††institutetext: Department of Physics and Astrophysics, University of Delhi,
Delhi 110007, India

# A closed clockwork theory: $\mathbb{Z}_{2}$ parity and more

Debajyoti Choudhury  and Suvam Maharana<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We develop a new class of clockwork theories with an augmented structure of
the near-neighbour interactions along a one-dimensional closed chain. Such a
topology leads to new and attractive features in addition to generating light
states with hierarchical couplings via the usual clockwork mechanism. For one,
there emerges a $\mathbb{Z}_{2}$ symmetry under the exchange of fields
resulting in a physical spectrum consisting of states, respectively even and
odd under the exchange parity with a two-fold degeneracy at each level. The
lightest odd particle, being absolutely stable, could be envisaged as a
potential dark matter candidate. The theory can also be obtained as a
deconstruction of a five-dimensional theory embedded in a geometry generated
by a linear dilaton theory on a $S^{1}/\mathbb{Z}_{2}$ orbifold with three
equidistant 3-branes. Analogous to the discrete picture, the $\mathbb{Z}_{2}$
symmetry in the bulk theory necessitates the existence of a KK spectrum of
even and odd states, with doubly degenerate modes at each KK level when
subject to certain boundary conditions.

##  1 Introduction

As successful as it might seem, the Standard Model (SM) of particle physics
is, by no means, the ultimate theory describing Nature in its totality. The
shortcomings, both theoretical and those pertaining to discrepancies with data
have led to continuing searches for realistic extensions. Over the years, a
plethora of phenomenologically relevant new physics models have been proposed
to address the shortcomings of the SM, but only for newer issues to emerge. A
recurrent problem pertains to the apparent hierarchy of mass scales and
couplings. For instance, models with axions or axion-like particles typically
require small axion-SM couplings Kawasaki:2013ae . The case of neutrino mass
models with hierarchically heavy right-handed neutrinos serves as another
example King:2003jb ; deGouvea:2016qpx . Even within the SM, the fermion
masses display a hierarchy among the different flavors. The issue, therefore,
warrants an underlying mechanism to generate such small masses or couplings,
the absence of which would require a fine-tuning in the corresponding
ultraviolet (UV)-complete theory.

The clockwork mechanism proposed in refs.Kaplan:2015fuy ; Choi:2015fiu ;
Giudice:2016yja  apparently offers an interesting solution to the
aforementioned conundrum. The original mechanism (applicable to particles of
any given spin), generically, entails $N+1$ copies of a field theory, each
involving a massless field with the masslessness being protected by some
global symmetry $G$. The entire construction can be pictured as a chain of
sites in a one-dimensional lattice in the field space with the full symmetry
being $G^{N+1}$. Furthermore, introducing near-neighbour interaction terms
with a strength parametrised by a dimensionless factor $q\,(>1)$ explicitly
breaks the global symmetry to a single factor $G^{\prime}$ corresponding to
which we obtain a single massless particle. This massless state, a linear
combination of the $N+1$ fields, interestingly, has a hierarchical
distribution along the sites with an exponential localisation towards one of
the boundary sites111While similar effects had been anticipated in
refs.Georgi:1989ic ; Choudhury:1991if  the goals of (and, hence, the field
assignments in) the clockwork scenarios are distinctly different..
Consequently, a coupling of the SM sector to the opposite boundary of the
clockwork space would have a coupling of magnitude $\sim 1/q^{N}$ with the
massless state. Thus, even without introducing any small coupling, just a
moderately large value of $N$ generates a very sizable suppression. The
massive modes (the so-called clockwork “gears”), on the other hand, have
nearly-flat distributions along the sites. This is the essence of the
clockwork mechanism which has found its applications in not only the original
motivations around axion physics Kaplan:2015fuy ; Choi:2015fiu ; Coy:2017yex ;
Agrawal:2017cmd ; Long:2018nsl ; Choi:2014rja ; Flacke:2016szy ;
Davidi:2018sii  but also in other contexts like flavor hierarchies in masses
and mixings Patel:2017pct ; Alonso:2018bcg ; Babu:2020tnf ;
vonGersdorff:2020ods , neutrino masses Ibarra:2017tju ; Banerjee:2018grm ;
Kitabayashi:2019qvi , dark matter Hambye:2016qkf ; Kim:2017mtc ; Kim:2018xsp ;
Goudelis:2018xqi ; Bernal:2020yqg , neutron-antineutron oscillation
Arun:2022eqs  as well as inflationary cosmology Kehagias:2016kzt ; Im:2017eju
; Joshipura:2020ibd . The mechanism, in its present form, largely accommodates
only Abelian groups, but discussions pertaining to non-Abelian clockwork
theories can be found in refs. Craig:2017cda ; Ahmed:2016viu ; Kang:2020cxo .

The formulation of the clockwork theories as described in ref.Giudice:2016yja
has another compelling attribute in that the one-dimensional lattice in the
theory space can be regarded as the deconstruction of a continuum 5D field
theory. The corresponding warped geometry turns out to be the one that is
generated by a 5D linear dilaton theory of gravity Antoniadis:2011qw ;
Cox:2012ee  with the fifth-dimension compactified on a $S^{1}/\mathbb{Z}_{2}$
orbifold augmented by a 3-brane at each of its boundaries. This setup, similar
to the RS1 model Randall:1999ee , also offers a possible solution to the
hierarchy problem along with a rich phenomenology of the dynamical graviton
and dilaton modes 2014 ; 2018 .

Since the mechanism relies on the specific structure of the near-neighbour
interactions, it is worth exploring whether there exist extensions of this
structure, with different topologies, that possess novel phenomenological
implications in addition to a similar clockwork mechanism. To this effect, we
attempt to construct a class of clockwork theories with a modified structure
compared to the original such that the resulting hierarchy of the massless
particle’s distribution along the sites first increases (decreases) from one
end towards an intermediate site $p$ and then decreases (increases) towards
the other end. Further, taking $p$ to be the centre-most site in the lattice
and identifying the two end-sites lead to a theory with a $\mathbb{Z}_{2}$
symmetry characterized by an exchange of fields in the two arms about the
$p$-th site. Another distinguishing feature of the construction is that it
also generates a spectrum of massive particles (CW gears) which are doubly
degenerate at each level with the exception of the heaviest mode222This is in
stark contrast to the two-sided theory considered in ref. Banerjee:2018grm  in
the context of neutrino masses.. This is but a consequence of the exchange
parity that emerges naturally and ensures that the lightest odd mode in the
spectrum remains absolutely stable and, therefore, could potentially be a dark
matter candidate.

We further identify the continuum counterpart of the new theory with the
geometry generated by a 5D linear dilaton (LD), albeit augmented by a third
3-brane at the centre of the orbifold, with the $\mathbb{Z}_{2}$ symmetry
being identified with the coordinate reflection in the fifth dimension about
the middle brane. The 4D spectrum, for a theory with a vanishing bulk mass,
consists of a purely massless mode and a tower of massive KK excitations with
a two-fold degeneracy at each KK level. The latter, automatically, are either
odd or even under the continuum version of the exchange parity which can be
regarded as a form of KK parity in a warped 5D scenario, a feature that is
typically absent in two-brane warped scenarios.

The primary goal of this paper is to present the generic structure of the
modified scenario and, therefore, we resort to discussing the phenomenological
implications only in passing, deferring dedicated studies thereof to future
works. The outline of the paper is as follows. In section 2, we review the
original one-sided clockwork theory and introduce our modified setup in
section 3. Here, we limit the discussion to bosonic field theories with
Abelian symmetries. Section 4 delineates the continuum limit and the
corresponding linear dilaton theory that generates the modified 5D metric. The
nature of bulk scalar and Abelian vector fields in the new background geometry
is evaluated individually in section 5. Finally, we conclude in section 6.

##  2 Prelude

We review here, briskly, the basic elements of the original clockwork
mechanism proposed in refs.Kaplan:2015fuy ; Giudice:2016yja , specifically the
case of the scalar clockwork. The idea essentially entails a low energy theory
of $(N+1)$ Nambu-Goldstone bosons corresponding to the spontaneous breaking of
an Abelian theory with a global symmetry $U(1)^{N+1}$ at some high scale $f$.
The theory below the scale $f$, therefore, has a Goldstone symmetry
$U(1)^{N+1}$ which is broken explicitly to a single (shift) symmetry
$U(1)_{CW}$ by introducing interaction terms among “adjacent” NGB species,
namely

$\begin{split}\mathcal{L}_{CW}&=-\frac{f^{2}}{2}\sum_{j=0}^{N}\partial_{\mu}U^{\dagger}_{j}\partial^{\mu}U_{j}-\frac{m^{2}f^{2}}{2}\Bigg{[}\sum_{j=0}^{N-1}U_{j}^{\dagger}U_{j+1}^{q}\Bigg{]}+\mbox{h.c.}\end{split}$
(1)

where $m$ is a mass parameter and $q$ is a real constant. Denoting the charge
operator corresponding to $U(1)_{j}$ by $Q_{j}$, (a simple choice is
$Q_{j}\\{U_{j}\\}=1,\forall j$), the generator corresponding to $U(1)_{CW}$ is

$\mathcal{Q}_{CW}=\sum_{j=0}^{N}\frac{Q_{j}}{q^{j}},$ (2)

From a different perspective, each $U_{j}$ defines a site on a discrete
landscape of scalar field theories and eq.1 simply denotes a chain of sites
with near-neighbour interactions. Representing
$U_{j}=e^{i\phi_{j}(x)/f}\,(j=0\dots N)$ with the $\phi_{j}$’s being the
pNGBs, the first term in eq.1 leads to canonical kinetic terms for the
$\phi_{j}$ while the second yields a $(N+1)\times(N+1)$ (mass)2 matrix apart
from quartic and higher order terms.

Diagonalizing the free Lagrangian through the unitary transformation

$\varphi_{n}=\sum_{j}a_{nj}\phi_{j},$ (3)

we expect one exactly massless mode (on account of the residual shift
symmetry) and $N$ massive modes. Indeed, the eigenvalues are

$m_{n}^{2}=m^{2}\left\\{\begin{array}[]{lcl}0&&n=0\\\\[8.61108pt]
\displaystyle 1+q^{2}-2q\cos\frac{n\pi}{N+1}&&n\neq 0\end{array}\right.$ (4)

with the corresponding eigenvectors being $(j=0\dots N,\quad n=1\ldots N)$

$\begin{array}[]{rclcrcl}a_{0j}&=&\displaystyle\frac{\mathcal{N}_{0}}{q^{j}},&&a_{nj}&=&\displaystyle\mathcal{N}_{n}\left[q\sin\frac{jn\pi}{N+1}-\sin\frac{(j+1)n\pi}{N+1}\right]\\\\[8.61108pt]
\mathcal{N}^{2}_{0}&\equiv&\displaystyle\frac{q^{2}-1}{q^{2}-q^{-2N}}&&\mathcal{N}_{n}^{2}&\equiv&\displaystyle\frac{2m^{2}}{(N+1)m^{2}_{n}}.\end{array}$
(5)

The clockwork(CW) mechanism can now be illustrated with a simple example.
Consider an external operator (i.e., composed of fields other than $\phi_{j}$)
$\mathcal{O}_{\rm ext}$ which couples to the clockwork sector at the $N$-th
site as

$\mathcal{L}_{int}=y\phi_{N}\mathcal{O}_{\rm ext}\ ,$ (6)

where $y$ is a coupling parameter. In the mass basis, this results in

$\mathcal{L}_{int}=y\left[\frac{\mathcal{N}_{0}}{q^{N}}\varphi_{0}+\sum_{n=1}^{N}\mathcal{N}_{n}q\sin\frac{Nn\pi}{N+1}\varphi_{n}\right]\,\mathcal{O}_{\rm
ext}.$ (7)

Assuming that all the dimensionless parameters in the theory are a priori
$\mathcal{O}(1)$, we see that $\varphi_{0}$ couples to the external operator
with an exponentially suppressed strength for $q>1$ and a sufficiently large
$N$. On the other hand, the heavy eigenstates couple to the external operator
with a nearly flat coupling profile. Thus, the clockwork mechanism generates
very small couplings naturally in a theory which does not contain any small
parameters whatsoever. This is the crux of the mechanism in a nutshell. In the
modified theory that we discuss in the subsequent sections we largely retain
this feature, albeit with small modifications. In addition, several new and
attractive features emanate automatically.

