###### Abstract

In this lectures, we give a review about the Minimal Supersymmetric Standard
Model (MSSM) with $R$-Parity Violation because it provides an attractive way
to generate neutrino masses, lepton mixing angles in acconcordance to present
neutrino data.

The Minimal Supersymmetric Standard Model (MSSM) with $R$-Parity Violation
M. C. Rodriguez

Grupo de Física Teórica e Matemática Física
Departamento de Física
Universidade Federal Rural do Rio de Janeiro - UFRRJ
BR 465 Km 7, 23890-000
Seropédica, RJ, Brazil,
email<EMAIL_ADDRESS>

PACS number(s): 12.60. Jv

Keywords: Supersymmetric Models

##  1 Introduction

Although the Standard Model (SM) describes the observed properties of charged
leptons and quarks it is not the ultimate theory. However, the necessity to go
beyond it, from the experimental point of view, comes at the moment only from
neutrino data [1].

On September of 2018 from 5-7, happend in Paris, France, the International
Conference “History of the Neutrino (1930-2018)”, where we get the following
intersting presentations:

  * •

Birth of the neutrino, from Pauli to the Reines-Cowan experiment [3];

  * •

Prehistory of neutrino oscillation [4];

  * •

The Nature of the Neutrino (Dirac/Majorana) and Double Beta Decay with or
without Neutrinos [5];

  * •

The Mikheyev-Smirnov-Wolfenstein (MSW) Effect [6];

  * •

Neutrino Mistakes: Wrong tracks and Hints, Hopes and Failures [7];

  * •

Neutrinos and Particle Physics Models [8];

  * •

Neutrino Masses and Mixing: A Little History for a Lot of Fun [9];

and many others interestings talks where it was presented in details that
neutrinos are massive particle, they can oscilate and they can be Dirac
particles [10] or Majorana fermions [11, 12]. A very nice review about the
status of Neutrinos can be found at [13].

Supersymmetry (SUSY) or symmetry between bosons (particles with integer spin)
and fermions (particles with half-integer spin) has been introduced in
theoretical papers nearly 30 years ago independently by Golfand and Likhtman
[14], Volkov and Akulov [15] and Wess and Zumino [16]. The supersymmetry
algebra was introduced in [14] 111They also constructed the first four-
dimensional field theory with supersymmetry, (massive) quantum electrodynamics
of spinors and scalars. There and in the Wess-Zumino article [16], the
supersymmetry generator $Q$ relates bosons with fermions in the usual sense.
The Volkov-Akulov article, however, deals only with fermions. The
supersymmetry generator acts in a non-linear way, turning a fermion field into
a composite bosonic one made of two fermion fields [15] 222They started the
foundations of supergravity. This illustrates that the supersymmetric algebra
by itself does not require superpartners – in contrast with what is commonly
said or thought now. Since that time there appeared thousands of papers. The
reason for this remarkable activity is the unique mathematical nature of
supersymmetric theories, possible solution of various problems of the SM
within its supersymmetric extentions as well as the opening perspective of
unification of all interactions in the framework of a single theory [18, 19,
20, 21, 22]. About the first version of the Supersymmetric Standard Model see
[17].

On this review we will present a short introduction for the Minimal
Supersymmetric Standard Model with $R$-Parity violation. First we present the
model at Sec.(2), then we present our results when the left handed stau-
neutrino get vev, Sec.(4). After it we present in a short way the same results
when all the left-handed sneutrinos get vev. Our conclusion are found in the
last section.

##  2 Review of the MSSM with $R$-Parity Interactions.

The Minimal Supersymmetric Standard Model (MSSM), known as MSSM, is a good
candidate to be the physics beyond the Standard Model, as presented in several
books and review about this subject [18, 19, 20, 21, 22]. It is the
supersymmetric extension of the SM that contains a minimal number of states
and interactions. The model has the gauge symmetry defined as

$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ (1)

extended by the supersymmetry to include the supersymmetric partners of the SM
fields which have spins that differ by $+1/2$ as required by the
supersymmetric algebra. The MSSM with $R$-Parity conservation has 124 free
parameters [19].

The partner of fermion masses and mixing constitues one of the most important
issues in modern physics, the only new fermion that does not mixing is the
gluinos and the only new bosons that does not mix is the sneutrinos.

However in the MSSM there are interactions that volates Lepton or Baryon
Number conservation [23, 24]. Some years ago it was proposed a model for the
structure of the lepton mixing which account for the atmospheric and the solar
anomalies. It is based on the simplest one-parameter extension of the MSSM
with bi-linear $R$-Parity violation [28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40].

We will introduce the following chiral superfields

$\displaystyle\hat{L}_{iL}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\hat{\nu}_{iL}\\\
\hat{l}_{iL}\end{array}\right)\sim\left(1,{\bf 2},-1\right),\,\
\hat{l}^{c}_{iR}\sim\left(1,{\bf 1},+2\right),$ (4) ,
$\displaystyle\hat{Q}_{iL}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\hat{u}_{iL}\\\
\hat{d}_{iL}\end{array}\right)\sim\left(3,{\bf 2},\frac{1}{3}\right),\,\
\hat{u}^{c}_{iR}\sim\left(3,{\bf 1},-\frac{4}{3}\right),\,\
\hat{d}^{c}_{iR}\sim\left(3,{\bf 1},\frac{2}{3}\right),$ (7)
$\displaystyle\hat{H}_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\hat{H}^{+}_{1}\\\
\hat{H}^{0}_{1}\end{array}\right)\sim\left(1,{\bf 2},+1\right),\,\
\hat{H}_{2}=\left(\begin{array}[]{c}\hat{H}^{0}_{2}\\\
\hat{H}^{-}_{2}\end{array}\right)\sim\left(1,{\bf 2},-1\right).$ (12)

It is possible to rotate the chiral superfields $\hat{H}_{2},\hat{L}_{3L}$ to
a new basis $\hat{H}^{\prime}_{2},\hat{L}^{\prime}_{3L}$ by a linear
transformation given by [18, 19, 39]

$\displaystyle\hat{H}^{\prime}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\mu^{2}+\mu^{2}_{3}}\left(\mu\hat{H}_{2}-\mu_{3}\hat{L}_{3L}\right),$
$\displaystyle\hat{L}^{\prime}_{3L}$ $\displaystyle=$
$\displaystyle\frac{1}{\mu^{2}+\mu^{2}_{3}}\left(\mu\hat{H}_{2}+\mu_{3}\hat{L}_{3L}\right).$
(13)

The non-zero vaccum expectation value of this model are denoted as

$\displaystyle\langle H_{1}\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\left[v_{1}+\sigma^{0}_{1}+\imath\varphi^{0}_{1}\right],\,\
\langle
H_{2}\rangle\equiv\frac{1}{\sqrt{2}}\left[v_{2}+\sigma^{0}_{2}+\imath\varphi^{0}_{2}\right],$
$\displaystyle\langle\tilde{L}_{i}\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\left[v^{L}_{i}+\tilde{\nu}^{R}_{i}+\imath\tilde{\nu}^{I}_{i}\right].$
(14)

