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Physics Letters B 757 (2016) 180–186

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Localization and mass spectrum of q -form ﬁelds on branes

Chun-E Fu

a,b

, Yuan Zhong

, Qun-Ying Xie

, Yu-Xiao Liu

d,∗

School of Science, Xi’an Jiaotong University, Xi’an 710049, PR China

Department of Physics, University of Toronto, M5S1A7 Toronto, Canada

School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, PR China

Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, PR China

a r t i c l e i n f o a b s t r a c t

Article history:

Received

28 January 2016

Received

in revised form 22 March 2016

Accepted

25 March 2016

Available

online 30 March 2016

Editor:

N. Lambert

Keywords:

q-form

ﬁeld

Kaluza–Klein

modes

Branes

In this paper, we investigate localization of a bulk massless q-form ﬁeld on codimension-one branes by

using a new Kaluza–Klein (KK) decomposition, for which there are two types of KK modes for the bulk

q-form ﬁeld, the q-form and (q − 1)-form modes. The ﬁrst modes may be massive or massless while

the second ones are all massless. These two types of KK modes satisfy two Schrödinger-like equations.

For a ﬁve-dimensional brane model with a ﬁnite extra dimension, the spectrum of a bulk 3-form ﬁeld

on the brane consists of some massive bound 3-form KK modes as well as some massless bound 2-form

ones with different conﬁguration along the extra dimension. These 2-form modes are different from those

obtained from a bulk 2-form ﬁeld. For a ﬁve-dimensional degenerated Bloch brane model with an inﬁnite

extra dimension, some massive 3-form resonant KK modes and corresponding massless 2-form resonant

ones are obtained for a bulk 3-form ﬁeld.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

In the 1920s, Kaluza and Klein presented a uniﬁed ﬁeld the-

ory

of gravitation and electromagnetism with the idea of a ﬁfth

dimension (one compact extra dimension) beyond the usual four

of space and time. The Kaluza–Klein (KK) theory was a purely

classical extension of general relativity to ﬁve dimensions. Sev-

eral

decades later, the idea that the space–time may be more than

four dimensions has attracted increasingly more attention, espe-

cially

when the braneworld models were proposed [1–9], which

were motivated by the classical problems such as cosmological

and hierarchy problems. Extra dimensions are usually assumed

to be compacted in order to not conﬂict with our visible four-

dimensional

world. However, they may also be inﬁnite if the brane

has a tension and the matter ﬁelds or particles are conﬁned on the

brane, which is the idea of the Randall–Sundrum (RS) thin brane

scenario [5]. In this thin brane scenario, the inﬁnite extra dimen-

sions

are invisible for a four-dimensional observer on the brane

and the four-dimensional Newtonian potential can be recovered at

large distances.

Subsequently,

it was found that the RS thin brane can be gen-

eralized

to more realistic thick branes generated dynamically by

Corresponding author.

E-mail

address: liuyx@lzu.edu.cn (Y.-X. Liu).

scalar ﬁelds and/or some other matter ﬁelds [10–28], or by pure

gravity [29,30]. In the thick brane scenario, one should not simply

assume that the matter ﬁelds are trapped on the brane since the

energy density of the brane smoothly distributed along the extra

dimensions; therefore, it is important and interesting to investigate

localization mechanisms for various bulk matter ﬁelds [31–55]. On

the other hand, investigations of the Kaluza–Klein (KK) modes of

bulk ﬁelds can provide a way to probe extra dimensions. For exam-

ple,

the massive KK modes for a metric and a vector ﬁeld can give

corrections to Newton’s law and Coulomb’s law, respectively (the

zero modes for them are responsible for the well-known Newton’s

law and Coulomb’s law at low energy) [5,56–58]. So it is neces-

sary

to get the mass spectra and eigenfunctions of these bulk ﬁelds

through a localization mechanism.

this paper, we focus on the localization of a massless q-form

ﬁeld on a codimension-one p-brane. The action of the q-form ﬁeld

[q]

in the bulk is written as [37]

S =−

2(q +1)!



√

−gY

···M

q+1

···M

q+1

, (1)

where the ﬁeld strength Y

[q+1]

is deﬁned as Y

[q+1]

= dX

[q]

...M

q+1

= ∂

...M

q+1

]

. The 0-form and 1-form ﬁelds are

just scalar and vector ﬁelds. The higher q-form ﬁelds with q ≥ 2

are

new types of ﬁelds in a higher-dimensional space–time with

dimension D > 4, which are used to solve some unknown prob-

lems

such as the cosmological constant problem or dark energy

http://dx.doi.org/10.1016/j.physletb.2016.03.069

0370-2693/

SCOAP

C.-E. Fu et al. / Physics Letters B 757 (2016) 180–186 181

problem [7,59]. For any q-form ﬁeld in the bulk without topologi-

cal

obstructions, there is the well-known Hodge duality, namely, a

bulk q-form ﬁeld X

[q]

is dual to a bulk (p − q)-form ﬁeld X

[p−q]

[37]. So there is a question how to make sure the q-form and its

dual ﬁelds are simultaneously localized on the brane. That is why

we proposed a new localization mechanism for the q-form ﬁeld in

Ref. [60] different from those in Refs. [37,61].

The

line element is assumed as

2A(z)



μν

)dx

+dz



(2)

where A(z) is the warp factor depending only on the extra dimen-

sion

coordinate z, and

μν

) is the induced metric on the brane.