##  3 The closed clockwork

As seen in the preceding section, the original clockwork scenario could be
thought of as an open chain of $U_{j}$’s with their charges, under the remnant
$U(1)_{CW}$, being given by a geometric progression. In contrast, we consider
a theory with a closed chain, restricting ourselves to a discussion of only
scalars and Abelian gauge fields333Incorporating non-Abelian groups in
discrete clockwork scenarios introduces certain complications. For example, in
the scalar theory, if the fields $U_{j}$ were to be bi-fundamental
representations of global non-Abelian groups $G_{j,L}\times G_{j,R}$, the non-
trivial case $q>1$ preserves (post explicit breaking of the full symmetry)
only a diagonal subgroup $H$. Consequently, this leads to a pseudo-Goldstone
multiplet susceptible to loop-level corrections to its mass Ahmed:2016viu . On
the other hand, non-Abelian gauge invariance would typically necessitate the
trivial case $q=1$ for which the clockwork mechanism does not exist. One may,
however, choose to invoke a clockwork hierarchy at the expense of losing gauge
invariance Craig:2017cda ; Giudice:2017suc ..

###  3.1 Scalar Clockwork

A closed clockwork chain implies, of course, that either the charge ratio $q$
has to be unity or that the geometric progression needs to invert at a point.
Since the first alternative does not lead to a hierarchy of couplings, we
adopt the second. To be very specific, we consider an even number $2p$ of
pseudo-NGBs and modify the Lagrangian of eq.1 to

$\begin{split}\mathcal{L}_{CW}&=-\frac{f^{2}}{2}\sum_{j=0}^{2p-1}\partial_{\mu}U^{\dagger}_{j}\partial^{\mu}U_{j}-\frac{m^{2}f^{2}}{2}\Bigg{[}\sum_{j=0}^{p-1}U_{j}^{\dagger
q}U_{j+1}\,+\,\sum_{j=p}^{2p-1}U_{j}^{\dagger}U_{j+1}^{q}\Bigg{]}+\mbox{h.c.}\\\
&=-\frac{1}{2}\sum_{j=0}^{2p-1}\partial^{\mu}\phi_{j}\partial_{\mu}\phi_{j}+\frac{m^{2}}{2}\Bigg{[}\sum_{j=0}^{p-1}(q\phi_{j}-\phi_{j+1})^{2}\,+\,\sum_{j=p}^{2p-1}(\phi_{j}-q\phi_{j+1})^{2}\Bigg{]}+\textit{O}(\phi^{4})\
.\end{split}$ (8)

Note that we have maintained the same charge ratio $q$ in both arms which
necessitates the pivoting to occur exactly midway, i.e., at the $p^{\rm th}$
site. Unequal ratios in the two arms could be chosen too, but only at the cost
of losing some very attractive features.

The identification $U_{2p}\equiv U_{0}$ (ensuring that $\phi_{0}$ couples to
both $\phi_{1}$ and $\phi_{2p-1}$) renders this a closed clockwork scenario
and distinguishes it from the two-sided but open-ended scenario presented in
ref. Banerjee:2018grm . This single difference turns out to have profound
implications as we would see below.

Before discussing the mass spectrum, it is worthwhile to enumerate the
symmetries of the theory. As before, the Lagrangian in eq.8 has a residual
shift symmetry $U(1)_{CW}$ which is now defined by the generator,

$\mathcal{Q}_{CW}=\sum_{j=0}^{p-1}\frac{Q_{j}}{q^{2p-j}}+\sum_{j=p}^{2p-1}\frac{Q_{j}}{q^{j}},$
(9)

where, as before, we have assumed that each $U_{j}$ has a unit charge under
the symmetry $U(1)_{j}$. Additionally, the theory has a
$\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$ symmetry, where the first
$\mathbb{Z}_{2}$ corresponds to the rigid exchanges of the fields, namely,
$\phi_{j}\to\phi_{2p-j}$ and the second to the exchanges
$\phi_{j}\to-\phi_{2p-j}$, both $\forall j$.444Equivalently, these can be
recast as the trivial rigid $\phi_{j}\to-\phi_{j}$ and
$\phi_{j}\to\phi_{2p-j}$. A consequence of these symmetries is that the
eigenmodes must be either even or odd under it. In other words, the
eigenstates may be represented as

$\varphi^{(\pm)}_{n}=\sum_{j}a^{(\pm)}_{nj}\phi_{j},$ (10)

where $(\pm)$ denote the even and odd modes respectively and $a^{(\pm)}_{nj}$
are the corresponding elements of the transformation matrix, satisfying the
eigenvalue equation,

$\sum_{j}[M_{\phi}^{2}]_{ij}a_{nj}=m^{2}\lambda_{n}a_{ni}\ ,\forall n,i\ .$
(11)

Reverting to the $(2p\times 2p)$ mass matrix, it is given by

$M_{\phi}^{2}=m^{2}\left(\begin{array}[]{cccccccc}\mbox{\tiny(row
$0$)}&2q^{2}&-q&0&0&...&0&-q\\\ &-q&q^{2}+1&-q&0&...&0&0\\\ &.&.&.&.&.&.&.\\\
&.&.&.&.&.&.&.\\\ &.&-q&q^{2}+1&-q&0&...&0\\\ \mbox{\tiny(row
$p$)}&0&...&-q&2&-q&...&0\\\ &0&0&...&-q&q^{2}+1&-q&...\\\ &.&.&.&.&.&.&.\\\
&.&.&.&.&.&.&.\\\ \mbox{\tiny(row $2p-1$)}&-q&0&0&0&...&-q&q^{2}+1\\\
\end{array}\right)_{2p\times 2p}.$ (12)

When compared to the mass matrices obtained in ref.Giudice:2016yja  from eq.1
or in ref.Banerjee:2018grm , differences appear only in the
$(0,0),(p,p),(0,2p-1),(2p-1,0)$ and $(2p-1,2p-1)$ elements and, together,
these would turn out to have very interesting consequences. For one, the
eigenvalues are

$m_{n}^{2}\equiv m^{2}\lambda_{n}=m^{2}\begin{cases}0&n=0\ ,\\\
2(1+q^{2})&n=p\ ,\\\ \displaystyle
1+q^{2}-2q\cos\frac{n\pi}{p}&\mbox{otherwise}\ .\end{cases}$ (13)

A few remarks on the physical spectrum are now in order

  * •

As is evident from eq.13, the spectrum consists of an exactly massless mode
$\varphi_{0}$ and is bounded from above by the massive mode $\varphi_{p}$.

  * •

In addition to the mass gap between the lightest mode $\varphi_{0}$ and the
level $1$ modes $\varphi_{1}/\varphi_{2p-1}$ ($\Delta M_{1}$) there exists
another one ($\Delta M_{2}$) between the heaviest state $\varphi_{p}$ and the
penultimate $\varphi_{p-1}/\varphi_{p+1}$ states555With $\Delta
M_{1,2}^{2}\approx m^{2}(q-1)^{2}$, the second gap ($\Delta M_{2}$) is
noticeable only for large $q$.. This is in contrast with the case in
Kaplan:2015fuy ; Giudice:2016yja  wherein only the lower gap exists.

  * •

The intermediate states are doubly degenerate at each level, with each
degenerate subspace being spanned by the eigenmodes $\varphi_{n}$ and
$\varphi_{2p-n}$, or, equivalently, $\varphi^{\pm}_{n}$. These fill a band of
width $\Delta m_{\hbox{\tiny band}}\sim 2m$ with mass-splittings $\delta
m_{n}/m_{n}$ of $\mathcal{O}(1/2p)$ for sufficiently large $p$. The mass-
splitting first increases in the band as we move up the spectrum and then
decreases progressively towards the end of the band. Fig.1 shows a schematic
of the physical modes emerging in the closed clockwork model along with that
of the original model for comparison. Clearly, the inter-level spacings in the
closed scenario are larger than those seen in the original one-sided clockwork
theories.

  * •

This degeneracy is not simply a consequence of the $\mathbb{Z}_{2}$ symmetry
alone666Were that to be the case, such a degeneracy could as well be seen in a
two-sided open chain which, evidently, we do not encounter.. A crucial role is
also played by the universality (i.e., the site-independence) of the
$U(1)_{j}$ charges $Q_{j}$ and the mass parameter $m^{2}$ as well as the
closed nature of the interactions in the sense described before. Any deformity
in the universal nature of the couplings and/or the structure of the
interactions near the pivots would lead to non-degeneracies in the masses even
if the $\mathbb{Z}_{2}$ symmetry exists, a notion which is further elaborated
in Appendix A.2. Nevertheless, for small deviations from uniformity we would
obtain quasi-degenerate states.

The discussion above should convince the reader that this setup is not merely
a trivial joining of two open-ended CW chains, but a distinct theory with
unique properties.

Figure 1: Schematic depiction of the physical states in the closed clockwork
theory (right) and in the original one-sided theory (left) for $N=20$.

The aforementioned transformation matrix elements $a^{(\pm)}_{nj}$ are given
by

$a^{(+)}_{0j}=\mathcal{N}^{+}_{0}\begin{cases}q^{j-2p},&0\leq j\leq p\\\
q^{-j},&j>p\end{cases},\qquad
a^{(+)}_{pj}=\mathcal{N}^{+}_{p}\begin{cases}(-q)^{-j},&0\leq j\leq p\\\
(-q)^{j-2p},&j>p\end{cases}$ (14)

$a_{nj}^{(+)}(n<p)=\mathcal{N}^{(+)}_{n}\left\\{\begin{array}[]{lcl}\displaystyle\sin\frac{jn\pi}{p}+\frac{2q}{q^{2}-1}\,\sin\frac{n\pi}{p}\,\cos\frac{jn\pi}{p},&&0\leq
j\leq p\\\\[8.61108pt]
\displaystyle-\sin\frac{jn\pi}{p}+\frac{2q}{q^{2}-1}\sin\frac{n\pi}{p}\,\cos\frac{jn\pi}{p},&&\displaystyle
p\leq j\leq 2p-1\end{array}\right.$ (15)

and,

$a_{nj}^{(-)}(n>p)=\mathcal{N}^{(-)}_{n}\\\ \sin{\frac{jn\pi}{p}}.$ (16)

The normalization factors $\mathcal{N}_{0,p}$ and $\mathcal{N}^{\pm}_{n}$ are
given by

$\begin{array}[]{rclcrcl}\mathcal{N}_{0}^{2}&=&\displaystyle\frac{q^{4p}(q^{2}-1)}{(q^{2p}-1)(q^{2}+1)}&&\mathcal{N}^{2}_{p}&=&\displaystyle\frac{q^{2p}(q^{2}-1)}{(q^{2p}-1)(q^{2}+1)}\\\\[10.76385pt]
\displaystyle\left(\mathcal{N}^{+}_{n}\right)^{2}&=&\displaystyle\frac{(q^{2}-1)^{2}}{p\left[1+q^{4}-2q^{2}\cos{\frac{2n\pi}{p}}\right]}&&\displaystyle\left(\mathcal{N}^{-}_{n}\right)^{2}&=&\displaystyle
p^{-1}\ .\end{array}$ (17)

Having established the identity of the physical states, let us now consider an
external operator $\mathcal{O}_{E}$ (obviously transforming trivially under
the $\mathbb{Z}_{2}$) coupling to the CW sector only at the $j$-th site, viz.,

$\mathcal{L}_{\rm
ext}=-\mathcal{O}_{E}\phi_{j}\,=\,-\left[\sum_{n=0}^{p}a^{+}_{jn}\varphi^{+}_{n}+\sum_{n=p+1}^{2p-1}a^{-}_{jn}\varphi^{-}_{n}\right]\,\mathcal{O}_{E}\
.$ (18)

The $\mathbb{Z}_{2}$ symmetry is, thus, explicitly broken777It would have
remained unbroken had $\mathcal{O}_{E}$ coupled with identical strength to
$\phi_{p+j}$ as well. as the external operator couples to both even and odd
modes. Note, though, that if the coupling site is chosen to be either of
$j=0,p$, the operator couples only to the even modes as the odd wavefunctions
vanish there. The mixing matrix elements in eq.14-16 specify that the light
mode ($\varphi_{0}$) is localised towards the $j=p$ site, away from which its
distribution falls exponentially, acquiring a weight $\sim q^{-p}$ at $j=0$.
In contrast, the heaviest mode ($\varphi_{p}$) is localised near $j=0$, with
its weight decreasing exponentially towards $j=p$ along both arms. Thus, at
$j=0$, $\mathcal{O}_{E}$ couples to $\varphi_{0}$ with a suppressed coupling
($\sim q^{-p}$) and to $\varphi_{p}$ with a strength $\sim\mathcal{O}(1)$.
This order of strengths reverses for an $\mathcal{O}_{E}$ coupling at the
$j=p$ site instead. On the other hand, the rest of the massive modes have a
nearly-flat distribution along the sites.

