# Lepton-flavor violating decays of neutral Higgs to muon and tauon in
supersymmetric economical 3-3-1 model

P.T.Giang L.T.Hue D.T.Huong H.N.Long Institute of Physics, VAST, 10 Dao Tan,
Ba Dinh, Hanoi, Vietnam

###### Abstract

We investigate Lepton-Flavor Violating (LFV) decays of Higgs to muon-tau in
the Supersymmetric Economical 3-3-1 (SUSYE331) model. In the presence of
flavor mixing in sleptons $\\{\tilde{\mu},\tilde{\tau}\\}$ and large values of
$v/v^{\prime}$, the ratio of
$Br(H\rightarrow\tau^{+}\mu^{-})/Br(H\rightarrow\tau^{+}\tau^{-})$ can reach
non-negligible values $\mathcal{O}(10^{-3})$, as in many known SUSY models. We
predict that for the Standard Model Higgs boson, the LHC may detect its decay
to muon and tauon. We also investigate the asymmetry between left and right
LFV values of corrections and prove that the LFV effects are dominated by the
left FLV term, which is $\mathcal{O}(10^{3})$ times larger than the right LFV
term in the limit of small values of $|\mu_{\rho}|/m_{SUSY}$. The
contributions of Higgs-mediated effects to the decay
$\tau\rightarrow\mu\mu\mu$ are also discussed.

###### keywords:

Supersymmetric models, Decays of taus, Supersymmetric Higgs bosons

###### PACS:

12.60.Jv , 13.35.Dx, 14.80.Da

††thanks: Email<EMAIL_ADDRESS>Email:
<EMAIL_ADDRESS>Email<EMAIL_ADDRESS>Email:
<EMAIL_ADDRESS>
##  1 Introduction

The experimental evidences of non-zero neutrino masses and mixing [1] have
shown that the Standard Model (SM) of fundamental particles and interactions
must be extended. Among many extensions of the SM known today, the models
based on gauge symmetry
$\mathrm{SU}(3)_{C}\otimes\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}$ (called
3-3-1 models) [2, 3] have interesting features. The model requires that the
number of fermion families $N$ be a multiple of the quark color in order to
cancel anomalies, which suggests an interesting connection between the number
of flavors and the strong color group. If further one uses the condition of
QCD asymptotic freedom, which is valid only if the number of families of
quarks is to be less than five, it follows that $N$ is equal to 3. In
addition, the third quark generation has to be different from the first two,
so this leads to the possible explanation of why top quark is
uncharacteristically heavy (see, for example, [4]). The 3-3-1 models can also
provide a solution of electric charge quantization observed in the nature [5].

In one of the 3-3-1 models [3] three $\mathrm{SU}(3)_{L}$ lepton triplets are
of the form $(\nu_{l},l,\nu_{l}^{c})_{L}$, where $\nu_{l}^{c}$ is related to
the right-handed component of the neutrino field $\nu_{l}$ (a model with
right-handed neutrinos). The scalar sector of this model requires three Higgs
triplets, and it is interesting to note that two Higgs triplets has the same
$\mathrm{U}(1)_{X}$ charge with two neutral components at their top and
bottom. Giving all neutral Higgs fields a vacuum expectation value (VEV), we
can remove one Higgs triplet. Hence the Higgs sector of the obtained model
becomes minimal, and it has been called the economical 3-3-1 model [6].

The lepton-flavor is absolutely conserved in the SM. Recently, experiments on
neutrino oscillations have proved that lepton flavor is not conserved. It
leads to motivation on a search for the signals of lepton flavor violations
(LFV) beyond the SM. Many versions of the extension of Minimal Supersymmetric
Standard Model (MSSM) with large $\tan\beta$ have been investigated in Higgs
LFV decay. The interesting here is there exists parameter space that predicts
the branching ratio of these types of decays are very sizable, enough to be
detected by present colliders such as CERN Large Hadron Collider (LHC) [7] or
International Linear Collider (ILC) [8]. For example, the SM [9] predicted
that branching ratio of $H\rightarrow\mu\tau$ is very suppressed. However, in
the beyond SM, this ratio can reach large values, more than $10^{-4}$. In
particular, Refs. [10, 11] showed that, in the MSSM,
$BR(H\rightarrow\mu^{+}\tau^{-})\sim 10^{-4}$ if $m_{H}/M_{SUSY}\sim 10^{-1}$
. The Minimal Supersymmetric Neutrino Seesaw Models ($\nu$MSSM) [12] predicted
the branching ratio of heavy Higgs LFV decay is of order $10^{-4}$ while that
of light Higgs LFV decay is of order $10^{-8}$. For more discussions in
details about the Higgs LFV decay, readers are referred to [13] for general
LFV framework, to [14] for two Higgs doublet models, to [13, 14, 15, 16, 17,
18, 19] for MSSM, and $\nu$MSSM and to [20] for little Higgs models (LTH).

The MSSM has shown that in the limit of large $\tan\beta$, the radiative
corrections become non-negligible in many Higgs LFV decay processes. For
example, refs. [15, 17] showed that the ratio $R_{b/t}\equiv
BR(H\rightarrow\bar{b}b)/BR(H\rightarrow\bar{t}t)$ can be distinguished
between the MSSM and non-supersymmetric models. The main reason is that the
Higgs boson couplings to down-type fermions receive a large corrections
enhanced by $\tan\beta$. It leads to many interesting decay processes in quark
sector such as $b\rightarrow s\gamma$ [21, 22]. The large value of $\tan\beta$
also leads to many interesting effects in the lepton sector, especially when
the LFV source in sleptons is included.

Recently, the Supersymmetric Economical 3-3-1 model (SUSYE331) has been
constructed [23]. Apart from interesting features that mentioned in refs. [23,
24, 25, 26], the scalar sector is minimal, and therefore it has been called
the economical. In a series of works [23, 24, 25, 26], we have developed and
proved that the non-supersymmetric version [6] and supersymmetric version are
consistent, realistic and very rich in physics. In the previous work [24], we
skip the LFV source in the soft sector. However, the model predicts more
interesting phenomenology if there exists LFV source in the soft breaking
terms. In this paper, we will concentrate on LFV Higgs decays to $\mu\tau$
with the presence of misalignment of sleptons $\\{\tilde{\mu},\tilde{\tau}\\}$
and their sneutrinos contained in soft breaking terms. In SUSYE331 model, for
generating fermion masses as well as canceling anomaly, one needs four Higgs
triplets. In particular, the ”up” $\rho^{0}$ Higgs gives mass for neutrinos
and the remain, ”down” $\rho^{\prime 0}$, gives mass for charged leptons [23,
25] and other Higgs give mass for quarks.The ratio of VEVs, namely
$\frac{<\rho^{0}>}{<\rho^{\prime 0}>}$, is denoted by $\tan\gamma$ which is
similar to $\tan\beta$ in MSMS. Hence,the $\rho^{0}$ and $\rho^{\prime 0}$
Higgs play very important roles if we consider effects of radiative correction
in lepton sector in the limit of large $\tan\gamma$. The corrections may cause
many non-negligible effects, such as the correction of lepton mass, branching
ratio of LFV Higgs decay…On the other hand, the model 331 is the extension of
SM based on extended gauge symmetry. Therefore, comparing to MSSM, the
SUSYE331 model contains new gauge bosons and new Higgses as well as their
superpartners. Because of appearing of new particles, the number of diagrams
contributing to LFV Higgs decay in SUSYE331 model is predicted more than that
in MSSM. It leads to LFV in Higgs decay effected in SUSYE331 may be larger
than in MSSM. Hence, in this work, we investigate the flavor violating Higgs
coupling in SUSYE331 model, specially we focus on the $\\{\mu,\tau\\}$
generations.

Our work is arranged as follows: In Section 2, we review the particle content
in SUSYE331 model. The analytic expressions of the Higgs effective couplings
are studied in Section 3. In Section 4, we study a numerical estimation on
decay $H\rightarrow\mu\tau$ at colliders and compare contribution from the
left and right LFV radiative corrections into the mentioned decay. In this
section, we also consider the contributions of Higgs exchange to branching
ratio of $\tau\rightarrow 3\mu$ decay. In the last section, we summarize our
main results.

##  2 Particle content

Let us give brief report on the particle content in SUSYE331 model [23]. The
superfields in the anomaly-free model are given by

$\widehat{L}_{aL}=\left(\widehat{\nu}_{a},\widehat{l}_{a},\widehat{\nu}^{c}_{a}\right)^{T}_{L}\sim(1,3,-1/3),\hskip
14.22636pt\widehat{l}^{c}_{aL}\sim(1,1,1),\hskip 14.22636pta=1,2,3$ (1)
$\widehat{Q}_{1L}=\left(\widehat{u}_{1},\ \widehat{d}_{1},\
\widehat{u}^{\prime}\right)^{T}_{L}\sim(3,3,1/3),$ $\widehat{u}^{c}_{1L},\
\widehat{u}^{\prime
c}_{L}\sim(3^{*},1,-2/3),\widehat{d}^{c}_{1L}\sim(3^{*},1,1/3),$ (2)
$\begin{array}[]{ccc}\widehat{Q}_{\alpha
L}=\left(\widehat{d}_{\alpha},-\widehat{u}_{\alpha},\widehat{d^{\prime}}_{\alpha}\right)^{T}_{L}\sim(3,3^{*},0),\hskip
14.22636pt\alpha=2,3,\end{array}$ (3) $\widehat{u}^{c}_{\alpha
L}\sim\left(3^{*},1,-2/3\right),\hskip 14.22636pt\widehat{d}^{c}_{\alpha L},\
\widehat{d}^{\prime c}_{\alpha L}\sim\left(3^{*},1,1/3\right),$ (4)
$\widehat{\chi}=\left(\widehat{\chi}^{0}_{1},\widehat{\chi}^{-},\widehat{\chi}^{0}_{2}\right)^{T}\sim(1,3,-1/3),\hskip
14.22636pt\widehat{\rho}=\left(\widehat{\rho}^{+}_{1},\widehat{\rho}^{0},\widehat{\rho}^{+}_{2}\right)^{T}\sim(1,3,2/3),$
(5) $\widehat{\chi}^{\prime}=\left(\widehat{\chi}^{\prime
o}_{1},\widehat{\chi}^{\prime+},\widehat{\chi}^{\prime
o}_{2}\right)^{T}\sim(1,3^{*},1/3),\,\widehat{\rho}^{\prime}=\left(\widehat{\rho}^{\prime-}_{1},\widehat{\rho}^{\prime
o},\widehat{\rho}^{\prime-}_{2}\right)^{T}\sim(1,3^{*},-2/3).$ (6)

Here we use some new notations as
$\widehat{\psi}^{c}_{L}=(\widehat{\psi}_{R})^{c}\equiv\widehat{\psi}_{R}^{\dagger}$
and exotic quarks are denoted by usual quarks with prime-superscripts
($u^{\prime}$ with the electric charge $q_{u^{\prime}}=2/3$ and $d^{\prime}$
with $q_{d^{\prime}}=-1/3$). The values in each parenthesis show corresponding
quantum numbers of the
$(\mathrm{SU}(3)_{c},\mathrm{SU}(3)_{L},\mathrm{U}(1)_{X})$ symmetry. In this
model, the $\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}$ gauge group is broken
via two steps:

$\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}\stackrel{{\scriptstyle
w,w^{\prime}}}{{\longrightarrow}}\
\mathrm{SU}(2)_{L}\otimes\mathrm{U}(1)_{Y}\stackrel{{\scriptstyle
v,v^{\prime},u,u^{\prime}}}{{\longrightarrow}}\mathrm{U}(1)_{Q},$ (7)

where the VEVs are defined by

$\sqrt{2}\langle\chi\rangle^{T}=\left(u,0,w\right),\hskip 14.22636pt\hskip
14.22636pt\sqrt{2}\langle\chi^{\prime}\rangle^{T}=\left(u^{\prime},0,w^{\prime}\right)$
(8) $\sqrt{2}\langle\rho\rangle^{T}=\left(0,v,0\right),\hskip 14.22636pt\hskip
14.22636pt\sqrt{2}\langle\rho^{\prime}\rangle^{T}=\left(0,v^{\prime},0\right).$
(9)

The vector superfields $\widehat{V}_{c}$, $\widehat{V}$ and
$\widehat{V}^{\prime}$ containing the usual gauge bosons are given in [23,
25]. The supersymmetric model possessing a general Lagrangian is studied in
[25]. In the following, only terms relevant to our calculations are displayed.

##  3  Higgs-muon-tauon effective interactions

In the SUSYE331 model [23, 25], at the tree level, the down-type leptons
($e,\mu,\tau$) only couple to the neutral Higgs $(\rho^{\prime 0})$ through
the Yukawa interaction given by

$\displaystyle\mathcal{L}_{llH}=-\frac{\lambda_{1ab}}{3}(L_{aL}l^{c}_{bL}\rho^{\prime
0}+\rm H.c.).$ (10)

In general case, $\lambda_{1ab}\neq 0$, the Lagrangian given in (10) not only
provides mass for the charged leptons but also gives the source of the lepton
flavor mixing at the tree level. It means that if the couplings
$\lambda_{1ab}\neq 0$ with $(a\neq b)$, the LFV processes, such as
Higgs$\rightarrow\mu\tau$, must be existed. In this case, our theory predicts
very large branching ratios of LFV processes which exceed to experimental
results discussed in [27]. Hence, in the following calculation, we skip the
$\lambda_{1ab}$ with $(a\neq b$) in (10) .

Let us consider another source of LFV which is caused by slepton mixing. More
details of slepton mixing, one can find in Appendix C. Because of slepton
mixing, the leading effective interactions of leptons with
$\rho^{o},\rho^{\prime o}$ Higgs can appeare at the one-loop order. In this
paper, we will concentrate only on the couplings of Higgs with
$\\{\mu,\tau\\}$ leptons.

In order to consider the $\mu,\tau$ flavor mixing at the one loop level, first
we rewrite the original Lagrangian (10) in terms of two component spinor
notations which are familiar to those in literature, namely

$\displaystyle-\mathcal{L}_{0\mu\tau}=\left(Y_{\mu}\mu^{c}_{L}\mu_{L}+Y_{\tau}\tau^{c}_{L}\tau_{L}\right)\rho^{\prime
0}+\mathrm{H.c.},$ (11)

where $Y_{\mu}\equiv\lambda_{122}/3,Y_{\tau}\equiv\lambda_{133}/3$.
At the one-loop level, if we skip all of the terms which are proportional to
$Y_{\mu}$ except terms contributing to mass of muon, then Yukawa interactions
containing Higgs-lepton-lepton couplings can be divided into two parts:

  * •

The lepton-flavor conversing (LFC) part given by

$\displaystyle-\Delta\mathcal{L}_{FC}$ $\displaystyle=$
$\displaystyle\left(Y_{\mu}\Delta^{1\rho}_{\mu}+Y_{\tau}\Delta^{2\rho}_{\mu}\right)\mu^{c}_{L}\mu_{L}\rho^{0*}+Y_{\tau}\Delta^{\rho}_{\tau}\tau^{c}_{L}\tau_{L}\rho^{0*}$
(12) $\displaystyle+$
$\displaystyle\left(Y_{\mu}\Delta^{1\rho^{\prime}}_{\mu}+Y_{\tau}\Delta^{2\rho^{\prime}}_{\mu}\right)\mu^{c}_{L}\mu_{L}\rho^{\prime
0}+Y_{\tau}\Delta^{\rho^{\prime}}_{\tau}\tau^{c}_{L}\tau_{L}\rho^{\prime
0}+\mathrm{H.c.},$

  * •

The lepton-flavor violating (LFV) part given as

$\displaystyle-\Delta\mathcal{L}_{FV}$ $\displaystyle=$ $\displaystyle
Y_{\tau}\left(\Delta^{\rho}_{L}\tau^{c}_{L}\mu_{L}+\Delta^{\rho}_{R}\mu^{c}_{L}\tau_{L}\right)\rho^{0*}$
(13) $\displaystyle+$ $\displaystyle
Y_{\tau}\left(\Delta^{\rho^{\prime}}_{L}\tau^{c}_{L}\mu_{L}+\Delta^{\rho^{\prime}}_{R}\mu^{c}_{L}\tau_{L}\right)\rho^{\prime
0}+\rm H.c.,$

where all of
$\Delta^{1\rho}_{\mu},\Delta^{2\rho}_{\mu},\Delta^{1\rho^{\prime}}_{\mu},\Delta^{2\rho^{\prime}}_{\mu},\Delta^{\rho}_{\tau},\Delta^{\rho^{\prime}}_{\tau},\Delta^{\rho}_{L},\Delta^{\rho^{\prime}}_{L},\Delta^{\rho}_{R}$
and $\Delta^{\rho^{\prime}}_{R}$ are the leading effective couplings.
From now on, for convenience, we use notation $\Delta$ to imply any radiative
correction of couplings appearing in (12) and (13). Note that $\Delta$ is a
dimensionless function of mass parameters and $\Delta^{\rho}_{\mu}$,
$\Delta^{\rho}_{\tau}$ are non-zero value even if we assume that there is no
flavor mixing in slepton sector. We emphasize that $\Delta^{\rho}_{\tau}$ is
one of quantities affecting on many observable quantities such as the ratio of
branching ratios $Br(H\rightarrow b\bar{b})/BR(H\rightarrow\tau\bar{\tau})$.
The contribution of $\Delta^{\rho}_{\tau}$ to that of branching ratios in the
SUSY model is studied in [15, 17]. The diagrams which contribute to all of
$\Delta$s are drawn in Appendix A.

