0)(z) is normalizable, i.e.,

[integraldisplay]

(0)(z)|2dz < , (39)

which implies that the zero mode is localized on the brane. Here we note that if fR(z) = 1 + 2R(z) = 0 has a solu

tion z = z0, then the effective potentials W would have

singularities at z = z0 and the corresponding zero mode

would vanish at that point. We note here that by making the KK reduction for the zero mode we will get the following relation between the effective four-dimensional Planck mass MPl and the fundamental ve-dimensional Planck mass M:

M2Pl M3/k, (40) which results in the effective four-dimensional general relativity. Hence, it is natural to set MPl, M, and k as the same

scale (as in the RS-2 model),

MPl M k, (41)

so that there is no hierarchy between them.

There is no analytical expression for the potential W with respect to the coordinate z. However, we can express it in the coordinate y as

W =

(x, y).

The perturbed Einstein equations for the TT tensor perturbations are given by [22]

(5)h =

f RfR yh, (29)

or, equivalently,

a2 (4)h + 4 a

a h + h [parenrightbigg] fR + h f R = 0, (30)

where a(y) = eA(y). By making the coordinate transforma

tion dz = a1dy, Eq. (30) becomes

2z + [parenleftbigg]3 za

a +

z + (4)[bracketrightbigg] h = 0. (31)

Then, by making the decomposition

h(x, z) = (a3/2 f 1/2R) (x)(z), (32) where (x) satises the TT condition = 0 =

, we can get from (31) the equation for the KK modes

z fR

a(y)a (y) a(y)

3 2

a(y)a 2(y) + a(y)

2a (y) a(y)

3a2(y)a (y)R (y) a(y)(1 + 2R(y))

2a2(y)R 2(y) (1 + 2R(y))

123

Eur. Phys. J. C (2015) 75:368 Page 5 of 10 368

a(y)a (y)R (y) + a 2(y)R (y)[bracketrightBig]

1 + 2R(y)

4 sech

2B(ky)

15B2 (2 + 3B)(4 + 5B)sech 2(ky)

128B(2 + 5B)(1 + 16Bk

2)k2

(1 + 8B(4 + 5B)k

2 + (1 40B

2k2) cosh(2ky))2

16(1 + 2B)(1 + 16Bk

+ 1 + 8B(4 + 5B)k

2 + (1 40B

2k2) cosh(2ky)

(42)

k2,

y ky, z kz, W W/k2, and m m/k. Then z is a

function of just y and Eq. (33) becomes

[bracketleftbig]

2z + W(z)[bracketrightbig](z) = m2

(z), (43) where the effective potential is

W(z( y)) =

14 sech

From now on, we dene dimensionless variables

15B2 (2 + 3B)(4 + 5B)sech 2

128B(2 + 5B)(1 + 16B

)

1 + 8B(4 + 5B) + (1 40B 2

) cosh(2 y)[bracketrightbig]2

16(1 + 2B)(1 + 16B

)

2 ) cosh(2 y) [bracketrightbigg]

+ 1 + 8B(4 + 5B)

+ (1 40B

(44)

Now the parameter k does not appear, which implies that we only need to consider different values of the dimensionless varieties B and

to search for resonant modes. For convenience, we remove the bars on all variables, which is equivalent to letting k = 1. So we have

1 =

3 32(1 + 4B)

, (45)

s =

3 + 9B

8(16 + 60B + 49B2)

, (46)

2 =

38(8 + 16B + 5B2)

. (47)

Note that fR = 1+2R(y) appears in the denominator of

the expression of W(z), (34), or W(z(y)), (42). For general relativity, we have fR = 1 and so the effective potential

W(z(y)) is always regular for a smooth solution of the thick brane. However, for the f (R) gravity with f (R) = R +

R2, fR may be vanishing and the effective potential will be divergent, i.e.,

1 + 2R(y) = 1 8B[2 + (2 + 5B) tanh2(y)] = 0(48)

will lead to the singularities of the effective potential W. This will result in two -like potential wells which are related to the appearance of the resonant KK modes of the tensor perturbations.