####  3.1.1 A possible dark matter candidate?

The preceding discussion establishes that, in the event of the SM coupling
with the CW sector at either of the sites at $j=0$ or $j=p$, the theory
preserves the exchange parity (i.e., the $\mathbb{Z}_{2}$ symmetry).
Consequently, the lightest odd CW particle (LOCP) is rendered absolutely
stable and, hence, can potentially be a dark matter (DM) candidate (and, by
the same token, lead to additional sources of missing-momentum signals at
colliders). There is, though, a key difference with most popular DM models.
The vanishing of the $\mathbb{Z}_{2}$-odd CW state wavefunctions at the
($\mathbb{Z}_{2}$) fixed points implies that these do not couple directly to
the SM sector. Consequently, the requisite relic abundance would be realised
primarily through a pair of DM particles annihilating into the even CW modes
which can further annihilate and/or decay into SM particles. Thus, the odd
tower would act as a secluded dark sector. In such models, co-annihilations
could play a significant role in enhancing the total cross-sections by virtue
of the degeneracy in the physical states. Of course, the extent of such
enhancements would depend on the mass-splittings dictated by the parameters of
the theory. Another mode of enhancement in the annihilation could be effected
by introducing couplings that break $U(1)_{CW}$ softly, e.g., a trilinear
coupling of CW fields that opens a new channel of pair annihilation and co-
annihilation via a mediator which could further introduce resonance
enhancements in $s$-channel processes and/or enhancements from the lightness
of the mediator in $t$-channel processes. Typically, for a small number of
sites ($p\lesssim 5$), the latter channel would prove to be the dominant one.
We limit the discussion here to only highlighting the possibility of the
aforementioned DM phenomenology and leave the detailed analysis to an
accompanying work.

###  3.2 Vector Clockwork

Having defined the scalar theory, one for vector fields proceeds quite
analogously. We start with a $U(1)^{2p}$ theory with identical gauge couplings
$g$ alongwith $2p$ copies of complex scalar fields $\phi_{j}$, such that the
Lagrangian is

$\mathcal{L}_{CW}=\sum_{j=0}^{2p-1}\left\\{-\frac{1}{4}F_{\mu\nu
j}F_{j}^{\mu\nu}-\left(D_{\mu}\phi_{j}\right)^{\ast}\left(D^{\mu}\phi_{j}\right)-\lambda\left(\left|\phi_{j}\right|^{2}-\frac{f^{2}}{2}\right)^{2}\right\\},$
(19)

with the identification $(F_{N},\phi_{N})\equiv(F_{0},\phi_{0})$. Although the
gauge symmetry does not require888Similarly, the terms
$\sum_{i,j}|\phi_{i}|^{2}|\phi_{j}|^{2}$ are also allowed by symmetry
considerations. However, since their introduction does not bring about a
qualitative change, we choose to omit them for the sake of simplicity. the
parameters $\lambda$ and $f$ to be site-independent, we, nonetheless, assume
universal values. This restriction not only leads to a mass matrix similar to
that obtained in the scalar case with the attendant phenomenology (see the
discussion in section 3.1 and Appendix A.2), but also facilitates an embedding
in a higher dimensional theory. The complex scalars $\phi_{j}$ have non-zero
charges under $U(1)_{j}\times U(1)_{j+1}$, viz.

$\begin{split}\left(j=0,\ldots,p-1\right)&:\left(q,\,-1\right)\\\
\left(j=p,\ldots,N-1\right)&:\left(-1,\,q\right)\ .\end{split}$

Following a spontaneous symmetry breaking, $U(1)^{2p}\to U(1)_{CW}$, the
complex scalars can be represented as,

$\phi_{j}=\frac{1}{\sqrt{2}}\left(\chi_{j}+f\right)e^{i\pi_{j}},$ (20)

where $\chi_{j}$ and $\pi_{j}$ are real. Integrating out the heavy radial
degrees of freedom $\chi_{j}$, we have a theory of massive $U(1)$ gauge fields
below the SSB scale $f$ defined by

$\begin{array}[]{rcl}\mathcal{L}_{\rm
eff}&=&\displaystyle-\sum_{j=0}^{2p-1}\frac{1}{4}\left(F_{\mu
j}\right)^{2}+\frac{g^{2}f^{2}}{2}\Bigg{\\{}\sum_{j=0}^{p-1}\left(qA_{\mu}j-A_{\mu
j+1}\right)^{2}+\sum_{j=p}^{2p-1}\left(A_{\mu j}-qA_{\mu
j+1}\right)^{2}\Bigg{\\}}\\\\[12.91663pt]
&+&\displaystyle\frac{f}{2}\sum_{j=0}^{2p-1}\left(\partial_{\mu}\pi_{j}\right)^{2}+\mathcal{L}_{GF}\
.\end{array}$ (21)

A convenient choice for the gauge-fixing term is given by

$\mathcal{L}_{GF}=\frac{-1}{2\xi}\left\\{\sum_{j=0}^{p-1}\left[\partial^{\mu}A_{\mu
j}+\xi
qgf\left(\pi_{j}-q\pi_{j+1}\right)\right]^{2}+\sum_{j=p}^{N-1}\left[\partial^{\mu}A_{\mu}j+\xi
qgf\left(-q\pi_{j}+\pi_{j+1}\right)\right]^{2}\right\\}.$ (22)

As eq.21 shows, the mass terms for the gauge fields resemble those for the
scalar CW. In the unitary gauge ($\xi\to\infty$), all of the $(2p-1)$ NGBs
decouple from the theory999The whole point of choosing the unitary gauge here
is to elucidate the existence of a massless scalar mode in the theory which is
not a Goldstone boson. One could, in general, choose a more convenient gauge
to facilitate computational simplicity.. One massless scalar mode
$\tilde{\pi}_{0}$, however, remains as a physical state in the spectrum, which
is now given by

$\mathcal{L}_{\rm
eff}^{\rm(mass)}=\sum_{n=0}^{2p-1}\left\\{-\frac{1}{4}\left(\mathcal{F}_{\mu\nu}^{n}\right)^{2}+\frac{1}{2}m_{n}^{2}\left(\mathcal{A}_{\mu}^{n}\right)^{2}-\frac{1}{2\xi}\left(a_{jn}\partial^{\mu}\mathcal{A}_{\mu}^{n}\right)^{2}\right\\}+\frac{1}{2}\left(\partial_{\mu}\tilde{\pi}_{0}\right)^{2}\
.$ (23)

where,

$m_{n}^{2}=\left\\{\begin{array}[]{lcl}0&&n=0\\\\[4.30554pt]
2g^{2}f^{2}(1+q^{2})&&n=p\\\\[4.30554pt] \displaystyle
g^{2}f^{2}\left(1+q^{2}-2q\cos\frac{n\pi}{p}\right)&&n\neq
0,p\end{array}\right.$ (24)

and the eigenstates $\tilde{\pi}_{0}$ and $\mathcal{A}_{\mu j}$ (corresponding
to $\mathcal{F}$) are defined as

$\tilde{\pi}_{0}=\sum_{j=0}^{2p-1}a_{0j}\pi_{j}\qquad{\rm
and}\qquad\mathcal{A}_{\mu n}=\sum_{j=0}^{2p-1}a_{nj}A_{\mu j}\ .$ (25)

In summary, therefore, there are $2p-1$ massive photons with a degeneracy
identical to that of the CW scalars, a massless photon and a massless scalar.

Similar to the scalar case in section (3.1) the complete spectrum obtained
here can be classified as even and odd eigenstates of clockwork parity.
Explicitly, the matrix elements $a_{nj}$ are the same as obtained in eq.14–17.
Consequently, the physical modes have the same nature of localised couplings
with an external sector as that of the scalar CW in eq.18. The distinction in
this case is that the external operator coupling to the $j$-th site is a
current charged under $U(1)_{j}$ of the CW sector.

##  4 The continuum perspective

Keeping the discrete theories in perspective, we now explore the possibility
of obtaining them as deconstructions of five dimensional bulk theories, an
advantage being that in doing this, we would encounter some new possibilities.
To identify the appropriate underlying 5D geometry, it is convenient to assume
a form of the metric that incorporates at least those symmetries encountered
in the clockwork theories along the discrete lattice. Since, in our case, the
latter includes a $\mathbb{Z}_{2}$ symmetry under field exchanges, a logical
ansatz for the 5D metric (with the fifth dimension being compactified to a
circle of radius $R$) would be

$ds^{2}=X(|z-z_{p}|)dx^{2}+Z(|z-z_{p}|)dz^{2},$ (26)

where, $X$ and $Z$ are, a priori, unknown functions of the extra
dimension101010The function $Z$ could, of course, be eliminated by a simple
redefinition of the coordinate $z$, but it is convenient to retain it. and, by
construction, are $\mathbb{Z}_{2}$ symmetric about $z_{p}=\pi R/2$. Here, $x$
denotes the 4D spacetime coordinates and $z$ is the fifth coordinate. The
action for a massless bulk scalar in this background geometry would, then, be
given by

$\begin{split}\mathcal{S}&=\int d^{4}x\,\int_{0}^{\pi
R}dz\sqrt{-g}\left[-\frac{1}{2}g^{MN}\partial_{M}\phi(x,z)\,\partial_{N}\phi(x,z)\right]\\\
&=-\frac{1}{2}\int d^{4}x\,\int_{0}^{\pi
R}dz\,X^{2}Z^{1/2}\left[X^{-1}(\partial_{\mu}\phi)^{2}+Z^{-1}(\partial_{z}\phi)^{2}\right]\
.\end{split}$ (27)

Re-scaling the scalar field $\phi\to X^{-1/2}Z^{-1/4}\phi$ in order to obtain
canonical kinetic terms post discretization, we have

$\begin{split}\mathcal{S}&=-\frac{1}{2}\int d^{4}x\,\int_{0}^{\pi
R}dz\left\\{(\partial_{\mu}\phi)^{2}+X^{2}Z^{-1/2}\left[\partial_{z}\left(X^{-1/2}Z^{-1/4}\phi\right)\right]^{2}\right\\}\end{split}.$
(28)

We, now, discretize the $z$-dimension into a lattice of $2p$ sites with
uniform inter-site spacing $a=\pi R/2p$ and specify a site $j$ as $z\to ja$.
The $z$-derivative in the second term of the action is to be replaced by the
difference111111Clearly, the derivative could, in principle, be mapped into
inequivalent differences, and these choices would, potentially, lead to
different discrete theories. This ambiguity is not unexpected, for it is well
known that several inequivalent discrete theories could flow into the same
continuum theory.

$\partial_{z}\left(X^{-1/2}Z^{-1/4}\phi\right)\to
a\left(X_{j+1}^{-1/2}Z_{j+1}^{-1/4}\phi_{j+1}-X_{j}^{-1/2}Z_{j}^{-1/4}\phi_{j}\right).$

Since the discretized derivative involves fields from adjacent sites, namely,
$\phi_{j+1}$ and $\phi_{j}$, we adopt a particular discretization
prescription, wherein the factor coupled to the derivative is replaced by

$X^{2}Z^{-1/2}\to\left(X_{j}Z_{j}^{-1/4}\right)\left(X_{j+1}Z_{j+1}^{-1/4}\right),$
(29)

which consists of equal powers of the functions $X$ and $Z$ evaluated at
adjacent sites $j$ and $j+1$. With this prescription we have,

$\mathcal{S}=\frac{-1}{2}\int
d^{4}x\,\sum_{j=0}^{2p-1}\Bigg{\\{}(\partial_{\mu}\phi_{j})^{2}+\frac{4p^{2}}{\pi^{2}R^{2}}\left(X_{j}^{1/2}Z_{j}^{-1/8}Z_{j+1}^{-3/8}\phi_{j+1}-X_{j+1}^{1/2}Z_{j+1}^{-1/8}Z_{j}^{-3/8}\phi_{j}\right)^{2}\Bigg{\\}}.$
(30)

The second term inside the braces is a simple square of a difference of the
form $(f_{1}[j+1]f_{2}[j]-f_{1}[j]f_{2}[j+1])^{2}$, and the sum is explicitly
invariant under $j\to 2p-j$, a property that would be absent without the
symmetrization encoded in eq.29. In other words, the (natural) prescription of
eq.29 preserves, on discretization, the $\mathbb{Z}_{2}$ symmetry of the
continuum theory121212Note that this discretization prescription is
categorically different from the one adopted in Giudice:2016yja  which would
break the $\mathbb{Z}_{2}$ symmetry if employed in our scenario. This also
demonstrates the fact that different prescriptions can lead to different
discrete theories which is reminiscent of what is usually encountered in the
lattice QCD framework, especially in the context of discretizing fermions on a
lattice. The distinct low-energy theories, however, lead to the same bulk 5D
theory in the $N\to\infty$ limit. It is worth stressing here that this does
not necessarily mean that the different discrete theories would also
correspond to the same effective 4D theory derived from the 5D action which
additionally depends on the particulars of the boundary conditions imposed.
Different boundary conditions would typically induce distinct 4D theories as
will be seen in a later section.. Specifically, the discrete theory defined in
eq.8 is obtained with the assignments

