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\newc{\cL}{{\cal L}}
\newc{\ra}{\rightarrow}
%\newc{\lam}{\lambda}
\newc{\eps}{\epsilon}
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\newc{\wino}{\widetilde{\cal W}}
\newc{\gluino}{\widetilde{\cal G}}
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\newc{\eg}{{\it e.g.}\ }
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\newc{\Dt}{\frac{d}{dt}}
\newc{\rpv}{$\not\!\!R_p$}
\newc{\bpv}{$\not\!\!B_p$}
\newc{\mpl}{$M_{Pl}$\ }
\newc{\mx}{$M_X$\ }
%\newc{\mgut}{{$M_{\rm GUT}$}
%\newcommand{\kapppa}{{\kappa^\prime}}
\newc{\tev}{\mbox{~TeV}}
\newc{\sect}[1]{\ref{sec:#1}}
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\newc{\eq}[1]{(\ref{eq:#1})}
\newc{\eqs}[2]{(\ref{eq:#1},\ref{eq:#2})}
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\newc{\ltau}{\lambda_\tau}
\newc{\lt}{\lambda_t}
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\newc{\lae}{{\Lam}_E}
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\newc{\lau}{{\Lam}_U}
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\newc{\lamhe}[1]{{\h}_{E^{#1}}}
\newc{\lamhed}[1]{{\h}_{E^{#1}}^\dagger}
\newc{\lamhd}[1]{{\h}_{D^{#1}}}
\newc{\lamhdd}[1]{{\h}_{D^{#1}}^\dagger}
\newc{\lamhu}[1]{{\h}_{U^{#1}}}
\newc{\lamhud}[1]{{\h}_{U^{#1}}^\dagger}
\newc{\lamd}[1]{{\Lam}_{D^{#1}}}
\newc{\lamu}[1]{{\Lam}_{U^{#1}}}
\newc{\lamet}[1]{{\Lam}_{E^{#1}}^T}
\newc{\lamdt}[1]{{\Lam}_{D^{#1}}^T}
\newc{\lamut}[1]{{\Lam}_{U^{#1}}^T}
\newc{\lames}[1]{{\Lam}_{E^{#1}}^*}
\newc{\lamds}[1]{{\Lam}_{D^{#1}}^*}
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\newc{\lamed}[1]{{\Lam}_{E^{#1}}^\dagg}
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\newc{\lsim}{\stackrel{<}{\sim}}
\newc{\gsim}{\stackrel{>}{\sim}}
\newc{\Tr}{{~\rm Tr}}
%soft masses
\newc{\me}{{(\bf m_{\tilde{E}}}^2)}
\newc{\mh}[1]{m_{H_{#1}}^2}
\newc{\ml}{{(\bf m_{\tilde{L}}}^2)}
\newc{\md}{{(\bf m_{\tilde{D}}}^2)}
\newc{\mup}{{(\bf m_{\tilde{U}}}^2)}
\newc{\mq}{{(\bf m_{\tilde{Q}}}^2)}
\newc{\mlh}[1]{({\bf m}_{ \tilde{L}_{#1} H_1}^2)}
\newc{\mhl}[1]{({\bf m}_{ H_1 \tilde{L}_{#1}}^2)}
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\newc{\del}{\partial}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{document}

%\preprint{hep-ph/yymmnnn}

\title{The  R-Parity Violating Minimal Supersymmetric Standard  Model}

\author{B.~Allanach}

\affiliation{CERN Theory Division }

\author{A.~Dedes and H.~K.~Dreiner}

\affiliation{Physikalisches Institut der Universitat Bonn, Nussallee 12,  D-53115 Bonn, Germany}



\begin{abstract}
We present a minimal version of supergravity with broken R-parity. 
We discuss the origins of lepton number violation in supersymmetry.
We have computed the full set of coupled one-loop renormalization group
equations for the gauge couplings, the superpotential parameters and
for all the soft supersymmetry breaking parameters. After making an
explicit choice of basis which we propose as a convention, we compute
the low-energy spectrum of the superparticles and the neutrinos as a
function of the small set of parameters at the unification scale. We
use the neutrino masses to set new bounds on the R-parity violating
couplings at the unification scale. We investigate the nature of the
lightest supersymmetric particle and find extensive regions in parameter
space, where it is {\it not} the neutralino. This leads to a novel set
of supersymmetric signatures, which we classify.
\end{abstract}


\maketitle

\section{Motivation : Why broken R-parity?}

{\it Theorem :} ``There are only two generalized parities 
which are discrete anomaly free with the minimal particle 
content of the Supersymmetric Stabdard Model. One is the 
Standard $Z_2$-parity (commonly called R-parity) and the other
is the $Z_3$ symmetry (commonly called Baryon-parity)
 which allows for lepton number violation and not
for  Baryon number violation (up to dimension-5 operators)~\cite{ir}''.
Since, is almost confirmed that neutrinos have masses this
is probably a good omen for lepton number violation. On the 
other hand, the proton is more stable than we expected within
the Minimal SUSY SU(5) Model to be. The later forced us
to forbid non-renormalizable B-number violating operators up
to dimension 5. The MSSM with baryon parity explaines both 
these observations for free. There is no reason to extend 
the particle content of the theory, as opposed to the R-parity
conserving MSSM with the see-saw
mechanism.  

\section{The minimal  R-parity violating Supersymmetric Standard model}

%The most general renormalizable and $SU(3)\times SU(2)\times U(1)$
%gauge invariant  
%superpotential 
We start  at some high scale \mx with the superpotential,
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%5
\barr 
W&=& \eps_{ab} \left[ (\ye)_{ij} L_i^a
H_1^b {\bar E}_j + (\yd)_{ij} Q_i^{ax} H_1^b {\bar D}_{jx} +
(\yu)_{ij} Q_i^{ax} H_2^b {\bar U}_{jx} \right]\nonr \\ &&+
\eps_{ab}\left[ \frac{1}{2} \lam_{ijk} L_i^a L_j^b{\bar E}_k +
\lamp_{ijk} L_i^a Q_j^{xb} {\bar D}_{kx} \right]+
\frac{1}{2}\eps_{xyz} \lampp_{ijk} {\bar U}_i^x{\bar
D}_j^y{\bar D}^z_k \nonr\\&& - \eps_{ab} \left[\mu H_1^a H_2^b
+\kap^i L_i^a H_2^b \right].  \label{superpot1} 
\earr 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
We denote an $SU(3)$ colour index of the fundamental representation by
$x,y,z=1,2,3$. The $SU(2)_L$ fundamental representation indices are
denoted by $a,b,c=1,2$ and the generation indices by $i,j,k=1,2,3$.
In the basis where the sneutrino vevs are zero (see below)
the $\kap_i$ term in the vector $\mu_\alpha$ should be small 
and of order of MeV in order to agree with the neutrino mass limits.
But what makes this parameter small? The natural solution
one can think is that this parameter is zero at some high
scale and is generated radiatively as we run down to low
energies. This is a solution we present here and seems to
work only in the minimal Supergravity scenario as we will see below.
In any case, the $\kap_i$ term in (\ref{superpot1}) can always be
rotated away in the Supersymmetric limit. One can shift the 
superfields by an orthogonal transformation\footnote{It is not hard
to generalize to a unitary transformation, Marc Thormeier, {\it
  unpublished}. In that case one can get rid of the phase of $\kap_i$
as well.}
%%%%%%%%%%%%%%%%%%%%5
\begin{widetext}
\barr
\cL_\alpha = {\cal O}_{\alpha\beta} \cL_\beta^\prime \Rightarrow
\left (\begin{array}{c} H_1 \\ L_1 \\ L_2 \\ L_3 \end{array}\right )=
\left ( \begin{array}{cccc} c_3 & -s_3 & 0 & 0 \\
                            c_2s_3 & c_2c_3 & -s_2 & 0 \\
                            c_1s_2s_3 &c_1s_2c_3 & c_1c_2 & -s_1 \\
                            s_1s_2s_3 & s_1s_2c_3 & s_1c_2 & c_1 
\end{array} \right ) \left ( \begin{array}{c} \cL_0^\prime \\
  \cL_1^\prime
 \\ \cL_2^\prime \\
\cL_3^\prime \end{array} \right ) \;, 
\earr
\end{widetext}
%%%%%%%%%%%%%
where 
$c_i=\cos\theta_i$ and $s_i=\theta_i$ are rotation angles.
They are given explicitly by 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
c_1&=&\frac{\kappa_2}{\sqrt{\kappa_2^2+\kappa_3^2}},\quad
s_1=\frac{\kappa_3}{\sqrt{\kappa_2^2+\kappa_3^2}}\nonumber\\
c_2&=&\frac{\kappa_1}{\sqrt{\vec{\kappa}^2}},\quad
\;\;\;\;\;\;\;\;s_2=\frac{\sqrt{\kappa_2^2+\kappa_3^2}}
{\sqrt{\vec{\kappa}^2}}\\
c_3&=&\frac{\mu}{\sqrt{\mu^2+\vec{\kappa}^2}},\quad
s_3=\frac{{\sqrt{\vec{\kappa}^2}}}
{\sqrt{\mu^2+\vec{\kappa}^2}} \nonumber
\label{angles}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where we have introduced the notation $\vec{\kappa}^2=\kappa_1^2+
\kappa_2^2+\kappa_3^2$.
After the above field redefinition the
only remaining superfield bi-linear term is
\begin{equation}
\mu''H_1'H_2
\end{equation}
with $\mu''=\sqrt{\mu^2+\vec{\kappa}^2}$ and $H_1'\equiv \cL_0'$.
It is well known~\cite{Nardi,add} that even if we set the billinear 
$\kappa$-term equal to zero this will be generated from loop 
corrections due to the appearence of the non-zero anomalous
dimensions $\gamma_{H_1}^{L_i}$. The same also will happens
for vanishing (at tree level) Yukawa couplings-this time due to
the non-vanishing anomalous dimension, $\gamma_{L_i}^{L_j}, i\ne j$
in addition to $\gamma_{H_1}^{L_i}$. Thus at low energies the most
general superpotential we can write down is (we drop the primes from
the lepton fields) 
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%5
\barr 
W&=& \eps_{ab} \left[ (\yu)_{ij} Q_i^a H_2^b \bar{U}_j+
\frac{1}{2} \lam_{\alpha\beta k}\cL_\alpha^a \cL_\beta^b \bar{E}_k
+\lam^\prime_{\alpha jk} \cL_\alpha^a Q_j^b \bar{D}_k -
\mu_\alpha \cL_\alpha^a H_2^b \right]+
\frac{1}{2}\epsilon_{xyz}\lam^{\prime\prime}_{ijk} \bar{U}_i^x
\bar{D}_j^y \bar{D}_k^z  \label{superpot2} 
\earr 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
where we have exploited the four vector
notation $\cL_\alpha=(H_1,L_i)$ and $\mu_\alpha=(\mu, \kappa_i)$.
It is also, $\lam_{0j k}\equiv (\ye)_{jk}$ and 
$\lam^\prime_{0j k}\equiv (\yd)_{jk}$. 


\section{Minimal Low energy R-parity violating Supergravity Model}

The soft supersymmetry breaking 
Lagrangian obatined at low energies (of the order of the Electroweak 
scale)
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\barr
-{\cal L}_{\rm SOFT} &=& (m_{\tilde{\cL}}^2)_{\alpha\beta}
{\tilde{\cL}}_\alpha^\dagger{\tilde{\cL}}_\beta + \mh{2} H_2^\dagger H_2  
+  \widetilde{Q}^\dagger \mq \widetilde{Q} 
+ \widetilde{\bar{E}} \me \widetilde{\bar{E}}^\dagger
+ \widetilde{\bar{D}} \md \widetilde{\bar{D}}^\dagger
+ \widetilde{\bar{U}} \mup \widetilde{\bar{U}}^\dagger \nonum \\[1mm]
&+ & \eps_{ab} \left[ (\hu)_{ij} Q_i^a H_2^b \bar{U}_j+
\frac{1}{2} h_{\alpha\beta k}\tilde{\cL}_\alpha^a \tilde{\cL}_\beta^b \bar{E}_k
+h^\prime_{\alpha jk} \tilde{\cL}_\alpha^a Q_j^b \bar{D}_k -
b_\alpha \tilde{\cL}_\alpha^a H_2^b~+~{\rm H.c} \right]+
\frac{1}{2}\epsilon_{xyz}h^{\prime\prime}_{ijk} \bar{U}_i^x
\bar{D}_j^y \bar{D}_k^z  ~+~{\rm H.c} \nonum \\[1mm]
&+& \biggl [ \frac{1}{2} M_1 \bino\bino + \frac{1}{2} M_2
\wino^{({\Gamma})}\wino^{({\Gamma})} + \frac{1}{2} M_3
\gluino^{(R)}\gluino^{(R)}~+~{\rm H.c} \biggr ] \;,
 \label{softhaber}
\earr
%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%
where $b_\alpha=(\widetilde{B},\widetilde{D}_i)$ is the bilinear
SUSY soft breaking term, and $(\hu)_{ij}, h_{\alpha\beta
  k},h^\prime_{\alpha jk},h^{\prime\prime}_{ijk}  $ the soft breaking
trilinear couplings. 

