Bulk scalar field in warped extra dimensional models

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by  Sumanta Chakraborty

Bulk scalar field in warped extra dimensional models
Sumanta Chakraborty
∗
IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India
Soumitra SenGupta
†
Department of Theoretical Physics, Indian Association for the Cultivation of Science, Kolkata-700032, India
(Dated: June5, 2014)
Thisworkpresentsageneralformalismtoanalyzeagenericbulkscalarfieldinamultiplewarped
extra-dimensionalmodelwitharbitrarynumberofextradimensions. TheKaluza-Kleinmassmodes
along with the self-interaction couplings are determined and the possibility of having lowest lying
KKmodemassesnearTeVscalearediscussed. Alsosomenumericalvaluesforlow-lyingKKmodes
4
hasbeenpresentedshowingexplicitlocalization aroundTeVscale. Itisarguedthattheappearance
1
of large number of closely spaced KK modes with enhanced coupling may prompt possible new
0
2 signatures in collider physics.

n
u
I. INTRODUCTION
J

4
Theories with extra spacetime dimensions have drawn considerable attention ever since the original

] proposal by Kaluza and Klein. There has been renewed interest in such theories since the emergence of
c
stringtheory. Severalnewideasinthiscontexthavebeenproposedandhaveinterestingconsequencesfor
q
particle phenomenology and cosmology [1–5]. In these higher-dimensional models, spacetime is usually
-
r taken to be a product of a four-dimensional spacetime and a compact manifold of dimension n. While
g
gravitycan propagate freely throughthe extra dimensions, StandardModel particles are confined to the
[
four dimensional spacetime. Observers in this three spatial-dimensional wall (a “3-brane”) will measure

2 an effective Planck scale M2 = Mn+2V , where V is the volume of the compact space. If V is large
pl n n n
v
enough it could make Planck scale of the order of TeV, thus removing the hierarchy between the Planck
9
and the weak scale.
7
Subsequently, Randall et al. [6, 7] proposed a higher-dimensional scenario that is based on nonfactor-
2
3 izable geometry and accounts for the hierarchy without introducing large extra dimensions. However,
. the braneworld model itself is not stable and it was shown in Ref. [8] that by introducing a scalar
1
field in the bulk, the modulus-namely the brane separation in the RS model-can be stabilized without
0
4 any fine-tuning. Assumption of negligibly small scalar backreaction on the metric in the GW approach
1 prompted further work in this direction, where the modification of the RS metric due to backreaction
: of the bulk fields has been derived (see [9]). The stability issues in such cases have been reexamined for
v
time-dependent cases [10, 11]; also the effect of gauge fields or higher form fields have been studied in
i
X several works (see [12]).
r In an effort to search for the signatures of extra dimensions, the roles of the Kaluza-Klein modes of
a
different bulk fields on the phenomenology at the standard model brane are of crucial importance.
For the five-dimensional RS model, [13] determined the bulk scalar KK modes and their self-
interactions. It is found that due to the exponential redshift factor in the RS model, KK scalar modes
in this spacetime have TeV scale mass splitting and inverse TeV couplings (see [7]). This is in sharp
contrast to the KK decomposition in product spacetimes, which for large compactified dimensions, give
rise to a large number of light KK modes (see [14]) with a very small coupling with brane fields. Due to
this very distinct feature, the RS model has interesting consequences [13, 15].
Motivated by string theory and other extra-dimensional models where one can have several extra
dimensions, in this paper we extend the results of the bulk scalar model in five-dimensional RS space-
time to arbitrary number of warped dimensions and have obtained the KK decomposition of the scalar
KK masses. We have shown that in these multiply warped models we have much larger number of KK
modes than the five-dimensional RS counterpart with effective couplings in the inverse TeV range. We
have also discussed possible numerical values for various parameters in our theory and have used them
to get possible numerical values of low-lying KK mode masses in our multiply warped model, showing
∗ [email protected]
† [email protected]
2
explicit localizationin TeV brane. Our results alsoestablish a generalformula for determining these KK
masses and couplings in the presence of any arbitrary number of extra dimensions.
The paper is organized as follows: We give a brief explanation for six-dimensional doubly warped
spacetime in Sec. II, the calculationfor bulk scalarfield has been done in this six-dimensionalspacetime
in Sec. III. The same calculations have been finally extended to higher-dimensional spacetime with
arbitrary number of extra dimensions in Secs. IV and V. The paper ends with a short discussion of our
results.
