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%                Neutrinos and Supersymmetry                         %
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%                file=book03c7.tex                                   %
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%                Last changed 23/8/2003 by JCR                       %
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\setcounter{chapter}{6}

\ignore{

I have put here my notes on the epsilon-model. I have made them
compatible with the notations of the SUSY chapter, avoiding
repetitions. 

}

\chapter{Bilinear Model for  R Parity Violation }

\section{The Model} 
 
In what follows we use the conventions of
Refs.~\cite{haber:1985rc,akeroyd:1998iq}. 
The supersymmetric Lagrangian is specified by the superpotential $W$ 
given by,
% 
\begin{equation}  
W^{\hbox{\scriptsize BRpV}}= W^{\hbox{\scriptsize MSSM}} +
\varepsilon_{ab} \epsilon_i\widehat L_i^a\widehat H_2^b 
\label{eq:WRPV} 
\end{equation} 
% 
where $W^{\hbox{\scriptsize MSSM}}$ was given in Eq.(\ref{eq:WMSSM}),
and $\epsilon_i$ are parameters with units of mass. The last term in  
Eq.~(\ref{eq:WRPV}) is the only $R$--parity violating term. 

Correspondingly the supersymmetry breaking is parameterized with a set of soft 
supersymmetry breaking terms,
%
\begin{eqnarray} 
\label{eq:VsoftRPV}
V_{\hbox{\scriptsize SB}}&=&
V^{\hbox{\scriptsize MSSM}}_{\hbox{\scriptsize SB}}
+\varepsilon_{ab} B_i\epsilon_i\widetilde L_i^a H_2^b\\
&\equiv&
V^{\hbox{\scriptsize MSSM}}_{\hbox{\scriptsize SB}}
+V^{\hbox{\scriptsize BRpV}}_{\hbox{\scriptsize SB}}
\end{eqnarray}
% 
and again, the last term in Eq.~(\ref{eq:VsoftRPV}) is the only R--parity 
violating term. 

The electroweak symmetry is broken when the two Higgs doublets $H_d$
and $H_u$, and the neutral component of the slepton doublets
$\widetilde L^1_i$ acquire vacuum expectation values. We introduce the
notation:
%
\begin{equation} 
H_d={{H^0_d}\choose{H^-_d}}\,,\qquad
H_u={{H^+_u}\choose{H^0_u}}\,,\qquad \widetilde L_i={{\tilde
L^0_i}\choose{\tilde\ell^-_i}}\,,
\label{eq:shiftdoub} 
\end{equation} 
%
where we shift the neutral fields with non--zero vevs as
%
\begin{equation}
H_d^0\equiv{1\over{\sqrt{2}}}[\sigma^0_d+v_d+i\varphi^0_d]\,,\quad
H_u^0\equiv{1\over{\sqrt{2}}}[\sigma^0_u+v_u+i\varphi^0_u]\,,\quad
\tilde{L}_i^0\equiv{1\over{\sqrt{2}}}[\tilde\nu^R_i+v_i+i\tilde\nu^I_i]\,.
\label{shiftedfields}
\end{equation}
%
Note that the $W$ boson acquires a mass $m_W^2=\quarter g^2v^2$, where
$v^2\equiv v_d^2 + v_u^2 + v_1^2+ v_2^2+ v_3^2 \simeq (246 \;
\rm{GeV})^2$.  We introduce the following notation in spherical
coordinates for the vacuum expectation values:
% 
\begin{eqnarray} 
v_d&=&v\sin\theta_1\sin\theta_2\sin\theta_3\cos\beta\nonumber\\
v_u&=&v\sin\theta_1\sin\theta_2\sin\theta_3\sin\beta\nonumber\\
v_3&=&v\sin\theta_1\sin\theta_2\cos\theta_3\label{eq:vevs}\\
v_2&=&v\sin\theta_1\cos\theta_2\nonumber\\
v_1&=&v\cos\theta_1\nonumber
\end{eqnarray} 
% 
which preserves the MSSM definition $\tan\beta=v_u/v_d$. In the MSSM
limit, where $\epsilon_i=v_i=0$, the angles $\theta_i$ are equal to
$\pi/2$.  In addition to the above MSSM parameters, our model contains
nine new parameters, $\epsilon_i$, $v_i$ and $B_i$. The three vevs are
determined by the one--loop tadpole equations, and we will assume
universality of the $B$--terms, $B=B_i$ at the unification scale.
Therefore, the only new and free parameters can be chosen as the
$\epsilon_i$.