Now let us construct the total effective Lagrangian for Higgs, muon and tauon
couplings in terms of physical eigenstates. First we write down the whole
Lagrangian coming from all of Eqs. (11), (12) and (13) in the matrix form

$\displaystyle-\mathcal{L}$ $\displaystyle=$ $\displaystyle
Y_{\tau}\left(\begin{array}[]{cc}\mu^{c}_{L}&\tau^{c}_{L}\\\
\end{array}\right)\mathcal{Y}_{l_{1}}\left(\begin{array}[]{c}\mu_{L}\\\
\tau_{L}\\\ \end{array}\right)~{}\rho^{\prime
0}+Y_{\tau}\left(\begin{array}[]{cc}\mu^{c}_{L}&\tau^{c}_{L}\\\
\end{array}\right)\mathcal{Y}_{l_{2}}\left(\begin{array}[]{c}\mu_{L}\\\
\tau_{L}\\\ \end{array}\right)\rho^{0*}+\mathrm{H.c.},$ (20)

where $\mathcal{Y}_{l_{1}}$ and $\mathcal{Y}_{l_{2}}$ are matrices defined by
the following formulas:

$\displaystyle\mathcal{Y}_{l_{1}}=\left(\begin{array}[]{cc}\Delta^{o\rho^{\prime}}_{\mu}&\Delta^{\rho^{\prime}}_{R}\\\
\Delta^{\rho^{\prime}}_{L}&1+\Delta^{\rho^{\prime}}_{\tau}\\\
\end{array}\right);\hskip 14.22636pt\hskip
14.22636pt\mathcal{Y}_{l_{2}}=\left(\begin{array}[]{cc}\Delta^{o\rho}_{\mu}&\Delta^{\rho}_{R}\\\
\Delta^{\rho}_{L}&\Delta^{\rho}_{\tau}\\\ \end{array}\right),$ (25)

with $y\equiv Y_{\mu}/Y_{\tau}$ , $\Delta^{o\rho}_{\mu}\equiv
y\Delta^{1\rho}_{\mu}+\Delta^{2\rho}_{\mu}$ and
$\Delta^{o\rho^{\prime}}_{\mu}\equiv
y+y\Delta^{1\rho^{\prime}}_{\mu}+\Delta^{2\rho^{\prime}}_{\mu}$.

Because of loop corrections, the mass matrix of the $\mu,\tau$ in (20) is no
longer diagonal. In order to find the physical eigenstates of muon and tauon,
we expand the Higgs $\rho$ and $\rho^{\prime}$ around the vacuum expectation
values. As a consequence, the mixing mass matrix for the muon and tauon are

$\displaystyle-\mathcal{L}_{mass}$ $\displaystyle=$ $\displaystyle
Y_{\tau}v^{\prime}\left(\begin{array}[]{cc}\mu^{c}_{L}&\tau^{c}_{L}\\\
\end{array}\right)\mathcal{Y}_{l}\left(\begin{array}[]{c}\mu_{L}\\\
\tau_{L}\\\ \end{array}\right)+\mathrm{H.c.},$ (29)

where

$\displaystyle\mathcal{Y}_{l}$ $\displaystyle\equiv$
$\displaystyle\mathcal{Y}_{l_{1}}+t_{\gamma}\mathcal{Y}_{l_{2}}=(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma})\left(\begin{array}[]{cc}\epsilon_{\mu}&\epsilon_{R}\\\
\epsilon_{L}&1\\\ \end{array}\right)$ (32) $\displaystyle=$
$\displaystyle(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma})Y_{\epsilon},$
(33)

with

$\displaystyle t_{\gamma}$ $\displaystyle\equiv$
$\displaystyle\tan\gamma=\frac{v}{v^{\prime}}=\frac{\langle\rho^{0}\rangle}{\langle\rho^{{}^{\prime}0}\rangle},\hskip
14.22636pt\epsilon_{\mu}\equiv\frac{\Delta^{o\rho^{\prime}}_{\mu}+\Delta^{o\rho}_{\mu}t_{\gamma}}{1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma}},$
$\displaystyle\epsilon_{L}$ $\displaystyle\equiv$
$\displaystyle\frac{\Delta^{\rho^{\prime}}_{L}+\Delta^{\rho}_{L}t_{\gamma}}{1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma}},\hskip
14.22636pt\epsilon_{R}\equiv\frac{\Delta^{\rho^{\prime}}_{R}+\Delta^{\rho}_{R}t_{\gamma}}{1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma}}$
(34)

and

$\displaystyle
Y_{\epsilon}=\left(\begin{array}[]{cc}\epsilon_{\mu}&\epsilon_{R}\\\
\epsilon_{L}&1\\\ \end{array}\right)$ (37)

It is easy to see that the mixing mass matrix of muon and tauon given in (37)
is a general matrix. Finding the mass eigenvalues of left-right leptons is
equivalent to finding a matrix $C$ satisfying:

$\displaystyle
C^{\dagger}\mathcal{Y}_{\epsilon}^{\dagger}\mathcal{Y}_{\epsilon}C$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}y_{\mu}^{2}&0\\\
0&y^{2}_{\tau}\\\ \end{array}\right)\equiv\mathcal{Y}^{2}_{d}.$ (40)

In our theory, the matrix $C$ can be found in a form

$\displaystyle C=\left(\begin{array}[]{cc}c_{\Lambda}&s_{\Lambda}\\\
-s_{\Lambda}&c_{\Lambda}\\\ \end{array}\right),$ (43)

where $c_{\Lambda}\equiv\cos\Lambda,s_{\Lambda}\equiv\sin\Lambda$ and
$\Lambda$ is the rotation angle given by

$\displaystyle
t_{2\Lambda}\equiv\tan(2\Lambda)=\frac{2(\epsilon_{\mu}\epsilon_{R}+\epsilon_{L})}{1+\epsilon^{2}_{R}-(\epsilon^{2}_{\mu}+\epsilon^{2}_{L})}.$
(44)

In addition, $\mathcal{Y}_{d}=\mathrm{diag}(y_{\mu},~{}y_{\tau})$ in which
$(y_{\mu},~{}y_{\tau})$ are defined as follows

$\displaystyle y^{2}_{\mu}$ $\displaystyle=$ $\displaystyle
r^{\prime}-rs^{2}_{\Lambda},\hskip
14.22636pty^{2}_{\tau}=r^{\prime}+rc^{2}_{\Lambda},$ (45)

where

$\displaystyle r^{2}$ $\displaystyle\equiv$ $\displaystyle
4(\epsilon_{\mu}\epsilon_{R}+\epsilon_{L})^{2}+\left[1+\epsilon^{2}_{R}-(\epsilon^{2}_{\mu}+\epsilon^{2}_{L})\right]^{2},\hskip
14.22636ptr^{\prime}\equiv\epsilon^{2}_{\mu}+\epsilon^{2}_{L}~{}.$ (46)

Note that the mass eigenvalues of muon and tauon are proportional to
$(y_{\mu},y_{\tau})$, namely

$\displaystyle
m_{\mu}=y_{\mu}Y_{\tau}v^{\prime}(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}),\hskip
14.22636ptm_{\tau}=y_{\tau}Y_{\tau}v^{\prime}(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}).$
(47)

On the other hand, the mass eigenstates of leptons $(\mu,\tau)$ and
$(\mu^{c},\tau^{c})$ are determined from two transformations

$\displaystyle L^{c}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\mu^{c}\\\ \tau^{c}\\\
\end{array}\right)=(U_{l})^{T}\left(\begin{array}[]{c}\mu^{c}_{L}\\\
\tau^{c}_{L}\\\ \end{array}\right)=U^{T}_{l}L^{c}_{L},$ (52) $\displaystyle L$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}\mu\\\ \tau\\\
\end{array}\right)=V_{l}\left(\begin{array}[]{c}\mu_{L}\\\ \tau_{L}\\\
\end{array}\right)=V_{l}L_{L},$ (57)

where $U_{l}$ and $V_{l}$ have come from (40), namely

$\displaystyle U^{\dagger}_{l}$ $\displaystyle=$
$\displaystyle\mathcal{Y}^{-1}_{d}C^{\dagger}\mathcal{Y}^{\dagger}_{\epsilon}=\left(\begin{array}[]{cc}\frac{1}{y_{\mu}}&0\\\
0&\frac{1}{y_{\tau}}\\\
\end{array}\right)\left(\begin{array}[]{cc}c_{\Lambda}&-s_{\Lambda}\\\
s_{\Lambda}&c_{\Lambda}\\\
\end{array}\right)\left(\begin{array}[]{cc}\epsilon_{\mu}&\epsilon_{L}\\\
\epsilon_{R}&1\\\ \end{array}\right),$ (64) $\displaystyle V_{l}^{\dagger}$
$\displaystyle=$ $\displaystyle
C=\left(\begin{array}[]{cc}c_{\Lambda}&s_{\Lambda}\\\
-s_{\Lambda}&c_{\Lambda}\\\ \end{array}\right).$ (67)

Next, we replace $\mathcal{Y}_{l_{1}}$ in Eq. (20) by a new form deduced from
Eq.(33)

$\mathcal{Y}_{l_{1}}=(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma})Y_{\epsilon}-\mathcal{Y}_{l_{2}}t_{\gamma}$

Now we have obtained a new expression of (20) as follows

$\displaystyle-\mathcal{L}$ $\displaystyle=$ $\displaystyle
Y_{\tau}(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma})\left(\begin{array}[]{cc}\mu^{c}_{L}&\tau^{c}_{L}\\\
\end{array}\right)\mathcal{Y}_{\epsilon}\left(\begin{array}[]{c}\mu_{L}\\\
\tau_{L}\\\ \end{array}\right)~{}\rho^{\prime o}$ (71) $\displaystyle+$
$\displaystyle Y_{\tau}\left(\begin{array}[]{cc}\mu^{c}_{L}&\tau^{c}_{L}\\\
\end{array}\right)\mathcal{Y}_{l_{2}}\left(\begin{array}[]{c}\mu_{L}\\\
\tau_{L}\\\ \end{array}\right)(\rho^{o*}-t_{\gamma}\rho^{\prime
o})+\mathrm{H.c.},$ (75)

In the basis of mass eigenstates of the muon and tauon given in Eq. (57), the
Lagrangian (75) transforms into

$\displaystyle-\mathcal{L}_{d}$ $\displaystyle=$ $\displaystyle
Y_{\tau}(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma})L^{cT}~{}\mathcal{Y}_{d}~{}L\rho^{\prime
o}$ (76) $\displaystyle+$ $\displaystyle
Y_{\tau}L^{cT}(U^{\dagger}_{l}\mathcal{Y}_{2}V_{l}^{\dagger})L(\rho^{o*}-t_{\gamma}\rho^{\prime
o})+\mathrm{H.c.}$

It is needed to emphasize that the first term in Eq. (76) generates only
masses for muon and tauon while the second creates masses as well as give rise
to the lepton flavor mixing. Sources of flavor mixing are two off-diagonal
elements of the matrix $(U^{\dagger}_{l}\mathcal{Y}_{2}V_{l}^{\dagger})$ :

$\displaystyle\left(U^{\dagger}_{l}\mathcal{Y}_{l_{2}}V^{\dagger}\right)_{12}$
$\displaystyle=$
$\displaystyle\frac{c^{2}_{\Lambda}\Delta^{\rho}_{R}\epsilon_{\mu}}{y_{\mu}}+\frac{(c^{2}_{\Lambda}\epsilon_{L}-c_{\Lambda}s_{\Lambda})\Delta^{\rho}_{\tau}}{y_{\mu}}$
(77) $\displaystyle+$
$\displaystyle\frac{c_{\Lambda}s_{\Lambda}(\Delta^{\rho}_{L}\epsilon_{L}+\Delta^{o\rho}_{\mu}\epsilon_{\mu}-\Delta^{\rho}_{R}\epsilon_{R})-s^{2}_{\Lambda}(\Delta^{\rho}_{L}+\epsilon_{R}\Delta^{\rho}_{R})}{y_{\mu}},$
$\displaystyle\left(U_{l}^{\dagger}\mathcal{Y}_{l_{2}}V^{\dagger}\right)_{21}$
$\displaystyle=$
$\displaystyle\frac{c^{2}_{\Lambda}\Delta^{\rho}_{L}}{y_{\tau}}+\frac{c^{2}_{\Lambda}\Delta^{o\rho}_{\mu}\epsilon_{R}-c_{\Lambda}s_{\Lambda}\Delta^{\rho}_{\tau}}{y_{\tau}}$
(78) $\displaystyle+$
$\displaystyle\frac{c_{\Lambda}s_{\Lambda}(\Delta^{\rho}_{L}\epsilon_{L}+\Delta^{o\rho}_{\mu}\epsilon_{\mu}-\Delta^{\rho}_{R}\epsilon_{R})-s^{2}_{\Lambda}(\Delta^{\rho}_{\tau}\epsilon_{L}+\epsilon_{\mu}\Delta^{\rho}_{R})}{y_{\tau}}$

In the further calculations, we consider a case of $(t_{\gamma}\Delta)\ll 1$
but large enough (as investigated in MSSM) to cause many interesting effects,
and we will comment more details after some numerical calculations. On the
other hand, the rotation angle given in Eq. (44) is very small, so we can set
$c_{\Lambda}\simeq 1,s_{\Lambda}\simeq\Lambda$. As a result, Eqs. (45), (77)
and (78) can be presented as very simple formulas:

$\displaystyle y_{\mu}$ $\displaystyle\simeq$
$\displaystyle\epsilon_{\mu},\hskip 14.22636pty_{\tau}\simeq 1,$
$\displaystyle\left(U^{\dagger}_{l}\mathcal{Y}_{l_{2}}V^{\dagger}\right)_{12}$
$\displaystyle\simeq$ $\displaystyle\Delta^{\rho}_{R},\hskip
14.22636pt\left(U^{\dagger}_{l}\mathcal{Y}_{l_{2}}V^{\dagger}\right)_{21}\simeq\Delta^{\rho}_{L},$
(79)

and the above LFV Lagrangian also appears in a simple form:

$\displaystyle-\mathcal{L}_{FV}$ $\displaystyle\simeq$ $\displaystyle
Y_{\tau}(\Delta^{\rho}_{R}\mu^{c}\tau+\Delta^{\rho}_{L}\tau^{c}\mu)(\rho^{0*}-t_{\gamma}\rho^{\prime
0})+\mathrm{H.c.}.$ (80)

Finally, in the mass-eigenstate basis for both lepton and Higgs, we obtain the
effective LFV Lagrangian:

$\displaystyle-\mathcal{L}_{FV}$ $\displaystyle\simeq$
$\displaystyle\sqrt{2}Y_{\tau}(\Delta^{\rho}_{R}\mu^{c}\tau+\Delta^{\rho}_{L}\tau^{c}\mu)\left(s_{\alpha}s_{\gamma}\phi_{S_{a36}}-c_{\alpha}s_{\gamma}\varphi_{S_{a36}}\right)+\mathrm{H.c.},$
(81)

where $\varphi_{S_{a36}}$ and $\phi_{S_{a36}}$ are the Higgs mass eigenstates
generated from the mixing of two original Higgs bosons $\rho^{0}$ and
$\rho^{\prime 0}$. The expressions of the Higgs mass eigenstates were
introduced in [25]. They are summarized in the Appendix C. The emphasis here
is that in the general supersymmetric model there exist both the leading
interactions of the muon, tauon with neutral scalar and pseudo scalar Higgs.
However, the SUSYE331 model contains only interactions among muon, tauon and
scalar Higgs.

The effective couplings given in (81) are widely investigated for many LFV
low-energy processes, specially in the MSSM [10, 18, 28]. In this paper we
first concentrate on some simple aspects of LFV in the SUSYE331 model. In
particular, we are going to consider the LFV in decays of the scalar Higgs,
i.e. $\Phi^{0}\rightarrow\tau^{\pm}\mu^{\mp}$, where
$\Phi^{0}=\varphi_{S_{a36}}$ or $\phi_{S_{a36}}$. First, we start with
studying the branching ratios of neutral Higgs decay into muon and tauon. The
SUSYE331 model predicts that the formula of these branching ratios is

$\displaystyle BR(\Phi^{0}\rightarrow\tau^{+}\mu^{-})$ $\displaystyle=$
$\displaystyle BR(\Phi^{0}\rightarrow\tau^{-}\mu^{+})$ $\displaystyle=$
$\displaystyle
2(1+\tan^{2}\gamma)\left(\mid\Delta^{\rho}_{L}\mid^{2}+\mid\Delta^{\rho}_{R}\mid^{2}\right)~{}BR(\Phi^{0}\rightarrow\tau^{+}\tau^{-}).$

This result is similar to that one given in [10], except the absence of angle
of mixing among Higgses. In the limit of appropriately large $\tan\gamma$, the
effects of LFV in the Higgs decay processes is not to be ignored. Hence, our
theoretical prediction is not much different from that of previous results
given in [10, 11, 15, 16, 28]. For details, we will study some numerical
calculations for the branching ratios indicated by Eq.(LABEL:Br1). In our
paper, we use the assumption for slepton mixing presented in Appendix C. The
diagrams giving contributions to $\Delta^{\rho}_{R}$ and $\Delta^{\rho}_{L}$
are shown in Fig.1. The relevant vertices to our calculation are presented in
Appendix B.