The superpotential is given by

$\displaystyle W$ $\displaystyle=$ $\displaystyle W_{2}+W_{3},$ (15)
$\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle\mu\;\left(\hat{H}_{1}\hat{H}_{2}\right)-\sum_{i=1}^{3}\mu_{i}\left(\hat{L}_{i}\hat{H}_{2}\right),$
$\displaystyle W_{3}$ $\displaystyle=$
$\displaystyle\sum_{i,j,k=1}^{3}\left[\,f^{l}_{ij}\left(\hat{H}_{1}\hat{L}_{i}\right)\hat{l}^{c}_{jR}+f^{d}_{ij}\left(\hat{H}_{1}\hat{Q}_{i}\right)\hat{d}^{c}_{jR}+f^{u}_{ij}\left(\hat{H}_{2}\hat{Q}_{i}\right)\hat{u}^{c}_{jR}+\lambda_{ijk}\left(\hat{L}_{i}\hat{L}_{j}\right)\hat{l}^{c}_{kR}\right.$
(16) $\displaystyle+$
$\displaystyle\left.\lambda^{\prime}_{ijk}\left(\hat{L}_{i}\hat{Q}_{j}\right)\hat{d}^{c}_{kR}\right],$

where

$\displaystyle\left(\hat{A}\hat{B}\right)\equiv\epsilon_{\alpha\beta}\hat{A}^{\alpha}\hat{B}^{\beta}$
(17)

and we introduce 12 new parameters in addition to the MSSM with $R$-Parity
conservation model.

We can add the following soft supersymmetry breaking terms to the MSSM [18,
19, 20]

$\displaystyle{\cal L}^{MSSM}_{Soft}$ $\displaystyle=$ $\displaystyle{\cal
L}^{MSSM}_{SMT}+{\cal L}^{MSSM}_{GMT}+{\cal L}^{MSSM}_{INT}\,\ ,$ (18)

where the scalar mass term ${\cal L}_{SMT}$ is given by the following relation

$\displaystyle{\cal L}^{MSSM}_{SMT}$ $\displaystyle=$
$\displaystyle-\sum_{i,j=1}^{3}\left[\,\tilde{L}^{\dagger}_{iL}\left(M_{L}^{2}\right)_{ij}\tilde{L}_{jL}+\tilde{l}^{c\dagger}_{iR}\left(M^{2}_{l}\right)_{ij}\tilde{l}^{c}_{jR}+M_{1}^{2}H^{\dagger}_{1}H_{1}+M_{2}^{2}H^{\dagger}_{2}H_{2}\right]\,\
,$ (19)

The $3\times 3$ matrices $M_{L}^{2}$ and $M^{2}_{l}$ are hermitian and
$M_{1}^{2}$ and $M_{2}^{2}$ are real. The gaugino mass term is written as

$\displaystyle{\cal L}^{MSSM}_{GMT}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\left(\,M_{3}\;\sum_{a=1}^{8}\lambda^{a}_{C}\lambda^{a}_{C}+M\;\sum_{i=1}^{3}\;\lambda^{i}\lambda^{i}+M^{\prime}\;\lambda\lambda\,\right)+hc\right]\,\
.$ (20)

Here, $M_{3},M$ and $M^{\prime}$ are complex. Finally, there is an interaction
term ${\cal L}_{INT}$ of the form

$\displaystyle{\cal L}^{MSSM}_{INT}$ $\displaystyle=$
$\displaystyle-B\mu\left(H_{1}H_{2}\right)-\sum_{i=1}^{3}\left\\{B_{i}\mu_{i}\left(H_{2}\tilde{L}_{i}\right)+\sum_{j,k=1}^{3}\left[\left(H_{1}\tilde{L}_{iL}\right)A^{E}_{ij}\tilde{l}^{c}_{jR}-\lambda^{\prime}_{ijk}\left(\tilde{L}_{iL}\tilde{Q}_{jL}\right)\tilde{d}^{c}_{kR}\right]\right\\}$
(21) $\displaystyle+$ $\displaystyle hc\,\ .$

The parameters $B\mu$, $B_{i}\mu_{i},A^{E}_{ij}$ and $\lambda^{\prime}_{ijk}$
are in general complex [18, 19].

##  3 Masses

Here we will present the masses of all particle of this model. The masses of
gluinos are the same as presented at MSSM with $R$-Parity conservation, due
this fact we will not present them here, for more detail see .

We first will present the results when $\tilde{L}_{3}\neq 0$, at Sec.(4), then
the results for the general case, when all the left-handed sneutrinos gain vev
at Sec.(5).

##  4 Results with only $\tilde{L}_{3}\neq 0$.

###  4.1 Bosons Masses

On the other hand, ${\cal L}_{Higgs}$ give mass to the gauge bosons, throught
the following expression:

$\displaystyle\left({\cal D}_{m}H_{1}\right)^{\dagger}\left({\cal
D}_{m}H_{1}\right)+\left({\cal D}_{m}H_{2}\right)^{\dagger}\left({\cal
D}_{m}H_{2}\right)+\sum_{i=1}^{3}\left({\cal
D}_{m}\tilde{L}_{3}\right)^{\dagger}\left({\cal D}_{m}\tilde{L}_{3}\right),$
(22)

where ${\cal D}_{m}$ is covariant derivates of the SM given by:

$\displaystyle{\cal D}_{m}H_{1}$ $\displaystyle\equiv$
$\displaystyle\partial_{m}H_{1}+\imath gT^{i}W^{i}_{m}H_{1}+\imath
g^{\prime}\left(\frac{Y_{H_{1}}}{2}\right)b_{Y^{\prime}}H_{1},$
$\displaystyle{\cal D}_{m}H_{2}$ $\displaystyle\equiv$
$\displaystyle\partial_{m}H_{2}+\imath gT^{i*}W^{i}_{m}H_{2}+\imath
g^{\prime}\left(\frac{Y_{H_{2}}}{2}\right)b_{Y^{\prime}}H_{2},$
$\displaystyle{\cal D}_{m}\tilde{L}_{i}$ $\displaystyle\equiv$
$\displaystyle\partial_{m}\tilde{L}_{i}+\imath
gT^{i*}W^{i}_{m}\tilde{L}_{i}+\imath
g^{\prime}\left(\frac{Y_{\tilde{L}_{i}}}{2}\right)b_{Y^{\prime}}\tilde{L}_{i}.$
(23)

Before calculate expression for the covariant derivatives it is important to
remember

$\displaystyle\sum_{i=1}^{3}\sigma^{i}W^{i}_{m}=\left(\begin{array}[]{cc}W^{3}_{m}&\sqrt{2}W^{+}_{m}\\\
\sqrt{2}W^{-}_{m}&-W^{3}_{m}\end{array}\right),\,\
\sum_{i=1}^{3}\sigma^{i*}W^{i}_{m}=\left(\begin{array}[]{cc}W^{3}_{m}&\sqrt{2}W^{-}_{m}\\\
\sqrt{2}W^{+}_{m}&-W^{3}_{m}\end{array}\right),$ (28)

where the charged gauge bosons are defined as

$W^{\pm}_{m}=\frac{1}{\sqrt{2}}\left(V^{1}_{m}\mp\imath V^{2}_{m}\right).$
(29)