\section{Radiative Electroweak Symmetry Breaking}

The full scalar potential can be derived from
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\barr
{\cal V}_{\rm SCALAR}={\cal V}_{\rm SUSY}+{\cal V}_{\rm SOFT}\;,
\label{vscalar}
\earr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
with 
\barr
{\cal V}_{\rm SUSY}&=&{\cal V}_{\rm F}+{\cal V}_{\rm D}= \nonumber \\
& &\sum_\Phi |\frac{d W}{d\Phi}|^2+
\sum_l \frac{g_l^2}{2}\sum_A\biggl (\sum_{\alpha,\beta}\Phi^*_\alpha
T_{l,A}^{\alpha\beta}\Phi_{\beta}\biggr )^2 \;, \label{vsusy}
\earr
and ${\cal V}_{\rm SOFT}=-{\cal L}_{\rm SOFT}$. In Eq.(\ref{vsusy})
$\Phi$'s are the scalar fields, $g_l$ are the gauge couplings
with $g$ for the $U(1)_{EM}$, $g_2$ for the $SU(2)_L$ and $g_3$ for
the $SU(3)$, 
$\alpha,\beta ..$ and $A,B...$ representation and generator 
indices respectively. The expressions for ${\cal V}_{\rm F}$ and 
${\cal V}_{\rm D}$ can be found in \cite{Haber}.


%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{table*}[t]
\caption{
The particle content of the Minimal Supergravity R-Parity
violating Model in terms of Superfields and their decomposition
into components with their
$SU(3)_c\times SU(2)_L\times U(1)_Y$ quantum numbers. $x,R$ are SU(3)
representation and generator indices, $a,\Gamma$ are SU(2)
representation and generator indices. $\alpha=0,..,3$ is the family
index of the Lepton superfield, and $i=1,..3$ the usual family
index of quarks, leptons and their superpartners. The fermionic 
components of superfields are Weyl two component spinors.}
\begin{tabular}{|c|c|c|}
\hline
Chiral Superfields & $SU(3)_c\times SU(2)_L\times U(1)_Y$ & Components 
\\ \hline
$Q^{a,x}_i$ & $(\bf{3},{\bf{2}},\frac{1}{6})$ & 
$\biggl ($ \begin{tabular}{c}  $\tilde{u}_L$ \\ $\tilde{d}_L$
\end{tabular}
$\biggr )$
$\; ,\;$ $\biggl ($ \begin{tabular}{c}  $u_L$ \\ $d_L$
\end{tabular}$\biggr )$
\\[4mm] 
$\bar{D}^{x}_i$ &  $(\bf{\bar{3}},{\bf{1}},\frac{1}{3})$ &
$\tilde{d}_R^* \;,\; d_R$ \\[2mm] 
$\bar{U}^{x}_i$ &  $(\bf{\bar{3}},{\bf{1}},-\frac{2}{3})$ &
$\tilde{u}_R^* \;,\; u_R$ \\[2mm] 
$\cL_\alpha^{a}=\{H_1^a,L_i^a\}$ & $(1,{\bf{2}},-\frac{1}{2})$ & 
$\biggl ($ \begin{tabular}{c}  $\tilde{\nu}_\alpha$ \\ $\tilde{e}_{L\alpha}$
\end{tabular}
$\biggr )= \biggl \{ \biggl ($ \begin{tabular}{c} 
$h_1^0$ \\ $h_1^-$ \end{tabular} 
$\biggr )\;,\; \biggl ($
\begin{tabular}{c} $\tilde{\nu}_i$ \\ $\tilde{e}_{Li}$ \end{tabular}
$\biggr ) \biggr \}$
 $\;,\;$ $ \biggl ($ 
\begin{tabular}{c}  $\nu_\alpha$ \\ $e_{L\alpha}$
\end{tabular}$\biggr )=\biggl \{ \biggl ($ \begin{tabular}{c} 
$\tilde{h}_1^0$ \\ $\tilde{h}_1^-$ \end{tabular} 
$\biggr )\;,\; \biggl ($
\begin{tabular}{c} $\nu_i$ \\ $e_{Li}$ \end{tabular}
$\biggr ) \biggr \}$
\\[4mm] 
$\bar{E}_i$ &  $(\bf{{1}},{\bf{1}},1)$ &
$\tilde{e}_R^* \;,\; e_R$ \\[2mm]
$h_2^a$ & $(\bf{1},{\bf{2}},\frac{1}{2})$ & 
$\biggl ($ \begin{tabular}{c}  $h_2^+$ \\ $h_2^0$
\end{tabular}
$\biggr )$
$\; ,\;$ $\biggl ($ \begin{tabular}{c}  $\tilde{h}_2^+$ \\ 
$\tilde{h}_2^0$ \end{tabular} $\biggr )$
\\[2mm] \hline
Vector Superfields & $SU(3)_c\times SU(2)_L\times U(1)_Y$ & Components 
\\ \hline 
$V_1$ & $(\bf{1},{\bf{1}},0)$ & $\bino$ $\;,\,$ $B_\mu$ \\[3mm]
$V_2$ & $(\bf{1},{\bf{3}},0)$ & $\wino^{(\Gamma)}$ 
$\;,\,$ $W^{(\Gamma)}_\mu$ \\[3mm]
$V_3$ & $(\bf{8},{\bf{1}},0)$ & $\gluino^{(R)}$ $\;,\,$ 
$G^{(R)}_\mu$ \\ \hline
\end{tabular}
\label{table1}
\end{table*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%
The scalar potential of the neutral scalar fields is
%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%
\barr
{\cal V}_{\rm
  Neutral}&=& (m_{H_2}^2+|\mu_\alpha|^2 )|h_2^0|^2 +
\biggl [(m_{\tilde{\cL}}^2)_{\alpha\beta}+
\mu_\alpha^*\mu_\beta \biggr ] \tilde{\nu}_\alpha^*\tilde{\nu}_\beta
-(b_\alpha \tilde{\nu}_\alpha h_2^0+b_\alpha^* \tilde{\nu}_\alpha^*
h_2^{0*}) \nonum \\
&+&\frac{1}{8}(g^2+g_2^2) \biggl (|h_2^0|^2-|\tilde{\nu}_\alpha|^2\biggr )^2
+ {\cal V}_{\rm 1loop}\;.
\label{vneutral}
\earr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Expanding the scalar fields $h_2^0=x_2+iy_2$, $\tilde{\nu}_\alpha=
x_\alpha+i y_\alpha$ and 
assuming that only the neutral scalar fields acquire a vacuum 
expectation values\footnote{Throught this paper we assume that the
vacuum expectation values are real.}, 
$<x_2>=v_2$, $<\tilde{x}_\alpha>=v_\alpha$ and
$<y_2>=<y_\alpha>=0$
the following minimization conditions can be derived from
(\ref{vneutral})
%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%
\barr
\Re  \biggl [ (m_{\tilde{\cL}}^2)_{\alpha\beta}+
\mu_\alpha^*\mu_\beta \biggr ] v_\alpha-\Re(b_\beta) v_2
-\frac{1}{4} (g^2+g_2^2) \biggl (|v_2|^2-|v_\alpha|^2\biggr ) v_\beta
+\frac{1}{2} \frac{\partial {\cal V}_{\rm 1loop}}{\partial v_\beta}
&=& 0 \nonum \\
(m_{H_2}^2+|\mu_\alpha|^2 )v_2 -\Re(b_\beta) v_\beta 
+\frac{1}{4} (g^2+g_2^2) \biggl (|v_2|^2-|v_\alpha|^2\biggr ) v_2
+\frac{1}{2} \frac{\partial {\cal V}_{\rm 1loop}}{\partial v_2} &=& 0
\;, \label{mins}
\earr
%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%
--------------------------------------------------------

We want to solve the system of these five minimization
conditions. We start by defining 
\barr
\tan\beta \ = \ \frac{v_u}{v_d} \;,
\label{tanbeta}
\earr
and
\barr
v^2 = v_u^2+v_d^2+\sum_i v_i^2 = \frac{2 M_W^2}{g_2^2} \;.
\label{vevtotal}
\earr
%%%%%%%%%%%%
Then the vev's $v_u$ and $v_d$ (I have diverted from the notation
of $v_1$ (now $v_d$) and $v_2$ (noe $v_u$) for obvious reason)
can be written 
\barr
v_d^2 &=& \cos^2\beta \biggl (v^2-\sum_i v_i^2 \biggr ) \;,
\label{vevd} \\
v_u^2 &=& \sin^2\beta \biggl (v^2-\sum_i v_i^2 \biggr ) \;, \label{vevu} 
\earr
The big advantage of using the definition (\ref{tanbeta},\ref{vevtotal}) is
that $\tan\beta$ remains invariant in the R-parity conserved (RPC)
and violating (RPV) model. This helps with  the direct comparison 
of the two models.

The minimization conditions (\ref{mins}) can be written as
%%%%%%%%%%%%%%%%%%%%%%5
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%
\barr
(m_{H_d}^2 +\mu^2) v_d + \biggl [ \mlh{i}+\kap_i^*\mu \biggr ] v_i
-\widetilde{B} v_u + \frac{1}{2} M_Z^2 \cos2\beta v_d +\frac{1}{2}
(g^2+g_2^2) \sin^2\beta v_d \sum_i v_i^2 + \frac{1}{2} 
\frac{\partial {\cal V}_{\rm 1loop}}{\partial v_d} &=& 0 \;,
\label{minI} \\
\biggl [ \mhl{i}+\mu^* \kap_i \biggr ] v_d + \biggl [ \ml_{ji}+
\kap_j^*\kap_i \biggr ] v_j -\widetilde{D}_i v_u
+\frac{1}{2} M_Z^2 \cos2\beta v_i +\frac{1}{2} (g^2+g_2^2)
\sin^2\beta v_i \sum_j v_j^2 +\frac{1}{2}
\frac{\partial {\cal V}_{\rm 1loop}}{\partial v_i} &=& 0 \;,
\label{minII} \\
(m_{H_u}^2+\mu^2+ |\kap_i|^2 )v_u -\widetilde{B} v_d
-\widetilde{D}_i v_i -\frac{1}{2} M_Z^2 \cos2\beta v_u
-\frac{1}{2} (g^2+g_2^2) \sin^2\beta v_u \sum_i v_i^2
+\frac{1}{2}\frac{\partial {\cal V}_{\rm 1loop}}{\partial v_u} &=& 0 \;,
\label{minIII}
\earr
%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%
We first derive $\mu$ and $\widetilde{B}$ 
from (\ref{minI}) and (\ref{minIII}). The $\mu$-parameter is obtained after
solving the quadratic equation 
\barr
A\mu^2+B\mu+ \Gamma =0 \;,
\earr
with 
\barr
A & \equiv & \tan^2\beta -1 \;, \\
B & \equiv & -\kap_i^* \frac{v_i}{v_d} \;, \\
\Gamma  & \equiv & \biggl \{ \biggl [ \bar{m}_{H_u}^2+|\kap_i|^2
-(g^2+g_2^2) \sum_i v_i^2 -\widetilde{D}_i \frac{v_i}{v_u} \biggr ]
\tan^2\beta \nonum \\
&-& \biggl [ \bar{m}_{H_d}^2 + \mlh{i} \frac{v_i}{v_d} \biggr
] \biggr \} +\frac{1}{2} M_Z^2 \;,. \label{minIV}
\earr
%%%%%%%%%%%%%%%%5
$\widetilde{B}$ takes a simple form 
\barr
\widetilde{B} &=& \frac{\sin2\beta}{2} \biggl \{\biggl [ \bar{m}_{H_d}^2
+\bar{m}_{H_u}^2 + 2 |\mu|^2 +|\kap_i|^2 \biggr ]\nonum \\ 
&+& \biggl [ \mlh{i}+\kap_i^*\mu \biggr ]\frac{v_i}{v_d}
 -\widetilde{D}_i \frac{v_i}{v_u}
\biggr \} \;, \label{minV}
\earr
%%%%%%%%%%%%
where in both (\ref{minIV}) and (\ref{minV}) is 
\barr
\bar{m}_{H_{u,d}}^2 &\equiv & m_{H_{u,d}}^2
+\frac{1}{2} \frac{1}{v_{u,d}} 
\frac{\partial {\cal V}_{\rm 1loop}}{\partial v_{u,d}} \;.
\earr
%%%%%%%%%%%%%%%%%%
Now Eq.(\ref{minII}) can be cast in a form,
%%%%%%%%%%%%%%
\barr
(M_{\tilde{\nu}}^2)_{ij} v_j = -\biggl [ \mhl{i}+\mu^*\kap_i \biggr ]
v_d + \widetilde{D}_i v_u -\frac{1}{2}
\frac{\partial {\cal V}_{\rm 1loop}}{\partial v_i} \;, \label{minVI}
\earr
%%%%%%%%%%%%%
where
\barr
(M_{\tilde{\nu}}^2)_{ij} &=& \ml_{ji}+\kap_i\kap_j^* +\frac{1}{2}
M_Z^2 \cos2\beta \delta_{ij} \nonum \\ &+& \frac{1}{2} (g^2+g_2^2) \sin^2\beta
(v^2-v_u^2-v_d^2) \delta_{ij} \;.
\earr
%%%%%%%%%%%%%%%%%
Now the numerical procedure is the following :
i) We solve (\ref{minIV}) and (\ref{minV}) first in the RPC
limit $v_i=0,\kap=\widetilde{D}=0$ and we define initial values
for $\mu$ and $\widetilde{B}$. ii) We pass $\mu$ into the system
(\ref{minV}) and we solve with respect to $v_i$ by treating $v_u$
and $v_d$ as constants (then the system (\ref{minV}) is linear
and even analytically you can find solution). iii) We correct
the vevs $v_u$ and $v_d$ from Eqs.(\ref{vevd},\ref{vevu}). iv) We iterate
the procedure until convergence of $\mu,\widetilde{B}, v_u,v_d,v_i$
is reached. I tryed extreme values of lambdas and kapas and I saw
that the system is always convergent apart from cases it simply
does not have a solution. So we achieved what we wanted : 
We start by giving as input $\tan\beta$ as in the MSSM, we define
$\mu$ and $\widetilde{B}$ from the mins and we have the sneutrino vevs.
Now we do not have to rotate to any basis. Just plug in those values
to the basis independent mass matrices below and find the spectrum.




