Describing the ADD model in a warped geometry

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by  Bin Guo

Describing the ADD model in a warped geometry
Bin Guo1,∗ Yu-Xiao Liu1†,‡ Ke Yang1,2,§ and Xin-He Meng3,4¶
1Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, People’s Republic of China
2IFAE, Universitat Auto`noma de Barcelona, 08193 Bellaterra, Barcelona, Spain
3Department of Physics, Nankai University, Tianjin 300071, People’s Republic of China
4State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
CAS, Beijing 100190, People’s Republic of China
Weproposeanewdescriptionofthe(4+N)-dimensionalArkani-Hamed-Dimopoulos-Dvali(ADD)
model in a (4+1)-dimensional warped geometry to solve the gauge hierarchy problem. It has the
5
same KK spectrum as in the ADD model and recovers its phenomenons that do not involve the
1
interaction among the graviton KK modes. There is no hierarchy between the fundamental length
0
2 and the size of the extra dimension. An explicit realization is constructed in the nonlocal gravity

n theorytogiveawarpeddescriptionofthesix-dimensionalADDmodel. Remarkably,theequivalent
a number N of the extra dimensions in this description may be non-integral, which provides a new
J
mechanism to escape theexperimental constrains.
2
1
PACSnumbers: 04.50.-h,04.50.Kd,11.27.+d

]
h
t I. INTRODUCTION R 1032/NM −1. Furthermore, there is no brane ten-
- ∗
∼
p sion in the model.
e A good inspiration to solve the remained hierarchy
h The idea of extra dimensions opens a new way to
problem in the ADD model comes from the Randall-
[ solve the gauge hierarchy problem—the large difference
Sundrum 1 (RS1) model [3] in five dimensions with the
1 Pbelatwncekensctahlee Melectrow1e0a1k6TsecVal.eOMneEWof ∼the1TfaemVouasndmothde- warped geometry given by
v Pl ∼
4 elsistheArkani-Hamed-Dimopoulos-Dvali(ADD)model ds2 = e−2k|y|η dxµdxν +dy2 (2)
µν
7 [1,2], alsocalledthe modeloflargeextradimensions. In
1
6 thismodel,thefundamentalscaleisM 1TeV,andthe = η dxµdxν +dz2 , (3)
2 hierarchybetweentheeffectivePlanck∗sc∼aleMPl andthe (1+k|z|)2 µν
0 fundamental scale M is generated by the large volume (cid:0) (cid:1)
∗ where k is the AdS curvature scale, y [ y ,y ] and
. b b
1 of the extra dimensions [1]: z [ zb,zb]withkzb =ekyb 1aretheph∈ysi−calandcon-
0 ∈ − −
formal coordinates of the fifth dimension, respectively.
5 M2 MN+2RN, (1) The effective Planck and electroweak scales read [3]
1 Pl ∼ ∗
:
v M3
where R and N are the size and number of the extra M2 = ∗(1 e−2kyb), M =M e−kyb, (4)
Xi dimensions, respectively. In this theory, the standard Pl k − EW ∗
r model particles are assumed to be confined on the brane where the fundamental scale is assumed as M∗ k
a and the extra dimensions are flat, so the electroweak 1016TeV. The gauge hierarchy problem is solved∼by∼a
OscnaeleoisftthheesafammeoaussthperefduinctdiaomnsenotfalthscealAe:DMDEWmo∼deMl ∗is. 3sm7Mall−p1h.yFsircoamlleanngotthheorftphoeinfitftohfdviimewen,saiolnarygbe∼co3n7fko−rm1 a∼l
∗
that it may break down the gravitational inverse-square size of the extra dimension generates the hierarchy be-
law in experimentally accessible regions, since the spac- tween the Planck scale and the electroweak scale via
ing of graviton KK modes is determined by the size of
large extra dimensions R: ∆E R−1. However, a re- MPl MEW(M∗zb). (5)
∼ ∼
mained hierarchy between the size of the extra dimen-
Furthermore, the spacing of graviton KK modes is di-
sions and the fundamental length M−1 is introduced as
∗ rectlydeterminedbytheconformalsizez : ∆E z−1
b ∼ b ∼
1TeV,ratherthanthe physicalsizey [3]. Thatistosay,
b
in warped extra dimension model, the new gravitational
effect of extra dimension appears at the conformal scale
†Correspondingauthor
∗Electronicaddress: [email protected] zb but not the physical scale yb. However,the conformal
‡Electronicaddress: [email protected] scale zb 1TeV−1 is too small in RS1 model.