Using those informations we can get

$\displaystyle{\cal D}_{m}\langle H_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{\imath
v_{1}}{2\sqrt{2}}\left(\begin{array}[]{c}g\sqrt{2}W^{-}_{m}\\\
-gW^{3}_{m}+g^{\prime}W^{\prime}_{m}\end{array}\right),$ (32)
$\displaystyle{\cal D}_{m}\langle H_{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{\imath
v_{2}}{2\sqrt{2}}\left(\begin{array}[]{c}-gW^{3}_{m}+g^{\prime}W^{\prime}_{m}\\\
-g\sqrt{2}W^{+}_{m}\end{array}\right),$ (35) $\displaystyle{\cal
D}_{m}\langle\tilde{L}_{3}\rangle$ $\displaystyle=$ $\displaystyle\frac{\imath
v^{L}_{3}}{2\sqrt{2}}\left(\begin{array}[]{c}-gW^{3}_{m}+g^{\prime}W^{\prime}_{m}\\\
-g\sqrt{2}W^{+}_{m}\end{array}\right),$ (38)

from those equations, we can show

$\displaystyle\left({\cal D}^{m}\langle
H_{1}\rangle\right)^{\dagger}\left({\cal D}_{m}\langle H_{1}\rangle\right)$
$\displaystyle=$
$\displaystyle\frac{v^{2}_{1}}{8}\left[2g^{2}W^{+m}W^{-}_{m}+g^{2}W^{3m}W^{3}_{m}-gg^{\prime}W^{3m}W^{\prime}_{m}-gg^{\prime}W^{\prime
m}W^{3}_{m}\right.$ $\displaystyle+$
$\displaystyle\left.(g^{\prime})^{2}W^{\prime m}W^{\prime}_{m}\right],$
$\displaystyle\left({\cal D}^{m}\langle
H_{2}\rangle\right)^{\dagger}\left({\cal D}_{m}\langle H_{2}\rangle\right)$
$\displaystyle=$
$\displaystyle\frac{v^{2}_{2}}{8}\left[2g^{2}W^{-m}W^{+}_{m}+g^{2}W^{3m}W^{3}_{m}-gg^{\prime}W^{3m}W^{\prime}_{m}-gg^{\prime}W^{\prime
m}W^{3}_{m}\right.$ $\displaystyle+$
$\displaystyle\left.(g^{\prime})^{2}W^{\prime m}W^{\prime}_{m}\right],$
$\displaystyle\left({\cal
D}^{m}\langle\tilde{L}_{3}\rangle\right)^{\dagger}\left({\cal
D}_{m}\langle\tilde{L}_{3}\rangle\right)$ $\displaystyle=$
$\displaystyle\frac{(v^{L}_{3})^{2}}{8}\left[2g^{2}W^{-m}W^{+}_{m}+g^{2}W^{3m}W^{3}_{m}-gg^{\prime}W^{3m}W^{\prime}_{m}-gg^{\prime}W^{\prime
m}W^{3}_{m}\right.$ (39) $\displaystyle+$
$\displaystyle\left.(g^{\prime})^{2}W^{\prime m}W^{\prime}_{m}\right].$

We get the following expression to the charged gauge bosons masses

$\displaystyle
M^{2}_{W}=\frac{g^{2}}{4}\left(v_{1}^{2}+v_{2}^{2}+(v^{L}_{3})^{2}\right).$
(40)

We can define the new angle $\beta$ in the following way

$\displaystyle\tan\beta$ $\displaystyle=$ $\displaystyle\frac{v_{2}}{v_{1}},$
(41)

and we can define a new angle $\theta$ in the following way

$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle v\cos\beta\sin\theta,$
$\displaystyle v_{2}$ $\displaystyle=$ $\displaystyle v\sin\beta\sin\theta,$
$\displaystyle v^{L}_{3}$ $\displaystyle=$ $\displaystyle v\cos\theta.$ (42)

This new angle $\theta$ tends to $(\pi/2)$ rad in the limit of the MSSM when
$v^{L}_{3}$ go to zero.

The neutral massive gauge boson ($Z^{0}$) get the following mass

$\displaystyle M^{2}_{Z}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4\cos^{2}\theta_{W}}\left(v_{1}^{2}+v_{2}^{2}+(v^{L}_{3})^{2}\right)=\frac{M^{2}_{W}}{\cos^{2}\theta_{W}},$
(43)

where $\theta_{W}$ is the Weinberg angle and it is defined as

$\displaystyle e=g\sin\theta_{W}=g^{\prime}\cos\theta_{W},$ (44)

and get a massless foton $A_{m}$. The rotation in this case is

$\displaystyle\left(\begin{array}[]{c}A_{m}\\\
Z_{m}\end{array}\right)=\left(\begin{array}[]{cc}\sin\theta_{W}&\cos\theta_{W}\\\
\cos\theta_{W}&-\sin\theta_{W}\end{array}\right)\left(\begin{array}[]{c}V^{3}_{m}\\\
V_{m}\end{array}\right)\,\ ,$ (51)

it is the exact expression we get in the SM. Therefore the neutral boson gauge
sector is exact the same as in the SM.

###  4.2 Charginos Masses

The supersymmetric partners of the $W^{\pm}$, together with the usual charged
leptons $l_{3L},l^{c}_{3R}$ and the $H^{\pm}$ mix to mass eigenstates called
charginos $\chi^{\pm}_{i}$ ($i=1,2,3$) which are four–component Dirac
fermions.

From our Lagrangian we get the following terms to get the mass matrix for
charginos

$\displaystyle
M\left(\imath\lambda^{-}\right)\left(\imath\lambda^{+}\right)+\frac{gv_{1}}{\sqrt{2}}\left(\imath\lambda^{+}\right)\tilde{H}^{-}_{1}+\frac{gv_{2}}{\sqrt{2}}\left(\imath\lambda^{-}\right)\tilde{H}^{+}_{2}+\frac{gv^{L}_{3}}{\sqrt{2}}\left(\imath\lambda^{+}\right)l_{3L}+\mu\tilde{H}^{-}_{1}\tilde{H}^{+}_{2}$
(52) $\displaystyle-$
$\displaystyle\mu_{3}l_{3L}\tilde{H}^{+}_{2}-\frac{f^{\tau}v_{1}}{\sqrt{2}}l_{3L}l^{c}_{3R}+\frac{f^{\tau}v^{L}_{3}}{\sqrt{2}}\tilde{H}^{-}_{1}l^{c}_{3R}+hc$

where

$\lambda^{\pm}=\frac{1}{\sqrt{2}}\,(\lambda^{1}\mp\imath\lambda^{2}),$ (53)

see definition of $W$ boson given at Eq.(29) and $f^{\tau}\equiv f^{l}_{33}$
and all the fermions fields are two-component Weyl-van der Waerden fermions
[25, 26, 27].

