# Flavour violating charged lepton decays in Little Randall-Sundrum Model

Akshay A<EMAIL_ADDRESS>Mathew Thomas Arun
<EMAIL_ADDRESS>School of Physics, Indian Institute of Science
Education and Research, Thiruvananthapuram, India

###### Abstract

The Little Randall-Sundrum (Little RS) model receives significantly stronger
constraints from the flavour observables in comparison to Randall-Sundrum (RS)
model. In this paper, we analyse the effect of the electro-weak sector in
Little RS on flavour changing decays of charged leptons. We compare the
predictions of the model with the current limits on the flavour violating
Branching Ratios of $\mu\rightarrow eee$, $\tau\rightarrow eee$,
$\tau\rightarrow\mu\mu\mu$, $\tau\rightarrow\mu ee$, $\tau\rightarrow e\mu\mu$
, $\mu\rightarrow e\gamma$, and $\mu Ti\rightarrow eTi$. And we show that the
dominant constraint arises from the $\mu Ti\rightarrow eTi$ process which
strongly limits the KK-1 gauge boson mass ($M_{KK}$) to be $\gtrsim 30.7TeV$.
We then derive and show that generalising the electro-weak gauge sector to
include the Brane Localised Kinetic Term (BLKT) relaxes this constraint to
$\gtrsim 12TeV$. Towards the conclusion, we comment on the possibility that
the large flavour violating currents can be mitigated by relaxing the
assumption regarding the unnatural thinness and rigidity of the UV-brane and
discuss the possibility of suppression of these currents in presence of fat
fluctuating branes.

##  I Introduction

The charged Lepton Flavour Violation (cLFV) has been in focus ever since
intermediate vector bosons were proposed[1, 2]. Even with the inclusion of the
neutrino oscillation phenomena in Standard Model (SM) of particle physics, the
cLFV processes are predicted to be very small. Hence, any evidence of cLFV
inevitably point towards New Physics (NP). As of now, these processes are
strongly constrained at 90% C.L. by the Branching Ratios $B_{\mu\rightarrow
e\gamma}<4.2\times 10^{-13}$[3], $B_{\mu\rightarrow eee}<1.0\times
10^{-12}$[4], $B_{\mu Ti\rightarrow eTi}<4.3\times 10^{-12}$[5],
$B_{\tau\rightarrow eee}<2.7\times 10^{-8}$[6], $B_{\tau\rightarrow
e\gamma}<3.3\times 10^{-8}$[7], $B_{\pi^{0}\rightarrow\mu e}<3.6\times
10^{-10}$[8] and $B_{Z\rightarrow\mu e}<7.5\times 10^{-7}$[9]. On the other
hand, various beyond SM scenarios do predict such flavour violating processes.

One of the successful extensions of SM in warped extra-dimensions has been the
Little Randall-Sundrum (Little RS) model [10, 11] with a fundamental scale of
$\sim 10^{3}TeV$. Recently, the author had studied [12] the correction to
$\epsilon_{K}$ in Little RS arising from the contribution of tree-level KK-1
gluon exchange in $K-\bar{K}$ oscillation. There, it was shown that the strong
bound on the imaginary part of the $C_{4}^{sd}$ operator in the effective
Hamiltonian [13] ruled out the mass scale lower than $\sim 32TeV$. This
constraint was shown to soften significantly in the presence of Brane
Localised Kinetic Terms (BLKTs) and Minimal Flavour Protection (MFP) [14]
flavour symmetry.

In this paper, we discuss the flavour violation in charged lepton sector of
Little RS. We choose to work in a basis that do not mix the left-handed
charged leptons. Rather, the entire mixing is taken to be in the right-handed
sector, while keeping the 5-D Yukawa coupling anarchic. There can, in
principle, exist mixing in the left-handed charged sector through the Tri-
bimaximal Cabibo mechanism[15], but such a choice will affect the neutrino
mixing matrix as well. For brevity, we do not consider such models here. Doing
this, we can study the flavour violations in the charged lepton sector
independent of the neutrino parameters.

We subject the lepton flavour violating decay predictions of anarchic Little
RS model to observations from the rare $\mu$ decays such as $\mu\to e\gamma$,
$\mu\to 3e$, and $\mu\to e$ conversion in the presence of $Ti$ nuclei and the
rare tri-lepton decays of $\tau$. Among these, we will show that $\mu\to e$
conversion proves to be the most constraining. And, this constraints the lower
limit of the KK-1 gauge boson mass to be $\gtrsim 30.7TeV$. For comparison,
the lower mass limit of gauge boson in Randall-Sundrum model [16, 17] that
satisfy the $\mu\rightarrow e$ conversion is $M_{KK}\gtrsim 5.9TeV$ [18]. In
line with the discussion on Kaon oscillation in [12], here, we study the
effect of Brane Localised Kinetic Term (BLKT) in the electro-weak sector of
the model, and in particular on the Z boson wave function. We show that this
modification relaxes the strong bounds on the KK-1 mass of the gauge boson to
$\gtrsim 12TeV$. Making the KK gauge boson available at the upcoming high
energy colliders.

For anarchic 5D Yukawa couplings, since the light fermions are localised close
to the Ultraviolet(UV)-brane, it is possible that the un-physical assumption
of rigid thin branes in Little RS might have a role to play in these large
flavour violations. If we replace this thin rigid UV-brane with a fat brane,
along with its fluctuations namely branons [19, 20, 21], then, we show that
the flavour violating couplings get suppressed, lowering the bound on KK-1
gauge boson mass to $\gtrsim 10TeV$.

The paper is organised as follows. In the next section, we briefly review the
Little RS model, gauge and lepton field KK decomposition and their
interactions. In sec.III we will derive the tri-lepton decay Branching Ratios
and $\mu\to e$ conversion rate in the model. Then, we will discuss the effect
of BLKT on the interactions and recompute the constraints in sec.IV. In sec.V,
we summarise the work and discuss on the effects of fat branes on the lepton
flavour violation.

##  II The Little RS model

In this section, we briefly recap the Little RS model. Our four-dimensional
space-time is assumed to emerge from a 5D AdS space-time, with a fundamental
scale $M=10^{3}TeV$ [10, 22], upon orbifolding on $M_{4}\times S^{1}/Z_{2}$.
The line element of this 5D space-time is given by,

$\displaystyle
ds^{2}=g_{MN}dx^{M}dx^{N}=e^{-2ky}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$ (1)

where M,N are 5 dimensional space-time indices,
$\eta_{\mu\nu}=diag(-1,+1,+1,+1)$ and $0\leq y\leq L$. The warp factor is
taken to be $kL\sim 7$ so that the warped down scale at the Infrared(IR)-brane
becomes $M_{5}e^{-kL}\sim{\cal O}(1\ \text{TeV})$.

The Higgs field, in order to stabilise its vacuum expectation value from
quantum fluctuations, is assumed to be localised on this IR-brane. To avoid
large localised flavour violating and proton decay operators on the brane, we
assume that the gauge fields and fermions propagate in the bulk. Moreover, we
consider these fields to transform under the adjoint and fundamental
representations of the Standard Model gauge group $SU(3)_{C}\times
SU(2)_{W}\times U(1)_{Y}$ respectively.

Below, we describe the five-dimensional gauge and fermion fields, relevant to
the process, and their Kaluza-Klein (KK) decompositions. We begin our
discussion with a review of the electroweak sector coupled with the Higgs
boson in 5-dimensional space-time.