%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Particle and Superparticle Masses}

The mass matrices for particle and superparticles are given in
both basis independent and basis dependent Eq.(\ref{basis}) choice
below. 
Note that we strictly follow the conventions of 
Haber and Grossman~\cite{Haber} which in the R-parity conserved limit
coincide with those of Haber and Kane~\cite{Kane}.

\subsection{CP-Even Higgs-Sneutrino masses}

Sneutrinos mix with Higgs bosons. If CP is conserved we can separate
them in CP-even and CP-odd types. Following Grossman and
Haber~\cite{Haber} let us denote with $\tilde{\nu}_+$ ($\tilde{\nu}_-$)
the CP-even(CP-odd) sneutrinos. If R-parity is broken the mass
of $\tilde{\nu}_+$ is in general different than the mass of
$\tilde{\nu}_-$ i.e., there is a sneutrino anti-sneutrino mass
spliting. The CP-even Higgs-sneutrinos $h,H,\tilde{\nu}_{+}^i$ are
obtained in a basis independent form after the diagonalization of
a $5\times 5$ matrix 
%%%%%%%%%%%%%%
\barr
{\cal L}=-\frac{1}{2} (x_2,x_\gamma) {\cal M}^2_{\rm CP-even}\left (
\begin{array}{c} x_2 \\  x_\delta \end{array} \right )
\earr  
%%%%%%%%%%%%
where 
%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%55
\barr
{\cal M}^2_{\rm CP-even} = \left ( \begin{array}{cc} 
\frac{b_\alpha v_\alpha}{v_2}+\frac{1}{2}(g^2+g_2^2) v_2^2 &
- b_\delta -\frac{1}{2}(g^2+g_2^2) v_2 v_\delta \\[3mm]
- b_\gamma -\frac{1}{2}(g^2+g_2^2) v_2 v_\gamma  &
(m_{\tilde{\nu}}^2)_{\gamma\delta}+\frac{1}{2}(g^2+g_2^2) v_\gamma
v_\delta  \end{array} \right ) \;,
\label{cpeven}
\earr
%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%
with
\barr
(m_{\tilde{\nu}}^2)_{\alpha\beta}=[(m_{\tilde{\cL}}^2)_{\alpha\beta}+
\mu^*_\alpha\mu_\beta
]-\frac{1}{4}(g^2+g_2^2)(v_2^2-v_\gamma^2)\delta_{\alpha\beta} \;.
\earr
%%%%%%%%%%%%%%
where $v_\gamma^2\equiv \sum_\gamma v_\gamma^2$.
In the basis choice Eq.(\ref{basis}), the CP-even squared mass matrix 
is written 
%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%5
\barr
{\cal M}^2_{\rm CP-even} = \left ( \begin{array}{ccccc}
\widetilde{B}\cot\beta+M_Z^2\sin^2\beta &
-\widetilde{B}-\frac{1}{2} M_Z^2\sin2\beta & -\widetilde{D}_1 &
-\widetilde{D}_2 & -\widetilde{D}_3 \\
-\widetilde{B}-\frac{1}{2} M_Z^2\sin2\beta &
\widetilde{B}\tan\beta +M_Z^2 \cos^2\beta & \widetilde{D}_1
\tan\beta & \widetilde{D}_2\tan\beta & \widetilde{D}_3\tan\beta
\\
-\widetilde{D}_1 & \widetilde{D}_1\tan\beta & & & \\
-\widetilde{D}_2 & \widetilde{D}_2\tan\beta & & 
(m_{\tilde{\nu}}^2)_{ij} & \\
-\widetilde{D}_3 & \widetilde{D}_3\tan\beta & & & \end{array}
\right ) \;, \label{bcpeven}
\earr
%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%
In deriving (\ref{bcpeven}),
we have made use of the minimization conditions Eqs(\ref{bmins})
and 
%%%%%%%%%%%%%%%%%%%%%5
\barr
(m_{\tilde{\nu}}^2)_{ij}=[\ml_{ij}+
\kap_i\kap_j
]+\frac{1}{2}M_Z^2 \cos2\beta \, \delta_{ij} \;.
\earr
%%%%%%%%%%%%%






\subsection{CP-Odd Higgs-Antisneutrino masses}

The CP-odd Higgs-sneutrinos $A,\tilde{\nu}_{-}^i$ (and the massless
Goldstone
boson in unitary gauge) are
obtained in a basis independent form after the diagonalization of
a $5\times 5$ matrix 
%%%%%%%%%%%%%%
\barr
{\cal L}=-\frac{1}{2} (y_2,y_\gamma) {\cal M}^2_{\rm CP-odd}\left (
\begin{array}{c} y_2 \\  y_\delta \end{array} \right )
\earr  
%%%%%%%%%%%%
where 
%%%%%%%%%%%%
\barr
{\cal M}^2_{\rm CP-odd} = \left ( \begin{array}{cc} 
\frac{b_\alpha v_\alpha}{v_2} &
b_\delta \\[3mm]
b_\gamma  &
(m_{\tilde{\nu}}^2)_{\gamma\delta}  \end{array} \right ) \;.
\label{cpodd}
\earr
%%%%%%%%%%%%%%%%%%
In the basis choice Eq.(\ref{basis}), the CP-odd squared mass matrix 
is written 
%%%%%%%%%%%%
\barr
{\cal M}^2_{\rm CP-even} = \left ( \begin{array}{ccccc}
\widetilde{B}\cot\beta &
\widetilde{B} & \widetilde{D}_1 &
\widetilde{D}_2 & \widetilde{D}_3 \\
\widetilde{B} &
\widetilde{B}\tan\beta & \widetilde{D}_1
\tan\beta & \widetilde{D}_2\tan\beta & \widetilde{D}_3\tan\beta
\\
\widetilde{D}_1 & \widetilde{D}_1\tan\beta & & & \\
\widetilde{D}_2 & \widetilde{D}_2\tan\beta & & 
(m_{\tilde{\nu}}^2)_{ij} & \\
\widetilde{D}_3 & \widetilde{D}_3\tan\beta & & & \end{array}
\right ) \;, \label{bcpodd}
\earr
%%%%%%%%%%%%%%
where again we made use of Eqs.(\ref{bmins}). For one generation 
we obtain two nonzero eigenvalues identified as the CP-odd Higgs
and the sneutrino,
%%%%%%%%%%%%%%%
\barr
m_{\tilde{\nu}_-}^2 &=& \frac{1}{2}\biggl [ m_{\tilde{\nu}}^2+\frac{2
  \tilde{B}}{\sin2\beta}+\sqrt{\biggl (m_{\tilde{\nu}}^2-\frac{2
    B}{\sin2\beta}\biggr )^2+4 \tilde{D}_1^2 \biggl
  (1+\tan^2\beta \biggr )} \biggr ] \nonum \\
m_{A}^2 &=& \frac{1}{2}\biggl [ m_{\tilde{\nu}}^2+\frac{2
  \tilde{B}}{\sin2\beta}-\sqrt{\biggl (m_{\tilde{\nu}}^2-\frac{2
    B}{\sin2\beta}\biggr )^2+4 \tilde{D}_1^2 \biggl
  (1+\tan^2\beta \biggr )} \biggr ]\;.
\label{cpodd1gen}
\earr
%%%%%%%%%%%%%%%%%%
Notice the $\tan\beta$ enhancement(reduction) of the sneutrino(higgs)
is due exclusively to a R-parity violating contribution. 
In order not to have negative CP-odd Higgs mass it must be
$m_{\tilde{\nu}}^2 > \frac{\tilde{D}_1^2 \tan\beta}{\tilde{B}}$ which
says that $\tilde{D}_1 << \tilde{B}$.
For
$\tilde{D}_1\to 0$ we have $m_{\tilde{\nu}_-}=m_{\tilde{\nu}}$ and 
$m_{A}^2=\frac{2
  \tilde{B}}{\sin2\beta}$ as it should be.


\subsection{Charged Higgs Bosons-Sleptons}

Charged Higgs Bosons mix with  sleptons. In a basis
independent way we find the $8\times 8$ mass matrix which upon
diagonilization gives the masse eigenstates : 
$G^\pm,H^\pm, \tilde{e}_{i=1,...6}$. 
%%%%%%%%%%%%%%
\barr
{\cal L}=- (h^-_2,\tilde{e}_{L_\gamma},\tilde{e}_{R_k})
 {\cal M}^2_{\rm Charged}\left (
\begin{array}{c} h_2^+ \\  \tilde{e}_{L_\delta}^{*} \\
\tilde{e}_{R_l}^* \end{array} \right )
\earr  
%%%%%%%%%%%%
where 
%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%
\barr
{\cal M}^2_{\rm Charged} = \left ( \begin{array}{ccc} 
(m^2)_{11} + D & b_\delta^*+ D_\delta &
\lam_{\beta\alpha l}\mu_\alpha^* v_\beta \\[3mm]
b_\gamma+D_\gamma^*  & (m^2)_{\delta\gamma}
 +\lam_{\alpha\gamma l}\lam_{\beta\delta l} 
v_\alpha v_\beta + D_{\gamma\delta} & 
h_{\alpha\gamma l} v_\alpha-\lam_{\alpha\gamma l}\mu_\alpha^* v_2
\\[3mm]
\lam^*_{\beta\alpha k} \mu_\alpha v_\beta &
h_{\alpha\delta k}^* v_\alpha -\lam_{\alpha\delta k}\mu_\alpha v_2 &
\me_{lk}+\lam_{\alpha\beta k}\lam_{\alpha\gamma l }v_\beta v_\gamma
+D_{lk}
\end{array} \right ) \;.
\label{Charged}
\earr
%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%
with 
%%%%%%%%%%%%%%%%
\barr
(m^2)_{11} &=& m_{H_2}^2+|\mu_\alpha|^2 \;, \nonum \\
D &=& \frac{1}{4}(g_2^2+g^2)(v_2^2-|v_\alpha|^2)+\frac{1}{2}g_2^2
|v_\alpha|^2 \;, \nonum \\
D_\delta &=& \frac{1}{2}g_2^2 v_2 v_\delta \;, \nonum \\
(m^2)_{\gamma\delta} &=& (m_{\tilde{\cL}}^2)_{\delta\gamma}+
\mu_\gamma \mu_\delta^* \;, \nonum \\
D_{\gamma\delta} &=& \frac{1}{4}(g_2^2-g^2)(v_2^2-v_\alpha^2)
\delta_{\delta\gamma}+\frac{1}{2}g_2^2 v_\gamma v_\delta \;, \nonum \\
(D)_{lk} &=& \frac{1}{2} g^2 (v_2^2-v_\alpha^2) \delta_{lk} \;. \\
\earr
%%%%%%%%%%%%%%%%%%%%%%%%
In the basis choice Eq.(\ref{basis}), the Charged Higgs Boson-Slepton
squared mass matrix  is written 
%%%%%%%%%%%
%%%%%%%%%%%%%%
\barr
{\cal L}=- (h^-_2,h_1^-,\tilde{e}_{L_i},\tilde{e}_{R_l})
 {\cal M}^2_{\rm Charged}\left (
\begin{array}{c} h_2^+\\ h_1^+ \\  \tilde{e}_{L_m}^{*} \\
\tilde{e}_{R_k}^* \end{array} \right )
\earr  
%%%%%%%%%%%%
with
%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%5
\barr
{\cal M}^2_{\rm Charged} = \left ( \begin{array}{cccc}
\tilde{B}\cot\beta + M_W^2 \cos^2\beta & \tilde{B} + 
\frac{1}{2}M_W^2 \sin2\beta & \tilde{D}_m & (\ye)_{jk} \kap_j^* v_1 \\[3mm]
* & \tilde{B}\tan\beta + M_W^2
\sin^2\beta & \widetilde{D}_m^*\tan\beta & (\ye)_{jk} \kap_j^* v_2 \\[3mm]
* & * & (\Omega^2_{LL})_{im} & (\Omega^2_{LR})_{ik} \\[3mm]
* & * & * & (\Omega^2_{RR})_{lk}
\end{array}\right ) \;, \label{bCharged}
\earr
%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%
where with ``*'' we denote the complex conjugate element and 
%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%555
\barr
(\Omega^2_{LL})_{im} &=& \biggl [ \ml_{im}+\mu_i\mu^*_m \biggr ]
+(\ye\yed)_{im} v_1^2+ \biggl (\frac{1}{2} M_Z^2-M_W^2 \biggr ) \cos2\beta
\delta_{im} \;, \nonum \\
(\Omega^2_{LR})_{ik} &=& (\he)_{ik} v_1 - \biggl [ (\ye)_{ik}\mu^*+
\lam_{lik} \kap_l^* \biggr ] v_2 \\
(\Omega^2_{RR})_{lk}&=& \me_{lk}+(\yed\ye)_{kl} v_1^2 +(M_W^2-M_Z^2) 
\cos2\beta \delta_{lk} \;.
\earr
%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%
In deriving (\ref{bCharged}) we have made use of the minimization
conditions (\ref{bmins}).
It is not hard to prove that the determinant of (\ref{Charged}) or
(\ref{bCharged}) is zero and Goldstone boson corresponds to the 
eigenvector $(-v_2,v_\alpha,0)$.