∼
§Electronicaddress: [email protected] Inthisletter,wewouldliketobuildnewwarpedbrane
¶Electronicaddress: [email protected] models thatsolvethe remainedhierarchyprobleminthe
2
ADD model and does not change its phenomenons. The where we have defined ψ 1 f(η)+ξ. To ensure the
≡ −
new model would have a large conformal size to recover stability,weshouldrequireψ >0. Byvaryingthe action
the phenomenons of ADD model and still keep a small (7) with respect to ξ and η respectively, one easily gets
physical size to solve the remained hierarchy problem.
(cid:3)η = , (cid:3)ξ = f , (8)
Further, in order to maintain the fundamental scale in R −R η
tphaertAicDleDs smhooudledlMbe∗c∼onMfinEeWd∼at1yTe=V0,tshinecsetatnhdeamrdamssopdae-l where fη ≡ ddfη. Now the choice of the definition of (cid:3)−1
becomes the choice of the solution of (cid:3)η = . Once we
rameters of fields confined at y = y in the background R
0 havechosena specific solutionofη, the originalnonlocal
(2) has a redshift e−ky0 [3, 4]. Thus, the graviton zero
theory is defined. It means that different solutions of
mode should be localizednear y =y to generatea large
b η correspond to different nonlocal theories, rather than
Planck mass in order to solve the gauge hierarchy prob-
different solutions of one nonlocal theory.
lem. This is opposite to the RS1 model constructed in
The modified Einstein equations are given by
general relativity, in which the graviton zero mode is lo-
1
calized near y = 0. We will realize the special property ψ ψ ∂ ξ∂Cη g
AB C AB
of the graviton zero mode and build the model in the R − 2 R−
nonlocal gravity theory, which is one of the simplest re- (cid:0) (cid:1)1
+ (g (cid:3) ) 1 ψ ∂ ξ∂ η+∂ η∂ ξ
AB A B A B A B
alization of this class of models. In fact, a special model −∇ ∇ − − 2
was constructed in the scalar-tensor gravity in Ref. [5]. = M1−dT(b), (cid:0) (cid:1) (cid:0) (9(cid:1))
∗ AB
However,there is no realbrane with tension constructed
by matter. We wouldlike to construct a brane with ten- where A,B,C denote the bulk indices 0,1,2, ,d,
sion. This is also one motivation of this letter. and T(b) repre·se·n·ts the energy-momentum tensor·o·f·the
AB
DifferentfromtheusualwaytomodifyEinsteingravity brane.
by local generalization, Deser and Woodard developed a The metric for the flat brane world model is assumed
modifiedgravitytheoryby addinganonlocalterm[6,7]. as [3]
Thisnonlocalgravitytheorywasproposedtoexplainthe
ds2 =a2(y)η dxµdxν +dy2, (10)
cosmic acceleration without the cosmological constant µν
[8–12]. It was found that this theory possesses the same where the brane coordinate indices µ,ν =0,1, ,d 1
··· −
gravitational degrees of freedom and initial value con- and η = diag( ,+, ,+). We consider an S /Z
µν 1 2
− ···
straints as general relativity, and has no ghost graviton orbifold extra dimension as in Ref. [3]; so the physical
mode [13]. extra dimensional coordinate y [ y ,y ]. The action
b b
∈ −
We will construct a five-dimensional warped brane describing one brane with tension σ located at y = 0
+
model in the nonlocal gravity. It will solve the re- and another brane with tension σ located at y = y is
− b
mained hierarchy and recover the phenomenons of the given by
six-dimensional ADD model that do not involve the in-
teraction between the graviton KK modes. We will also Sb = ddx q(y =0)σ++ q(y =yb)σ− ,(11)
− −
propose a possible way to construct a class of general- Z hp p i
ized five-dimensional warped models that describe the whereq (y =0)andq (y =y )aretheinducedmetrics
µν µν b
(4+N)-dimensional ADD model. at y =0 and y =y , respectively.
