#### Clockwork/linear dilaton: structure and phenomenology

Accepted: May
Clockwork/linear dilaton: structure and phenomenology
Gian F. Giudice 0 1 2 4
Yevgeny Kats 0 1 2 3 4
Matthew McCullough 0 1 2 4
Riccardo Torre 0 1 2 4
Alfredo Urbano 0 1 2 4
Geneva 0 1 2
Switzerland 0 1 2
0 Via Bonomea 256 , 34136 Trieste , Italy
1 Via Dodecaneso 33 , 16146 Genova , Italy
2 Beer-Sheva 8410501 , Israel
3 Department of Physics, Ben-Gurion University
4 Theoretical Physics Department , CERN
5 Sezione di Trieste , SISSA
The linear dilaton geometry in ve dimensions, rediscovered recently in the continuum limit of the clockwork model, may o er a solution to the hierarchy problem which is qualitatively di erent from other extra-dimensional scenarios and leads to distinctive signatures at the LHC. We discuss the structure of the theory, in particular aspects of naturalness and UV completion, and then explore its phenomenology, suggesting novel strategies for experimental searches. In particular, we propose to analyze the diphoton and dilepton invariant mass spectra in Fourier space in order to identify an approximately periodic structure of resonant peaks. Among other signals, we highlight displaced decays from resonantly-produced long-lived states and high-multiplicity nal states from cascade
Phenomenology of Field Theories in Higher Dimensions
1 Introduction 2
Properties of the model
Basic setup
2.3 Impact of cosmological constant terms
KK mode mass spectrum and couplings
nal states, branching fractions, lifetimes
KK mode production cross sections at the LHC
3
LHC signatures
3.1
Standard signatures
3.2
Novel signatures
3.1.1
3.1.2
3.1.3
3.2.1
3.2.2
3.2.3
3.2.4
Continuum s-channel e ects at high invariant mass
Continuum t-channel e ects in dijets at high invariant mass
Distinct
and e+e resonances
Periodicity in the diphoton and dilepton spectra
Turn-on of the spectrum at low invariant mass
Cascades within the KK graviton and KK dilaton towers
Resonant production of particles with displaced decays
4
Conclusions
A Background solutions
B Radius stabilization and comparison with RS C
Graviton decays to SM particle pairs
D Graviton decays to gravitons E F
Dilaton/radion spectrum and couplings with SM elds
Dilaton/radion decays to SM particle pairs
G Trilinear graviton/dilaton/radion decays
G.1 KK graviton decays into two scalars
G.2 KK graviton decays into a KK graviton and a scalar
G.3 KK graviton decays: branching ratios
G.4 KK scalar decays into two scalars G.5 KK scalar decays into two KK gravitons { i {
Representing the KK graviton tower by an e ective propagator
An exact solution to Einstein's equations
Introduction
59
60
60
61
HJEP06(218)9
It has been shown recently [1] that the clockwork mechanism, introduced in refs. [2{4] to
reconcile super-Planckian eld excursions with renormalizable quantum
eld theories, is a
much broader tool with many possible applications and generalisations, some of which have
been explored in refs. [1, 5{25]. A particularly interesting result is the observation that
discrete clockworks have a non-trivial continuum limit that singles out a
ve-dimensional
theory with a special geometry. This geometry coincides with the one obtained in theories
with a
ve-dimensional dilaton which acquires a background pro le linearly varying with
the extra-dimensional coordinate. This theory can address the Higgs naturalness problem
in setups where the Standard Model (SM) lives on a brane embedded in a truncated
version of the ve-dimensional linear-dilaton space. Earlier, the same setup was proposed
in ref. [26] motivated by the seven-dimensional gravitational dual [27, 28] of Little String
Theory [29, 30], which is a six-dimensional strongly-coupled non-local theory that arises on
a stack of NS5 branes. Additionally, several recent studies have examined in more detail
how the linear dilaton setup can be embedded in supergravity [31, 32].
The clockwork interpretation of the linear dilaton theory has helped in elucidating
its relation with Large Extra Dimension (LED) [33{35] and Randall-Sundrum (RS) [36]
theories, thus providing a coherent map of approaches to the hierarchy problem in extra
dimensions. Because of the double interpretation of the same theory either as a linear
dilaton setup emerging from an e ective description of non-critical string theory, including
duals of Little String Theory, or as the continuum version of a clockwork model, we will
refer to this theory as Clockwork / Linear Dilaton (CW/LD).
So far, CW/LD has received very little experimental attention, in spite of its attractive
and distinguishing features. Nonetheless, some phenomenological aspects of CW/LD have
already been discussed in the literature. The distinctive KK graviton spectrum, with a
mass gap followed by a narrowly spaced spectrum of modes, and some of its associated
collider signatures have been pointed out in ref. [37]. A more detailed study of the KK
graviton phenomenology, including in particular the case of a small mass gap, has been
undertaken in ref. [38]. Finally, the KK dilaton / radion collider signatures have been
studied in ref. [39].
In the current paper we aim at providing a comprehensive picture of the collider
phenomenology of CW/LD, extending previous studies, deriving new constraints from present
{ 1 {
LHC data, and suggesting new characteristic signatures, in the hope of motivating
dedicated experimental searches.
We start by reviewing in section 2 the structure of CW/LD and the consequences of
its UV embedding in string theory for the low-energy parameters. We also discuss the
e ect of adding cosmological constant terms to the e ective theory, arguing that the bulk
theory should be supersymmetric to avoid destabilising the setup. Then, we explain the
salient features of CW/LD for collider applications. The theory describes a tower of
massive spin-two particles which, depending on the point of view, can be interpreted either
as the Kaluza-Klein (KK) excitations of the
ve-dimensional graviton or as the
continuum version of the clockwork gears. Their mass spectrum and couplings are completely
xed in terms of only two parameters: the fundamental gravity scale M5 and the mass
k which characterizes the geometry of CW/LD. We encounter also a tower of spin-zero
particles obtained from the combination of the single radion state with the KK excitations
of the dilaton. In CW/LD the same scalar
eld both induces the non-trivial geometry
and stabilises the extra dimension (thus playing the role of the Goldberger-Wise eld [40]
in RS). The mass spectrum and couplings of the scalar modes are not fully determined
by M5 and k alone, but also depend on the brane-localised stabilising potential and on a
possible Higgs-curvature coupling. However, many features of the scalar phenomenology
are independent of these details.
The rest of the paper is devoted to the collider phenomenology of CW/LD. In
section 3.1 we study the e ect of s-channel and t-channel exchange of KK gravitons in
diphoton, dilepton, and dijet distributions at high invariant mass. An interesting feature of
CW/LD is that these processes can be reliably computed within the e ective theory,
unlike LED where these processes are dominated by incalculable UV contributions. We also
analyse the resonant production of new states decaying into diphoton and dilepton
nal
states. All signatures discussed in section 3.1 correspond to already existing searches
performed by the LHC collaborations and results can be adapted to the case of CW/LD
using studies of continuum distributions (previously done for LED) or resonant production
(previously done for RS).
In section 3.2 we explore new strategies that can be used at the LHC to discover or
constrain CW/LD. The near-periodicity of invariant mass distributions with characteristic
separations in the 1-5% range (at the edge of experimental resolution) has prompted us
to suggest a data analysis based on a Fourier transform, similarly to what is routinely
encountered in other elds, for example in analyses of CMB temperature uctuations. We
show that such an analysis is competitive with other searches as a discovery mode, as well
as being e ective for extracting model parameters.
A distinguishing feature of CW/LD with respect to LED or RS is the mass gap of
the KK tower, followed by a near-continuum of modes. We suggest that such a turn-on
of the spectrum may be observable even when it occurs at low invariant mass, where the
individual resonances are di cult to see due to the experimental resolution. Such searches
would require the use of trigger-level analysis / data scouting or ISR-based triggers.
An important new result presented in this paper is the calculation of the decay chains
of graviton or scalar excited modes into lighter KK modes. We nd that such cascades
{ 2 {
are the dominant decay mode for most of the scalar KK tower, and in certain parameter
regions also a signi cant decay mode for part of the graviton KK tower. We study the
properties of the high-multiplicity nal states that arise from such decays. We also explore
the possibility of displaced vertices originating from resonant particle production, which is
another signature characteristic of CW/LD.
Finally we collect in the appendices a compendium of formul for the production
cross sections and decay rates of the new particles in CW/LD. This should be a useful
resource for those interested in experimental or phenomenological simulations of CW/LD in
collider environments. Other appendices are dedicated to additional details and discussions
concerning the structure of the theory.