In order to get the mass matrix for charginos, we start with the basis

$\psi^{-}=\left(\imath\lambda^{-}\\!,\,\tilde{H}^{-}_{1}\\!,\,l^{-}_{3L}\right)^{T},\hskip
17.07164pt\psi^{+}=\left(\imath\lambda^{+}\\!,\,\tilde{H}^{+}_{2}\\!,\,l^{c}_{3R}\right)^{T}.$
(54)

The mass terms of the lagrangian of the charged gaugino–higgsino system can
then be written as

${\cal
L}_{m}=\frac{1}{2}\,\left(\psi^{+T}\\!,\,\psi^{-T}\right)\,\,Y^{\pm}\,\left(\begin{array}[]{c}\psi^{+}\\\
\psi^{-}\end{array}\right)+hc$ (55)

where

$Y^{\pm}=\left(\begin{array}[]{cc}0&X\\\ X^{T}&0\end{array}\right),$ (56)

therefore we can rewrite Eq.(55) in the following way

${\cal
L}_{m}=\frac{1}{2}\left[\psi^{+T}X\psi^{-}+\psi^{-T}X^{T}\psi^{+}\right]+hc$
(57)

taken into account Eq.(52), we get the following expression

$X=\left(\begin{array}[]{ccc}M&\frac{gv_{2}}{\sqrt{2}}&0\\\
\frac{gv_{1}}{\sqrt{2}}&\mu&-\frac{f^{\tau}v^{L}_{3}}{\sqrt{2}}\\\
\frac{gv^{L}_{3}}{\sqrt{2}}&-\mu_{3}&\frac{f^{\tau}v_{1}}{\sqrt{2}}\end{array}\right).$
(58)

The $3\times 3$ matrix $X$ of Eq.(107) can be put into diagonal form in the
following way [18, 19, 28, 30]

$XX^{T}=X^{T}X=\left(\begin{array}[]{ccc}M^{2}+\frac{g^{2}v^{2}_{2}}{2}&\frac{g}{\sqrt{2}}\left(Mv_{1}+\mu
v_{2}\right)&\frac{g}{\sqrt{2}}\left(Mv^{L}_{3}+\mu_{3}v_{2}\right)\\\
\frac{g}{\sqrt{2}}\left(Mv_{1}+\mu
v_{2}\right)&\frac{1}{2}\left(g^{2}v^{2}_{1}+2\mu^{2}+(f^{\tau})^{2}(v^{L}_{3})^{2}\right)&A\\\
\frac{g}{\sqrt{2}}\left(Mv^{L}_{3}+\mu_{3}v_{2}\right)&A&\frac{1}{2}\left((f^{\tau})^{2}v^{2}_{1}+g^{2}(v^{L}_{3})^{2}+2\mu^{2}_{3}\right),\end{array}\right),$
(59)

where

$A=\frac{1}{2}\left((f^{\tau})^{2}v_{1}v^{L}_{3}+g^{2}v_{1}v^{L}_{3}-2\mu\mu_{3}\right).$
(60)

From Eq.(59) it is easy to get the following analytical results

$\displaystyle det(XX^{T})$ $\displaystyle=$ $\displaystyle
det(X^{T}X)=\frac{(f^{\tau})^{2}}{8}\left[g^{2}v_{2}\left(v^{2}_{1}-(v^{L}_{3})^{2}\right)-2M\left(\mu
v_{1}+\mu_{3}v^{L}_{3}\right)\right]^{2},$ (61) $\displaystyle Tr(XX^{T})$
$\displaystyle=$ $\displaystyle
Tr(X^{T}X)=\frac{1}{2}\left[2(M^{2}+\mu^{2}+\mu^{2}_{3})+g^{2}(v^{2}_{1}+v^{2}_{2}+(v^{L}_{3})^{2})+(f^{\tau})^{2}(v^{2}_{1}+(v^{L}_{3})^{2})\right].$

Therefore, the smallest eigenvalues of $XX^{T}$ and $X^{T}X$ with tau mass
$m_{\tau}$.

Using $\tan\beta=1$333See Eq.(41). and $M_{Z}=91.187$ GeV,
$s^{2}_{W}=0.223$444$\theta_{W}$ is the weak mixing angle and see Eq.(44).,
$\mu_{3}=0.9$ GeV, $v^{L}_{3}=0$ GeV (this value is consistent with that of
Ref. [37, 40]), $\mu=150$ GeV, $M=250$ GeV we obtain from Eq.(59) the masses
$1.777$ (in GeV) for the tau, and 124.3 and 275.7 GeV for the charginos.

###  4.3 Neutralinos Masses

In this review we choose the basis [mssm]

$\psi^{0}_{MSSMRPV}=\left(\begin{array}[]{ccccccc}\imath\lambda^{\prime}&\imath\lambda^{3}&\tilde{H}^{0}_{1}&\tilde{H}^{0}_{2}&\nu_{3L}\end{array}\right)^{\\!\rm
T}.\hskip 17.07164pt$ (63)

From our Lagrangian we get the following terms to get the mass matrix for
neutralinos

$\displaystyle\frac{M}{2}\left(\imath\lambda^{3}\right)\left(\imath\lambda^{3}\right)+\frac{M^{\prime}}{2}\left(\imath\lambda^{\prime}\right)\left(\imath\lambda^{\prime}\right)+\frac{gv_{1}}{\sqrt{2}}\left(\imath\lambda^{3}\right)\tilde{H}^{0}_{1}-\frac{gv_{2}}{\sqrt{2}}\left(\imath\lambda^{3}\right)\tilde{H}^{0}_{2}+\frac{gv^{L}_{3}}{\sqrt{2}}\left(\imath\lambda^{3}\right)\nu_{3L}$
(64) $\displaystyle-$
$\displaystyle\frac{g^{\prime}v_{1}}{\sqrt{2}}\left(\imath\lambda^{\prime}\right)\tilde{H}^{0}_{1}+\frac{g^{\prime}v_{2}}{\sqrt{2}}\left(\imath\lambda^{\prime}\right)\tilde{H}^{0}_{2}+\frac{g^{\prime}v^{L}_{3}}{\sqrt{2}}\left(\imath\lambda^{\prime}\right)\nu_{3L}-\mu\tilde{H}^{0}_{1}\tilde{H}^{0}_{2}+\mu_{3}\nu_{3L}\tilde{H}^{0}_{2}+hc$

The mass terms of the neutral gaugino–higgsino system can then be written as

${\cal L}_{m}=\frac{1}{2}\,(\psi^{0}_{MSSMRPV})^{\rm
T}\,Y^{neutralino}_{MSSMRPV}\,\psi^{0}_{MSSMRPV}+\,\ hc$ (65)

with

$Y^{neutralino}_{MSSMRPV}=\left(\begin{array}[]{ccccc}M^{\prime}&0&-\frac{g^{\prime}v_{1}}{2}&\frac{g^{\prime}v_{2}}{2}&\frac{g^{\prime}v^{L}_{3}}{2}\\\
0&M&\frac{gv_{1}}{2}&-\frac{gv_{2}}{2}&\frac{gv^{L}_{3}}{2}\\\
-\frac{g^{\prime}v_{1}}{2}&\frac{gv_{1}}{2}&0&-\mu&0\\\
\frac{g^{\prime}v_{2}}{2}&-\frac{gv_{2}}{1}&-\mu&0&\mu_{3}\\\
\frac{g^{\prime}v^{L}_{3}}{2}&\frac{gv^{L}_{3}}{2}&0&\mu_{3}&0\end{array}\right)\,\
.$ (66)

We can get numerical results for Eq.(66), using the values presented after
Eq.(LABEL:valorescharginosantesdadiag) together with $M^{\prime}=-200$ GeV, we
obtain a massive neutrino and its mass is $m_{\nu_{3}}=0.051$ eV555The
atmospheric neutrino mass scalae is $m^{2}=3\cdot 10^{-3}\mbox{eV}^{2}$., and
four heavy neutralinos with masses $269.30,-202.88,-100.55,84.2$ GeV.