Fig. 2 The structure of the parameter space of (, B) for the f (R)-brane model. Areas I (1 < < s for a xed B), II (s < < k for a xed B), and III (k < < 2 for a xed B) correspond to 2 0. Area I corresponds to the split branes. Area III corresponds to

the singular W

Now we analyze the parameter space of (B, ) where Eq. (48) has solutions. Since 1 + 2R(y) is an even func

tion of y, we only consider the region y (0, +). We

rst discuss the case of > 0. It is clear that 1 + 2R(y)

decreases with y and 1 + 2R(0) = 1 + 16B > 0. So the

sufcient and necessary condition that Eq. (48) has a solution is 1 + 2R(y +) = 1 40B2 < 0. Therefore, the

relation between and B is

> 1

40B2 > 0. (49)

On the other hand, according to the constraint on (16), we should have 1

40B2 < 2, which results in B > 2. Next, we discuss the case of < 0, for which 1 + 2R(y) increases

with y and 1 + 2R(y +) = 1 40B2 > 0. So

the sufcient and necessary condition that Eq. (48) has a solution is 1 + 2R(0) = 1 + 16B 0. Therefore, the

relation between and B is

116B < 0. (50)

Furthermore, the constraint on (16) requires

1 16B

namely B 25, which contradicts with B > 0. So Eq. (48)

has no solution for negative . In conclusion, only when the constraint conditions

B > 2 and k

140B2 < 2 (51) are satised, does Eq. (48) have a solution. The relations among 1, s, k, 2 are shown in Fig. 2. Because of the monotony of it, fR (the derivative of f (R) with respect of R) is negative in some range of y as long as Eq. (48) has a solution. This will lead to the existence of ghosts.

In order to judge whether there are resonant modes, we consider the partner equation of the Schrdinger-like equation (35): KK(z) = m2(z), for which the corresponding

123

368 Page 6 of 10 Eur. Phys. J. C (2015) 75:368

Fig. 3 The effective potential W and the dual one Ws as functions of y and (1 2). The parameters are set to B = 2 (a, b), and B = 4

(c, d)

potential is given by

Ws(z) =

(see Fig. 3). Furthermore, it can be seen from Fig. 3 that there is also another interesting quasiwell with singularity for large B and satisfying the constraint conditions (51).

To get the numerical solution of Eq. (33), we impose the following conditions:

even(0) = 1, zeven(0) = 0; (57) odd(0) = 0, zodd(0) = 1. (58)

Here even and odd denote the even and odd parity modes of (z), respectively. The solution under this imposition does not affect the relative probability dened below.

The function |(z)|2 can be interpreted as the probability

of nding the massive KK modes at the position z along the extra dimension [37]. In order to nd the resonant modes, we denite the relative probability [37],

P(m2) = [integraltext]

zb |(z)|2dz

[integraltext]

(za)2

2za

a +

15 a2

3 2

zaz fR

a fR

(z fR)2

2z fR

fR . (52)

Similar to W(z), there is no analytical expression for Ws(z). In the y coordinate, we have

Ws(z(y)) =

14 sech

f 2R

2B y

3B2 (2B + 3B2)sech2 y

384B(2 + 5B)(1 + 16B)

+ [1 + 8B(4 + 5B) + (1 40B

2) cosh(2y)]

16B 128B(2 + 3B)

+ 1 + 8B(4 + 5B) + (1 40B

2) cosh(2y)

(53)

The structure of W and Ws is shown in Fig. 3.

Because dzdy = a(y) is positive, the shape of Ws(z) is the

same as that of Ws(z(y)). Thus, in order to estimate whether there are resonant modes we just need to consider the shape of Ws(z(y)). The appearance of the quasiwell of the dual potential may lead to resonances of the graviton KK modes. Noticing

Ws(0) = W(0) > 0, (54)

0 < Ws|z 0, (55) 0 < W|z 0, (56)

it is convenient to judge whether there are resonant modes for Eq. (33) from Ws. Solving 2yW|y=0 = 0 and 2yWs|y=0 = 0,

respectively, results in

8 36B 46B

, (59)

where 2zb is approximately the width of the brane, and zmax is taken as zmax = 10zb. It is clear that large relative proba

bilities P(m2) of nding massive KK modes within a narrow range zb < z < zb around the brane location indicate the

existence of resonant modes.

