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Physics Letters B 746 (2015) 266–275

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Graviton modes in multiply warped geometry

Mathew Thomas Arun

, Debajyoti Choudhury

, Ashmita Das

, Soumitra SenGupta

b,∗

Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India

Department of Theoretical Physics, Indian Association for the Cultivation of Sciences, 2A&B R.S.C. Mullick Road, Kolkata 700 032, India

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

Article history:

Received

30 March 2015

Accepted

5 May 2015

Available

online 7 May 2015

Editor:

J. Hisano

The negative results in the search for Kaluza–Klein graviton modes at the LHC, when confronted with

the discovery of the Higgs, have been construed to have severely limited the eﬃcacy of the Randall–

Sundrum

model as an explanation of the hierarchy problem. We show, though, that the presence

of multiple warping offers a natural resolution of this conundrum through modiﬁcations in both the

graviton spectrum and their couplings to the Standard Model ﬁelds.

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

1. Introduction

Despite the spectacular success of the Standard Model (SM) of

elementary particles, the search for new physics beyond the SM

continues. One of the primary motivations for this is to resolve

the well-known gauge hierarchy/naturalness problem in connec-

tion

with the ﬁne tuning of the higgs mass against large radia-

tive

corrections. Among several proposals to address this problem,

models with extra spatial dimensions draw special attention. In

this context, the warped geometry model proposed by Randall

and Sundrum (RS) [1] turned out to be particularly successful for

(i) it resolves the gauge hierarchy problem without bringing in any

other intermediate scale in the theory in contrast to the large ex-

tra

dimensional models; (ii) the modulus of the extra dimensional

model can be stabilized to a desired value by the Goldberger–Wise

mechanism [2], and (iii) a similar warped solution can be ob-

tained

from a more fundamental theory like string theory where

extra dimensions appear naturally [3]. As a result, several search

strategies at the LHC were designed speciﬁcally [4–7] to detect the

indirect/direct signatures of these warped extra dimensions e.g.

through the dileptonic decays of Kaluza–Klein (KK) excitations of

the graviton which appear in these models at the TeV scale.

The

original RS model was deﬁned as a slice of AdS

space with

an S

orbifolding and a pair of three-branes located at the

orbifold ﬁxed points, viz. y = 0, π (with the SM ﬁelds being lo-

calized

on the last mentioned). The parameters characterizing the

Corresponding author.

E-mail

addresses: thomas.mathewarun@gmail.com (M.T. Arun),

debajyoti.choudhury@gmail.com (D. Choudhury), ashmita.phy@gmail.com (A. Das),

soumitraiacs@gmail.com (S. SenGupta).

theory are the 5-dimensional fundamental (gravitational) scale M

and the bulk cosmological constant 

. The solution to Einstein’s

equations, on demanding a (1 +3)-dimensional Lorentz symmetry,

then leads to a warp-factor in the metric of the form exp(−k

where r

is the compactiﬁcation radius and k



−

/24 M

Clearly, the applicability of the semiclassical treatment (as opposed

to a full quantum gravity calculation) requires that the bulk cur-

vature

be substantially smaller than M

. An analogous string

theoretic argument [8] relating the D3 brane tension to the string

scale (related, in turn, to M

through Yang–Mills gauge couplings)

demands the same, leading to k

 0.1. On the other hand, too

small a value for this ratio would, typically, necessitate a consid-

erable

hierarchy between r

−1

and M

, thereby taking away from

the merits of the scenario. Thus, it is normally accepted that one

should consider only 0.01 ≤ k

≤ 0.1. Indeed, this constraint

plays a crucial role in most of the phenomenological studies of this

scenario, and certainly for the aforementioned results reported by

the ATLAS and the CMS groups. Throughout our analysis we shall

impose an analogous condition on the bulk curvature as an im-

portant

restriction to ensure the applicability of our semiclassical

calculations.

In the context of the original RS model, the large exponential

warping is held responsible for the apparent lightness of the Higgs

vacuum expectation value v (and its mass), as perceived on our

brane, related as it is to some naturally high scale



v ∼ O(M

applicable at the other brane, through the relation

v =



−π k

. (1)

Here



v is determined by the natural scale of higher dimensional

model ∼ ﬁve dimensional Planck scale M

and k

≈ 12 would

explain the hierarchy with r

being stabilized to this value by

http://dx.doi.org/10.1016/j.physletb.2015.05.008

0370-2693/

SCOAP

M.T. Arun et al. / Physics Letters B 746 (2015) 266–275 267

some mechanism [2]. The compactiﬁcation leads to a non-trivial

tower of gravitons with the levels being given by

= x

−π k

(2)

where x

’s are the roots of the Bessel function of order one. With

only the lowest (massless) graviton wavefunction being localized

away from our brane, its coupling to the SM ﬁelds is small, viz.