II. SIX-DIMENSIONAL DOUBLY WARPED SPACETIME AND EINSTEIN EQUATIONS
Inthissectionweshalldiscussdoublycompactifiedsix-dimensionalspacetimewithZ orbifoldingalong
2
each compactified direction. For a detailed discussion we refer the reader to [15]. The manifold under
consideration is given by, M1,5 = M1,3 S1/Z S1/Z [15].
2 2
× ×
Weletthecompactifieddimensionstoyandz,respectively. Thenoncompactifieddimensionsaretaken
to be, xµ(µ=0,1,2,3). The modu(cid:2)li along the co(cid:3)mpact dimensions are givenby R and r , respectively.
y z
The corresponding metric ansatz is taken as
ds2 =b2(z) a2(y)η dxµdxν +R2dy2 +r2dz2, (1)
µν y z
with η = diag( 1,1,1,1). Thus we(cid:2)have four orbifold fixed(cid:3)points, which are given by (y,z) =
µν
−
(0,0),(0,π),(π,0),(π,π), respectively.
The total bulk-brane action could be given by
S =S +S +S (2)
6 5 4
S = d4xdydz√ g (R Λ ) (3)
6 6 6 6
− −
Z
S = d4xdydz[V δ(y)+V δ(y π)]+ d4xdydz[V δ(z)+V δ(z π)] (4)
5 1 2 3 4
− −
Z Z
2 2
S = d4xdydz√ g (L V)δ(y y )δ(z z ). (5)
4 vis i j
− − − −
i=1j=1Z
XX
Here the brane potentials in generalhave the particular functional dependence V =V (z) and V =
1,2 1,2 3,4
V (y). Finally the full six-dimensional Einstein’s equation is given by,
3,4
R
M4√ g R g = Λ √ g g
6 MN MN 6 6 MN
− − − 2 −
(cid:18) (cid:19)
+ √ g V (z)g δα δβδ(y)
− 5 1 αβ M N
+ √ g V (z)g δα δβδ(y π)
− 5 2 αβ M N −
+ g˜V (y)g˜ δα˜ δβ˜δ(z)
− 5 3 α˜β˜ M N
+ p g˜V (y)g˜ δα˜ δβ˜δ(z π) (6)
− 5 4 α˜β˜ M N −
p
In the aboveexpressionM, N are bulk indices, α, β run over the usual four spacetime coordinatesgiven
by xµ. The quantities g and g˜ are the metric in y = textrmconstant and z = constant hypersurfaces,
respectively. The line element derived from the above Einstein’s equation turns out to be [15]
cosh2(kz)
ds2 = exp( 2cy )η dxµdxν +R2dy2 +r2dz2. (7)
cosh2(kπ) − | | µν y z
(cid:2) (cid:3)
In the above line element we have the following identification for the constants k and c given by
c Ryk
≡ rzcosh(kπ) (8)
k ≡rz 1−0MΛ64.
q
3
TheboundarytermsleadtothebranetensionsandusingtheEinstein’sequationacrossthetwoboundaries
at y =0, y =π, respectively, thus we readily obtain
Λ
V (z)= V (z)=8M2 − 6sech(kz). (9)
1 2
− 10
r
Thus the two 4-branes situated at y = 0 and y = π would have a z-dependent brane tension. The fact
that the two tensions are equal and opposite is reminiscent of the originalRS-form. Similarly we get the
brane tensions for other two 4-branes as
8M4k
V (y)=0;V (y)= tanh(kπ). (10)
3 4
− r
z
HereV wereintroducedtoaccountorbifoldingalongthe z-directionandwithg beingaconstant,the
3,4 zz
resultinghypersurfaceshouldhaveonly aconstantenergydensity. The factthatg is dependentonthe
yy
coordinate z makes the two hypersurfaces for y orbifolding to have a z-dependent energy density.
The 3-brane located at (y =0,z =π) suffers no warping and can be identified with the Planck brane.