$\mu$$\tau^{c}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$\small\rho^{0*}$$\tilde{l}_{L_{\alpha}}$$(a)$

$\mu$$\tau^{c}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda^{3}_{A}$$\lambda^{8}_{A}$$\rho^{0*}$$\tilde{l}_{L_{\alpha}}$$(b)$

$\mu$$\tau^{c}$$\tilde{\rho}_{1}^{\prime-}$$\tilde{\rho}_{1}^{+}$$\tilde{W}^{-}$$\tilde{W}^{+}$$\rho^{0*}$$\tilde{\nu}_{L\alpha}$$(c)$

$\mu$$\tau^{c}$$\tilde{\rho}_{2}^{\prime-}$$\tilde{\rho}_{2}^{+}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$\rho^{0*}$$\tilde{\nu}_{R\alpha}$$(d)$

$\mu$$\tau^{c}$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\rho^{0*}$$\tilde{\nu}_{L_{\alpha}}$$\tilde{\nu}_{R_{\beta}}$$(e)$

$\mu$$\tau^{c}$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\rho^{0*}$$\tilde{\nu}_{R_{\alpha}}$$\tilde{\nu}_{L_{\beta}}$$(f)$

$\tau$$\mu^{c}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$\rho^{0*}$$\tilde{l}_{R_{\alpha}}$$(i)$

$\mu$$\tau^{c}$$\lambda_{B}$$\rho^{*0}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$$(k)$

$\tau$$\mu^{c}$$\lambda_{B}$$\rho^{*0}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$$(l)$

Figure 1: Diagrams contributing to $\Delta^{\rho}_{L}$
[$(a),(b),(c),(d),(e),(f),(k)$] and $\Delta^{\rho}_{R}$ [$(i),(l)$].

Using Feynman rules, we can obtain the expression $\Delta_{L}^{\rho}$ from the
diagrams in Fig.1, namely:

$\displaystyle\Delta^{\rho}_{L}$ $\displaystyle=$
$\displaystyle\Delta^{\rho}_{La}+\Delta^{\rho}_{Lb}+\Delta^{\rho}_{Lc}+\Delta^{\rho}_{Ld}+\Delta^{\rho}_{Le}+\Delta^{\rho}_{Lf}+\Delta^{\rho}_{Lk},$
(83)

where

$\displaystyle\Delta^{\rho}_{La}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
2}}{216\pi^{2}}\mu_{\rho}m^{\prime}c_{L}s_{L}\left[I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})-I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right],$
$\displaystyle\Delta^{\rho}_{Lb}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{24\pi^{2}}\mu_{\rho}m_{\lambda}c_{L}s_{L}\left[I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})-I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right],$
$\displaystyle\Delta^{\rho}_{Lc}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}c_{\nu_{L}}s_{\nu_{L}}\left[I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}})-I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}})\right],$
$\displaystyle\Delta^{\rho}_{Ld}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}c_{\nu_{R}}s_{\nu_{R}}\left[I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R2}})-I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R3}})\right],$
$\displaystyle\Delta^{\rho}_{Le}$ $\displaystyle=$
$\displaystyle\frac{(Y_{\nu_{\mu\tau}})h_{\mu\tau}-h_{\tau\mu}\mu_{\rho}}{8\pi^{2}}$
$\displaystyle\times$
$\displaystyle\left[s_{\nu_{(L-R)}}\left(s_{\nu_{L}}s_{\nu_{R}}I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R2}})+c_{\nu_{L}}c_{\nu_{R}}I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R3}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.c_{\nu_{(L-R)}}\left(s_{\nu_{R}}c_{\nu_{L}}I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R2}})-s_{\nu_{L}}c_{\nu_{R}}I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R3}})\right)\right],$
$\displaystyle\Delta^{\rho}_{Lf}$ $\displaystyle=$
$\displaystyle-\Delta^{\rho}_{Le},$ $\displaystyle\Delta^{\rho}_{Lk}$
$\displaystyle=$ $\displaystyle\frac{g^{\prime
2}}{288\pi^{2}}\mu_{\rho}m^{\prime}s_{L}c_{L}\left[s^{2}_{R}\left(I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})-I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})\right)\right.$ (84)
$\displaystyle+$ $\displaystyle\left.c^{2}_{R}\left(I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})-I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right)\right].$

Also, the $\Delta^{\rho}_{R}$ receives contributions from two diagrams ($i$)
and ($l$) of Fig.1 too,

$\displaystyle\Delta^{\rho}_{R}$ $\displaystyle=$
$\displaystyle\Delta^{\rho}_{Ri}+\Delta^{\rho}_{Rl},$ (85)

where

$\displaystyle\Delta^{\rho}_{Ri}$ $\displaystyle=$
$\displaystyle-\frac{g^{\prime
2}}{72\pi^{2}}\mu_{\rho}m^{\prime}c_{R}s_{R}\left[I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{R_{2}})-I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{R_{3}})\right],$
$\displaystyle\Delta^{\rho}_{Rl}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
2}}{288\pi^{2}}\mu_{\rho}m^{\prime}s_{R}c_{R}\left[s^{2}_{L}\left(I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})-I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right)\right.$ (86)
$\displaystyle+$ $\displaystyle\left.c^{2}_{L}\left(I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})-I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right)\right]$

Here we have used some new notations

$\displaystyle s_{\nu_{(L-R)}}\equiv
s_{\nu_{L}}c_{\nu_{R}}-s_{\nu_{R}}c_{\nu_{L}},\hskip
14.22636ptc_{\nu_{(L-R)}}\equiv
s_{\nu_{L}}s_{\nu_{R}}+c_{\nu_{L}}c_{\nu_{R}},$ (87)

where $s_{L}$, $c_{L}$ and $s_{R}$, $c_{R}$ are deduced from mixing angles for
left and right handed sleptons (for details, see Appendix C.2). The same
relations hold for sneutrino sector, with corresponding notations for mixing
angles $s_{\nu_{L}},s_{\nu_{R}},c_{\nu_{L}}$ and $c_{\nu_{R}}$. The function
$I_{3}(x,y,z)$ is similar to that mentioned in literature [11],

$\displaystyle
I_{3}(x,y,z)=\frac{xy\log(x/y)+yz\log(y/z)+zx\log(z/x)}{(x-y)(y-z)(z-x)}.$
(88)

The analytical results appearing in (84) show that contributions from two
diagrams (e) and (f) to the $\Delta^{\rho}_{L}$ always are the same magnitude
but opposite in sign. Therefore the total contribution of these two diagrams
to $\Delta^{\rho}_{L}$ vanishes. On the other hand, results obtained from the
Eq.(84) show that if we neglect the terms of the slepton mixing, namely
$s_{L}=s_{R}=s_{\nu_{L}}=s_{\nu_{R}}=0$, the amounts collected from the
$\Delta^{\rho}_{L,R}$ class diagrams given in Fig. 1 are all zero. This
corresponds to the case of lepton flavor conservation:
$\Delta^{\rho}_{L}=\Delta^{\rho}_{R}=0.$

We also remind that analytical expressions of other $\Delta$ functions can be
found in Appendix A. These results demonstrate that the values
$\Delta^{1\rho}_{\mu},\Delta^{2\rho}_{\mu}$ and $\Delta^{\rho}_{\tau}$ given
by expressions (94), (95) and (96) obtained at one loop approximation, do not
vanish even if we assume no mixing of the sleptons. These quantities create
non-negligible effects of the lepton masses. They are widely discussed in many
previous papers. Another feature of the SUSYE331 model that we would like to
remind here: there are two independent sources (Yukawa coupling at tree level)
to create masses of slepton and neutrino sectors. Hence, contributions to LFV
corrections come from two independent sources: mixing of lepton and sneutrino
sectors. We assume that the model contains both LFV sources.

Before coming to numerical computation section, it is necessary to note that
the formulas of LFV corrections, such as $\Delta$s in this case, have not been
established for the SUSYE331 model before. So let us give some general
comments on the formulas of $\Delta$s which discriminate against those in MSSM
versions:

  * •

At the one loop approximation, the effective couplings $\Delta$s are obtained
from the diagrams such as those listed in Figs. (7) and (8). We can
distinguish two types of diagram which give contribution to $\Delta$s. The
first type of diagram does not include any Higgsino propagators, for example
Fig. 1 (k) and (l), and they are known as pure gaugino-mediated diagrams. The
second, containing at least one Higgsino propagator like remaining diagrams,
is Higgsino-mediated type. In general case, each of these kinds of diagram may
give main contribution to the $\Delta$s depending on regions of mass parameter
space. If each Higgsino-mediated diagram gives the dominated contribution to
$\Delta$s that reach single maximum value. In contrast, each $\Delta$ that
gains values from pure gaugino-mediated diagrams, $\Delta^{\rho}_{Lk}$ for
example, is proportional to $|\mu_{\rho}|$. Additionally, we can see the
analytic expressions of $\Delta$s given in (84), (86) and Appendix A. It is
well known in beyond MSSM theories [10], all of effective couplings $\Delta$s
are obtained from both types of diagrams, except $\Delta^{\prime}_{\mu}$. In
the limit of large values of $|\mu_{\rho}|$, the dominated contributions of
$\Delta$s are caused by pure gaugino-mediated diagrams. This conclusion also
is happened in the SUSYE331. However, in the SUSYE331, there are the
additional $SU(3)_{L}$ gaugino-mediated diagrams. Hence the values of
$\Delta^{\rho}_{L}$ can be changed in comparison with other models. Details of
this difference are discussed in section 4.

  * •

The difference between the predictions of the model under consideration and
other ones due to hypercharge structure of particle content. For example, let
us compare our expressions of $\Delta$s with those of $\Delta$s in MSMS [10].
All contributions to the $\Delta$s obtained from Fig.1, are proportional to
$I_{3}$ functions. Rate coefficients in both models are the same level for
diagrams of Higgsino-mediated type whereas the rate coefficients in the model
under consideration are smaller than that in the MSSM model for diagrams of
pure gaugino-mediated type. As a consequence, the large contribution to the
$\Delta$s from the pure gaugino-mediated type will happen if mass parameters
are large. Furthermore, in this limit of mass parameters, the pure gaugino-
mediated diagrams are the only source giving contribution to radiative
corrections $\Delta^{2\rho}_{\mu}$ of muon mass. It is nature to keep the
ratio $Y_{\tau}/Y_{\mu}$, at one loop correction, to be the same as those at
tree level. This leads to the limit of the mass parameters, which does not
exceed $10$ TeV .

In the next section we will investigate some numerical results. On that basis,
we will compare the effects of the LFV origin in the left- and right-slepton
sectors as well as sneutrino sectors. In order to investigate numerically, we
are going to use results from [25] such as:
$g^{\prime}/g=\frac{3\sqrt{2}s_{W}}{\sqrt{4c^{2}_{W}-1}}$, $s^{2}_{W}=0.2312$
and $\alpha^{-1}_{em}=128$ at the weak scale.

##  4  Numerical results

|
---|---
Figure 2: $|\Delta^{\rho}_{R}|^{2}$ as a function of $|\mu_{\rho}|/\tilde{m}_{R}$ with four different choices of masses ratios: 1) blue curve–$m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 2) green curve–$3m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 3) yellow curve- $m^{\prime}=\tilde{m}_{R}=3\tilde{m}_{L}$; 4) red curve–$m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}/3$. Two black lines correspond to two values $10^{-5}$ and $10^{-3}$ of $|50\Delta^{\rho}_{R}|^{2}$ . |
---|---
Figure 3: $|\Delta^{\rho}_{L}|^{2}$ as a function of
$|\mu_{\rho}|/\tilde{m}_{L}$ with four different choices of masses ratios: 1)
blue curve–$m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 2) green
curve–$3m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 3) yellow curve–
$m^{\prime}=\tilde{m}_{L}=3\tilde{m}_{R}$; 4) red
curve–$m^{\prime}=\tilde{m}_{L}=\tilde{m}_{R}/3$. A black line corresponds to
value $10^{-3}$ of $|50\Delta^{\rho}_{L}|^{2}$.

In this section we firstly discuss some numerical results that relate to any
signals of LFV decays $H\rightarrow\mu\tau$. Let us start with the maximum LFV
in both left and right sectors, especially
$s_{L}c_{L}=s_{R}c_{R}=s_{\nu_{L}}c_{\nu_{L}}=s_{\nu_{R}}c_{\nu_{R}}=0.5$. It
means that we can assign values of mass parameters like:

$\displaystyle\tilde{m}^{2}_{(\tau_{L},\tau_{R},\nu_{\tau_{L}},\nu_{\tau_{R}})}$
$\displaystyle=$
$\displaystyle\tilde{m}^{2}_{(\mu_{L},\mu_{R},\nu_{\mu_{L}},\nu_{\mu_{R}})}$
$\displaystyle=$ $\displaystyle\tilde{m}^{2}_{L,R,\nu_{L},\nu_{R}},$
$\displaystyle\tilde{m}^{2}_{(L_{2},R_{2},\nu_{L_{2}},\nu_{R_{2}})}$
$\displaystyle=$ $\displaystyle 0.2\,\tilde{m}^{2}_{L,R,\nu_{L},\nu_{L}}$

and

$\tilde{m}^{2}_{(L_{3},R_{3},\nu_{L_{3}},\nu_{R_{3}})}=1.8~{}\tilde{m}^{2}_{(L,R,\nu_{L},\nu_{R})},$

where $\tilde{m}^{2}_{(L,R,\nu_{L},\nu_{R})}$ are mass parameters used to
compare with SUSY mass scale $m_{SUSY}$. We would like to emphasize that
branching ratios of Higgs decays to muon and tauon are sizable if $\tan\gamma$
is large enough. Therefore, in the following calculations, we take
$\tan\gamma\sim 50$.

The Fig.2 displays the quantity $|50\Delta^{\rho}_{R}|^{2}$ as a function of
$|\mu_{\rho}|/\tilde{m}_{R}$ while Fig.3 displays the
$|50\Delta^{\rho}_{L}|^{2}$ as a function of $|\mu_{\rho}|/\tilde{m}_{L}$
where all other relevant parameters are fixed. Each curve presented in Figs.2,
3 contains a single peak. All the peaks of the curves are obtained at mass
parameters at which the contribution of the Higgs-mediated diagrams to
$\Delta$s are dominated. Corresponding to each curve, there are two regions of
mass parameter space separated by deep wells. Deep wells, which divide the
parameter space into two parts. The first part, the mass parameters are
located in the right hand side of deep wells. In this region of parameter
space,the pure gaugino-mediated type can give main contribution to $\Delta$s.
The second part, the mass parameters are located in the left handed side of
deep wells at which the dominated contribution to $\Delta$s is obtained by the
Higgs-mediated. All of the maximum points of the curves in the Fig.2 and Fig.3
are reached at $|\mu_{\rho}|/\tilde{m}_{L,R}\sim\mathcal{O}(10^{-1})$ and
these values depend weekly on the changes of values of $\tilde{m}_{L}$ and
$\tilde{m}_{R}$. On the other hand, the maximum values of the
$\Delta^{\rho}_{L}$ is $\mathcal{O}(10^{-3})$, as concerned in the MSSM [10,
12, 18, 29] while the maximum values of the $\Delta^{\rho}_{R}$ are much
smaller than those of $\Delta^{\rho}_{L}$, specifically
max$(|\Delta^{\rho}_{R}|)^{2}/\mathrm{max}(|\Delta^{\rho}_{L}|)^{2}\sim
10^{-3}$. This large difference comes from the symmetry of
$SU(3)_{L}\times\mathrm{U}(1)_{X}$ model. In particular, in the left side of
wells the main contributions to $\Delta^{\rho}_{L}$ of SUSYE331 model come
from the $SU(3)_{L}$ gaugino-mediated diagrams, namely diagrams ((b), (c),
(d)) in Fig.1. In contrast, the main contributions to $\Delta^{\rho}_{R}$ come
from only $\mathrm{U}(1)$ gaugino-mediated diagram. Figs.2 and 3 also show
that both $\Delta^{\rho}_{L}$ and $\Delta^{\rho}_{R}$ are very sensitive with
the changes of $\tilde{m}_{L}$ and $\tilde{m}_{R}$. More details, Fig.4 draws
the dependence of $|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ on
$|\mu_{\rho}|/\tilde{m}_{L}$, where
$m^{\prime}=m_{\lambda}=\tilde{m}_{L}\equiv m_{SUSY}$ and four different fixed
values of $\tilde{m}_{R}$. The maximal and minimal values of the ratio
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ on all the curves in Fig.4
have the same value at different values of $|\mu_{\rho}|/\tilde{m}_{SUSY}$. In
the parameter region where the Higgs-mediated diagrams give dominated
contribution to $\Delta$s, the ratio
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ is very small $(<10^{-3})$.
But in the remaining parameters, that ratio is increased. In the limit
$|\mu_{\rho}|/\tilde{m}_{SUSY}\geq 30$, the ratio
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ reaches a constant value.
More general, we can investigate the influence of
$\tilde{m}_{R}/\tilde{m}_{L}$ on the ratio
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ through contour plots drawn
in Fig.5. On the drawing results showed that
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}\leq\mathcal{O}(10^{-2})$
whenever $|\mu_{\rho}|/m_{SUSY}\leq 5$ and that ratio does not depend too much
on the ratio $\tilde{m}_{R}/\tilde{m}_{L}$. However, in the limit
$|\mu_{\rho}|/m_{SUSY}\geq 7$ and $\tilde{m}_{R}<0.5\tilde{m}_{L}$, the ratio
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ changes very rapidly if
small changes $\tilde{m}_{L}$ and $\tilde{m}_{R}$. It means that chirality
effects of phenomena relating with $\Delta^{\rho}_{L}$ and $\Delta^{\rho}_{R}$
are sensitive with the change of ratio $\tilde{m}_{R}/\tilde{m}_{L}$ at large
values of $\mu_{\rho}$. On the other hand, the left picture in Fig.5 indicates
that when the ratio $|\mu_{\rho}|/m_{SUSY}\geq 7$, it will exist in some
regions of parameter space of $\tilde{m}_{R},\tilde{m}_{L}$ at which the
contributions of left- and right-lepton sectors into the $H\rightarrow\mu\tau$
decay process are of the same order. In this case, the pure gaugino-mediated
diagrams give the dominated contribution to both $\Delta^{\rho}_{L}$ and
$\Delta^{\rho}_{R}$, and also $\Delta^{2\rho}_{\mu}$ ( see (95) in Appendix
A). Recalling that large values of $\Delta^{2\rho}_{\mu}$ can strongly affect
directly on the ratio $Y_{\mu}/Y_{\tau}$. The results presented in Fig.5 again
confirm that whenever $|\mu_{\rho}|/m_{SUSY}\geq 7$ and
$\tilde{m}_{R}<0.5\tilde{m}_{L}$, the right-lepton sector gives dominated
contribution to the branching ratio of $H\rightarrow\mu\tau$ decay process.