Properties of the model
Basic setup
We consider a 5D space in which the extra dimension is a circle, with the circumference
parameterized by a coordinate y in the range
R
y
R. The SM lives on a brane
(TeV brane) at y = yT = 0, another brane (Planck brane) is at y = yP =
R, and a Z2
orbifold symmetry identi es y $
y. The full action in the Einstein frame is
where
is a bulk potential and
Sbulk = M53
SB =
2
X
i=T;P
S = Sbulk + SB + SGHY + SSM ;
Z
d x
SGHY = 4M53 X
d x
dyp
Ki (y
SSM = 2
yT) ;
g R
dyp
3
V (S) =
4k2e 2S=3
T(S) = e S3 M53
4k +
P(S) = e S3 M53 + 4k +
2
2
T (S
P (S
ST)2 ;
SP)
2
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
parameterise the scalar potentials at the TeV and Planck branes near their local minima
at S = ST;P. Here S is the dilaton eld, M5 is the ve-dimensional reduced Planck mass,
{ 3 {
and LSM is the SM Lagrangian.1
in appendix A) are solved by
k is a mass parameter, R is the 5D Ricci scalar, g is the determinant of the 5D metric,
= g=g55 is the determinant of the induced metric at the boundaries, i are two masses that
determine the strength of the dilaton boundary potentials, Ki are the extrinsic curvatures
of the two boundaries, which determine the Gibbons-Hawking-York (GHY) term [41, 42],
Einstein's equations and the dilaton equation of motion (which are presented in detail
is assumed to be not much higher than the electroweak scale, while the exponentially
greater scale MP is an illusion, created by the exponential factor in (2.10). To account for
the hierarchy, one needs
kR '
1
ln
MP r k !
M5
M5
potentials in eqs. (2.7){(2.8) naturally allow for this, one may argue that since the dilaton
realises dilatations non-linearly one expects it would enter the action as an analytic function
of
eS. The boundary potentials in eqs. (2.7){(2.8) would correspond to non-analytic
functions, since S
log . However, it has been argued that such logarithmic terms can
actually arise in string-theoretic setups [26] and we will assume this to be the case.
To understand the origin of the hierarchy in di erent extra-dimensional setups it is
useful to introduce the proper size of the extra dimension L5, given by
L5
Z R
R
dy pg55 ;
w
R d5x p
R d5x p
g (y
g (y
R)=pg55
0)=pg55
1=4
;
1=p8 G
and the warp factor w, which can be de ned through the ratio of the total spacetime
volumes on the two di erent branes
where the extra pg55 comes from the de nition of the covariant -function. Both L5 and
w are purely geometrical quantities that characterise the extra-dimensional compacti ed
1This is one possible choice for the dilaton coupling to the SM elds, however in general the coupling is
model-dependent. For instance, one could have taken a coupling eSLSM in the Jordan frame, which would
result in a di erent Einstein frame action.
{ 4 {
space and their de nitions are explicitly invariant under coordinate reparametrization. The
quantity L5 corresponds to the proper length of the compacti ed space (in the special case
of a single extra dimension). The warp factor w encodes the information of how much each
of the four spacetime dimensions is stretched between one end of the compacti ed space
and the opposite end.
In the case of a single at extra dimension of radius RLED,
L5 = 2 RLED ;
w = 1 ;
MP2 = L5 M53
(LED):
(2.14)
We recognize in eq. (2.14) the familiar LED result that the largeness of MP2 =M52 comes
entirely from the e ect of a large extra-dimensional volume (L5).
HJEP06(218)9
For RS one nds
L5 = 2 RRS ;
w = ekRS RRS ;
RRS is the location of the brane (in the coordinate
system where g55 = 1). In RS the proper length L5 is a number of order one, in natural
units, while the hierarchy MP =M5 comes entirely from the warp factor w. Notice also that
RS reproduces LED when kRS ! 0 and RRS ! RLED.
For CW/LD, we obtain
L5 =
3
k
e 32 k R
1 ;
w = e 32 k R
;
larger than its natural value 1=M5, as illustrated in
gure 1, but not as extreme as in
LED with one extra dimension. As a result, in CW/LD the hierarchy MP =M5 is explained
by a combination of volume (as in LED) and warping (as in RS). While in RS the warp
factor depends exponentially on the proper size, the warp factor of CW/LD is linearly
proportional to L5, so that MP ' k(L5M5=3)3=2. However, MP is still exponentially
sensitive to the parameters k and R which, as shown in section 2.5, determine the physical
mass spectrum of the graviton excitations.
The di erence between RS and CW/LD lies in the geometry of the corresponding
compacti ed spaces. To appreciate this di erence it is useful to work in both cases in the
coordinate basis where the line element is ds2 = A2(z)dx2 + dz2, such that the function
A(z) measures the warping of four-dimensional spacetime as a function of the physical
distance along the extra dimension. For RS one nds A(z) = ekRSz and, because of the steep
exponential behavior, an order-one separation between the branes is su cient to obtain
the large warp factor needed to explain the hierarchy. For CW/LD one nds A(z) = 2kz=3
and, because of the slower linear dependence on the coordinate z, an exponentially large
separation between the branes is needed to obtain the required warping factor.
Nonetheless, the stabilization of the compacti cation radius is naturally obtained with order-one
parameters, and the KK masses and interaction scales are set by the typical size of k, R,
and M5 and not by the size of L5, which corresponds to a much larger distance.
Unlike the case of at geometry, where a single extra dimension would need to have
a Solar System size and is therefore excluded, CW/LD makes the possibility of a single
{ 5 {
D
E
L
M
5 =
1
0
0
M
5 =
T
e
V
1
T
e
V
k [GeV]
(blue) and 100 TeV (red). The horizontal dotted lines indicate the LED limit. The plot on the right
zooms in on the phenomenologically most relevant range of k.
relatively large extra dimension viable again. This is illustrated in gure 1, which shows
the transition between the regime of roughly constant kR (and L5 / k 2=3) at xed MP to
the regime at small k, in which the proper size of the extra dimension is frozen at 2 R and
CW/LD turns into LED. This feature of CW/LD is important because the phenomenology
of a single extra dimension di ers signi cantly from that of multiple extra dimensions, as
we will discuss. (A similar possibility arises in the low-curvature RS model, as pointed out
and analyzed in refs. [43, 44].)
A remarkable feature of CW/LD is that the same eld S that determines the spacetime
geometry also stabilizes the compacti cation radius by
xing the factor k R in terms of
the boundary conditions of S on the branes, set by the brane potentials for the dilaton eld
appearing in SB [15, 39]. In this way the model automatically leads to radion stabilization.
This is to be contrasted with the case of RS. The original RS model su ers from a
stabilization problem, especially pressing because a massless radion with TeV-scale interactions
is experimentally ruled out. Luckily, a simple solution in RS is readily found by adding
a bulk scalar eld with a small mass term, which does not signi cantly perturb the AdS
metric, but generates a stabilizing potential for the radion [40]. CW/LD has this feature
already built in its structure. A more detailed discussion of the stabilisation is given in
appendix B.
2.2
Stringy origins of the linear dilaton eld theory
Since the linear dilaton action is rather peculiar from a eld theory perspective, it is
instructive to consider the UV motivation for this setup. Let us brie y review the stringy
setting for the
string theory.2
eld theory we are considering, beginning with the worldsheet CFT in
We will only consider the spacetime metric and dilaton in the Polyakov
2For more details see e.g. refs. [45{49] and the recent papers [31, 32].
{ 6 {
the superpartner elds that are also present in the massless spectrum.3 The worldsheet
action is
S =
1 Z
+
4
0
S(X)R(2) + : : : :
(2.17)
Here XM are the target space coordinates of an as-yet un xed spacetime dimension D,
0 is the string tension, and h is the worldsheet metric. To determine if the action is
Weyl-invariant at the quantum level we may consider the -functions that describe the renormalization of the massless elds, which are in superstring theory
In matching to the e ective theory, these -functions arise as the equations of motion for
massless elds in the target spacetime. Thus the relevant action is given by [50{52]4
S =
MDD 2 Z
2
dDxp
g eS
D
10
This is analogous to the 5D linear dilaton action in the Jordan frame
S =
M53 Z
2
d x
5 p
g eS
from which, with a Weyl transformation g ! e 2S=3g, one obtains the Einstein frame bulk
action (2.2). From this matching we see that k2 is in fact related to the string tension.
The action (2.20) is also often called the string frame action. It describes the low energy
e ective theory for the dilaton and graviton below the scale of string excitations.
It is known that the string e ective action is classically scale-invariant.5 To see this
the dilaton eld as
= e 2S=(D 2). The e ective action becomes
symmetry realised linearly we may make a Weyl transformation g ! e 2S=(D 2)g and write
S =
MDD 2 Z
2
dDxp
g
D
10
+ : : : + O( 0) :
(2.22)
rescaled S !
S
In this basis we see that under the transformation gMN !
2gMN ,
!
= 2, the action is
D 2 . Thus, since the overall coe cient factors out of the classical equations
of motion, they are invariant under this transformation. Alternatively, in the standard
Jordan-frame action eq. (2.21) the transformation is realised non-linearly as S ! S + ,
where
is a constant, which again only rescales the total action.
3We do not consider the bosonic string theory due to its inherent problems with tachyons.
4Note that this e ective action may be written in numerous forms, all related by eld rede nitions. We
5Discussions of this point are found in refs. [54, 55], however we will essentially follow the discussion of
will use the convention of ref. [53].
ref. [46] section 13.2, albeit in a di erent basis.