From Fig. 3, we can see that there is a quasiwell when 1, which is consistent with our analysis. So, it seems

that we may nd resonant modes for small . When 2

and B 4, which satises the condition (51), the effective

potential has two -like potential wells and there are found resonant modes.

Firstly, we discuss the case of B 2. When approaches

1, the dual potential Ws(z) has a quasiwell in the middle, so resonant modes of Eq. (33) may appear. We use the numerical method to search for resonant modes for B 2, but no

resonant mode is found. The reason is that the quasiwell is not deep or wide enough. The potential W(z) for B = 2

is shown in Fig. 4a, b. The relations between the relative probability P and the mass square m2 for B = 2 are shown

in Fig. 5.

Secondly, we discuss the case of B > 2. Similar to the case of B 2, there is a quasiwell in the middle of the

zmax

zmax |(z)|2dz

2 + (2 + 5B)[radicalbig](16 +

52B + 67B

16B(2 + 3B)(4 + 7B)

2 (2 + 5B)[radicalbig](16 +

16B + 13B

8 + 24B 2B

16B2(38 + 107B)

This implies that both the effective potential and the corresponding dual one have an interval structure in the range of 1 < < and the range of 1 < < , respectively

123

Eur. Phys. J. C (2015) 75:368 Page 7 of 10 368

4 2

2 4 z

4 2

2 4 z

4 2

(a)

(b)

(c)

10 5

5 10 z

6 4 2

2 4 6 z

(d)

(e)

(f)

Fig. 4 The shapes of the effective potential W(z). The parameters are set to B = 2 (a, b) and B = 4 (c, d), = 1 (black line), = s (dashed

red line), and = 0 (dotted blue line) (a, c)

1.0P

0.8

0.6

0.4

0.2

0.0 0.2 0.4 0.6 0.8 m2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 m2

0.0 0.5 1.0 1.5 2.0 2.5 m2

(a)

(b)

(c)

(d)

Fig. 5 The relative probability P(m2) of the KK modes for Eq. (33) with B = 2. The dashed red and solid blue lines are for odd and even parity

KK modes, respectively

dual potential Ws(z), but no resonant modes are found when 1 k (see Figs. 4c, d, and 6ad). When k < 2,

the potential W(z), which has Z2 symmetry and is shown in Fig. 4e, f, will have two -like potential wells. A series of resonant modes appears (see Fig. 6e, f) and their wave functions are shown in Fig. 7 for B = 4.

The last point of interest for this section is to calculate the contribution of the massive graviton KK modes including the resonances to Newtons law. The correction to Newtons law between two mass points m1 and m2 localized at z = 0, with

a distance r from the KK modes, is given by [6,8]

U(r) GN m1m2 r

1 + [integraldisplay]

dmk emr2m(0)[bracketrightbigg] , (60)

where m(z) is normalized such that m(z)|z cos(z),

and the effective four-dimensional Newtons constant GN is given by

GN M2Pl. (61)

Next, we rst calculate the expression of m(0). As examples, we just discuss two representative cases: B = 4, = 0,

and B = 4, = 2. The corresponding numerical results are

shown in Fig. 8. As we can see from Fig. 8a, b, the trend of m(0) with m changing is analogous to the relative probability P(m2) (see Fig. 6c, f). Secondly, in order to do the integration in Eq. (60) we need to obtain analytical expression of m(0) by using some approximate methods. One of the simplest methods is to simulate the original function with a few simple linear functions. For example, for the case of B = 4, = 0, the original function can be divided into two

parts (see the blue line in Fig. 8a):

m(0) =

m4k , 0 m 4k, m(0) = 1, m > 4k.

(62)

123

368 Page 8 of 10 Eur. Phys. J. C (2015) 75:368

1.0

0.8

0.6

0.4

0.2

0.0 0.5 1.0 1.5 m2

0.0 0.5 1.0 1.5 2.0 m2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 m2

(a)

(b)

(c)

1.0

0.8

0.6

0.4

0.2

0.0 0 1 2 3 4 m2

2 4 6 8 10 m2

10 20 30 40 50 60 m2

(d)

(e)

(f)