|
---|---
Figure 4: $\frac{|\Delta^{\rho}_{R}|^{2}}{|\Delta^{\rho}_{L}|^{2}}$ as a function of $|\mu_{\rho}|/\tilde{m}_{L}$ with four different choices of masses ratios: 1) blue curve-$m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 2) green curve-$3m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 3) yellow curve- $m^{\prime}=\tilde{m}_{L}=3\tilde{m}_{R}$; 4) red curve-$m^{\prime}=\tilde{m}_{L}=\tilde{m}_{R}/3$. A black line in the left side of figure corresponding to the value $\frac{|\Delta^{\rho}_{R}|^{2}}{|\Delta^{\rho}_{L}|^{2}}$ equals 1. Both black lines in the right side of figure presenting $\frac{|\Delta^{\rho}_{R}|^{2}}{|\Delta^{\rho}_{L}|^{2}}$ are $2\times 10^{-3}$ and $0.1$. |
---|---
Figure 5:  Contour plot of $\frac{|\Delta^{\rho}_{R}|^{2}}{|\Delta^{\rho}_{L}|^{2}}$, $\tilde{m}_{R}/\tilde{m}_{L}$ vs $|\mu_{\rho}|/m_{SUSY}$, where $\tilde{m}_{R}=\tilde{m}_{\nu_{R}}$, $m^{\prime}=m_{\lambda}=\tilde{m}_{L}=\tilde{m}_{\nu_{L}}=m_{SUSY}$. The red region corresponds to the values of $\frac{|\Delta^{\rho}_{R}|^{2}}{|\Delta^{\rho}_{L}|^{2}}\geq 0.5$.  |
---|---
Figure 6:  Contour plot of $BR(H\rightarrow\mu\tau)/BR(H\rightarrow\tau\tau)$,
$\tilde{m}_{g}$ vs $|\mu_{\rho}|/m_{SUSY}$, where
$m^{\prime}=m_{\lambda}=\tilde{m}_{g}$ and
$\tilde{m}_{R}=\tilde{m}_{\nu_{R}}=\tilde{m}_{L}=\tilde{m}_{\nu_{L}}=m_{SUSY}$.
In the left picture, the green and yellow regions correspond to the values of
$BR(H\rightarrow\mu\tau)/BR(H\rightarrow\tau\tau)\geq\mathcal{O}(10^{-3})$ .

Now we investigate more details the region of parameter space where Higgsino-
mediated diagrams give a dominated contribution. In this region of parameter
space, both $\Delta^{2\rho}_{\mu}$ and $\Delta^{\rho}_{R}$ are much smaller
than $\Delta^{\rho}_{L}$ so we just focus on $\Delta^{\rho}_{L}$. From Fig.6,
we can estimate the ratio of
$Br(H\rightarrow\mu\tau)/Br(H\rightarrow\tau\tau)$ that can reach the order of
$10^{-3}$ in the limit $0.1\leq|\mu_{\rho}|/M_{SUSY}\leq 6$ and
$0.1\leq|\tilde{m}_{g}|/M_{SUSY}\leq 7$ where $\tilde{m}_{g}$ is mass of
gauginos. Let us briefly review the decay properties of neutral Higgs bosons
in the SUSYE331 model. At the tree level, the couplings of neutral Higgs
bosons to up-fermions, down-fermion are modified with respect to the SM
coupling by factors which are given in Table 1.

Table 1:  Coupling of neutral Higgs bosons to fermion. Particles | Up-fermion | Down-fermion | Exotic up-quark | Exotic down-quark
---|---|---|---|---
SM Higgs | 1 | 1 | 0 | 0
$\varphi_{Sa36}$ | $c_{\alpha}$ | $c_{\alpha}$ | $s_{\alpha}/s_{\gamma}$ | $c_{\alpha}/s_{\gamma}$
$\phi_{Sa36}$ | $s_{\alpha}$ | $s_{\alpha}$ | $c_{\alpha}$ | $s_{\alpha}$

We assume that all exotic quarks have masses heavier than that of all neutral
Higgses. It means that the neutral Higgs cannot decay into the exotic quarks.
The neutral Higgs bosons may decay mainly into the pairs of fermions. This
prediction depends on the mass of the neutral Higgs. For neutral Higgs
$\varphi_{Sa36}$, its mass depends on the vacuum expectation values
$v,v^{\prime}$. So it should be predicted SM Higgs with mass smaller than
about $130$ GeV. Decay of $\varphi_{Sa36}$ to $b\overline{b}$ and
$\tau\overline{\tau}$ are dominated, the branching ratios of $90$ percent and
$8$ percent, respectively. Combined with the results in Fig.6, the branching
ratio $Br(\varphi_{Sa36}\rightarrow\mu\tau)$ is $8\times 10^{-3}$ percent.
This may be a good signification of new physics in the present limits of
colliders. For neutral Higgs $\phi_{Sa36}$, it is heavy Higgs, the main
productions of decay are the the gauge bosons such as $W^{+}W^{-}$,
$ZZ$,…Hence, the branching ratio $\phi\rightarrow\mu\tau$ is very suppressed.
We would like to note that the effective interactions of the muon, tauon and
Higgs given in (81) not only leads to the LFV of Higgs decay process, but also
affects the other physical processes with lepton-flavor violations. Some of
these processes which are looked seriously by present experiments are, for
instance, $\tau\rightarrow\mu\mu\mu$ and $\tau\rightarrow\mu\gamma$. Let us
apply the effective couplings given in (81) to the $\tau\rightarrow\mu\mu\mu$
decay process. In a general way-regardless of the model, the general effective
Lagrangian describing decay of $\tau\rightarrow\mu\mu\mu$ was studied in [30].
However, in this work we focus on the effect of the Higgs-mediated LFV
interactions on the $\tau\rightarrow\mu\mu\mu$ decay process. Hence, the four
dimensional effective Lagrangian which is built through Higgs exchange is
formulated by

$\displaystyle\mathcal{L}^{eff}_{\tau\mu\mu\mu}$ $\displaystyle=$
$\displaystyle-2\sqrt{2}G_{F}m_{\mu}m_{\tau}\tan\gamma\left(\frac{s^{2}_{\alpha}}{m^{2}_{\phi_{Sa36}}}+\frac{c^{2}_{\alpha}}{m^{2}_{\varphi_{Sa36}}}\right)(\mu^{c}\mu+\bar{\mu}\bar{\mu}^{c})$
(89) $\displaystyle\times$
$\displaystyle(\Delta^{\rho}_{L}\tau^{c}\mu+\Delta^{\rho}_{R}\mu^{c}\tau)+\mathrm{H.c.}.$

We would like to remind that the decay process of $\tau\rightarrow\mu\mu\mu$
were investigated, by [11, 30] for examples, in a general model-independent
way. The predicted results show that when Higgs exchange effects are much
smaller than other ones, the ratio $Br(\tau\rightarrow
3\mu)/BR(\tau\rightarrow\mu\gamma)$ becomes constant with a value
$\sim\mathcal{O}(10^{-3})$. Now, we will discuss in more details whether
Higgs-mediated effects can make any significations to the ratio
$Br(\tau\rightarrow 3\mu)/BR(\tau\rightarrow\mu\gamma)$ in the SUSYE331 model.
We can divide our results into two cases, namely $\phi^{*}_{S_{a36}}$ and
$\varphi^{*}_{S_{a36}}$ Higgs-mediated effects. The results can be written in
two respective forms:

$\displaystyle
BR(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})_{\phi^{*}_{S_{a36}}}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\tan^{2}\gamma~{}\frac{m^{2}_{\mu}m^{2}_{\tau}}{m^{4}_{\phi_{S_{a36}}}}s^{4}_{\alpha}\left(|\Delta^{\rho}_{L}|^{2}+|\Delta^{\rho}_{R}|^{2}\right)$
$\displaystyle\simeq$ $\displaystyle 7\times
10^{-11}\left(\frac{\tan\gamma}{40}\right)^{2}\left(\frac{100\mathrm{GeV}}{m_{\phi_{S_{a36}}}}\right)^{4}$
$\displaystyle\times$
$\displaystyle\left(\frac{|\Delta^{\rho}_{L}|^{2}+|\Delta^{\rho}_{R}|^{2}}{10^{-3}}\right)s^{4}_{\alpha}$

and

$\displaystyle
BR(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})_{\varphi^{*}_{S_{a36}}}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\tan^{2}\gamma~{}\frac{m^{2}_{\mu}m^{2}_{\tau}}{m^{4}_{\varphi_{S_{a36}}}}c^{4}_{\alpha}\left(|\Delta^{\rho}_{L}|^{2}+|\Delta^{\rho}_{R}|^{2}\right)$
(91) $\displaystyle\simeq$ $\displaystyle 7\times
10^{-11}\left(\frac{\tan\gamma}{40}\right)^{2}\left(\frac{100\mathrm{GeV}}{m_{\varphi_{S_{a36}}}}\right)^{4}$
$\displaystyle\times$
$\displaystyle\left(\frac{|\Delta^{\rho}_{L}|^{2}+|\Delta^{\rho}_{R}|^{2}}{10^{-3}}\right)c^{4}_{\alpha}.$

These results immediately lead to a consequence: the maximum contribution of
Higgs exchange processes can be estimated through the formula:

$\displaystyle BR(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})_{H^{*}}$
$\displaystyle\simeq$ $\displaystyle 7\times
10^{-11}\left(\frac{\tan\gamma}{40}\right)^{2}\left(\frac{100\mathrm{GeV}}{m_{\mathrm{H}}}\right)^{4}$
(92) $\displaystyle\times$
$\displaystyle\left(\frac{|\Delta^{\rho}_{L}|^{2}+|\Delta^{\rho}_{R}|^{2}}{10^{-3}}\right).$

The values of the branching ratios decrease rapidly corresponding to the
enhancement of Higgs masses . We stress that in the model under consideration,
the $\varphi_{S_{a36}}$ is identified with the SM Higgs boson and the remain,
$\phi_{S_{a36}}$, is heavy one. Overall, in our model, the SM Higgs-mediated
gives larger contribution to the branching ratio of the
$\tau\rightarrow\mu\mu\mu$ decay process than that of heavy Higgs. That kind
of branching increases if the $\tan\gamma$ increases. The branching ratio
estimated in (92) is $\simeq 10^{-11}$ in the limit of $\tan\gamma\simeq 50$
and the Higgs mass is of the order of $100$ GeV. However, the branching ratio
of $\tau\rightarrow\mu\mu\mu$ can be reached at the present limits of
experiment $BR(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})\leq 3.2\times
10^{-8}$ [31]. It means that the contribution to $\tau\rightarrow\mu\mu\mu$ is
suppressed in the limit of $\tan\gamma\simeq 50$. This result is different
from that predicted in the MSSM model [16, 10, 11]. In particular, for the
MSSM, the dominant contributions to $BR(\tau\rightarrow\mu\mu\mu)$ are induced
by the dipole term and the Higgs-mediated term at the limit
$\tan\beta=50,m_{A}=100$ GeV.

Because of sub-dominated contribution of Higgs-mediated to
$BR(\tau\rightarrow\mu\mu\mu)$, the dominated contribution to that branching
ratio is still obtained from the photon-penguin couplings. This result leads
to the values of well-known ratios such as

$\displaystyle\frac{Br(\tau\rightarrow
3\mu)}{BR(\tau\rightarrow\mu\gamma)}\simeq\mathcal{O}(10^{-3})$ (93)

The predicted result is the concerned result given in [16, 11, 30, 32].

We emphasize that in the SUSYE331 model, in order to get the dominated
contribution to the $B(\tau\rightarrow\mu\mu\mu)$, the values of $\tan\gamma$
must be $10^{2}$. In this limit of $\tan\gamma$, the result given in (93) is
not holden.

##  5 Conclusions

In this paper, we have studied the LFV interactions of Higgs bosons in the
SUSYE331 model. We have the unique existence of the lepton-number violation in
the slepton sector at the tree level. On the basis of this assumption we have
examined the lepton-number violating interactions of Higgs bosons at the one-
loop level. Specially we have concentrated our study on the LFV couplings of
Higgs bosons with muon and tauon. The analytical expressions of the effective
Higgs-muon-tauon couplings are established at the one-loop level. One of the
features is that the model does not contain the LFV interactions of neutral
pseudo-scalar Higgs bosons. For the neutral Higgs scalars, the model contains
two types of radiative interactions that violate lepton number, namely,
$\phi_{Sa36}\mu\tau$ and $\varphi_{Sa36}\mu\tau$. These effective couplings
depend on ratios of SUSY mass parameters and $\tan\gamma$. There is an exactly
similar to the other SUSY models, all LFV couplings are built from two types
of diagrams as Higgs-mediated diagrams and pure gaugino-mediated ones.
Depending on the SUSY parameters, each type of diagram gives the main
contribution to LFV couplings. In this work, we have also studied the
branching ratio of the neutral Higgs decay into muon and tauon. In the limit
$|\mu_{\rho}|/m_{SUSY}\leq 7$, the ratio of
$BR(H\rightarrow\tau\mu)/BR(H\rightarrow\tau\tau)$ in this region can reach
values that can be observed by near future experiments and the contributions
from both left and right LFV sectors to $Br(H\rightarrow\mu\tau)$ are of the
same order. Outside this region the effects of left and right LFV terms mix in
different ways in different regions of mass parameter space. We predicted that
for the SM Higgs boson, LHC may detect the decay of SM Higgs boson to muon and
tauon. For heavy Higgs bosons, the branching of LFV decay is very suppressed.
We have also studied the contribution of Higgs exchange to decay $H\rightarrow
3\mu$. In the limit $\tan\gamma=50$, the $Br(H\rightarrow 3\mu)$ is very
small, out of direct detection of present searching of experiment and it leads
to predicted results such as the ratio of $BR(\tau\rightarrow
3\mu)/BR(\tau\rightarrow\mu\gamma)$.

## Acknowledgments

L.T.H would like to thank the Organizers of KEK-Vietnam Visiting Program 2011
(Exchange Program for East Asia Young Researchers, JSPS), especially Prof. Y.
Kurihara, for the support for his initial work at KEK. This work was supported
in part by the National Foundation for Science and Technology Development
(NAFOSTED) of Vietnam under Grant No. 103.01-2011.63.