\section{The Scalar Potential}
\label{ThePot}

The electroweak symmetry is broken when the Higgs and lepton fields
acquire non--zero vevs. These are calculated via the minimization of
the effective potential or, in the diagramatic method, via the tadpole
equations.  The full scalar potential at tree level is
% 
\begin{equation} 
V_{\hbox{\scriptsize Total}}^0 
= \sum_i \left| { \partial W \over \partial z_i} \right|^2
+ V_D + V_{\hbox{\scriptsize SB}}^{\hbox{\scriptsize MSSM}} 
+V_{\hbox{\scriptsize SB}}^{\hbox{\scriptsize BRpV}} 
\label{V}
\end{equation} 
% 
where $z_i$ is any one of the scalar fields in the superpotential in
Eq.~(\ref{eq:WRPV}), $V_D$ are the $D$-terms, and
$V^{\hbox{\scriptsize BRpV}}_{\hbox{\scriptsize SB}}$ 
is given in Eq.~(\ref{eq:VsoftRPV}).

The tree level scalar potential contains the following linear terms
%
\begin{equation}
V_{linear}^0=t_d^0\sigma^0_d+t_u^0\sigma^0_u+t_1^0\tilde\nu^R_1
+t_2^0\tilde\nu^R_2+t_3^0\tilde\nu^R_3\,,
\label{eq:Vlinear}
\end{equation}
%
where the different $t^0$ are the tadpoles at tree level. They are
given by
%
\begin{eqnarray}
t_d^0&=&\Big(m_{H_d}^2+\mu^2\Big)v_d+v_dD-\mu\Big(Bv_u+v_i\epsilon_i\Big)
\nonumber\\ t_u^0&=&-B\mu
v_d+\Big(m_{H_u}^2+\mu^2\Big)v_u-v_uD+v_iB_i\epsilon_i +v_u\epsilon^2
\nonumber\\ t_1^0&=&v_1D+\epsilon_1\Big(-\mu
v_d+v_uB_1+v_i\epsilon_i\Big)+\half\Big( v_iM^2_{Li1}+M^2_{L
1i}v_i\Big)
\label{eq:tadpolesBRpV}\\
t_2^0&=&v_2D+\epsilon_2\Big(-\mu
v_d+v_uB_2+v_i\epsilon_i\Big)+\half\Big(
v_iM^2_{Li2}+M^2_{L2i}v_i\Big) \nonumber\\
t_3^0&=&v_3D+\epsilon_3\Big(-\mu v_d+v_uB_3+v_i\epsilon_i\Big)
+\half\Big(v_iM^2_{Li3}+M^2_{L3i}v_i\Big) \nonumber
\end{eqnarray}
%
where we have defined
$D=\eighth(g^2+g'^2)(v_1^2+v_2^2+v_3^2+v_d^2-v_u^2)$ and
$\epsilon^2=\epsilon_1^2+\epsilon_2^2+\epsilon_3^2$. A repeated index
$i$ in Eq.~(\ref{eq:tadpolesBRpV}) implies summation over $i=1,2,3$. The
five tree level tadpoles $t_{\alpha}^0$ are equal to zero at the
minimum of the tree level potential, and from there one can determine
the five tree level vacuum expectation values.

It is well known that in order to find reliable results for the
electroweak symmetry breaking it is necessary to include the one--loop
radiative corrections. The full scalar potential at one loop level,
called effective potential, is
%
\begin{equation} V_{total} = V_{total}^0 + V_{RC} \end{equation}
%
where $V_{RC}$ include the quantum corrections. In our approach we use
the diagramatic method, which incorporates the radiative corrections
through the one--loop corrected tadpole equations. The one loop
tadpoles are
%
\begin{equation}
t_{\alpha}=t^0_{\alpha} -\delta t^{\overline{DR}}_{\alpha}
+T_{\alpha}(Q)=t^0_{\alpha} +\widetilde T^{\overline{DR}} _{\alpha}(Q)
\label{tadpoles}
\end{equation}
%
where $\alpha=d,u,1,2,3$ and $\widetilde T^{\overline{DR}}
_{\alpha}(Q) \equiv-\delta t^{\overline{MS}}_{\alpha}+T_{\alpha}(Q)$
are the finite one loop tadpoles. At the minimum of the potential we
have $t_{\alpha}=0$, and the vevs calculated from these equations are
the renormalized vevs.