###  II.1 Bulk Gauge Fields

####  II.1.1 Action of the 5D Theory

Let us consider the bulk gauge fields $W_{M}^{a}$ and $B_{M}$, where
$M=\\{0,1,2,3,5\\}$, of $SU(2)_{L}\times U(1)_{Y}$, coupled to the scalar
localised on the IR-brane. The 5D action for this system is given by,

$S_{\rm gauge}=\int d^{4}x\int_{0}^{L}\\!dy\,\Big{(}{\cal L}_{\rm W,B}+{\cal
L}_{\rm Higgs}\Big{)}\ .$ (2)

The kinetic part of the 5D gauge theory (${\cal L}_{\rm W,B}$) and the Higgs-
sector Lagrangian (${\cal L}_{\rm Higgs}$) are given by,

${\cal L}_{\rm
W,B}=\sqrt{g}\,g^{KM}g^{LN}\left(-\frac{1}{4}\,W_{KL}^{a}W_{MN}^{a}-\frac{1}{4}\,B_{KL}B_{MN}\right),$
(3)

and

$\displaystyle{\cal L}_{\rm Higgs}$ $\displaystyle=$
$\displaystyle{\delta(|y|-L)}\left[(D_{M}\Phi)^{\dagger}\,(D^{M}\Phi)-V(\Phi)\right]$
$\displaystyle V(\Phi)$ $\displaystyle=$
$\displaystyle-\mu^{2}\Phi^{\dagger}\Phi+\lambda\left(\Phi^{\dagger}\Phi\right)^{2}.$
(4)

The field strength tensor, of the gauge field is denoted by $W_{MN}$, and the
covariant derivative by,

$\displaystyle
D_{M}=\partial_{M}-ig_{5}\tau_{a}W_{M}^{a}-ig^{\prime}_{5}IB_{M}\ ,$ (5)

where $g_{5}$ and $g_{5}^{\prime}$ are the 5D gauge couplings of W and B
bosons, respectively and the Higgs doublet can be decomposed as,

$\Phi(x)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}-i\sqrt{2}\,\varphi^{+}(x)\\\
v+h(x)+i\varphi^{3}(x)\end{array}\right),$ (6)

with the Higgs vacuum expectation value denoted by $v$, and
$\varphi^{\pm}=(\varphi^{1}\mp i\varphi^{2})/\sqrt{2}$.

The $\delta-$function in Eq.II.1.1 ensures that the Higgs vacuum expectation
value is stabilised to $\sim\mathcal{O}(1TeV)$. For simplicity, we also
perform the usual redefinitions of the gauge fields,

$\displaystyle W_{M}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(W_{M}^{1}\mp iW_{M}^{2}\right),$
$\displaystyle Z_{M}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{g_{5}^{2}+g_{5}^{\prime
2}}}\left(g_{5}W_{M}^{3}-g_{5}^{\prime}B_{M}\right),$ $\displaystyle A_{M}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{g_{5}^{2}+g_{5}^{\prime
2}}}\left(g_{5}^{\prime}W_{M}^{3}+g_{5}B_{M}\right)\ .$ (7)

####  II.1.2 KK-Decomposition

After compactification, the Kaluza-Klein decomposition of 5D gauge field
becomes,

$\displaystyle V_{\mu}(x,y)$
$\displaystyle=\sum_{n}V_{\mu}^{(n)}(x)\,f^{(n)}_{V}(y)\ ,$ (8)

where
$V_{\mu}^{(n)}(x)=\\{A_{\mu}^{(n)}(x),Z_{\mu}^{(n)}(x),W^{\pm(n)}_{\mu}(x)\\}$
are the 4-dimensional KK-modes of the photon, Z-boson and the $W^{\pm}$ boson
and $f^{(n)}_{V}(y)$ their wave profiles in the bulk. The Euler-Lagrange
equation of motion of these bulk modes are given by,

$\displaystyle-\partial_{5}\left(e^{-2ky}\partial_{5}f^{(n)}_{V}\right)=m^{2}_{n}f^{(n)}_{V}\
.$ (9)

In this equation, we have used
$\partial_{\mu}\partial^{\mu}V_{\mu}^{(n)}(x)=m_{n}^{2}V_{\mu}^{(n)}(x)$.
These fields are set to satisfy the boundary condition, $(\delta
V^{\mu}\partial_{y}V_{\mu})\big{|}_{0,L}=0$ and the ortho-normality condition,

$\int_{0}^{L}dyf^{(n)}_{V}f^{(m)}_{V}=\delta_{nm}.$ (10)

Solving Eq.9, for $m_{n}=0$, we find that the zero-mode profile of the gauge
boson is flat and is given by,

$\displaystyle f^{(0)}_{V}(y)=\frac{1}{\sqrt{L}}$ (11)

For the higher KK-modes ( $m_{n}\neq 0$), the solution to Eq.9 is given in
terms of the Bessel $J$ and Bessel $Y$ functions and is of the form,

$f^{(n)}_{V}(y)=N^{(n)}_{V}e^{ky}\left[J_{1}\left(\frac{m_{n}e^{ky}}{k}\right)+b^{(n)}_{V}Y_{1}\left(\frac{m_{n}e^{ky}}{k}\right)\right],$
(12)

where $N_{V}^{(n)}$ and $b_{V}^{(n)}$ are the two constants of integration.
While $N_{V}^{(n)}$ is fixed by the orthonormality condition, $b^{(n)}_{V}$ is
determined using the boundary condition. Demanding Neumann boundary condition
at $y=0$, and $y=L$ we get,

$\displaystyle b_{V}^{(n)}$ $\displaystyle=$
$\displaystyle-\frac{J_{0}(\frac{m_{n}}{k})}{Y_{0}(\frac{m_{n}}{k})}\ \ \text{
at }y=0,$ $\displaystyle b_{V}^{(n)}$ $\displaystyle=$
$\displaystyle-\frac{J_{0}(\frac{m_{n}}{k}e^{kL})}{Y_{0}(\frac{m_{n}}{k}e^{kL})}\
\ \text{ at }y=L\ .$ (13)

The mass spectrum of the KK-modes ($m_{n}=x_{n}ke^{-kL}$) can be computed from
the solutions to the equation
$J_{0}(x_{n})Y_{0}\left(x_{n}e^{-kL}\right)-Y_{0}(x_{n})J_{0}\left(x_{n}e^{-kL}\right)=0$,
obtained by equating the relations given above. For future convenience we
denote the KK-1 gauge boson mass by $M_{KK}=m_{1}$.

###  II.2 Bulk Fermion Fields

The Clifford algebra in 5-dimensional space-time is defined by gamma matrices,
$\Gamma^{A}=\\{\gamma^{0},\gamma^{1},\gamma^{2},\gamma^{3},\gamma^{5}\\}$,
that satisfy $\\{\Gamma^{A},\Gamma^{B}\\}=2\eta^{AB}$, where $\eta^{AB}$ is
the flat metric defined on the 5D tangent space. Since the algebra is
irreducible, one cannot construct a chirality projection operator in 5
dimensions. Thus the fermion fields constructed in this geometry have 4
complex degrees of freedom, which on compactification leads to vector-like
4-dimensional fermions. Moreover, due to the lack of chiral symmetry, the
geometry do not prohibit a mass term in the bulk for the fermions. These mass
terms will become crucial for the geometric Froggatt-Nielsen mechanism to
generate 4-dimensional fermion mass hierarchy. Let us start our discussion by
constructing the 5-dimensional fermionic action.

####  II.2.1 5D Fermionic Action

The 5D action for doublet ($\widehat{\ell}$) and singlet ($\widehat{e}$)
leptons can be written as,

$S_{\text{fermion}}=S_{\text{kin}}+S_{\text{yuk}}$ (14) $\displaystyle
S_{\text{kin}}$ $\displaystyle=$ $\displaystyle\int
d^{5}x\sqrt{-g}\,\bar{\widehat{\ell}}\left(\Gamma^{A}E_{A}^{M}D_{M}+m_{\ell}\right)\widehat{\ell}$
$\displaystyle+\int
d^{5}x\sqrt{-g}\,\sum_{e=e,\mu,\tau}\bar{\widehat{e}}\left(\Gamma^{A}E_{A}^{M}D_{M}+m_{e}\right)\widehat{e}$
$\displaystyle S_{\text{Yuk}}$ $\displaystyle=$ $\displaystyle\int
d^{5}x\sqrt{-g}~{}\bar{\widehat{\ell}}_{i}\left(\tilde{Y}_{5D}\right)^{ij}\widehat{e}_{j}H(x^{\mu})\delta(y-L)$
(15) $\displaystyle+h.c.,$

where $m_{\ell}$ and $m_{e}$ are the bulk masses for the doublet and singlet
lepton fields and $E^{M}_{A}$ the inverse fünfbeins. We denote the tangent
space indices with $A,B$ and the 5-dimensional space-time index with $M,N$.
Since the fünfbeins satisfy the condition,
$e^{A}_{M}\eta_{AB}e^{B}_{N}=g_{MN}$, for the geometry given in Eq.1, they
become $e^{A}_{M}=(e^{-ky}\delta^{\alpha}_{\mu},1)$. The five dimensional
anarchic Yukawa matrix is denoted as $\Big{(}\tilde{Y}_{5D}\Big{)}^{ij}$, with
$i,j$ representing the generational indices. In the above equation, $D_{M}$
represent the covariant derivative in 5-dimensions given by,
$D_{M}=\partial_{M}+\omega_{M}$, where
$\omega_{M}=\frac{1}{8}\omega_{MAB}[\Gamma^{A},\Gamma^{B}]$ and the spin
connection given by,

$\omega_{MAB}=g_{RN}E^{N}_{A}\Big{(}\partial_{M}E^{R}_{B}+\tilde{\Gamma}^{R}_{MT}E^{T}_{B}\Big{)}\
,$ (16)

where $E^{A}_{M}$ are the inverse fünfbeins and $\tilde{\Gamma}^{R}_{MS}$ are
the Christoffel connections.