b
Then the µν and dd components of the modified Ein-
stein equations (9) and the equations (8) for η and ξ are
II. THE MODEL given by
1
ψ′′+(d 1)Hψ′+ (d 1)H′+ d(d 1)H2 ψ
Westartwiththefollowingactionofthenonlocalgrav- − − 2 −
ity theory proposed in Ref. [6] in (d + 1)-dimensional (cid:20) (cid:21)
1
spacetime: + ξ′η′ M1−d σ δ(y)+σ δ(y y ) =0, (12)
2 − ∗ + − − b
S = M∗2d−1 dd+1x√−gR 1+f((cid:3)−1R) +Sb, (6) 12ξ′η′−(cid:2)dHψ′− 21d(d−1)H2ψ(cid:3) =0, (13)
Z (cid:16) (cid:17) and
whereM is thefundamental(d+1)-dimensionalPlanck
∗
mass, is the curvaturescalar,(cid:3)−1 is the inverseofthe η′′+dHη′ = 2dH′ d(d+1)H2, (14)
R − −
d’Alembertian operator, and Sb is the action describing ξ′′+dHξ′ = [2dH′+d(d+1)H2]f , (15)
η
thebraneconfiguration. Notethatinordertodefinethis
theory, we should specify what definition of (cid:3)−1 we use. respectively, where H a′/a and the prime denotes the
≡
Thismodelhasalocalformbyaddingtwoauxiliaryfields derivative with respect to the coordinate y. Subtracting
η and ξ [10, 11, 14]: (13) from (12) we get
ψ′′+(2d 1)Hψ′+ (d 1)H′+d(d 1)H2 ψ
S = M∗2d−1 Z dd+1x√−g(ψR+ξ(cid:3)η)+Sb, (7) = M∗1−d σ+−δ(y)+σ−δ((cid:2)y−−yb) , − (cid:3)(16)
(cid:2) (cid:3)
3
which can be rewritten as where (cid:3)(d) denotes the d-dimensionald’Alembertian op-
erator. After changing to the conformal coordinate z
∂y +dH ∂y+(d 1)H ψ with adz = dy and defining A ad−1ψ, the linearized
−
≡
= (cid:16)M1−d σ δ(cid:17)((cid:16)y)+σ δ(y y(cid:17)) . (17) equation (22) becomes
∗ + − − b
∂ A
Now it can be checked(cid:2) that the system supp(cid:3)orts the fol- ∂ ∂ h + z ∂ h +(cid:3)(d)h =0. (23)
z z µν z µν µν
lowingsolutionforthe warpfactora(y)andthe function A
ψ(y):
We decompose the tensor perturbation h as
µν
a(y)=e−k|y|, ψ =c edk|y|+c e(d−1)k|y|. (18)
1 2 hµν(x,z)=εµν(x)A−21(z)Ψ(z), (24)
SotheRS1geometryisreproduced. Theconditionψ >0
becomes c > c ek|y|. After integrating Eq. (17) at wherethefour-dimensionalpartofthegravitonKKmode
2 1
y =0andy =y−brespectively,wecanobtaintherelations εµν(x) satisfies (cid:3)(d)εµν(x) = m2εµν(x). Then we ob-
tainaSchr¨odinger-likeequationfortheextradimensional
σ
c = + , σ = σ edkyb. (19) part:
1 2kMd−1 − − +
∗
1∂ ∂ A 1(∂ A)2
We assume that the brane located at y = 0 has positive ∂ ∂ Ψ+ z z z Ψ=m2Ψ. (25)
tension, i.e., σ > 0. Then the brane located at y = y − z z 2 A − 4 A2
+ b (cid:18) (cid:19)
isanegativetensionbrane. Therelationbetweenthe σ
+ Itcanberewrittenas ∂ +1∂zA ∂ +1∂zA Ψ=m2Ψ,
andσ− isdifferentfromthecaseintheRS1model,where z 2 A − z 2 A
σ = σ . It is interesting to consider the tensionless which implies that the eigenvalues m2 are nonnegative
+ − (cid:0) (cid:1)(cid:0) (cid:1)
limit σ− = 0. In this limit c =0 and ψ e(d−1)k|y|. In and the brane system is stable under the tensor pertur-
+ 1
the process similar to the following, we ∝will find it has bation.
