# Majorana Neutrinos and Clockworked Yukawa Couplings contribution to non-
observation of the rare leptonic decay $l_{i}\rightarrow l_{j}\gamma$,
Clockwork Photon and Clockwork Graviton.

Gayatri Ghosh<EMAIL_ADDRESS>Department of Physics, Gauhati
University, Jalukbari, Assam-781015, India  Department of Physics, Pandit
Deendayal Upadhayay Mahavidyalaya, Karimganj, Assam-788720, India

###### Abstract

The clockwork is an extra-dimensional set-up for generating light particles
with exponentially suppressed or hierarchical couplings of light particles
with N massive states having comparable masses near the threshold scale of the
mechanism in theories which contain no small parameters at the fundamental
level. We explore the prospect of charged lepton flavour violation (cLFV) in a
clockwork framework which encompasses Dirac mass terms as well as Majorana
mass terms for the new clockwork fermions. We deive the masses of the non zero
clockwork Majorana masses, and new particles in a clockwork framework and for
their Yukawa couplings to the lepton doublets, in the framework where the
clockwork parameters are universal. When the new clockwork Majorana masses are
non zero, neutrino masses are generated as a result from the exchange of heavy
messenger particles such as right handed iso$-$singlet neutrinos or
iso$-$triplet scalar bosons known as the seesaw mechanism. In the case of non
zero clockwork Majorana masses, owing to the sizable effective Yukawa
couplings of the higher mass modes neutrino masses can only be made tiny by
conjecturing large Majorana mass in the teraelectron volt range for all the
clockwork gears. This is apparent from the constraints on the mass scale of
the clockwork fermions due to the non-observation of the rare cLFV decay
$\mu\rightarrow e\gamma$, $\tau\rightarrow\mu\gamma$, $\tau\rightarrow
e\gamma$. A general description of the clockwork mechanism valid for fermions,
gauge bosons, and gravitons is discussed here. This mechanism can be
implemented with a discrete set of new fields or, in its continuum version,
through an extra spatial dimension. In both cases the clockwork emerges as a
useful tool for model-building applications. Notably, the continuum clockwork
offers a solution to the Higgs naturalness problem, which turns out to be the
same as in linear dilaton duals of Little String Theory.

##  I Introduction

Beyond Standard Model Physics can be explained at low energy as non
renormalisable field operators involving Standard Model fields. There is a
great syndicate between the interaction scales and mass scales like the
Weinberg operator which may be generated via two right handed neutrinos
oppositely charged under $U(1)^{{}^{\prime}}$. Interaction scales and mass
scales are totally distinct quantities. When measured in natural units, the
difference betweeen mass and scales has little fall out if the couplings are
of the order 1. But the dynamics gets totally changed in presence of tiny
yukawa couplings q ; w ; e  which are small in natural units since now the
interaction scales enter low scale energies. In this work we discuss lepton
flavor violation in a generalisation of clockwork models where large effective
interaction scales or full UV completion enters at energies exponentially
smaller than intimated by a given interaction strength .

Neutrino’s tiny mass is one of the most challenging open questions in
Fundamental Physics and points to new laws of particle physics and reshape
theories of cosmology. One probable solution to this enigma is provided by the
seesaw mechanism, in which the smallness of neutrino masses is unravelled by
the violation of the lepton number at a very high energy scale by two units
$\Delta_{L}=2$ a ; b ; c ; d . Tiny yukawa couplings of the neutrino on the
other hand, couples to the Standard Model Higgs in models with lepton number
conservation. Occurrence of tiny Yukawa couplings is a phenomenologically
feasible possibility, and can be worked out in further extensions of the
model. A new mechanism of producing small couplings in theories coupled to the
Standard Model has been introduced q ; w ; e ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14
; 15 . The implicative mechanism, of deconstruction models, is a linear model
with no large hierarchies in the site dependent field theory parameters, which
essentially arouses site-dependent suppressed couplings to the zero$-$mode 16
. Originally, the clockwork theory was proposed for a quiver of axions, it is
now generalized to spin-0 scalar, spin-$\frac{1}{2}$ fermion, spin-1 boson, or
spin-2 graviton and other fields 17 . A clockwork fermion elucidates a very
small Dirac neutrino mass or explain the hierarchical pattern of quark and
lepton masses. With a clockwork gauge boson the presence of tiny gauge charges
is justified. Clockwork graviton, solves the naturalness problem of the
electroweak scale, syndicating an effective explanation for the weakness of
gravity. Independently of the specific implementation of the clockwork, the
theory proposes the existence of $N$ particles at the univesal mass scale $m$,
which are known as the ‘clockwork gears’, as they are the degrees of freedom.
The gears have universal properties and their exploration of the clockwork
will also lead us to consider the limit $N$ tends to infinity eventually the
continuum limit, in which the 1$-$D space lattice in field space is elucidated
as a physical spatial dimension. Some studies on the applications and
generalizations of clockwork mechanism is done 18 ; 19 ; 20 ; 21 ; 22 ; 23 ;
24 ; 25 ; 26 ; 27 ; 28 ; 29 ; 30 ; 31 ; 32 ; 33 ; 34 .