##  Appendix A  Analytic formulas and diagrams contributing to $\Delta$s

Let us write down all expressions of $\Delta$s given as following

$\displaystyle\Delta^{1\rho}_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{g^{\prime
2}}{108\pi^{2}}\mu_{\rho}m^{\prime}\left[c^{2}_{L}I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})+s^{2}_{L}I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right]$ (94) $\displaystyle-$
$\displaystyle\frac{g^{2}}{24\pi^{2}}\mu_{\rho}m_{\lambda}\left[c^{2}_{L}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})+s^{2}_{L}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}\left[c^{2}_{\nu_{L}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}})+s^{2}_{\nu_{L}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}\left[c^{2}_{\nu_{R}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R2}})+s^{2}_{\nu_{R}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R3}})\right]$
$\displaystyle+$
$\displaystyle\frac{Y_{\nu_{\mu\tau}}(h_{\mu\tau}-h_{\tau\mu})\mu_{\rho}}{8\pi^{2}}$
$\displaystyle\times$
$\displaystyle\left[s^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R2}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R3}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.c^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R2}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R3}})\right)\right]$
$\displaystyle+$ $\displaystyle\frac{g^{\prime
2}}{288\pi^{2}}\mu_{\rho}m^{\prime}\left[c^{2}_{L}\left(c^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})+s^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right)\right.$
$\displaystyle+$ $\displaystyle\left.s^{2}_{L}\left(c^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})+s^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right)\right].$
$\displaystyle\Delta^{2\rho}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
2}}{288\pi^{2}}\mu_{\rho}m^{\prime}s_{L}c_{L}s_{R}c_{R}\left[I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})-I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right.$ (95) $\displaystyle-$
$\displaystyle\left.I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})+I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right].$
$\displaystyle\Delta^{\rho}_{\tau}$ $\displaystyle=$
$\displaystyle-\frac{g^{\prime
2}}{108\pi^{2}}\mu_{\rho}m^{\prime}\left[s^{2}_{L}I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})+c^{2}_{L}I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right]$ (96) $\displaystyle-$
$\displaystyle\frac{g^{2}}{24\pi^{2}}\mu_{\rho}m_{\lambda}\left[s^{2}_{L}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})+c^{2}_{L}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}\left[s^{2}_{\nu_{L}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}})+c^{2}_{\nu_{L}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}\left[s^{2}_{\nu_{R}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R2}})+c^{2}_{\nu_{R}}I_{3}(m^{2}_{\lambda}\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R3}})\right]$
$\displaystyle+$
$\displaystyle\frac{Y_{\nu_{\mu\tau}}(h_{\mu\tau}-h_{\tau\mu})\mu_{\rho}}{8\pi^{2}}$
$\displaystyle\times$
$\displaystyle\left[s^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R2}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R3}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.c^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{3L}},\tilde{m}^{2}_{\nu_{R2}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R3}})\right)\right]$
$\displaystyle+$ $\displaystyle\frac{g^{\prime
2}}{288\pi^{2}}\mu_{\rho}m^{\prime}\left[s^{2}_{L}\left(s^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})+c^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right)\right.$
$\displaystyle+$ $\displaystyle\left.c^{2}_{L}\left(s^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})+c^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right)\right].$

$\displaystyle\Delta^{1\rho^{\prime}}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{Y^{2}_{\nu_{\mu\tau}}}{4\pi^{2}}\mu^{2}_{\rho}\left[s^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R2}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R3}})\right)\right.$
(97) $\displaystyle+$
$\displaystyle\left.c^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R3}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R2}})\right)\right].$
$\displaystyle\Delta^{2\rho^{\prime}}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime 2}m^{\prime}}{144\pi^{2}}$ $\displaystyle\times$
$\displaystyle\left[\frac{h^{\prime}_{\mu}c_{L}c_{R}+h^{\prime}_{\tau}s_{L}s_{R}+h^{\prime}_{\mu\tau}c_{L}s_{R}+h^{\prime}_{\tau\mu}s_{L}c_{R}}{Y_{\tau}}c_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{h^{\prime}_{\mu}s_{L}s_{R}+h^{\prime}_{\tau}c_{L}c_{R}-h^{\prime}_{\mu\tau}s_{L}c_{R}-h^{\prime}_{\tau\mu}c_{L}s_{R}}{Y_{\tau}}s_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}s_{L}c_{R}+h^{\prime}_{\tau}c_{L}s_{R}-h^{\prime}_{\mu\tau}s_{L}s_{R}+h^{\prime}_{\tau\mu}c_{L}c_{R}}{Y_{\tau}}s_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}c_{L}s_{R}+h^{\prime}_{\tau}s_{L}c_{R}+h^{\prime}_{\mu\tau}c_{L}c_{R}-h^{\prime}_{\tau\mu}s_{L}s_{R}}{Y_{\tau}}c_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right].$
$\displaystyle\Delta^{\rho^{\prime}}_{\tau}$ $\displaystyle=$
$\displaystyle\Delta^{1\rho^{\prime}}_{\mu}+\frac{g^{\prime
2}m^{\prime}}{144\pi^{2}}$ $\displaystyle\times$
$\displaystyle\left[\frac{h^{\prime}_{\mu}c_{L}c_{R}+h^{\prime}_{\tau}s_{L}s_{R}+h^{\prime}_{\mu\tau}c_{L}s_{R}+h^{\prime}_{\tau\mu}s_{L}c_{R}}{Y_{\tau}}s_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{h^{\prime}_{\mu}s_{L}s_{R}+h^{\prime}_{\tau}c_{L}c_{R}-h^{\prime}_{\mu\tau}s_{L}c_{R}-h^{\prime}_{\tau\mu}c_{L}s_{R}}{Y_{\tau}}c_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}s_{L}c_{R}+h^{\prime}_{\tau}c_{L}s_{R}-h^{\prime}_{\mu\tau}s_{L}s_{R}+h^{\prime}_{\tau\mu}c_{L}c_{R}}{Y_{\tau}}c_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}c_{L}s_{R}+h^{\prime}_{\tau}s_{L}c_{R}+h^{\prime}_{\mu\tau}c_{L}c_{R}-h^{\prime}_{\tau\mu}s_{L}s_{R}}{Y_{\tau}}s_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right]$

In the case of $\Delta^{\rho^{\prime}}_{L}$, it also receives contributions
from two diagrams (similar to diagrams (e) and (f) in Fig.1 ) which cancel
each other. Therefore we do not repeat them in Figs.and 9 and 10 . Formula of
$\Delta^{\rho^{\prime}}_{L}$ then is

$\displaystyle\Delta^{\rho^{\prime}}_{L}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime 2}m^{\prime}}{144\pi^{2}}$ $\displaystyle\times$
$\displaystyle\left[\frac{h^{\prime}_{\mu}c_{L}c_{R}+h^{\prime}_{\tau}s_{L}s_{R}+h^{\prime}_{\mu\tau}c_{L}s_{R}+h^{\prime}_{\tau\mu}s_{L}c_{R}}{Y_{\tau}}c_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{h^{\prime}_{\mu}s_{L}s_{R}+h^{\prime}_{\tau}c_{L}c_{R}-h^{\prime}_{\mu\tau}s_{L}c_{R}-h^{\prime}_{\tau\mu}c_{L}s_{R}}{Y_{\tau}}s_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}s_{L}c_{R}+h^{\prime}_{\tau}c_{L}s_{R}-h^{\prime}_{\mu\tau}s_{L}s_{R}+h^{\prime}_{\tau\mu}c_{L}c_{R}}{Y_{\tau}}s_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}c_{L}s_{R}+h^{\prime}_{\tau}s_{L}c_{R}+h^{\prime}_{\mu\tau}c_{L}c_{R}-h^{\prime}_{\tau\mu}s_{L}s_{R}}{Y_{\tau}}c_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{\tilde{L}_{2}},\tilde{m}^{2}_{\tilde{R}_{3}})\right].$
$\displaystyle\Delta^{\rho^{\prime}}_{R}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime 2}m^{\prime}}{144\pi^{2}}$ $\displaystyle\times$
$\displaystyle\left[\frac{h^{\prime}_{\mu}c_{L}c_{R}+h^{\prime}_{\tau}s_{L}s_{R}+h^{\prime}_{\mu\tau}c_{L}s_{R}+h^{\prime}_{\tau\mu}s_{L}c_{R}}{Y_{\tau}}s_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{h^{\prime}_{\mu}s_{L}s_{R}+h^{\prime}_{\tau}c_{L}c_{R}-h^{\prime}_{\mu\tau}s_{L}c_{R}-h^{\prime}_{\tau\mu}c_{L}s_{R}}{Y_{\tau}}c_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}s_{L}c_{R}+h^{\prime}_{\tau}c_{L}s_{R}-h^{\prime}_{\mu\tau}s_{L}s_{R}+h^{\prime}_{\tau\mu}c_{L}c_{R}}{Y_{\tau}}c_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}c_{L}s_{R}+h^{\prime}_{\tau}s_{L}c_{R}+h^{\prime}_{\mu\tau}c_{L}c_{R}-h^{\prime}_{\tau\mu}s_{L}s_{R}}{Y_{\tau}}s_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right].$

$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$\rho^{0*}$$\tilde{l}_{L_{\alpha}}$$(a)$

$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda^{3}_{A}$$\lambda^{8}_{A}$$\rho^{0*}$$\tilde{l}_{L_{\alpha}}$$(b)$

$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}_{1}^{\prime-}$$\tilde{\rho}_{1}^{+}$$\tilde{W}^{-}$$\tilde{W}^{+}$$\rho^{0*}$$\tilde{\nu}_{L\alpha}$$(c)$

$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}_{2}^{\prime-}$$\tilde{\rho}_{2}^{+}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$\rho^{0*}$$\tilde{\nu}_{R\alpha}$$(d)$

$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\rho^{0*}$$\tilde{\nu}_{L_{\alpha}}$$\tilde{\nu}_{R_{\beta}}$$(e)$

$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\rho^{0*}$$\tilde{\nu}_{R_{\alpha}}$$\tilde{\nu}_{L_{\beta}}$$(f)$

$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$\rho^{0*}$$\tilde{l}_{R_{\alpha}}$$(i)$

$Y_{\mu(\tau)}$$\mu$$\mu^{c}$$\lambda_{B}$$\rho^{*0}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$$(k)$$(\tau)$$(\tau^{c})$

Figure 7: Diagrams contributing to $\Delta^{1\rho}_{\mu}$ ( or
$\Delta^{\rho}_{\tau})$.
$Y_{\tau}$$\mu$$\mu^{c}$$\lambda_{B}$$\rho^{0*}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$
Figure 8: Diagram contributing to $\Delta^{2\rho}_{\mu}$.

$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\rho^{\prime
0}$$\tilde{\nu}_{L_{\alpha}}$$\tilde{\nu}_{R_{\beta}}$$(a)$

$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\rho^{\prime
0}$$\tilde{\nu}_{R_{\alpha}}$$\tilde{\nu}_{L_{\beta}}$$(b)$

$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\lambda_{B}$$\rho^{\prime
0}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$$(c)$

Figure 9: Diagrams contributing to $\Delta^{1\rho^{\prime}}_{\mu}$
$[(a),(b)]$, $\Delta^{2\rho^{\prime}}_{\mu}$ $[(c)]$ ( or
$\Delta^{\rho^{\prime}}_{\tau}$ $[(a),(b),(c)]$).

$\mu$$\tau^{c}$$\lambda_{B}$$\rho^{\prime
0}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$$(a)$

$\tau$$\mu^{c}$$\lambda_{B}$$\rho^{\prime
0}$$\tilde{e}_{L_{\alpha}}$$\tilde{e}_{R_{\beta}}$$(b)$

Figure 10: Diagrams contributing to $\Delta^{\rho^{\prime}}_{L}$ [(a)] and
$\Delta^{\rho^{\prime}}_{R}$ [(b)].

##  Appendix B Lagrangian

We have denoted by $f_{L,R}(\bar{f}_{L,R}^{c})$ two component spinor of the
generic matter left-handed and right-handed fermion, respectively. The
$\tilde{f}_{L}(\tilde{f}_{L}^{c})$ are their superpartners which satisfy
$(\bar{f}_{L}^{c}=(f_{R})^{\dagger T},\tilde{f}_{L}^{c}=\tilde{f}^{*}_{R}$).
The four-component Dirac spinor can be represented through two-component
spinor such as: $\mu\equiv(\mu_{L},\;\mu_{R})=(\mu_{L},\;\bar{\mu}_{L}^{c})$
and
$\widetilde{{\mu}}\equiv(\tilde{\mu}_{L},\;\tilde{\mu}_{R})=(\widetilde{\mu}_{L},\;\widetilde{\mu}_{L}^{c*})$.
We also emphasize that three-left handed leptons contained in the triplet
$L_{aL}$ of $\mathrm{SU}(3)_{L}$ are $(f_{a1L},f_{a2L},f_{a3L})$:

$\displaystyle f_{a1L}$ $\displaystyle\in$ $\displaystyle\\{\nu_{eL},\nu_{\mu
L},\nu_{\tau L}\\}\equiv\\{\nu_{1L},\nu_{2L},\nu_{3L}\\},$ $\displaystyle
f_{a2L}$ $\displaystyle\in$ $\displaystyle\\{e_{L},\mu_{L},\tau_{L}\\},$
$\displaystyle f_{a3L}$ $\displaystyle\in$
$\displaystyle\\{\nu^{c}_{eL},\nu^{c}_{\mu L},\nu^{c}_{\tau
L}\\}\equiv\\{\nu^{c}_{1L},\nu^{c}_{2L},\nu^{c}_{3L}\\}\equiv\\{(\nu^{c}_{1})_{L},(\nu^{c}_{2})_{L},(\nu^{c}_{3})_{L}\\}$
(102)

while $f_{aL}^{c}$ is singlet under $\mathrm{SU}(3)_{L}$. Conventions for two
component spinors used in our paper are the same as those given in [11] except
the lower index $L$, which is used to distinguish between Dirac spinors $f$
and left-handed Weyl spinors $f_{L}$.

Next, let us find the interactional vertices relating with our calculation. We
start to collect related terms from the Lagrangian given in [23, 33] into the
following ones:

  1. 1.

Gaugino and Higgsino mass terms:

$\displaystyle\mathcal{L}_{gh}$ $\displaystyle=$
$\displaystyle-\left[\frac{1}{2}m_{\lambda}\sum_{b=1}^{8}\left(\lambda^{b}_{A}\lambda^{b}_{A}\right)+\frac{1}{2}m^{\prime}\lambda_{B}\lambda_{B}+\mu_{\chi}\widetilde{\chi}\widetilde{\chi}^{\prime}+\mu_{\rho}\widetilde{\rho}\widetilde{\rho}^{\prime}\right]+\mathrm{H.c.}$
(103) $\displaystyle=$
$\displaystyle-\frac{1}{2}m^{\prime}\lambda_{B}\lambda_{B}-m_{\lambda}\tilde{W}^{+}\tilde{W}^{-}-\frac{1}{2}m_{\lambda}\lambda_{A}^{3}\lambda_{A}^{3}$
$\displaystyle-$ $\displaystyle
m_{\lambda}\tilde{Y}^{\prime+}\tilde{Y}^{\prime-}-m_{\lambda}\tilde{X^{0*}}\tilde{X^{0}}-\frac{1}{2}m_{\lambda}\lambda_{A}^{8}\lambda_{A}^{8}-\mu_{\chi}\left(\widetilde{\chi}^{0}_{1}\widetilde{\chi}^{\prime
0}_{1}+\widetilde{\chi}^{-}\widetilde{\chi}^{\prime+}+\widetilde{\chi}^{0}_{2}\widetilde{\chi}^{\prime
0}_{2}\right)$ $\displaystyle-$
$\displaystyle\mu_{\rho}\left(\widetilde{\rho}^{+}_{1}\widetilde{\rho}^{\prime-}_{1}+\widetilde{\rho}^{0}\widetilde{\rho}^{\prime
0}+\widetilde{\rho}^{+}_{2}\widetilde{\rho}^{\prime-}_{2}\right)+\mathrm{H.c.}.$

where

$\displaystyle\tilde{W}^{\pm}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}(\lambda_{A}^{1}\mp i\lambda_{A}^{2}),\hskip
14.22636pt\tilde{Y}^{\pm}\equiv\frac{1}{\sqrt{2}}(\lambda_{A}^{6}\pm
i\lambda_{A}^{7}),$ $\displaystyle\tilde{X}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}(\lambda_{A}^{4}+i\lambda_{A}^{5}),\hskip
14.22636pt\tilde{X}^{*}\equiv\frac{1}{\sqrt{2}}(\lambda_{A}^{4}-i\lambda_{A}^{5})$
(104)

where we have used the results of mass eigenstate states for Higgsinos and
gauginos given in [24] and [26].

  2. 2.