The other three can be valid choices for Standard Model (visible) brane. However, if we assume that
there is no brane having lower energy scale than ours, we are forced to identify the SM brane to be
located at (y =π,z =0). The suppression factor on the TeV brane can be given by
exp( cπ)
f = − . (11)
cosh(kπ)
The desired suppression of 10 16 on the TeV brane can be obtained by choosing different combinations
−
of the parameters c and k. However we also have an extra relation as presented in Eq. (8), which shows
that in order to avoid large hierarchy between the two moduli R and r , either of the two parameters
y z
c and k must be large and the other should be small. For example, we can easily assume c 11.4 and
∼
k 0.1. However a small hierarchy also exists in the originalRS model, where there exists an one order
∼
ofmagnitudehierarchybetweenr andk,satisfyingPlanck-to-TeVscalewarpingbykr 11.5. Anatural
∼
question that arises with this discussion is whether stabilization of these moduli to the desired values is
possible. For the five-dimensional RS model this has been shown in [8] by introducing a bulk stabilizing
scalar field and tuning the VEV of the scalar field at the boundaries. The modulus in the theory is
stabilized near Planck length without any fine-tuning.
In this case as well we can adopt a similar procedure by introducing a bulk stabilizing scalar field.
Again choosing appropriate VEV at the boundary, we can stabilize R and r to desired values [16]. In
y z
our six-dimensional braneworld scenario with y and z dependence, the action for the scalar field can be
expressed such that
1
S = d4xdydz√ g ∂ φ∂Mφ V(φ)
6 M
− 2 −
Z (cid:18) (cid:19)
2
+ d4x g λ (φ) φ2 v2 2δ(y y )δ(z z ), (12)
− ij ij − ij − i − j
i,j=1Z
X p (cid:0) (cid:1)
where the coupling parameters, λ tend to infinity as the scalar field approach to the following values,
ij
φ(0,0) = v , φ(0,π) = v , φ(π,0) = v and φ(π,π) = v . We take V(φ) = m2φ2. Now following Ref.
0 1 3 4
[8, 17] we can obtain the equation of motion in the separable form as
ψ (y) 4cψ (y)=pψ(y)
′′ ′
−
b2R2 5b˙
y χ¨(z)+ χ˙(z) = R2b2m2 p χ(z), (13)
rz2 " b # y −
(cid:0) (cid:1)
where we have made the following decomposition, φ(y,z)=ψ(y)χ(z) and p is the separability constant.
Alsointheaboveexpressionprimedenotedifferentiationwithrespecttoy anddotdenotesdifferentiation
with respect to z. Finally, the above equations with appropriate boundary condition [17] can be solved,
which when substituted into the action leads to an effective potential for the moduli as
(1 2v+v2) k
V = πv2 − + (1+2α 8ν+2ν2)
eff 2 2kνπ 12ν −
h (cid:16)
+ v(22+2α 8ν+2ν2)+(1+2α 8ν+2ν2) , (14)
− −
(cid:17)i
4
where we have used the following shorthand notations, v =v /v , α= 10m2M4/Λ and ν = 4+ p.
1 2 − 6 c2
Then solving the equations, ∂ V = 0 and ∂ V = 0 and then through the second derivatives with
ν eff k eff
p
respect to ν and k we can arrive at the minimum values of c and k. As an illustrative example we can
startwith v =0.43andp 1,leadingto c 11.24andk 0.422. Note thatthesevalues canresolvethe
∼ ∼ ∼
gauge hierarchyproblem. Thus along this line any higher-dimensional braneworldmodels can have their
modulistabilized. Fromnowonweshallassumethatsuchastabilizationhasbeenperformedandallthe
modulihence forthwill havethose stabilizedvalues. Inthis analysis,followingthe stabilizingbulk scalar
model, we have assumed that the backreaction of the bulk stabilizing field is negligibly small. Moreover
from the action of the bulk stabilizing scalar it may be observed that at the boundaries the stabilizing
scalar tends to their VEVs v when the coupling λ (φ) tends to infinity. This is exactly similar to the
ij ij
five-dimensional counter part of the Goldberger-Wise calculation of modulus stabilization. As a result
at the boundaries, the stabilizing scalaris frozento different values v and hence does not contribute to
ij
the dynamics of the model.
III. BULK FIELD IN SIX-DIMENSIONAL DOUBLY WARPED SPACETIME
In this section we carry out the Kaluza-Klein decomposition of a nongravitational bulk scalar field
propagatinginthespacetimedescribedbyEq. (7)inthespiritofthework[13]withbulkscalarfield. We
find that in these multiply warped spacetime the SM brane contains larger number of TeV scale scalar
KKmodesthanthefive-dimensionalRSmodel. Thishassignificantphenomenologicalconsequences[18].