O(M

−1

). As the couplings of the others to the SM ﬁelds suffer

no such suppression, they are, presumably, accessible to collider

searches. The ATLAS Collaboration [5], though, has reported neg-

ative

results ruling out a level-1 KK graviton in the mass range

below 1.03 (2.23) TeV, with the exact lower bound depending on

the value chosen for k

This

result immediately brings forth a potential problem for the

model, for Eqs. (1) and (2) together demand that

∼

= x



= x



(3)

Since k

 0.1, it is immediately apparent that, unless



v is at

least two orders of magnitude smaller than M

, a 126 GeV Higgs

[9,10] would cry out for a KK graviton below a TeV. Indeed, this

argument has been inverted in the literature [11] to argue for a

much lower cutoff (in other words



v) in the theory. In other words,

some new physics would need to appear at least two orders of

magnitude below the fundamental scale M

, which, in the RS sce-

nario

is very close to the four-dimensional Planck scale itself.

Let

us remind ourselves of the nature of cutoffs in the effective

four-dimensional theory, considered as a theory of the SM ﬁelds

augmented by the RS gravitons. While the SM is operative below

the scale of the ﬁrst KK graviton, the new four-dimensional the-

ory

is operative all the way up to the compactiﬁcation scale ∼ r

−1

when each of the KK graviton is expected to take part in the am-

plitude

estimation as the beam energy is increased. Beyond the

energy ∼ r

−1

, we indeed encounter new physics by probing into

the extra dimension where the theory can no longer be deﬁned as

an effective theory in four dimensions deﬁned by standard model

and KK gravitons.

is important to realize, at this stage, that part of the afore-

mentioned

problem lies in the very restrictive nature of the RS

model as it is impossible to lower r

−1

by two orders without dis-

turbing

the value of the warped factor signiﬁcantly. This, in turn,

would introduce a little hierarchy necessitating a ﬁne tuning of

2–3 orders so that the Higgs mass may be kept ∼ 125 GeV. This

feature would worsen further if a graviton KK mode continues to

elude us in the forthcoming runs of the LHC, as well as in future

collider experiments.

the other hand, within the context of a generalization of

the RS model with additional warped extra dimensions, a lower

cutoff appears naturally, in the form of a larger compactiﬁcation

radius. In other words, the problem is circumvented without the

need for any additional (small) ﬁne tuning. Indeed, once we admit

more than four dimensions, there is no particular reason to restrict

the number to ﬁve, especially with constructs such as string theo-

retic

models arguing in favour of many more. Such variants of the

RS model have been proposed earlier [12–15,28] with these, typ-

ically,

considering several independent S

orbifolded dimen-

sions

along with M

(1,3)

. For example, codimension-2 brane models

[16] have been invoked to address aspects like Hubble expansion

and inﬂation [17–19], Casimir densities [20,21], little RS hierarchy

[22], gravity and matter ﬁeld localizations [23,24], fermion mass

generations [25,26], moduli stabilization [27], etc.

begin our study, with a brief discussion of the basic features

of warped geometry model in 6-dimension with two successive

orbifoldings.

2. Multiply warped brane world model in 6D

Consider a doubly warped compactiﬁed six-dimensional space–

time

with successive Z

orbifolding in each of the extra di-

mensions,

viz. M

1,5

→[M

1,3

× S

] × S

. Demanding four-

dimensional

) Lorentz symmetry within the set up, requires the

line element to be given by [28]

(z)[a

(y)η

μν

+ R

]+r

, (4)

where the compact directions are represented by the angular co-

ordinates

y, z ∈[0, π] with R

and r

being the corresponding

moduli. Just as in the RS case, non-trivial warp factors a(y) and

b(z), when accompanied by the orbifolding necessitates the pres-

ence

of localized energy densities at the orbifold ﬁxed points, and

in the present case, these appear in the form of tensions associated

with the four end-of-the-world 4-branes.

The

total bulk-brane action for the six dimensional space time

is, thus,

S = S



xdydz

√

−g

−)



xdydz

√

−g

(z)δ(y) + V

(z)δ(y −π )]



xdy, dz



−

(y)δ(z) + V

(y)δ(z −π )] , (5)

where  is the (six dimensional) bulk cosmological constant and

is the natural scale (quantum gravity scale) in six dimensions.

The ﬁve-dimensional metrics in S

are those induced on the ap-

propriate

4-branes, which accord a rectangular box shape to the

space. Furthermore, the SM (and other) ﬁelds may be localized on

additional 3-branes located at the four corners of the box, viz.



=0,π



xdydz

√

−g

δ(y − y

)δ(z − z

These terms, however, are not germane to the discussions of this

paper, and we shall not discuss S

any further.

For

a negative bulk cosmological constant , the solutions for

the 6-dimensional Einstein ﬁeld equations are given by [28]

a(y) = e

−c|y|

c =

cosh kπ

b(z) =

cosh (kz)

cosh (kπ)

k = r



−

10M

≡r



. (6)

The Israel junction conditions specify the brane tensions. The

smoothness of the warp factor at z = 0implies V

(y) be vanish-

ing,

while the ﬁxed point at z = π necessitates a negative tension,

viz.

(y) = 0, V

(y) =

−

tanh (kπ). (7)

With the warping in the y-direction being similar to that in the

5D RS model, the two 4-branes sitting at y = 0 and y = π have

equal and opposite energy densities. However, the z-warping dic-

tates

that, rather than being constants, these energy densities must

be z-dependent, viz.

(z) =−V

(z) = 8M



−

sech

(kz). (8)

Such a z-dependence can arise from a scalar ﬁeld distribution con-

ﬁned

on the brane. For a detailed discussion on this we refer our

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