{ 7 {
(2.18)
(2.19)
(2.20)
(2.21)
This scale invariance is only classical, and will not be respected quantum mechanically
as the normalisation of the action is physical. Nonetheless, if the action only contains the
dilaton and metric there are selection rules on how the quantum corrections enter. They
may be determined by considering ~ to transform as ~ ! ~e (essentially, one can think of
e S as being ~). As fractional powers of ~ will not arise in the perturbative expansion, we see
that the only additional scalar potential terms in the bulk action will be proportional to the
dimensionless quantity V / (~e S)n, where n is an integer. Essentially the perturbative
series is an expansion in e S, since this plays the role of a coupling constant [55]. This
argument shows that the classical scale invariance can persist perturbatively if only the
dilaton and metric are present.
However, non-perturbative e ects, or the presence of
additional elds with couplings that do not follow the classical scale invariance, may spoil
any selection rules.
An additional structural aspect motivated by the UV picture is that, since we are
considering a superstring origin for this 5D action, the full e ective action should also
be supersymmetric. Due to the supersymmetry of the e ective action, the classical scale
invariance of the action in fact survives quantum corrections. This means that additional
terms, such as a cosmological constant, are not expected to arise perturbatively in the
bulk action.
We note that in superstring theory for D 6= 10 neither the dilaton Weyl anomaly nor
the dilaton potential vanish. However, one may consider the linear dilaton background
D
10
for which the Weyl anomalies vanish, at the expense of a curved metric in the Einstein
frame. Since this background allows for a vanishing
-function, it describes a worldsheet
CFT. Such theories are known as non-critical string theories, in that the critical number of
dimensions has not been chosen, however they still describe string worldsheet CFTs [53, 56].
This describes the basic features of the non-critical string UV motivation for the linear
dilaton eld theory.
In addition to the non-critical string motivation, the linear dilaton theory in 7D has
been shown to arise in the Little String Theory (LST) limit of critical superstring theory
with a stack of NS5 branes. This limit corresponds to vanishing string coupling [27]. It
has been argued that LST-like theories are dual to backgrounds which asymptote to string
theory in the linear dilaton background (see section 3 of ref. [57]). However, in this case
one has two extra dimensions. The linear dilaton form of the e ective action persists to 5D
when two of the additional dimensions are compacti ed [26]. In this framework, M5 and k
are determined by the string scale Ms, the number of the NS5 branes N , and the volume
of the six compacti ed dimensions V6 as M5
we have M5=k
Ms2(N V6)1=3, hence the ratio depends on the UV parameters.
Ms3V61=3=N 1=6 and k
p
Ms=
N [37]. Thus
In this work we will not consider the phenomenology of the additional RR two-form
eld, nor the additional states required by supersymmetry. All of these states would likely
have the usual LD spectrum, with a mass gap and densely packed states. Furthermore, the
superpartners, such as the dilatino and gravitino, may be charged under remnants of the
{ 8 {
R-symmetries that may make the lightest states stable. These neutral fermions will only
be pair-produced in colliders and thus their greatest e ect will likely be to contribute to the
decay channels of the KK gravitons. Furthermore, we do not include the e ects of genuine
string excitations which, if entering below the cuto
M5, could lead to additional signatures.
Having considered the stringy origins of the action, including the bulk supersymmetry, we
can also take a viewpoint that is agnostic of the UV completion, and consider the action
from a purely non-supersymmetric eld theory perspective. There are two terms that enter
Einstein's equations in the form of a cosmological constant (CC): one from the bulk and
one from the branes. These terms may be written in the Jordan frame action as
SCC =
Z
dDxp
g
D e D2 2 S ; SBCC =
Z
d
D 1 p
x
B
D 1 e 2(DD 21) S ;
(2.24)
where
is the determinant of the induced metric. These terms violate classical scale
invariance, which is why they did not arise in the string e ective action at tree level.
However, from a purely eld theory perspective classical scale invariance is impotent unless
the UV theory and non-perturbative corrections also respect it. If we are being agnostic
as to the UV, including considering non-supersymmetric theories, then we cannot rely on
such arguments to forbid these cosmological constant terms. Continuing in this vein, let
us determine how large these terms can be before they signi cantly modify the solution.
Since the inclusion of the cosmological constant terms generally leads to Einstein's
equations that cannot in general be solved analytically,6 we will instead perform a
perturbative analysis valid for small cosmological constants, which we will parameterise as "
(bulk) and "B (brane). We consider a bulk potential which, in Einstein frame, is given by
Sbulk =
Z
d x
5 p
g
M53 "
2
R(g)
1
#
(2.25)
(2.26)
3 "B 4
4
(y
R) ;
and a boundary potential given by7
SB =
Z
d4x dyp
1
3 "
40
(
4kM53
2
(1 + "B)e S=3 +
3) + "
B
3 e S=3 +
3 "
32
3 "
20
3 "B
4
1
(y)
= e 2k R=3. The speci c form of the terms at the boundaries has been chosen
to give a vanishing 4D cosmological constant, thus that particular ne-tuning, present in
6For its novelty, in appendix J we include an isolated exact solution that satis es a di erent set of
boundary conditions, and is thus not of interest here.
7The junction conditions, which follow immediately from integrating the equations of motion following
eq. (2.25) at the branes give, for a given solution to the bulk equations of motion, conditions on the absolute
value and gradient of the brane potentials i(S) and d i(S)=dS. Thus there are an in nite class of brane
potentials that can satisfy these constraints. We present only those that correspond to the usual linear
dilaton brane potential and a cosmological constant on the branes. The quadratic brane terms proportional
to T;P are irrelevant for this discussion.
{ 9 {
the boundary conditions
S(0) = 0 ;
(0) = 0 ;
and the junction conditions are all satis ed for the following dilaton and metric pro les
all extra-dimensional models, has already been performed. This means that any further
tuning of " or "B is now in addition to the tuning for a vanishing 4D CC.
Since
is exponentially small in our scenario, the limit "
B ! 0 corresponds to the
solution whenever the bulk CC is non-vanishing, which is one possibility of interest here,
and the limit " ! 0 corresponds to only having a CC on the 0-brane, up to tiny corrections
on the
R-brane, which is the second possibility of interest. We present both solutions
together for convenience.
Working in the conformally at metric ds2 = e2 (y)(
dx dx + dy2), we nd that
the bulk Einstein's equations and S equation of motion
HJEP06(218)9
S = 2ky +
2
3
=
ky +
3"
32
"
80
1
2ky
6e4ky=3
e 2ky
+ B 5
2ky
5e 2ky
;
5(1 + ky)
e 2ky
+ B 1
ky
e 2ky
:
"
6
The physical impact of these modi cations can be expressed in numerous ways. One
way is to derive from the expression of the 4D Planck mass the value of kR, which
characterizes the physical mass spectrum of the KK modes. We nd that, at leading order in "
and "B and for the same input values of M5 and k, the usual CW/LD value of R given by
eq. (2.11) is modi ed as follows
R
R";"B = 1 + "
B
3 "
32
18
25 kR
e4 kR=3
Let us consider the e ect of ". For the linear dilaton solution in the conformally at
coordinates we have kR
10, thus e4 kR=3
1018. This means that unless j"j . 10 16,
"
j j
the value of kR and, consequently, the mass spectrum will be signi cantly di erent from
the usual CW/LD prediction. Similarly, the metric will have deviated signi cantly from
the linear dilaton form in moving from the IR brane to the UV brane. Of course, nothing
radical will happen, with the metric
owing smoothly from the linear dilaton form at
10 16, to a dS or AdS-like solution. We see this re ected in the reduced (comoving)
radius required to achieve the desired hierarchy of Planck scale. In the AdS case the mass
spectrum and wavefunctions would correspond to kRS
kp"=12, at larger ". Thus one
needs to require an extremely small value of " to retain the linear dilaton solution. From a
eld theory perspective this is an enormous tuning, however by considering the superstring
motivation the bulk theory is supersymmetric and this protects against the generation
1
3
3
2
"
4
(2.27)
(2.28)
(2.29)
(2.30)
(2.31)
(2.32)
(2.33)
of
D if the action started out with classical scale invariance, which is the case for the
string e ective action. The requirement is actually a little stronger than requiring solely
supersymmetry, since the vacuum expectation value of the superpotential must also vanish
to avoid supersymmetric AdS solutions. It has recently been shown that the CW/LD
background is indeed consistent with a bulk supersymmetry [31, 32]. Supersymmetry
must be broken on the SM brane, however locality still protects against the generation of
supersymmetry breaking in the bulk action (see e.g. ref. [58] for related discussions). Thus,
given that the motivation for the linear dilaton setup is coming from string theory, the
consequent bulk supersymmetry naturally protects against the generation of ".
B
Now consider " . The story for this parameter is very di erent. The reason is that
supersymmetry breaking must exist at least at the TeV scale on the SM brane, and thus
there is no symmetry, including supersymmetry, that can protect against the generation
of "B
M5=k. Studying the perturbation to the metric we see that the metric retains
B
the linear dilaton form for "
B not too large. Similarly, the comoving radius required is
only signi cantly modi ed for "
1. Thus, although we have no control over the size of the "
B brane cosmological constant term on the TeV brane, the setup is robust to its
presence as long as "B is not too large, which in turn does not require signi cant ne-tuning
unless k
M5.