$\begin{split}&X_{j}\propto Z_{j}\propto\exp\left(\frac{-4k\pi
R}{6p}|j-p|\right)\\\\[4.30554pt] \mbox{and}\qquad&q\equiv\exp\left(\frac{k\pi
R}{2p}\right),\qquad m^{2}\equiv\frac{4p^{2}}{\pi^{2}R^{2}q}\ .\end{split}$
(31)

Therefore, in the continuum limit, $p\to\infty$, the 5D metric has the form,

$ds^{2}=\exp\left(\frac{-4}{3}k|z-z_{p}|\right)\,(dx^{2}+dz^{2}),$ (32)

where $z_{p}=\pi R/2$. Furthermore, writing the parameters $q$ and $m^{2}$
defined in eq.LABEL:eqn:xydefn as functions of the lattice spacing $a$, which
acts as a natural UV regulator here, we have

$q(a)=e^{ka}\quad\mbox{and}\quad m^{2}(a)=\left[a^{2}q(a)\right]^{-1}\ .$ (33)

In the far UV ($a\to 0$) limit, an expansion in $a$, leads to a simple
functional form for the mass eigenvalues of the discrete theory, namely,

$\begin{split}m^{2}\left[1+q^{2}-2q\cos\Big{(}\frac{n\pi}{p}\Big{)}\right]&\longrightarrow
k^{2}+\frac{4n^{2}}{R^{2}}+\mathcal{O}(1/2p)\\\
2m^{2}\left(1+q^{2}\right)&\longrightarrow\infty\ .\end{split}$ (34)

In other words, these evolve into the KK spectrum of a scalar with an apparent
bulk mass $k^{2}$. However, as can be easily ascertained, the scalar is
actually massless, with this apparent “bulk mass” being an artefact of the
warping. Whether we obtain the exact same KK spectrum as above for the bulk
theory defined in eq.27 will be ascertained in a subsequent section.

###  4.1 The extended linear dilaton theory

Although the preceding treatment looks somewhat similar to that in
ref.Giudice:2016yja , the precise outcomes of eq.32 and eq.34 specify a
distinct geometry and a consequent mass spectrum that is distinctively
different. However, until now we have just assumed the background geometry and
it is contingent on us to obtain the metric in eq.32 as a solution of the
five-dimensional Einstein equations. To this end, we consider a 5D linear
dilaton scenario Antoniadis:2011qw ; Cox:2012ee , with the fifth dimension
compactified on a $S^{1}/\mathbb{Z}_{2}$ orbifold of radius $R$. Furthermore,
three rigid 3-branes are placed at $z=0,\,\pi R/2$ and $\pi R$ respectively.
The bulk and brane actions in this setup are, in the Einstein frame,

$\begin{split}\mathcal{S}_{Bulk}&=2M_{5}^{3}\int
d^{4}x\,dz\sqrt{-g}\left(\frac{1}{4}\mathcal{R}-\frac{1}{12}g^{MN}\partial_{M}S\partial_{N}S-V(S)\right)\\\
\mathcal{S}_{Brane}&=-2M_{5}^{3}\int
d^{4}x\,dz\frac{\sqrt{-g}}{\sqrt{g_{55}}}\sum_{\alpha=1}^{3}\lambda_{\alpha}(S)\delta(z-z_{\alpha})\
,\end{split}$ (35)

where $z_{\alpha}\equiv\\{0,\,\pi R/2,\,\pi R\\}$. Here, $\mathcal{R}$ is the
5D Ricci scalar, $S(x,z)$ is the dilaton field and $M_{5}$ is the 5D
fundamental mass scale. The bulk and brane potentials, $V(S)$ and
$\lambda_{\alpha}(S)$ are given, respectively, by

$V(S)=-e^{-2S/3}k^{2}\
,\qquad\lambda_{\alpha}(S)=\frac{e^{-S/3}}{2}\frac{\Lambda_{\alpha}}{M_{5}^{3}}\qquad,$
(36)

where the $\Lambda$’s ($\Lambda_{2}=4kM_{5}^{3}=-\Lambda_{1}=-\Lambda_{3}$)
are the vacuum energies on the branes. The dimensionful parameter $k$ provides
a measure of the vacuum energy in the bulk and, thereby, the extent of the
warping in the resulting geometry131313A similar 5D setup was assumed in
Kogan:1999wc ; Agashe:2007jb  in the context of RS-like warped scenarios,
albeit without a linear dilaton.. The most general metric with 4D Poincare
invariance corresponding to the action in eq.35, written in terms of the
conformal coordinates is DeWolfe:1999cp ,

$ds^{2}=e^{2\sigma(z)}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}\right),\qquad\eta_{\mu\nu}=diag\\{-1,+1,+1,+1\\}.$
(37)

The corresponding equations of motion are given by

$\begin{split}\left(S^{\prime\prime}+3\sigma^{\prime}S^{\prime}\right)e^{-2\sigma(z)}&=4k^{2}e^{-S/3}-M_{5}^{-3}\sum_{\alpha}e^{-(\sigma+S/3)}\Lambda_{\alpha}\delta(z-z_{\alpha})\\\
9\left(\sigma^{\prime\prime}-\sigma^{\prime
2}\right)e^{-2\sigma(z)}&=-3M_{5}^{-3}\sum_{\alpha}e^{-(\sigma+S/3)}\Lambda_{\alpha}\delta(z-z_{\alpha})\\\
36\sigma^{\prime 2}-S^{\prime 2}&=12k^{2}e^{2(\sigma-S/3)}\ .\end{split}$ (38)

These are most easily solved by writing the bulk potential $V(S)$ in terms of
a superpotential $W(S)$ DeWolfe:1999cp , viz.,

$V(S)=\frac{3}{4}\left(\frac{\partial W(S)}{\partial
S}\right)^{2}-\frac{1}{3}W(S)^{2}\,,$ (39)

such that the solutions to the equations

$S^{\prime}(z)=3e^{\sigma(z)}\frac{\partial W(S)}{\partial S}\
,\qquad\sigma^{\prime}(z)=-\frac{1}{3}e^{\sigma(z)}W(S)$ (40)

subject to the junction conditions

$W(S)\Bigg{|}^{z_{\alpha}+\epsilon}_{z_{\alpha}-\epsilon}=e^{-S/3}\frac{\Lambda_{\alpha}}{M_{5}^{3}}\qquad\mbox{and}\qquad\frac{\partial
W}{\partial
S}\Bigg{|}^{z_{\alpha}+\epsilon}_{z_{\alpha}-\epsilon}=-\frac{1}{3}e^{-S/3}\frac{\Lambda_{\alpha}}{M_{5}^{3}}$
(41)

(with $\epsilon$ being an infinitesimally small positive quantity) are also
solutions to eq.38. For our case, this implies

$W(S)=\left\\{\begin{array}[]{rcl}\displaystyle
e^{-S/3}\sum_{\alpha=1}^{3}\frac{\Lambda_{\alpha}}{2M_{5}^{3}}\left[\theta(z-z_{\alpha})-\theta(z_{\alpha}-z)\right]&&z\in(0,\pi
R),\\\\[12.91663pt] \displaystyle
e^{-S/3}\left[-\frac{\Lambda_{0}}{2M_{5}^{3}}\theta(-z)+\frac{\Lambda_{\pi}}{2M_{5}^{3}}\theta(z-\pi
R)\right]&&z\not\in(0,\pi R)\ .\end{array}\right.$ (42)

With the orbifolding dictating that $\sigma(z)$ be symmetric about the
orbifold fixed points, the second of eqs.40 implies that $W$ must be an odd
function across the fixed points. This, alongwith the conditions imposed by
eq.41, fixes the form of $W(S)$ in the second line of eq.42. With these
ingredients in order, we obtain the following solutions to eq.38,

$\begin{split}\sigma(z)&=-\frac{2}{3}k|z-z_{p}|e^{(\sigma_{o}-\frac{1}{3}S_{o})}+\sigma_{o}\\\
S(z)&=-2k|z-z_{p}|e^{(\sigma_{o}-\frac{1}{3}S_{o})}+S_{o}\ ,\end{split}$ (43)

where, $\sigma_{o}$ and $S_{o}$ are constants of integration, and we use the
notation $z_{p}=\pi R/2$. Choosing, for convenience, $\sigma_{o}=S_{o}/3$, the
solutions reduce to141414This can also be regarded as a rescaling of the
parameter $k$ by defining $k=\tilde{k}e^{-(\sigma_{o}-\frac{1}{3}S_{o})}$.
Relating $k$ to the vacuum energy in the bulk demands that
$\tilde{k}e^{-(\sigma_{o}-\frac{1}{3}S_{o})}<M_{5}$. While $S_{o}$ is related
to the VEVs of the dilaton at the branes as we will see in the next
subsection, $\tilde{k}$ and $\sigma_{o}$ are free parameters which can be
suitably configured to get optimum values for $k$.

$\begin{split}\sigma(z)&=-\frac{2}{3}k|z-z_{p}|+\sigma_{o}\\\
S(z)&=-2k|z-z_{p}|+S_{o}\ ,\end{split}$ (44)

The metric in eq.37, therefore, has the explicit form 151515Note that we drop
the overall constant factor $e^{2\sigma_{o}}$ here as it does not affect the
form of the geometry.

$ds^{2}=\exp\left(\frac{-4}{3}k|z-z_{p}|\right)(dx^{2}+dz^{2}).$ (45)

So, the extended linear dilaton (LD) theory defined in eq.35 successfully
generates the required clockwork geometry of eq.32. Evidently, the symmetry of
the triple brane setup about $z=z_{p}$ in terms of the physical parameters in
the theory has induced $\mathbb{Z}_{2}$ symmetric solutions for $\sigma(z)$
and $S(z)$ and, hence, for the resulting metric, under the transformation
$z\to\pi R-z$.

###  4.2 Addressing the hierarchy problem

The role of the linear dilaton model as a solution to the hierarchy problem
has been elucidated in detail in refs.Antoniadis:2011qw ; Giudice:2016yja  and
analogous arguments readily follow for our three 3-brane setup. For brevity’s
sake, we refrain from discussing this in detail and, instead, highlight only
the key differences. To begin with, starting with the fundamental scale
$M_{5}$, we may obtain, on dimensional reduction, the Planck mass $(M_{P}\sim
10^{19})$ as the effective mass scale in the 4D Einstein-Hilbert action, viz.,

$M_{P}^{2}=2M_{5}^{3}\int_{0}^{\pi
R}dz\,e^{3\sigma(z)}=\frac{2M_{5}^{3}}{k}\left(1-e^{-k\pi R}\right).$ (46)

The above relation resembles the one obtained in the RS model Randall:1999ee .

The major difference from the two-brane theory lies in the emergence of two
distinct physical scenerios that ameliorate the hierarchy issue. For $k>0$, we
have two negative tension (IR) branes at the orbifold boundaries and a
positive tension (UV) brane at the central $\mathbb{Z}_{2}$ fixed point. We
shall call it the IR-UV-IR setup. In this scenario, we have $M_{P}\sim M_{5}$
for a nominally large value of $k\pi R\sim\mathcal{O}(10)$. For a given value
of $k$, though, the required magnitude of $k\pi R$ is roughly twice of that
necessary in two-brane models. This difference stems from the fact that in our
three brane model the hierarchy is explained by the warping profile operative
between the boundary branes and the brane present in the middle of the
orbifold. A standard model (SM) sector localised on the IR brane would have
its effective mass parameter161616For the SM this means the independent mass
parameter $\mu$ in the Higgs sector or, equivalently, the symmetry breaking
scale $v$. ($m_{IR}$) derived from the parameters in the fundamental 5D theory
via relations like $m_{IR}=e^{-ak\pi R}m_{0}$, thereby lowering it to the TeV
scale. Here, $a$ is an $\mathcal{O}(1)$ numerical constant arising from a
combination of the exponential in the (induced) 4D metric determinant and that
emanating from field redefinitions. $m_{0}$ denotes a parameter of the brane
localised sector in the full 5D theory, presumably near the Planck scale. Note
that this argument is similar to the one presented in the original RS scenario
Randall:1999ee .