the same spectrum as in the (d+1)-dimensional ADD From now on we restrict our discussion in the case of
model with codimension one. d = 4. Setting m2 = 0 in Eq. (25), we can obtain the
The solutions of the auxiliary fields η and ξ are some- normalized zero mode
nwex-htaahmtspceloec.tmioWpnleicidanteentdho.eterWtehgeeiorwneiglyilonusye[0∈,E+[qn.yb)(,1(an4n)+df1od)reynbηo]taaessttahhnee Ψ0(z)= AN21 = σ+2kσM+(2∗3z(1++kkz|z2)|)++2cc2z 12 . (26)
solution of the field η(y) in th∈is sect∞ion as η (y). Then 0 2kM∗3 b b 2 b!
n
the solution of Eq. (14) is given by Even though the RS1 background is reproduced, the
graviton zero mode is localized near the negative ten-
η (y)=( 1)n(d+1)ky+λ e(−1)ndk(y−nyb)+ρ , (20)
n n n sion brane, rather than near the positive one in the RS1
−
model [3]. So in our model we should confine the stan-
where the coefficients λ are determined by the recur-
n
sion relation λ1 = 2(d 1)/d edkybλ0, λn+2 = 1 dard model particles on the positive tension brane to
e−(−1)ndkyb 2(d−1)/d+−λn. T−he coefficients ρn ca(cid:0)n b−e szoerlvoemthoedgeaiusgleochaielirzaerdchnyeaprrotbhleemn.egBaetcivaeusteenthsieongrbavraitnoen,
determined by the continuity condition of η(y). The ex-
(cid:1) we need the compactification of the extra dimension to
act solutions of the auxiliary fields η and ξ are unnec-
recover the effective four-dimensional gravity.
essary since they do not influence the spectrum of the
Substituting the zero mode into the action (7)
massive KK gravitons and the couplings of these gravi-
and making the reduction, we get the effective four-
tons to matter.
dimensional Planck mass:
σ
M2 =M3 dza3ψ =M3z + (2+kz )+2c . (27)
III. PHYSICAL IMPLICATIONS Pl ∗ ∗ b 2kM3 b 2
Z (cid:20) ∗ (cid:21)
Because a(0) = 1, the vacuum expectation value of
Since the metric (10) has the d-dimensional Poincare
the Higgs scalar, which sets all mass parameters of the
symmetry, we can consider the transverse and traceless
standard model, is not warped down or redshifted. So
(TT) part of the perturbation separately, which corre-
all the mass scales are not suppressed on the positive
spondstothespin-2gravitoninddimensions. Thetensor
tensionbraneandthe electroweakscaleremains the fun-
perturbation is parameterized as
damental scale M M [3]. In order to solve the
EW ∗
ds2 =a2(ηµν +hµν)dxµdxν +dy2, (21) gauge hierarchy proble∼m, we set the dimensionless pa-
rameter c as c 1 and all the fundamental scales as
where the perturbation hµν(x,y) satisfies the TT condi- 1TeV, wh2ich me2a∼ns we set M and k as 1TeV and σ
tions ∂µh = 0 = ηµνh . The µν component of the ∗ +
µν µν as (1TeV)4. Then the effective four-dimensional Planck
linearized perturbation equations reads
mass M 1016TeV leads to the condition kz 1016,
Pl b
ψ′ which corr∼espondsto ky 37 (or y 37TeV−1)∼in the
a2h′′ + daa′+a2 h′ +(cid:3)(d)h =0, (22) b ∼ b ∼
µν ψ µν µν physical coordinate.