Upon compactification, the 5D Dirac fermions decompose to two 4D Weyl spinors.
To ensure that only the correct chiral projections survives at the zero-mode,
we use boundary conditions,

$\widehat{\ell}_{L}(++),~{}\widehat{\ell}_{R}(--),~{}\widehat{e}_{L}(--),~{}\widehat{e}_{R}(++),$
(17)

at the orbifold fixed points $(y=0,\ y=L)$. Here, $L,R$ stand for the left and
right chiral fields under the 4-dimensional chiral projection operator and
$+(-)$ stands for the Neumann (Dirichlet) boundary conditions. For example,
$\widehat{\ell}_{L}(++)$ means that we apply the Neumann boundary conditions
at both $y=0$ & $y=L$.

####  II.2.2 The Kaluza Klein decompositions

After compactification, the KK expansion of a generic fermion field ($\Psi$)
becomes,

$\Psi(x,y)_{L,R}=\sum_{n=0}^{\infty}\frac{1}{\sqrt{L}}\psi^{(n)}_{L,R}(x)f^{(n)}_{L,R}(y,c),$
(18)

where $\psi^{(n)}_{L,R}(x)$ denotes the corresponding four dimensional KK-
modes and $f_{L,R}(y)$ their extra-dimensional profiles in the bulk. These
wave profiles are set to satisfy the ortho-normality condition,

$\int_{0}^{L}dy\,e^{-3ky}\,f_{L,R}^{(n)}f_{L,R}^{(m)}=\delta_{n,m}.$ (19)

The normalised zero mode profile for doublets and singlets, with their
respective bulk mass parameters $c_{\ell_{i}}=m_{\ell_{i}}/k$ and
$c_{e_{i}}=-m_{e_{i}}/k$, can be derived as [13, 12],

$\displaystyle f^{(0)}_{L}(y,c_{\ell_{i}})=\sqrt{k}f^{0}(c_{\ell_{i}})\
e^{ky(2-c_{\ell_{i}})}\ e^{(c_{\ell_{i}}-0.5)kL},$ (20) $\displaystyle
f^{(0)}_{R}(y,c_{e_{i}})=\sqrt{k}f^{0}(c_{e_{i}})\ e^{ky(2-c_{e_{i}})}\
e^{(c_{e_{i}}-0.5)kL},$ (21)

where,

$f^{0}(c)=\sqrt{\frac{(1-2c)}{1-e^{-(1-2c)kL}}}\ .$ (22)

Using the boundary conditions given in Eq.17, the lightest and next to
lightest modes of chiral leptons in , 4 dimensions, become,

$\displaystyle\Psi_{L}=(\widehat{\ell}_{L}^{i(0)},\widehat{\ell}_{L}^{i(1)},\widehat{e}_{L}^{i(1)}),$
$\displaystyle\Psi_{R}=(\widehat{e}_{R}^{i(0)},\widehat{e}_{R}^{i(1)},\widehat{\ell}_{R}^{I(1)}).$
(23)

####  II.2.3 Yukawa interaction

Using the action in Eq.15 and the wave function for the zero mode leptons
given in Eq.21, the 4-dimensional Yukawa matrix can be derived in terms of the
5D anarchic Yukawa as,

$\displaystyle Y_{4D}^{ij}$ $\displaystyle=$ $\displaystyle
f^{(0)}_{L}(L,c_{\ell_{i}})Y_{5D}^{ij}f^{(0)}_{R}(L,c_{e_{j}})$
$\displaystyle=$
$\displaystyle\sqrt{\frac{(1-2c_{\ell_{i}})(1-2c_{e_{j}})}{(e^{(1-2c_{\ell_{i}})kL}-1)(e^{(1-2c_{e_{j}})kL}-1)}}$
$\displaystyle\times$ $\displaystyle
e^{(1-(c_{\ell_{i}}+c_{e_{j}}))kL}Y_{5D}^{ij}\ .$

For the fermion KK-modes given in Eq.23, the above mass matrix becomes,

$\mathcal{M}=\left(\begin{array}[]{ccc}M_{0}&M_{0}F_{R}&0\\\
F_{L}M_{0}&F_{L}M_{0}F_{R}&M_{KK}\\\ 0&M_{KK}&0\end{array}\right)\ ,$ (25)

where the
$F_{L,R}^{i}=\frac{f^{(1)}_{L,R}(L,c_{\ell_{i},e_{i}})}{f^{(0)}_{L,R}(L,c_{\ell_{i},e_{i}})}$
and $M_{0}^{ij}=\frac{v}{\sqrt{2}}Y_{4D}^{ij}$. Since, the fermions are in the
flavour basis, we need to rotate this mass matrix to obtain the physical
states. In order to do that, it will be easier if we first diagonalise the SM
part, $M_{0}$, with a biunitary transformation $(U_{L},U_{R})$. Acting on the
mass matrix $\mathcal{M}$ with diag($U_{L},1,1$) on the left and on the right
with diag($U_{R}^{\dagger},1,1$) we get,

$\mathcal{M}=\left(\begin{array}[]{ccc}M_{D}&\frac{v}{\sqrt{2}}\Delta_{R}&0\\\
\frac{v}{\sqrt{2}}\Delta_{L}&\Delta_{1}&M_{KK}\\\
0&M_{KK}&0\end{array}\right)\ ,$ (26)

where $M_{D}=U_{L}M_{0}U_{R}^{\dagger}$,
$\frac{v}{\sqrt{2}}\Delta_{R}=U_{L}M_{0}F_{R}=M_{D}U_{R}F_{R}$,
$\frac{v}{\sqrt{2}}\Delta_{L}=F_{L}M_{0}U_{R}^{\dagger}=F_{L}U_{L}^{\dagger}M_{D}$
and $\Delta_{1}=F_{L}M_{0}F_{R}=F_{L}U_{L}^{\dagger}M_{D}U_{R}F_{R}$.

Moreover, since the mixing of higher KK-modes with the zero-mode are
suppressed by $\frac{v}{M_{KK}}$, it is convenient and informative to
diagonalise the lower $2\times 2$ part of the mass matrix. With that, to the
leading order expansion in $x=\frac{\Delta_{1}}{M_{KK}}$, the diagonal mass
matrix becomes [18],

$\mathcal{M}_{D}=\left(\begin{array}[]{ccc}M_{D}&x_{R}\frac{1}{2}(1+\frac{x}{4})&x_{R}\frac{1}{2}(1-\frac{x}{4})\\\
x_{L}\frac{1}{2}(1+\frac{x}{4})&M_{KK}+\frac{\Delta_{1}}{2}&0\\\
x_{L}\frac{1}{2}(1-\frac{x}{4})&0&-M_{KK}+\frac{\Delta_{1}}{2}\end{array}\right)\
,$ (27)

where $x_{L,R}=\frac{v}{\sqrt{2}}\Delta_{L,R}$. Note that the off diagonal
elements in this matrix are very small compared to $M_{KK}$. Now, the
degeneracy in the KK-1 mode is lifted to,

$\displaystyle M_{KK}^{(1)}=M_{KK}+\frac{\Delta_{1}}{2}\ ,$ $\displaystyle
M_{KK}^{(2)}=-M_{KK}+\frac{\Delta_{1}}{2}\ .$ (28)

####  II.2.4 Couplings with Z boson

Before discussing the flavour violating effects, we need to identify the
relevant couplings of the gauge boson. For simplicity, we will consider the
couplings of fermion bilinear with an abelian gauge field in the bulk of AdS.
Generalisation to flavour violating interactions of the non-Abelian gauge
field is then straightforward. A 5-dimensional action for the $U(1)$ gauge
field can be written as,

$\displaystyle\mathcal{S}=-\frac{1}{4g_{5}^{2}}\int
d^{5}x\sqrt{-g}\left(g^{CM}g^{DN}F_{CD}F_{MN}\right)\ ,$ (29)

where $F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}$ is the field strength
tensor and $g_{5}$ the 5D gauge coupling.
In the unitary gauge ($A_{5}=0$), the vector field can be Fourier expanded as,