This model break lepton number and therefore necessarily generate non-zero
Majorana neutrino masses [18, 19, 31, 32]. At tree-level only one of the
neutrinos pick up a mass by mixing with neutralinos [33, 34], leaving the
other two neutrinos massless [35]. The LSP in this case is the neutralino
$\tilde{\chi}^{0}_{1}$ and some of two body decay are [18, 19, 28, 30]

$\displaystyle\tilde{\chi}^{0}_{1}\rightarrow\tau^{\pm}W^{\mp},\,\
\tilde{\chi}^{0}_{1}\rightarrow\nu_{\tau}Z,$ (67)

etc.

More realistic neutrino masses require radiative correction and it is shown at
Fig.(1) and in this way the solar mass is explained and more details about it
can be found at [18, 19, 24, 28, 36, 37, 38, 39, 40].

Figure 1: One-loop contribution to neutrino masses, coming from the
interaction in our Superpotential defined at Eq.(16). This figure was taken
from [24].

We can explain the mixing angle as discussed [30].

###  4.4 Higgs masses

The Higgs potential of our model has the following form

$V^{H}_{MSSMRPV}=V_{F}+V_{D}+V_{soft}.$ (68)

The first term at Eq.(68) is

$V_{F}=\sum_{l}F_{l}^{\dagger}F_{l},$ (69)

where $l=H_{1,2},\tilde{L}_{3L}$ and $\tilde{l}^{c}_{3R}$; the $F$ terms are

$\displaystyle F^{\dagger}_{H_{1}}$ $\displaystyle=$
$\displaystyle-\frac{\mu}{2}H_{2}-\frac{\mu_{3}}{2}\tilde{L}_{3L},$
$\displaystyle F^{\dagger}_{H_{2}}$ $\displaystyle=$
$\displaystyle-\frac{\mu}{2}H_{1}-\frac{f^{\tau}}{3}\tilde{L}_{3L}\tilde{l}^{c}_{3R},$
$\displaystyle F^{\dagger}_{L_{3L}}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{3}}{2}H_{1}-\frac{f^{\tau}}{3}\tilde{L}_{3L}H_{2},$
$\displaystyle F^{\dagger}_{l_{3R}}$ $\displaystyle=$
$\displaystyle-f^{\tau}\left(H_{1}\tilde{L}_{3}\right).$ (70)

where we have defined $f^{\tau}$ after Eq.(52) and we get

$\displaystyle V_{F}$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}}{4}\left(|H_{1}|^{2}+|H_{2}|^{2}\right)+\frac{\mu^{2}_{3}}{4}|\left(\tilde{L}_{3L}|^{2}+|H_{2}|^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{(f^{\tau})^{2}}{9}\left[\left(|\tilde{L}_{3L}|^{2}+|H_{1}|^{2}\right)|\tilde{l}^{c}_{3R}|^{2}+|H^{\dagger}_{1}\tilde{L}_{3L}|^{2}-|H_{1}|^{2}|\tilde{L}_{3L}|^{2}\right]$
$\displaystyle+$
$\displaystyle\frac{\mu_{H}\mu_{3}}{4}\left(H^{\dagger}_{2}\tilde{L}_{3L}+\tilde{L}^{\dagger}_{3L}H_{2}\right)+\frac{\mu_{H}f^{\tau}}{6}\left(H_{1}\tilde{L}^{\dagger}_{3L}\tilde{l}^{c*}_{3R}+H^{\dagger}_{1}\tilde{L}_{3L}\tilde{l}^{c}_{3R}\right)$
$\displaystyle+$
$\displaystyle\frac{\mu_{3}f^{\tau}}{6}\left(H^{\dagger}_{2}H_{1}\tilde{l}^{c*}_{3R}+H^{\dagger}_{1}H_{2}\tilde{l}^{c}_{3R}\right),$
(71)

where $|H_{1}|^{2}\equiv H_{1}^{\dagger}H_{1}$ as usual.

The soft term that contribute to the scalar potential is given by

$\displaystyle V_{soft}$ $\displaystyle=$ $\displaystyle-{\cal L}_{SMT}-{\cal
L}_{Int}.$ (72)

Therefore

$\displaystyle V_{soft}$ $\displaystyle=$ $\displaystyle
M^{2}_{H_{1}}|H_{1}|^{2}+M^{2}_{H_{2}}|H_{2}|^{2}+M^{2}_{L}|\tilde{L}_{3L}|^{2}+M^{2}_{l}|\tilde{l}_{3R}|^{2}+B\mu\left(H_{1}H_{2}\right)+B_{3}\mu_{3}\left(H_{2}\tilde{L}_{3L}\right)$
(73) $\displaystyle+$ $\displaystyle
A^{E}_{33}\left(H_{1}\tilde{L}_{3L}\right)\tilde{l}^{c}_{3R}+hc.$

The third term at Eq.(68) is

$V_{D}=\frac{1}{2}\left[D^{i}D^{i}+\left(D^{\prime}\right)^{2}\right],$ (74)

where $i=1,2,3$. There is one $D$-term came from ${\cal L}_{Scalar}$ and from
superpotential for each of the four gauge groups

$\displaystyle SU(2)_{L}:\,\ D^{i}$ $\displaystyle=$
$\displaystyle-\frac{g}{2}\left[H_{1}^{\dagger}\sigma^{i}H_{1}+H_{2}^{\dagger}\sigma^{i}H_{2}+\tilde{L}_{3L}^{\dagger}\sigma^{i}\tilde{L}_{3L}\right],$
$\displaystyle U(1)_{Y}:\,\ D^{\prime}=$ $\displaystyle-$
$\displaystyle\frac{g^{\prime}}{2}\left[|H_{1}|^{2}-|H_{2}|^{2}-|\tilde{L}_{3L}|^{2}+2|\tilde{l}^{c}_{3R}|^{2}\right].$
(75)

Using the following relation

$\sigma^{i}_{ab}\sigma^{i}_{cd}=2\delta_{ad}\delta_{bc}-\delta_{ab}\delta_{cd},$
(76)

we finally get

$\displaystyle V_{D}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{8}\left[(|H_{1}|^{2}-|H_{2}|^{2})^{2}+4(|H^{\dagger}_{2}H_{1}|^{2}+|H^{\dagger}_{1}\tilde{L}_{3L}|^{2}+|H^{\dagger}_{2}\tilde{L}_{3L}|^{2})-2(|H_{1}|^{2}+|H_{2}|^{2})|\tilde{L}_{3L}|^{2}+(|\tilde{L}_{3L}|^{2})^{2}\right]$
(77) $\displaystyle+$
$\displaystyle\frac{(g^{\prime})^{2}}{8}\left[|H_{1}|^{2}-|H_{2}|^{2}-|\tilde{L}_{3L}|^{2}+2|\tilde{l}^{c}_{3R}|^{2}\right].$