To summarise, bulk supersymmetry protects against the presence of a bulk
cosmological constant that would otherwise radically alter the setup.8 On the other hand, the
lack of supersymmetry on the SM brane does not imperil the solution as the brane
cosmological constant does not feed into a bulk cosmological constant, due to locality, and it
does not generate a signi cant deviation from the linear dilaton solution unless "B is O(1).
Thus, if the UV motivations are taken seriously, in particular the bulk supersymmetry,
then the theory does not exhibit signi cant tuning beyond that required for a vanishing
4D cosmological constant. On the other hand, if one were agnostic as to the UV
structure one could only conclude that the necessity for j"j
additional tuning.
10 16 corresponds to an extreme
At this stage it is worth commenting on the deconstruction of the continuum model,
which gives the discrete clockwork graviton model presented in section 2.5 of ref. [1]. While
this is a perfectly valid, di eomorphism invariant, multi-graviton model, since it follows
from a deconstruction of a solution of Einstein's equations accompanied by the dilaton,
the question of the cosmological constant for the continuum model is equally relevant for
the discrete. There is nothing to forbid a cosmological constant at each site. Translation
invariance enforces that all cosmological constants must be equal, just as " is
positionindependent in the continuum, however no symmetry protects the overall magnitude of
each term and the cosmological constant could imperil the clockwork wavefunctions. As
with the continuum model, the only plausible solution to this problem is supersymmetry.
Thus, if one wants the discrete setup to be safe from large quadratic corrections, one really
requires an entire 4D copy of Supergravity, with the associated gravitino at each site. Thus
the discrete model of ref. [1] alone is not su cient to address the hierarchy problem.
8This discussion has focussed on the presence of cosmological constant terms, however in the absence of
supersymmetry the dilaton potential itself is also not protected from dangerous radiative corrections, the
presence of which may lead to similarly signi cant corrections to the form of the bulk geometry.
2.4
In CW/LD, M5 and k are two independent input parameters. As the collider
phenomenology changes signi cantly by varying the ratio k=M5, we think it is interesting to explore
a wide range of this ratio and to study the corresponding experimental features, some of
which are novel and characteristic. However, we rst want to discuss in this section which
values of k=M5 are reasonable.
The CW/LD action in eq. (2.25) has an enhanced symmetry (a shift in S) for k ! 0.
This suggests that small k=M5 is technically natural. On the other hand, in the string
theory setup, both M5 and k might be expected to arise from the single dimensionful
string tension parameter 0, although their relation is not uniquely determined. In the
little string theory limit a hierarchy of O(N 1=3) is possible, where N is the number of
NS5-branes in the theory.
While the form of the bulk action is motivated by string theory in the UV, the boundary
action has no such motivation. In particular, we have no guidance as to the parameters ST,
SP, which determine the form of the dilaton solution to the equations of motion. We may
thus write y=0 =
0 and Sy=0 = S0, in which case 2k R = jST
SPj and S0 = ST. On the
equations of motion the overall factors can be absorbed into a rede nition of the 5D Planck
scale M5;e
= M5e 0 , ke
= ke 0 S0=3, thus the only e ect of taking general boundary
conditions on these constants is to modify the relationship between the observables M5;e
and ke . As a result, since we do not have guidance on the scale of these parameters from
the UV, in this paper we are instead open minded as to considering relatively small k=M5,
especially because this possibility gives rise to a variety of interesting collider phenomena.
It should also be kept in mind, however, that to have the linear dilaton solution with
k
M5 does require tuning of the brane CC at that order.
In this work, instead of using lower bounds on k based on theoretical considerations,
we will be guided by the phenomenological limits from beam dump experiments,
supernova emission, and nucleosynthesis. For M5 in the domain of interest to present and
future collider experiments, the lower limits on k are in the 10 MeV / 1 GeV range [38],
taking into account that astrophysical and cosmological bounds are subject to large
numerical uncertainties.
2.5
KK mode mass spectrum and couplings
The KK gravitons have masses
m0 = 0 ;
and couple to the SM stress-energy tensor T
L
1
(n) h~(n) T
G
(0)2 = MP2 ;
G
(n)2 = M53 R
G
1 +
= M53 R
1
k
2
m2n
1
:
n
2
R2
;
as
;
k2R2
n2
(2.34)
(2.35)
(2.36)
50
150
200
for M5 = 10 TeV, k = 200 GeV, however the result does not strongly depend on these parameters.
The blue band indicates the typical ATLAS and CMS resolutions in the diphoton and dielectron
channels (
1%).
The zero mode is the usual massless graviton, while the rest of the KK modes appear after
a mass gap of order k and their couplings to the SM are not suppressed by MP . A unique
property of this scenario is that the KK modes form a narrowly-spaced spectrum above
the mass map. At the beginning of the spectrum, the relative mass splitting is
m2
m1
m1
3
' 2 (kR)2
1:5% ;
i.e., comparable to the diphoton and dielectron invariant mass resolutions in ATLAS and
CMS, which are typically around 1% [59{64]. The splittings then increase, as shown in
gure 2, reaching a maximum value
Eventually they start decreasing, becoming asymptotically
mn+1
mn
mn
max
1
' 2kR
5%
for n
kR :
mn+1
mn
mn
1
' n
for n
kR ;
(2.37)
(2.38)
(2.39)
thus dropping below the experimental resolution at n
The physics of the KK dilatons (including the radion degree of freedom) depends
on the details of the brane potentials that stabilize the extra dimension [39, 65]. We rst
focus on the limit of rigid boundary conditions for the dilaton eld, obtained when the mass
parameters T;P in the brane-localized potential in eq. (2.3) are in nitely large. The general
case is presented in detail in appendix E. The dominant features of the phenomenology of
the model as a whole are largely independent of the details of the brane-localized potential.
For rigid boundary conditions, the KK dilatons have masses
m0 =
r 8
9
k ;
n
2
R2
;
(2.41)
1
:
(2.42)
(2.43)
= 0 corresponds to rigid boundary conditions (see appendix E for details). The horizontal
dotted gray lines correspond to the spectrum m2n = k2 + n2=R2. The dashed black lines correspond
to the analytical approximations discussed in appendix E, eqs. (E.31){(E.32).
and couple to SM particles via the trace of T
as9
L
interaction scale of dilatons with respect to gravitons, when compared at equal masses.
If the boundary conditions are relaxed by lowering the mass parameters
T;P in the
brane potential, the n = 0 and n = 1 modes can become signi cantly lighter, and both are
massless in the
T;P ! 0 limit (unstabilized limit). At the same time, all the higher KK
dilaton modes shift down by one mode (as in \Hilbert's Hotel") so that the massive mode
spectrum in the unstabilized limit ( T;P = 0) is again described by the expression for m2n
in eq. (2.40) with n starting from 1. This is shown in
gure 3. The coupling strengths to
T
get modi ed as well. In the unstabilized limit, modes with mn
k have
(n)2
3
' 4
M53 R
m4n 2
k4
;
9These expressions follow from eqs. (3), (4) and (39) of ref. [65], and we have con rmed them. However,
while the expression for
(n)2 agrees with the result given in eq. (41) of ref. [65] in the limit n
taken there, our expression for (0)2 is larger than theirs by a factor of 2. Our expressions for both
and
(n)2 (in the same limit) agree with those in eqs. (4.4){(4.5) of ref. [39] after taking into account that
kR
(0)2
the four- and
ve-dimensional Planck masses, MP l and M , of ref. [39] are related to our MP and M5 as
MP2 l = MP2 =2, M 3 = M53=2. We thank the authors of ref. [39] for clarifying this to us.
gg
34%
Pi qiqi
38%
W +W
hh
3.2%
where the dimensionless parameter , which vanishes in the rigid limit ( T;P !
de ned in appendix E. By comparing this with eq. (2.42), one can see that the interactions
with T
get signi cantly suppressed as one goes away from the rigid limit. In addition,
there appear couplings to LSM, the SM Lagrangian,
where in the unstabilized limit for mn
k
1
'
L
(n) nLSM ;
(n) 2
'
' 3M53 R :
nal states, branching fractions, lifetimes
Since the KK gravitons couple to the SM via T , the relative branching fractions into the
various SM particles are the same as in any 5D model with the SM on a brane. These
are shown in table 1 for KK gravitons that are much heavier than the SM particles. The
detailed expressions, including phase space e ects, which must be taken into account for
lighter KK gravitons, are given in appendix C. The total decay rate of a mode-n KK
graviton into SM particles in this limit is
In the absence of other types of decays, the resulting lifetime is
c n
6:6
m2n
1
:
Gn!SM =
M5
We see that the KK graviton decays can be prompt, but it is also possible, especially if
M5 & 10 TeV, that KK modes below a certain mass will be displaced, or even stable on
detector scale. The last factor in eq. (2.47) gives a further enhancement to the lifetimes
of the lightest modes besides the mn 3 factor. For instance, for the rst mode we
nd
(1
k2=m21) 1
k2R2
100. As a result, it is possible that, within the same theory at a
given M5 and k, some KK gravitons decay promptly, while others lead to displaced vertices.
Importantly, we nd that decays of KK gravitons to SM particles are not the full story. Graviton self-interactions allow a heavy KK graviton to decay to a pair of lighter
KK gravitons.10 The method of calculation and the detailed expressions for the rates of
these decays are presented in appendix D. Their total rate, in the limit n
kR
1, is
given by
Gn!P G`Gm ' 3
595
m7n=2
214 2 k1=2RM53
;
10Numerical results for graviton-to-graviton decays were also considered for RS models in [66].