We consider a free scalar field in the bulk for which the action is given by
1
S = d4x dy dz√ G GAB∂ Φ∂ Φ+m2Φ2 , (15)
A B
2 −
Z Z Z
(cid:0) (cid:1)
where G with A,B = µ,y,z is given by Eq. (7), and m is of order of M . After an integration by
AB pl
parts, this can be written as
1 cosh3(kz)
S = d4x dy dz R r e 2σ η ∂ Φ∂ Φ
2 y z − cosh3(kπ) µν µ ν
Z Z Z h
cosh5(kz)
+ R r e 4σ m2Φ
y z − cosh5(kπ) 2
r cosh3(kz)
z Φ∂ e 4σ ∂ Φ
− Ry y(cid:18) − cosh3(kπ) y (cid:19)
R cosh5(kz)
yΦ∂ e 4σ ∂ Φ , (16)
− rz z(cid:18) − cosh5(kπ) z (cid:19)i
where σ =cy . Now we make the following substitution for KK decomposition,
| |
α (y)β (z)
n m
Φ(x,y,z)= φ (x) . (17)
nm
n,m Ry √rz
X
p
The following normalization conditions are imposed on the fields α and β,
dye 2σα α =δ (18)
− n m nm
Z
cosh3(kz)
dz β β =δ . (19)
cosh3(kπ) p q pq
Z
The differential equation satisfied by the function α (y) is
n
1 d dα
e 4σ m =A2 e 2σα , (20)
− R2 dy − dy m − m
y (cid:18) (cid:19)
5
50
Α
0
-50
0.0 0.5 1.0 1.5 2.0 2.5 3.0
y
FIG. 1: The figure shows variation of the quantity αm with extra-dimension parameter y. The vertical line
representsthey=π lineshowingthefactthatthequantityαm ismaximumaty=π,thepositionofTeVbrane.
whereA standsforKKmodemasseigenvalue. Theabovedifferentialequationcanbefurthersimplified
m
and cast to the following form,
d2α dα
m 4c m +A2 R2e2σα =0. (21)
dy2 − dy m y m
The above equation can be solved in terms of Bessel functions of first and second order as
e2σ A eσR A eσR
m y m y
α = J +b Y , (22)
m 2 m 2
N c c
m (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
with N representing an overall normalization. Now we can proceed much further. The mass modes
m
determinedbyA mustbereal. Thisrealityconditionimposedonthemassmodesrequiresthedifferential
m
operator on the left hand side of Eq. (22) to be self-adjoint. This self-adjointness imply that derivatives
ofα (y)shouldbe continuousatthe orbifoldfixedpoints. Thesegivestwoconditionsontheparameters
m
A and b , expressed as,
m m
2J AmRy + AmRyJ AmRy
2 c c 2′ c
b = (23)
m −2Y (cid:16)AmRy(cid:17)+ AmRyY (cid:16)AmRy(cid:17)
2 c c 2′ c
0 = [2J (x(cid:16) )+x(cid:17)J (x )] 2Y(cid:16) (x e(cid:17)cπ)+x e cπY x e cπ
2 m m 2′ m 2 m − m − 2′ m −
− [2Y2(xm)+xmY2′(xm)](cid:2)2J2(xme−cπ)+xme−cπJ2′(cid:0)xme−cπ(cid:1)(cid:3) (24)
where xm =AmecπRy/c. Since to make Planck s(cid:2)cale down to TeV scale we(cid:0)should h(cid:1)(cid:3)ave ecπ 1. Then
≫
theaboveequationreducestothefollowingform,2J (x )+x J (x )=0. Thenforlightmodemasses
2 m m 2′ m
we have x to be order of unity [13]. This keeps A R /c also order of unity. Then from Eq. (22) as
1 m y
well as from Fig.1 we observe that the modes α (y) are larger near the 3-brane at y = π, which makes
m
these low-mass Kaluza-Klein modes to be found preferentially near the y = π region (see Fig.1). Thus,
with the TeV brane being situated at y =π, we observe that the low-mass KK modes are exponentially
suppressedandhence confinedto the TeVbrane. Alsoforthese low-lyingKKmassmodes the coefficient
b is of the order of e 4cπ, which shows that we can ignore the Y (y ) part compared to J (y ), while
m − 2 m 2 m
performing integrals involving α .