On the other hand, for $k<0$, the negative tension brane is located at the
central fixed point and the positive tension branes at the orbifold
boundaries. Hence, we refer to this as the UV-IR-UV setup. The hierarchy is
addressed in a similar manner as before for $|k\pi R|\sim\mathcal{O}(10)$,
only that in this case the fundamental scale, $M_{5}$, is near the TeV scale.

The triple brane theory, therefore, proposes the plausibility of two distinct
scenarios in terms of their role in explaining the Planck-EW hierarchy
(Fig.2). However, one needs to check for the stability of both the scenarios
in terms of the physicality of the emerging dilaton modes. This is necessary
as the coefficients of their kinetic terms depend on the parameters of the
theory and, hence, may lead to unphysical degrees of freedom for certain
values of the parameter space. For a related discussion see Appendix B.

Phenomenologically, the two scenarios could have distinct, albeit rich,
consequences. For instance, the dynamical dilaton (see Appendix B) and
graviton modes emerging in the two setups would lead to their different
couplings with the SM confined to the IR brane. A more detailed inspection of
these aspects, though, is beyond the scope of the current exercise and would
be pursued later.

Figure 2: Graphical depiction of the warp factor $e^{2\sigma(z)}$ in the IR-
UV-IR (blue) and the UV-IR-UV (green) scenario.

###  4.3 Radius stabilisation

Modulus stabilisation in warped scenarios like the RS model typically involves
the addition of a bulk scalar field (the modulus field) whose VEVs at the
branes stabilise the size of the extra dimension $\grave{a}$ la the
Goldberger-Wise mechanism Goldberger:1999uk . In a linear dilaton model the
dilaton field itself can accomplish the role of a modulus field171717Even if
the dilaton field is not considered to be responsible for the stabilisation, a
GW-like mechanism can be consistently adopted for the purpose. See
Giudice:2016yja  for a discussion. Cox:2012ee . In this section we extend
those arguments to our triple brane model181818Stabilization of triple-brane
systems have been studied before in Choudhury:2000wc  in the context of the RS
scenario. by invoking additional potentials on the branes. These potentials
could, in principle, arise even as quantum corrections on the brane resulting
from the dilaton’s interactions with brane localised matter. Rather than posit
a complicated potential, we limit ourselves to the simplest choice, viz.

$\lambda_{\alpha}(S)=\frac{e^{-S/3}}{2}\left[\frac{\Lambda_{\alpha}}{M_{5}^{3}}+\mu_{\alpha}\left(S-\mathcal{V}_{\alpha}\right)^{2}\right],$
(47)

where, $\mu_{\alpha}$ and $\mathcal{V}_{\alpha}$ are constants. For this to be
consistent with the junction conditions in eq.41 we must have

$S(z_{\alpha})=\mathcal{V}_{\alpha}.$ (48)

In other words, the junction conditions impose boundary conditions on the
dilaton profile such that the dilaton field acquires nonzero vacuum
expectation values at the branes. This further ensures that the augmented
brane potentials do not generate any backreaction in the system.

Now, taking the general form of the solutions in eq.44 and imposing the
appropriate boundary conditions, viz.,

$S(z_{\alpha})=-k\pi R+S_{o}=\mathcal{V}_{\alpha}\ ,$ (49)

we obtain the size of the extra dimension in terms of the vacuum expectation
values of the dilaton field at the branes

$\pi
R=\frac{\left(\mathcal{V}_{2}-\mathcal{V}_{1}\right)}{k}\qquad\mbox{and}\qquad\pi
R=\frac{\left(\mathcal{V}_{2}-\mathcal{V}_{3}\right)}{k}\ .$ (50)

However, note that, together, the two conditions stipulate that the mechanism
is consistent only when the VEVs on the branes located at the orbifold
boundaries are equal. While this can be ensured by imposing an exact
$\mathbb{Z}_{2}$ symmetry in the full (bulk–brane) theory about $z=z_{p}$, it
would be interesting to find a mechanism or a more fundamental theory in which
such a symmetry would emerge naturally. An immediate discussion, though, is
beyond the scope of this work.

##  5 Bulk fields in clockwork/linear dilaton geometry

Having determined the configuration of the 5D theory which generates the
modified clockwork geometry, it is imperative to study the nature of bulk
field theories in the background specified by the metric in eq.45, so as to
ascertain and verify the correspondence drawn earlier between the discrete and
the 5D continuum theories. Although it is straightforward to generalise the
ensuing discussion to non-Abelian theories we choose to study only bulk
Abelian theories here in order to directly compare with the discrete clockwork
theories. As before, we limit the discussion to only bosonic fields. We note
that, in eq.35, the terms proportional to $M_{5}^{3}k^{2}$ are the ones that
primarily influence the geometry. In the following bulk field actions neither
do we assume any large mass parameters nor do we assume vacuum expectation
values for the scalar fields. Consequently, the backreaction due to these
fields can be entirely neglected.

###  5.1 Bulk scalar in LDG

Assuming the metric of eq.37—a valid assumption as long as the backreaction
due to the new field is negligible—the action for a scalar field $\phi$ is
given by

$\mathcal{S}=\frac{-1}{2}\int d^{4}x\,\int_{0}^{\pi
R}dz\,e^{3\sigma(z)}\left[(\partial_{\mu}\phi)^{2}+(\partial_{z}\phi)^{2}\right]$
(51)

which is invariant under the $\mathbb{Z}_{2}$ transformation ($z\to\pi R-z$)
with $z=z_{p}$ being a fixed point under this transformation191919This, of
course, is in addition to the orbifold fixed points.. The existence of this
symmetry stipulates that the solutions of the equation of motion should be
classifiable as representations of the corresponding parity operator
$\mathcal{Z}$, namely202020Note that we treat the fields $\phi^{(\pm)}(x,z)$
as two independent degrees of freedom which enables us to simultaneously
impose different sets of boundary conditions for the even and odd fields. This
follows from the fact that, in the two patches of the orbifold, the fields can
be deemed to be a priori disconnected (and, hence, independent) with support
only on the individual patches. Demanding continuous solutions to the equation
of motion, the $\mathbb{Z}_{2}$ symmetry of the system dictates the existence
of two (even and odd) fields $\phi^{(\pm)}(x,z)$ as linear combinations of the
piecewise fields. Hereinafter, we resort to discussing the physical properties
of the theory directly in terms of the even and odd bulk fields.

$\mathcal{Z}\phi^{(\pm)}(x,z)\equiv\phi^{(\pm)}(x,\pi
R-z)=\pm\phi^{(\pm)}(x,z)\ .$ (52)

Clearly, this is analogous to the $\mathbb{Z}_{2}$ parity in the discrete
case. The $U(1)_{CW}$ symmetry in the discrete theory—see eq.9— is reflected
by the invariance of the action under a constant shift of the bulk fields,
$\phi^{(\pm)}(x,z)\to\phi^{(\pm)}(x,z)+c$. Effecting the KK decompositions

$\phi^{(\pm)}(x,z)=\frac{1}{\sqrt{\pi
R}}\,\sum_{n=0}^{\infty}\,\varphi_{n}^{(\pm)}(x)f^{(\pm)}_{n}(z)\ ,$ (53)

with the obvious normalisation

$\frac{1}{\pi R}\int_{0}^{\pi
R}dz\,e^{3\sigma(z)}f^{(\pm)}{(n)}f^{(\pm)}{(n^{\prime})}=\delta^{nn^{\prime}},$
(54)

the equations of motion for $\varphi^{(\pm)}(x)$ and $f^{(\pm)}(z)$ are

$\begin{array}[]{rcl}\displaystyle\left(\partial_{x}^{2}-m_{n}^{2}\right)\varphi^{(\pm)}_{n}(x)&=&0\\\\[8.61108pt]
\displaystyle\left(\partial_{z}^{2}-k^{2}+m_{n}^{2}\right)\left(e^{3\sigma(z)/2}f^{(\pm)}_{n}(z)\right)&=&0\
.\end{array}$ (55)

At this stage, the masses $m_{n}$ for the KK modes $\phi_{n}$ are arbitrary,
and are to be determined only by solving for the eigenvalues of the operator
in the second equation. This, in turn, needs the boundary conditions (BC) on
the bulk field to be specified. With the $\mathbb{Z}_{2}$ stipulating that the
BC at $z=2z_{p}$ must be given by that at the origin, we choose to impose the
BCs at $z=0$ and $z=z_{p}$. Given that an individual BC could either be
Dirichlet-like ($\phi(x,z)=0$) or Neumann-like ($\partial_{z}\phi(x,z)=0$),
various combinations are possible, each with its particular set of
consequences, and we examine below the two broad classes:

  * •

Case I:  Imposing identical BCs at $z=0,z_{p}$, we have two further choices,
namely both Dirichlet-like (DD) or both Neumann-like (NN) Explicitly, we
obtain the following solutions:
Neumann BC at both $z=0\,\mbox{and}\,z_{p}$ (NN)
This admits only the even solutions, namely (here, $n\geq 1$)

$\begin{array}[]{rcl}\displaystyle
f_{0}^{+}(z)&=&\displaystyle\mathcal{N}_{0}\\\ m_{0}&=&0\\\
f_{n}^{+}(z)&=&\displaystyle\mathcal{N}_{n}^{+}e^{-\frac{3}{2}\sigma(z)}\left\\{-\frac{kR}{2n}\sin\left[\frac{2n}{R}|z-z_{p}|\right]+\cos\left[\frac{2n}{R}(z-z_{p})\right]\right\\}\\\
m_{n}^{2}&=&\displaystyle k^{2}+\frac{4n^{2}}{R^{2}}\ .\end{array}$ (56)

Dirichlet BC at $z=0\,\mbox{and}\,z_{p}$ (DD)
Understandably, these lead only to the odd solutions, namely

$f_{n}^{-}(z)=\mathcal{N}_{n}^{-}e^{-\frac{3}{2}\sigma(z)}\sin\left[\frac{2n}{R}(z-z_{p})\right]\
,\qquad m_{n}^{2}=k^{2}+\frac{4n^{2}}{R^{2}}\ .$ (57)

The normalisation factors are given by

$\mathcal{N}_{0}=\sqrt{\frac{k\pi R}{1-e^{-k\pi
R}}},\qquad\mathcal{N}^{+}_{n}=2\sqrt{2}\frac{n}{m_{n}R},\qquad\mathcal{N}^{-}_{n}=\sqrt{2}$
(58)

Thus, the physical spectrum consists of an exactly massless mode
($\varphi_{0}$) and, beyond a mass gap of magnitude $\sim k$, two mass-
degenerate KK-towers, with one each of $\mathbb{Z}_{2}$-even
($\varphi_{n}^{(+)}$) and odd ($\varphi_{n}^{(-)}$) mode at each level. In a
limited sense, this mimics the spectrum of the discrete theory (see eq.34).
Just as in the discrete case, the occurrence of a mass degeneracy can be
attributed to the $\mathbb{Z}_{2}$ symmetry of the action as well as to the
boundary conditions assumed. (For further elucidation of the mass degeneracy,
see Appendix A.1.) Note that, given the BCs, the equation of motion has no
non-trivial solution for $m^{2}<k^{2}$.

The mass eigenvalues are graphically shown in Fig.3 for a benchmark set of
values for the parameters $k$ and $R$. Evidently, the inter-level spacing
increases progressively with $n$ and then tends to saturate to $\Delta
m_{n}\sim 2\pi/R$ for large $n$. While this might seem at variance with the
discrete case, this was only to be expected. If the gravitational interaction
could be neglected, the five-dimensional theory could, notionally, be treated
as a free and, hence, UV-complete one with the masses defined by $k$ and
$R^{-1}$. The spectrum of the discrete theory, on the other hand, must depend
on both the lattice spacing $a$ and the UV-scale $f$. Yet, in the event of $p$
being large, an analogous behaviour is seen for $n\ll p$ with an opposite
behaviour (namely, the subsequent spacings decreasing) being seen for $n>p$
(see eq.34). In other words, the mass eigenvalues therein assume the form of
the KK spectrum when the RG evolution of the parameters, with respect to the
lattice spacing $a$, is taken into account in the sense indicated in eq.34.