Fig. 6 The relative probability P(m2) of the KK modes for Eq. (33) with B = 4 and k = 1. The dashed red and solid blue lines are for odd and

even parity KK modes, respectively

20 10

0.4

10 20 z

0.2

20 10

10 20 z

20 10

0.2

0.4

1.5

0.3

1.0

0.2

0.5

0.1

0.5

20 10

10 20 z

0.5

20 10

10 20 z

0.1

1.0

0.5

1.5

0.2

2.0

0.3

1.0

Fig. 7 The resonant wave function n(z) for Eq. (33) with B = 4, k = 1, and = 2. The red and solid blue lines are for odd and even parity

KK modes, respectively. The coordinate axis n denotes the wave function for the nth resonant mode

Note that m(0) is dimensionless. Substituting this result into Eq. (60), we get the approximate expression for the gravity potential U(r):

U(r) = GN

m1m2

1 +1kr

83 + 4kr + O[(kr)

][parenrightbigg] ,

1 +e4kr(4kr + e4kr 1)

8(kr)3 + O [bracketleftbigg]

1 (kr)4

. (63)

We expand U(r) in terms of kr in two extreme situations:

U(r 1/k)

GN m1m2 r

U(r 1/k)

GN m1m2 r

1 +1

[bracketrightbigg][parenrightbigg] . (64)

8(kr)3

123

Eur. Phys. J. C (2015) 75:368 Page 9 of 10 368

Fig. 8 m(0) for resonant wave function (z) with B = 4,

= 0, and B = 4, = 2. The

red lines are the corresponding numerical result and the blue line is an approximate linear tting

m 0

1.0

3.5

0.8

3.0

2.5

0.6

2.0

1.5

0.4

1.0

0.2

0.5

m k

5 10 15 20 25

m k

5 10 15 20

(a)

(b)

For the case without resonance, the correction to Newtons law has the following obvious characteristic. The main correction occurs at a short distance r of r < 1/k 1.6

1033 cm and it is proportional to 1

r2 . So, in the case of the current accuracy of the experiment [36], such a correction is unobservable and undetectable in this brane model. At a large distance (r 1/k), the correction can be neglected.

For the case of B = 4, = 2 with multiple resonances,

the analysis and method are similar. Because the value of m(0) inevitably tends 1 when the parameter m approaches innity, the leading correction to Newtons law at a large distance r 1/k is also proportional to

1r4 . At a short distance r < 1/k, the leading term is GNm1m2kr2 . The effect of the reso

nant modes occurs at a few Planck lengths. It is very difcult to give the expression of this correction.

4 Conclusion

In this paper, we investigated localization and resonant KK modes of the tensor uctuation for the f (R)-brane model with f (R) = R + R2. This investigation was based on

the interesting analytical brane solution found in Ref. [33], where the warp factor was given by eA(y) = sechB(ky) and

the parameter is constrained as 1 2 (see Eq.

(16)).

It was found that, when 1 s(< 0), the brane has

an internal structure, but the effective potential W(z) for the KK modes of the tensor uctuation has only a simple structure (volcano-like potential with a single or double well) and no resonances were found. The reason is that the quasiwell of the effective potential is not deep enough. This is different from the brane models in general relativity, for which there are usually resonant modes when the brane has an internal structure [1517].

However, the effective potential may have a rich structure with singularities and it will support a series of resonant KK modes when B is large enough (B > 2) and > 0, although the brane has no inner structure anymore. The reason is that the effective potential for the f (R)-brane model is decided by both the warp factor and the function f (R). It was found

that, when B > 2 and 1

40B2k2 < 2, the effective potential has two -like potential wells and there are a series of resonant modes on the brane.

The existence of resonant modes contributes a correction to the Newtonian graviton potential at short distance of the Planck scale. But fR is negative in some range so long as the effective potential has singularities because of the monotony of fR. However, this anomaly will result in the appearance of ghosts. So we hope to construct a f (R)-brane model with positive fR and graviton resonances in the future.

Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11375075), and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2015-jl01).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

Web End =http://creativecomm http://creativecommons.org/licenses/by/4.0/

Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP3.