\subsection{Squarks}

\subsubsection{Down Squarks}
The down squark mass eigenstates $\tilde{d}_i, i=1,...6$ are 
given by diagonalizing the following mass matrix written here on
a basis independent form
%%%%%%%%%%%%%%
\barr
{\cal L}=- (\tilde{d}_{L_i}^*,\tilde{d}_{R_{i+3}}^*) {\cal M}^2_{\rm Down
  } \left ( \begin{array}{c}  \tilde{d}_{L_j} \\
\tilde{d}_{R_{j+3}} \end{array} \right ) \;,
\earr
%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%55
\begin{widetext}
%%%%%%%%%%%%%%%55 
\barr
{\cal M}^2_{\rm Down } = \left (\begin{array}{cc}
\mq_{ij}+\lam^{\prime *}_{\alpha il}\lam^{\prime}_{\gamma jl} 
v_\alpha v_\gamma +
\biggl (\frac{1}{4}g_2^2+\frac{1}{12}g^2 \biggr ) (v_2^2-v_\alpha^2
)\delta_{ij} & 
h_{\alpha ij}^{\prime *} v_\alpha - \lam^{\prime *}_{\alpha
  ij}\mu_\alpha v_2 \\[3mm]
* & \md_{ij}+\lam^{\prime *}_{\alpha l j}\lam^{\prime}_{\beta l i}
v_\alpha v_\beta +\frac{1}{6} g^2 (v_2^2 -v_\alpha^2 )\delta_{ij} 
\end{array} \right ) \;, \nonum \\[3mm] \label{downsquarks} 
\earr
%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%
and in the basis choice (\ref{basis}) 
%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%55
\barr
{\cal M}^2_{\rm Down } = \left (\begin{array}{cc}
\mq_{ij}+(\yd\ydd)_{ji} v_1^2 + D^d_{L_ij} & 
(\hd^*)_{ij} v_1 -  [ (\yds)_{ij} \mu + \lam^{\prime *}_{mij} 
\kap_m  ] v_2\\[3mm]
* & \md_{ij}+ (\ydd\yd)_{ji} v_1^2 + D^d_{R_ij}
\end{array} \right ) \;, \nonum \\[3mm] \label{bdownsquarks} 
\earr
%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%55
where
%%%%%%%%%%%%%%
\barr
D^d_{L_ij} &=& - \frac{1}{3} \biggl ( M_W^2+\frac{1}{2}
M_Z^2 \biggr )\cos2\beta \delta_{ij} \;, \nonum \\
D^d_{R_ij} &=& \frac{1}{3} \biggl ( M_W^2-M_Z^2
\biggr ) \cos2\beta \delta_{ij}\;,
\earr
%%%%%%%%%%%%%%%%%%%%%%%
are the D-term contributions.


\subsubsection{Up Squarks}
The up squark mass eigenstates $\tilde{u}_i, i=1,...6$ are 
given by diagonalizing the following mass matrix written here on
a basis independent form
%%%%%%%%%%%%%%
\barr
{\cal L}=- (\tilde{u}_{L_i}^*,\tilde{u}_{R_{i+3}}^*) {\cal M}^2_{\rm Up
  } \left ( \begin{array}{c}  \tilde{u}_{L_j} \\
\tilde{u}_{R_{j+3}} \end{array} \right ) \;,
\earr
%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%5
\begin{widetext}
%%%%%%%%%%%%%%5 
\barr
{\cal M}^2_{\rm Up } = \left (\begin{array}{cc}
\mq_{ij}+(\yu\yud)_{ji} v_2^2-
\frac{1}{4}\biggl (g_2^2-\frac{1}{3}g^2 \biggr ) (v_2^2-v_\alpha^2
)\delta_{ij} & 
(\hu^*)_{ij} v_2-(\yu^*)_{ij}\mu_\alpha v_1 \\[3mm]
* & \mup_{ij}+(\yud\yu)_{ji} v_2^2
-\frac{1}{3} g^2 (v_2^2 -v_\alpha^2 )\delta_{ij} 
\end{array} \right ) \;, \nonum \\[3mm] \label{upsquarks} 
\earr
%%%%%%%%%%%%%%%%%%%%%5
\end{widetext}
%%%%%%%%%%%%%%%%%%5
%%%%%%%%%%%%%%%%%%%%%%
and in the basis choice (\ref{basis}) 
%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%555
\barr
{\cal M}^2_{\rm Up } = \left (\begin{array}{cc}
\mq_{ij}+(\yu\yud)_{ji} v_2^2 + D^u_{L_ij} & 
(\hu^*)_{ij} v_2 -   (\yus)_{ij} \mu v_\alpha \\[3mm]
* & \mup_{ij}+ (\yud\yu)_{ji} v_2^2 + D^u_{R_ij}
\end{array} \right ) \;, \nonum \\[3mm] \label{bupsquarks} 
\earr
%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%
\barr
D^u_{L_ij} &=&  \biggl ( \frac{2}{3} M_W^2-\frac{1}{6}
M_Z^2 \biggr )\cos2\beta \delta_{ij} \;, \nonum \\
D^u_{R_ij} &=& \frac{2}{3} \biggl ( M_Z^2-M_W^2
\biggr ) \cos2\beta \delta_{ij}\;,
\earr
%%%%%%%%%%%%%%%%%%%%%%%
are the D-term contributions.
Notice that the form of the Up Squark mass matrix does 
remains as in the R-parity conserved case in the basis (\ref{basis}).


\subsection{Quarks}

The Down quark masses are ,
%%%%%%%%%%%%%%
\barr
(m_d)_{ij}=\lam^\prime_{\alpha i j} v_\alpha \;,
\label{downquarks}
\earr
%%%%%%%%%%%%
and in basis (\ref{basis}) 
%%%%%%%%%%%%%%
\barr
(m_d)_{ij}=(\yd)_{ij} v_1 \;.
\label{bdownquarks}
\earr
%%%%%%%%%%%%
The Up quark masses are always 
%%%%%%%%%%%%%%
\barr
(m_u)_{ij}=(\yu)_{ij} v_2 \;.
\label{upquarks}
\earr
%%%%%%%%%%%%


\subsection{Neutrinos-Neutralinos}

Neutrinos mix with neutralinos resulting one massive neutrino at
tree level and four massive neutralinos.
The Lagrangian contains the ($7 \times 7$ for three generations of
 neutrinos)  neutrino-neutralino mass matrix as
%%%%%%%%%%%
\barr
{\cal L}= -\frac{1}{2} (-i \bino, -i \wino^{(3)}, \tilde{h_2^0},
\nu_\alpha)
{\cal M}_{\rm N} \left ( \begin{array}{c}
-i \bino \\  -i \wino^{(3)} \\ \tilde{h_2^0} \\
\nu_\beta \end{array} \right ) \;,
\earr
%%%%%%%%%%%%%%
where the eigenvalues $\tilde{\chi}^0_{1,...4}, \nu_{i=1,...3}$ are
given upon the diagonalization of the matrix\footnote{We have
  corrected
the sign error for the $\mu_\alpha$ parameter in Eq.(3.2) of
\cite{Haber}. We would like to thank H. Haber for communication.}
%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%55
\barr
{\cal M}_{\rm N} = \left ( \begin{array}{cccc}
M_1 & 0 & M_Z s_w \frac{v_2}{\sqrt{v_\gamma^2}} & -M_Z s_w 
\frac{v_\beta}{\sqrt{v_\gamma^2}} \\[3mm]
0 & M_2 &  -M_Z c_w  \frac{v_2}{\sqrt{v_\gamma^2}} & M_Z c_w
\frac{v_\beta}{\sqrt{v_\gamma^2}} \\[3mm]
M_Z s_w \frac{v_2}{\sqrt{v_\gamma^2}} 
& -M_Z c_w  \frac{v_2}{\sqrt{v_\gamma^2}} & 0 & -\mu_\beta  \\[3mm]
-M_Z s_w 
\frac{v_\alpha}{\sqrt{v_\gamma^2}} &M_Z c_w
\frac{v_\alpha}{\sqrt{v_\gamma^2}}  & -\mu_\alpha & 0_{\alpha\beta} 
\end{array} \right ) \;,\label{neutralino}
\earr
%%%%%%%%%%%%%%%%%%%%%%5
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%
with $M_Z^2=\frac{1}{2} (g^2+g_2^2) (v_2^2+v_\alpha^2)$,
$M_W^2=\frac{1}{2} g_2^2 (v_2^2+v_\alpha^2)$ and $s_W=\sin\theta_W$
the weak mixing angle. In the basis (\ref{basis}) the above matrix is
written
%%%%%%%%%%%
\barr
{\cal L}= -\frac{1}{2} (-i \bino, -i \wino^{(3)}, \tilde{h_2^0},
\tilde{h_1^0},\nu_i)
{\cal M}_{\rm N} \left ( \begin{array}{c}
-i \bino \\  -i \wino^{(3)} \\ \tilde{h_2^0} \\ \tilde{h_1^0} \\
\nu_j \end{array} \right )
\earr
%%%%%%%%%%%%%%
with
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%%
\barr
{\cal M}_{\rm N} = \left ( \begin{array}{ccccc}
M_1 & 0 &  M_Z s_w\sin\beta & -M_Z s_w \cos\beta & 0_j \\
0 & M_2 & -M_Z c_w \sin\beta & M_Z c_w \cos\beta & 0_j \\
M_Zs_w \sin\beta & -M_Z c_w \sin\beta & 0 & -\mu & -\kap_j \\
-M_Z s_w \cos\beta & M_Z c_w \cos\beta & -\mu & 0 &  0_j \\
0_i & 0_i & -\kap_i & 0_i & 0 \end{array} \right ) \;.
\label{bneutralino}
\earr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The matrices (\ref{neutralino},\ref{bneutralino}) have five non-zero
eigenvalues, i.e., 4 neutralinos + 1 neutrino. 




\subsection{Leptons-Charginos}

Leptons mix with charginos.
The Lagrangian contains the ($5 \times 5$ for three generations of
 leptons)  lepton-chargino mass matrix as\footnote{ We used the
 notation  $(\wino^\pm =\frac{1}{\sqrt{2}}
(\wino^{(1)}\mp i \wino^{(2)})$.}
%%%%%%%%%%%
\barr
{\cal L}= - (-i \wino^-, 
e_{L_\alpha}^-)
{\cal M}_{\rm C} \left ( \begin{array}{c}
-i \wino^+ \\ \tilde{h}_2^+ \\
e_{R_k}^+ \end{array} \right ) + {\rm H.c}
\earr
%%%%%%%%%%%%%%
where the eigenvalues $\tilde{\chi}^\pm_{1,...,2}, l=(e,\mu,\tau)$ are
given upon the diagonalization of the matrix
%%%%%%%%%%%%%%%%%%%%%
\barr
{\cal M}_{\rm C} = \left ( \begin{array}{ccc}
M_2 & g_2 v_2 & 0_k \\
g_2 v_\alpha & \mu_\alpha & \lam_{\beta\alpha k} v_\beta \end{array}
\right ) \;, \label{chargino}
\earr
%%%%%%%%%%%%%%%%%%%
whereas in basis (\ref{basis})
%%%%%%%%%%%
\barr
{\cal L}= - (-i \wino^-, \tilde{h}_1^-,
e_{L_i}^-)
{\cal M}_{\rm C} \left ( \begin{array}{c}
-i \wino^+ \\ \tilde{h}_2^+ \\
e_{R_j}^+ \end{array} \right ) + {\rm H.c}
\earr
%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%
\barr
{\cal M}_{\rm C} = \left ( \begin{array}{ccccc}
M_2 & \sqrt{2} M_W \sin\beta & 0 & 0 & 0 \\
\sqrt{2} M_W cos\beta & \mu & 0 & 0 & 0 \\
0 & \kap_1 & & & \\
0 & \kap_2 & & (\ye)_{ij} v_1 & \\
0 & \kap_3 & & & \end{array} \right ) \;. \label{bchargino}
\earr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\section{Boundary Conditions at \mx}


\section{Results} 

\subsection{No-Scale case}

\subsection{General Case}

\section{Conclusions}

The ultimate paper!

%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{widetext}
%%%%%%%%%%%%%%%%%%%%





%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix

\vspace*{1cm}

\section{Notation and Anomalous Dimensions}
The chiral superfields of the $R_p$-MSSM and the \rpv-MSSM have the 
following $G_{SM}=SU(3)_c\times SU(2)_L\times U(1)_Y$ quantum numbers
\barr
L:&(1,2,-\half),\quad {\bar E}:&(1,1,1),\qquad\, Q:\,(3,2,\frac{1}{6}),\quad
{\bar U}:\,(3,1,\frac{2}{3}),\nonr\\ {\bar D}:& ~(3,1,-\frac{1}{3}),\quad
H_1:&(1,2,-\half),\quad  H_2:\,(1,2,\half).
\label{fields}
\earr
 The \rpv-MSSM superpotential is then given by
\barr 
W&=& \eps_{ab} \left[ (\ye)_{ij} L_i^a
H_1^b {\bar E}_j + (\yd)_{ij} Q_i^{ax} H_1^b {\bar D}_{jx} +
(\yu)_{ij} Q_i^{ax} H_2^b {\bar U}_{jx} \right]\nonr \\ &&+
\eps_{ab}\left[ \frac{1}{2}(\lame{k})_{ij} L_i^a L_j^b{\bar E}_k +
(\lamd{k})_{ij} L_i^a Q_j^{xb} {\bar D}_{kx} \right]+
\frac{1}{2}\eps_{xyz} (\lamu{i})_{jk} {\bar U}_i^x{\bar
D}_j^y{\bar D}^z_k \nonr\\&& - \eps_{ab} \left[\mu H_1^a H_2^b
+\kap^i L_i^a H_2^b \right].  \label{superpot} 
\earr 
We denote an $SU(3)$ colour index of the fundamental representation by
$x,y,z=1,2,3$. The $SU(2)_L$ fundamental representation indices are
denoted by $a,b,c=1,2$ and the generation indices by $i,j,k=1,2,3$.
We have introduced the twelve $3\times3$ matrices
\beq 
\ye,\quad \yd,\quad
\yu,\quad \lame{k},\quad \lamd{k},\quad \lamu{i}, 
\label{matrices} 
\eeq
for all the Yukawa couplings. This implies the following conventions
in the Martin and Vaughn~\cite{mv} notation
\barr 
Y^{L_i^a Q^{bx}_j \bar{D}_{ky}}
&=& Y^{L_i^a \bar{D}_{ky} Q^{bx}_j} = Y^{\bar{D}_{ky} L_i^a Q^{bx}_j}
= Y^{Q^{bx}_j L_i^a \bar{D}_{ky}} \label{a4}\nonum  \\ 
&=& Y^{Q^{bx}_j
\bar{D}_{ky} L_i^a} = Y^{\bar{D}_{ky} Q^{bx}_j L_i^a} =
(\lamd{k})_{ij} \eps_{ab} \delta^{y}_{x},\\ 
Y^{L_i^a L_j^b\bar{E}_k} &=& Y^{L_i^a \bar{E}_k L_j^b} = 
Y^{\bar{E}_k L_i^a L_j^b} =
(\lame{k})_{ij} \eps_{ab} = -(\lame{k})_{ji} \eps_{ab},
\\ 
Y^{{\bar U}_{ix}{\bar D}_{jy}{\bar D}_{kz}}&=&Y^{{\bar
D}_{jy}{\bar U}_{ix}{\bar D}_{kz}}= Y^{{\bar D}_{jy}{\bar D}_{kz}{\bar
U}_{ix}}= \eps_{x y z} \left({\bf\Lambda}_{U^i}\right)_{jk} =
-\eps_{x y z} \left({\bf\Lambda}_{U^i}\right)_{kj},
\label{a6} 
\earr 