Fermion-sfermion-gaugino interaction terms:

$\displaystyle\mathcal{L}_{l\tilde{l}\tilde{V}}$ $\displaystyle=$
$\displaystyle-\frac{ig^{\prime}}{\sqrt{3}}\left[-\frac{1}{3}(\bar{L}\tilde{L}\bar{\lambda}_{B}-\tilde{L}^{\dagger}L\lambda_{B})+(\bar{l}^{c}\tilde{l}^{c}\bar{\lambda}_{B}-\tilde{l}^{c*}l^{c}\lambda_{B})\right]$
(105) $\displaystyle-$
$\displaystyle\frac{ig}{\sqrt{2}}(\bar{L}\lambda^{i}\tilde{L}\bar{\lambda}^{i}_{A}-\tilde{L}^{\dagger}\lambda^{i}L\lambda^{i}_{A}),$

where $i=1,2,...,8$ is a color index and
$L\equiv(L_{1L},\;L_{2L},\;L_{3L})^{T}$,
$\tilde{L}\equiv(\tilde{L}_{1L},\;\tilde{L}_{2L},\;\tilde{L}_{3L})^{T}$,
$\bar{l}^{c}=(l_{1R},\;l_{2R},\;l_{3R})^{T}\equiv(\bar{l}^{c}_{1L},\;\bar{l}^{c}_{2L},\;\bar{l}^{c}_{3L})^{T}\equiv(e_{R},\mu_{R},\tau_{R})^{T}\equiv(\bar{e}^{c}_{L},\;\bar{\mu}^{c}_{L},\;\bar{\tau}^{c}_{L})^{T}$,
and
$\tilde{l}^{c}=(\tilde{l}^{*}_{1R},\;\tilde{l}^{*}_{2R},\;\tilde{l}^{*}_{3R})^{T}\equiv(\tilde{e}^{*}_{R},\;\tilde{\mu}^{*}_{R},\;\tilde{\tau}^{*}_{R},)^{T}$.
In this paper we just focus on interactions relating with two fermions, namely
$l_{aL}=\\{\mu_{L},\tau_{L}\\}$. All interested terms are given as

$\displaystyle\mathcal{L}_{l\tilde{l}\tilde{V}}$ $\displaystyle=$
$\displaystyle\left[\bar{\mu}_{L}\left(\frac{ig^{\prime}}{3\sqrt{3}}\bar{\lambda}_{B}+\frac{ig}{\sqrt{2}}(\bar{\lambda}^{3}_{A}-\frac{1}{\sqrt{3}}\bar{\lambda}^{8}_{A})\right)\tilde{\mu}_{L}\right.$
(106) $\displaystyle-$
$\displaystyle\left.\mu_{L}\left(\frac{ig^{\prime}}{3\sqrt{3}}\lambda_{B}+\frac{ig}{\sqrt{2}}(\lambda^{3}_{A}-\frac{1}{\sqrt{3}}\lambda^{8}_{A})\right)\tilde{\mu}^{*}_{L}\right.$
$\displaystyle-$
$\displaystyle\left.\frac{ig^{\prime}}{\sqrt{3}}(\bar{\mu}^{c}_{L}\tilde{\mu}^{c}_{L}\bar{\lambda}_{B}-\mu^{c}_{L}\tilde{\mu}^{c*}_{L}\lambda_{B})+\frac{ig}{\sqrt{2}}\left(\bar{\mu}_{L}\tilde{\mu}_{L}\bar{\lambda}^{3}_{A}-\tilde{\mu}^{*}_{L}\mu_{L}\lambda^{3}_{A}\right)\right.$
$\displaystyle+$
$\displaystyle\left.\frac{ig}{\sqrt{2}}\left(\bar{\mu}_{L}\tilde{\mu}_{L}\bar{\lambda}^{8}_{A}-\tilde{\mu}^{*}_{L}\mu_{L}\lambda^{8}_{A}\right)\right.$
$\displaystyle-$
$\displaystyle\left.ig\left((\bar{\mu}_{L}\overline{\tilde{W}}^{+}\tilde{\nu}_{\mu
L}-\mu_{L}\tilde{W}^{+}\tilde{\nu}^{*}_{\mu
L})+(\bar{\mu}_{L}\overline{\tilde{Y}}^{+}\tilde{\nu}^{c}_{\mu
L}-\mu_{L}\tilde{Y}^{+}\tilde{\nu}^{c*}_{\mu L})\right)\right]$
$\displaystyle+$ $\displaystyle[\mu\rightarrow\tau].$

From this, we list the related vertices in Table 2

Vertex | Factor | Vertex | Factor
---|---|---|---
$\bar{\mu}_{L}\bar{\lambda}_{B}\tilde{\mu}_{L}$ | $-\frac{g^{\prime}}{3\sqrt{3}}$ | $\mu_{L}\lambda_{B}\tilde{\mu}^{*}_{L}$ | $\frac{g^{\prime}}{3\sqrt{3}}$
$\bar{\mu}^{c}_{L}\tilde{\mu}^{c}_{L}\bar{\lambda}_{B}$ | $\frac{g^{\prime}}{\sqrt{3}}$ | $\mu^{c}_{L}\tilde{\mu}^{c*}_{L}\lambda_{B}$ | $\frac{-g^{\prime}}{\sqrt{3}}$
$\bar{\mu}_{L}\tilde{\mu}_{L}\bar{\lambda}^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$ | $\tilde{\mu}^{*}_{L}\mu_{L}\lambda^{3}_{A}$ | $\frac{g}{\sqrt{2}}$
$\bar{\mu}_{L}\tilde{\mu}_{L}\bar{\lambda}^{8}_{A}$ | $\frac{g}{\sqrt{6}}$ | $\tilde{\mu}^{*}_{L}\mu_{L}\lambda^{8}_{A}$ | $\frac{-g}{\sqrt{6}}$
$\bar{\mu}_{L}\overline{\tilde{W}}^{+}\tilde{\nu}_{\mu L}$ | $g$ | $\mu_{L}\tilde{W}^{+}\tilde{\nu}^{*}_{\mu L}$ | $-g$
$\bar{\mu}_{L}\overline{\tilde{Y}}^{+}\tilde{\nu}^{c}_{\mu L}$ | $g$ | $\mu_{L}\tilde{Y}^{+}\tilde{\nu}^{c*}_{\mu L}$ | $-g$
$\bar{\tau}_{L}\bar{\lambda}_{B}\tilde{\tau}_{L}$ | $-\frac{g^{\prime}}{3\sqrt{3}}$ | $\tau_{L}\lambda_{B}\tilde{\tau}^{*}_{L}$ | $\frac{g^{\prime}}{3\sqrt{3}}$
$\bar{\tau}^{c}_{L}\tilde{\tau}^{c}_{L}\bar{\lambda}_{B}$ | $\frac{g^{\prime}}{\sqrt{3}}$ | $\tau^{c}_{L}\tilde{\tau}^{c*}_{L}\lambda_{B}$ | $\frac{-g^{\prime}}{\sqrt{3}}$
$\bar{\tau}_{L}\tilde{\tau}_{L}\bar{\lambda}^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$ | $\tilde{\tau}^{*}_{L}\tau_{L}\lambda^{3}_{A}$ | $\frac{g}{\sqrt{2}}$
$\bar{\tau}_{L}\tilde{\tau}_{L}\bar{\lambda}^{8}_{A}$ | $\frac{g}{\sqrt{6}}$ | $\tilde{\tau}^{*}_{L}\tau_{L}\lambda^{8}_{A}$ | $\frac{g}{-\sqrt{6}}$
$\bar{\tau}_{L}\overline{\tilde{W}}^{+}\tilde{\nu}_{\tau L}$ | $g$ | $\tau_{L}\tilde{W}^{+}\tilde{\nu}^{*}_{\tau L}$ | $-g$
$\bar{\tau}_{L}\overline{\tilde{Y}}^{+}\tilde{\nu}^{c}_{\tau L}$ | $g$ | $\tau_{L}\tilde{Y}^{+}\tilde{\nu}^{c*}_{\tau L}$ | $-g$
Table 2: Vertices of lepton-slepton-gaugino interaction at tree level.

  3. 3.

Higgs-Higgsino-gaugino:

$\displaystyle\mathcal{L}_{H\tilde{H}\tilde{V}}$ $\displaystyle=$
$\displaystyle-\frac{ig^{\prime}}{\sqrt{3}}\left[-\frac{1}{3}(\bar{\tilde{\chi}}\chi\bar{\lambda}_{B}-\chi^{\dagger}\tilde{\chi}\lambda_{B})+\frac{1}{3}(\bar{\tilde{\chi}}^{\prime}\chi^{\prime}\bar{\lambda}_{B}-\chi^{\prime\dagger}\tilde{\chi}^{\prime}\lambda_{B})\right.$
(107) $\displaystyle+$
$\displaystyle\left.\frac{2}{3}(\bar{\tilde{\rho}}\rho\bar{\lambda}_{B}-\rho^{\dagger}\tilde{\rho}\lambda_{B})-\frac{2}{3}(\bar{\tilde{\rho}}^{\prime}\rho^{\prime}\bar{\lambda}_{B}-\rho^{\prime\dagger}\tilde{\rho}^{\prime}\lambda_{B})\right]$
$\displaystyle-$
$\displaystyle\frac{ig}{\sqrt{2}}\left[\bar{\tilde{\rho}}\lambda^{a}\rho\bar{\lambda}^{a}_{A}-\rho^{\dagger}\lambda^{a}\tilde{\rho}\lambda^{a}_{A}+\bar{\tilde{\chi}}\lambda^{a}\chi\bar{\lambda}^{a}_{A}-\chi^{\dagger}\lambda^{a}\tilde{\chi}\lambda^{a}_{A}\right.$
$\displaystyle-$
$\displaystyle\left.\bar{\tilde{\rho}}^{\prime}\lambda^{*a}\rho^{\prime}\bar{\lambda}^{a}_{A}+\rho^{\prime\dagger}\lambda^{*a}\tilde{\rho}^{\prime}\lambda^{a}_{A}-\bar{\tilde{\chi}}^{\prime}\lambda^{*a}\chi^{\prime}\bar{\lambda}^{a}_{A}+\chi^{\prime\dagger}\lambda^{*a}\tilde{\chi}^{\prime}\lambda^{a}_{A}\right]$

Vertices of neutral Higgs-Higgsino-gaugino interactions are shown in table 3

Table 3: Vertices of the neutral Higgs-Higgsino-gaugino interactions. Vertex | Factor | Vertex | Factor
---|---|---|---
$\bar{\tilde{\chi}}^{0}_{1}\chi^{0}_{1}\bar{\lambda}_{B}$ | $\frac{-g^{\prime}}{3\sqrt{3}}$ | $\tilde{\chi}^{0}_{1}\chi^{0\dagger}_{1}\lambda_{B}$ | $\frac{g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\chi}}^{0}_{2}\chi^{0}_{2}\bar{\lambda}_{B}$ | $\frac{-g^{\prime}}{3\sqrt{3}}$ | $\tilde{\chi}^{0}_{2}\chi^{0\dagger}_{2}\lambda_{B}$ | $\frac{g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\chi}}^{0}_{1}\chi^{0}_{1}\bar{\lambda}^{3}_{A}$ | $\frac{g}{\sqrt{2}}$ | $\tilde{\chi}^{0}_{1}\chi^{0\dagger}_{1}\lambda^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$
$\bar{\tilde{\chi}}^{0}_{1}\chi^{0}_{1}\bar{\lambda}^{8}_{A}$ | $\frac{g}{\sqrt{6}}$ | $\tilde{\chi}^{0}_{1}\chi^{0\dagger}_{1}\lambda^{8}_{A}$ | $\frac{-g}{\sqrt{6}}$
$\bar{\tilde{\chi}}^{0}_{2}\chi^{0}_{2}\bar{\lambda}^{8}_{A}$ | $\frac{-g\sqrt{2}}{\sqrt{3}}$ | $\tilde{\chi}^{0}_{2}\chi^{0\dagger}_{2}\lambda^{8}_{A}$ | $\frac{g\sqrt{2}}{\sqrt{3}}$
$\bar{\tilde{\chi}}^{\prime 0}_{1}\chi^{\prime 0}_{1}\bar{\lambda}_{B}$ | $\frac{g^{\prime}}{3\sqrt{3}}$ | $\tilde{\chi}^{\prime 0}_{1}\chi^{\prime 0\dagger}_{1}\lambda_{B}$ | $\frac{-g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\chi}}^{\prime 0}_{2}\chi^{\prime 0}_{2}\bar{\lambda}_{B}$ | $\frac{g^{\prime}}{3\sqrt{3}}$ | $\tilde{\chi}^{\prime 0}_{2}\chi^{\prime 0\dagger}_{2}\lambda_{B}$ | $\frac{-g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\chi}}^{\prime 0}_{1}\chi^{\prime 0}_{1}\bar{\lambda}^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$ | $\tilde{\chi}^{\prime 0}_{1}\chi^{\prime 0\dagger}_{1}\lambda^{3}_{A}$ | $\frac{g}{\sqrt{2}}$
$\bar{\tilde{\chi}}^{\prime 0}_{1}\chi^{\prime 0}_{1}\bar{\lambda}^{8}_{A}$ | $\frac{-g}{\sqrt{6}}$ | $\tilde{\chi}^{\prime 0}_{1}\chi^{\prime 0\dagger}_{1}\lambda^{8}_{A}$ | $\frac{g}{\sqrt{6}}$
$\bar{\tilde{\chi}}^{\prime 0}_{2}\chi^{\prime 0}_{2}\bar{\lambda}^{8}_{A}$ | $\frac{g\sqrt{2}}{\sqrt{3}}$ | $\tilde{\chi}^{\prime 0}_{2}\chi^{\prime 0\dagger}_{2}\lambda^{8}_{A}$ | $\frac{-g\sqrt{2}}{\sqrt{3}}$
$\bar{\tilde{\rho}}^{0}\rho^{0}\bar{\lambda}_{B}$ | $\frac{2g^{\prime}}{3\sqrt{3}}$ | $\tilde{\rho}^{0}\rho^{0\dagger}\lambda_{B}$ | $\frac{-2g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\rho^{0}}}\rho^{0}\bar{\lambda}^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$ | $\tilde{\rho}^{0}\rho^{0\dagger}\lambda^{3}_{A}$ | $\frac{g}{\sqrt{2}}$
$\bar{\tilde{\rho}}^{0}\rho^{0}\bar{\lambda}^{8}_{A}$ | $\frac{g}{\sqrt{6}}$ | $\tilde{\rho}^{0}\rho^{0\dagger}\lambda^{8}_{A}$ | $\frac{-g}{\sqrt{6}}$
$\bar{\tilde{\rho}}^{\prime 0}\rho^{\prime 0}\bar{\lambda}_{B}$ | $\frac{-2g^{\prime}}{3\sqrt{3}}$ | $\tilde{\rho}^{\prime 0}\rho^{\prime 0\dagger}\lambda_{B}$ | $\frac{2g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\rho^{\prime 0}}}\rho^{\prime 0}\bar{\lambda}^{3}_{A}$ | $\frac{g}{\sqrt{2}}$ | $\tilde{\rho}^{\prime 0}\rho^{\prime 0\dagger}\lambda^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$
$\bar{\tilde{\rho}}^{\prime 0}\rho^{\prime 0}\bar{\lambda}^{8}_{A}$ | $\frac{-g}{\sqrt{6}}$ | $\tilde{\rho}^{\prime 0}\rho^{\prime 0\dagger}\lambda^{8}_{A}$ | $\frac{g}{\sqrt{6}}$
$\bar{\tilde{\chi}}^{-}\bar{\tilde{W}}^{+}\chi^{0}_{1}$ | $g$ | $\chi^{0*}\tilde{W}^{+}\tilde{\chi}^{-}$ |  -$g$
$\bar{\tilde{\chi}}^{-}\bar{\tilde{Y}}^{+}\chi^{0}_{2}$ | $g$ | $\chi^{0*}_{2}\tilde{Y}^{+}\tilde{\chi}^{-}$ | $-g$
$\bar{\tilde{\chi}}^{0}_{2}\bar{\tilde{X}}\chi^{0}_{1}$ | $g$ | $\chi^{0*}_{1}\tilde{X}\tilde{\chi}^{0}_{2}$ | $-g$
$\bar{\tilde{\chi}}^{0}_{1}\bar{\tilde{X}}^{*}\chi^{0}_{2}$ | $g$ | $\chi^{0*}_{2}\tilde{X}^{*}\tilde{\chi}^{0}_{1}$ | $-g$
$\bar{\tilde{\rho}}^{+}_{1}\bar{\tilde{W}}^{-}\rho^{0}$ | $g$ | $\rho^{0*}\tilde{W}^{-}\tilde{\rho}^{+}_{1}$ |  -$g$
$\bar{\tilde{\rho}}^{+}_{2}\bar{\tilde{Y}}^{-}\rho^{0}$ | $g$ | $\rho^{0*}\tilde{Y}^{-}\tilde{\rho}^{+}_{2}$ |  -$g$
$\bar{\tilde{\chi}}^{\prime+}\bar{\tilde{W}}^{-}\chi^{\prime 0}_{1}$ | $-g$ | $\chi^{0*}_{1}\tilde{W}^{-}\tilde{\chi}^{\prime+}$ | $g$
$\bar{\tilde{\chi}}^{\prime+}\bar{\tilde{Y}}^{-}\chi^{\prime 0}_{2}$ | $-g$ | $\chi^{\prime 0*}_{2}\tilde{Y}^{-}\tilde{\chi}^{\prime+}$ | $g$
$\bar{\tilde{\chi}}^{\prime 0}_{2}\bar{\tilde{X}}^{*}\chi^{\prime 0}_{1}$ | $-g$ | $\chi^{\prime 0*}_{1}\tilde{X}\tilde{\chi}^{\prime 0}_{2}$ | $g$
$\bar{\tilde{\chi}}^{\prime 0}_{1}\bar{\tilde{X}}\chi^{\prime 0}_{2}$ | $-g$ | $\chi^{\prime 0*}_{2}\tilde{X}\tilde{\chi}^{\prime 0}_{1}$ | $g$
$\bar{\tilde{\rho}}^{\prime-}_{1}\bar{\tilde{W}}^{+}\rho^{\prime 0}$ | $-g$ | $\rho^{\prime 0*}\tilde{W}^{+}\tilde{\rho}^{\prime-}_{1}$ | $g$
$\bar{\tilde{\rho}}^{\prime-}_{2}\bar{\tilde{Y}}^{+}\rho^{\prime 0}$ | $-g$ | $\rho^{\prime 0*}\tilde{Y}^{+}\tilde{\rho}^{\prime-}_{2}$ | $g$
  4. 4.