In this work we explore the impact of Majorana neutrinos in case of charged
lepton flavour violation in the fermionic clockwork model and the constraints
on the masses of the clockwork gears. Concretely, we identify the masses of
the clockwork gears with the zero modes of a clockwork sector, which fits the
neutrino masses and charged lepton masses, such that tiny yukawa couplings are
naturally generated and therefore tiny neutrino masses. We establish the
clockwork framework for the right handed neutrinos by counting Majorana mass
terms. We show how in the clockwork theory the suppression of the Yukawa
couplings by field dependent exponential factors, is not altered by the
manifestation of the Majorana mass terms. In fact, the combination of the
clockwork “suppression” and the Majorana “seesaw” sets now the mass
constraints on the clockwork gears. The clockwork mechanism suppresses the the
zero mode couplings, the yukawa couplings of the higher modes induce, via
loops, potentially large sizable rates for the leptonic rare decays. The rest
of the paper is organized as follows. In section 2, we present the most
general phenomenology for clockwork neutrinos with Dirac and Majorana mass
terms. In section 3, we discuss the impact of Majorana neutrinos on lepton
flavour violation in the clockwork scenario and calculate constraints on the
clockwork gear masses. We close with a summary.

##  II Clockwork Fermion

Fermions may be massless due to a chiral symmetry. Fermion realisation of the
clockwork model requires a single chiral symmetry which is splitted amongst a
number of field site in the underlying model. The left out chiral symmetry
drags the massless fermion exponentially to one edge of the clockwork. To this
climax, one introduces $N+1$ chiral fermions $\psi_{Rj}(j=0,...,N)$ together
with N fermions $\psi_{Li}(i=0,...,N-1)$ of opposite chirality. The global
chiral symmetry is spontanously broken down by N mass parameters $m_{i}$ which
pair up the fields in N massive Dirac fermions, leaving a single massless
chiral component or zero mode. Let us signify $U(1)_{Rj}$ and $U(1)_{Li}$ the
Abelian factors under which $N+1$ chiral fermions $\psi_{Rj}$ and N fermions
$\psi_{Li}(i=0,...,N-1)$ have charge 1, respectively. Then $m_{j}$ get charges
$(1,-1)$ under $U(1)_{Lj}\times U(1)_{Rj}$ , and $mq_{j}$ get charges $(1,-1)$
under $U(1)_{Lj}\times U(1)_{Rj}$ . One abelian factor of the chiral symmetry
or the zero mode in the $\psi_{R}$ sector remains unbroken by both $m$ and
$mq$.

Under the Standard Model gauge group, we expand the Standard Model with $n$
left-handed and $n+1$ right-handed chiral fermions, singlets, that we
represent as $\psi_{Li}(i=0,...,n-1)$ and $\psi_{Ri}(i=0,...,n)$ respectively.
The Lagrangian of the model interprets as:

$\textit{L}=\textit{L}_{SM}+\textit{L}_{CLOCKWORK}+\textit{L}_{INT}$ (1)

. Here $\textit{L}_{SM}$ is the Standard Model Lagrangian,
$\textit{L}_{CLOCKWORK}$ is the part of the Lagrangian involving only the new
right-handed chiral fermions, singlets, and $\textit{L}_{INT}$ is the
interaction term of the new fields, right-handed chiral fermions, singlets
with the Standard Model fields. The Standard Model fields only share coupling
to the last one of the fermionic clockwork, right-handed chiral fermion.