The soft SUSY breaking Lagrangian is given by,
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
-{\cal L} &=& \mh{1} H_1^\dagger H_1 + \mh{2} H_2^\dagger H_2
+ \widetilde{L}^\dagger \ml \widetilde{L}  
+ \widetilde{L_i}^\dagger  \mlh{i} H_1
+ H_1^\dagger \mhl{i} \widetilde{L}\nonum \\
&+&  \widetilde{Q}^\dagger \mq \widetilde{Q} 
+ \widetilde{\bar{E}} \me \widetilde{\bar{E}}^\dagger
+ \widetilde{\bar{D}} \md \widetilde{\bar{D}}^\dagger
+ \widetilde{\bar{U}} \mup \widetilde{\bar{U}}^\dagger \nonum \\
&+& \biggl [(\he)_{ij} \widetilde{L_i} H_1 \widetilde{\bar{E}_j}
+ (\hd)_{ij} \widetilde{Q_i} H_1 \widetilde{\bar{D}_j}
+ (\hu)_{ij} \widetilde{Q_i} H_2 \widetilde{\bar{U}_j} \nonum \\
&+& 
(\lamhe{k})_{ij} \widetilde{L_i} \widetilde{L_j} \widetilde{\bar{E}_k}
+(\lamhd{k})_{ij} \widetilde{L_i} \widetilde{Q_j} \widetilde{\bar{D}_k}
+(\lamhu{i})_{jk} \widetilde{\bar{U}_i} \widetilde{\bar{D}_j} 
\widetilde{\bar{D}_k}
+ \; \; {\rm H.c} \biggr ]
\nonum \\
&-& \biggl [\widetilde{B} H_1 H_2 + \widetilde{D_i} 
\widetilde{L_i} H_2\; \; + {\rm H.c} \biggr ] \;, \label{soft}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%
where we have introduce the soft SUSY breaking trilinear couplings
%%%%%%%%%%%%%%%%%%%%%%%5
\beq 
\he,\quad \hd,\quad
\hu,\quad \lamhe{k},\quad \lamhd{k},\quad \lamhu{i}, 
\label{trilsoft} 
\eeq
%%%%%%%%%%%%%%%%%%
defined analogously as the Yukawa couplings in (\ref{a4}-\ref{a6}).


In general the one-loop renormalisation group equations for the Yukawa
couplings are given by \cite{mv}
\beq
\frac{d}{dt} Y^{ijk}=Y^{ijp}\left[\frac{1}{16\pi^2}\gamma_p^{k}
\right]+(k\lra i)+(k\lra j),
\label{rgyukawa}
\eeq
and the  anomalous dimensions are
\barr
\gamma_i^{j}&=& \half Y_{ipq}Y^{jpq}-2\delta_i^j\sum_ag_a^2C_a(i),\\
 \label{gamone}
\earr
We have denoted by $C_a(f)$ the quadratic Casimir of the 
representation $f$ of the gauge group $G_a$. 
For details see the Appendix A of ~\cite{add}.

The one-loop anomalous dimensions are given by~\cite{add}\footnote{A typo 
in $\gamma^{(1)E_j}_{E_i}$ of \cite{add} has been corrected.}
\barr
\gamma^{L_i}_{L_j} &=&\left(\ye \ye^\dagg \right)_{ij}
+(\lame{q}\lame{q}^\dagg)_{ij} +3 (\lamd{q}\lamd{q}^\dagg)_{ij}
-\delta^i_j(\frac{3}{10}g_1^2+\frac{3}{2}g_2^2), \label{gamll1}\\
%
\gamma^{E_i}_{E_j} &=& 2 \left(\ye^\dagg \ye \right)_{ji}
+ \mbox{Tr}(\lame{i}\lame{j}^\dagg) -\delta^i_j(\frac{6}{5}g_1^2),
\label{gamee1}\\
%
\gamma^{Q_i}_{Q_j} &=& \left(\yd \yd^\dagg \right)_{ij}
+ \left(\yu \yu^\dagg \right)_{ij}
+ (\lamd{q}^\dagg\lamd{q})_{ji}
-\delta^i_j(\frac{1}{30}g_1^2+\frac{3}{2}g_2^2+\frac{8}{3}g_3^2),\\
%
\gamma^{D_i}_{D_j} &=& 2 \left(\yd^\dagg \yd \right)_{ji}
+2 \mbox{Tr}(\lamd{j}^\dagg\lamd{i})
+2 (\lamu{q}\lamu{q}^\dagg)_{ij}
-\delta^i_j(\frac{2}{15}g_1^2+\frac{8}{3}g_3^2),\\
%
\gamma^{U_i}_{U_j} &=& 2\left(\yu^\dagg \yu \right)_{ji}
+ \mbox{Tr}(\lamu{i}\lamu{j}^\dagg)
-\delta^i_j(\frac{8}{15}g_1^2+\frac{8}{3}g_3^2),\\
%
\gamma^{H_1}_{H_1} &=& \mbox{Tr}\left(3\yd\yd^\dagg+\ye\ye^\dagg \right)
-(\frac{3}{10}g_1^2+\frac{3}{2}g_2^2),\\
%
\gamma^{H_2}_{H_2} &=& 3 \mbox{Tr}\left( \yu\yu^\dagg\right)
-(\frac{3}{10}g_1^2+\frac{3}{2}g_2^2),\\
%
\gamma^{H_1}_{L_i} &=& {\gamma^{(1)L_i}_{H_1}}^* =-3 (\lamd{q}^*\yd)_{iq}
- (\lame{q}^*\ye)_{iq}.\label{mixedad}
\earr
Note that here, $H_{1,2},L,Q$ represent the fields $H_{1,2}^a$, $L^a$,
$Q^{a}$ where $a$ is the index of the fundamental representation of
$SU(2)$ (i.e.\ no factors of $\eps_{ab}$ are factored in). 
 The $\beta$-functions for the Yukawa couplings~\cite{add} as well
for the billinear superpotential couplings are just 
combinations of the above anomalous dimensions
(\ref{gamll1}-\ref{mixedad}). The two loop anomalous 
dimensions in the \rpv -MSSM can be found in 
\cite{add}. We present the one-loop beta functions for the
superpotential couplings and masses for complemetarity reasons.

The RGEs for the Yukawa couplings (including full family dependence)
are  given by
\barr
16\pi^2 \Dt (\ye)_{ij} &=& (\ye)_{ik}\ggam{E_j}{E_k}
+(\ye)_{ij}\ggam{H_1}{H_1}
-(\lame{j})_{ki}\ggam{H_1}{L_k}
+(\ye)_{kj}\ggam{L_i}{L_k},\label{eq:ye}\\
%
16\pi^2\Dt (\yd)_{ij} &=& (\yd)_{ik}\ggam{D_j}{D_k}
  +(\yd)_{ij}\ggam{H_1}{H_1}
  -(\lamd{j})_{ki}\ggam{H_1}{L_k}
  +(\yd)_{kj}\ggam{Q_i}{Q_k},\label{eq:yd}\\
%
16\pi^2\Dt (\yu)_{ij} &=& (\yu)_{ik}\ggam{U_j}{U_k}\label{yu}
+(\yu)_{ij}\ggam{H_2}{H_2}
+(\yu)_{kj}\ggam{Q_i}{Q_k}, 
\\
%         
16\pi^2\Dt (\lame{k})_{ij} &=& (\lame{l})_{ij}\ggam{E_k}{E_l}
+(\lame{k})_{il}\ggam{L_j}{L_l}
+(\ye)_{ik}\ggam{L_j}{H_1}
-(\lame{k})_{jl}\ggam{L_i}{L_l}-(\ye)_{jk}\ggam{L_i}{H_1}, \label{eq:lame}\\
%
16\pi^2\Dt (\lamd{k})_{ij} &=& (\lamd{l})_{ij}\ggam{D_k}{D_l}\lab{lamd}
+(\lamd{k})_{il}\ggam{Q_j}{Q_l}
+(\lamd{k})_{lj}\ggam{L_i}{L_l}
-(\yd)_{jk}\ggam{L_i}{H_1},\\
%
16\pi^2\Dt (\lamu{i})_{jk} &=& (\lamu{i})_{jl}\ggam{D_k}{D_l} \lab{lamu}
+(\lamu{i})_{lk}\ggam{D_j}{D_l}
+(\lamu{l})_{jk}\ggam{U_i}{U_l}.
%
\earr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The RGE's for the bilinear terms are 
%%%%%%%%%%%%%%%%%%
\barr
16\pi^2\Dt\mu&=&\mu\left\{ \ggam{H_1}{H_1}+\ggam{H_2}{H_2}\right\}
+\kap^i\ggam{H_1}{L_i},\\[3mm]
16\pi^2\Dt\kap^i&=&\kap^i\ggam{H_2}{H_2}+\kap^p\ggam{L_i}{L_p}
+\mu\ggam{L_i}{H_1}. \lab{kappa}
\earr
%%%%%%%%%%%%%%%%%%%%


\section{A method to derive the soft SUSY breaking RGE's}

A straightforword way to derive the RGEs for the soft
SUSY breaking couplings and masses is by a direct use
of the explict formulae at 1-loop given in~\cite{mv}. 
This is a somehow tedious job. A very 
elegant method which is also very helpful for numerical
calculations is that described in~\cite{jones}. All the soft SUSY
RGE's can be derived by the knoweldge of just the anomalous
dimensions (\ref{gamll1}-\ref{mixedad}) when usually an action
of an operator on them is taking place\footnote{For the gaugino masses
a knowledge of the $\beta$-function of the gauge couplings is
also neccecary. Since the \rpv-couplings do not affect the gaugino masses
at one loop one can use the gaugino  RGEs directly from~\cite{mv}.}.
 The method not only works at
one loop but it has been proven to all orders in perturbation 
theory~\cite{jones}. Thus the soft $\beta$-functions for the
bilinear ${\bf b}^{ij}$, trilinear ${\bf h}^{ijk}$  
and scalar masses  $({\bf m}^2)^i_j$ soft SUSY breaking terms can be read from
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
16\pi^2 \frac{d {\bf b}^{ij}}{dt} &=& \gamma^i_l {\bf b}^{jl}+\gamma^j_l
{\bf b}^{il}-2 (\gamma_1)^i_l {\bf \mu}^{jl} -2 (\gamma_1)^j_l 
{\bf \mu}^{il} \;, \label{bmass}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d {\bf h}^{ijk}}{dt} &=& \gamma^i_l {\bf h}^{jkl}+
\gamma^j_l {\bf h}^{ikl} + \gamma^k_l {\bf h}^{jil} 
-2 (\gamma_1)^i_l {\bf Y}^{jkl} -2 (\gamma_1)^j_l {\bf Y}^{ikl}
-2 (\gamma_1)^k_l {\bf Y}^{jil} \;, \label{tril}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d ({\bf m}^2)^i_j}{dt} &=& \biggl ( 2 {\cal O} {\cal O}^*
+2 M M^* g_a^2 \frac{\partial}{\partial g^2_a} + 
\widetilde{{\bf Y}}_{lmn} \frac{\partial}{\partial {{\bf Y}}_{lmn}}
+
\widetilde{{\bf Y}}^{lmn} \frac{\partial}{\partial {{\bf Y}}^{lmn}}
+ X_a \frac{\partial}{\partial g_a} \biggr )\gamma^i_j \;,\label{smass}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
where 
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
(\gamma_1)_j^i &=& {\cal O} \gamma_j^i \;, \label{b4} \\[5mm]
{\cal O} &=& \biggl ( M_a g_a^2 \frac{\partial}{\partial g_a^2} - 
{\bf h}^{lmn} \frac{\partial}{\partial {\bf Y}^{lmn}} \biggr ) \;,
\label{b5}\\[5mm]
\widetilde{{\bf Y}}^{ijk} &=& {\bf Y}^{ljk} ({\bf m}^2)^i_l +
{\bf Y}^{lik} ({\bf m}^2)^j_l+{\bf Y}^{lji} ({\bf m}^2)^k_l \;, \label{b6}
\end{eqnarray}
%%%%%%%%%%%%%%%
and repeated indices are summed over and $t=\ln(Q)$, $Q$ is the
renormalization scale. At one loop the last term in 
(\ref{smass}) is irrelevant, see \cite{jones} for details. The
Renormalization Group Equations (\ref{bmass}-\ref{smass}) valid
as long as we do not eliminate the U(1) Fayet-Iliopoulos D-term.
In the case we do so an extra term proportional to the U(1) gauge
coupling appears ( see the  ${\cal S}$-term in the RGEs
of the Appendix C.).