(2.44)
(2.45)
(2.46)
(2.47)
(2.48)
G→ GG0.010
→
G
Γ G
Γ
to the asymptotic expression given in eq. (2.48). Here we have taken M5 = 10 TeV, k = 10 GeV,
however the dependence on these parameters is weak.
while for smaller values of n it is reduced relative to this expression in the fashion shown
in gure 4. This rate can be quite sizable, and even dominate over decays to SM particles
for large n:
Gn!P G`Gm
Gn!SM
4:1
r mn :
k
This has signi cant e ects on the phenomenology. First, decays to SM particles are diluted,
as shown for the example of the diphoton channel in gure 5 (left). Second, signatures due
to decays to lighter KK gravitons can be important because the branching fraction for such
decays can be large, as shown in gure 5 (right).
Even though the contribution to the total width ( n) from decays to lighter KK
gravitons can be signi cant, note from (2.48) that all modes within the range of validity of the
theory (mn . M5) have n
mn. Even the stricter condition
n < mn
mn 1, which
allows us to treat the KK excitations as individual resonances, is ful lled as long as
This is satis ed by all modes with mn . M5 if
mn . 6:8
k
M5
1=7
M5 :
k & 1:5
Note also that the extra contribution to the decay rate does not preclude displaced decays,
since only high-n modes are a ected (see gure 4).
A KK graviton can also decay to a KK scalar and a KK graviton, or to a pair of KK
scalars.11 The branching fractions for such decays are usually small, as shown in gure 6.
11Cascade decays of gravitons to KK scalars were also considered for RS models in [67].
k = 0.1 GeV
10 GeV 1 TeV
k = 0.1 GeV
10 GeV
1 TeV
G
G
m [GeV]
m [GeV]
for k = 0:1, 10 and 1000 GeV, as a function of the KK graviton mass. In the left plot, the thick
black curve shows the result that would be obtained without accounting for decays to lighter KK
k = 0.1 GeV
10 GeV
1 TeV
k = 0.1 GeV
10 GeV
1 TeV
R( 1
B
G( 1
0.1
0.1
1
10
m [GeV]
or a pair of KK scalars (right) for k = 0:1, 10 and 1000 GeV, as a function of the KK graviton
mass, assuming rigid boundary conditions for the dilaton
eld. We show separately contributions
involving two scalar zero modes (dotted), one scalar zero mode (dashed) and no zero modes (solid).
Let us now consider the decays of KK scalars. As we have already mentioned, their
couplings to the SM are model dependent. The decay rates have been computed in ref. [39]
and we present the full expressions in appendix F. For rigid boundary conditions (which
implies couplings to T
only), assuming the Higgs-curvature interaction (which we will
discuss below) to be absent, and neglecting the SM particle masses relative to the KK
scalar mass, the total partial width to SM particles is dominated by W +W , ZZ and hh,
where the last expression applies for n 6= 0. The corresponding lifetime is
and is given by
c n
0 1
8mk2n2 A ;
9m2n
1 GeV
k
3
1 TeV
mn
kR
10
0
1
8k2 1
9m2n
k2 A :
m2n
For large n, the last factor in eq. (2.53) is equal to one, while for n = 1 it is equal to
11, showing a smaller enhancement than in the graviton case. However, there
is a higher overall tendency for the decays to be displaced than in the KK graviton case,
at least for the low modes, where KK tower cascades are irrelevant. The decay rate into
SM particles is even smaller below the W +W
threshold, where gg becomes the dominant
SM decay channel, with the rate approximately given by eq. (2.52) times 49 ( s=2 )2. In
that regime
c n
66 cm
M5
10 TeV
3
3
1 GeV
k
9m2n
k2 A :
m2n
(2.52)
(2.53)
(2.54)
Relaxed boundary conditions, which allow KK scalars to decay via a direct coupling to
LSM, eq. (2.44), rather than only through T , generate sizable branching ratios to gg and
relative to the total rate to pairs of SM particles, of O(30%) and O(3%), respectively.
The partial widths to W +W , ZZ, hh are modi ed by O(1) factors, so the total SM rates
of the heavy modes stay in the same ballpark. The light modes, on the other hand, can
become signi cantly shorter lived.
Because the couplings of the scalar KK modes to the SM are somewhat suppressed,
cascade decays within the KK tower play an important role. We compute these decays in
appendix G. We nd that the KK scalar decay rates, with the exception of the rst few
modes, are dominated by the cascades, as shown in gure 7.
the KK mode spectrum for di erent values of M5 and k. Interestingly, even for a single
choice of the parameters M5 and k, the model typically contains particles with a very wide
range of lifetimes. For xed M5 and k, the variation of the KK mode lifetimes with mn is
more evident for gravitons than for scalars.
Since no symmetry prevents its appearance, it is natural to introduce on the SM brane
a renormalizable curvature-Higgs interaction
L =
RHyH ;
(2.55)
where
is a dimensionless coupling. After electroweak symmetry breaking, this
interaction induces a kinetic mixing, proportional to v=M5, between the Higgs and the radion
component of the KK scalars. Through a
eld rede nition (see ref. [39] for the explicit
calculation) the kinetic mixing can be turned into a mass mixing. It is not di cult to
where in the last term we integrated by parts. Thus, the scalar uctuations satisfying the
constraint equation
3
2
(G0 + 2 0G) =
do not mix with the vector
and are therefore the physical scalar uctuations that are not
eaten by the massive graviton modes. One may be concerned that we may have omitted
an important term in performing integration by parts, however the vector eld carries odd
parity under the orbifold symmetry, thus the boundary term vanishes.
If the previous constraint is employed we may continue without consideration of the
massive graviton degrees of freedom as we have isolated the scalar
uctuations that are
not kinetically mixed with them. To proceed we will consider the following general metric
ds2 = e2 (y) (1 + 2F (x; y))
dx dx + (1 + 2G(x; y)) dy2 ;
S(x; y) = S0(y) + '(x; y) ;
T~MN
TMN
with (y) = 2ky=3. We consider the linearized (i.e. at the rst order in the scalar
uctuations) Einstein's equations, which we recast in the form RMN = T~MN =M53 + BMN , where
13 gMN TP QgP Q and BMN accounts for boundary contributions at the two
branes. As customary, some equations are dynamical, some equations provide constraints.
In particular, the o -diagonal
components enforce the condition
2F (x; y) = G(x; y) ;
as found in eq. (E.4), while the 5 component can be straightforwardly integrated, and we
nd eq. (E.6). As for the rest of the Einstein's equation, we nd
(E.6)
(E.7)
(E.8)
(E.9)
(E.10)
(E.11)
(E.12)
(E.13)
(E.15)
R
=
R55 = 2
T~ =M53 =
T~55=M53 =
Bi=T;P =
B5i=5T;P =
e
2
3
e
4e
23 S00'0 +
e
2
3
9M53 (6F
F + F 00 + 9 0F 0 + 6F 3( 0)2 + 00 ;
F
2F 00 6 0F 0 ;
2F V (S0) + 'V 0(S0)
;
3M53 G i=T;P(S0) + ' 0i=T;P(S0) (y
yi=T;P) :
2GV (S0) + 'V 0(S0) ;
3G) i=T;P(S0) + 3' 0i=T;P(S0) (y
yi=T;P) ; (E.14)
In addition, we need the equation of motion for the uctuations of the dilaton eld. We nd
' + '00 + 3 0'0 + 6S00F 0 + 4F 3 0S00 + S000
2
3 V 00(S0)'e2
e 2
X
3
=
i=T;P e M53 2F 0i(S0) + ' 0i0(S0) (y
yi) : (E.16)
Finally, we need to impose junction conditions at the boundaries. There is only one relevant
junction condition, corresponding to the boundary term of eq. (E.16). We nd
'0 i=T;P + 2F S00 i=T;P =
3' 0i0=T;P(S0(yi))e
M53
;
where, for a generic rst derivative A0(y), the jump [A0] at the generic point y is de ned as
). Using this de nition, we can rewrite eq. (E.17) in the form
We can now extract a dynamical equation for F (x; y). Following ref. [117], we consider in
the bulk the combination
R55
4
R
=
1
M53
~
T55
4
~
T
;
and we nd
we nd
(E.17)
(E.18)
(E.19)
(E.20)
(E.22)
(E.23)
3F 00 3 0F 0 +6F 3( 0)2 + 00 =
6 ( 0)2 + 00
Using the constraint in eq. (E.6), and the explicit expressions for the background solution,
1
3
(S00)2
: (E.21)
23 S00'0 2F
F + F 00 + 2kF 0 = 0 :
This equation can be conveniently rewritten in the form
+
d
2
dy2
k
2 ekyF (x; y) = 0 :
Because of the relation
2F (x; y) = G(x; y), the 5D eld G(x; y) satis es the same equation
of motion. Similarly, starting from the eq. (E.16) in the bulk, and using the constraint in
eq. (E.6), it is possible to show that also the scalar eld
uctuation '(x; y) respects the
same equation
' + '00 + 2k'0 = 0. However, only a certain combination of these elds
has a canonical kinetic term in the bulk. By direct diagonalization of the quadratic action,
as illustrated in ref. [65], one nds it to be v(x; y)
it also satis es the 5D equation of motion
v + v00 + 2kv0 = 0. This is the equation we
have to solve together with the appropriate boundary conditions. To this end, we need an
explicit expression for the brane potential. The latter can be obtained by means of the
solution-generating method championed in ref. [113], and we nd the explicit expressions
reported in eqs. (2.7){(2.8). The mass parameters
T=P do not a ect the background
solution (since they do not contribute to the background junction conditions) but they
enter in the junction conditions for the perturbations in eq. (E.17). Notice the presence
of a massless dilaton in the limit
T=P ! 0 since in this limit a shift in S (in the Jordan
p6ekyM53=2[F (x; y)
'(x; y)=3], and
1
X
n=0
frame) corresponds to an overall rescaling of the action. We now have all the ingredients
needed to solve the scalar equation of motion. Using
2F = G, F =
, G =
, we adopt
the notation of ref. [39]. We introduce the KK decompositions
(x; y) =
n(y) n(x) ; '(x; y) =
X 'n(y) n(x) ; v(x; y) =
X vn(y) n(x) : (E.24)
1
n=0
1
n=0
Notice that the 4D part of the three KK decompositions is equal. This is a direct
consequence of the de nition of v(x; y) and the constraint in eq. (E.6).