Using Eqs.(71,73,75) at Eq.(68) we get

$\displaystyle V^{H}_{MSSMRPV}$ $\displaystyle=$ $\displaystyle
m^{2}_{1h}|H_{1}|^{2}+\left(m^{2}_{2h}+\mu^{2}_{3}\right)|H_{2}|^{2}+\left(M^{2}_{L}+\mu^{2}_{3}\right)|\tilde{L}_{3L}|^{2}+M^{2}_{l}|\tilde{l}^{c}_{3R}|^{2}+\left[B\mu\left(H_{1}H_{2}\right)\right.$
$\displaystyle+$
$\displaystyle\left.B_{3}\mu_{3}\left(H_{2}\tilde{L}_{3L}\right)-A^{E}_{33}\left(H_{1}\tilde{L}_{3L}\right)\tilde{l}^{c}_{3R}+hc\right]$
$\displaystyle+$
$\displaystyle\frac{(f^{\tau})^{2}}{9}\left[\left(|H_{1}|^{2}+|\tilde{L}_{3L}|^{2}\right)|\tilde{l}^{c}_{3R}|^{2}-|H_{1}|^{2}|\tilde{L}_{3L}|^{2}+|H^{\dagger}_{1}\tilde{L}_{3L}|^{2}\right]$
$\displaystyle+$
$\displaystyle\frac{\mu_{H}\mu_{3}}{4}\left(H^{\dagger}_{1}\tilde{L}_{3L}+\tilde{L}^{\dagger}_{3L}H_{1}\right)+\frac{\mu_{H}f^{\tau}}{6}\left[(H_{2}\tilde{L}^{\dagger}_{3L})\tilde{l}^{c*}_{3R}+(H^{\dagger}_{2}\tilde{L}_{3L})\tilde{l}^{c}_{3R}\right]$
$\displaystyle+$
$\displaystyle\frac{\mu_{3}f^{\tau}}{6}\left[(H^{\dagger}_{2}H_{1})\tilde{l}^{c*}_{3R}+(H^{\dagger}_{1}H_{2})\tilde{l}^{c}_{3R}\right]+\frac{g^{2}}{8}\left[(|H_{1}|^{2}-|H_{2}|^{2})^{2}+4(|H^{\dagger}_{2}H_{1}|^{2}+|H^{\dagger}_{1}\tilde{L}_{3L}|^{2}\right.$
$\displaystyle+$
$\displaystyle\left.|H^{\dagger}_{2}\tilde{L}_{3L}|^{2})-2(|H_{1}|^{2}+|H_{2}|^{2})|\tilde{L}_{3L}|^{2}+(|\tilde{L}_{3L}|^{2})^{2}\right]$
$\displaystyle+$
$\displaystyle\frac{(g^{\prime})^{2}}{8}\left[|H_{1}|^{2}-|H_{2}|^{2}-|\tilde{L}_{3L}|^{2}+2|\tilde{l}^{c}_{3R}|^{2}\right],$

where we have defined the following new parameters

$\displaystyle m^{2}_{1h}$ $\displaystyle=$ $\displaystyle
M^{2}_{H_{1}}+\frac{\mu^{2}}{4},\,\
m^{2}_{2h}=M^{2}_{H_{2}}+\frac{\mu^{2}}{4},$

From Eq.(LABEL:totalpotential) it is easy to reproduce the minimum of our
potential scalar and it is given by [18]

$\displaystyle V^{min}_{MSSMRPV}$ $\displaystyle=$
$\displaystyle\frac{(g^{2}+(g^{\prime})^{2})}{32}\left(v^{2}_{1}-v^{2}_{2}+(v^{L}_{3})^{2}\right)^{2}+\frac{m^{2}_{1h}}{2}v^{2}_{1}+\frac{m^{2}_{2h}}{2}v^{2}_{2}-B\mu
v_{1}v_{2}+B_{3}\mu_{3}v_{2}v^{L}_{3}+\frac{M^{2}_{L}}{2}(v^{L}_{3})^{2}$ (80)
$\displaystyle+$
$\displaystyle\frac{\mu^{2}_{3}}{2}(v^{2}_{2}+(v^{L}_{3})^{2})-\mu\mu_{3}v_{1}v^{L}_{3},$

and the constraint equataions are

$\displaystyle m^{2}_{1h}v_{1}-B\mu
v_{2}-\mu\mu_{3}v^{L}_{3}+\frac{(g^{2}+(g^{\prime})^{2})}{8}v_{1}\left(v^{2}_{1}-v^{2}_{2}+(v^{L}_{3})^{2}\right)$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left(m^{2}_{2h}+\mu^{2}_{3}\right)v_{2}-B\mu
v_{1}-B_{3}\mu_{3}v^{L}_{3}+\frac{(g^{2}+(g^{\prime})^{2})}{8}v_{2}\left(v^{2}_{1}-v^{2}_{2}+(v^{L}_{3})^{2}\right)$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left(M^{2}_{L}+\mu^{2}_{3}\right)v^{L}_{3}-\mu\mu_{3}v_{1}-B_{3}\mu_{3}v_{2}+\frac{(g^{2}+(g^{\prime})^{2})}{8}v^{L}_{3}\left(v^{2}_{1}-v^{2}_{2}+(v^{L}_{3})^{2}\right)$
$\displaystyle=$ $\displaystyle 0,$ (81)

those equations are in agremment as presented at [18].

In the bases

$\displaystyle\left(\begin{array}[]{cccc}H^{+}_{1}&H^{+}_{2}&\tilde{l}^{c}_{3R}\end{array}\right),$
(83)
$\displaystyle\left(\begin{array}[]{cccc}H^{-}_{1}&H^{-}_{2}&\tilde{l}_{3L}\end{array}\right),$
(85)

the mass matrix is given by

$\displaystyle M^{2}_{H^{\pm}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}M^{2}_{hh}&M^{2}_{\tilde{l}h}\\\
M^{2}_{\tilde{l}h}&M^{2}_{\tilde{l}\tilde{l}}\end{array}\right),$ (88)

where

$\displaystyle M^{2}_{hh}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}A&B\mu+\frac{g^{2}}{4}v_{1}v_{2}\\\
\frac{g^{2}}{4}v_{1}v_{2}&C\end{array}\right),$ (91) $\displaystyle
M^{2}_{\tilde{l}h}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}D&E^{*}\\\ E&F\end{array}\right),\,\
M^{2}_{\tilde{l}\tilde{l}}=\left(\begin{array}[]{cc}G&H\\\
I&J\end{array}\right),$ (95)