References

1. I. Antoniadis, Phys. Lett. B 246, 377 (1990)2. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 436, 257 (1998). http://arxiv.org/abs/hep-ph/9804398

Web End =arXiv:hep-ph/9804398 [hep-ph]

3. N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429, 263 (1998). http://arxiv.org/abs/hep-ph/9803315

Web End =arXiv:hep-ph/9803315 [hep-ph]

4. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999). http://arxiv.org/abs/hep-ph/9905221

Web End =arXiv:hep-ph/9905221

5. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999). http://arxiv.org/abs/hep-th/9906064

Web End =arXiv:hep-th/9906064

6. J.D. Lykken, L. Randall, JHEP 0006, 014 (2000). http://arxiv.org/abs/hep-th/9908076

Web End =arXiv:hep-th/9908076

7. G.R. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 484, 112 (2000). http://arxiv.org/abs/hep-th/0002190

Web End =arXiv:hep-th/0002190

8. C. Csaki, J. Erlich, T.J. Hollowood, Y. Shirman, Nucl. Phys. B 581, 309 (2000). http://arxiv.org/abs/hep-th/0001033

Web End =arXiv:hep-th/0001033

9. C. Csaki, J. Erlich, T.J. Hollowood, Phys. Rev. Lett. 84, 5932 (2000). http://arxiv.org/abs/hep-th/0002161

Web End =arXiv:hep-th/0002161

10. M. Gremm, Phys. Lett. B 478, 434 (2000). http://arxiv.org/abs/hep-th/9912060

Web End =arXiv:hep-th/9912060 11. A. Kehagias, K. Tamvakis, Phys. Lett. B 504, 38 (2001). http://arxiv.org/abs/hep-th/0010112

Web End =arXiv:hep-th/0010112 [hep-th]

123

368 Page 10 of 10 Eur. Phys. J. C (2015) 75:368

12. C. Csaki, J. Erlich, T.J. Hollowood, Y. Shirman, Nucl. Phys. B 581, 309 (2000). http://arxiv.org/abs/hep-th/0001033

Web End =arXiv:hep-th/0001033

13. K. Hinterbichler, Rev. Mod. Phys. 84, 671 (2012). http://arxiv.org/abs/1105.3735

Web End =arXiv:1105.3735 [hep-th]

14. C. de Rham (2014). http://arxiv.org/abs/1401.4173

Web End =arXiv:1401.4173 [hep-th]15. H. Guo, Y.-X. Liu, Z.-H. Zhao, F.-W. Chen, Phys. Rev. D 85, 124033 (2012). http://arxiv.org/abs/1106.5216

Web End =arXiv:1106.5216 [hep-th]

16. Q.-Y. Xie, J. Yang, L. Zhao, Phys. Rev. D 88, 105014 (2013). http://arxiv.org/abs/1310.4585

Web End =arXiv:1310.4585 [hep-th]

17. W. Cruz, L. Sousa, R. Maluf, C. Almeida, Phys. Lett. B 730, 314 (2014). http://arxiv.org/abs/1310.4085

Web End =arXiv:1310.4085 [hep-th]

18. Y. Zhong, Y.-X. Liu, Z.-H. Zhao (2014). http://arxiv.org/abs/1404.2666

Web End =arXiv:1404.2666 [hep-th]19. T.P. Sotiriou, V. Faraoni, Rev. Mod. Phys. 82, 451 (2010). http://arxiv.org/abs/0805.1726

Web End =arXiv:0805.1726 [gr-qc]

20. A. De Felice, S. Tsujikawa, Living Rev. Rel. 13, 3 (2010). http://arxiv.org/abs/1002.4928

Web End =arXiv:1002.4928 [gr-qc]

21. S. Nojiri, S.D. Odintsov, Phys. Rept. 505, 59 (2011). http://arxiv.org/abs/1011.0544

Web End =arXiv:1011.0544 [gr-qc]

22. Y. Zhong, Y.-X. Liu, K. Yang, Phys. Lett. B 699, 398 (2011). http://arxiv.org/abs/1010.3478

Web End =arXiv:1010.3478 [hep-th]

23. M. Parry, S. Pichler, D. Deeg, JCAP 0504, 014 (2005). http://arxiv.org/abs/hep-ph/0502048

Web End =arXiv:hep-ph/0502048

24. V.I. Afonso, D. Bazeia, R. Menezes, A.Y. Petrov, Phys. Lett. B 658, 71 (2007). http://arxiv.org/abs/0710.3790

Web End =arXiv:0710.3790 [hep-th]