$A_{\mu}(x,y)=\sum_{n}A_{\mu}^{(n)}(x)f^{(n)}_{A}(y)\ ,$ (30)

and the coupling of the zero-mode lepton bi-linear with the gauge KK-modes can
be computed from the overlap integral,

$g_{L,R}^{(n)}(c_{\ell,e})=\frac{g_{5}}{L}\int_{0}^{L}dy\,e^{-3ky}f_{A}^{(n)}(y)f_{L,R}^{(0)}(y,c_{\ell,e})f_{L,R}^{(0)}(y,c_{\ell,e})$
(31)

In the above equation, $f_{A}^{(n)}$ and $f_{L,R}^{(0)}$ are as given in Eq.12
and Eq.22 respectively. Since the geometric Froggatt-Nielsen mechanism
requires distinct bulk mass values ‘$c_{\ell,e}$’ for the leptons, the
couplings $g^{(n)}_{L,R}$ are different depending on the localisation of the
fermion zero mode. Replacing the abelian gauge field with the Z-boson, we can
explicitly write these interactions as,

$\mathcal{L}=g^{(n)}_{L}(c_{\ell_{i}})\bar{\widehat{\ell}}_{L}^{i(0)}\gamma_{\mu}Z^{\mu(n)}{\widehat{\ell}}_{L}^{i(0)}+g^{(n)}_{R}(c_{e_{i}})\bar{\widehat{e}}_{R}^{i(0)}\gamma_{\mu}Z^{\mu(n)}{\widehat{e}}_{R}^{i(0)}\
.$ (32)

These couplings, being depended on the bulk mass parameter ‘$c_{\ell,e}$’,
generate flavour violations in the interactions of the gauge boson KK-modes on
rotating the fermions to their mass basis. With this understanding, we can now
address the consequences of such terms in Little RS.

##  III Flavour violating decays in Little RS

In this section, we focus on the charged lepton flavour violating decay
processes such as $\mu\rightarrow e\gamma$, $\mu^{-}\rightarrow
e^{+}e^{-}e^{-}$, $\mu\ Ti\to e\ Ti$, $\tau^{-}\to e^{-}e^{-}e^{+}$,
$\tau^{-}\to\mu^{-}e^{-}e^{+}$ and $\tau^{-}\to e^{-}\mu^{-}\mu^{+}$, in the
Little RS framework. Among these, the only loop process is the $\mu\to
e\gamma$. Though this decay, is divergent in Randall-Sundrum model with brane
localised Higgs, since Little RS is an effective theory with much lower cut
off $\sim 10^{3}TeV$, we expect a need to re-analyse this decay. The dominant
contribution to the process $\mu\rightarrow e\gamma$ proceeds through a 1-loop
Feynman diagram with brane localised Higgs and KK-fermions fields as shown in
Fig.1.

Figure 1: The diagram that generates the process $\mu\to e\gamma$.

The amplitude of the process, assuming $M_{KK}^{(i)}$ to be much greater than
the energy scales involved, thus becomes,

$\displaystyle A_{\mu\rightarrow e\gamma}$ $\displaystyle=$
$\displaystyle\displaystyle\bar{u}(p^{\prime})\\{eA_{\mu}\sum_{i}\frac{1}{M_{KK}^{(i)4}}Y_{ei}$
(33) $\displaystyle\times$
$\displaystyle\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{(\not{p}^{\prime}+\not{k}+M_{KK}^{(i)})\gamma^{\mu}(\not{p}^{\prime}+\not{k}+M_{KK}^{(i)})}{k^{2}-m_{H}^{2}}$
$\displaystyle\times$ $\displaystyle\displaystyle Y_{i\mu}\\}u(p)$
$\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2m_{\mu}}\bar{u_{L}}(p^{\prime})\sigma^{\mu\nu}F_{\mu\nu}u_{R}(p)C_{L}(q^{2})$
$\displaystyle+$
$\displaystyle\displaystyle\frac{1}{2m_{\mu}}\bar{u_{R}}(p^{\prime})\sigma^{\mu\nu}F_{\mu\nu}u_{L}(p)C_{R}(q^{2})\
.$

The divergence in this amplitude come through the large number of fermion KK-
modes that contribute to this loop
$\frac{C_{L,R}}{m_{\mu}^{2}}\sim\frac{1}{16\pi^{2}}log(N_{KK})$. Their
contributions worsen at two-loops. Thus, the cut-off dependent part of the
Wilson Coefficient has the form [18],

$\displaystyle\frac{C_{L,R}}{m_{\mu}^{2}}$ $\displaystyle\sim$
$\displaystyle\displaystyle\frac{1}{16\pi^{2}}\left(\frac{Y_{5D}}{M_{KK}}\right)^{2}\left\\{log\left(\frac{\Lambda_{5D}}{k}\right)\right.$
(34) $\displaystyle+$
$\displaystyle\displaystyle\left.\frac{1}{16\pi^{2}}Y_{5D}\frac{\Lambda_{5D}^{2}}{k^{2}}+....\right\\}\
,$

where the first and second terms are the 1-loop and 2-loop contributions
respectively.

Randall-Sundrum model, being UV complete, requires
$\Lambda_{5D}=M_{pl}=10^{16}TeV$ to avoid hierarchy. Moreover, to avoid
quantum gravity effects the curvature should satisfy the condition
$k/M_{pl}\ll 0.1$. Hence, assuming $k\lesssim 10^{15}TeV$, the number of KK
modes that contribute to the process becomes $N_{KK}\gtrsim 10$. At 2-loop,
dimension analysis suggests that the amplitude becomes $\sim
N_{KK}^{2}=(\frac{\Lambda_{5D}}{k})^{2}\gtrsim 100$ [18]. Which means that the
1-loop and 2-loop contributions in Eq.34 are of the same order
($Log(\frac{\Lambda_{5D}}{k})\sim\frac{Y_{5D}}{16\pi^{2}}\frac{\Lambda_{5D}^{2}}{k^{2}}\sim
1$). And the higher loops contribute strongly and the result is not
convergent. This feature threatens any reliable calculations of the $\mu\to
e\gamma$ process in Randall-Sundrum model.

In comparison, Little Randall-Sundrum model is an effective theory with a
scale $10^{3}TeV$, such that quantum gravity effects are insignificant. It is
straightforward to find a parameter space, for example $\Lambda_{5D}\sim
10^{3}TeV$ and $k\sim 800TeV$, in which the number of KK-fermions contributing
to the loop can be much smaller
($N_{KK}\sim\frac{1000}{800}\sim\mathcal{O}(1)$), and the cut-off dependent
part of the Wilson Coefficient becomes,

$\displaystyle\frac{C_{L,R}}{m_{\mu}^{2}}$ $\displaystyle\sim$
$\displaystyle\displaystyle\frac{1}{16\pi^{2}}\left(\frac{Y_{5D}}{M_{KK}}\right)^{2}\left\\{0.2+0.008+....\right\\}\
,$ (35)

where the 2-loop term is much smaller than the 1-loop with the higher loop
terms further suppressed. Though the amplitude is calculable now, still, the
result is sensitive to the cut-off scale of the model, but this can be cured
if we can dynamically stabilize this scale. Being phenomenological in nature,
Little RS requires new physics to do this. One way this could be achieved is
by embedding the model in a six-dimensional $S^{1}/Z_{2}\times S^{1}/Z_{2}$
scenario, for which all the radii are stabilised [23, 24].