$\textit{L}_{INT}=-Y\tilde{H}\bar{L_{l}}\psi_{Rn}$ (2)

, where $\tilde{H}=i\tau_{2}H^{*}$. H is the Standard Model Higgs doublet and
$L_{l}$ is the Standard Model left handed leptonic fields.

The Lagrangian for the fermion fields is

$\textit{L}=\textit{L}_{kin}-\sum^{n-1}_{j=0}(m_{i}\bar{\psi}_{\textit{Lj}}{\psi}_{\textit{Rj}}-m_{i}^{{}^{\prime}}\bar{\psi}_{\textit{Lj}}{\psi}_{\textit{Rj+1}}+h.c)-\sum^{n-1}_{j=0}(\frac{1}{2}M_{Li}\bar{\psi^{c}_{Li}}\psi_{Li})-\sum^{n}_{j=0}(\frac{1}{2}M_{Ri}\bar{\psi^{c}_{Ri}}\psi_{Ri})\equiv\textit{L}_{kin}-(\bar{\psi}_{\textit{L}}M_{{\psi}}{\psi}_{\textit{R}}+h.c)$
(3)

where $\textit{L}_{kin}$ is the kinetic term for all fermions. We take
universal values for $m$ and $q$. $m$ is real with a chiral rotation of the
fermions, the parameters q are in general complex, but we choose them as real
for simplicity. $M$ is a ($2n$ \+ 1) × ($2n$ \+ 1) mass matrix.
$\textit{L}_{kin}$ is symmetric with respect to the global gauge group
$U(n)_{L}\times U(n+1)_{R}$. The mass terms $m_{i}$ break the global symmetry
$U(n)_{L}\times U(n+1)_{R}\rightarrow\prod^{n-1}_{i=0}U(1)_{i}$, where
$U(1)_{i}$ acts as $\psi_{L,i}\rightarrow e^{i\beta_{i}}\psi_{L,i}$,
$\psi_{L,i}\rightarrow e^{i\beta_{i}}\psi_{L,i}$, which when combined with the
mass terms $m_{i}$, automatically breaks the global symmetry $U(n)_{L}\times
U(n+1)_{R}\rightarrow U(1)_{CW}$, where under $U(1)_{CW}$ clockwork fermion
fields tansform as $\psi_{L,i}\rightarrow e^{i\beta}\psi_{L,i}$,
$\psi_{L,i}\rightarrow e^{i\beta}\psi_{L,i}$ for all i. $M_{Li}$ and $M_{Ri}$
are the Majorana masses for the left and right handed singlet fields. For
simplicity we assume universal Dirac masses, Majorana masses and nearest
neighbor interactions as $m_{i}=m,m_{i}^{{}^{\prime}}=mq$,
$M_{Li}=M_{Ri}=m\tilde{q}$ for all i. This conjecture leads to the mass
matrix:

$M_{\psi}=m\begin{pmatrix}\tilde{q}&0&...&0&1&-q&..&0\\\
0&\tilde{q}&...&0&0&1&...&0\\\ &...........................................\\\
0&0&........&\tilde{q}&0&0&0&-q\\\ 1&0&..........&0&\tilde{q}&0&...&0\\\
-\tilde{q}&1&...&0&0&\tilde{q}&......&0\\\
&..............................................\\\
&...............................................\\\
0&0&.......&-\tilde{q}&0&0&0&\tilde{q}\\\ \end{pmatrix}.$ (4)

, The eigen values of the mass matrix are:

$M_{0}=m\tilde{q}$ (5)

,

$M_{k}=m\tilde{q}-m\sqrt{\kappa_{r}},\hskip 2.84544ptr=1......n$ (6)

,

$M_{n+k}=m\tilde{q}+m\sqrt{\kappa_{r}},\hskip 2.84544ptr=1......n$ (7)