(cid:16) (cid:17)
4
In the following discussion, we analyze the spectrum Equation(35)showsthatthecouplingsofthefirstfew
of the massive KK modes and their couplings to the KKmodestothematterhavethesimilarstrengthtothe
matter confined on the positive tension brane. With coupling of the massless graviton to the matter. How-
the Neumann boundary condition at the positive ten- ever, the higher KK modes have stronger couplings to
sion brane: ∂ h = 0, we can obtain the solutions the matter. To estimate their effects, we first consider
z µν z=0
|
for the Schr¨odinger-like equation (25): the process that involves emission of gravitons, which
could be observedas missing energy [1, 2, 15]. The total
1
1 1 2 1 cross section for this process is of order
Ψ (z)= z+ J m (z+ )
n 0 n
N β β
n (cid:18) (cid:19) (cid:20) (cid:16) (cid:17) 1 E4
1 Σ (ξ(n))2 = . (36)
+αY m (z+ ) , (28) n E2(1TeV)4
0 n
β
(cid:16) (cid:17)(cid:21) So the cross sectieon would have a significant increase
wmhaelirzeatβ1io≡n ck1o+nst2aMσn+∗3tc.2,Tαhe=s−peYJc11t((rmmuββnnm)),mandoNf tnhiesgthraevnitoorn- wmhaesnsivteheKeKnemrgoydeaspwpritohacthhe1TmeaVss.Oganpt1h0e−o4tehVerwhoaunldd,cothne-
n
KK modes can be obtained by the Neumann boundary tribute an obvious deviation to Newton potential when
condition at zb: ∂zhµν|z=zb =0, which reads as disTtahnecetolteaslsctrhoasns tshecetciornitic(3a6l)diasntadnctheeRccri∼tic2amlmdi.stance
J m (z + 1) Y m (z + 1) are the same as in the six-dimensionalADD model. The
1 n b β 1 n b β
= . (29) mechanismcanbeclearlyseenfromEq. (27),inourfive-
(cid:16)J1(mn/β) (cid:17) (cid:16)Y1(mn/β) (cid:17) dimensional warped model M2 M3kz2 M4z2 and
Since kzb 1016 and c2 1, the first few KK modes in the six-dimensional ADD mPold∼el M∗P2l ∼bM∼∗4R2∗, wb hich
satisfy the c∼ondition mn ∼ 1 and α π(mn)2 0. implies zb R.
β ≪ ≈ 8 β → Further,∼in the six-dimensionalADD model, the num-
Then the solution (28) becomes
ber of KK modes with same mass m is about m R
n n
1 and they have the same couplings of 1 to matter.
Ψn(z) 1 z+ 1 2 J0 mn(z+ 1) . (30) In our five-dimensional warped model, thMePrle is only one
≈ N β β
n (cid:18) (cid:19) (cid:18) (cid:19) KK mode with mass m and the coupling to matter is
n
In this approximation,the spectrum reads (mnzb)21. Because m R( 1 )2 ((mnzb)21)2, the sum of
xn theMpPlropagatorsof alnl theMKPlK m∼odesMinPlthe ADD model
m , (31)
n ≈ z isequaltothepropagatorwiththesamemassinourfive-
b
dimensional model. So the phenomenons that do not in-
where x satisfies J (x )=0, and x =3.83, x =7.02,
n 1 n 1 2 volvethe interactionamong the KK modes are the same
x = 10.17, . The normalization constant N can be
3 ··· n in these two models.
determined as follows:
zb 2 zb
1= Ψ2dz z[J (m z)]2dz
n ≈ N2 0 n IV. THE WARPED DESCRIPTION OF THE
Z−zb n Z0
1 (4+N)-DIMENSIONAL ADD MODEL
z2[J (m z )]2. (32)
≈ N2 b 0 n b
n
In this section, we point out a possible way to further
Using the approximation J0(x) ≈ π2xcos(x− 41π) and generalizethefive-dimensionalwarpedmodeltodescribe
theapproximateformulaofthezeroqpointofJ1(xn),xn the (4+N)-dimensional ADD model. We also assume
≈
(n+1)π,wehaveN 2zb2. ThenthenormalizedKK that the fundamental scale is M∗ ∼ 1TeV and the stan-
4 n ≈ πxn dard model particles are confined on the brane at y =0.