The cut-off independent part of the Wilson Coefficient, $C_{L,R}(q^{2}=0)$,
can be derived from Eq.33 as,

$C_{L,R}(q^{2}=0)=\frac{em_{\mu}}{32\pi^{2}}\sum_{i}Y_{ei}\frac{m_{H}^{2}}{M_{KK}^{(i)3}}Y_{i\mu}\
,$ (36)

where $Y=\mathcal{M_{D}}/(v/\sqrt{2})$ is the rotated Yukawa coupling and
$M_{KK}^{(i)}$ are the masses of the first KK-mode given in Eq.27 and Eq.28
respectively. The branching ratio for this process now becomes [25, 26],

$B(\mu\rightarrow
e\gamma)=\frac{12\pi^{2}}{(G_{F}m_{\mu}^{2})^{2}}\Big{[}|C_{L}(0)|^{2}+|C_{R}(0)|^{2}\Big{]}\
,$ (37)

where, $C_{L}(0)$, and $C_{R}(0)$ can be derived by using Eq.27 and Eq.28 in
Eq.36,

$\displaystyle C_{L}(0)$ $\displaystyle=$ $\displaystyle\displaystyle
e\frac{m_{\mu}m_{H}^{2}}{32\pi^{2}}[Y_{e1}Y_{1\mu}\frac{1}{M_{KK}^{(1)3}}+Y_{e2}Y_{2\mu}\frac{1}{M_{KK}^{(2)3}}]$
$\displaystyle=$ $\displaystyle\displaystyle
e\frac{m_{\mu}m_{H}^{2}}{32\pi^{2}M_{KK}^{4}}[\Delta_{R}\Delta_{1}\Delta_{L}]_{e\mu}\
,$ $\displaystyle C_{R}(0)$ $\displaystyle=$ $\displaystyle\displaystyle
e\frac{m_{\mu}m_{H}^{2}}{32\pi^{2}}[Y_{e1}Y_{1\mu}\frac{1}{M_{KK}^{(1)3}}+Y_{e2}Y_{2\mu}\frac{1}{M_{KK}^{(2)3}}]^{\dagger}$
(38) $\displaystyle=$ $\displaystyle\displaystyle
e\frac{m_{\mu}m_{H}^{2}}{32\pi^{2}M_{KK}^{4}}[\Delta_{R}\Delta_{1}\Delta_{L}]^{\dagger}_{e\mu}\
.$

$\Delta_{L}$ and $\Delta_{R}$ are the off-diagonal terms in the fermion mass
matrix and $\Delta_{1}$ is the Yukawa mass of the KK-1 leptons given in Eq.26.

This finite part can be computed and on comparing it with the experimental
bound on the Branching ratio $B_{expt}(\mu\rightarrow e\gamma)\lesssim
0.042\times 10^{-11}$ [27], we obtain the lower limit on the mass scale
$M_{KK}\gtrsim 1.4TeV$.

###  III.1 Tri-lepton decays and $\mu-e$ conversions

Before we discuss the tri-lepton decays in Little RS model, to study the
flavour violating effects, it is important and insightful if we identify the
model independent four-fermion interactions that contribute to the processes.
Adopting the parametrisation in [28, 25], the most general low-energy
effective, dimension-6, Lagrangian responsible for these processes can be
written as,

$\begin{split}-\mathcal{L_{{\rm
eff}}}=&\frac{4G_{F}}{\sqrt{2}}\left[g_{3}^{ij}(\bar{e}_{iR}\gamma^{\mu}e_{jR})(\bar{e}_{kR}\gamma_{\mu}e_{kR})\right.\\\
&+\left.g_{4}^{ij}(\bar{\ell}_{iL}\gamma^{\mu}\ell_{jL})(\bar{\ell}_{kL}\gamma_{\mu}\ell_{kL})\right.\\\
&+\left.g_{5}^{ij}(\bar{e}_{iR}\gamma^{\mu}e_{jR})(\bar{\ell}_{kL}\gamma_{\mu}\ell_{kL})\right.\\\
&+\left.g_{6}^{ij}(\bar{\ell}_{iL}\gamma^{\mu}\ell_{jL})(\bar{e}_{kR}\gamma_{\mu}e_{kR})\right]+{\rm
h.c.}\ ,\end{split}$ (39)

where $g_{3,4,5,6}$ are dimensionless Wilson coefficients. Note that, we have
only considered vector operators and not scalar or pseudoscalar ones. This is
because the Higgs contribution to the flavour violating process is suppressed
by small masses of the fermions involved. Moreover, the next to leading order
effects from KK-fermions mixing are further suppressed.

Like fermions, the electro-weak symmetry breaking with brane localised Higgs
boson also mix the KK-levels of the Z-boson. Details of the symmetry breaking
and the mass matrices of electro-weak gauge bosons are given in in Appendix
VI.1. To diagonalise this mass matrix, we need to rotate the gauge basis
$(Z^{(0)},Z^{(1)})$ to the physical basis $(Z_{(0)},Z_{(1)})$, wherein the
admixture enters as,

$Z_{(0)}=Z^{(0)}+f\frac{m_{Z}^{2}}{M_{KK}^{2}}Z^{(1)},\quad
Z_{(1)}=Z^{(1)}-f\frac{m_{Z}^{2}}{M_{KK}^{2}}Z^{(0)}\ ,$ (40)

where $f~{}(\sim\sqrt{2kL})$ parameterises the mixing between the zero and
first KK level. From Eq.31 it is clear that, since the extra-dimensional wave
profile of $Z^{(0)}$ is flat, it couples democratically to all the lepton
generations. Whereas, couplings of the fermion zero-mode bilinear with the
$Z^{(1)}$ is determined by the appropriate overlap integral and is dependent
on the bulk mass parameter $c_{\ell_{i},e_{i}}$,

$\alpha_{i}=2\sqrt{2\pi}\int_{0}^{L}dy\,e^{-3ky}f^{(1)}_{Z}(y)[f^{(0)}_{L,R}(y,c_{\ell_{i},e_{i}})]^{2},$
(41)

In the above equation, $\alpha_{e}$, $\alpha_{\mu}$, and $\alpha_{\tau}$
denote the ratios of couplings given in Eq.32 to the SM ones. On rotating to
the mass basis of leptons, the matrix which describes the $Z^{(1)}_{\mu}$
couplings take the form,

$\mathcal{L}_{int}=g_{L,R}\bar{\Psi}U^{\dagger}_{L,R}\alpha
U_{L,R}\gamma^{\mu}\Psi Z^{(1)}_{\mu}\ ,$ (42)

where $g_{L,R}$ are the SM gauge couplings, $U_{L,R}$ are the unitary mixing
matrices for charged SM leptons, $\alpha={\rm
diag}(\alpha_{e},\alpha_{\mu},\alpha_{\tau})$ and

$\Psi=\left(\begin{array}[]{l}e_{L,R}\\\ \mu_{L,R}\\\
\tau_{L,R}\end{array}\right)$ (43)

It can easily be seen that the generational dependence of the $Z^{(1)}$ boson
couplings, in Eq.41, conspire to create off-diagonal elements that generate
flavour violations. Moreover, this property of the KK-1 mode of the Z-boson is
inherited by the physical $Z_{(0)}$ boson due to the admixture. Thus the
dominant flavour violation in the electro-weak sector is mediated by the zero-
mode of the physical Z-boson and integrating them out from low-energy
processes results in the Wilson Coefficients shown in Eq.39. A detailed
discussion is given in AppendixVI.2.

Now, using the operators in Eq.39 in Eq.64, the branching ratio for the
process $\mu\to 3e$ can be written as,

$BR(\mu\rightarrow 3e)=2\left(|g^{\mu e}_{3}|^{2}+|g^{\mu
e}_{4}|^{2}\right)+|g^{\mu e}_{5}|^{2}+|g^{\mu e}_{6}|^{2}\ ,$ (44)

where we have assumed $BR(\mu\to e\nu\nu)=1$.

And, the $\mu-e$ conversion rate [28] becomes,

$B_{conv}=\frac{2p_{e}E_{e}G_{F}^{2}m_{\mu}^{3}\alpha_{QED}^{3}Z_{eff}^{4}Q_{N}^{2}}{\pi^{2}Z\Gamma_{capt}}\left[|g_{R}^{\mu
e}|^{2}+|g_{L}^{\mu e}|^{2}\right]\ ,$ (45)

where the couplings $g_{3,4,5,6}$, $g_{L,R}$ are given in Appendix(VI.2),
$\alpha_{QED}$ is the QED coupling strength and the remaining atomic physics
constants are given in [28].

In order to constraint the Little RS model, we impose the following current
PDG limits: $BR(\mu\rightarrow 3e)<10^{-12}$ [4, 29]; $B_{\mu Ti\to
eTi}<4.3\times 10^{-12}$ [5, 29]. And for the rare tau decays, we employ the
constraints $BR(\tau\rightarrow e^{+}e^{-}e^{+})<2.7\times 10^{-8}$,
$BR(\tau\rightarrow\mu^{+}\mu^{-}\mu^{+})<2.1\times 10^{-8}$,
$BR(\tau\rightarrow\mu^{-}e^{-}e^{+})<1.8\times 10^{-8}$ [29]. For computing
the $\mu\to e$ conversion, we use the numerical values for Titanium from [30].

Comparing these limits with Eq.44 and Eq.45, we can derive the lower bound on
the mass scale in the model. We present the constraints on $M_{KK}$ for Little
RS and compare it with the bounds obtained in Randall-Sundrum model in Table
1. The details regarding our numerical analysis is given in AppendixVI.3.