, where $\kappa_{r}$ is defined as

$\kappa_{r}=q^{2}+1-2qcos\frac{r\pi}{n+1}$ (8)

. Some studies on different aspects of Lepton Flavour Violation has been
extensively done in GG . Also $A_{4}$ symmetry which gives the corrections to
the TBM form for the leading order neutrino mixing matrix is studied in GG1 .
The predictions of vanishing $\theta_{13}$ by TBM is owing to its invariance
under $\mu-\tau$ exchange symmetry GG . Small explicit breaking of $\mu-\tau$
symmetry can generate large Dirac CP violating phase in the context of
neutrino oscillations GC .

##  III Lepton Flavour Violation in case of $M_{Li},M_{Ri}\neq 0$

The Yukawa couplings for the zero mode of the clockwork gears are suppressed,
thereby explains why the neutrino masses are small, but the Yukawa couplings
for the higher modes of the clockwork gears are unsuppressed which leads to
observable effects like lepton flavor violation at low energies. We find in
this work that if the clockwork gear masses are low then signatures of lepton
flavor violation existing in the Yukawa couplings of the higher modes through
quantum effects generates rare leptonic decays such as$l_{i}\textrightarrow
l_{j}+\gamma$ or $\mu-e$ conversion processes in nuclei, induced by clockwork
fermions, with decay rates which could be accessible at next run of LHC.

We calculate the rate $l_{i}\rightarrow l_{j}+\gamma$ from a . For N clockwork
generations, we have

$B\left(\mu\rightarrow
e\gamma\right)\simeq\frac{3\alpha_{em}\upsilon^{4}}{8\pi}|\sum^{N}_{\alpha=1}\sum^{n_{\alpha}}_{k=1}\frac{Y_{k}^{e\alpha}Y_{k}^{\mu\alpha}}{M_{k}^{\alpha^{2}}}F\left(x_{k}^{\alpha}\right)|^{2}$
(9)

where $\alpha_{em}$ signifies the fine stucture constant. $n_{\alpha}$is the
number of clockwork gears in the $\alpha$ $-$th generation. $M_{k}^{\alpha}$
is the mass of the $k-$th mode in the $\alpha$ $-$th generation
$\left(k=1,...,n_{\alpha}\right)$. Here

$\displaystyle x_{k}^{\alpha}\equiv\frac{M_{k}^{\alpha^{2}}}{M_{W}^{2}}$ (10)

We define the loop function F(x) as

$F(x)\equiv\frac{1}{6\left(1-x\right)^{4}}\left(10-43x+78x^{2}-49x^{3}+4x^{4}-18x^{3}logx\right)$
(11)

, which has limits $F(0)=\frac{5}{3}$ and $F(\propto)=\frac{2}{3}$

For N clockwork generations of clockwork gears we calculate the rate for
$l_{i}\rightarrow l_{j}+\gamma$ from 1 ; 2 ; 3 .

Figure 1: Predicted value of $BR(\mu\rightarrow e+\gamma)$ for points of the
parameter space reproducing the observed neutrino oscillation parameters, as a
function of the mass of the first clockwork gear and the Yukawas satisfying
the charged lepton masses. Figure 2: Predicted value of $BR(\mu\rightarrow
e+\gamma)$ for points of the parameter space reproducing the observed neutrino
oscillation parameters as a function of the mass of the first clockwork gear
and the Yukawas satisfying the charged lepton masses. It follows from the
figure that the clockwork gears must be larger than $\sim 10$ TeV in order to
evade the experimental constraints, unless very fine cancellations occurs
among all contributions to the process $BR(\mu\rightarrow e+\gamma)$.