with

$\displaystyle A$ $\displaystyle=$ $\displaystyle
B\mu\frac{v_{2}}{v_{1}}+\frac{g^{2}}{4}(v^{2}_{2}-(v^{L}_{3})^{2})+\mu\mu_{3}\frac{v^{L}_{3}}{v_{1}}+\frac{|f^{\tau}|^{2}}{2}(v^{L}_{3})^{2},$
$\displaystyle C$ $\displaystyle=$ $\displaystyle
B\mu\frac{v_{1}}{v_{2}}+\frac{g^{2}}{4}(v^{2}_{1}+(v^{L}_{3})^{2})-\mu\mu_{3}\frac{v^{L}_{3}}{v_{2}},$
$\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{|f^{\tau}|^{2}}{2}v^{2}_{1}-\frac{g^{2}}{4}(v^{2}_{1}-v^{2}_{2})+\mu\mu_{3}\frac{v_{1}}{v^{L}_{3}}-B_{3}\mu_{3}\frac{v_{2}}{v^{L}_{3}}+M^{2}_{L},$
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{f^{\tau}}{\sqrt{2}}\mu_{3}v_{2}\left(A^{E}_{33}+\mu^{*}\tan\beta\right),$
$\displaystyle F$ $\displaystyle=$ $\displaystyle
M^{2}_{l}+\frac{|f^{\tau}|^{2}}{2}(v^{2}_{1}+(v^{L}_{3})^{2})-\frac{g^{2}}{4}(v^{2}_{1}-v^{2}_{2}+(v^{L}_{3})^{2}),$
$\displaystyle G$ $\displaystyle=$
$\displaystyle-\mu\mu_{3}-\frac{|f^{\tau}|^{2}}{2}v_{1}v^{L}_{3}+\frac{g^{2}}{4}v_{1}v^{L}_{3},$
$\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\left(f^{\tau}\mu
v_{2}-f^{\tau*}A^{E*}_{33}v^{L}_{3}\right),$ $\displaystyle I$
$\displaystyle=$ $\displaystyle-B_{3}\mu_{3}+\frac{g^{2}}{4}v_{2}v^{L}_{3},$
$\displaystyle J$ $\displaystyle=$
$\displaystyle-\frac{f^{\tau*}}{\sqrt{2}}\left(\mu
v^{L}_{3}+\mu_{3}v_{1}\right).$ (96)

those equations are in agremment as presented at [18].

We can show the following analytical results

$\displaystyle det\left(M^{2}_{H^{\pm}}\right)$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle Tr\left(M^{2}_{H^{\pm}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{v_{1}v_{2}v^{L}_{3}}\left\\{v_{1}v_{2}v^{L}_{3}\left[4M^{2}_{l}+2g^{2}v^{2}_{2}-(g^{\prime})^{2}\left(v^{2}_{1}-v^{2}_{2}+(v^{L}_{3})^{2}\right)\right]+4B\mu
v^{L}_{3}\left(v^{2}_{1}+v^{2}_{2}\right)\right.$ (97) $\displaystyle-$
$\displaystyle\left.4\mu_{3}\left[B_{3}v_{1}\left(v^{2}_{2}+(v^{L}_{3})^{2}\right)+\mu
v_{2}\left(v^{2}_{1}+(v^{L}_{3})^{2}\right)\right]\right\\}.$

The pseudoscalar squared mass matrix is given

$\displaystyle
M^{2}_{A}=\left(\begin{array}[]{ccc}B\mu\frac{v_{2}}{v_{1}}+\mu\mu_{3}\frac{v^{L}_{3}}{v_{1}}&B\mu&-\mu\mu_{3}\\\
B\mu&B\mu\frac{v_{1}}{v_{2}}-B_{3}\mu_{3}\frac{v^{L}_{3}}{v_{2}}&-B_{3}\mu_{3}\\\
-\mu\mu_{3}&-B_{3}\mu_{3}&\mu\mu_{3}\frac{v_{1}}{v^{L}_{3}}-B_{3}\mu_{3}\frac{v_{2}}{v^{L}_{3}}\end{array}\right),$
(101)

and we get

$\displaystyle det\left(M^{2}_{A}\right)$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle Tr\left(M^{2}_{A}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{v_{1}v_{2}v^{L}_{3}}\left\\{4B\mu
v^{L}_{3}\left(v^{2}_{1}+v^{2}_{2}\right)-\mu_{3}\left[B_{3}v_{1}\left(v^{2}_{2}+(v^{L}_{3})^{2}\right)+\mu
v_{2}\left(v^{2}_{1}+(v^{L}_{3})^{2}\right)\right]\right\\}.$ (102)

those equations are in agremment as presented at [18].

We can also get the mass squared mass matrix for the scalar sector but we will
not show it here.

##  5 Results with all $\tilde{L}_{i}\neq 0$.

###  5.1 Bosons Masses

In this case, we continue to using the Eq.(103) plus the following terms

$\displaystyle\left({\cal D}_{m}\tilde{L}_{1}\right)^{\dagger}\left({\cal
D}_{m}\tilde{L}_{1}\right)+\left({\cal
D}_{m}\tilde{L}_{2}\right)^{\dagger}\left({\cal D}_{m}\tilde{L}_{2}\right),$
(103)

same the same procedure presented at Sec.(4.1) we get the following expression
to the charged gauge bosons masses

$\displaystyle
M^{2}_{W}=\frac{g^{2}}{4}(v_{1}^{2}+v_{2}^{2}+(v^{L}_{1})^{2}+(v^{L}_{2})^{2}+(v^{L}_{3})^{2}).$
(104)

then we can rewrite

$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle
v\sin\theta_{1}\sin\theta_{2}\sin\theta_{3}\sin\beta,$ $\displaystyle v_{2}$
$\displaystyle=$ $\displaystyle
v\sin\theta_{1}\sin\theta_{2}\sin\theta_{3}\cos\beta,$ $\displaystyle
v^{L}_{3}$ $\displaystyle=$ $\displaystyle
v\sin\theta_{1}\sin\theta_{2}\cos\theta_{3},$ $\displaystyle v^{L}_{2}$
$\displaystyle=$ $\displaystyle v\sin\theta_{1}\cos\theta_{2},$ $\displaystyle
v^{L}_{1}$ $\displaystyle=$ $\displaystyle v\cos\theta_{1},$ (105)

and the neutral massive gauge boson ($Z^{0}$) get the following mass

$\displaystyle M^{2}_{Z}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4\cos^{2}\theta_{W}}(v_{1}^{2}+v_{2}^{2}+(v^{L}_{1})^{2}+(v^{L}_{2})^{2}+(v^{L}_{3})^{2}).$
(106)

###  5.2 Charginos Masses

In this case the $X$ non diagonal mass matrix for charginos, defined at
Eq.(56), is given by

$X=\left(\begin{array}[]{cc}{\cal M}^{MSSM}_{\tilde{\chi}}&{\cal
M}_{\tilde{V}L}\\\ {\cal M}_{\tilde{H}L}&{\cal
M}^{Yukawa}_{lepton}\end{array}\right).$ (107)

where

${\cal
M}^{MSSM}_{\tilde{\chi}}=\left(\begin{array}[]{cc}M&\frac{gv_{1}}{\sqrt{2}}\\\
\frac{gv_{2}}{\sqrt{2}}&\mu_{H}\end{array}\right),\,\ {\cal
M}^{Yukawa}_{lepton}=\frac{v_{1}}{\sqrt{2}}\left(\begin{array}[]{ccc}f^{l}_{11}&f^{l}_{12}&f^{l}_{13}\\\
f^{l}_{21}&f^{l}_{22}&f^{l}_{23}\\\
f^{l}_{31}&f^{l}_{32}&f^{l}_{33}\end{array}\right).$ (108)