Now that we have obtained the solutions, the theory in eq.51 can be rewritten
in terms of the even and odd bulk fields. However, this would tend to make the
action look cumbersome as one adds higher order terms. Since, all that we
expect after dimensional reduction is a 4D theory with the $\mathbb{Z}_{2}$
parity intact, it is simpler to consider the equivalent, namely a single bulk
field $\phi(x,z)$ and ascribe it with the expansion

$\phi(x,z)\equiv\frac{1}{\sqrt{\pi
R}}\sum_{n=0}^{\infty}\left\\{\varphi_{n}^{(+)}(x)f_{n}^{(+)}(z)+\varphi_{n}^{(-)}(x)f_{n}^{(-)}(z)\right\\}$
(59)

when the 4D theory is to be determined.

With this definition, we examine the coupling profile of the bulk sector with
an operator $\mathcal{O}_{E}$ confined to one of the branes, i.e., at
$z=z_{\alpha}\equiv\\{0,\pi R/2,\pi R\\}$. To this end, we introduce the
interaction action

$\begin{split}\mathcal{S}_{ext}=-\int
d^{4}x\,dz\,\frac{\sqrt{-g}}{\sqrt{g_{55}}}\exp\left(\frac{-5}{6}S(z)\right)\mathcal{O}_{E}\phi(x,z)\delta(z-z_{\alpha})\end{split}.$
(60)

The dilaton factor $e^{-5\,S(z)/6}$ has been introduced to induce localisation
of the $0$-th mode towards one of the fixed points. The numerical constant
($-5/6$) is fixed by demanding that the heavy KK modes are de-localised, i.e.,
the exponential factor in their wavefunctions is eliminated. This choice,
although not strictly necessary in the context of the 5D theory alone, is
adopted to mimic the coupling profile typical of the clockwork mechanism in
the corresponding discrete theory212121In general we can also introduce a
dilaton factor $e^{cS(z)}$ coupling to the overall Lagrangian in the bulk
action of eq.51, where $c$ is a constant which determines the coupling
strength. Such a construction has been studied in Kang:2020cxo  in the context
of the original CW/LD geometry. We assume $c=0$ in our modified theory only to
simplify matters in hand and generalisation with respect to dilaton couplings
would be an interesting aspect to explore.. Using eq.56-57, the above action
simplifies to

$\begin{array}[]{rcl}\displaystyle\mathcal{S}_{ext}&=&\displaystyle\frac{-1}{\sqrt{\pi
R}}\int
d^{4}x\,dz\,\frac{\sqrt{-g}}{\sqrt{g_{55}}}\,\exp\left(\frac{-5}{6}S(z)\right)\delta(z-z_{\alpha})\\\\[6.45831pt]
&&\displaystyle\hskip
60.00009pt\left\\{\sum_{n=0}^{\infty}f^{+(n)}\mathcal{O}_{E}\varphi^{+(n)}(x)+\sum_{n=0}^{\infty}f^{-(n)}\mathcal{O}_{E}\varphi^{-(n)}(x)\right\\}\\\\[12.91663pt]
&=&\displaystyle\frac{-1}{\sqrt{\pi R}}\int
d^{4}x\Bigg{\\{}\mathcal{N}_{0}e^{-k|z_{\alpha}-z_{p}|}\mathcal{O}_{E}\varphi^{+(0)}(x)\\\
&&\displaystyle\hskip
40.00006pt+\sum_{n=1}^{\infty}\mathcal{N}^{+}_{(n)}\mathcal{O}_{E}\varphi^{+(n)}(x)\\\\[8.61108pt]
&&\displaystyle\hskip
60.00009pt\left[-\frac{kR}{2n}\sin\left(\frac{2n}{R}|z_{\alpha}-z_{p}|\right)+\cos\left(\frac{2n}{R}(z_{\alpha}-z_{p})\right)\right]\Bigg{\\}}.\end{array}$
(61)

The fact that the odd wavefunctions vanish at the fixed points and that the
external operator is trivially invariant under the bulk $\mathbb{Z}_{2}$
tranformation ensure that the brane localised interactions preserve the
$\mathbb{Z}_{2}$ parity. Evidently, the coupling of the zero mode at the
boundary branes ($z_{\alpha}=0,\pi R$) is exponentially suppressed, whereas,
that at the central brane ($z_{\alpha}=z_{p}$) is $\sim\mathcal{O}(1)$. In
contrast, the massive modes have nearly similar couplings, of magnitude
$\sim\mathcal{O}(1)$, at every brane. This, clearly, is analogous to what we
obtain in the discrete case discussed in sec.3.1.

  * •

Case II: An interesting possibility is afforded by the imposition of an
asymmetric set of BCs, e.g., Neumann BC at $z=0$ and Dirichlet BC at $z=z_{p}$
or vice-versa. These BCs do not admit an exactly massless mode which is in
stark contrast with what we encounter in case I. Explicitly,

Neumann BC at $z=0$ and Dirichlet BC at $z=z_{p}$ (ND) leads to the odd
solutions

$f_{n}^{-}(z)=\tilde{\mathcal{N}}_{n}^{(-)}e^{-\frac{3}{2}\sigma(z)}\sin\left[\beta_{n}(z-z_{p})\right]$
(62)

where $\beta_{n}$ are the solutions to the trascendental equation

$\tan{\frac{\beta_{n}\pi R}{2}}=-\frac{\beta_{n}}{k}.$ (63)

and $m_{n}^{2}=k^{2}+\beta_{n}^{2}$. On the other hand,

Dirichlet BC at $z=0$ and Neumann BC at $z=z_{p}$ (DN), gives the even
solutions

$f_{n}^{+}(z)=\tilde{\mathcal{N}}_{n}^{(+)}e^{-\frac{3}{2}\sigma(z)}\left\\{-\frac{k}{\beta_{n}}\sin\left[\beta_{n}|z-z_{p}|\right]+\cos\left[\beta_{n}(z-z_{p})\right]\right\\}$
(64)

Once again, $\beta_{n}$ are given by a very similar equation, namely,

$\tan{\frac{\beta_{n}\pi R}{2}}=\frac{\beta_{n}}{k}.$ (65)

with $m_{n}^{2}=k^{2}+\beta_{n}^{2}$. That the (ND) and (DN) cases admit odd
and even solutions, respectively, is dictated by the boundary conditions and
the continuity conditions at $z=z_{p}$ in each case, along with the fact that
the theory is $\mathbb{Z}_{2}$ symmetric. Clearly, the KK modes are non-
degenerate with the even modes being heavier than the odd ones for a
particular KK level, and, hence, incommensurable with the discrete CW mass
spectrum in the sense shown222222It remains to be examined whether this
scenario has a different (non-CW) discrete analogy that could be derived as a
deconstruction of the 5D theory, presumably by invoking a different discrete
form for the $z$-derivatives than what we have adopted here. in eq.34. Similar
to case I, the inter-level spacings in the individual towers increase as we go
higher in the spectrum and tend to saturate for large values of $n$ for which
$\beta_{n}\to(2n-1)/R\gg k$. Therefore, any quasi-degeneracy seemingly present
in the low lying KK states gets disturbed in the heavier levels.

In addition to the KK spectrum, there exists a single non-trivial
$\mathbb{Z}_{2}$ even solution (satisfying the DN BCs) for $m^{2}<k^{2}$ given
by

$\begin{split}f_{0}^{+}(z)&=\tilde{\mathcal{N}}_{0}^{(+)}e^{-\frac{3}{2}\sigma(z)}\left\\{-\frac{k}{\tilde{\beta}}\sinh\left[\tilde{\beta}|z-z_{p}|\right]+\cosh\left[\tilde{\beta}(z-z_{p})\right]\right\\}\end{split}.$
(66)

where, $\tilde{\beta}\equiv\sqrt{k^{2}-m_{0}^{2}}$ is obtained as a solution
of the equation

$\tan{\frac{\tilde{\beta}\pi R}{2}}-\frac{\tilde{\beta}}{k}=0.$ (67)

This is unlike case I which admits a trivial massless mode. For values of
$|k\pi R|$ that potentially address the EW-Planck hierarchy, the preceding
equation has the solution $\tilde{\beta}\sim k$, with
$f_{0}^{+}(z)\to\tilde{\mathcal{N}}_{0}^{(+)}$, i.e., an almost flat profile
along the fifth dimension similar to what was obtained for the exactly
massless state in case I. In other words, the erstwhile flat (and, hence,
massless) solution acquires a slight non-triviality in its wavefunction and,
thereby, is very slightly lifted in mass. A part of the full spectrum is shown
schematically in Fig.3 for a specific set of values for $k$ and $R$.

We now turn to the question of the coupling with a brane localised operator.
If an interaction of the form shown in eq.60 is assumed, the nature of
coupling with the light mode remains quite similar to what we encountered in
case I for $|k\pi R|\gg 1$. The heavier KK modes, too, have similar
($\mathcal{O}(1)$) couplings at the branes as previously seen. The major
distinction, however, appears in the case of the odd modes in that they now
have vanishing couplings only at the central fixed point. Couplings with a
sector confined to one of the boundary branes, therefore, would break the
$\mathbb{Z}_{2}$ parity explicitly unless there exist couplings with a mirror
sector in the opposite brane as well.

Figure 3: Schematic representation of the first few KK levels in the case of
identical (left) and asymmetric (right) boundary conditions for a benchmark
parameter set.

At this juncture, an interesting distinction between the two sets of BCs is
worth mentioning. Note that the (NN) BC preserves the symmetry under a
constant shift of the 4D field $\varphi_{0}$, i.e., under $\varphi_{0}(x)+c$,
due to the $z$-independence of its wavefunction. This, in turn, justifies its
masslessness. The (DD) BC clearly breaks this symmetry which is why we do not
obtain an odd massless mode. The (DN) BC, too, breaks this symmetry for the
zeroth mode, thereby, leading to it being massive.

###  5.2 Bulk vector in LDG

The discussion of the bulk theory for a massless vector boson runs, to a large
extent, parallel to that for the scalar theory. The free field action is, of
course, given by

$\mathcal{S}=\int
d^{4}x\,dz\sqrt{-g}\left\\{-\frac{1}{4}g^{MM^{\prime}}g^{NN^{\prime}}F_{MN}F_{M^{\prime}N^{\prime}}\right\\}$
(68)

where $M=\\{\mu,z\\}$ etc. Choosing the generalized $R_{\xi}$ gauge-fixing
term

$\mathcal{S}_{GF}=\int
d^{4}x\,dz\sqrt{-g}e^{-4\sigma(z)}\left\\{-\frac{1}{2\xi}\left[\partial^{\mu}A_{\mu}+\xi
e^{-\sigma(z)}\partial_{z}\left(e^{\sigma(z)}A_{z}\right)\right]^{2}\right\\}$
(69)

allows us to eliminate all the $A_{\mu}-A_{z}$ mixing terms upto total
derivatives along the $z$ dimension. Defining the field decomposition and
normalisation condition

$A^{(\pm)}_{M}=\frac{1}{\sqrt{\pi
R}}\sum_{n=0}^{\infty}\mathcal{A}^{(\pm)}_{M\,n}(x)f^{(\pm)}_{n}(z),\qquad\frac{1}{\pi
R}\int
dz\,e^{\sigma(z)}f^{(\pm)}_{n}f^{(\pm)}_{n^{\prime}}=\delta_{nn^{\prime}}$
(70)

and dropping the $(\pm)$ superscripts for brevity, we obtain the following
equations of motion for $\mathcal{A}_{\mu}^{(n)}(x)$, $\mathcal{A}^{(n)}_{z}$
and $f^{(n)}(z)$ (for both even and odd modes),

$\begin{split}\partial_{\mu}\mathcal{F}^{\mu\nu(n)}+m_{n}^{2}\mathcal{A}^{\nu(n)}+\frac{1}{\xi}\partial^{\nu}\left(\partial^{\mu}\mathcal{A}_{\mu}^{(n)}\right)&=0\\\\[4.30554pt]
\left(\partial_{x}^{2}+\xi
m_{n}^{2}\right)\mathcal{A}_{z}^{(n)}(x)&=0\\\\[4.30554pt]
\left(\partial_{z}^{2}-\frac{1}{9}k^{2}+m_{n}^{2}\right)\left(e^{\sigma(z)/2}f^{(n)}(z)\right)&=0\
,\end{split}$ (71)

where the field strength tensors $\mathcal{F}^{(n)}$ correspond to the 4D
fields $\mathcal{A}_{\mu}^{(n)}$. The equation for $f(z)$ is similar to the
one for the bulk scalar with the only difference being in the coefficient of
the $k^{2}$ term. Following the terminology therein, the solutions for the
identical BC case are:

NN boundary conditions

$\begin{split}f^{+(0)}(z)&=\mathcal{N}_{0}^{(+)}\\\
f^{+(n)}(z)&=\mathcal{N}_{n}^{(+)}e^{-\sigma(z)/2}\left\\{-\frac{kR}{6n}\sin\left[\frac{2n}{R}|z-z_{p}|\right]+\cos\left[\frac{2n}{R}(z-z_{p})\right]\right\\}\\\
m_{0}^{2}&=0\\\ m_{n}^{2}&=\frac{1}{9}k^{2}+\frac{4n^{2}}{R^{2}}\end{split}$
(72)

DD boundary conditions

$f^{-(n)}(z)=\mathcal{N}_{n}^{(-)}e^{-\sigma(z)/2}\sin\left[\frac{2n}{R}(z-z_{p})\right]\
,\qquad m_{n}^{2}=\frac{1}{9}k^{2}+\frac{4n^{2}}{R^{2}}\ ,$ (73)

where, as before, $n\geq 1$ and $(\pm)$ denote $\mathbb{Z}_{2}$ even and odd
wavefunctions respectively. The normalisation factors, $\mathcal{N}$, are
given by

$\begin{split}\mathcal{N}_{0}=\sqrt{\frac{k\pi R}{3\left(1-e^{-\frac{1}{3}k\pi
R}\right)}},\qquad\mathcal{N}^{+}_{n}=2\sqrt{2}\frac{n}{m_{n}R},\qquad\mathcal{N}^{-}_{n}=\sqrt{2}\end{split}.$
(74)

As in the scalar case, and as expected, we have exactly one massless mode
$A^{(+)}_{0\mu}$ which is to be identified with the four-dimensional gauge
field. The 4D masses of the vector bosons can be attributed to the Goldstone-
like degrees of freedom $\mathcal{A}_{z}^{(n)}$ $(n\geq 1)$ which, in the
unitary gauge ($\xi\to\infty$), vanish from the effective 4D theory. In
addition, there remains a massless scalar mode in the spectrum. Therefore,
following the treatment in the scalar theory, the spectrum of even and odd KK
modes can be classified in terms of the $\mathbb{Z}_{2}$ parity,

$\mathcal{Z}\mathcal{A}_{\mu}^{(\pm)(n)}(x)=\pm\mathcal{A}_{\mu}^{(\pm)(n)}(x)\
,\qquad\mathcal{Z}\mathcal{A}_{z}^{(+)(0)}(x)=\mathcal{A}_{z}^{(+)(0)}(x)\ .$
(75)

In other words, the $\mathbb{Z}_{2}$ assignments as well as the degeneracy
within a level is exactly the same as that for the spectrum obtained in the
discrete scenario in sec.3.2. The coupling profile of the KK modes with a
brane localised operator can be seen by assuming that the operator is a
current ($\mathcal{J}_{M}=\\{\mathcal{J}_{\mu},\mathcal{O}_{S}\\}$) composed
of fields charged under the bulk $U(1)$ gauge group and, therefore, couples to
the bulk gauge field. The appropriate interaction is defined by

$\begin{split}\mathcal{S}_{ext}&=-\int
d^{4}x\,dz\,\frac{\sqrt{-g}}{\sqrt{g_{55}}}e^{-\frac{7}{6}S(z)}\mathcal{J}_{M}A^{M}(x,z)\delta(z-z_{\alpha}),\end{split}$
(76)

where, $\mathcal{J}_{\mu}$ is a vector current 232323Identifying the external
sector with the SM, this can be envisaged as the current associated with some
of the popular and well-motivated scenarios with an Abelian gauge group like
$U(1)_{B-L}$, $U(1)_{L_{\mu}-L_{\tau}}$, etc., or with a dark sector group
$U(1)_{D}$ with the light vector being a $Z^{\prime}$ or a dark photon-like
particle. The light scalar, on the other hand, can be identified as an axion-
like particle. Prospective models based on such bulk scenarios, therefore,
would simultaneously induce couplings of a $Z^{\prime}$ (or $A^{\prime}$) and
an ALP with the SM sector., $\mathcal{O}_{S}$ is a scalar operator, and,
similar to the scalar theory, we assume

$A_{M}(x,z)\equiv\frac{1}{\sqrt{\pi
R}}\sum_{n=0}^{\infty}\left\\{\mathcal{A}^{(+)}_{M\,n}(x)f_{n}^{(+)}(z)+\mathcal{A}^{(-)}_{M\,n}(x)f_{n}^{(-)}(z)\right\\}.$
(77)

With this expansion, the action simplifies to (in the unitary gauge)

$\begin{split}\mathcal{S}_{ext}&=-\frac{1}{\sqrt{\pi R}}\int
d^{4}x\,dz\,\frac{\sqrt{-g}}{\sqrt{g_{55}}}e^{-\frac{7}{6}S(z)}\Bigg{\\{}\sum_{n=0}^{\infty}f^{+(n)}\mathcal{J}_{\mu}\mathcal{A}^{\mu+}_{(n)}(x)+\sum_{n=0}^{\infty}f^{-(n)}\mathcal{J}_{\mu}\mathcal{A}^{\mu-}_{(n)}(x)\\\
&+f^{+(0)}\mathcal{O}_{S}\mathcal{A}^{(+)}_{z(0)}(x)\Bigg{\\}}\delta(z-z_{\alpha})\\\
&=-\frac{1}{\sqrt{\pi R}}\int
d^{4}x\Bigg{\\{}\mathcal{N}^{(+)}_{0}e^{-\frac{1}{3}k|z_{\alpha}-z_{p}|}\mathcal{J}_{\mu}\mathcal{A}^{\mu+}_{(0)}(x)+\mathcal{N}^{(+)}_{0}e^{-\frac{1}{3}k|z_{\alpha}-z_{p}|}\mathcal{O}_{S}\mathcal{A}^{(+)}_{z(0)}(x)\\\
&+\sum_{n=1}^{\infty}\mathcal{N}_{n}^{(+)}\Bigg{[}-\frac{kR}{6n}\sin\left[\frac{2n}{R}|z_{\alpha}-z_{p}|\right]+\cos\left[\frac{2n}{R}(z_{\alpha}-z_{p})\right]\Bigg{]}\mathcal{J}_{\mu}\mathcal{A}^{\mu+}_{(n)}(x)\Bigg{\\}},\end{split}$
(78)

Analogous to the scalar case, and going by the argument mentioned therein, the
dilaton factor $e^{-7S(z)/6}$ has been introduced to induce localisation of
the massless modes (both the vector and the scalar) alone towards one of the
branes. Further, from eq.72 \- 73, we see that the coupling profiles of the KK
modes at the branes is of the same nature as obtained for the scalar theory
and reflects the clockworking attribute of the corresponding discrete theory.

By analogy with the scalar case, again, it is straightforward to see that the
EOMs in eq.71 have solutions corresponding to asymmetric BCs as well with no
mass degeneracy. Of course, in this case, there exists a light vector mode
with a nonzero mass, with the corresponding light scalar acting as a Goldstone
boson which vanishes altogether in the unitary gauge. This can also be
understood by noting that the (DN) BC does not preserve the 4D gauge
invariance of the zeroth mode as opposed to that in the (NN) case.

##  6 Summary

The clockwork mechanism provides a novel way of generating hierarchical
couplings without resorting to large fine tunings. The fact that the original
class of clockwork theories also finds a correspondence with the 5D linear
dilaton model makes it a compelling avenue to be explored further. In this
paper, we substitute the original open ended construction by a closed chain
structure of the near-neighbour interactions. The resulting $\mathbb{Z}_{2}$
symmetry under field exchanges about the centre-most site gives rise to two
sets of physical states, one even and the other odd under an exchange parity.
The even massless state is localised towards one of the $\mathbb{Z}_{2}$ fixed
points (the pivot sites) reflecting the clockwork mechanism. Keeping this
symmetry intact then renders the lightest odd state absolutely stable and,
hence, a potential dark matter candidate.

This new class of theories can also be obtained as a deconstruction of a bulk
5D theory in a linear dilaton background invariant under a $\mathbb{Z}_{2}$
transformation about a fixed point identified with the centre of the compact
extra dimension. The discretization prescription, however, needs to be amended
so as to preserve the $\mathbb{Z}_{2}$ symmetry. The underlying
$\mathbb{Z}_{2}$ invariant warped background metric is obtained as the exact
(i.e., including backreaction) solution of an extended linear dilaton theory
on an $S^{1}/\mathbb{Z}_{2}$ orbifold with three equidistant 3-branes. The
theory, therefore, potentially provides two physically inequivalent setups,
namely, the IR-UV-IR and the UV-IR-UV scenarios, both admitting different, yet
viable solutions to the EW-Planck hierarchy problem.

Bulk field theories (whether scalar or vector) in this background, on
compactification, are characterised by a physical spectrum consiting of two KK
towers with states that are even and odd under the $\mathbb{Z}_{2}$ parity,
mimicking the notion of KK parity in typical UED models with a flat metric.
With identical boundary conditions at the orbifold fixed points (NN and DD),
the theory admits a massless mode and the two KK towers (beyond a mass gap
$\sim k$) turn out to be doubly degenerate at each level with mass eigenvalues
that match with those in the discrete theory in the large $N$ limit at leading
order in $(1/N)$. This scenario, therefore, elucidates the correspondence
between the discrete and the continuum clockwork theories. On the other hand,
with non-identical boundary conditions, we obtain a non-degenerate spectrum of
heavy KK states along with a light mode that has a small, albeit nonzero,
mass. While this picture does not immediately relate to the discrete theories
that we introduced, it is potentially interesting in its own right. In
particular, it demonstrates that the choice of boundary conditions is germane
to the aspect of establishing a correspondence between a discrete CW theory
and a bulk 5D theory in the linear dilaton background.

In short, this study demonstrates that the clockwork mechanism and its
correspondence with the linear dilaton model need not be associated with only
a specific topology of the discrete and continuum theories and that extensions
within this CW/LD framework are possible, like the one we have discussed in
this work which projects a richer phenomenology and, presumably, new and
interesting model building applications.

##  Appendix A Mass Degeneracy

###  A.1 Degeneracy in the continuum CW

Reexamining the equation of motion (eq.55) for the bulk modes of the scalar
theory, namely,

$\left(\partial_{z}^{2}-k^{2}+m_{n}^{2}\right)\left(e^{3\sigma(z)/2}f_{n}(z)\right)=0\
,$ (79)

we immediately see that, if $f_{n}(z)$ satisfies the equation, so does
$f^{\prime}_{n}(z)$ and for the same $m_{n}$. In other words, the two
wavefunctions, if admissible, are degenerate, and if they are $\mathbb{Z}_{2}$
eigenstates, they must correspond to opposite eigenvalues. It is easy to
verify that the odd (DD) solutions are indeed derivatives of the even (NN)
solutions, viz.

$f^{(-)}_{n}(z)=-m_{n}^{-1}\partial_{z}f^{(+)}_{n}(z).$ (80)

Naively, it might seem that further derivatives would produce more candidate
eigenstates for a given $m_{n}$. However, note that the boundary conditions
restrict the admissible solutions. As for the remaining, these can be written
as linear combinations of the (fundamental) solutions $f^{(\pm)}_{n}$ for each
KK level $n$ and, hence, are not independent. This is expected as the
solutions $f^{(\pm)}_{n}$ span the two-dimensional degenerate space that is
stipulated by the $\mathbb{Z}_{2}$ symmetry. Clearly, the same argument also
holds for the vector field theory.

In contrast, for the mixed BC case (DN and ND), the BCs at $z=0$ corresponding
to the even and odd modes of the same KK level $n$ are incompatible with a
relation like eq.80 which serves to explain the non-degeneracy encountered in
that case.

We end this discussion by reexamining the source of this degeneracy. Consider
any deformation along the $z$-dimension, e.g., a different value of the
parameter $k$ over a patch in the bulk (and, to ensure $\mathbb{Z}_{2}$
symmetry, a corresponding patch in the opposite region). This would introduce
continuity conditions on the derivatives of the solutions that, e.g., are
$\mathbb{Z}_{2}$-odd for even solutions. The mass eigenvalues would have a
dependence on these derivative conditions, thereby, breaking the degeneracy
between the even and the odd modes. Thus, it is the $\mathbb{Z}_{2}$ symmetry,
the boundary conditions and the universality of the parameters in the theory
that, together, facilitate the emergence of a degeneracy in the physical mass
spectrum.