The present stringent upper bound $BR(\mu\rightarrow e+\gamma)\leq$ $4.2\times
10^{-13}$ from the MEG experiment poses firm constraints on the mass scale of
the clockwork gears. In Fig. 1, 2, 3, 4, 5, 6, 7 we present the branching
ratio expected for points reproducing the measured neutrino masses as well as
charged lepton masses, assuming two clockwork generations, as earned in the
figures presented in section 3, as a function of the mass of the first
clockwork gear. It is observed from Fig.1, 2 that if the yukawa modes
$Y_{\mu\alpha}^{\nu}$ are of the order $0$ to $6$, then the clockwork gears
must be larger than $10$ TeV then the clockwork gears must be larger than $10$
TeV in order to elude the experimental constraints on $BR(\mu\rightarrow
e+\gamma)$ as apparent from Fig.1, 2, except very fine cancellations occurs
among all contributions to this process. Similarly for observation of
$BR(\tau\rightarrow e+\gamma)$ for a large number of clockwork generations we
look forward to even more restricted lower limits on the lightest gear mass,
due to more number of particles in the loop. As evident from Fig.3, 4, it
follows from the figure that the clockwork gears must be larger than $\sim
0.2$ TeV and must be less than 1 TeV for Yukawas corresponding to 0.5 as
restricted by the experimental constraints on $BR(\tau\rightarrow
e+\gamma)\leq 3.3\times 10^{-8}$, unless very fine cancellations occurs among
all contributions to the process $BR(\tau\rightarrow e+\gamma)$. Also for
Yukawas resembling the values around 2.42338 the values of the mass of the
first clockwork gear must be less than 2.5 TeV and greater than 0.2 TeV, if
the yukawa coupling modes $Y_{\tau\alpha}^{\nu}$ are of the order $0.4$ to
$2.4238$. It follows from Fig.5, that the clockwork gears must be larger than
$2$ TeV and must be less than 5 TeV as restricted by the experimental
constraints $BR(\tau\rightarrow\mu+\gamma)\leq 4.4\times 10^{-8}$,for Yukawa
$Y_{\nu}^{\tau\alpha}$ corresponding to 1 unless very fine cancellations
occurs among all contributions to the process $BR(\tau\rightarrow\mu+\gamma)$
and the clockwork gears must be larger than $\sim 2$ TeV and must be less than
$40$ TeV restricted by the experimental constraints
$BR(\tau\rightarrow\mu+\gamma)\leq 4.4\times 10^{-8}$, for Yukawa
$Y_{\nu}^{\tau\alpha}$ corresponding to 7.67. It is found from Fig.7 that the
clockwork gears must be larger than $\sim 90$ TeV and must be less than 200
TeV as constained by the experimental upper bound on $BR(\tau\rightarrow
e+\gamma)\leq 4.4\times 10^{-8}$,for Yukawa $Y_{\nu}^{\tau\alpha}$
corresponding to $1$ to $2$ unless very fine cancellations occurs among all
contributions to the process $BR(\tau\rightarrow\mu+\gamma)$ and the clockwork
gears must be greatee than $\sim 90$ TeV for Yukawa $Y_{\nu}^{\tau\alpha}$
corresponding to $2$ to $7.67$ as constrained by experimental limit.

Figure 3: Predicted value of $BR(\tau\rightarrow e+\gamma)$ for points of the
parameter space reproducing the observed neutrino oscillation parameters, as a
function of the mass of the first clockwork gear. It follows from the figure
that the clockwork gears must be larger than $\sim 0.2$ TeV and must be less
than 1 TeV for Yukawas corresponding to 0.5 as restricted by the experimental
constraints on $BR(\tau\rightarrow e+\gamma)\leq 3.3\times 10^{-8}$, unless
very fine cancellations occurs among all contributions to the process
$BR(\tau\rightarrow e+\gamma)$. Also for Yukawas resembling the values around
2.42338 the values of the mass of the first clockwork gear must be less than
2.5 TeV and greater than 0.2 TeV.