25. N. Deruelle, M. Sasaki, Y. Sendouda, Prog. Theor. Phys. 119, 237 (2008). http://arxiv.org/abs/0711.1150

Web End =arXiv:0711.1150 [gr-qc]

26. A. Balcerzak, M.P. Dabrowski, Phys. Rev. D 77, 023524 (2008). http://arxiv.org/abs/0710.3670

Web End =arXiv:0710.3670 [hep-th]

27. V. Dzhunushaliev, V. Folomeev, B. Kleihaus, J. Kunz, J. High Energy Phys. 04, 130 (2010). http://arxiv.org/abs/0912.2812

Web End =arXiv:0912.2812 [gr-qc]

28. Y.-X. Liu, Y. Zhong, Z.-H. Zhao, H.-T. Li, J. High Energy Phys. 06, 135 (2011). http://arxiv.org/abs/1104.3188

Web End =arXiv:1104.3188 [hep-th]

29. J. Hoff da Silva, M. Dias, Phys. Rev. D 84, 066011 (2011). http://arxiv.org/abs/1107.2017

Web End =arXiv:1107.2017 [hep-th]

30. H. Liu, H. Lu, Z.-L. Wang, JHEP 1202, 083 (2012). http://arxiv.org/abs/1111.6602

Web End =arXiv:1111.6602 [hep-th]

31. T. Carames, M. Guimaraes, J. H. da Silva (2012). http://arxiv.org/abs/1205.4980

Web End =arXiv:1205.4980 [gr-qc]

32. D. Bazeia, R. Menezes, A.Y. Petrov, A. da Silva, Phys. Lett. B 726, 523 (2013). http://arxiv.org/abs/1306.1847

Web End =arXiv:1306.1847 [hep-th]

33. Bazeia D, Lobo J, A.S., R. Menezes, A. Y. Petrov, A. da Silva, Phys. Lett. B 729, 127 (2014). http://arxiv.org/abs/1311.6294

Web End =arXiv:1311.6294 [hep-th]

34. O. DeWolfe, D.Z. Freedman, S.S. Gubser, A. Karch, Phys. Rev. D 62, 046008 (2000). http://arxiv.org/abs/hep-th/9909134

Web End =arXiv:hep-th/9909134

35. V.I. Afonso, D. Bazeia, L. Losano, Phys. Lett. B 634, 526 (2006). http://arxiv.org/abs/hep-th/0601069

Web End =arXiv:hep-th/0601069

36. J.C. Long, H.W. Chan, A.B. Churnside, E.A. Gulbis, M.C.M. Varney, J.C. Price, Nature 421, 922 (2003). http://arxiv.org/abs/hep-ph/0210004

Web End =arXiv:hep-ph/0210004

37. Y.X. Liu, J. Yang, Z.H. Zhao, C.E. Fu, Y.S. Duan, Phys. Rev. D 80, 065019 (2009). http://arxiv.org/abs/0904.1785

Web End =arXiv:0904.1785 [hep-th]

123

字數: 6408

顯示更少內容

The Author(s) 2015

摘要

翻譯

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Recently, a family of interesting analytical brane solutions were found in f(R) gravity with ...... in Bazeia et al. (Phys Lett B 729:127 2014 ). In these solutions, the inner brane structure can be turned on by tuning the value of the parameter ....... In this paper, we investigate how the parameter ...... affects the localization and the quasilocalization of the tensorial gravitons around these solutions. It is found that, in a range of ......, despite the brane having an inner structure, there is no graviton resonance. However, in some other regions of the parameter space, although the brane has no internal structure, the effective potential for the graviton Kaluza-Klein (KK) modes has a singular structure, and there exist a series of graviton resonant modes. The contribution of the massive graviton KK modes to Newton's law of gravity is discussed briefly.

詳細資料

標題

The structure of f(R)-brane model

作者

Xu, Zeng-guang; Zhong, Yuan; Yu, Hao; Liu, Yu-xiao

頁面

1-10

出版年份

2015

出版日期

Aug 2015

出版者

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

來源類型

學術期刊

出版物語言

English

DOI

https://doi.org/10.1140/epjc/s10052-015-3597-0

ProQuest 文件識別碼

1732878418

著作權

The Author(s) 2015

The structure of f(R)-brane model

Jump to:

全文

摘要

詳細資料

建議來源