Yukawa interaction terms:

$\displaystyle\mathcal{L}_{l\tilde{l}\tilde{H}}$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{1ab}}{3}\left(L_{aL}\tilde{\rho}^{\prime}\tilde{l}^{c}_{bL}+\tilde{L}_{aL}\tilde{\rho}^{\prime}l^{c}_{bL}\right)-\frac{\lambda_{3ab}}{3}\left(L_{aL}\tilde{\rho}\tilde{L}_{bL}+\tilde{L}_{aL}\tilde{\rho}L_{bL}\right),$
(108) $\displaystyle\mathcal{L}_{llH}$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{1ab}}{3}L_{aL}l^{c}_{bL}\rho^{\prime}-\frac{\lambda_{3ab}}{3}\epsilon^{\alpha\beta\gamma}(L_{aL})_{\alpha}(L_{bL})_{\beta}(\rho)_{\gamma}+\mathrm{H.c.}.$
(109)

Our work needs only terms which include leptons $\mu$ or $\tau$, such as:

$\displaystyle\mathcal{L}_{l\tilde{l}\tilde{H}}$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{1ab}}{3}\left[l_{aL}\tilde{\rho}^{\prime
0}\tilde{l}^{c}_{bL}+(\tilde{\nu}_{aL}\tilde{\rho}^{\prime-}_{1}+\tilde{l}_{aL}\tilde{\rho}^{\prime
0}+\tilde{\nu}^{c}_{aL}\tilde{\rho}^{\prime-}_{2})l^{c}_{bL}\right]$ (110)
$\displaystyle-$
$\displaystyle\frac{\lambda_{3ab}}{3}\left[l_{aL}\tilde{\rho}^{+}_{2}\widetilde{\nu}_{bL}-l_{aL}\widetilde{\rho}^{+}_{1}\widetilde{\nu}^{c}_{bL}+\widetilde{\nu}^{c}_{aL}\widetilde{\rho}^{+}_{1}l_{bL}-\widetilde{\nu}_{aL}\widetilde{\rho}^{+}_{2}l_{bL}\right],$

From now we just note that because the conversation of lepton flavor in the
lepton sector at tree level then $\lambda_{1ab}=0$ with $a\neq b$ and
$\lambda_{3cd}=0$ with $c=d$. For simplicity, we use new notations:
$Y_{e}\equiv\lambda_{111}/3,Y_{\mu}\equiv\lambda_{122}/3,Y_{\tau}\equiv\lambda_{133}/3$
and
$Y_{\nu_{e\mu}}\equiv\lambda_{312}/3,Y_{\nu_{\mu\tau}}\equiv\lambda_{323}/3,Y_{\nu_{e\tau}}\equiv\lambda_{313}/3$.
Eq.(110) now can be written in the common form:

$\displaystyle\mathcal{L}_{l\tilde{l}\tilde{H}}$ $\displaystyle=$
$\displaystyle-Y_{\mu}\left[\mu_{L}\tilde{\mu}^{c}_{L}\tilde{\rho}^{\prime
0}+(\tilde{\nu}_{\mu
L}\tilde{\rho}^{\prime-}_{1}+\tilde{\mu}_{L}\tilde{\rho}^{\prime
0}+\tilde{\nu}^{c}_{\mu L}\tilde{\rho}^{\prime-}_{2})\mu^{c}_{L}\right]$ (111)
$\displaystyle-$ $\displaystyle
Y_{\tau}\left[\tau_{L}\tilde{\tau}^{c}_{L}\tilde{\rho}^{\prime
0}+(\tilde{\nu}_{\tau
L}\tilde{\rho}^{\prime-}_{1}+\tilde{\tau}_{L}\tilde{\rho}^{\prime
0}+\tilde{\nu}^{c}_{\tau L}\tilde{\rho}^{\prime-}_{2})\tau^{c}_{L}\right]$
$\displaystyle-$ $\displaystyle
Y_{\nu_{ab}}[l_{aL}\tilde{\rho}^{+}_{2}\tilde{\nu}_{bL}-l_{aL}\tilde{\rho}^{+}_{1}\tilde{\nu}^{c}_{bL}+\tilde{\nu}^{c}_{aL}\tilde{\rho}^{+}_{1}l_{bL}-\tilde{\nu}_{aL}\tilde{\rho}^{+}_{2}l_{bL}]$

Corresponding vertices are shown in Fig. 4

Vertex | Factor | vertex | Factor
---|---|---|---
$\mu_{L}\tilde{\mu}^{c}_{L}\tilde{\rho}^{\prime 0}$ | $-iY_{\mu}$ | $\tilde{\nu}_{\mu L}\tilde{\rho}^{\prime-}_{1}\mu^{c}_{L}$ | $-iY_{\mu}$
$\tilde{\mu}_{L}\tilde{\rho}^{\prime 0}\mu^{c}_{L}$ | $-iY_{\mu}$ | $\tilde{\nu}^{c}_{\mu L}\tilde{\rho}^{\prime-}_{2}\mu^{c}_{L}$ | $-iY_{\mu}$
$\tau_{L}\tilde{\tau}^{c}_{L}\tilde{\rho}^{\prime 0}$ | $-iY_{\tau}$ | $\tilde{\nu}_{\tau L}\tilde{\rho}^{\prime-}_{1}\tau^{c}_{L}$ | $-iY_{\tau}$
$\tilde{\tau}_{L}\tilde{\rho}^{\prime 0}\tau^{c}_{L}$ | $-iY_{\tau}$ | $\tilde{\nu}^{c}_{\tau L}\tilde{\rho}^{\prime-}_{2}\tau^{c}_{L}$ | $-iY_{\tau}$
$\mu_{L}~{}\tilde{\rho}^{+}_{2}~{}\tilde{\nu}_{\tau_{L}}$ | $-2iY_{\nu_{\mu\tau}}$ | $\mu_{L}~{}\tilde{\rho}^{+}_{1}~{}\tilde{\nu}^{c}_{\tau}$ | $2iY_{\nu_{\mu\tau}}$
$\tau_{L}~{}\tilde{\rho}^{+}_{2}~{}\tilde{\nu}_{\mu_{L}}$ | $-2iY_{\nu_{\tau\mu}}$ | $\tau_{L}~{}\tilde{\rho}^{+}_{1}~{}\tilde{\nu}^{c}_{\mu_{L}}$ | $2iY_{\nu_{\tau\mu}}$
Table 4: Higgsino-lepton-slepton interacions

  5. 5.

In the soft Lagrangian, the mass term of sleptons is given by

$\displaystyle\mathcal{L}_{\tilde{f}mass}$ $\displaystyle=$
$\displaystyle-\tilde{m}^{2}_{Lab}\tilde{L}_{aL}^{\dagger}\tilde{L}_{bL}-\tilde{m}^{2}_{Rab}\tilde{l}_{aL}^{c*}\widetilde{l}_{bL}^{c}-\left[h^{\prime}_{ab}\tilde{L}^{T}_{aL}\rho^{\prime}\tilde{l}^{c}_{bL}\right.$
(112) $\displaystyle+$
$\displaystyle\left.h_{ab}\varepsilon^{\alpha\beta\gamma}(\tilde{L}_{aL})_{\alpha}(\tilde{L}_{bL})_{\beta}(\rho)_{\gamma}+\frac{\lambda_{1ab}}{3}\mu_{\rho}\rho^{*}\tilde{L}_{aL}\tilde{l}^{c}_{bL}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{\lambda_{3ab}}{3}\mu_{\rho}\varepsilon^{\alpha\beta\gamma}(\rho^{\prime*})_{\alpha}(\tilde{L}_{aL})_{\beta}(\tilde{L}_{bL})_{\gamma}+\mathrm{H.c.}\right],$

here $a,b$ are flavor indices $\\{a,b=1,2,3\\}$ or $a,b=\\{e,\mu,\tau\\}$ and
$\alpha,\beta,\gamma$ are component indices of $SU(3)_{L}$. The
$\varepsilon^{\alpha\beta\gamma}$ is the antisymmetric tensor. In this paper
we focus on the mixing of slepton $\widetilde{\mu}$ and $\widetilde{\tau}$.
This mixing makes mass-eigenstate basis of slepton is different from [24]. For
more detail, please see in Appendix C. The Lagrangian relating with Higgs-
lepton-slepton interactions has the form

$\displaystyle\mathcal{L}_{\tilde{\mu}\tilde{\tau}H^{0}}$ $\displaystyle=$
$\displaystyle-\left[(h^{\prime}_{\mu\tau}\tilde{\mu}_{L}\rho^{\prime
0}\tilde{\tau}^{c}_{L}+h^{\prime}_{\tau\mu}\tilde{\tau}_{L}\rho^{\prime
0}\tilde{\mu}^{c}_{L}+h^{\prime}_{\tau}\tilde{\tau}^{c}_{L}\rho^{\prime
0}\tilde{\tau}_{L}+h^{\prime}_{\mu}\tilde{\mu}^{c}_{L}\rho^{\prime
0}\tilde{\mu}_{L})\right.$ (113) $\displaystyle+$
$\displaystyle\left.\rho^{0}(h_{\mu\tau}-h_{\tau\mu})(\tilde{\nu}^{c}_{\mu
L}\tilde{\nu}_{\tau L}-\tilde{\nu}_{\mu L}\tilde{\nu}^{c}_{\tau
L})+\frac{1}{2}Y_{\tau}\mu_{\rho}\rho^{0*}\tilde{\tau}_{L}\tilde{\tau}^{c}_{L}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{2}Y_{\mu}\mu_{\rho}\rho^{0*}\tilde{\mu}_{L}\tilde{\mu}^{c}_{L}+Y_{\nu_{\mu\tau}}\mu_{\rho}\rho^{\prime
0*}\left(\tilde{\nu}^{c}_{\tau L}\tilde{\nu}_{\mu L}-\tilde{\nu}^{c}_{\mu
L}\tilde{\nu}_{\tau L}\right)+\mathrm{H.c.}\right]$

Vertices of Higgs- slepton-slepton interactions are listed in Table 5.

Vertex | Factor | Vertex | Factor
---|---|---|---
$\tilde{\mu}^{c}_{L}\tilde{\mu}_{L}\rho^{\prime 0}$ | $-ih^{\prime}_{\mu}$ | $\tilde{\tau}^{c}_{L}\tilde{\tau}_{L}\rho^{\prime 0}$ | $-ih^{\prime}_{\tau}$
$\tilde{\mu}_{L}\tilde{\tau}^{c}_{L}\rho^{\prime 0}$ | $-ih^{\prime}_{\mu\tau}$ | $\tilde{\tau}_{L}\tilde{\mu}^{c}_{L}\rho^{\prime 0}$ | $-ih^{\prime}_{\tau\mu}$
$\rho^{0}\tilde{\nu}^{c}_{\mu L}\tilde{\nu}_{\tau L}$ | $-i(h_{\mu\tau}-h_{\tau\mu})$ | $\rho^{0}\tilde{\nu}_{\mu L}\tilde{\nu}^{c}_{\tau L}$ | $i(h_{\mu\tau}-h_{\tau\mu})$
$\rho^{0}\tilde{\tau}^{*}_{L}\tilde{\tau}^{c*}_{L}$ | $-\frac{i}{2}Y_{\tau}\mu_{\rho}$ | $\rho^{0}\tilde{\mu}^{*}_{L}\tilde{\mu}^{c*}_{L}$ | $-\frac{i}{2}Y_{\mu}\mu_{\rho}$
$\rho^{\prime 0}\tilde{\nu}^{*}_{\tau_{L}}\tilde{\nu}^{c*}_{\mu_{L}}$ | $-iY_{\nu_{\mu\tau}}\mu_{\rho}$ | $\rho^{\prime 0}\tilde{\nu}^{*}_{\mu_{L}}\tilde{\nu}^{c*}_{\tau_{L}}$ | $iY_{\nu_{\mu\tau}}\mu_{\rho}$

Table 5:  Slepton-slepton-Higgs vertices.

.

##  Appendix C Mass eigenstates of particles in the SUSYE331 model

###  C.1  Neutral Higgs

The physical states of Higgs (mass eigenstates) have been studied in [24]. For
convenience we review the main results in this appendix. First we expand the
neutral Higgs components around the VEVs by

$\displaystyle\chi^{T}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{u+S_{1}+iA_{1}}{\sqrt{2}},&\chi^{-},&\frac{w+S_{2}+iA_{2}}{\sqrt{2}}\\\
\end{array}\right),\hskip
14.22636pt\rho^{T}=\left(\begin{array}[]{ccc}\rho^{+}_{1},&\frac{v+S_{5}+iA_{5}}{\sqrt{2}},&\rho^{+}_{2}\\\
\end{array}\right)$ (116) $\displaystyle\chi^{\prime T}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{u^{\prime}+S_{3}+iA_{3}}{\sqrt{2}},&\chi^{\prime+},&\frac{w+S_{4}+iA_{4}}{\sqrt{2}}\\\
\end{array}\right),\hskip 14.22636pt\rho^{\prime
T}=\left(\begin{array}[]{ccc}\rho^{\prime-}_{1},&\frac{v^{\prime}+S_{6}+iA_{6}}{\sqrt{2}},&\rho^{\prime-}_{2}\\\
\end{array}\right)$ (119)

where $\\{u,w,u^{\prime},w^{\prime},v,v^{\prime}\\}$,
$\\{S_{1},S_{2},S_{3},S_{4},S_{5},S_{6}\\}$ and
$\\{A_{1},A_{2},A_{3},A_{4},A_{5},A_{6}\\}$ are VEV, scalar, and pseudo scalar
parts of neutral Higgs, respectively. The Higgs mass spectrum and the Higgs
mass eigenstates given in [24] showed that:

Scalar Higgs: Mass eigenstates of six original scalar Higgs
$\\{S_{1},S_{2},S_{3},S_{4},S_{5},S_{6}\\}$ are defined as three massless
eigenstates $\\{S^{\prime}_{1a},S^{\prime}_{5},\varphi_{S_{24}}\\}$ and three
massive ones $\\{\phi_{S_{24}},\varphi_{S_{a36}},\phi_{S_{a36}}\\}$. The
relations between the original and the mass-eigenstate base are 111 There are
some different definitions for $\gamma$ in [23, 24, 25]. In this paper we use
notations identifying with those of [23].:

$\displaystyle\left(\begin{array}[]{c}S_{1}\\\ S_{2}\\\ S_{3}\\\ S_{4}\\\
S_{5}\\\ S_{6}\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}c_{\beta}s_{\theta}&-s_{\beta}c_{\theta}&-c_{\beta}c_{\theta}&-s_{\alpha}s_{\beta}s_{\theta}&-c_{\alpha}s_{\beta}s_{\theta}&0\\\
c_{\beta}c_{\theta}&s_{\beta}s_{\theta}&c_{\beta}s_{\theta}&-s_{\alpha}s_{\beta}c_{\theta}&-c_{\alpha}s_{\beta}c_{\theta}&0\\\
s_{\beta}s_{\theta}&-c_{\beta}c_{\theta}&s_{\beta}c_{\theta}&s_{\alpha}c_{\beta}s_{\theta}&c_{\alpha}c_{\beta}s_{\theta}&0\\\
s_{\beta}c_{\theta}&c_{\beta}s_{\theta}&-s_{\beta}s_{\theta}&s_{\alpha}c_{\beta}c_{\theta}&c_{\alpha}c_{\beta}c_{\theta}&0\\\
0&0&0&-c_{\alpha}s_{\gamma}&s_{\alpha}s_{\gamma}&c_{\gamma}\\\
0&0&0&c_{\alpha}c_{\gamma}&-s_{\alpha}c_{\gamma}&s_{\gamma}\\\
\end{array}\right)\left(\begin{array}[]{c}S^{\prime}_{1a}\\\
\varphi_{S_{24}}\\\ \phi_{S_{24}}\\\ \varphi_{S_{a36}}\\\ \phi_{S_{a36}}\\\
S^{\prime}_{5}\\\ \end{array}\right)$ (138)

where some new notations are defined as follows:

$t_{\theta}\equiv\tan\theta\equiv\frac{u}{w}=\frac{u^{\prime}}{w^{\prime}},\;c_{\theta}\equiv\cos\theta,\;s_{\theta}\equiv\sin\theta,$
$t_{\beta}\equiv\tan\beta\equiv\frac{w}{w^{\prime}},\;c_{\beta}\equiv\cos\beta,\;s_{\beta}\equiv\sin\beta,$

and

$t_{\gamma}\equiv\tan\gamma\equiv\frac{v}{v^{\prime}},\;s_{\gamma}\equiv\sin\gamma,\;c_{\gamma}\equiv\cos\gamma,$

and $\alpha$ is determined through relations:

$\tan{2\alpha}\equiv\frac{-2m^{2}_{36a}}{m^{2}_{66a}-m^{2}_{33a}},\;c_{\alpha}\equiv\cos\alpha,\;s_{\alpha}\equiv\sin\alpha$
$m^{2}_{33a}=\frac{18g^{2}+g^{\prime 2}}{54c^{2}_{\theta}}(w^{2}+w^{\prime
2})=\frac{(18g^{2}+g^{\prime 2})w^{2}}{54c^{2}_{\theta}s^{2}_{\beta}}$
$m^{2}_{66a}=\frac{9g^{2}+2g^{\prime 2}}{27}(v^{2}+v^{\prime
2})=\frac{(9g^{2}+2g^{\prime 2})v^{2}}{27s^{2}_{\gamma}}$
$m^{2}_{36a}=\frac{9g^{2}+2g^{\prime 2}}{54}\sqrt{\frac{(v^{2}+v^{\prime
2})(w^{2}+w^{\prime 2})}{c^{2}_{\theta}}}=\frac{9g^{2}+2g^{\prime
2}}{54}\frac{vw}{|c_{\theta}s_{\gamma}s_{\beta}|}$

The mass eigenvalues of three physical Higgses
$\phi_{S_{24}},\varphi_{S_{a36}}$ and $\phi_{S_{a36}}$ are:

$\displaystyle m^{2}_{\phi_{S_{24}}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4}(1+t^{2}_{\theta})(w^{2}+w^{\prime
2})=\frac{g^{2}w^{2}}{c^{2}_{\theta}s^{2}_{\beta}}$ (139) $\displaystyle
m^{2}_{\varphi_{S_{a36}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{33a}+m^{2}_{66a}-\sqrt{(m^{2}_{33a}-m^{2}_{66a})^{2}+4m^{4}_{36a}}\right]$
(140) $\displaystyle m^{2}_{\phi_{S_{a36}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{33a}+m^{2}_{66a}+\sqrt{(m^{2}_{33a}-m^{2}_{66a})^{2}+4m^{4}_{36a}}\right]$
(141)

Pseudo-scalar Higgs: There are five Goldstone bosons
$\\{A_{5},A_{6},A^{\prime}_{1},A^{\prime}_{2},\varphi_{A}\\}$ and one massive
physical Higgs $\varphi_{A}$. Because the $\varphi_{A}$ does not receive any
contributions from the $A_{5},A_{6}$ and sleptons as well as their sneutrinos
only couple to $\rho,\rho^{\prime}$, hence there is no coupling of pseudo
scalar $\varphi_{A}$ to muon and tauon at the one loop approximation. This is
the difference between MSMS and our model.