Figure 4: Predicted value of $BR(\tau\rightarrow e+\gamma)$ for points of the
parameter space reproducing the observed neutrino oscillation parameters, as a
function of the mass of the first clockwork gear and the Yukawas satisfying
the charged lepton masses. Figure 5: Predicted value of
$BR(\tau\rightarrow\mu+\gamma)$ for points of the parameter space reproducing
the observed neutrino oscillation parameters, as a function of the mass of the
first clockwork gear. It follows from the figure that the clockwork gears must
be larger than $2$ TeV and must be less than 5 TeV restricted by the
experimental constraints $BR(\tau\rightarrow\mu+\gamma)\leq 4.4\times
10^{-8}$,for Yukawa $Y_{\nu}^{\tau\alpha}$ corresponding to 1 unless very fine
cancellations occurs among all contributions to the process
$BR(\tau\rightarrow\mu+\gamma)$ and the clockwork gears must be larger than
$\sim 2$ TeV and must be less than $40$ TeV restricted by the experimental
constraints $BR(\tau\rightarrow\mu+\gamma)\leq 4.4\times 10^{-8}$,for Yukawa
$Y_{\nu}^{\tau\alpha}$ corresponding to 7.67. Figure 6: Predicted value of
$BR(\tau\rightarrow\mu+\gamma)$ for points of the parameter space reproducing
the observed neutrino oscillation parameters, as a function of the mass of the
first clockwork gear. Figure 7: Predicted value of
$BR(\tau\rightarrow\mu+\gamma)$ for points of the parameter space reproducing
the observed neutrino oscillation parameters, as a function of the mass of the
first clockwork gear. It follows from the figure that the clockwork gears must
be larger than $\sim 90$ TeV and must be less than 200 TeV as constained by
the experimental upper bound on $BR(\tau\rightarrow e+\gamma)\leq 4.4\times
10^{-8}$,for Yukawa $Y_{\nu}^{\tau\alpha}$ corresponding to $1$ to $2$ unless
very fine cancellations occurs among all contributions to the process
$BR(\tau\rightarrow\mu+\gamma)$ and the clockwork gears must be greatee than
$\sim 90$ TeV for Yukawa $Y_{\nu}^{\tau\alpha}$ corresponding to $2$ to $7.67$
as constrained by experimental limit.

##  IV Clockwork Photon

A clockwork photon can be acquired by clockworking the invariance of gauge
symmetry. Let us take $N+1$ $U(1)$ gauge groups with same gauge coupling $g$,
and N complex scalars, $\psi_{j}$, where $j=0,......,N-1$ having charge
$\left(1,-q\right)$. The underlying gauge symmetry is $U(1)_{j}\times
U(1)_{j+1}$. Given these scalars acquire a negative mass$-$squared, which
generates vacuum expectation values which are at the same scale $\lambda$ .
The Lagrangian is

$\textit{L}=-\sum^{N}_{j=0}\frac{1}{4}F_{\alpha\beta}^{j}F^{j\alpha\beta}-\sum^{N-1}_{j=0}[|D_{\alpha}\psi_{j}|^{2}+\lambda(|\psi_{j}|^{2}-\frac{1}{2}\lambda^{2})]$
(12)

where

$D_{\alpha}\psi_{j}=[\delta_{\alpha}-ig(A_{\alpha}^{j}-qA_{\alpha}^{j+1})]\psi_{j}$
(13)

The spontaneous symmetry breaking is of the form $U(1)^{N+1}\rightarrow U(1)$

$\textit{L}=-\sum^{N}_{j=0}\frac{1}{4}F_{\alpha\beta}^{j}F^{j\alpha\beta}-\sum^{N-1}_{j=0}\frac{g^{2}f^{2}}{2}(A_{\alpha}^{j}-qA_{\alpha}^{j+1})^{2}$
(14)

The above implicit Lagrangian is below the scale $\lambda$ while working in
unitary gauge field. The mass matrix is same as of the clockwork form, where
the heavy gauge bosons are the photon gear $A_{\alpha}$ and one photon
remaining massless. The clockwork photon has important implications. When
matter gets charge only under the Abelian factor identical to the last site of
the clockwork gear, then the clockwork phenomenon will produce exponentially
small couplings to the massless photon which generates visible particles with
millicharges. Heavy photon gears could be attainable in next run of collider
searches.

##  V Clockwork Graviton

For $N+1$ copies of general relativity, there are $N+1$ analogous massless
gravitons. Each graviton can be descrribed through an extension of the metric
around flat space$-$time,

$g^{\mu\nu}_{i}=\varepsilon^{\mu\nu}_{i}+\frac{2}{M^{2}_{i}}h^{\mu\nu}_{i}$
(15)

The clockwork mechanism breaks $N+1$ copies of diffeomorphism invariance to
one diffeomorphism invariance at the linear level through near-neighbour
Pauli-Fierz term attire for massive gravitons.