modes are q A possible class of actions in five dimensions are
1
πxn 1 2 1 M3
Ψn(z)≈ 2z2 z+ β J0 mn(z+ β) . (33) S = ∗ d5x√ g ψ + (gAB,ψ,Φi) +Sb, (37)
r b (cid:18) (cid:19) (cid:18) (cid:19) 2 − R L
Z h i
TheinteractionbetweentheseKKmodesandthematter
where only the field ψ nonminimally couples to the cur-
confined on the positive tension brane is
vature scalar, and Φ in the Lagrange density denote
i
L
L(n) =M−23Tµν(x)h(n)(x,0)=ξ(n)Tµν(x)ε(n)(x),(34) other dynamic and/or nondynamic fields. Assume that
int ∗ µν µν the system supports a family of solutions:
where Tµν is thee symmetric conseervedeMinkowski space a(y)=e−k|y|, ψ(y) eαk|y|, (38)
energy-momentum tensor. Then it is simple to get ∝
e ξ(n) M−23 xn xn12 . (35) wrahmereetekr α∼isMd∗etaenrdmitnheedvbayluteheofdtehteaildiomfetnhseioLnalegsrsanpgae-
∝ ∗ βz2 ∝ 1016TeV density. For our model (7), α=4.
r b
e
5
The graviton zero mode can be written as M−1 to the matter. In the warped description, there
Pl
Ψ0 a3/2ψ1/2 e21(α−3)k|y| = (1 +k z )(α−3)/2, which is is only one graviton KK mode with mass mn and its
loc∝alized near∝y = yb and y = 0 whe|n|α > 3 and α < 3, coupling to matter is about MP−l1(mnzb)N2−1. Because
risesdpeetcetrimveilnye.dTbhye effective four-dimensionalPlanck mass mNn−1RN−1(MP−l1)2 = (MP−l1(mnzb)N2−1)2, the sum of
the propagatorsof allthe gravitonKK modes with mass
m in the ADD model is equal to the propagator with
M2 M3(kz )α−3z . (39) n
Pl ∼ ∗ b b the same mass in the five-dimensional warped descrip-
tion. Soourdescriptionrecoversthephenomenonsofthe
Comparing with the corresponding relation in the (4+
N)-dimensional ADD model, M2 MN+2RN, if α = ADD model that do not involve the interaction among
Pl ∼ ∗ the gravitonKK modes.
N +2, then z R and it can be shown that the two
b
∼ ThisclassofmodelsgiveanewrealizationoftheADD
models will share the same KK spectrum. Note that
the physical size of the extra dimension is y 74M−1, model: the effects of multi extra dimensions may origi-
b ∼ N ∗ nate from the warped geometry with only one extra di-
which indicates that the remained hierarchy problem in
mension. How to distinguish these two models and mea-
the ADD model is solved in this generalized warped de-
sure the number of the extra dimensions are interesting
scription. Inthefollowing,weconsiderthecaseofN 2.
≥ theoretical and experimental problems. More interest-
Having reproduced the graviton KK spectrum in the
ingly, the equivalent number N of the extra dimensions
ADD model, we turn to the question of the couplings
in the new models may be non-integral, which provides
betweenthegravitonKKmodesandthematterconfined
a new mechanismto escape the experimentalconstrains.
onthe braneaty =0. With the similarproceduresasin
Setting the fundamentalscale M k 1TeV,the criti-
section III, we can write the graviton KK modes as ∗
cal distance is given by Rc =2 1∼0α3−22∼−16mm. Then to
1 1 12 1 satisfy the experimental constra×in Rc . 0.1mm [16, 17],
Ψn(z)≈ Nn (cid:18)z+ k(cid:19) JN2−1(cid:18)mn(z+ k)(cid:19). (40) weFounrtlyhenremedorαe,&sin4c.e18thoergNeom&e2t.r1y8i.sasliceofAdS ,this
5
The spectrum is given by m xn, where x are deter- new model may have a close relation to the AdS/CFT
n ≈ zb n correspondenceandhasaholographicdescription[4,18].
mined by J (x )= 0. The normalization constants are
N n
2
N−1 1−N −N
Nn ∝ zb/√xn and then Ψn(0) ∝ xn2 k 2 zb 2. The
couplings of these KK modes to the matter at y =0 are
V. ACKNOWLEDGEMENT
given by
ξ(n) M−23Ψ (0) M−1xN2−1. (41) This workwassupportedbythe NationalNaturalSci-
∝ ∗ n ∝ Pl n ence Foundation of China (Grants No. 11075065, No.
In the (e4 + N)-dimensional ADD model, the num- 11075078, and No. 11375075). K. Yang was supported
ber of the graviton KK modes with the same mass m by the scholarship granted by the Chinese Scholarship
n
is about mN−1RN−1 and each mode has a coupling of Council (CSC).
n
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