###  C.2  Mass eigenstates of sleptons

The masses of sleptons in SUSYE331 models were studied in details in [24]. In
the work, they assumed that lepton numbers are conserved even in the slepton
sector. This assumption leaded to the absence of mixing terms in slepton
sector. Our work is interested in studying the source of LFV caused by the
mixing between slepton $\tilde{\mu}$ and $\tilde{\tau}$ and ignore all other
sources of FLV . So with two assumptions of R-parity conversation and the
small left-right mixing in slepton sector, we can base on [24] to write the
mass terms of charged sleptons in the form:

$\displaystyle-\mathcal{L}_{\tilde{l}\tilde{l}^{*}}$ $\displaystyle=$
$\displaystyle\sum_{\tilde{l}_{L_{a}}}\tilde{m}^{2}_{\tilde{l}_{L_{a}}}\tilde{l}^{*}_{L_{a}}\tilde{l}_{L_{a}}+\left(\tilde{m}^{2}_{L_{\mu\tau}}\tilde{\mu}^{*}_{L}\tilde{\tau}_{L}+\mathrm{H.c.}\right)$
(142) $\displaystyle+$
$\displaystyle\sum_{\tilde{l}_{R_{a}}}\tilde{m}^{2}_{\tilde{l}_{R_{a}}}\tilde{l}^{*}_{R_{a}}\tilde{l}_{R_{a}}+\left(\tilde{m}^{2}_{R_{\mu\tau}}\tilde{\mu}^{*}_{R}\tilde{\tau}_{R}+\mathrm{H.c.}\right)$

where
$\tilde{l}_{L_{a}}=\\{\tilde{e}_{L},\tilde{\mu}_{L},\tilde{\tau}_{L}\\}$,
$\tilde{l}_{R_{a}}=\\{\tilde{e}_{R},\tilde{\mu}_{R},\tilde{\tau}_{R}\\}$ and

$\displaystyle\tilde{m}^{2}_{\tilde{l}_{L_{a}}}\equiv B_{aa}$ $\displaystyle=$
$\displaystyle M^{2}_{aa}+\frac{1}{4}\mu_{0a}^{2}+\frac{v^{\prime
2}}{18}\lambda^{2}_{1aa}+\frac{1}{18}\lambda^{2}_{2a}(u^{2}+w^{2})$
$\displaystyle-$
$\displaystyle\frac{g^{2}}{2}\left(H_{3}-\frac{1}{\sqrt{3}}H_{8}-\frac{2t^{2}}{3}H_{1}\right),$
$\displaystyle\tilde{m}^{2}_{\tilde{l}_{R_{a}}}\equiv C_{aa}$ $\displaystyle=$
$\displaystyle m^{2}_{aa}+\frac{v^{\prime
2}}{18}\lambda^{2}_{1aa}+g^{2}t^{2}H_{1},$
$\displaystyle\tilde{m}^{2}_{L_{\mu\tau}}\equiv B_{23}$ $\displaystyle=$
$\displaystyle
M^{2}_{23}+\frac{1}{4}\mu_{02}\mu_{03}+\frac{1}{18}\lambda_{22}\lambda_{23}(u^{2}+w^{2}),$
$\displaystyle\tilde{m}^{2}_{R_{\mu\tau}}\equiv C_{23}$ $\displaystyle=$
$\displaystyle m^{2}_{23},$ $\displaystyle H_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{6}\left[(u^{2}+w^{2})\frac{\cos
2\beta}{s^{2}_{\beta}}-2v^{2}\frac{\cos 2\gamma}{s^{2}_{\gamma}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{6}\left[(u^{2}+w^{2})\left(\cot^{2}\beta-1\right)-2v^{2}\left(\cot^{2}\gamma-1\right)\right]$
$\displaystyle H_{3}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{4}\left[u^{2}\frac{\cos
2\beta}{s^{2}_{\beta}}-2v^{2}\frac{\cos 2\gamma}{s^{2}_{\gamma}}\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\left[u^{2}\left(\cot^{2}\beta-1\right)-2v^{2}\left(\cot^{2}\gamma-1\right)\right]$
$\displaystyle H_{8}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{4\sqrt{3}}\left[v^{2}\frac{\cos
2\gamma}{s^{2}_{\gamma}}+(u^{2}-2w^{2})\frac{\cos
2\beta}{s^{2}_{\beta}}\right],$ $\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{3}}\left[v^{2}\left(\cot^{2}\gamma-1\right)+(u^{2}-2w^{2})\left(\cot^{2}\beta-1\right)\right],$
$\displaystyle t^{2}$ $\displaystyle\equiv$
$\displaystyle\left(\frac{g^{\prime}}{g}\right)^{2}=\frac{6s^{2}_{W}}{3-4s^{2}_{W}}$
(143)

Note that here we use notations $M^{2}_{aa}$ and $m^{2}_{aa}$ in stead of
$m^{2}_{aL}$ and $m^{2}_{la}$ in the soft breaking term of [23]. We assume
that the mixing matrix of the $\tilde{\mu}_{L,R}$ and $\tilde{\tau}_{L,R}$
slepton masses is given by

$\displaystyle-\mathcal{L}_{\tilde{\mu}\tilde{\tau}}$ $\displaystyle=$
$\displaystyle\left(\tilde{\mu}_{L}^{*},\;\tilde{\tau}_{L}^{*}\right)\left(\begin{array}[]{cc}\tilde{m}^{2}_{\mu_{L}}&\tilde{m}^{2}_{L_{\mu\tau}}\\\
\tilde{m}^{*2}_{L_{\mu\tau}}&\tilde{m}^{2}_{\tau_{L}}\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{\mu_{L}}\\\
\tilde{\tau_{L}}\\\
\end{array}\right)+\left(\tilde{\mu}^{c*}_{L},\;\tilde{\tau}^{c*}_{L}\right)\left(\begin{array}[]{cc}\tilde{m}^{2}_{\mu_{R}}&\tilde{m}^{2}_{R_{\mu\tau}}\\\
\tilde{m}^{*2}_{R_{\mu\tau}}&\tilde{m}^{2}_{\tau_{R}}\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{\mu}^{c}_{L}\\\
\tilde{\tau}^{c}_{L}\\\ \end{array}\right)$ (152)

This form is the same as that given in [10, 11] ( for detail, see [10],
Appendix A.1). Hence, the mass eigenstates and eigenvalues of sleptons in our
model are similar to that in the MSSM [10, 11]. In particular, the mass mixing
matrix of the left handed and right handed of sleptons given in (113) produce
the mass eigenstates such as $\\{\tilde{l}_{L_{2}},\tilde{l}_{L_{3}}\\}$ and
$\\{\tilde{l}_{R_{2}},\tilde{l}_{R_{3}}\\}$. The corresponding mass
eigenvalues are $\\{\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{L_{3}}\\}$ and
$\\{\tilde{m}^{2}_{R_{2}},\tilde{m}^{2}_{R_{3}}\\}$.

From now we adopt conventions the flavor states of sleptons are
$\tilde{\mu}_{L},\tilde{\tau}_{L}$ and
$\tilde{\mu}^{c}_{L},\tilde{\tau}^{c}_{L}$ while the mass eigenstates are
$\tilde{l}_{L_{2}},\tilde{l}_{L_{3}}$ and
$\widetilde{l}_{R_{2}},\widetilde{l}_{R_{3}}$, respectively. The relations
between these two kinds of basics are:
$\tilde{\mu}_{L}=c_{L}\tilde{l}_{L_{2}}-s_{L}\tilde{l}_{L_{3}}$,
$\tilde{\tau}_{L}=s_{L}\tilde{l}_{L_{2}}+c_{L}\tilde{l}_{L_{3}}$, with
$c_{L}=\cos\theta_{L}$, $s_{L}=\sin\theta_{L}$;
$\widetilde{\mu}^{c}_{L}=c_{R}\widetilde{l}_{R_{2}}-s_{R}\widetilde{l}_{R_{3}}$,
$\widetilde{\tau}^{c}_{L}=s_{R}\widetilde{l}_{R_{2}}+c_{R}\widetilde{l}_{R_{3}}$,
with $c_{R}=\cos\theta_{R}$, $s_{R}=\sin\theta_{R}$. The mixing parameters
satisfy the following relations:

$\displaystyle
s_{L}c_{L}=\frac{\tilde{m}^{2}_{L\mu\tau}}{\tilde{m}^{2}_{L_{3}}-\tilde{m}^{2}_{L_{2}}},\hskip
14.22636pts_{R}c_{R}=\frac{\tilde{m}^{2}_{R\mu\tau}}{\tilde{m}^{2}_{R_{3}}-\tilde{m}^{2}_{R_{2}}}.$
(154)

###  C.3  Sneutrinos

The general Lagrangian which gains masses for sneutrinos is given in [24] as
follows

$\displaystyle\mathcal{L}_{\tilde{\nu}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\tilde{\nu}^{*}_{aL},&\tilde{\nu}^{*}_{aR}\\\
\end{array}\right)\left(\begin{array}[]{cc}A_{ab}&E_{ab}\\\ E_{ab}&G_{ab}\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{\nu}_{bL}\\\
\tilde{\nu}_{bR}\\\ \end{array}\right),$ (160)

where

$\displaystyle\nu_{aL}\equiv(\nu_{1L},~{}\nu_{2L},~{}\nu_{3L})^{T},\hskip
14.22636pt\nu_{aR}\equiv\nu^{c*}_{aL}=(\nu^{c*}_{1L},\nu^{c*}_{2L},\nu^{c*}_{3L}),$
(161)

and

$\displaystyle A_{ab}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}\delta_{ab}\left(H_{3}+\frac{1}{\sqrt{3}}H_{8}-\frac{2t^{2}}{3}H_{1}\right)+M^{2}_{ab}+\frac{1}{4}\mu_{0a}\mu_{0b},$
$\displaystyle+$
$\displaystyle\frac{1}{18}\lambda_{2a}\lambda_{2b}(v^{2}+w^{2})+\frac{2}{9}\lambda_{3ca}\lambda_{3cb}v^{2},$
$\displaystyle G_{ab}$ $\displaystyle=$
$\displaystyle-g^{2}\delta_{ab}\left(\frac{1}{\sqrt{3}}H_{8}+\frac{t^{2}}{3}H_{1}\right)+M^{2}_{ab}+\frac{1}{4}\mu_{0a}\mu_{0b},$
$\displaystyle+$
$\displaystyle\frac{1}{18}\lambda_{2a}\lambda_{2b}(v^{2}+u^{2})+\frac{2}{9}\lambda_{3ca}\lambda_{3cb}v^{2},$
$\displaystyle E_{ab}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\left[(\varepsilon_{ab}-\varepsilon_{ba})v+\frac{1}{6}\mu_{\rho}v^{\prime}(\lambda_{3ab}-\lambda_{3ba})\right].$
(162)

If the LFV happens only in the $\\{\tilde{\nu}_{\mu},\tilde{\nu}_{\tau}\\}$
sector, we can rewrite the non-vanishing terms given in (162) in more explicit
formulas:

$\displaystyle m^{2}_{\tilde{\nu}_{aL}}\equiv A_{aa}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}\left(H_{3}+\frac{1}{\sqrt{3}}H_{8}-\frac{2t^{2}}{3}H_{1}\right)+M^{2}_{aa}+\frac{1}{4}\mu^{2}_{0a}$
$\displaystyle+$
$\displaystyle\frac{1}{18}\lambda^{2}_{2a}(v^{2}+w^{2})+\frac{2}{9}v^{2}\sum_{c}\lambda^{2}_{3ca},$
$\displaystyle m^{2}_{\tilde{\nu}_{aR}}\equiv G_{aa}$ $\displaystyle=$
$\displaystyle-g^{2}\left(\frac{1}{\sqrt{3}}H_{8}+\frac{t^{2}}{3}H_{1}\right)+M^{2}_{aa}+\frac{1}{4}\mu^{2}_{0a}$
$\displaystyle+$
$\displaystyle\frac{1}{18}\lambda^{2}_{2a}(v^{2}+u^{2})+\frac{2}{9}v^{2}\sum_{c}\lambda^{2}_{3ca},$
$\displaystyle m^{2}_{\tilde{\nu}_{L\mu\tau}}\equiv A_{23}$ $\displaystyle=$
$\displaystyle
M^{2}_{23}+\frac{1}{4}\mu_{02}\mu_{03}+\frac{1}{18}\lambda_{22}\lambda_{23}(v^{2}+w^{2})+\frac{2}{9}v^{2}\lambda^{2}_{3c2}\lambda_{3c3},$
$\displaystyle m^{2}_{\tilde{\nu}_{R\mu\tau}}\equiv A_{23}$ $\displaystyle=$
$\displaystyle
M^{2}_{23}+\frac{1}{4}\mu_{02}\mu_{03}+\frac{1}{18}\lambda_{22}\lambda_{23}(v^{2}+u^{2})+\frac{2}{9}v^{2}\lambda_{3c2}\lambda_{3c3}.$
(163)

Similar to the charged sleptons sector, we denote by
$\\{\tilde{\nu}_{\mu_{L}},~{}\tilde{\nu}_{\tau_{L}},~{}\tilde{\nu}_{\mu_{R}},~{}\tilde{\nu}_{\tau_{R}}\\}$
the flavor eigenstates while by
$\\{\tilde{\nu}_{L2},~{}\tilde{\nu}_{L3},~{}\tilde{\nu}_{R2},~{}\tilde{\nu}_{R3}\\}$
the mass eigenstates. Also, notations
$\\{\tilde{m}^{2}_{\nu_{L2}},~{}\tilde{m}^{2}_{\nu_{L3}},$
$~{}\tilde{m}^{2}_{\nu_{R2}},\tilde{m}^{2}_{\nu_{R3}}\\}$ denote the mass
eigenstates of sneutrinos. Here the relations between two bases are:

$\displaystyle\widetilde{\nu}_{\mu_{L}}$ $\displaystyle=$ $\displaystyle
c_{\nu_{L}}\tilde{\nu}_{L2}-s_{\nu_{L}}\tilde{\nu}_{L3},\hskip
14.22636pt\widetilde{\nu}_{\tau_{L}}=s_{\nu_{L}}\tilde{\nu}_{L2}+c_{\nu_{L}}\tilde{\nu}_{L3},$
$\displaystyle\widetilde{\nu}_{\mu_{R}}$ $\displaystyle=$ $\displaystyle
c_{\nu_{R}}\tilde{\nu}_{R2}-s_{\nu_{R}}\tilde{\nu}_{R3},\hskip
14.22636pt\widetilde{\nu}_{\tau_{R}}=s_{\nu_{R}}\tilde{\nu}_{R2}+c_{\nu_{R}}\tilde{\nu}_{R3},$
$\displaystyle s_{\nu_{L}}c_{\nu_{L}}$ $\displaystyle=$
$\displaystyle\frac{\tilde{m}^{2}_{\nu_{L\mu\tau}}}{\tilde{m}^{2}_{\nu_{L3}}-\tilde{m}^{2}_{\nu_{L2}}},\hskip
14.22636pts_{\nu_{R}}c_{\nu_{R}}=\frac{\tilde{m}^{2}_{\nu_{R\mu\tau}}}{\tilde{m}^{2}_{\nu_{R3}}-\tilde{m}^{2}_{\nu_{R2}}}.$
(164)

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