We de ne
n to
be the solution of the equation of motion of a free 4D scalar eld of mass mn, namely
m2n n(x) = 0. At this level of the analysis we are interested in the mass
spectrum mn. We can therefore solve the equation of motion focusing on a speci c mode |
equivalently,
n(y), 'n(y) or vn(y) | and imposing the corresponding boundary
conditions. We consider the mode
n, for which the equation of motion is
d
2
dy2 + m2n
k
2
eky
n
= 0 :
We can write a generic solution in the form
n(y) = Nne ky [sin( ny) + wn cos( ny)] ;
with
n2 = m2n
k2, and Nn, wn constants.
eqs. (E.18){(E.19). Using the constraint in eq. (E.6), we nd
The boundary conditions are given by
9
9
4k 0n0 + 3 0n
4k 0n0 + 3 0n
2k n =
2k n =
4k + 9 T
4k + 9 P
2
2
n +
n +
3
3
4k n
4k n
0
0
We start considering the rigid limit i ! 1. The boundary condition at the TeV brane
xes wn =
3 n=k. The boundary condition at the Planck brane can be solved analytically.
We nd
with n = 1; 2; 3; : : : . The lowest state in the KK tower corresponds to the radion. Notice
that the mass of the radion vanishes in the absence of warping, k = 0.
The identi cation of the radion with the lowest state in the KK tower can be
understood by means of the following argument. In the rigid limit i ! 1, if k = 0 we have
from eq. (E.29) a massless state and an ordinary KK tower with spectrum m2n = n2=R2,
n = 1; 2; 3; : : : . Note that there is no zero mode in the KK tower. The point is that in
this setup it is straightforward to identify the states m2n with the KK decomposition of the
scalar S. Indeed, in this situation we just have a free scalar eld in the bulk of a at extra
dimension (since k = 0) with an in nite mass term on the two branes. After KK reduction,
the equation of motion for the y-dependent variable reduces to an ordinary Schrodinger
equation for a free particle in a one-dimensional well 0 6 y 6
R with in nite potential
(E.25)
(E.26)
(E.27)
(E.28)
(E.29)
on the two walls. As well known from Quantum Mechanics, the quantization condition
precisely implies m2n = n2=R2, n = 1; 2; 3; : : : , thus making possible the identi cation of
these states with the k = 0 limit of m2n in eq. (E.29). Consequently, the remaining scalar
degree of freedom can be identi ed with the radion eld. The second state in the KK tower
corresponds to the dilaton, with mass m2dil = k2 + 1=R2.
For a generic value of T;P, the boundary condition at the TeV brane gives
wn =
3 n T
2(k2 + n2) + k T
:
(E.30)
HJEP06(218)9
The boundary condition at the Planck brane can be solved numerically. We show our result
in gure 3, with T;P
2k= T;P, and taking for simplicity T = P
At the rst order in T | and for generic value of n | we nd
.
1
8k2
' 9
2 T
9
1
2
:
2 i
T
; (E.31)
(E.32)
(E.33)
(E.34)
(E.35)
(E.36)
We now move to discuss the couplings of the scalar perturbations with SM
elds.
To that end, we need to determine the proper normalization factor in eq. (E.26). By
expanding the 5D action up to the quadratic order in the scalar
uctuations, and using
the KK decompositions in eq. (E.24), we nd the following normalization condition in 4D
Z
Sn4D kin = Cn
d4x n(x)
m2n
n(x) ;
Cn =
27M53 Z R
16k2
0
dye2ky
0 2 +
n
8k2
3
2n +
3
8k 0n n
The coupling of the radion and the scalar KK modes with the SM Lagrangian at the TeV
brane are described by the action
Sint =
Z
d x
4 p
Te '=3LSM :
We can now extract the value of Nn by plugging in Cn the solution found in eq. (E.26).
T
4k3 2Ti sin2 R n
3 h4k2 k2 + n2 2
+ 4k k2 + n2 2
T + k4 + 2k2 n2 + 9 n4
(2 R n sin 2 R n)+3 n k4 + n4 + 2k2 n2 + 2T (2 R n + sin 2 R n) :
2
= p g g
S
S =
Z
d4x n(x)Tr[T ]
d x
4 p gT
g ;
2
;
'n(0)
3
2
1
Silnint =
De ning the coupling constants
T
we nd | at the rst order in T=P, and for generic n | the following couplings with the
trace of the energy-momentum tensor
1
6
k 1 + coth k 3R
M53
1 +
9
' 6
1
s k
M53
1 +
9
;
2knR
(n) = p3 M53R (n2 + k2R2) (9n2 + k2R2) (1
T) :
Expanding at the rst order in the scalar
uctuation, and using the de nition of the
energy-momentum tensor (here for a generic metric g)
(E.37)
(E.38)
(E.39)
(E.40)
(E.41)
(E.42)
(E.43)
while for the direct coupling with the SM Lagrangian we have
1
'
1
'
(0) =
(n) =
n T
p
k 3 R3=2
k 1 + coth k 3R
M53
s
n2 + k2R2
M53(9n2 + k2R2) :
2 T
' 27
s k
M53
;
In the rigid limit the direct coupling with the SM Lagrangian vanishes. This is a
consequence of the junction conditions in eqs. (E.18){(E.19) in which a large value of 0T0 ;P forces
the scalar eld uctuation ' on the two branes to vanish. We note that the analytical
approximations in eqs. (E.40){(E.43) do not feature an explicit P-dependence. However, at
higher orders, an explicit P-dependence appears since it unavoidably enters via the scalar
masses. For this reason, in
gure 24 we show the full numerical results for the couplings
in eq. (E.39) under the assumption T = P
already adopted in
gure 3.
In light of the discussion we put forward in section 2.5 about the KK mode production
cross sections, it is also interesting to present a useful analytical limit for the couplings in
eq. (E.39) for a generic value of T and large n. By approximating the KK scalar spectrum
with m2n
n2=R2, we nd, at the leading order in a kR=n expansion
1
'
Equipped with these results, we can now move to compute the dilaton/radion decays
to SM particle pairs.
)
( Φ
HJEP06(218)9
SM Lagrangian (right panel). The dashed black lines (for simplicity, only for radion and dilaton)
represent the analytical approximations in eqs. (E.40){(E.43). We only show the rst few modes of
the KK tower.
F
Dilaton/radion decays to SM particle pairs
Using the results of ref. [39] (accounting for the di erence in conventions described in
footnote 9), the partial decay widths of a KK scalar of mass mn are
ZZ =
1
xZ +
fh(mn)
n 1 +
fh(mn) +
W +W
1 xW + 4 xW
3 2
fh(mn)
n 1+
fh(mn) +
)
( φ
xZ
2
xW
2
1
8 2
x
2
h
4
m2 )2
h
2
+
bQCD + n
2
2
0 ;
bQED + n
0 ;
34 2n (1
34 2n (1
xZ )1=2 0
;
xW )1=2 0
;
4
2
(F.1)
(F.2)
36 m2h
18 m2h
m2n
m2n
m2
h
m2
h
72 m4h
m2n(m2n
14 2n (1
m2 )
h
m2
h
xh)1=2 0
;
4
(F.3)
(F.4)
(F.5)
(F.6)
where
where
0
8
4mm2ni2 ;
fh(mn) = 1
m2n
6 m2n ;
m2
(n) is given in eqs. (2.41){(2.42), n
(n)= ('n),
is the Higgs-curvature coupling as
de ned in eq. (2.55), Nf = 3 for each quark, 1 for each charged lepton and 1=2 for each
(Majorana) neutrino, bQCD = 11
2nQ=3, where nQ is the number of quarks lighter than
mn, and bQED =
(4=3) P
f Nf Qf2 , where the summation is over fermions lighter than mn
and Qf is the electric charge. In the region of the W and top masses, threshold e ects are
accounted for by taking
HJEP06(218)9
with b2 = 19=6 and bY =
41=6, and (relevant also for production)
bQED = b2 + bY
fh(mn)(2 + 3xW + 3xW (2
xW )f (xW ))
8
+ 3 fh(mn)xt(1 + (1
xt)f (xt))
bQCD = 7 + fh(mn) xt (1 + (1
xt)f (xt)) ;
f (x) = <
8
>
>
>
>
:
>
>>> sin 1
1
4
ln
1
p
1 + p
1
p
1
1
2
x
i
2
(F.7)
(F.8)
(F.9)
(F.10)
(G.1)
G
Trilinear graviton/dilaton/radion decays
In this section we discuss trilinear decays involving scalars and gravitons. The interactions
responsible for these processes are encoded in the action obtained by expanding the metric
including both tensor and scalar perturbations
ds2 = e2 (y) h(1 + 2F (x; y)) (
+ 2h (x; y)=M53=2)dx dx + (1
4F (x; y)) dy2i ;
S(x; y) = S0(y) + '(x; y) :
Although technically involved, the computation does not pose particular complications.