Model |  $BR(\mu\to 3e)$ |  $B_{conv}$ |  $BR(\mu\to e\gamma)$
---|---|---|---
RS |  $2.5TeV$ |  $5.9TeV$ |  ${}^{*}8TeV$
Little RS |  $20.8TeV$ |  $30.7TeV$ |  $1.4TeV$

Model |  $BR(\tau\to 3e)$ |  $BR(\tau\to 3\mu)$ |  $BR(\tau\to\mu ee)$
---|---|---|---
RS |  $0.1TeV$ |  $0.4TeV$ |  $0.36TeV$
Little RS |  $2.48TeV$ |  $2.43TeV$ |  $2.50TeV$

Table 1: Constraints on the first KK-mode mass, $M_{KK}$, coming from various
measurements for a brane Higgs field in both the RS and the Little RS. Except
for the $BR(\mu\to e\gamma)$ in the RS model where we have used bulk Higgs,
since the brane Higgs is not computable in the RS.

##  IV Brane Localised Gauge Kinetic Terms

Going beyond the simplest possible extension of SM in Little RS, the action of
the gauge field in 5-dimensions, given in Eq.29, can be generalised [31, 32,
12, 33] as,

$\begin{split}\mathcal{S}=&-\frac{1}{4g_{5}^{2}}\int
d^{5}x\sqrt{-g}\left\\{\left(g^{AM}g^{BN}F_{AB}F_{MN}\right)\right.\\\
&+\left.\left(l_{UV}\delta(y)+l_{IR}\delta(y-L)\right)g^{\alpha\mu}g^{\beta\nu}F_{\alpha\beta}F_{\mu\nu}\right\\}\end{split}$
(46)

where $l_{UV}$ and $l_{IR}$ are the localised kinetic term strengths at the UV
and the IR branes respectively. The origin of these terms are for the time
being unknown, but it is understood that for correct re-normalisation of the
model such terms are necessary [34, 35]. These terms can be hypothesised to
have their origin in the unperturbative effects of brane localised matter
coupled to the gauge field.

Varying the action in Eq.46 with respect to the field, the equation of motion
becomes,

$-\partial_{5}(e^{-2ky}\partial_{5}f^{(n)}_{A})=(1+l_{IR}\delta(y-L)+l_{UV}\delta(y))m^{2}_{n}f^{(n)}_{A}\
,$ (47)

where $f_{A}^{(n)}(y)$ are the wavefunctions that satisfy the ortho-normality
condition,

$\int_{0}^{L}dy\left[1+l_{IR}\delta(y)+l_{UV}\delta(y-L)\right]f^{(n)}_{A}f^{(m)}_{A}=\delta_{nm}\
.$ (48)

The solution to Eq.47 is given by,

$f^{(n)}_{A}(y)=N^{(n)}_{A}e^{ky}\left[J_{1}\left(\frac{m_{n}}{ke^{-ky}}\right)+b^{(n)}_{A}Y_{1}\left(\frac{m_{n}}{ke^{-ky}}\right)\right]\
,$ (49)

where $N^{(n)}_{A}$ is the normalisation constants and $b_{A}^{(n)}$ the
integration constant fixed by the modified boundary conditions,

$\displaystyle\partial_{y}f^{(n)}_{A}|_{y=0}$ $\displaystyle=$ $\displaystyle-
l_{UV}m_{n}^{2}f^{(n)}_{A}(0)\ ,$
$\displaystyle\partial_{y}f^{(n)}_{A}|_{y=L}$ $\displaystyle=$
$\displaystyle+e^{2kL}l_{IR}m_{n}^{2}f^{(n)}_{A}(L).$ (50)

Using the above relations, $b_{A}^{(n)}$ becomes,

$\displaystyle b_{A}^{(n)}$ $\displaystyle=$
$\displaystyle-\frac{J_{0}(\frac{m_{n}}{k})+m_{n}l_{UV}J_{1}(\frac{m_{n}}{k})}{Y_{0}(\frac{m_{n}}{k})+m_{n}l_{UV}Y_{1}(\frac{m_{n}}{k})}\
\ \text{at}\ y=0,$ $\displaystyle b_{A}^{(n)}$ $\displaystyle=$
$\displaystyle-\frac{J_{0}(\frac{m_{n}}{k}e^{kL})-m_{n}l_{IR}e^{kL}J_{1}(\frac{m_{n}}{k}e^{kL})}{Y_{0}(\frac{m_{n}}{k}e^{kL})-m_{n}l_{IR}e^{kL}Y_{1}(\frac{m_{n}}{k}e^{kL})}\
\ \text{at}\ y=0$

where $m_{n}=x_{n}ke^{-kL}$, and $x_{n}$ are the roots of the master equation
obtained by equating the two relations in Eq.LABEL:eq:gaugebcconsts.

From Eq.47 and Eq.48, the zero mode wave-function can be derived as,

$\displaystyle f^{(0)}_{A}(y)=\frac{1}{\sqrt{L+l_{IR}+l_{UV}}}\ .$ (52)

(a)

(b)

Figure 2:  (a) The coupling of KK-1 Z boson, $|g_{1}/g_{0}|$, with leptons
having bulk mass parameters $c=0.8$(red solid) and $c=0.7$(black dotted) as a
function of $kl_{IR}$ (assuming $kl_{UV}=0$ ). (b) The coupling of KK-1 Z
boson, $|g_{1}/g_{0}|$,with fermions having bulk mass parameters $c=0.8$(red
solid) and $c=0.7$(black dotted) as a function of $kl_{UV}$ (assuming
$kl_{IR}=0$).

.

Since the BLKT modifies the bulk gauge field wave profile, their overlap with
the lepton bilinear becomes,

$g_{L,R}^{(n)}(c_{\ell,e})=\frac{g_{5}}{L}\int_{0}^{L}dy\,e^{-3ky}f_{A}^{(n)}(y)f_{L,R}^{(0)}(y,c_{\ell,e})f_{L,R}^{(0)}(y,c_{\ell,e})\
,$ (53)

where $g_{5}=g_{0}\sqrt{L+l_{IR}+l_{UV}}$, with $g_{0}$ denoting the coupling
of $Z^{(0)}$ boson with the fermion. This generalization of the gauge field
does not affect the gauge zero-mode couplings. On the other hand, the higher
KK-mode couplings are modified significantly. Since the cLFV is mediated by
the $Z_{\mu}^{(1)}$, to capture the effect of BLKT, it is instructive to
define the quantity

$\Big{|}\frac{\Delta
g_{1}}{g_{0}}\Big{|}=\Big{|}\frac{g_{L,R}^{(1)}(c_{2})-g_{L,R}^{(1)}(c_{1})}{g_{0}}\Big{|}\
,$ (54)

where $g_{L,R}^{(1)}(c)$ is given in Eq.53.

For illustration, in Fig.2 we display the coupling strengths of KK-1 partner
of the Z boson, $\Big{|}\frac{g_{1}}{g_{0}}\Big{|}$(we denote
$g_{L,R}^{(1)}(c)=g_{1}$), and lepton bilinears, with the bulk mass parameters
$c_{1}=0.8$ and $c_{2}=0.7$, as a function of the BLKT strengths. The values
of $\Big{|}\frac{\Delta g_{1}}{g_{0}}\Big{|}$ for different BLKTs are given in
Table 2.

BLKT | $kl_{IR}=kl_{UV}=0$ |  $kl_{IR}=-5$,$kl_{UV}=0$ |  $kl_{IR}=0$,$kl_{UV}=-5$
---|---|---|---
$\Big{|}\frac{\Delta g_{1}}{g_{0}}\Big{|}$ | 0.013 | 0.015 | 0.008

BLKT |  $kl_{IR}=5$, $kl_{UV}=0$ |  $kl_{IR}=0$, $kl_{UV}=5$ |  $kl_{IR}=5$, $kl_{UV}=-5$
---|---|---|---
$\Big{|}\frac{\Delta g_{1}}{g_{0}}\Big{|}$ | 0.003 | 0.0123 | 0.002

Table 2: $\Big{|}\frac{\Delta g_{1}}{g_{0}}\Big{|}$ values for the three cases
of BLKTs computed with $c_{1}=0.8$ and $c_{2}=0.7$.