###  A.2 Degeneracy in discrete CW

While the degeneracy in this case could be motivated from that for the
continuum, we examine the issue on its own. As previously mentioned, the
degeneracy in the spectrum emerges by virtue of the three characteristic
features of the theory, namely, the universality of the parameters along the
sites, the $\mathbb{Z}_{2}$ symmetry and the specific form of interactions
involving the pivot sites. It is, perhaps, easier to trace the consequences of
each of the properties individually in order to verify the preceding
assertion. To start with, the universal nature of the parameters essentially
leads to a (pseudo-)tridiagonal mass matrix with deviations only in the
$0$-th, $p$-th and $(2p-1)$-th rows (which, for brevity, we address as the
pivot rows) with these being a consequence of the $\phi_{0}-\phi_{2p-1}$ and
the $\phi_{p-1}-\phi_{p}$ mixings. The resulting eigenvalue equations, barring
those involving the pivot ones, are second order linear difference equations
of the form

$-qa_{j-1,n}+(1+q^{2}-\lambda_{n})a_{j,n}-qa_{j+1,n}=0,$ (81)

with equal coefficients for $a_{j-1,n}$ and $a_{j+1,n}$. Here, $\lambda_{n}$
denotes the $n$-th eigenvalue. This equation is nothing but a second-order
$q$-difference equation or, in other words, the discretized version of the
harmonic equation with ordinary derivatives replaced by $q$-derivatives.
Consequently, the solutions (away from the pivot points) are given by

$a_{nj}=\mathcal{N}_{n}\begin{cases}\sin{j\theta_{n}}+B\cos{j\theta_{n}},&j\leq
p\\\ C\sin{j\theta_{n}}+D\cos{j\theta_{n}},&j>p\end{cases},$ (82)

where, $B$, $C$ and $D$ are constant coefficients, $\mathcal{N}_{n}$ are the
normalisation factors and $\theta_{n}$ is an a priori undetermined function of
$n$. Reverting to the eigenvalue equation for the pivot rows, we have

$\begin{split}2q^{2}a_{n,0}-qa_{n,1}-qa_{n,N-1}&=\lambda_{n}a_{n,0}\\\
-qa_{n,p-1}+2a_{n,p}-qa_{n,p+1}&=\lambda_{n}a_{n,p}\ .\end{split}$ (83)

Furthermore, the eigenvalues have the form,

$\lambda_{n}=1+q^{2}-2q\cos{\theta_{n}}.$ (84)

Thus, in effect, the universal parametrisation fixes the form of the
eigensystem to the simple expressions given in eq.82 and 84.

Applealing, now, to the $\mathbb{Z}_{2}$ symmetry, it stipulates the existence
of even and odd solutions. Consequently, one obtains for the odd eigenvectors
$a^{-}_{k,p}=0$ which implies $B=-\tan{p\theta_{n}}$.

Finally, with the odd solutions satisfying the second equation in 83
trivially, we see that the first pivot equation determines the exact form of
the eigenvalues by specifying $\theta_{n}=n\pi/p$, or, alternatively,

$\lambda_{n}=1+q^{2}-2q\cos{\frac{n\pi}{p}},$ (85)

which clearly are the doubly-degenerate eigenvalues discussed before in
sec.3.1. The pivot equations, therefore, play the role of boundary conditions
about the $\mathbb{Z}_{2}$ fixed points as encountered in the continuum
picture. In summary, the preceding arguments establish that the structural
properties of the theory intricately conspire to give rise to the degenerate
mass spectrum and that a deformity in any of the three properties would break
the degeneracy.

##  Appendix B Dynamical dilaton modes and IR-UV-IR/UV-IR-UV

In sec.4.2 we had discussed briefly the emergence of two kinds of scenarios in
our triple brane theory, namely, the IR-UV-IR and the UV-IR-UV setups242424We
specify an IR brane as one where any localised theory, as viewed in the
effective 4D metric, would have its physical mass parameters set by the TeV
scale. Similarly a UV brane is one where a theory would have all its mass
parameters, ostensibly, near the Planck scale. Hence, for $k>0$ the IR branes
are situated at the boundaries of the orbifold with the UV brane located in
the middle, and the other way around for $k<0$., corresponding to $k>0$ and
$k<0$ respectively. Each of these provides possible explanations of the
hierarchy problem. Here, we delineate the stability of the two cases with
regards to the physicality of the dynamical modes generated therein.
Concentrating on scalars, the said modes can be identified by considering, at
the linear order, the scalar fluctuations252525Note that the vector
fluctuations of the metric, alongwith one combination of the two scalar
flucuations, disappear in the unitary gauge to resurface as the longitudinal
modes of the massive tensor modes. $\Psi(x,z)$ and $\Phi(x,z)$ in our CW
metric Kofman:2004tk , viz.

$ds^{2}=F(z)^{2}\left[(1+2\Psi(x,z))\eta_{\mu\nu}dx^{\mu}dx^{\nu}+(1+\Phi(x,z))\right],\qquad
F(z)\equiv e^{\sigma(z)}.$ (86)

Furthermore, the perturbation of the linear dilaton field $S$ in eq.35 is
denoted by $\delta S(x,z)$. On linearization of the equations of motion, the
aforementioned separation of the scalar mode from the others is encapsulated
in the constraint equations

$\begin{split}&\Phi-\frac{F}{F^{\prime}}\Psi^{\prime}-\frac{F}{9F^{\prime}}S^{\prime}\delta
S=0\\\ &\Phi=-2\Psi\end{split},$ (87)

where the primes denote $z$-derivatives. On using these, only one independent
bulk scalar remains, and its dynamical equation reduces to

$\left(\partial_{x}^{2}+\partial_{z}^{2}-k^{2}\right)\left[e^{3\sigma(z)/2}\Phi(x,z)\right]=0.$
(88)

On effecting the decomposition

$\Phi(x,z)=\sum_{n}Q_{n}(x)\tilde{\Phi}_{n}(z)\ ,$ (89)

equations of motion are decoupled, and

$\begin{split}&\left(\partial_{x}^{2}-m_{n}^{2}\right)Q_{n}(x)=0\\\
&\left(\partial_{z}^{2}-k^{2}+m_{n}^{2}\right)\left[e^{3\sigma(z)/2}\tilde{\Phi}(z)\right]=0.\end{split}$
(90)

Additionally, by integrating the linearised Einstein equations (including the
boundary terms) over a small interval about the fixed points, we obtain two
junction conditions (JC) at each brane for the perturbations. One of these is
equivalent to eq.87 when evaluated at the branes and, hence, imposes no new
constraints. The remaining condition, namely,

$\delta
S^{\prime}-S^{\prime}\Phi\Bigg{|}^{z_{\alpha}+\epsilon}_{z_{\alpha}-\epsilon}=6e^{\sigma(z)}\lambda_{\alpha}^{\prime\prime}(S)\delta
S\Bigg{|}_{z_{\alpha}},$ (91)

relates the perturbation $\delta S$ to the parameters at the branes. This, in
turn, can be substituted in eq.87 to obtain JCs operative directly on $\Phi$,
one at each fixed point262626Note that on account of the $\mathbb{Z}_{2}$
symmetric structure of the theory about the central fixed point, the junction
condition at one of the orbifold boundaries trivially fixes that on the
opposite boundary.. Here, $\lambda_{\alpha}(S)$ are the brane potentials, as
defined in eq.47 and the primes on $\lambda_{\alpha}$ denote differentiations
with respect to $S$. The JC at the $z=z_{p}$ brane generates the solutions

$\begin{array}[]{rcl}\displaystyle\tilde{\Phi}_{0}(z)&\propto&\displaystyle
e^{-3\sigma/2}\left\\{\sinh{\left[\tilde{\beta}|z-z_{p}|\right]}+\tilde{\gamma}\cosh{\left[\tilde{\beta}(z-z_{p})\right]}\right\\}\\\\[8.61108pt]
\displaystyle\tilde{\Phi}_{n>0}(z)&\propto&\displaystyle
e^{-3\sigma/2}\left\\{\sin{\left[\beta_{n}|z-z_{p}|\right]}+\gamma_{n}\cos{\left[\beta_{n}(z-z_{p})\right]}\right\\},\end{array}$
(92)

where $n=0$ corresponds to the radion mode and

$\begin{array}[]{rclcrcl}\displaystyle\tilde{\beta}^{2}&\equiv&\displaystyle
k^{2}-m_{0}^{2}&&\displaystyle\tilde{\gamma}&\equiv&\displaystyle\frac{3\tilde{\beta}\mu_{p}}{(k\mu_{p}-k^{2}+\tilde{\beta}^{2})}\\\\[8.61108pt]
\displaystyle\beta_{n}^{2}&\equiv&\displaystyle
m_{n}^{2}-k^{2}&&\gamma_{n}&\equiv&\displaystyle\frac{3\beta_{n}\mu_{p}}{(k\mu_{p}-k^{2}-\beta_{n}^{2})}\
.\end{array}$ (93)

Here, $\mu_{\alpha}$ are the mass parameters in the brane potentials
$\lambda_{\alpha}(S)$, as detailed in eq.47. Solving the equations emerging
from the JC at $z=0$, on the other hand, specifies the mass spectrum which, in
the limit $\epsilon_{\alpha}\equiv|k|/\mu_{\alpha}\ll 1$ and upto linear order
in $\epsilon_{\alpha}$, are given by

$\begin{array}[]{rcl}m_{0}^{2}&\simeq&\displaystyle\left\\{\begin{array}[]{rclcl}\displaystyle\frac{8k^{2}}{9}\left(1-\frac{1}{9}\left[(\epsilon_{0}-\epsilon_{p})+(\epsilon_{0}+\epsilon_{p})\coth{\frac{k\pi
R}{6}}\right]\right)&\simeq&\displaystyle\frac{8k^{2}}{9}\left(1-\frac{2\epsilon_{0}}{9}\right)&&k>0\\\\[12.91663pt]
\displaystyle\frac{8k^{2}}{9}\left(1-\frac{1}{9}\left[(\epsilon_{p}-\epsilon_{0})+(\epsilon_{0}+\epsilon_{p})\coth{\frac{k\pi
R}{6}}\right]\right)&\simeq&\displaystyle\frac{8k^{2}}{9}\left(1-\frac{2\epsilon_{p}}{9}\right)&&k<0\end{array}\right.\\\\[30.1388pt]
m_{n}^{2}&\simeq&\displaystyle
k^{2}+\frac{4n^{2}}{R^{2}}\left[1-\frac{12(4n^{2}+k^{2}R^{2})}{|k|\pi
R(36n^{2}+k^{2}R^{2})}(\epsilon_{0}+\epsilon_{p})\right]\ .\end{array}$ (94)

At the second order in the scalar perturbations the dynamical field, after
diagonalisation of the action, is specified by a combination of $\Phi$ and
$\delta S$, namely,

$\xi(x,z)\equiv\left(\frac{\Phi}{2}-\frac{\delta
S}{3}\right)=\sum_{n}Q_{n}(x)\tilde{\xi}_{n}(z),$ (95)

which satisfies the dynamical equation given in eq.LABEL:eqn:appeom.

The free 4D actions for the physical modes are, then, given by

$S_{n}=\mathcal{C}_{n}\int
d^{4}x\,Q_{n}\left(\partial_{x}^{2}-m_{n}^{2}\right)Q_{n}\,,$ (96)

where

$\mathcal{C}_{n}\equiv\frac{3M_{5}^{3}}{2}\int_{0}^{\pi
R}dz\,F^{-1}\left[\frac{9}{2S^{\prime
2}}\left(F^{2}\tilde{\Phi}_{n}\right)^{\prime
2}+\left(F^{2}\tilde{\Phi}_{n}\right)^{2}\right]>0.$ (97)

The positivity of $\mathcal{C}_{n}$ ensures that the theory is free of
negative kinetic terms (and, hence, ghosts) for either sign of $k$. It is easy
to ascertain that, for, $\epsilon_{\alpha}\ll 1$, the theory is free of
tachyonic modes (as exemplified by a negative mass-squared) as well. For large
values of $\epsilon_{\alpha}$, however, one needs to perform the stability
analysis numerically as closed-form analytical expressions for the KK masses
are not straightforward. Moreover, invoking non-minimalities, such as assuming
a Higgs-curvature interaction on a brane, could lead to a ghost radion field
depending on the value of the mixing parameter Cox:2012ee . In the spirit of
the RS gravity models Davoudiasl:2003zt ; George:2011sw , adding brane-
localised curvature terms could potentially introduce a ghost radion as well
and, hence, would constrain their coefficients. Thus, in such cases, the
stability of the two scenarios ought to be inspected again thoroughly. It
should also be borne in mind that a possible embedding of linear dilaton
models in a string theoretic setting Antoniadis:2011qw  offers a way to
ameliorate putative instabilities arising on account of negative tension
branes.

###### Acknowledgements.

S.M. acknowledges research Grant No. CRG/2018/004889 of the SERB, India.

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