Eective action of a self-interacting scalar eld on brane Cheng-Chen Liac Zheng-Quan Cuiac Tao-Tao Suiac and Yu-Xiao Liuabc aInstitute of Theoretical Physics Research Center of Gravitation

2025-05-03 0 0 541.01KB 10 页 10玖币

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Eﬀective action of a self-interacting scalar ﬁeld on brane

Cheng-Chen Liac, Zheng-Quan Cuiac, Tao-Tao Suiac, and Yu-Xiao Liuabc∗

aInstitute of Theoretical Physics &Research Center of Gravitation,

Lanzhou University, Lanzhou 730000, China

bKey Laboratory for Magnetism and Magnetic of the Ministry of Education,

Lanzhou University, Lanzhou 730000, China

cJoint Research Center for Physics,

Lanzhou University and Qinghai Normal University,

Lanzhou 730000, China

In extra dimensional theories, the four-dimensional ﬁeld theory is reduced from a fundamental

ﬁeld theory in the bulk spacetime by integrating the extra dimensional part. In this paper we

investigate the eﬀective action of a self-interacting scalar ﬁeld on a brane in the ﬁve-dimensional

thick braneworld scenario. We consider two typical thick brane solutions and obtain the P¨oschl-

Teller and harmonic potentials of the Kaluza-Klein (KK) modes, respectively. The analytical mass

spectra and wave functions along extra dimension of the KK modes are obtained. Further, the

eﬀective coupling constant between diﬀerent KK particles, cross section, and decay rate for some

processes of the KK particles are related to the fundamental coupling in ﬁve dimensions and the

new physics energy scale. Some interesting properties of these interactions are found with these

calculations. The KK particles with higher mode have longer lifetime, and they almost do not

interact with ordinary matter on the brane if their mode numbers are large enough. Thus, these

KK particles with higher modes might be a candidate of dark matter.

PACS numbers:

I. INTRODUCTION

After it was proposed in the 1920s, Kaluza-Klein (KK)

theory [1,2] was discussed repeatedly over the past 100

years. The essential idea of this theory is that the

observable physical phenomena of our four-dimensional

world can be reduced from a ﬁve-dimensional funda-

mental theory. According to string theory, the exis-

tence of extra dimensions is inevitable. Various mod-

els containing extra dimensions were proposed and dis-

cussed seriously as eﬀective theories at certain energy

level. In the braneworld scenario [3–6], our world is

described as a four-dimensional brane embedded in a

higher-dimensional spacetime called bulk and extra di-

mensions are suggested to be extended far more larger

than they were thought before [4], or even inﬁnite [3,6].

The development of extra dimension theories gives new

insight into fundamental physics. Nevertheless, the rea-

sons that introduce the concept of extra dimensions are

quite diﬀerent. In KK theory, the ﬁfth dimension was

proposed in order to unify the four-dimensional grav-

ity and electromagnetic force by regarding the electro-

magnetic ﬁeld as the ﬁfth component of the metric. In

Arkani-Hamed-Dimopoulos-Dvali (ADD) model [4] and

Randall-Sundrum (RS)-1 model [5], the existence of ex-

tra dimensions provides an alternative mechanism to re-

solve the gauge hierarchy problem in particle physics.

On the other hand, some eﬀorts have been made to ex-

∗liuyx@lzu.edu.cn, corresponding author

plore the possibility of inﬁnite extra dimensions, such as

RS-2 model [6] and domain wall model [3]. Combining

the above two models [3,6], one can obtain the model

of a brane with thickness (called thick brane) in a ﬁve-

dimensional curve spacetime with an inﬁnite extra di-

mensions. Such brane is called thick brane. In contrast,

the branes in the RS-1 and RS-2 models are called thin

branes since they are a pure four-dimensional hypersur-

face without thickness.

We may regard thin brane as an approximation to thick

brane if the thickness of a brane can be neglected. How-

ever, there is a signiﬁcant diﬀerence between thin brane

and thick brane. In a thick brane model, there are no

special dimensions as “extra dimensions” since the four-

dimensional ﬁelds in the Standard Model, which are de-

scribed by zero modes of some higher-dimensional ﬁelds,

are lived in the bulk. However, those zero modes are

localized around the thick brane and they can not prop-

agate along extra dimensions. For a thin brane model,

there are some very special extra dimensions and all the

matter ﬁelds are conﬁned on a hypersurface, i.e., the

brane. Both thin and thick branes have their respec-

tive advantages. For example, thin brane resolves the

hierarchy problem, while thick brane has its application

in holographical QCD.

For all kinds of extra dimension theories, an important

question is why we stay in a four-dimensional world and

do not observe these extra dimensions. In the original

KK theory it was explained by compacting the extra di-

mension into a tiny circle with size of Planck scale so

that we can not ﬁnd it in current experiments even if

we actually occupy the whole volume of the ﬁve dimen-

sions. In ADD model and RS-1/RS-2 model, it is a prior

arXiv:2210.15298v2 [hep-th] 11 Mar 2023

hypothesis that all matter ﬁelds are conﬁned on a four-

dimensional hypersurface embedded in ﬁve-dimensional

spacetime. In the thick brane model which we mainly

concern in this paper, all ﬁelds live in the bulk and the

extra dimension extends inﬁnitely, but the matter ﬁelds

is trapped in a very narrow region along the extra dimen-

sion. This mechanism is called as localization. There are

a lot of references for the thick brane models and localiza-

tion of gravity and matter ﬁelds [7–20], see Refs. [21–26]

for reviews.