m
Similar analysis for β yields
m
d2β dβ cosh2(kπ)
m +5ktanh(kz) m +r2B2 β =0. (25)
dz2 dz z mcosh2(kz) m
6
The solution, apart from an overallnormalization, can be expressed as
5 10k2+B2 r2(1+cosh(2kπ))
β (z) = exp k2z2 H − m z
m −2 √5/2kz 10k2
(cid:20) (cid:21) (cid:18) (cid:19)
5 10k2+B2 r2(1+cosh(2kπ)) 1 5k2z2
+ E exp k2z2 F − m z , , , (26)
m −2 1 1 − 10k2 2 2
(cid:20) (cid:21) (cid:18) (cid:19)
where F istheKummerconfluenthypergeometricfunctionandH istheHermitepolynomialofdegree
1 1 n
n. Then from Fig.2 we observe that this function is also maximum at z = 0. Hence the low-lying KK
mass modes are confined to the TeV brane located at z = 0. From the behavior of both α and β we
m n
find that all the low-lying KK mass modes are confined to y = π,z = 0 brane, i.e., the TeV brane. A
possibleexperimentalsignatureofthebulkscalarKKmodescanoriginateviacouplingofthebulkscalar
to diHiggs in the form Φ(x)h2(x). For m m the dominant decay channels are gg and b¯b which leads
Φ h
∼
to multijets as final product which though may be difficult to differentiate from the QCD back ground
[19–21]. Also when the mass of bulk scalar is in the range of 250 to 350 GeV then enhanced production
of Φ hh occurs. Moreover for bulk scalar mass in the range 160 to 250 GeV we have a relatively
→
larger cross sections for the diphoton channel. In this region due to small mixing the branching ratio is
dominated by gg and b¯b. The diphoton channel is a very promising search channel as branching ratio
remains more or less at constantleveleven up to tt¯threshold [22, 23]. This might become possible if the
LHC runs extends the diphoton searches for invariant masses above existing m =150GeV.
γγ
Todeterminetheparametersofthesolutionweproceedasfollows: wewantB tobereal,asitappears
m
in the mass modes. Thus self-adjointness also applies in this case. This implies that derivatives of β
m
to be continuous around the orbifold fixed points. At z = 0, this is trivially satisfied irrespective of the
quantity E . Howeveratz =π allthe termsare suppressedby exp( 5k2z2/2),thus the self-adjointness
m
−
there leads to
H (a)
√5/2kπ
E = (27)
m −1F a,1,5k2z2 +2a F a+1,3, 5k2z2
1 − 2 2 1 1 2 −2
10k2+B2 r2(1+cosh(2kπ))
a = − (cid:0) m z (cid:1) (cid:0). (cid:1) (28)
10k2
At large values of z, confluent hypergeometric series have a large value. Being in the denominator the
term can be neglected for practical purposes. Using the above equations we readily obtain the following
action for the field φ(x) as,
1
S = d4x ηµν∂ φ ∂ φ + M φ φ (29)
µ nm ν nm nmpq nm pq
2
Z (cid:16)Xn,m n,Xm,p,q (cid:17)
M = A2δ δ +B2δ P +m2P Q (30)
nmpq n np mq n np mq np mq
where we have the following ex(cid:8)pression for the element Q , (cid:9)
nm
cosh5(kz)
Q = dz β β (31)
nm cosh5(kπ) n m
Z
and P as,
mn
P = dye 4σα α (32)
nm − n m
Z
Now from the previous discussion we have the solution for these two sets of functions α (y) and β (z),
n n
which can be used to determine Q and P in order to obtain the masses of the KK modes by
nm nm
evaluating the quantity M . In contrastto the five-dimensionalsituation (see [13]) where the masses
nmpq
of the bulk fields appear as a diagonalized mass matrix, in this case the bulk field Φ(x,y,z) manifests
itself to some four-dimensional observer as an infinite KK tower with mass being determined by the
the quantity M such that a scalar φ has a mass M after an appropriate diagonalization
nmpq nm nmnm
procedure.
7
Β
1.0
0.5
z
0.5 1.0 1.5 2.0 2.5 3.0
-0.5
-1.0
FIG.2: Thefigureshowsvariationofthequantityβn withextra-dimensionparameterz. Thegraphclearlyshows
thefact that thequantity βn is maximum at z =0, theposition of TeV brane.