\section{Mass Matrices in the BRpV Model} 
\label{MassMatrices}

\subsection{Scalar Mass Matrices}

\subsubsection{Charged Scalars}
 
The mass matrix of the charged scalar sector follows from the
quadratic terms in the scalar potential
% 
\begin{equation} 
V_{quadratic}= S'^- \bold{M_{S^{\pm}}^2} S'^+
\label{eq:Vquadratic} 
\end{equation} 
%
where the unrotated charged scalars are $S'^+=(H_d^+, H_u^+,
\widetilde{e}_L^+, \widetilde{\mu}_L^+
\widetilde{\tau}_L^+,\widetilde{e}_R^+, \widetilde{\mu}_R^+
\widetilde{\tau}_R^+)$.  For convenience we will divide this
$(8\times8)$ matrix into blocks in the following way:
\begin{equation} 
\bold{M_{S^{\pm}}^2}=\left[\matrix{ {\bold M_{HH}^2} & {\bold
M_{H\tilde\ell}^{2T}} \cr {\bold M_{H\tilde\ell}^2} & {\bold
M_{\tilde\ell\tilde\ell}^2} }\right] + \xi m_W^2 \left[\matrix{ {\bold
M_{A}^2} & {\bold M_{B}^{2T}} \cr {\bold M_{B}^2} & {\bold M_{C}^2}
}\right]
\label{eq:subdivM} 
\end{equation} 
% 
where the charged Higgs block is
% 
\begin{equation} 
{\bold M_{HH}^2}
=
\left[\matrix{ B\mu{{v_u}\over{v_d}}+\quarter
g^2(v_2^2-\sum_{i=1}^3 v_i^2) +{{t_d}\over{v_d}} & B\mu+\quarter g^2
v_d v_u \cr +\mu \sum_{i=1}^3 \epsilon_i {{v_i}\over{v_d}} +\half
\sum_{i,j=1}^3 v_i \left(h_E h_E^{\dagger}\right)_{ij} v_j & \cr
\vb{30} B\mu+\quarter g^2v_d v_u & B\mu{{v_d}\over{v_u}}+\quarter
g^2(v_d^2+\sum_{i=1}^3 v_i^2) \cr &- \sum_{i=1}^3 B_i\epsilon_i
{{v_i}\over{v_u}} +{{t_u}\over{v_u}} }\right]
%\nonumber 
\label{eq:subMHH} 
\end{equation} 
% 
This matrix reduces to the usual charged Higgs mass matrix in the
MSSM, Eq.~(\ref{eq:chargedHiggs}), when we set $v_i=\epsilon_i=0$.
The slepton block is given by,
% 
\begin{eqnarray} 
\label{eq:subtautau}
{\bold M_{\tilde\ell\tilde\ell}^2}&=& \left[\matrix{ {\bold M^2_{LL}}&
{\bold M^2_{LR}}\cr {\bold M^2_{RL}}& {\bold M^2_{RR}} }\right]
\end{eqnarray} 
%
where
%
\begin{eqnarray} \left({\bold M^2_{LL}}\right)_{ij} &=&\half v_d^2 \left(h_E^*
h_E^T\right)_{ij} +\quarter g^2\left(-\sum_{k=1}^3
v_k^2-v_d^2+v_u^2\right)\delta_{ij} +\quarter g^2 v_i v_j
-\frac{v_u}{v_i} B_i \epsilon_i \delta_{ij} +\frac{t_i}{v_i}
\delta_{ij}\cr \vb{18} &&+\mu \frac{v_d}{v_i}\epsilon_i \delta_{ij}
-\epsilon_i \left(\sum_{k=1}^3 \frac{v_k}{v_i} \epsilon_k\right)
\delta_{ij} +\epsilon_i \epsilon_j +M^2_{L ji}\cr \vb{18} &&-\half
\sum_{k=1}^3 \frac{v_k}{v_i} \left(M^2_{L ik}+M^2_{L
ki}\right)\delta_{ij} \end{eqnarray}
%
\begin{eqnarray}
  \label{eq:MchLR}
  {\bold M^2_{LR}} &=&\frac{1}{\sqrt{2}}\, \left(v_d A_E -\mu v_u
h_E \right)\\
{\bold M^2_{RL}}&=&\left({\bold M^2_{LR}}\right)^{\dagger} 
\end{eqnarray}
%
\begin{eqnarray} 
\left({\bold M_{RR}^2}\right)_{ij}&=& \quarter g'^2\left(-\sum_{k=1}^3
v_k^2-v_d^2+v_u^2\right)\delta_{ij} +\half\, v_d^2 \left( h_E^T
h_E^*\right)_{ij} \cr \vb{18} &&+\left(\sum_{k=1}^3
\left(h_E^T\right)_{ik} v_k\right) \left(\sum_{s=1}^3
\left(h_E^*\right)_{sj} v_s\right) +M^2_{R ji}
\end{eqnarray} 
% 
We recover the usual slepton mass matrices of the MSSM,
Eq.~(\ref{eq:MassSleptons}), by replacing
$v_i=\epsilon_i=0$ (note that we need to replace the expression of the
tadpole $t_i$ in Eq.~(\ref{eq:tadpolesBRpV}) before taking the limit).
The mixing between the charged Higgs sector and the slepton sector is
given by the following $6\times2$ block (repeated indices are not
summed unless an explicit sum appears):
% 
\begin{equation} 
\phantom{\hskip -3mm}
{\bold M_{H\tilde\ell}^2}\hskip 0mm=\hskip -1mm 
\left[\hskip -1mm
\matrix{ \ds -\mu\epsilon_i-\half v_d
\sum_{k=1}^3 \left(h_E^* h_E^T\right)_{ik} v_k +\quarter g^2v_d v_i &
-B_i\epsilon_i+\quarter g^2v_uv_i \cr \vb{20} \ds -\sqrthalf v_u
\sum_{k=1}^3\, \left(h_E^T\right)_{ik}\epsilon_k -\sqrthalf
\sum_{k=1}^3\, \left(A_E^T\right)_{ik}v_k &\ds
 -\sqrthalf \sum_{k=1}^3 \left(h_E^T\right)_{ik}(\mu
v_k+\epsilon_kv_d) }\right]
\label{eq:subHtau} 
\end{equation} 
% 
and as expected, this mixing vanishes in the limit $v_i=\epsilon_i=0$.
The charged scalar mass matrix in Eq.~(\ref{eq:subdivM}), after
setting $t_u=t_d=t_i=0$, has determinant equal to zero for $\xi=0$,
since one of the eigenvectors corresponds to the charged Goldstone
boson with zero eigenvalue.