In Ref. [61], based on the following ansatz for the zero mode:

···μ

, z) =

(n)

···μ

), (3a)

···μ

q−1

, z) =0, (3b)

where the ˆ indicates the effective quantities on the brane, it was

concluded that the zero (massless) mode of a massless q-form

ﬁeld with q ≥ (p − 1)/2in the bulk cannot be localized on the RS

p-brane, which indicates that only the zero mode of the scalar (the

0-form ﬁeld) is trapped to the brane in the case of p = 3 (while

the zero mode of the 3-from ﬁeld cannot be localized). This result

conﬂicts with the well-known Hodge duality in the bulk without

topological obstructions [62] .

order to solve the above contradiction, Duff and Liu sug-

gested

to keep the ansatz (3) for q <(p − 1)/2but modify it for

q ≥ (p −1)/2as [37]

···μ

, z) =0, (4a)

···μ

q−1

, z) =

(n)

···μ

q−1

). (4b)

With this new KK decomposition, Hodge duality in the bulk im-

plies

Hodge duality on the brane for the zero modes. However,

Hodge duality on the brane for other KK modes cannot be ob-

tained.

order to further complete the investigation of localization

and Hodge duality of q-form ﬁelds on the p-brane, we recently

proposed a new general KK decomposition for a bulk q-form ﬁeld

without choosing a gauge in advance: [60]

···μ

, z) =



(n)

···μ

) U

(n)

(z) e

(2q−p) A/2

(5a)

···μ

q−1

, z) =



(n)

···μ

q−1

) U

(n)

(z) e

(2q−p) A/2

(5b)

where the index n marks dif ferent KK modes. There are two types

of KK modes, the q-form modes

(n)

[

and the (q − 1)-form ones

(n)

[

q−1]

. The functions U

(n)

(z) and U

(n)

(z) give respectively the dis-

tributions

of the two types of KK modes along the extra dimension.

They satisfy two non-independent Schrödinger-like equations [60],

which will lead to some interesting results about the localization

of these KK modes if we consider explicit brane models. With this

KK decomposition new results for Hodge dualities were discovered.

It was found that the Hodge duality in the bulk naturally results

in two dualities on the brane. One is the duality between the

zero modes

(0)

[

and

(0)

[

p−q−1]

, or between

(0)

[

q−1]

and

(0)

[

p−q]

. The

other is between two group KK modes, (

(n)

[

(n)

[

q−1]

) and (

(n)

[

p−q]

(n)

[

p−q−1]

), where

(n)

[

and

(n)

[

p−q]

are massive while

(n)

[

q−1]

and

(n)

[

p−q−1]

are massless. The effective ﬁeld theories on the brane for

these dual KK modes are physically equivalent [60].

this paper, we would like to investigate localization and

mass spectra for bulk q-form ﬁelds in two typical brane models

by following the general KK decomposition (5). We ﬁrst consider

a p-brane model with a ﬁnite extra dimension [45]. Using nu-

merical

method we will ﬁnd that for a bulk 3-form ﬁeld, there

are some massive bound 3-form KK modes

(n)

[

and a series of

massless bound 2-form ones

(n)

[

with different extra dimensional

conﬁgurations U

(n)

on the 3-brane. These 2-form KK modes are

different from those got from a bulk 2-form ﬁeld, because they

come from the 2-form parts of a bulk 3-form ﬁeld, as we can

see from (5b). Besides, there are continuous unbound KK modes.

Then we move to a degenerated 3-Bloch brane model [48]. For the

bulk 3-form ﬁeld, some resonant massive 3-form modes and the

corresponding massless resonant 2-form ones appear. The n-level

(n > 0) bound/resonant massive 3-form KK mode always has op-

posite

parity with the n-level (n > 0) massless 2-form one.

This

paper is organized as follows. We ﬁrst review the new lo-

calization

mechanism for a bulk q-form ﬁeld on codimension-one

p-branes in Sec. 2. Then we investigate the bound KK modes and

the resonant ones for a bulk 3-form ﬁeld in different p-brane mod-

els

in Sec. 3. Finally, a brief conclusion is given in Sec. 4.

2. Review of the new localization mechanism

In this section, we brieﬂy review the new localization mech-

anism

for a bulk q-form ﬁeld on codimension-one p-branes pro-

posed

in Ref. [60]. The action and line element are given in Eqs. (1)

and

(2), respectively. Substituting the general KK decomposition (5)

into

the action (1) for the q-form ﬁeld, we get [60]

eff

=−

2(q +1)!



n,n





p+1



−



(1)



···μ

q+1

(n)



)

···μ

q+1

+ I

(2)



···μ

(n)



)

···μ

+ I

(3)



···μ

(n)



)

···μ

+2I

(4)



···μ

(n)



)

···μ



(6)

where we have assumed that U

(n)

1,2

(z) satisfy the following or-

thonormality

conditions:

(1)



≡



dz U

(n)



)

=δ



, (7a)

(2)



≡

q +1



dz U

(n)



)

=(q +1)δ



, (7b)

and I

(3,4)



are deﬁned as

(3)



≡

q +1



dz e

(p−2q ) A

∂

(n)

(2q−p) A/2

)∂



)

(2q−p) A/2

(8)

(4)



≡

q +1



dz e

(p−2q ) A/2

(n)

∂



)

(2q−p) A/2

). (9)

From the KK decomposition, we see that there will be two

types of KK modes for the bulk q-form ﬁeld, i.e., the q-form KK

modes

(n)

[

and the (q − 1)-form ones

(n)

[

q−1]

. Further with the

effective action (6), it is known that if the orthonormality condi-

tions

(7a) and (7b) can be satisﬁed, these two types of KK modes

will be localized on the brane, and they couple with each other

(in fact they can be decouple d if we choose a suitable gauge [60]).

We also note that the (q −1)-form KK modes are always massless,

while the q-form ones may be massless or massive.

can introduce a mass parameter m

for the massive q-form

KK modes for simplicity, i.e., I

(3)



= m



. For different massive

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