Thesolutionforα (y)ispresentedinequation(22). AsimilarsolutionwasobtainedbyWiseet. al(see
n
[13]) except for the fact that we have Besselfunctions of secondorder. Following their discussionwe can
argueinasimilarmannerthatthelightestKKmodeshavemassparameterA suppressedexponentially
m
with respecanct to the the scale m appearing in Eq. (15). Since we have taken m to be order of Planck
scale and c to be around 12, by stabilization these mass modes A are in the TeV range. Also from the
m
solution we could observe that the modes α (y) are larger in the region y =π. This has been explained
n
earlier through graphical presentation of the function α .
m
The solutionforβ (z)hasbeenpresentedinEq. (26). Thoughwehavearguedfollowingthe graphical
n
presentationofthefunctionβ ,wenowprovideatheoreticalmotivationforourabove-mentionedresults.
n
The solution has an overall factor of exp 5k2z2 and we see that the solution has maximum value
−2
around z = 0. Hence the bulk field being a product of these two functions α (y) and β (z) as shown
m n
(cid:2) (cid:3)
in Eq. (16) has mass parameter in the TeV range and has maximum value around (y = π,z = 0). Now
from Sec. II this is precisely the SM brane. Hence, the bulk field has a maximum in the SM brane; i.e.,
the KK modes are most likely to be found in that region where A and B are in the TeV range. This
m m
sets the stage for KK excitations to have TeV scale mass splitting on the SM brane.
Now we would like to compute some low-lying KK mode masses numerically. For that we need to fix
some parameters, k, c, r and the bulk mass of the scalar field m. We shall take the bulk mass to be in
z
Planck scale. Then we can determine the remaining parameters, by making the following demands: (a)
ifwe haveagaugebosonfieldinthis multiply warpedscenario,its lowestmassiveKKmodesshouldlead
to W andZ bosonmasses,(b) the suppressionf as presentedinEq. (11) shouldbe 10 16, andfinally
−
∼
(c) the hierarchybetween R and r should be small. The KK mode of the gauge boson in this multiply
y z
warped spacetime can be obtained from Ref. [24].
ThisdesiredmassforW andZ boson 100GeVcanbeobtainedwithf 10 16and 1 =7 1017GeV,
∼ ∼ − rz ×
about14timessmallercomparedtoPlanckscale. Theotherparametersk andccanbedeterminedusing
small hierarchy between R and r along with desired warping of f 10 16. This finally leads to, the
y z −
∼
following estimation: k =0.25, c=11.52 and the ratio between moduli being Ry =61. The suppression
rz
factor turns out to be f = 1.45 10 16. Thus we will calculate the low-lying KK masses for our bulk
−
×
scalar field with these sets of parameters (see Table.I).
We now present the self-interactions of the bulk scalar field. From the four-dimensional point of view
theseself-interactionscaninducecouplingsbetweentheKKmodes. Inthiscaseself-couplingsofthelight
modes are suppressedby the warpfactorand hence if the Planckscale setthe six-dimensionalcouplings,
the low-lying KK modes have TeV range self-interactions. We present the interaction term in the action
8
TABLE I:The masses of theKK modes of thescalar field are given in GeV units. Wehavechosen thefollowing
values, 1 =7×1017 GeV, k=0.25, c=11.52. Some representative masses of low-lying KK modes are given.
rz
m1111 =99.513 m1212 =99.651 m1313 =99.709 m1414 =99.743
m2121 =178.614 m2222 =178.866 m2323 =178.965 m2424 =179.026
m3131 =257.714 m3232 =258.069 m3333 =258.228 m3434 =258.309
m4444 =337.592 m5555 =416.957 m6666 =501.445 m7777 =583.371
with coupling parameter λ such that
π π λ
S = d4x dy dz√G Φ2m, (33)
int M4m 6
Z Z−π Z−π −
where the coupling λ is of the order of unity. Then we can expand in modes and the self-interactions of
light KK states become
2m
π π cosh5(kz) λ α β
Sint =Z d4xZ−πdyZ−πdzRyrze−4σcosh5(kπ)M4m−6φ2pmq Rpy √rqz! . (34)
Thus the effective four dimensional coupling constants are p
4λ π π cosh5(kz)
λ = dye 4σα2m dz β2m, (35)
eff (MRy)m−1(Mrz)m−1M2m−4 Z0 − p Z0 cosh5(kπ) q
which reduces to,
λ 4λ c m−1 1 m−1 Me cπ 1 4−2m 1r4m 5dr J2 Apkeσr 2m π(β )2mdz
eff ≃ MR Mr − cosh(kπ) −  (cid:16)A (cid:17) q
(cid:18) y(cid:19) (cid:18) z(cid:19) (cid:18) (cid:19) Z0 p Z0
  (36)
in the large kR and kr limit. Hence we observe that the relevant scale for four-dimensional physics
y z
is not the scale set by Planck scale, i.e., M, but this is Me cπ 1 . Hence the KK reduction has
− cosh kπ
lead the couplings from Planck scale to the TeV scale by the warp factor on the SM brane located at
(y =π,z =0).