\ni For our one loop calculations one has to add the gauge fixing.
The part of the mass matrix in Eq.~(\ref{eq:subdivM}) that comes from
the gauge fixing reads for the $(2\times2)$ $A$ block

\begin{equation} 
{\bold M_{A}^2}= \left[\matrix{ \ds \frac{v_d^2}{v^2}&\ds
\frac{-v_uv_d}{v^2}\cr \vb{18} \ds \frac{-v_uv_d}{v^2}&\ds
\frac{v_u^2}{v^2} }\right]
\end{equation} 

\ni
for the $(6\times2)$ $B$ and the $(6\times6)$ $C$ blocks

\begin{equation} 
{\bold M_{B}^2}= \left[\matrix{ \ds \frac{v_iv_d}{v^2}&\ds
\frac{-v_iv_u}{v^2}\cr \vb{18} 0 & 0 }\right]
\hskip 10mm ; \hskip 10mm {\bold M_{C}^2}= \left[\matrix{ {\bold
M_{D}^2}&0\cr \vb{18} 0 & 0 }\right]
\end{equation} 

\ni
where the $(3\times 3)$ $D$ block is

\begin{equation} 
{\bold M_{D}^2}= \left[\matrix{ \ds\frac{v_1^2}{v^2}&\ds
\frac{v_1v_2}{v^2}&\ds \frac{v_1v_3}{v^2}\cr \vb{18}
\ds\frac{v_2v_1}{v^2}&\ds \frac{v_2^2}{v^2}&\ds \frac{v_2v_3}{v^2}\cr
\vb{18} \ds\frac{v_3v_1}{v^2}&\ds \frac{v_2v_3}{v^2}&\ds
\frac{v_3^2}{v^2} }\right]
\end{equation} 
% 
The charged scalar mass matrices are diagonalized by the following
rotation matrices,
% 
\begin{equation} 
S_i^{\pm}= {\bold R}^{S^{\pm}}_{ij} S^{\pm'}_j
\end{equation} 
%
with the eigenvalues $\rm{diag}(m^2_{S_1},\ldots,m^2_{S_8})= {\bold
R^{S^{\pm}}}\, {\bold M_{S^{\pm}}^2}\, \left({\bold
R^{S^{\pm}}}\right)^{\dagger}$.