After expanding the 5D action up to the cubic order in the
uctuations, one just needs
to apply the KK reduction | already introduced in appendix D and appendix E for,
respectively, tensor and scalar perturbations | in order to extract the 4D Lagrangian
interactions. In this process a non-trivial check is the exact cancellation of the
GibbonsHawking-York terms against the total derivatives coming from the Ricci scalar.
In addition to the trilinear graviton vertex already studied in appendix D, the inclusion
of scalar perturbations gives rise to three additional structures that we shall now discuss
in detail. Let us start from the trilinear 5D action involving one graviton and two scalar
elds (labelled `Tensor-Scalar-Scalar', TSS in the following). We nd
d4xdye3 h
(G.2)
2
3
X
+ e5
X
i=T;P
Z
This trilinear action is responsible for KK graviton decay into two scalars, Gl !
and KK scalar decay into a graviton and a scalar, l ! Gm +
n. Next, we consider
the trilinear 5D action involving two graviton and one scalar elds (labelled TTS in the
following). We nd
STTS =
d4xdy6e3
h h
12F 0 2 + S00(@y')=9 + 4S002F=3
+ e5
'V 0(S0)=6 + F V (S0)=3
Z
d x
4 e
4
M53
F
4
h h
' 0i[S0(yi)] +
[S0(yi)]
:
(G.3)
This trilinear action is responsible for KK graviton decay into one scalar and one graviton,
Gl ! Gm + n, and KK scalar decay into two gravitons, l ! Gm + Gn. Finally, we have
the trilinear 5D action involving three scalar elds (labelled SSS in the following). We nd
SSSS =
d4xdyM53 e
3
1536F 3 0 2
12F 2S00'0
2F '0 2 + 240(@yF )F 2 0 + 12F 2'S00 0 + 4F 2'S000
F '2V 00(S0)
'3V 000(S0)=6
6
(G.4)
This trilinear action is responsible for KK scalar decay into two scalars, l !
m + n. In
the following, we shall compute and discuss each one of these processes. We describe the
computations in full generality | that is for an arbitrary choice of
| but we specialize
all our numerical results as well as analytical approximations in the rigid limit
= 0.
G.1
KK graviton decays into two scalars
The 4D Lagrangian density for a xed triad (n; m; l) of KK states is
TSS
L(n;m;l) =
1
M53=2p
R
C(n;m;l)(k; R) h~(n) (@
TSS
l) ;
(G.5)
TSS
where the dimensionless function C(n;m;l)(k; R) encodes the integral over the extra
dimendye3 k n(y) 3 ~ m(y) ~ l(y) +
'~m(y)'~l(y) :
(G.6)
2
3
Notice that, compared with eq. (E.26), we de ned here the dimensionless scalar
wavefunction ~
n
pM53=k n. Furthermore, from the constraint in eq. (E.6) we get
3
2k
'~n(y) =
2k ~ n(y) +
32 ~ 0n(y) :
(G.7)
Gl! m n (left panel) and
Gl!Gm n (right panel)
nal state masses. The example shown is M5 = 10 TeV, k = 10 GeV, for mode
l = 1000. The corresponding mass of the decaying particle is m1000 ' 1 TeV.
From eq. (G.5) we nd the decay width for the process Gl !
m + n
Gl! m n (k; R) =
jC(l;m;n)(k; R)j2
TSS
480 2(1 + mn)ml7RM53
5=2(ml2; m2m; m2n) ;
(G.8)
where (x; y; z)
x2 + y2 + z2
the density plot of the decay width
2xy
2xz
2yz. In the left panel of gure 25 we show
Gl! m n as a function of the nal state masses. From
this plot it is evident that the decay width is dominated by states close to the phase space
boundary. If compared with the trilinear graviton decay computed in appendix D, it is
easy to see that the KK graviton decay into two scalars is largely sub-dominant. The main
reason is that in the process Gl !
m + n conservation of angular momentum requires a
nal state in d-wave, L = 2. By direct computation, it is possible to show that the decay
width
the two
Gl! m n is indeed proportional to the fourth power of the relative velocity between
nal state particles, de ned as vrel = jp~m=Em
p~n=Enj, in agreement with the
usual correspondence between total orbital angular momentum L and velocity vrel, which
implies the scaling vr2eLl in the non-relativistic limit. Consequently, close to the phase space
boundary the decay width receives a further suppression due to the fact that nal state
particles are produced almost at rest with very small relative velocity.
Finally, we provide an analytical approximation for the sum over the kinematically
allowed nal state particles, in parallel with the discussion presented for the graviton decay
into gravitons. We nd
Gl!P m n =
Gl!P GmGn
6k2
595ml2 :
(G.9)
where
Gl!P GmGn is the inclusive width computed in eq. (D.13). This result clearly shows
the relative suppression between the two channels.
KK graviton decays into a KK graviton and a scalar
The 4D Lagrangian density for a xed triad (n; m; l) of KK states is
TTS
where C(n;m;l)(k; R) encodes the integration over the extra dimension, and we nd
h TTS
TTS
C(n;m;l)(k; R) + T(n;m;l)(k; R)
P(n;m;l)(k; R)i
TTS
nh~(m)h~(l)
(G.10)
dye3 pk h48k2 ~
'~n m l +
HJEP06(218)9
8k2
3
4k
3
TTS
L(n;m;l) =
1
M53=2( R)
TTS
C(n;m;l)(k; R) =
Z R
0
TTS TTS
The two functions T(n;m;l)(k; R) and P(n;m;l)(k; R) arise, respectively, from interactions
localized on the TeV and Planck brane. We
nd
T(Tn;TmS;l)(k; R) = 4k3=2 h'~n(0) m(0) l(0) + 2 ~ n(0) m(0) l(0)i ;
P(n;m;l)(k; R) = 4k3=2e2k R h'~n( R) m( R) l( R) + 2 ~ n( R) m( R) l( R)i : (G.13)
TTS
We can now compute the decay width for the process Gl ! Gm + n. We nd
Gl!Gm n =
jS(TnT;mS;l) + S(n;l;m)j
TTS
2
2880M53ml7m4m 3R2
1=2(ml2; m2m; m2n) F (ml2; m2m; m2n) ;
(G.14)
where F (x; y; z)
TTS
P(n;m;l)(k; R).
x4 + x3(26y
4z) + 2x2(63y2
28yz + 3z2) + 2x(13y
2z)(y
z)2 + (y
z)4, and we introduced the short-hand notation S(n;m;l)
TTS
TTS TTS
C(n;m;l)(k; R) + T(n;m;l)(k; R)
In the right panel of gure 25 we show the density plot of the decay width
Gl!Gm n
nal state masses. From this plot we see that the decay width is
dominated by heavy gravitons and light scalars in the nal state.
We provide an analytical approximation for the sum over the kinematically allowed
nal state particles, in parallel with the discussion presented for the graviton decay into
gravitons. We nd
(G.11)
(G.12)
Gl!P Gm n =
Gl!P GmGn
(G.15)
1524k3=2
595ml3=2 :
G.3
KK graviton decays: branching ratios
We can now summarize our results.
In gure 26 we show the branching ratios of the graviton KK modes. For clarity, in
the nal states involving KK scalars we plotted separately the case with a radion in the
nal state. All the decay widths are inclusive, meaning that we consider the sum over
all kinematically allowed
nal state particles. For the inclusive processes, in the limit
l
kR
1 we nd the simple scaling relations
k
l
M53
m2k2
l
M53 ;
HJEP06(218)9
=
= ϕ
=
= ϕ ϕ
= ϕ ϕ
[
m3k
l
M53 :
=
]
= ϕ ϕ
= ϕ
Branching ratios of the graviton KK modes for k = 10 GeV and rigid boundary
conditions. We take M5 = 10 TeV.
that should be compared with the decay width into SM states
Gl!SM
(G.17)
As shown in gure 26, and already discussed in the main text, graviton decays into
gravitons competes with the SM decay channels at small k since the ratio of their decay widths
parametrically grows as
(ml=k)1=2. On the contrary, KK graviton decays into,
respectively, a KK graviton and a scalar and two scalars feature the suppressions
and (k=ml)2 if compared to the graviton into SM decay.