From the above discussion we now try to obtain some bounds on the parameters in our model, e.g.,
R , r from the requirement of precision electroweak test. For that purpose we can use the same setup
y z
and put a bulk gauge boson whose KK modes can be detected in precision electroweak tests. We define
a quantity denoted by
∞ g2 M2
V = n W , (37)
g2 M2
n=1(cid:18) 0 n (cid:19)
X
where M is the mass of W gauge boson and M is the mass of higher KK modes of the bulk gauge
W n
bosonandg is the effective four-dimensionalgaugecoupling alongwithg to be the gaugecouplingsfor
0 n
higherKKmodes. ThenfromRef. [25]wecouldarguethatforprecisionelectroweaktestweshouldhave
V < 0.0013 with 95 percent confidence level. From this result we can get the following bounds on the
parameters of this model, 1/R < 5.95 1017GeV. This leads to a bound on 1/r as well by assuming
y z
a small hierarchy between the two mod×uli as, 1/r < 3.63 1019GeV. From Ref. [24] it can be easily
z
×
verified that this bound is respected by gauge couplings and KK mode masses. Thus these multiply
warped models indeed satisfy precision electroweak tests.
IV. SEVEN-AND-HIGHER-DIMENSIONAL SPACETIME WITH MULTIPLE WARPING
With an aim to arrive at a generic result we shall now try to extend our analysis with one more
extra dimension. For that purpose we start with a seven-dimensional spacetime where the space-
like dimensions are successively warped. In other words the manifold of interest could be given by
9
M(1,3) S1/Z S1/Z S1/Z . Then the total brane-bulk action can be given by
2 2 2
× × ×
(cid:2)(cid:8) (cid:2) (cid:3)(cid:9) (cid:3)S = S +S +S +S (38)
7 6 5 4
S = d4xdydzdw√ g (R Λ ) (39)
7 7 7 7
− −
Z
S = d4xdydzdw[V δ(w)+V δ(w π)]
6 1 2
−
Z
+ d4xdydzdw[V δ(z)+V δ(z π)]
3 4
−
Z
+ d4xdydzdw[V δ(y)+V δ(y π)], (40)
5 6
−
Z
with appropriate actions (S ) for 12 possible 4-branes at the edges (z,w) = (0,0),(0,π),(π,0),(π,π),
5
(z,y) = (0,0),(0,π),(π,0),(π,π) and (y,w) = (0,0),(0,π),(π,0),(π,π). We also have eight possible
3-branes at the corners (y,z,w) = (0,0,0),(0,0,π),(0,π,0),(π,0,0),(π,π,0),(0,π,π),(π,0,π),(π,π,π).
By natural extension of the method as illustrated in the previous section we get the line element and
other parameters such that [15],
cosh2(ℓw) cosh2(kz)
ds2 = exp( 2cy )η dxµdxν +R2dy2 +r2dz2 + 2dw2
cosh2(ℓπ) cosh2kπ − | | µν y z ℜw
(cid:26) (cid:27)
Λ 2 (cid:2) (cid:3)
ℓ2 = − 7ℜw
15
ℓr
z
k =
cosh(ℓπ)
w
ℜ
ℓR kR
y y
c = = (41)
cosh(kπ)cosh(ℓπ) r cosh(kπ)
w z
ℜ
It may be of interest that the 5-brane at w =π does not represent a flat metric (y and z dependencies).
In order to obtain substantial warping along the w direction (from w = π to w = 0), one need to make
ℓπ substantial(same orderof magnitude as RS scenario). The seven-dimensionalortriply warpedmodel
has a structure analogous to that of six-dimensional one, not only in the, form of functional dependence
but also on the nature of warping. This method can easily be extended to even higher dimensions. Also
note that the orbifolding requires branes situated at edges of n-dimensional hypercube with 3-branes at
the corners. If one of the directionsuffers a large warpingany other directionshould havesmall warping
so that there is no large hierarchy coming from the moduli. In this case also we have several candidates
for our SM brane. However applying the fact that no brane should have less energy than ours, leads to
(y =π,z =0,w=0) to be SM brane.