\subsubsection{CP--Even Neutral Scalars}

\ni The quadratic scalar potential includes
% 
\begin{equation} 
V_{quadratic}= \half[\sigma^0_d,\sigma^0_u,\tilde\nu_{i}^R] {\bold
M^2_{S^0}}\left[\matrix{ \sigma^0_d \cr \sigma^0_u \cr \tilde\nu_{i}^R
}\right]+ \cdots
\label{eq:NeutScalLag} 
\end{equation} 
% 
where the neutral CP-even scalar sector mass matrix in
Eq.~(\ref{eq:NeutScalLag}) is given by
%
\begin{equation} {\bold M_{S^0}^2} = \left[\matrix{ {\bold M_{SS}^2} & {\bold
M_{S\widetilde{\nu}_R}^2} \cr \vb{16} {\bold
M_{S\widetilde{\nu}_R}^2}^T& {\bold M_{\widetilde{\nu}_R
\widetilde{\nu}_R}^2} }\right] \end{equation}
%
where
%
\begin{equation} {
\hskip -3mm
\bold M_{SS}^2}= 
\left[\hskip -1mm
\matrix{ \ds
B\mu{{v_u}\over{v_d}}+\quarter g_Z^2v_d^2+\mu \sum_{k=1}^3\epsilon_k
{{v_k}\over{v_1}}+{{t_d}\over{v_d}} & -B\mu-\quarter g^2_Zv_dv_u \cr
\vb{18} -B\mu-\quarter g^2_Zv_dv_u &\ds
 B\mu{{v_d}\over{v_u}}+\quarter g^2_Zv_u^2-\sum_{k=1}^3\,
B_k\epsilon_k {{v_k}\over{v_2}}+{{t_u}\over{v_u}} }\right]
\label{eq:MSS}
\end{equation}
%
\begin{equation} {\bold M_{S\widetilde{\nu}_R}^2}= \left[\matrix{
-\mu\epsilon_i+\quarter g^2_Zv_dv_i \cr \vb{18} B_i\epsilon_i-\quarter
g^2_Zv_uv_i }\right] \nonumber \end{equation}
%
and
%
\begin{eqnarray} \left({\bold M_{\widetilde{\nu}_R
\widetilde{\nu}_R}^2}\right)_{ij}
\hskip -10pt&=\hskip -10pt& \left(\mu\epsilon_i{{v_d}\over{v_i}} -
B_i\epsilon_i{{v_u}\over{v_i}} -\epsilon_i \sum_{k=1}^3 \epsilon_k
\frac{v_k}{v_i} - \half \sum_{k=1}^3 \frac{v_k}{v_i} \left(M^2_{L
ik}+M^2_{L ki}\right) + {{t_i}\over{v_i}}\right)\delta_{ij} +\quarter
g^2_Zv_iv_j \cr \vb{20} &&+ \epsilon_i \epsilon_j +\half \left(M^2_{L
ij}+M^2_{L ji}\right) \end{eqnarray}
%
where we have defined $g_Z^2\equiv g^2+g'^2$. In the upper--left
$2\times2$ block, in the limit $v_i=\epsilon_i=0$, the reader can
recognize the MSSM mass matrix corresponding to the CP--even neutral
Higgs sector, Eq.~(\ref{eq:PseScalM}).  To define the rotation matrices let us define the
unrotated fields by
%
\begin{equation} S'^0=(\sigma_d^0,\sigma_u^0,\widetilde{\nu}_1^R,
\widetilde{\nu}_2^R, \widetilde{\nu}_3^R) \end{equation}
%
Then the mass eigenstates are $S^0_i$ given by
%
% 
\begin{equation} 
\label{eq:weakeigenrotS0} 
S^0_i= {\bold R}^{S^0}_{ij} S'^0_j
\end{equation} 
%
with the eigenvalues $\rm{diag}(m^2_{S_1},\ldots,m^2_{S_5})= {\bold
R^{S^0}}\, {\bold M_{S^0}^2}\, \left({\bold R}^{S^0}\right)^{\dagger}$.




\subsubsection{CP--Odd Neutral Scalars}

\ni The quadratic scalar potential includes
% 
\begin{equation} 
V_{quadratic}=\half[\varphi^0_d,\varphi^0_u,\tilde\nu_{i}^I] {\bold
M^2_{P^0}}\left[\matrix{ \varphi^0_d \cr \varphi^0_u \cr
\tilde\nu_{i}^I }\right] + \cdots
\label{eq:PseudoScalLag} 
\end{equation} 
% 
where the CP-odd neutral scalar mass matrix is
%
\begin{equation} {\bold M_{P^0}^2} = \left[\matrix{ {\bold M_{PP}^2} & {\bold
M_{P\widetilde{\nu}_I}^2} \cr \vb{16} {\bold
M_{P\widetilde{\nu}_I}^2}^T& {\bold M_{\widetilde{\nu}_I
\widetilde{\nu}_I}^2} }\right] + \xi m_Z^2 \left[\matrix{ {\bold
M_{E}^2} & {\bold M_{F}^{2T}} \cr {\bold M_{F}^2} & {\bold M_{G}^2}
}\right]
\label{eq:MP0}
\end{equation}
%
where

\begin{equation} {\bold M_{PP}^2}= \left[\matrix{ \ds
B\mu{{v_u}\over{v_d}}+\mu\sum_{k=1}^3
\epsilon_k{{v_k}\over{v_d}}+{{t_d}\over{v_d}} & B\mu \cr \vb{20} B\mu
& \ds B\mu{{v_d}\over{v_u}}-\sum_{k=1}^3
B_k\epsilon_k{{v_k}\over{v_u}}+{{t_u}\over{v_u}} }\right]
\label{eq:MPP}
\end{equation}

\begin{equation} {\bold M_{P\widetilde{\nu}_I}^2}= \left[\matrix{ -\mu\epsilon_i
\cr \vb{18} -B_i\epsilon_i }\right] \nonumber \end{equation}