Now from (\ref{bmass})
the RGEs for the bilinear soft SUSY breaking masses  in the 
\rpv-MSSM are 
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
16 \pi^2 \frac{d\widetilde{B}}{dt} &=& \widetilde{B}
\biggl [\gamma^{H_1}_{H_1}+
\gamma^{H_2}_{H_2} \biggr ] +\widetilde{D}_i \gamma^{H_1}_{L_i}
-2 \mu \biggl [(\gamma_1)^{H_1}_{H_1}+(\gamma_1)^{H_2}_{H_2}
\biggr ]-2 \kappa_i (\gamma_1)^{H_1}_{L_i} \;,\label{b7}\\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16 \pi^2 \frac{d\widetilde{D}_i}{dt} &=&
\biggl [\gamma^{L_i}_{L_l}\widetilde{D}^l
+\gamma_{H_2}^{H_2}\widetilde{D}^i
 \biggr]
+\widetilde{B} \gamma_{H_1}^{L_i} 
-2 \biggl [ (\gamma_1)^{L_i}_{L_l} \kappa^l +  (\gamma_1)^{H_2}_{H_2}\kappa^i
\biggr ] -2 \mu (\gamma_1)_{H_1}^{L_i} \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The RGEs for the trilinear  soft SUSY breaking masses  in the 
\rpv-MSSM can be read from (\ref{tril})
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
16 \pi^2 \frac{d (\he)_{ik}}{d t}  &=& 
\gamma_{L_l}^{L_i} (\he)_{lk} +\gamma_{H_1}^{H_1} (\he)_{ik}
+\gamma^{H_1}_{L_l} (\lamhe{k})_{il} + \gamma^{E_k}_{E_l} (\he)_{il}
\nonumber \\
&-&
2 (\gamma_1)_{L_l}^{L_i} (\ye)_{lk} -2 (\gamma_1)_{H_1}^{H_1} (\ye)_{ik}
-2 (\gamma_1)^{H_1}_{L_l} (\lame{k})_{il}
-2 (\gamma_1)^{E_k}_{E_l} (\ye)_{il} \;, \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16 \pi^2 \frac{d (\hd)_{ik}}{d t}  &=& 
\gamma_{Q_l}^{Q_i} (\hd)_{lk} +\gamma_{H_1}^{H_1} (\hd)_{ik}
-\gamma^{H_1}_{L_l} (\lamhd{k})_{li} + \gamma^{D_k}_{D_l} (\hd)_{il}
\nonumber \\
&-&
2 (\gamma_1)_{Q_l}^{Q_i} (\yd)_{lk} -2 (\gamma_1)_{H_1}^{H_1} (\yd)_{ik}
+2 (\gamma_1)^{H_1}_{L_l} (\lamd{k})_{li}
-2 (\gamma_1)^{D_k}_{D_l} (\yd)_{il} \;, \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16 \pi^2 \frac{d (\hu)_{ik}}{d t}  &=&
\gamma_{Q_l}^{Q_i} (\hu)_{lk} +\gamma_{H_2}^{H_2} (\hu)_{ik}
 + \gamma^{U_k}_{U_l} (\hu)_{il} \nonumber \\
&-&
2 (\gamma_1)_{Q_l}^{Q_i} (\yu)_{lk} -2 (\gamma_1)_{H_2}^{H_2} (\yu)_{ik}
-2 (\gamma_1)^{U_k}_{U_l} (\yu)_{il} \;, \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16 \pi^2 \frac{d (\lamhe{k})_{ij}}{d t}  &=&
\gamma^{L_i}_{L_l} (\lamhe{k})_{lj}-\gamma_{H_1}^{L_i}(\he)_{jk}+
\gamma_{L_l}^{L_j}(\lamhe{k})_{il}
+\gamma_{H_1}^{L_j} (\he)_{ik}+\gamma^{E_k}_{E_l}(\lamhe{l})_{ij}
\nonumber \\
&-& 
2 (\gamma_1)^{L_i}_{L_l} (\lame{k})_{lj}+2 (\gamma_1)_{H_1}^{L_i}(\ye)_{jk}
-2 (\gamma_1)_{L_l}^{L_j}(\lame{k})_{il} -2 (\gamma_1)_{H_1}^{L_j} (\ye)_{ik}
-2 (\gamma_1)^{E_k}_{E_l}(\lame{l})_{ij} \;, \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16 \pi^2 \frac{d (\lamhd{k})_{ij}}{d t}  &=&
\gamma^{L_i}_{L_l} (\lamhd{k})_{lj}-\gamma_{H_1}^{L_i}(\hd)_{jk}+
\gamma_{Q_l}^{Q_j}(\lamhd{k})_{il}
+\gamma^{D_k}_{D_l}(\lamhd{l})_{ij}
\nonumber \\
&-& 
2 (\gamma_1)^{L_i}_{L_l} (\lamd{k})_{lj}+2 (\gamma_1)_{H_1}^{L_i}(\yd)_{jk}
-2 (\gamma_1)_{Q_l}^{Q_j}(\lamd{k})_{il}
-2 (\gamma_1)^{D_k}_{D_l}(\lamd{l})_{ij} \;, \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16 \pi^2 \frac{d (\lamhu{i})_{jk}}{d t}  &=&
\gamma^{U_i}_{U_l} (\lamhu{l})_{jk}
+\gamma_{D_l}^{D_j}(\lamhu{i})_{lk}
+\gamma^{D_k}_{D_l}(\lamhu{i})_{jl}
\nonumber \\
&-& 
2 (\gamma_1)^{U_i}_{U_l} (\lamu{l})_{jk}
-2 (\gamma_1)_{D_l}^{D_j}(\lamu{i})_{lk}
-2 (\gamma_1)^{D_k}_{D_l}(\lamu{i})_{jl} \;.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{eqnarray}
The RGEs for the   soft SUSY breaking masses  in the 
\rpv-MSSM can be written from (\ref{smass}) 
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
16\pi^2 \frac{d \me^{E_i}_{E_j}}{dt}\equiv 
16\pi^2 \frac{d \me_{ji}}{dt}  &=& 4 (\hed\he)_{ji} 
+2 \Tr (\lamhe{i}\lamhed{j})-\delta_{ij} \biggl (\frac{24}{5} g_1^2
|M_1|^2 \biggr ) \nonumber \\ 
&+& 2 (\yed \widetilde{\ye})_{ji}
+\Tr (\widetilde{\lame{i}}\lamed{j}) +2 (\widetilde{\yed}\ye)_{ji}
+\Tr (\lame{i}\widetilde{\lamed{j}}) \;,\\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d \ml^{L_i}_{L_j}}{dt}\equiv 
16\pi^2 \frac{d \ml_{ij}}{dt}  &=& 
2 (\he\hed+\lamhe{q}\lamhed{q}+3\lamhd{q}\lamhdd{q})_{ij}
-\delta_{ij}\biggl (\frac{6}{5} g_1^2 |M_1|^2+6 g_2^2 |M_2|^2\biggr )
\nonumber \\
&+& (\widetilde{\ye}\yed)_{ij}
+(\widetilde{\lame{q}}\lamed{q})_{ij}
+3 (\widetilde{\lamd{q}}\lamdd{q})_{ij} \nonumber \\
&+& (\ye\widetilde{\yed})_{ij}
+ (\lame{q}\widetilde{\lamed{q}})_{ij}
+3 (\lamd{q}\widetilde{\lamdd{q}})_{ij} \; \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d ({\bf m^2})^{H_1}_{L_i}}{dt}\equiv 
16\pi^2 \frac{d \mhl{i}}{dt}  &=&
-6 (\lamhd{q}^*\hd)_{iq}-2(\lamhe{q}^*\he)_{iq} \nonumber \\
&-& 3 (\lamds{q}\widetilde{\yd})_{iq}-(\lames{q}\widetilde{\ye})_{iq}
-3 (\widetilde{\lamds{q}}\yd)_{iq} - (\widetilde{\lames{q}}\ye)_{iq}
\; \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d \mq^{Q_i}_{Q_j}}{dt}\equiv 
16\pi^2 \frac{d \mq_{ij}}{dt}  &=&
2 ( \hd \hdd+\hu\hud)_{ij}+2 (\lamhdd{q}\lamhd{q})_{ji}
-\delta_{ij} \biggl (\frac{2}{15}g_1^2 |M_1|^2+6 g_2^2 |M_2|^2
+\frac{32}{3}g_3^2 |M_3|^2 \biggr ) \nonumber \\
&+&(\widetilde{\yd}\ydd)_{ij} + (\widetilde{\yu}\yud)_{ij}
+(\lamdd{q}\widetilde{\lamd{q}})_{ji}  \nonumber \\
&+& (\yd\widetilde{\ydd})_{ij} + (\yu\widetilde{\yud})_{ij}
+(\widetilde{\lamdd{q}}\lamd{q})_{ji} \;, \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d \md^{D_i}_{D_j}}{dt}\equiv 
16\pi^2 \frac{d \md_{ji}}{dt}  &=&
4(\hdd\hd)_{ji}+4\Tr(\lamhdd{j}\lamhd{i})+4(\lamhu{q}\lamhud{q})_{ij}
\nonumber \\
&-& \delta_{ij} \biggl (\frac{8}{15}g_1^2 |M_1|^2+\frac{32}{3}g_3^2
|M_3|^2 \biggr ) \nonumber \\
&+& 2 (\ydd\widetilde{\yd})_{ji}+2\Tr(\lamdd{j}\widetilde{\lamd{i}})
+2 (\widetilde{\lamu{q}}\lamud{q})_{ij} \nonumber \\
&+& 2(\widetilde{\ydd}\yd)_{ji}+2\Tr(\widetilde{\lamdd{j}}\lamd{i})
+2 (\lamu{q}\widetilde{\lamud{q}})_{ij}\;, \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d \mup^{U_i}_{U_j}}{dt}\equiv 
16\pi^2 \frac{d \mup_{ji}}{dt}  &=&
4(\hud\hu)_{ji}+2\Tr(\lamhu{i}\lamhud{j})
-\delta_{ij} \biggl (\frac{32}{15}g_1^2 |M_1|^2+\frac{32}{3}g_3^2
|M_3|^2 \biggr ) \nonumber \\
&+& 2 (\yud\widetilde{\yu})_{ji} + \Tr(\widetilde{\lamu{i}}\lamud{j})
+ 2 (\widetilde{\yud}\yu)_{ji} + \Tr(\lamu{i}\widetilde{\lamud{j}})
\;, \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d \mh{1}}{dt}  &=&
\Tr (6 \hd\hdd +2\he\hed)-\biggl ( \frac{6}{5} g_1^2 |M_1|^2 +
6 g_2^2 |M_2|^2 \biggr ) \nonumber \\
&+& 3 \Tr(\widetilde{\yd}\ydd) +\Tr(\widetilde{\ye}\yed)
+3 \Tr(\yd\widetilde{\ydd}) + \Tr(\ye\widetilde{\yed}) \;,  \\[3mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d \mh{2}}{dt}  &=&
6 \Tr (\hu\hud)-\biggl ( \frac{6}{5} g_1^2 |M_1|^2 +
6 g_2^2 |M_2|^2 \biggr ) \nonumber \\
&+& 3 \Tr(\widetilde{\yu}\yud) + 3 \Tr(\yu\widetilde{\yud}) \;, \label{b22}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where from (\ref{b4}-\ref{b5}) 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\barr
(\gamma_1)^{L_i}_{L_j} &=&-\left(\he \ye^\dagg \right)_{ij}
-(\lamhe{q}\lame{q}^\dagg)_{ij} -3 (\lamhd{q}\lamd{q}^\dagg)_{ij}
-\delta^i_j(\frac{3}{10}M_1g_1^2+\frac{3}{2}M_2g_2^2), \label{g1gamll1}\\
%
(\gamma_1)^{E_i}_{E_j} &=& -2 \left(\ye^\dagg \he \right)_{ji}
- \mbox{Tr}(\lamhe{i}\lame{j}^\dagg) -\delta^i_j(\frac{6}{5}M_1 g_1^2),
\label{g1gamee1}\\
%
(\gamma_1)^{Q_i}_{Q_j} &=& -\left(\hd \yd^\dagg \right)_{ij}
- \left(\hu \yu^\dagg \right)_{ij}
- (\lamd{q}^\dagg\lamhd{q})_{ji}
-\delta^i_j(\frac{1}{30}M_1 g_1^2+\frac{3}{2} M_2g_2^2+
\frac{8}{3}M_3 g_3^2),\\
%
(\gamma_1)^{D_i}_{D_j} &=& -2 \left(\yd^\dagg \hd \right)_{ji}
-2 \mbox{Tr}(\lamd{j}^\dagg\lamhd{i})
-2 (\lamhu{q}\lamu{q}^\dagg)_{ij}
-\delta^i_j(\frac{2}{15}M_1 g_1^2+\frac{8}{3}M_3g_3^2),\\
%
(\gamma_1)^{U_i}_{U_j} &=& -2\left(\yu^\dagg \hu \right)_{ji}
- \mbox{Tr}(\lamhu{i}\lamu{j}^\dagg)
-\delta^i_j(\frac{8}{15}M_1 g_1^2+\frac{8}{3}M_3g_3^2),\\
%
(\gamma_1)^{H_1}_{H_1} &=& -\mbox{Tr}\left(3\hd\yd^\dagg+\he\ye^\dagg \right)
-(\frac{3}{10}M_1g_1^2+\frac{3}{2}M_2g_2^2),\\
%
(\gamma_1)^{H_2}_{H_2} &=& -3 \mbox{Tr}\left( \hu\yu^\dagg\right)
-(\frac{3}{10}M_1g_1^2+\frac{3}{2}M_2g_2^2),\\
%
(\gamma_1)^{H_1}_{L_i} &=& {(\gamma_1)^{L_i}_{H_1}}^* =3 (\lamd{q}^*\hd)_{iq}
+ (\lame{q}^*\he)_{iq} \;,\label{g1mixedad}
\earr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and from (\ref{b6})
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
(\widetilde{\ye})_{ik} &=& (\ye)_{lk}\ml_{il}+(\ye)_{ik} \mh{1}
+(\lame{k})_{il}\mhl{l}+(\ye)_{il}\me_{lk} \;, \label{b31}\\[2mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(\widetilde{\yd})_{ik} &=&
(\yd)_{lk}\mq_{il}-(\lamd{k})_{li}\mhl{l}
+(\yd)_{ik}\mh{1}+(\yd)_{il} \md_{lk} \;, \\[2mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(\widetilde{\yu})_{ik} &=&
(\yu)_{lk}\mq_{il}+(\yu)_{ik}\mh{2}+(\yu)_{il}
\mup_{lk} \;, \\[2mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(\widetilde{\lame{k}})_{ij} &=&
(\lame{k})_{lj}\ml_{il}-(\ye)_{jk}\mlh{i}+(\lame{k})_{il}\ml_{jl}
+(\ye)_{ik}\mlh{j}+(\lame{l})_{ij}\me_{lk} \;,\\[2mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(\widetilde{\lamd{k}})_{ij} &=&
(\lamd{k})_{lj}\ml_{il}-(\yd)_{jk}\mlh{i}+(\lamd{k})_{il}\mq_{jl}
+(\lamd{l})_{ij}\md_{lk} \;, \\[2mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(\widetilde{\lamu{i}})_{jk} &=&
(\lamu{l})_{jk}\mup_{li}+(\lamu{i})_{lk}\md_{lj}+(\lamu{i})_{jl}\md_{lk} \;.
\label{b36}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{eqnarray}