$\textit{L}=-\frac{m^{2}}{2}\sum^{N-1}_{i=0}([h^{\mu\nu}_{i}-qh^{\mu\nu}_{i+1}]-[\varepsilon_{\mu\nu}(h^{\mu\nu}_{i}-qh^{\mu\nu}_{i+1})]^{2})$
(16)

The massive lagrangian in Eq.(16) is invariant under the gauge symmetry

$h^{\mu\nu}_{i}\rightarrow
h^{\mu\nu}_{i}+\frac{1}{q^{i}}(\delta^{\mu}A^{\nu}+\delta^{\nu}A^{\mu})$ (17)

where $A^{\mu}$ is a space-time vector. This gauge symmetry imposes the
masslessness of the clockwork graviton and is maintained by the clockwork form
of the mass terms. The mass matrix is of the clockwork structure, with the
massive gravitons being the gears and one remaining as zero mass graviton. If
the energy$-$momentum tensor $T^{\mu\nu}$ couples only to the last site of the
clockwork gear at Planck alike scale $M_{N}$, then the coupling to the zero
mass graviton is,

$-\frac{1}{M_{N}}h^{\mu\nu}_{N}T_{\mu\nu}\rightarrow-\frac{1}{M_{P}}\tilde{h}^{\mu\nu}_{0}T_{\mu\nu},$
(18)

where, $M_{P}=\frac{q^{N}M^{N}}{N_{0}}$. The effective Planck scale $M_{P}$,
that realises the strength of gravity in the low-energy scale of the theory,
is exponentially larger than $M_{N}$ amplified by a factor $q_{N}$ . This
provides the clockwork mechanism a result to the hierarchy problem in where
new physics, like quantum gravity, sets near the weak scale, in total
resemblance as afforded by Large Extra Dimensions x  and the warped extra
dimensional model of Randall$-$Sundrum y . The metric of the clockwork
space$-$time is similar to a 5D metric known as the linear dilaton model which
resembles the dual of Little String Theory z , after compressing additional
dimensions. This counts for a signature of hint towards an untrodden
connection between string theory and the continuum clockwork theory, as it
distinguishes the clockwork solution of the hierarchy nature with the Little
String Theory solution.

##  VI Conclusion

The clockwork is a significant mechanism for promoting exponentially
suppressed interactions within a quantum field theory containing only $O(1)$
parameters, in natural units, and a definite number of clockwork fields. As a
result, one can stir up exponentially large interaction scales, even when no
new physics materializes at this high energy scale. Clockwork holds in models
of inflation or relaxation to prompt apparent super Planckian field
excursions.

We have scutinised in detail the impact of the Majorana masses in the
clockwork sector in the generation of lepton favour violation. When the
Majorana masses are non-vanishing, the zero mode of the clockwork gear is no
longer massless. But, the corresponding Yukawa coupling still inspite of that
has the clockwork structure. In this case, small neutrino masses arises of the
interaction between the standard seesaw mechanism and the “clockworked” Yukawa
couplings, and needs very large Majorana masses to generate the small neutrino
mass scale inferred as concluded from oscillation experiments. The Standard
Model leptons couple to the fermions of the clockwork sector with a site
dependent field strength, which prompts lepton flavour violating charged
current, neutral current and Higgs boson interactions. From the non findings
of the rare leptonic decay scale of the clockwork fermions due to the non-
observation of the rare cLFV decay $\mu\rightarrow e\gamma$,
$\tau\rightarrow\mu\gamma$, $\tau\rightarrow e\gamma$ our results stir up that
the lightest particle of the clockwork sector must have a mass greater than 10
TeV, 0.2 TeV to 2.5 TeV and 2 TeV to 40 TeV respectively if the Yukawa
couplings of exponentially suppressed interactions within a quantum field
theory contains only $O(1)$ parameters.

##  VII Acknowledgement

GG would like to thank would like to thank University Grants Commission RUSA,
MHRD, Government of India for financial support. GG would also like to thank
Prof. Sudhir Vempati for useful discussion on this topic.

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