Another principal problem need to be considered is

how to recover ordinary four-dimensional theory such

as electroweak gauge theory and general relativity from

the underlying ﬁve-dimensional theory. It leads to the

idea that ordinary four-dimensional physics corresponds

to some ﬁve-dimensional physics. From the very popu-

lar AdS/CFT perspective, our ordinary four-dimensional

physics, as a ﬁeld theory living on the boundary of a ﬁve-

dimensional spacetime, is totally equivalent to a gravita-

tional theory in the bulk. On the other hand, in KK

theory, domain wall model and brane model, it is ac-

complished by reducing the high-dimensional fundamen-

tal theory to the four-dimensional eﬀective one, which is

realized by integrating the high-dimensional action over

extra dimensions.

Most of the works about localization of a scalar ﬁeld

on a thick brane focus on free ﬁeld and few works dis-

cuss the self-interaction of a scalar ﬁeld in the bulk as

well as its eﬀective theory on a thick brane. In this

paper we will investigate the eﬀective action of a self-

interacting scalar ﬁeld on a thick brane. We ﬁrst as-

sume a ﬁve-dimensional fundamental action of a self-

interacting scalar ﬁeld. Then we employ the KK reduc-

tion procedure to derive the four-dimensional eﬀective

actions of the scalar KK modes. The eﬀective action

should be coincident with the action in current four-

dimensional theory. Furthermore, the ﬁve-dimensional

interaction will bring new four-dimensional particles and

new interaction terms between them. This will make

some signiﬁcant prophecies on particle collision at higher

energy level.

The organization of this paper is as follows. In Sec. II,

we give a general formulation for the KK reduction of a

ﬁve-dimensional action of a self-interacting scalar ﬁeld.

In Sec. III, we consider two brane solutions and obtain

two kinds of typical potentials which demonstrate some

signiﬁcant properties of the four-dimensional eﬀective ac-

tion. Finally, in Sec. IV we devote to the conclusions and

discussions.

II. KK REDUCTION OF ACTION

Let us ﬁrst consider a free massless scalar ﬁeld in ﬁve-

dimensional spacetime:

S=Z√−g d5x−1

2gMN ∂Mφ∂Nφ.(1)

The ﬁve-dimensional metric is proposed as

ds2=gMN dxMdxN=e2A(z)˜gµν (xλ)dxµdxν+dz2,(2)

where M, N and µ, ν are the ﬁve-dimensional and four-

dimensional coordinate indices, z=x5is the extra di-

mensional coordinate, and ˜gµν (xλ) is the reduced four-

dimensional metric at any ﬁxed position of the extra di-

mensional coordinate z=z0by a constant factor e2A(z0).

Usually, we can set e2A(0) = 1 by using the degree of

freedom of the coordinate transformation xµ→¯xµ=

eA(0)xµ.

In order to reduce the ﬁve-dimensional fundamental

action (1) to a four-dimensional eﬀective one, we sepa-

rate the variables of extra dimension from ordinary four

dimensions

φ(xµ, z) = X

ϕn(xµ)fn(z),(3)

which is called the KK decomposition. Substituting the

above decompsition into the action (1), we have

S=−1

2Zp−˜gd4xZe5A(z)dzX

n,mfnfm∂µϕn∂νϕm

(4)

+ϕnϕm∂5fn∂5fm.

The ﬁeld equation corresponding to the action (1) is

the ﬁve-dimensional Klein-Gordon equation:

gMN ∇M∇Nφ= 0.(5)

By virtue of the KK decomposition (3), the above Klein-

Gordon equation (5) can be converted to two equations.

One is the familiar four-dimensional Klein-Gordon equa-

tion of four-dimensional modes ϕn(xµ) and another is an

eigenvalue equation of the extra-dimensional part fn(z)

((4) −ζn)ϕn(xµ)=0,(6)

(∂2

z+ 3(∂zA)∂z)fn(z) = ζnfn(z),(7)

where (4) = ˜gµν ∇µ∇ν. With the redeﬁnition of the

ﬁeld

χn(z) = e3

2Afn(z),(8)

Eq. (7) becomes a one-dimensional Schr¨odinger-like

equation

−∂2

z+V(z)χn(z) = ζnχn(z),(9)

where the eﬀective potential is

V(z) = 3

2∂2

zA+9

4(∂zA)2.(10)

If the eigenstate χn(z) of the Schr¨odinger-like equa-

tion is a normalizable bound state, the corresponding

KK mode is localized on a brane, in the sense that the

energy density of the ﬁeld actually distributes in a ﬁnite

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Eectiveactionofaself-interactingscalareldonbraneCheng-ChenLiac,Zheng-QuanCuiac,Tao-TaoSuiac,andYu-XiaoLiuabcaInstituteofTheoreticalPhysics&ResearchCenterofGravitation,LanzhouUniversity,Lanzhou730000,ChinabKeyLaboratoryforMagnetismandMagneticoftheMinistryofEducation,LanzhouUniversity,Lanzhou730000...

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Eective action of a self-interacting scalar eld on brane Cheng-Chen Liac Zheng-Quan Cuiac Tao-Tao Suiac and Yu-Xiao Liuabc aInstitute of Theoretical Physics Research Center of Gravitation

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