Numericaly we would like to follow the following procedure :\\
a) Define the anomalous dimensions (\ref{gamll1}-\ref{mixedad})\\
b) Define $(\gamma_1)^i_j$ from (\ref{g1gamll1}-\ref{g1mixedad})\\
c) Define (\ref{b31}-\ref{b36}).\\
d) Plug (a,b,c) into (\ref{b7}-\ref{b22}).

This is much simbler from putting the explicit formulae of the
Appendix C below. 

 

\section{Explicit RGE's for the soft supersymmetric breaking terms}

We arrange here the explicit formulae of the RGEs. 
As a cross check we have calculated them by first using
the explicit formulae of the Ref.~\cite{mv} and second by
using the method described in Appendix B. We found agreement 
using both methods.  

Thus the RGE for the bilinear $\mu$ and $\kap_i$  terms of the
superpotential  parameters is given by
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
16 \pi^2 \frac{d \mu}{d t} \ &=& \ \mu \biggl [ 3 \Tr (\yu \yud)+
\Tr (3 \yd \ydd + \ye \yed)-  \frac{3}{5} g_1^2 - 3 g_2^2 
\biggr ] \nonumber \\
&-& \kap_p \biggl [ \lames{n} \ye + 3 \lamds{n} \yd \biggr ]_{pn} \;,
 \\[4mm]
16 \pi^2 \frac{d \kap_i}{d t} \ &=& \ \kap_i \biggl [ 3 \Tr ( \yu
\yud) - \frac{3}{5} g_1^2 - 3 g_2^2 \biggr ] \nonum \\  
&+& \kap_p \biggl [\ye \yed + \lame{n} \lamed{n} +3 \lamd{n} \lamdd{n}
\biggr ]_{ip}
- \mu \biggl [ \lame{n} \yes + 3 \lamd{n}{\yds} \biggr ]_{in} \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
Similar, the RGE's for the soft SUSY breaking bilinear terms can
be read from,
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
16 \pi^2 \frac{d \widetilde{B}}{d t} \ &=& \ \widetilde{B}
\biggl [ 3 \Tr (\yud \yu)+3 \Tr (\ydd \yd)+\Tr (\yed \ye) 
-\frac{3}{5} g_1^2 -3 g_2^2 \biggr ] \nonum \\
&+& \mu \biggl [ 6 \Tr(\yud \hu)+6 \Tr(\ydd \hd)+2 \Tr(\yed \he) +
\frac{6}{5} g_1^2 M_1 + 6 g_2^2 M_2 \biggr ] \nonum \\
&-& \widetilde{D}_l \biggl [ \lames{n} \ye + 3 \lamds{n} \yd \biggr ]_{ln}
- \kap_l \biggl [2 \lames{n} \he+6 \lamds{n} \hd \biggr ]_{ln} \; ,
\\[4mm]
16 \pi^2 \frac{d \widetilde{D}_i}{d t} \ &=& \ \widetilde{D}_i \biggl [
3 \Tr(\yu \yud)-\frac{3}{5}g_1^2 -3 g_2^2 \biggr ]
%\nonum \\
+ 
\kap_i \biggl [ 6 \Tr(\hu \yud)+ \frac{6}{5} g_1^2 M_1 +6 g_2^2 M_2 \biggr ]
\nonum \\
&+& \widetilde{D}_l \biggl [ \ye \yed + \lame{n} \lamed{n} + 3 \lamd{n}
\lamd{n} \biggr ]_{il} 
%\nonum \\
+ 2 \kap_l \biggl [ \he \yed + \lamhe{n} \lamed{n} +3 \lamhd{n} \lamdd{n}
\biggr ]_{il} \nonum \\
&-& 2 \mu \biggl [ \lamhe{n} \yes+3 \lamhd{n}\yds \biggr ]_{in}
-\widetilde{B} \biggl [ \lame{n} \yes+3 \lamd{n}\yds \biggr ]_{in} \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%
The RGE's for the soft SUSY trilinear couplings are given by
\begin{eqnarray}
16 \pi^2 \frac{d (\he)_{ij}}{d t} \ &=& \ (\he)_{il} \biggl [
2 (\yed \ye)_{lj}+ \Tr(\lamed{l}\lame{j}) \biggr ]
\nonum \\ 
&+& (\he)_{lj} \biggl [\ye \yed+\lame{n}\lamed{n}+3 \lamd{n}\lamdd{n}
\biggr ]_{il} \nonum \\
&+& (\he)_{ij} \biggl [ \Tr(\yed \ye)+3 \Tr(\ydd  \yd) -\frac{9}{5}g_1^2
-3 g_2^2 \biggr ]
 \nonum \\ &+& (\lamhe{j})_{il} \biggl [-\lames{n} \ye - 3 \lamds{n} \yd
\biggr ]_{ln} \nonum \\
&+& (\ye)_{il} \biggl [ 4 (\yed \he)_{lj}+2\Tr(\lamed{l}\lamhe{j}) \biggr ]
\nonum \\ 
&+&(\ye)_{lj} \biggl [ 2\he\yed+2\lamhe{n}\lamed{n}+6 \lamhd{n}\lamdd{n}
\biggr ]_{il} \nonum \\
&+& (\ye)_{ij} \biggl [ 2 \Tr(\yed \he)+ 6 \Tr(\ydd \hd)+\frac{18}{5}g_1^2
M_1+6 g_2^2M_2 \biggr ]\nonum \\ &+&
(\lame{j})_{il} \biggl [ -2 (\lames{n}\he)-6(\lamds{n}\hd)\biggr ]_{ln} \;,
\\[4mm]
16 \pi^2 \frac{d (\hd)_{ij}}{d t} \ &=& \ (\hd)_{il} \biggl [
2 (\ydd \yd)_{lj}+2 \Tr(\lamdd{l}\lamd{j})+2(\lamu{n}\lamud{n})_{jl} \biggr
] \nonum \\
%%%%%%%%%%%%%%%%%%%%%CHANGE%%%%%%%%%%%%%%%%
&+& (\hd)_{lj} \biggl [ \yd \ydd+\yu\yud \biggr ]_{il}+
(\hd)_{lj} \biggl [\lamdd{n}\lamd{n} \biggr ]_{li}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonum \\ 
&+& (\hd)_{ij} \biggl [ \Tr(\yed \ye)+3 \Tr(\ydd\yd) -\frac{7}{15}g_1^2
-3 g_2^2 -\frac{16}{3}g_3^2 \biggr ] \nonum \\
&+& (\lamhd{j})_{li}\biggl [(\lames{n}\ye)+3 (\lamds{n}\yd) \biggr ]_{ln}
\nonum \\ &+& (\yd)_{il} \biggl [ 4 (\ydd\hd)_{lj}+4\Tr(\lamdd{l}\lamhd{j})+
4(\lamhu{n}\lamud{n})_{jl} \biggr ]\nonum \\
%%%%%%%%%%%%%%%CHANGE%%%%%%%%%%%%%%%%%%%%%%
&+& (\yd)_{lj} \biggl [ 2 \hd \ydd+2 \hu\yud \biggr ]_{il}
 +(\yd)_{lj} \biggl [ 2 \lamdd{n}\lamhd{n} 
\biggr ]_{li} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonum \\ 
&+& (\yd)_{ij} \biggl [ 2 \Tr(\yed \he)+6 \Tr(\ydd \hd)
+\frac{14}{15}g_1^2 M_1+6 g_2^2M_2+\frac{32}{3}g_3^2M_3  \biggr ]
\nonum \\
&+& (\lamd{j})_{li} \biggl [ 2 (\lames{n}\he)+
6 (\lamds{n}\hd) \biggr ]_{ln} \;,
\\[4mm]
16 \pi^2 \frac{d (\hu)_{ij}}{d t} \ &=& \ 
(\hu)_{il} \biggl [ 2 (\yud \yu)_{lj}+\Tr(\lamud{l}\lamu{j} \biggr ]
\nonum \\
&+& (\hu)_{lj} \biggl [ (\yu \yud)_{il}+(\yd \ydd)_{il}+(\lamdd{n}
\lamd{n})_{li} \biggr ] \nonum \\
&+& (\hu)_{ij} \biggl [ 3 \Tr(\yud \yu) - \frac{13}{15}g_1^2-3 g_2^2 - 
\frac{16}{3}g_3^2 \biggr ] \nonum \\
&+& (\yu)_{il} \biggl [ 4 (\yud \hu)_{lj}+2 \Tr (\lamud{l}\lamhu{j}) \biggr ]
\nonum \\
&+& (\yu)_{lj} \biggl [ 2 (\hu \yud)_{il} +2 (\hd \ydd)_{il}+
2 (\lamdd{n}\lamhd{n})_{li} 
\biggr  ]
\nonum \\
&+& (\yu)_{ij} \biggl [ 6\Tr(\yud \hu)+\frac{26}{15}g_1^2M_1+6g_2^2M_2
+\frac{32}{3}g_3^2M_3 \biggr ] \;,
\\[4mm]
16 \pi^2 \frac{d (\lamhe{k})_{ij}}{d t} \ &=& \ 
(\lamhe{l})_{ij}\biggl [ 2 (\yed\ye)_{lk}
+ \Tr(\lamed{l}\lame{k})\biggr ] \nonum \\
&+&
(\he)_{jk}\biggl [\lame{n}\yes +3 \lamd{n}\yds \biggl ]_{in}
\nonum \\ &+& (\lamhe{k})_{jl} \biggl [ -\ye\yed -\lame{n}\lamed{n}-
3\lamd{n}\lamdd{n} \biggr ]_{il} \nonum \\
&+& (\lamhe{k})_{il}\biggl [ \ye \yed+\lame{n}\lamed{n}+3\lamd{n}\lamdd{n}
\biggr ]_{jl} \nonum \\ &+& (\he)_{ik} \biggl [-\lame{n}\yes-3\lamd{n}\yds
\biggr ]_{jn} \nonum \\
&+& (\lame{l})_{ij} \biggl [4 (\yed\he)_{lk}+2\Tr(\lamed{l}\lamhe{k}) \biggr ]
\nonum \\
&+&(\ye)_{jk} \biggl [2 \lamhe{n} \yes+6\lamhd{n}\yds \biggr ]_{in}
\nonum \\
&+& (\lame{k})_{jl} \biggl [-2 \he\yed-2\lamhe{n}\lamed{n}-6\lamhd{n}\lamdd{n}
\biggr ]_{il} 
\nonum \\
&+& (\lame{k})_{il} \biggl [2 \he\yed+2\lamhe{n}\lamed{n}+6\lamhd{n}\lamdd{n}
\biggr ]_{jl} 
\nonum \\
&+& (\ye)_{ik} \biggl [-2\lamhe{n}\yes-6\lamhd{n}\yds \biggr ]_{jn}
\nonum \\
&-&(\lamhe{k})_{ij} \biggl [\frac{9}{5}g_1^2+3g_2^2 \biggr] +
(\lame{k})_{ij}\biggl [\frac{18}{5}g_1^2M_1+6g_2^2M_2 \biggr ] \;,
\\[4mm]
16 \pi^2 \frac{d (\lamhd{k})_{ij}}{d t} \ &=& \ 
(\lamhd{l})_{ij}\biggl [2(\ydd\yd)_{lk}+2\Tr(\lamdd{l}\lamd{k})+
2(\lamud{n}\lamu{n})_{lk} \biggr ]
\nonum \\
&+& (\lamhd{k})_{lj} \biggl [\ye\yed+\lame{n}\lamed{n}+3\lamd{n}\lamdd{n}
\biggr ]_{il} %CHANGE
\nonum \\
&+& (\hd)_{jk} \biggl [ \lame{n}\yes+3 \lamd{n}\yds \biggr]_{in}
\nonum \\
&+& (\lamhd{k})_{il} \biggl [(\yd\ydd)_{jl}+(\yu\yud)_{jl}+
(\lamdd{n}\lamd{n})_{lj} \biggr ]
\nonum \\
&+& (\lamd{l})_{ij} \biggl [4 (\ydd\hd)_{lk}+4\Tr(\lamdd{l}\lamhd{k})+
4(\lamud{n}\lamhu{n})_{lk} \biggr ] 
\nonum \\
&+& (\lamd{k})_{lj} \biggl [2 \he\yed+2\lamhe{n}\lamed{n}+6 \lamhd{n}
\lamdd{n} \biggr ]_{il} %CHANGE
\nonum \\
&+& (\yd)_{jk} \biggl [2\lamhe{n}\yes+6\lamhd{n}\yds\biggr ]_{in}
\nonum \\
&+& (\lamd{k})_{il} \biggl [2(\hd\ydd)_{jl}+2(\hu\yud)_{jl}+
2(\lamdd{n}\lamhd{n})_{lj} \biggr ]
\nonum \\
&-& (\lamhd{k})_{ij} \biggl [\frac{7}{15}g_1^2+3g_2^2+\frac{16}{3}g_3^2 
\biggr ] 
\nonum \\
&+& (\lamd{k})_{ij} \biggl [
\frac{14}{15}g_1^2M_1+6g_2^2M_2+\frac{32}{3}g_3^2M_3
\biggr ]\;,
\\[4mm]
16 \pi^2 \frac{d (\lamhu{i})_{jk}}{d t} \ &=& \ 
(\lamhu{i})_{jl} \biggl [ 2 (\ydd \yd)_{lk}+2\Tr(\lamdd{l}\lamd{k})
+2(\lamud{m}\lamu{m})_{lk} \biggr]
\nonum \\
&+& (\lamhu{l})_{jk} \biggl [2(\yud\yu)_{li}+\Tr(\lamud{l}\lamu{i}) 
\biggr ]
\nonum \\
&+& (\lamhu{i})_{kl} \biggl [-2(\ydd\yd)_{lj}-2\Tr(\lamdd{l}\lamd{j})
-2(\lamud{n}\lamu{n})_{lj} \biggr ]
\nonum \\
&+&
(\lamu{i})_{jl} \biggl [ 4(\ydd\hd)_{lk}+4\Tr(\lamdd{l}\lamhd{k})+
4(\lamud{m}\lamhu{m})_{lk} \biggr ]
\nonum \\
&+&
(\lamu{l})_{jk} \biggl [4 (\yud \hu)_{li}+2\Tr(\lamud{l}\lamhu{i})\biggr ]
\nonum \\
&+&
(\lamu{i})_{kl} \biggl [ -4(\ydd\hd)_{lj}-4\Tr(\lamdd{l}\lamhd{j})-
4(\lamud{n}\lamhu{n})_{lj}\biggr]
\nonum \\
&-&
(\lamhu{i})_{jk} \biggl [\frac{4}{5}g_1^2+8g_3^2\biggr ]+
(\lamu{i})_{jk} \biggl [\frac{8}{5}g_1^2M_1+16g_3^2M_3 \biggr ].
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%