V. BULK FIELDS IN SEVEN-AND-HIGHER-DIMENSIONAL SPACETIME
Following the methods of previous sections, we shall carry out the Kaluza-Klein decomposition of a
bulk scalar field propagating in the spacetime given by Eq. (41). As in the previous section in this
case as well we can write the bulk scalar field in terms of product of four functions. By making KK
decompositionweagainendupwithKKmassmodeshavingTeVscalemassesandsplittings. Theaction
for the bulk scalar field in this seven-dimensional spacetime can be given as
1
S = d4x dy dz dw√ G G ∂AΦ∂BΦ+m2Φ2 . (42)
AB
2 −
Z Z Z Z
(cid:2) (cid:3)
From the line element as given by Eq. (38), we readily obtain the following form for the action
1 cosh4(ℓw)cosh3(kz)
S = d4x dy dz dw R r e 2σ η ∂ Φ∂ Φ
2 y zℜw − cosh4(ℓπ) cosh3(kπ) µν µ ν
Z Z Z Z h
1 r cosh4(ℓw)cosh3(kz)
+ 2ℜRwyz cosh4(ℓπ) cosh3(kπ)e−4σ(∂yΦ)2
10
1 R cosh4(ℓw)cosh5(kz)
+ 2ℜwrz y cosh4(ℓπ) cosh5(kπ)e−4σ(∂zΦ)2
1r R cosh6(ℓw)cosh5(kz)
+ z y e 4σ(∂ Φ)2
2 w cosh6(ℓπ) cosh5(kπ) − w
ℜ
1 cosh6(ℓw)cosh5(kz)
+ m2 r R e 4σΦ2 , (43)
2 ℜw z ycosh6(ℓπ) cosh5(kπ) −
i
where σ =cy . We make the following substitution for the bulk field:
| |
α β γ
p q r
Φ= φ (x) (44)
pqr
Xpqr Ry √rz √ℜw
p
We also impose the following normalization for the functions α , β and γ ,
p q r
e 2α α dy = δ (45)
− m n mn
Z
cosh3(kz)
β β dz = δ (46)
cosh3(kπ) m n mn
Z
cosh4(ℓw)
γ γ dw = δ . (47)
cosh4(ℓπ) m n mn
Z
Now applying integration by parts to the integral as presented in Eq. (43) we readily obtain
1 cosh4(ℓw)cosh3(kz)
S = d4x dy dz dw e 2σ (ηµν∂ φ ∂ φ )α α β β γ γ
2  − cosh4(ℓπ) cosh3(kπ) µ pqr ν abc p a q b r c
Z Z Z Z n pqXrabc
 
1 1 cosh4(ℓw)cosh3(kz)
φ φ β β γ γ α ∂ e 4σ∂ α
− 2R2  cosh4(ℓπ) cosh3(kπ) pqr abc q b r c p y − y a 
y pqrabc
X (cid:0) (cid:1)
 
1 1 cosh4(ℓw) cosh5(kz)
e 4σφ φ α α γ γ β ∂ ∂ β
− 2r2  cosh4(ℓπ) − pqr abc p a r c q z cosh5(kπ) z b 
z pqrabc (cid:18) (cid:19)
X
1 1 cosh5(kz) cosh6(ℓw) 
e 4σφ ψ α α β β γ ∂ ∂ γ
− 2 2 cosh5(kπ) − pqr abc p a q b r w cosh6(ℓπ) w c
ℜw (cid:20) (cid:18) (cid:19)(cid:21)
X
1 cosh6(ℓw)cosh5(kz)
+ m2 e 4σφ φ α α β β γ γ . (48)
2 cosh6(ℓπ) cosh5(kπ) − pqr abc p a q b r c
(cid:20)X (cid:21)o
Thenwemakethe followingchoiceforthe differentialequationssatisfiedby the functionsα , β andγ ,
n n n
1
∂ e 4σ∂ α = A2e 2σα (49)
− R2 y − y n n − n
y
1 cosh5(cid:0)(kz) (cid:1) cosh3(kz)
∂ ∂ β = B2 β (50)
−r2 z cosh5(kπ) z n ncosh3(kπ) n
z (cid:18) (cid:19)
1 cosh6(ℓw) cosh4(ℓw)
∂ ∂ γ = C2 γ . (51)
− 2 w cosh6(ℓπ) w n ncosh4(ℓπ) n
ℜw (cid:18) (cid:19)
The first equation as presented in (49) can be solved and has an identical solution as that obtained in
the previous section. However, for convenience we rewrite the solution,
e2σ A eσR A eσR
p y p y
α = J +b Y (52)
p 2 p 2
N c c
p (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)