Let us consider these cases in more detail. It is useful to remember that in the graviton
case the wavefunction integrals prefer small mass splittings, which is physically understood
from the view of extra-dimensional momentum conservation in the k ! 0 limit.
For the decay of graviton into a graviton and a scalar there is no wavefunction
suppression. This is because in the sti limit the boundary conditions for the dilaton are
Dirichlet-Dirichlet and, when integrated over the wavefunction for two gravitons, one no
longer requires that ml
mm + mn as the integrand is y-parity odd in the limit k ! 0.
Naively one would expect the total width should then scale as
m4=M53.
l
However, in the limit k ! 0 the dilaton acquires an additional Z2 symmetry that would
completely forbid such decays. Thus one expects the amplitude should still scale
proportional to k due to the vertex factor itself. Thus, when squared, this explains the form of the
decay width in eq. (G.16), and the extra suppression relative to the Gl !
P GmGn decays.
The suppression for Gl !
P
m n decays can be understood from momentum
conservation. The wavefunction integral suppression is still in action as in the k ! 0 limit the
integrand is y-parity even. In addition to this the decay of a spin-2 graviton to two scalars
will be d-wave suppressed, hence the matrix element is itself proportional to the nal state
momenta, which are proportional to the mass splitting. When squared this leads to an
additional factor of (k=ml)2 relative to the Gl !
KK scalar decays into two scalars
The 4D Lagrangian density for a xed triad (n; m; l) of KK states is
LSSS
(n;m;l) =
1
M53=2 hC(SnS;Sm(;1l))(k; R) n(@
SSS
l) + B(n;m;l)(k; R) n m l ;
i
where the coe cient of the derivative interaction comes from the extra-dimensional integral
Z R
0
C(SnS;Sm(;1l))(k; R) = 6
dye3 k3=2 ~ n(y) ~ m(y) ~ l(y) ;
C(SnS;Sm(;2l))(k; R) + T(n;m;l)(k; R) + P(n;m;l)(k; R). The rst term arises from the following
extra
SSS SSS
dimensional integral
C(SnS;Sm(;2l))(k; R) =
Z R
0
dye3 k3=2
3 ~ n ~ 0m ~ 0l + 64k2 ~ n ~ m ~ l
6k ~ n ~ m'~0l + ~ n'~0m'~0l
20k ~ 0n ~ m ~ l + 4k2 ~ n'~m'~l
81
16 k2'~n'~m'~l ;
SSS SSS
while T(n;m;l)(k; R) and P(n;m;l)(k; R) describe interactions localized, respectively, on the
TeV and Planck branes. We (G.18) (G.19) (G.20)
SSS
T(n;m;l)(k; R)
SSS
P(n;m;l)(k; R)
nd
1
6
4k
9
4k
4k
'~n(0)'~m(0) ~ l(0)
'~n(0) ~ m(0) ~ l(0) ; (G.21)
[(27 P + 4k)'~n( R)'~m( R)'~l( R)
(G.22)
+(72k + 162 P)'~n( R)'~m( R) ~ l( R)
+216k'~n( R) ~ m( R) ~ l( R)i :
Notice that both TeV and Planck brane contributions vanish in the rigid limit. Equipped
with this result, we can compute the decay width for the process l !
m + n. We nd
l! m n (k; R) =
1=2(ml2; m2m; m2n)
16 (1 + mn)ml3M53
ml2 + m2m + m2n C(SnS;Sm(;1l))(k; R)
(G.23)
+
X h SSS 2
C(n;m;l)(k; R) + T(n;m;l)(k; R) + P(n;m;l)(k; R)i 2
SSS SSS
:
perm:
In the left panel of gure 27 we show the density plot of the decay width
l! m n as a
function of the nal state masses. From this plot we see that the decay width is dominated
by the region close to the phase space boundary.
l!GmGn =
jS(Tl;TnS;m) + S(l;m;n)j
TTS 2
2880(1 + mn)M53ml3m4nm4m 3R2
1=2(ml2; m2m; m2n) F (M m2; Mn2; Ml2) :
In the central panel of gure 27 we show the density plot of the decay width
function of the nal state masses. From this plot we see that the decay width is dominated
by the region close to the phase space boundary with an additional sizable contribution
coming from the decay into two light gravitons.
G.6
KK scalar decays into a KK graviton and a scalar
The decay process l !
nd the width
m + Gn is controlled by the Lagrangian density in eq. (G.5). We
l! mGn(k; R) = jC(n;m;l)(k; R)j2
TSS
96M53ml3m4n 2R
5=2(ml2; m2m; m2n) :
In the right panel of gure 27 we show the density plot of the decay width
function of the nal state masses. From this plot we see that the decay width prefers the
presence of a light scalar in the nal state.
G.7
KK scalar decays: branching ratios
We can now summarize our results. We already discussed in gure 7 the branching ratios
of the scalar KK modes for k = 10 GeV, rigid boundary conditions and
= 0. For the
inclusive processes, in the limit l
kR
1 we nd the simple scaling relations
l!GmGn (central panel) and
l! mGn (right panel) as a function of the nal state masses. The example shown is M5 = 10 TeV,
k = 10 GeV, for mode l = 1000. The corresponding mass of the decaying particle is m1000 ' 1 TeV.
G.5
KK scalar decays into two KK gravitons
The decay process l ! Gm + Gn is controlled by the Lagrangian density in eq. (G.10).
We nd the width
m2k2
l
M53 ;
l!P m n
l!P mGn
l!P GmGn
m2k2
l
M53 ;
(G.24)
l!GmGn as a
(G.25)
l! mGn as a
mlk3
M53 ;
(G.26)
l!SM
3
mlk
M53 :
As already noticed and discussed in the main text, the suppression of the SM decay modes
makes all decay channels in eq. (G.26) important for the phenomenology.
H
Dijet angular distributions
Analyses of the dijet angular distributions (such as refs. [74{76]) use the variable , de ned
HJEP06(218)9
in terms of the rapidities of the two jets as
which can be expressed in terms of Mandelstam variables as
(G.27)
(H.1)
(H.2)
(H.3)
:
(H.4)
(I.1)
Therefore, the angular distributions of the various dijet processes can be obtained from the
expressions for
d
dt
(s; t; u)
available in appendix A of ref. [43] as
d
d
(s; ) =
( + 1)2
d
dt
+ 1
s
1= + 1
1= + 1
s
+ 1
The e ective graviton propagators in our model are di erent from those of ref. [43] and are
derived in appendix I.
I
Representing the KK graviton tower by an e ective propagator
For computing matrix elements involving KK gravitons, it is sometimes useful to sum the
propagators (times couplings) of the whole KK graviton tower and work in terms of the
e ective propagator
exp(jy1
y2j) ;
8
t = <
s
+ 1
for 0 <
t <
for
t < s
Se (s)
'
=
1
X
n=1 G
1
(n)2 s
1
M53 k
1
dm
1
m2n + i
1
r
1
s + k2 + k
k2
m2 s
m2 + i
the sum of squared propagators,
Se (s) !
for s
for jsj
k
2
k
2
1
1
X
n=1 G
1
(n)4 (s
2M56R k
1
1
' 4M56s
m2n)2 + s 2n(s)
2
R (s)
k
2 3=2
for s
k2 ;
1
m2)2 + s 2(s)
Another useful quantity, relevant for on-shell KK graviton production (at p
s > k), is
(I.2)
(I.3)
(I.4)
where (s) is the width of KK gravitons with mass near ps. For s
k2, there is an extra
suppression factor of approximately
1
k
2 3=2
as can be seen by using m2
' s in the k-dependent prefactor in the integrand.
Note the important di erence between the result of eq. (I.3) and the nave square of
eq. (I.2), which has been also discussed in refs. [44, 83] in the context of the low-curvature
RS model. When using the formulas of ref. [43] as envisioned in appendix H, one should
use jS(s)j2e
For S(s) with p
whenever jS(s)j2 with p
s > k appears, but Se (s) when S(s) is not squared.
s < k, S(t), or S(u), one should use Se (s), Se (t), or Se (u), respectively.
J
An exact solution to Einstein's equations
Working in the comoving coordinate ds2 = e2 (z)
dx dx + dz2 we nd that a solution
of Einstein's equation and the dilaton equation of motion from the action in eq. (2.25) is
S(z) =
log
+ 1 ;
(z) =
kz +
log
+ 1 ;
(J.1)
3
2
4kz
1
4kz
p3"
where we set the integration constants such that (0) = 0 and S(0) = 0. Note that there
is no smooth limit to either AdS or CW/LD geometry, suggesting this is a branch of the
general solution to the equations of motion that is rather unique. If one reparameterises
k = p3"k~ then in the limit " ! 0 one appears to recover an LD-like solution, however
inspecting eq. (2.25) one sees that in this limit the bulk potential is vanishing. Thus this
limit corresponds to having a massless bulk scalar with non-trivial boundary conditions on
S and S0 at the branes, due to a brane potential. { 61 {
Rev. D 90 (2014) 023545 [arXiv:1404.6209] [INSPIRE].
HJEP06(218)9
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