(a)

(b)

Figure 3:  Probability distribution function satisfying (a) the experimental
branching ratio for the process $\mu\rightarrow eee$ with
$kl_{IR}=5,kl_{UV}=-5$ (Black solid) $kl_{UV}=0,kl_{IR}=0$ (Red dashed) (b)
the experimental $\mu Ti\to eTi$ conversion rate $kl_{IR}=5,kl_{UV}=-5$(Black
solid)$kl_{UV}=0,kl_{IR}=0$ (Red dashed)

The numerical analysis is similar to the scenario without BLKT, but using the
modified couplings in Eq.53. The probability distribution function of
simulated data, for the processes $\mu\to eee$ and $\mu Ti\to eTi$, is
presented in Fig.3. And for comparison, we have also shown the scenario
without BLKT. These plots clearly show that the branching ratio and conversion
rate constraints relax on imposing BLKT, bringing down the lower limit on the
KK-1 gauge boson mass scale to $\sim 12TeV$. Thus, making the model relevant
at the upcoming hadronic collider searches. The bounds on these processes in
presence of BLKT are summarised in Table 3.

| $kl_{IR}=kl_{UV}=0$ |  $kl_{IR}=5$,$kl_{UV}=-5$
---|---|---
$B_{(\mu\rightarrow 3e)}$ | 20.8 TeV | 7.49 TeV
$B_{\mu Ti\to eTi}$ | 30.7 TeV | 12.03 TeV
Table 3: Bounds on the $M_{KK}$ for the different BLKTs considered.

##  V Summary

(a)

(b)

Figure 4:  Probability distribution function satisfying (a) Constraints on the
branching ratio of $\mu\rightarrow eee$ with fat brane of size ratio 0.2
(black solid) and 0.1 (blue dot-dashed), and thin brane (red dashed)
(b)Constraints on the $\mu-e$ conversion with fat brane of size ratio 0.2
(black solid) and 0.1 (blue dot-dashed), and thin brane (red dashed)

As an effective theory below $10^{3}TeV$, the Little Randall-Sundrum model has
been quite successful in relaxing the strong constraints from the electro-weak
precision observables without introducing custodial symmetry. As an added
advantage, it also predicts an enhanced signal at LHC[12] compared to its UV
complete counterpart. On the other hand, the flavour predictions turn for the
worse. In a previous article by the author[12], the effect of Little RS on
Kaon oscillation were discussed, where they observed that the contribution to
the $\epsilon_{K}$ parameter was enhanced by tree level KK-1 gluon exchange
diagram. The lower limit on compactification scale was computed to be $\sim
32TeV$. On the other hand, these limits were relaxed on including the BLKT for
gluons and the scale was lowered to $\sim 5TeV$. The relaxation of the
constraints were also achieved by imposing Minimal Flavour Protection, $U(3)$
flavour symmetry. This brings us to believe that the simplistic structure of
gauge kinetic terms, considered so far in the literature, may not be the
correct nature of the universe. Instead, we need to introduce BLKT. These
generalised gauge kinetic terms are also necessary to correctly re-normalise
the gauge sector [34, 35, 36, 37, 38].

In this paper, we extend the study of flavour violation to the charged lepton
sector. The anarchic Little RS model was subjected to a set of experimental
constraints from the rare decays of, $\mu\rightarrow eee$, $\tau\rightarrow
eee$, $\tau\rightarrow\mu\mu\mu$, $\tau\rightarrow\mu ee$, $\tau\rightarrow
e\mu\mu$ , $\mu\rightarrow e\gamma$, and $\mu Ti\rightarrow eTi$. Unlike the
case for hadrons, here, the flavour violations are mediated by the SM Z-boson
through mixing with the KK-1 partners in gauge basis. We note that the Little
RS suffer stronger bounds from the lepton flavour violating sector, which
constrains the lower limit on the KK-1 gauge boson mass to $M_{KK}\sim
30.7TeV$. To mitigate such large constraints and make the model viable at
colliders, in this article, we proposed Brane Localised Kinetic Terms for
electroweak gauge bosons. Here, we considered non-minimal kinetic terms in
both UV and IR branes and found that positive values of BLKT on IR-brane and
negative BLKT on UV-brane relax the bounds, effectively reducing the lower
limit to $M_{KK}\geq 12TeV$. The electro-weak sector in Little RS exhibits a
very rich flavour phenomenology. These interactions, discussed in our paper,
can also contribute significantly to processes involving Lepton Universality
violation.

Before concluding, we note that that the flavour violations, discussed here,
are generated due to the difference in the localisating wave profiles of
fermion zero modes at the UV-brane. If the wave profiles were degenerate, the
KK modes of the $Z$ boson would have coupled democratically to all the
leptons. It is interesting to investigate whether these large corrections can
be mitigated by relaxing our assumption on the unnatural thinness and rigidity
of the brane. Though beyond the scope of this paper, if we consider fat
branes, we can show that the lower limit on $M_{KK}$ softens significantly.
For demonstration, we choose two scenarios with the ratio of brane width to
compactification radius 0.1 and 0.2, and we have plotted the Probability
Distribution Function satisfying the constraints on the branching ratio of
$\mu\rightarrow eee$ and $\mu\to e$ conversion in Fig.4. For ratio 0.2, it can
be seen that the constraints from $\mu\rightarrow eee$ limits $M_{KK}\gtrsim
8TeV$, while $\mu Ti\rightarrow eTi$ limits $M_{KK}\gtrsim 12TeV$. Moreover,
for flavour violations arising on the brane, the pseudo-Nambu-Goldstone bosons
of the spontaneously broken translational symmetry (branons) are understood
[19, 39] to suppress the coupling of gauge boson KK-modes with the fermion
bilinear. Hence, a major part of the flavour violation appears outside the
brane. This interesting feature disappears as the brane gets thinner and more
rigid.

## Acknowledgements

M.T.A. acknowledges financial support of DST through INSPIRE Faculty grant
[DST/INSPIRE/04/2019/002507].

##  VI Appendix

###  VI.1 Mass Matrix of Gauge Fields

After spontaneous symmetry breaking of the Higgs, the mass terms from the
Lagrangian given in Eq.3 and in Eq.II.1.1 becomes,

$\begin{split}&\mathcal{L}_{m}=\sum_{n}m^{(n)2}_{w}W^{+(n)}_{\mu}W^{-\mu(n)}+m_{A}^{(n)2}A^{(n)}_{\mu}A^{\mu(n)}\\\
&+m_{z}^{(n)2}Z^{(n)}_{\mu}Z^{\mu(n)}+\frac{v^{2}}{2}\int
dy\delta(y-L)g_{5}^{2}e^{-2ky}\times\\\
&\left(\sum_{m,n}W^{+(m)}_{\nu}W^{-\mu(n)}f_{w}^{(m)}(y)f_{w}^{(n)}(y))\right)+\frac{v^{2}}{2}(g_{5}^{2}+g_{5}^{\prime
2})\times\\\ &\int
e^{-2ky}\sum_{m,n}Z^{(m)}_{\mu}Z^{\mu(n)}f_{z}^{(m)}(y)f_{z}^{(n)}(y)\delta(y-L)dy,\end{split}$
(55)

where $m^{(n)}_{w}$, $m^{(n)}_{A}$ and $m^{(n)}_{z}$ is the nth KK masses of
W-boson, photon, Z-boson respectively.
Mass matrix of these gauge fields computed from the Lagrangian Eq.55:
The mass term of $A_{\mu}$ is,

$\begin{bmatrix}A_{\mu}^{(0)}&A_{\mu}^{(1)}\end{bmatrix}\begin{bmatrix}0&0\\\
0&m^{(1)2}_{A}\end{bmatrix}\begin{bmatrix}A_{\mu}^{(0)}&A_{\mu}^{(1)}.\end{bmatrix}$
(56)

Zero-mode of the photon does not couple with the Higgs and hence it is
massless.

The mass term of $W^{\pm}_{\mu}$ is,

$\begin{split}&M_{w}=m^{(0)2}_{w}W^{+(0)}_{\mu}W^{-\mu(0)}+m^{(1)2}_{w}W^{+\mu(1)}_{\mu}W^{-\mu(1)}+\frac{v^{2}}{2}\times\\\
&\int
g_{5}^{2}e^{-2ky}\sum_{m,n}W^{+(m)}_{\mu}W^{-\mu(n)}f_{w}^{(m)}(y)f_{w}^{(n)}(y)\delta(y-L)dy.\end{split}$
(57)

This can be represented by the matrix form,

$M_{W}=\begin{bmatrix}W_{\mu}^{+(0)}&W_{\mu}^{+(1)}\end{bmatrix}\begin{bmatrix}a_{00}&a_{01}\\\
a_{10}&a_{11}+m^{(1)2}_{w}\end{bmatrix}\begin{bmatrix}W_{\mu}^{-(0)}\\\
W_{\mu}^{-(1)},\end{bmatrix}$ (58)

where $a_{m,n}=g_{5}^{2}\frac{v^{2}}{2}e^{-2kL}f_{w}^{(m)}(L)f_{w}^{(n)}(L)$.