\ni and


\begin{eqnarray} \left({\bold M_{\widetilde{\nu}_I
\widetilde{\nu}_I}^2}\right)_{ij}&=&
\left(\mu\epsilon_i{{v_d}\over{v_i}} - B_i\epsilon_i{{v_u}\over{v_i}}
-\epsilon_i \sum_{k=1}^3 \epsilon_k \frac{v_k}{v_i} - \half
\sum_{k=1}^3 \frac{v_k}{v_i} \left(M^2_{L ik}+M^2_{L ki}\right) +
{{t_i}\over{v_i}}\right)\delta_{ij} \cr \vb{20} &&+ \epsilon_i
\epsilon_j +\half \left(M^2_{L ij}+M^2_{L ji}\right) \end{eqnarray}



Finally the part of the mass matrix in Eq.~(\ref{eq:MP0}) that comes
from the gauge fixing reads for the $(2\times2)$ $E$ block
%
\begin{equation} 
{\bold M_{E}^2}= \left[\matrix{ \ds \frac{v_d^2}{v^2}&\ds
\frac{-v_uv_d}{v^2}\cr \vb{18} \ds \frac{-v_uv_d}{v^2}&\ds
\frac{v_u^2}{v^2} }\right]
\end{equation} 
%
for the $(3\times2)$ $F$ block
%
\begin{equation} 
{\bold M_{F}^2}= \left[\matrix{ \ds \frac{v_iv_d}{v^2}&\ds
\frac{-v_iv_u}{v^2} }\right]
\end{equation} 
%
and for the $(3\times3)$ $G$ block
%
\begin{equation} 
{\bold M_{G}^2}= \left[\matrix{ \ds\frac{v_1^2}{v^2}&\ds
\frac{v_1v_2}{v^2}&\ds \frac{v_1v_3}{v^2}\cr \vb{18}
\ds\frac{v_2v_1}{v^2}&\ds \frac{v_2^2}{v^2}&\ds \frac{v_2v_3}{v^2}\cr
\vb{18} \ds\frac{v_3v_1}{v^2}&\ds \frac{v_3v_2}{v^2}&\ds
\frac{v_3^2}{v^2}\cr }\right]
\end{equation} 
% 
The charged pseudo--scalar mass matrices are diagonalized by the
following rotation matrices,
% 
\begin{equation}
\label{eq:weakeigenrotP0} 
P_i= {\bold R}^{P^0}_{ij} P'_j
\end{equation} 
%
with the eigenvalues $\rm{diag}(m^2_{A_1},\ldots,m^2_{A_5})= {\bold
R}^{P^0}\, {\bold M_{P^0}^2}\, \left({\bold R}^{P^0}\right)^{\dagger}$,
%
where the unrotated fields are
%
\begin{equation} P'^0=(\varphi_d^0,\varphi_u^0,\widetilde{\nu}_1^I,
\widetilde{\nu}_2^I, \widetilde{\nu}_3^I) \end{equation}