The RGE's for the gaugino masses are not affected by the 
R-parity violating couplings up to 1-loop. 

The RGE's for the SUSY soft breaking masses are given by,

%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
16\pi^2 \frac{d \me_{ij}}{dt} \ &=& \ 2 (\yed \ye)_{in} \me_{nj} +
\Tr (\lamed{i}\lame{n})\me_{nj} \nonum \\
&+& 2 \me_{in} (\yed \ye)_{nj} + \me_{in}\Tr(\lamed{n}\lame{j}) \nonum \\
&+& 4(\yed\ye)_{ij}\mh{1} + 4(\yed \lame{j})_{ir} \mlh{r} \nonum \\
&+& 4 \Tr[\ml \lamed{j}\lame{i}]+4\mlh{q} (\lamed{i}\ye)_{qj} \nonum \\
&+& 4 [\yed \ml \ye]_{ij}+4(\hed\he)_{ij}+2\Tr(\lamhed{i}\lamhe{j})\nonum \\
&-& \biggl (\frac{24}{5}|M_1|^2 g_1^2-\frac{6}{5}g_1^2 {\cal S} \biggr )
\delta_{ij}\;, \\[4mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d \ml_{ij}}{dt} \ &=& \
\ml_{in}(\ye\yed)_{nj}-\mlh{i}(\lames{q}\ye)_{jq} \nonum \\
&+& \ml_{in} (\lame{q}\lamed{q})_{nj}+(\ye\yed)_{in}\ml_{nj} \nonum \\
&-& (\lame{q}\yes)_{iq}\mhl{j}+(\lame{q}\lamed{q})_{in}\ml_{nj} \nonum \\
&+& 2(\ye\yed)_{ij}\mh{1}+2(\lame{p})_{ir}(\yed)_{pj}\mhl{r} \nonum \\
&-& 3(\lamd{q}\yds)_{iq}\mhl{j}
+3(\ml\lamd{q}\lamdd{q})_{ij} \nonum \\
&-& 3(\lamds{q}\yd)_{jq}\mlh{i}+3(\lamd{q}\lamdd{q}\ml)_{ij} \nonum \\
&+& 2(\ye)_{ip}(\lamed{p})_{qj}\mlh{q}+
2 (\lame{p})_{ir}\ml_{qr}(\lamed{p})_{qj}
\nonum \\
&+& 2 (\ye)_{ir}\me_{rq}(\yed)_{qj}+2(\lame{r})_{ip}\me_{rq}(\lamed{q})_{pj}
\nonum \\
&+& 6 (\lamd{r})_{ip}\md_{rq} (\lamdd{q})_{pj}+
6(\lamd{p})_{ir}\mq_{qr}(\lamdd{q})_{pj} \nonum \\
&+&
2(\he\hed)_{ij}+2(\lamhe{q}\lamhed{q})_{ji}+6(\lamhd{q}\lamhdd{q})_{ij}
 \nonum \\
&-&\biggl (\frac{6}{5}g_1^2 |M_1|^2 + 6 g_2^2|M_2|^2 +
\frac{3}{5}g_1^2 {\cal S} \biggr )\delta_{ij} \;,
\\[4mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2 \frac{d\mlh{i}}{dt} \ &=& \ 
(\lamed{q}\ye)_{iq}\mh{1}+(\lamed{q}\lame{q})_{in}\mhl{n} \nonum \\
&-& \mhl{n}(\ye\yed)_{ni}-3(\lamds{q}\yd)_{iq}\mh{1} \nonum \\
&+&3(\lamd{q}\lamdd{q})_{ni}\mhl{n}+ [ \Tr(\ye\yed)+3\Tr(\ydd\yd)]\mhl{i}
\nonum \\
&+& \ml_{ni} ( \lamed{q}\ye-3\lamds{q}\yd)_{nq} + 2(\lamed{p})_{iq}
\ml_{qr}(\ye)_{rp} \nonum \\
&+& 2 (\lamed{q}\ye)_{ir}\me_{rq}-6(\lamds{q}\yd)_{ir}\md_{rq}\nonum \\
&-& 6(\lamds{p})_{iq}\mq_{qr}(\yd)_{rp} - 
[ 2\lamhe{q}^*\he+6\lamhd{q}^*\hd ]_{iq} \;,
\\[4mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2\frac{d \mq_{ij}}{dt} \ &=& \ [\yd\ydd+\yu\yud]_{nj}\mq_{in} 
+(\lamdd{q}\lamd{q})_{jn}\mq_{in}\nonum \\
&+&[\yd\ydd+\yu\yud]_{in}\mq_{nj}+\mq_{nj}(\lamdd{q}\lamd{q})_{ni} \nonum \\
&+&2(\yd)_{ir}\md_{rq}(\ydd)_{qj}+2(\yd\ydd)_{ij}\mh{1} \nonum \\
&+&2(\lamdd{p})_{jq}\ml_{qr}(\lamd{p})_{ri}+2(\yu)_{ir}\mup_{rq}(\yud)_{qj}
\nonum \\
&+&2(\yu\yud)_{ij}\mh{2}+2(\lamdd{q}\lamd{r})_{ji}\md_{rq} \nonum \\
&-& 2 \mhl{l} (\lamd{k})_{li}(\ydd)_{kj}
-2 (\yd)_{iq} (\lamdd{q})_{jk}\mlh{k} \nonum \\
&+&2[\hd\hdd+\hu\hud]_{ij}+2(\lamhdd{q}\lamhd{q})_{ji} \nonum \\
&-&\biggl (\frac{2}{15}g_1^2|M_1|^2+6g_2^2|M_2|^2+\frac{32}{3}g_3^2|M_3|^2-
\frac{1}{5}g_1^2 {\cal S} \biggr )\delta_{ij} \;,
\\[4mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2\frac{d \md_{ij}}{dt} \ &=& 2(\ydd\yd)_{in}\md_{nj}
+2\Tr(\lamdd{i}\lamd{n})\md_{nj} \nonum \\
&+&2(\lamu{q}\lamud{q})_{ni}\md_{nj}+2(\ydd\yd)_{nj}\md_{in}\nonum \\
&+&2\Tr(\lamd{j}\lamdd{n})\md_{in}+2(\lamu{p}\lamud{p})_{jn}\md_{in}\nonum \\
&+&4(\ydd)_{iq}\mq_{qr}(\yd)_{rj}+4(\ydd)_{ip}(\yd)_{pj}\mh{1}\nonum \\
&+&4(\lamd{j}\lamdd{i})_{rq}\ml_{qr}+4(\lamdd{i}\lamd{j})_{qr}\mq_{qr}\nonum \\
&+&4(\lamud{p})_{qi}(\lamu{p})_{jr}\md_{rq}
+4(\lamu{l}\lamud{q})_{ij}\mup_{lq}\nonum \\
&+&4(\hdd\hd)_{ij}+4\Tr(\lamhdd{i}\lamhd{j})
+4(\lamhu{p}\lamhud{p})_{ji} \nonum \\
&-&4 (\lamd{i}\yds)_{lj}\mhl{l}-4(\lamds{j}\yd)_{li}\mlh{l} \nonum \\
&-&\biggl (\frac{8}{15}g_1^2|M_1|^2+\frac{32}{3}g_3^2|M_3|^2
-\frac{2}{5}g_1^2{\cal S} \biggr )\delta_{ij} \;,
\\[4mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2\frac{d \mup_{ij}}{dt} \ &=& \
2(\yud\yu)_{in}\mup_{nj}+\Tr(\lamud{i}\lamu{n})\mup_{nj} \nonum \\
&+&2(\yud\yu)_{nj}\mup_{in}+\Tr(\lamu{j}\lamud{n})\mup_{in}\nonum \\
&+&4(\yud)_{iq}\mq_{qr}(\yu)_{rj}+4(\yud\yu)_{ij}\mh{2}\nonum \\
&+&4(\lamud{i}\lamu{j})_{qr}\md_{rq}+4(\hud\hu)_{ij}+2\Tr(\lamhud{i}
\lamhu{j})\nonum \\
&-&\biggl (\frac{32}{15}g_1^2|M_1|^2+\frac{32}{3}g_3^2|M_3|^2
+\frac{4}{5}g_1^2{\cal S} \biggr )\delta_{ij} \;,
\\[4mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2\frac{d\mh{1}}{dt} \ &=& \
2\Tr(\yed\ye)\mh{1}+6\Tr(\ydd\yd)\mh{1}+(\yed\lame{q})_{qn}\mhl{n} \nonum \\
&+&3(\ydd\lamd{q})_{qn}\mhl{n}+3(\lamdd{q}\yd)_{nq}\mlh{n} \nonum \\
&+&(\lamed{q}\ye)_{nq}\mlh{n}+2(\yed\ye)_{qr}\me_{rq} \nonum \\
&+&2(\ye\yed)_{rq}\ml_{qr}+6(\ydd\yd)_{qr}\md_{rq}\nonum \\
&+&6(\yd\ydd)_{rq}\mq_{qr}
+2\Tr(\hed\he)+6\Tr(\hdd\hd) \nonum \\
&-&\biggl (\frac{6}{5}g_1^2|M_1|^2+6g_2^2|M_2|^2+\frac{3}{5}g_1^2{\cal S}
\biggr ) \delta_{ij} \;,
\\[4mm]
%%%%%%%%%%%%%%%%%%%%%%%%
16\pi^2\frac{d\mh{2}}{dt} \ &=& \
6\Tr(\yud\yu)\mh{2}+6(\yud\yu)_{qr}\mup_{rq} \nonum \\
&+& 6(\yu\yud)_{rq}\mq_{qr}+6\Tr(\hud\hu) \nonum \\
&-&\biggl (\frac{6}{5}g_1^2|M_1|^2+6g_2^2|M_2|^2-\frac{3}{5}g_1^2{\cal S}
\biggr )\delta_{ij} \;,
\end{eqnarray} 
where 
%%%%%%%%%%%%%%%%
\begin{eqnarray}
\mhl{i}=\mlh{i}^* \;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%
and 
%%%%%%%%%%%%%%%%
\begin{eqnarray}
{\cal S} = m_{H_2}^2-m_{H_1}^2+\Tr[{\bf m_{\tilde{Q}}}^2-
{\bf m_{\tilde{L}}}^2-2{\bf m_{\tilde{U}}}^2+{\bf m_{\tilde{D}}}^2
+{\bf m_{\tilde{E}}}^2] \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%

\vspace*{2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%




\begin{thebibliography}{99}

\bibitem{ir}
L.~E.~Ibanez and G.~G.~Ross,
%``Discrete gauge symmetries and the origin of baryon and lepton number conservation in supersymmetric versions of the standard model,''
Nucl.\ Phys.\ B {\bf 368}, 3 (1992).
%%CITATION = NUPHA,B368,3;%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%CITATION = HEP-PH 9610540;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{jones}
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