The second matrix at Eq.(107) is

${\cal M}_{\tilde{V}L}=\left(\begin{array}[]{ccc}0&0&0\\\
A&B&C\end{array}\right),$ (109)

where we have defined

$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\left(f^{l}_{11}v^{L}_{1}+f^{l}_{21}v^{L}_{2}+f^{l}_{31}v^{L}_{3}\right),$
$\displaystyle B$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\left(f^{l}_{12}v^{L}_{1}+f^{l}_{22}v^{L}_{2}+f^{l}_{32}v^{L}_{3}\right),$
$\displaystyle C$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\left(f^{l}_{13}v^{L}_{1}+f^{l}_{23}v^{L}_{2}+f^{l}_{33}v^{L}_{3}\right),$
(110)

while the last matrix at Eq.(107) has the following expression

${\cal
M}_{\tilde{H}L}=\left(\begin{array}[]{cc}\frac{gv^{L}_{1}}{\sqrt{2}}&-\mu_{1}\\\
\frac{gv^{L}_{2}}{\sqrt{2}}&-\mu_{2}\\\
\frac{gv^{L}_{3}}{\sqrt{2}}&-\mu_{3}\end{array}\right),$ (111)

It is easy to realize that, from Eqs.(109,111), chargino sector decouple from
the usual lepton sector in the limit

$\displaystyle\mu_{1}=\mu_{2}=\mu_{3}=0,\,\ v^{L}_{1}=v^{L}_{2}=v^{L}_{3}=0.$
(112)

Using the parameters given before Eq.(LABEL:valorescharginosantesdadiag), plus
the following parameters $f^{l}_{11}=6.3\cdot 10^{-6}$, $f^{l}_{22}=1.21\cdot
10^{-3}$, $f^{l}_{33}=2.3\cdot 10^{-2}$,
$f^{l}_{12}=f^{l}_{13}=f^{l}_{23}=f^{l}_{21}=f^{l}_{31}=f^{l}_{32}=10^{-9}$ we
obtain from Eq.(107) the masses $0.0005,0.105,1.777$ (in GeV) for the usual
leptons, and 105.36 and 294.65 GeV for the charginos.

###  5.3 Neutralinos Masses

Using Eq.(65) we can write in this case the following mass matrix for
neutralinos at non-diagnal expression

$Y^{neutralino}_{MSSMRPV}=\left(\begin{array}[]{cc}{\cal
M}_{\tilde{\chi}^{0}}&m^{T}\\\ m&0\end{array}\right)\,\ .$ (113)

where

${\cal
M}_{\tilde{\chi}^{0}}=\left(\begin{array}[]{cccc}M_{1}&0&-\frac{g^{\prime}v_{2}}{2}&-\frac{g^{\prime}v_{1}}{2}\\\
0&M_{2}&\frac{gv_{2}}{2}&-\frac{gv_{1}}{2}\\\
-\frac{g^{\prime}v_{2}}{2}&\frac{gv_{2}}{2}&0&-\mu_{H}\\\
\frac{g^{\prime}v_{1}}{2}&-\frac{gv_{1}}{1}&-\mu_{H}&0\end{array}\right)\,\
m=\left(\begin{array}[]{cccc}-\frac{g^{\prime}v^{L}_{1}}{2}&\frac{gv^{L}_{1}}{2}&0&\mu_{1}\\\
-\frac{g^{\prime}v^{L}_{2}}{2}&\frac{gv^{L}_{2}}{2}&0&\mu_{2}\\\
-\frac{g^{\prime}v^{L}_{3}}{2}&\frac{gv^{L}_{3}}{2}&0&\mu_{3}\end{array}\right)\,\
.$ (114)

Using the same parameters are given after Eq.(66), but only changing the
values of $\mu_{3}$ for $\mu_{3}=11.4$,, we obtain a massive neutrino and its
mass is $m_{\nu_{3}}=0.051$ eV, and four heavy neutralinos with masses
$259.05,-208.42,-101.23,100.65$ GeV.

###  5.4 Higgs masses

The $F$ terms are

$\displaystyle F^{\dagger}_{H_{1}}$ $\displaystyle=$
$\displaystyle-\frac{\mu}{2}H_{2}-\sum_{i=1}^{3}\frac{\mu_{i}}{2}\tilde{L}_{iL},$
$\displaystyle F^{\dagger}_{H_{2}}$ $\displaystyle=$
$\displaystyle-\frac{\mu}{2}H_{1}-\sum_{i,j=1}^{3}\frac{f^{l}_{ij}}{3}\tilde{L}_{iL}\tilde{l}^{c}_{jR},$
$\displaystyle F^{\dagger}_{L_{iL}}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{i}}{2}H_{1}-\sum_{j=1}^{3}\frac{f^{l}_{ij}}{3}\tilde{l}^{c}_{jR}H_{2},$
$\displaystyle F^{\dagger}_{l_{iR}}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{3}\left(f^{l}_{ij}\right)^{2}\left(H_{1}\tilde{L}_{jL}\right).$
(115)

The $D$-term are

$\displaystyle SU(2)_{L}:\,\ D^{i}$ $\displaystyle=$
$\displaystyle-\frac{g}{2}\left[H_{1}^{\dagger}\sigma^{i}H_{1}+H_{2}^{\dagger}\sigma^{i}H_{2}+\sum_{j=1}^{3}\tilde{L}_{jL}^{\dagger}\sigma^{i}\tilde{L}_{jL}\right],$
$\displaystyle U(1)_{Y}:\,\ D^{\prime}=$ $\displaystyle-$
$\displaystyle\frac{g^{\prime}}{2}\left[|H_{1}|^{2}-|H_{2}|^{2}-\sum_{j=1}^{3}|\tilde{L}_{jL}|^{2}+2\sum_{j=1}^{3}|\tilde{l}^{c}_{jR}|^{2}\right].$
(116)

The soft term that contribute to the scalar potential is given by

$\displaystyle V_{soft}$ $\displaystyle=$ $\displaystyle
M^{2}_{H_{1}}|H_{1}|^{2}+M^{2}_{H_{2}}|H_{2}|^{2}+\sum_{i,j=1}^{3}\tilde{L}_{iL}\left(M^{2}_{L}\right)_{ij}\tilde{L}_{jL}+\sum_{i,j=1}^{3}\tilde{l}^{c}_{iR}\left(M^{2}_{l}\right)_{ij}\tilde{l}^{c}_{jR}+B\mu\left(H_{1}H_{2}\right)$
(117) $\displaystyle+$
$\displaystyle\sum_{i=1}^{3}B_{i}\mu_{i}\left(H_{2}\tilde{L}_{3L}\right)+\sum_{i,j=1}^{3}A^{E}_{ij}\left(H_{1}\tilde{L}_{iL}\right)\tilde{l}^{c}_{jR}+hc.$

Using Eqs.(115,115,117), we can reproduce at tree-level the scalar mass
matrices presented at [28, 30].

##  6 Conclusions

In this article we have presented the MSSM with $R$-Parity violation. We hope
this review can be useful to all the people wants to learn about
Supersymmetry.

Acknowledgments

The author would like to thanks to Instituto de Física Teórica (IFT-Unesp) for
their nice hospitality during the period I developed this review about SUSY.

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