\subsubsection{Squark Mass Matrices}


In the unrotated basis $\widetilde{u}'_i=(\widetilde{u}_{Li},
\widetilde{u}_{Ri}^*)$ and $\widetilde{d}'_i=(\widetilde{d}_{Li},
\widetilde{d}^*_{Ri})$ we get
%
\begin{equation} V_{quadratic}=\half \widetilde{u}'^{\dagger}\,
\bold{M_{\widetilde{u}}}^2\, \widetilde{u}' +\half
\widetilde{d}'^{\dagger}\, \bold{M_{\widetilde{d}}}^2\, \widetilde{d}'
\end{equation}
%
where
%
\begin{equation} \bold{M_{\widetilde{q}}}^2= \left( \matrix{
\bold{M^2_{\widetilde{q}LL}}&\bold{M^2_{\widetilde{q}LR}}\cr \vb{18}
\bold{M^2_{\widetilde{q}RL}}&\bold{M^2_{\widetilde{q}RR}} } \right)
\label{eq:MassSquarks}
\end{equation}
%
with $\widetilde{q}=(\widetilde{u},\widetilde{d})$. The blocks are
different for up and down type squarks. We have
%
\begin{eqnarray} \bold{M^2_{\widetilde{u}LL}}&=&\smallfrac{1}{2}\, v_u^2\, h_U^*
h_U^T + M^2_Q +\smallfrac{1}{6}\, (4 m_W^2 -m_Z^2)\,
\frac{1}{v^2}\,
\left(v_d^2-v_u^2+\sum_i v_i^2\right) \cr
\vb{18} \bold{M^2_{\widetilde{u}RR}}&=& \smallfrac{1}{2}\, v_u^2\,
h_U^T h_U^* + M^2_U + \smallfrac{2}{3}( m_Z^2 -m_W^2) \,
\frac{1}{v^2}\,
\left(v_d^2-v_u^2+\sum_i v_i^2\right) \cr
\vb{18} \bold{M^2_{\widetilde{u}LR}}&=&\frac{v_u}{\sqrt{2}}\, A_U^*
-\mu \frac{v_d}{\sqrt{2}}\, h_U^* + \sum_{i=1}^3 \frac{
v_i}{\sqrt{2}}\, \epsilon_i\, h_U^* \cr \vb{18}
\bold{M^2_{\widetilde{u}RL}}&=&\bold{M^2_{\widetilde{u}LR}}^{\dagger}
\label{eq:M2squ}\end{eqnarray}
%
and
%
\begin{eqnarray} \bold{M^2_{\widetilde{d}LL}}&=&\frac{1}{2}\, v_d^2\, h_D^* h_D^T
+ M^2_Q -\smallfrac{1}{6}\, (2 m_W^2 + m_Z^2) \,
\frac{1}{v^2}\,
\left(v_d^2-v_u^2+\sum_i v_i^2\right) \cr \vb{18}
\bold{M^2_{\widetilde{d}RR}}&=& \smallfrac{1}{2}\, v_d^2\, h_D^T h_D^*
+ M^2_D - \smallfrac{1}{3}( m_Z^2 -m_W^2) \,
\frac{1}{v^2}\,
\left(v_d^2-v_u^2+\sum_i v_i^2\right)  \cr \vb{18}
\bold{M^2_{\widetilde{d}LR}}&=&\frac{v_d}{\sqrt{2}}\, A_D^* -\mu
\frac{v_u}{\sqrt{2}}\, h_D^* \cr \vb{18}
\bold{M^2_{\widetilde{d}RL}}&=&\bold{M^2_{\widetilde{d}LR}}^{\dagger}
\label{eq:M2sqd}
\end{eqnarray}
%
We define the mass eigenstates
%
\begin{equation} \widetilde{q}=\bold{R^{\widetilde{q}}}\, \widetilde{q}' \end{equation}
%
which implies
%
\begin{equation}
\label{eq:RotSquarks} 
\widetilde{q}'_i=\bold{R^{\widetilde{q}}}_{ji}^{\ *} \,
\widetilde{q}_j \end{equation}
%
The rotation matrices are obtained from
%
\begin{equation} 
\bold{R^{\widetilde{q}}}^{\ \dagger}
\left(\bold{M_{\widetilde{q}}^{diag}}\right)^2
\bold{R^{\widetilde{q}}}= \bold{M_{\widetilde{q}}}^2 .
\end{equation}
%

\subsection{Chargino Mass Matrix}
 

The charginos mix with the charged leptons forming a set of five
charged fermions $F_i^{\pm}$, $i=1,\ldots,5$ in two component spinor
notation. In a basis where $\psi^{+T}=(-i\lambda^+,\widetilde
H_u^+,e_R^+,\mu_R^+,\tau_R^+)$ and $\psi^{-T}=(-i\lambda^-,\widetilde
H_d^-,e_L^-,\mu_L^-,\tau_L^-)$, the charged fermion mass terms in the
Lagrangian are
% 
\begin{equation} 
{\cal L}_m=-{1\over 2}(\psi^{+T},\psi^{-T}) \left(\matrix{{\bold 0} &
\bold M_C^T \cr {\bold M_C} & {\bold 0} }\right) \left(\matrix{\psi^+
\cr \psi^-}\right)+h.c.
\label{eq:chFmterm} 
\end{equation} 
% 
where the chargino/lepton mass matrix is given by