Similiarly mass term of the Z boson is,

$\begin{split}&M_{Z}=m^{(0)2}_{z}Z^{\mu(0)}_{\mu}Z^{\mu(0)}+m^{(1)2}_{z}Z^{(1)}_{\mu}Z^{\mu(1)}+\frac{v^{2}}{2}(g_{5}^{2}+g_{5}^{\prime
2})\times\\\ &\int
e^{-2ky}\sum_{m,n}Z^{(m)}_{\mu}Z^{\mu(n)}f_{z}{(m)}(y)f_{z}^{(n)}(y)\delta(y-L)dy.\end{split}$
(59)
$M_{Z}=\begin{bmatrix}Z_{\mu}^{(0)}&Z_{\mu}^{(1)}\end{bmatrix}\begin{bmatrix}b_{00}&b_{01}\\\
b_{10}&b_{11}+m^{(1)2}_{z}\end{bmatrix}\begin{bmatrix}Z_{\mu}^{(0)}\\\
Z_{\mu}^{(1)}\end{bmatrix},$ (60)

where $b_{m,n}=g_{5}^{2}\frac{v^{2}}{2}e^{-2kL}f_{z}^{(m)}(L)f_{z}^{(n)}(L)$
and $Z^{\mu(0)}$, $Z^{\mu(1)}$ are in the gauge basis of the Z boson. This is
a feature of IR-brane localised electro-weak symmetry breaking.

###  VI.2 Couplings and $l_{i}\to 3l_{j}$ branching ratio

On going to mass basis, the flavour-basis coupling matrices
$C^{F}_{L,R}=g_{L,R}\,{\rm diag}(\alpha_{e},\alpha_{\mu},\alpha_{\tau})$ gets
rotated to $C_{L,R}=U_{L,R}\,C^{F}_{L,R}\,U^{{\dagger}}_{L,R}$. Since
$\alpha_{e}\neq\alpha_{\mu}\neq\alpha_{\tau}$, in the rotated basis $C_{L,R}$
generate off-diagonal entries that lead to flavour violation. Using the
unitarity of $U_{L,R}$, we get,

$g_{L,R}^{(1)\mu
e}=g_{L,R}(U_{1,2}^{L,R}U_{2,2}^{L,R*}(\alpha_{\mu}-\alpha_{e})+U_{1,3}^{L,R}U_{2,3}^{L,R*}(\alpha_{\tau}-\alpha_{e}))\
,$
$g_{L,R}^{(1)\tau\mu}=g_{L,R}(U_{2,1}^{L,R}U_{3,1}^{L,R*}(\alpha_{e}-\alpha_{\mu})+U_{2,3}^{L,R}U_{3,3}^{L,R*}(\alpha_{\tau}-\alpha_{e}))\
,$ $g_{L,R}^{(1)\tau
e}=g_{L,R}(U_{1,2}^{L,R}U_{3,2}^{L,R*}(\alpha_{\mu}-\alpha_{e})+U_{1,3}^{L,R}U_{3,3}^{L,R*}(\alpha_{\tau}-\alpha_{e}))\
.$ (61)

where $g_{L,R}$ are the usual SM couplings. Using Eq.40, the couplings to
$Z_{0}$ are obtained via multiplication by $f\frac{m_{Z}^{2}}{M_{KK}^{2}}$,
that is

$g^{l_{i}l_{j}}_{L,R}=\frac{-fm_{z}^{2}}{M_{KK}^{2}}g^{(1)l_{i}l_{j}}_{L,R}\
,$ (62)

With the above equations, it is now relatively simple to derive the Effective
Lagrangian is given in Eq.39. The Wilson coefficients $g_{3,4,5,6}$ given in
the Effective Lagrangian arise from the processes with the exchange of
$Z_{(0)}$ and $Z_{(1)}$ bosons. These can be derived as,

$\displaystyle g^{l_{i}l_{j}}_{3}$ $\displaystyle=$ $\displaystyle
2g_{R}\left[f-\alpha_{j}\right]\frac{m_{Z}^{2}}{M_{KK}^{2}}g_{R}^{(1)l_{i}l_{j}},$
$\displaystyle g^{l_{i}l_{j}}_{4}$ $\displaystyle=$ $\displaystyle
2g_{L}\left[f-\alpha_{j}\right]\frac{m_{Z}^{2}}{M_{KK}^{2}}g_{L}^{(1)l_{i}l_{j}},$
$\displaystyle g^{l_{i}l_{j}}_{5}$ $\displaystyle=$ $\displaystyle
2g_{L}\left[f-\alpha_{j}\right]\frac{m_{Z}^{2}}{M_{KK}^{2}}g_{R}^{(1)l_{i}l_{j}},$
$\displaystyle g^{l_{i}l_{j}}_{6}$ $\displaystyle=$ $\displaystyle
2g_{R}\left[f-\alpha_{j}\right]\frac{m_{Z}^{2}}{M_{KK}^{2}}g_{L}^{(1)l_{i}l_{j}},$
(63)

where $i$ and $j$ are $(e,\mu,\tau)$. The first term in the above equation are
computed from the $Z_{0}$ exchange, while the second is from $Z_{1}$ exchange.
The relevant branching fractions for the process $\l_{i}\rightarrow 3\l_{j}$
now become [25, 18],

$BR(\l_{i}\rightarrow
3l_{j})=2\left(|g^{\l_{i}\l_{j}}_{3}|^{2}+|g^{\l_{i}\l_{j}}_{4}|^{2}\right)+|g^{\l_{i}\l_{j}}_{5}|^{2}+|g^{\l_{i}\l_{j}}_{6}|^{2}\
.$ (64)

###  VI.3 Numerical Analysis

Here we present the numerical analysis of the Little RS parameter space, to
determine how accurately the Little RS geometric origin of flavour can be
tested in current and future experiments. We will discuss the full parameter
scan of the bulk mass parameters, namely ‘c’ values, that fit the lepton mass
within the experimental error. We have assumed anarchic Yukawa in the lepton
sector. We choose the basis in which the left-handed mixing matrix $U_{L}=I$
and $U_{R}$, the right-handed mixing matrix, contains the mixing elements.
This means that the flavour violation are generated in the right-handed sector
and hence the model is independent from fitting the $U_{PMNS}$ matrix, which
should happen once neutrino phenomenology is modelled in Little RS.

The $3\times 3$ complex matrices of 5-dimensional Yukawa couplings $Y_{e}$
contains 9 real and 9 complex elements. Since we have assumed left-handed
sector to be aligned with SM, the 5-dimensional Yukawa should also be assumed
to have some symmetry. Which reduces the total free parameters to 6 real and 3
complex phases. For simplicity, we also choose the basis in which the bulk
mass parameters are diagonal and real.

We restrict the 5-dimensional Yukawa couplings to the range
$0.1\leq|Y_{i,j}|\leq 3$ so that the values do not introduce unnatural
hierarchies and remain below the perturbative limit in the model. Ignoring the
mixing of fermionic KK-modes, for the time being, we can write,

$\zeta=U_{L}^{\dagger}Y_{4D}U_{R}\ ,$ (65)

where, $Y_{4D}$ are the 4D Yukawa coupling defined in Eq.II.2.3 and $\zeta$ is
defined as

$\zeta=\frac{\sqrt{2}}{v}diag(m_{e},m_{\mu},m_{\tau}).$ (66)

Using this, we compute the 5-dimensional Yukawa coupling as

$(Y_{5D})_{i,j}=f^{(0)-1}(c_{\ell i})(U_{L}\zeta
U_{R}^{\dagger})f^{(0)-1}(c_{ej})\ .$ (67)

Where $U_{L}$ and $U_{R}$ are the lepton mixing matrices.

$c_{\ell 1}$ | $c_{\ell 2}$ | $c_{\ell 3}$ | $c_{e1}$ | $c_{e2}$ | $c_{e3}$
---|---|---|---|---|---
2.15-2.25 | 1.1-1.2 | 1-1.05 | 0.8-0.95 | 0.65-0.8 | 0.55-0.70
Table 4: The of bulk mass parameter $`c^{\prime}$ used in our scan.

We have run the scan over $10^{6}$ iterations and collected 5000 points in the
Little RS parametric space satisfying the anarchic Yukawa conditions. The
range of $c$ values satisfying the above condition given in Table4 Using these
points, we calculated the bounds on $M_{KK}$ for all decays mentioned in
sec.III.

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