\begin{equation}
M_{C}=\left[ \begin{array}{cc}
M_{\chi^\pm} & E^\prime\\
E & M_E
\end{array}\right] 
\end{equation}
%
where $M_E=\frac{1}{\sqrt{2}}v_{d}h_{E}$ is the charged leptons mass 
matrix and $M_{\chi^\pm}$ is the usual MSSM chargino mass matrix, 
Eq.~(\ref{eq:ChaM2x2}). The sub-matrix  \( E \) is 
%
\begin{equation}
%\label{#}
E=\left[ \begin{array}{cc}
\frac{1}{\sqrt{2}}gv_{1} & -\epsilon _{1}\\
\frac{1}{\sqrt{2}}gv_{2} & -\epsilon _{2}\\
\frac{1}{\sqrt{2}}gv_{3} & -\epsilon _{3}
\end{array}\right] 
\end{equation}
%
and \( E' \) can be written as \( E'=-v.h_{E} \), where \( v \) is defined 
as:
%
\begin{equation}
%\label{#}
v=\left[ \begin{array}{ccc}
0 & 0 & 0\\
\frac{v_{1}}{\sqrt{2}} & \frac{v_{2}}{\sqrt{2}} & \frac{v_{3}}{\sqrt{2}}
\end{array}\right] 
\end{equation}
% 
We note that chargino sector decouples from the lepton sector in the
limit $\epsilon_i=v_i=0$. As in the MSSM, the chargino mass matrix is
diagonalized by two rotation matrices $\bold U$ and $\bold V$ defined
by
%
\begin{equation} F_i^-=U_{ij}\, \psi_j^- \quad ; \quad F_i^+=V_{ij}\, \psi_j^+
\end{equation}
%
Then
%
\begin{equation} 
\label{eq:CharMdiag}
\bold{U}^* \bold{M_C} \bold{V}^{-1} = \bold{M_{CD}} 
\end{equation}
%
where $\bold{M_{CD}}$ is the diagonal charged fermion mass matrix.  To
determine $\bold{U}$ and $\bold{V}$ we note that
%
\begin{equation} \bold{M^2_{CD}}=\bold{V} \bold{M_C^{\dagger}} \bold{M_C}
\bold{V}^{-1} = \bold{U^*} \bold{M_C} \bold{M_C^{\dagger}}
(\bold{U^*})^{-1} \end{equation}
%
implying that $\bold{V}$ diagonalizes $\bold{M_C^{\dagger} M_C}$ and
$\bold{U^*}$ diagonalizes $\bold{M_C M_C^{\dagger}}$. For future
reference we note that
%
\begin{equation} \psi^-_j=\bold{U^*}_{kj}\, F_k^- \quad ; \quad
\psi^+_j=\bold{V^*}_{kj}\, F_k^+ \end{equation}
%

\ni In the previous expressions the $F_i^{\pm}$ are two component
spinors. We construct the four component Dirac spinors out of the two
component spinors with the conventions\footnote{Here we depart from
the conventions of Ref.\cite{haber:1985rc} because we want the $e^-$,
$\mu^-$ and $\tau^-$ to be the particles and not the
anti--particles.},
%
\begin{equation}
\label{eq:chi-} 
\chi_i^-=\left(\matrix{ F_i^- \cr \vb{18} \overline{F_i^+}\cr}
\right) \end{equation}
%
The chargino masses coming from \Eq{eq:CharMdiag} can have a negative
sign. We fix that by defining a sign parameter
%
\begin{equation}
\eta^-_i=\frac{m_{\chi^-_i}}{|m_{\chi^-_i}|}
\end{equation}
%
This corresponds to the following substitution
%
\begin{equation}
\label{etasign}
\chi^-_i\rightarrow \gamma_5\, \chi_i^-
\end{equation}
%
and therefore
%
\begin{eqnarray}
&&\chi^-_i \rightarrow \left( \eta^-_i P_L + P_R \right) \chi^-_i
\quad ; \quad \overline{\chi^-}_i \rightarrow \overline{\chi^-}_i\, 
\left(P_L +\eta^-_i  P_R \right) \cr
\vb{18}
&&\chi^+_i \rightarrow \left( P_L +  \eta^-_i P_R \right) \chi^+_i
\quad ; \quad \overline{\chi^+}_i \rightarrow \overline{\chi^+}_i\, 
\left( \eta^-_i P_L + P_R \right) 
\end{eqnarray}




\subsection{Neutralino Mass Matrix}


In our model, the neutrinos acquire
mass\cite{hirsch:2000ef,romao:1999up}, and this is due to a mixing
between the neutralino sector and the neutrinos, forming a set of
seven neutral two component fermions $F^0_i$, $i=1,...7$.  In the
basis $\psi^{0T}=
(-i\lambda',-i\lambda^3,\widetilde{H}_d^1,\widetilde{H}_u^2,\nu_{e},
\nu_{\mu},\nu_{\tau})$ the neutral fermions mass terms in the
Lagrangian are given by
% 
\begin{equation} 
{\cal L}_m=-\frac 12(\psi^0)^T{\bold M}_N\psi^0+h.c.
\label{eq:NeuMLag} 
\end{equation} 
% 
where the neutralino/neutrino mass matrix is
% 
\begin{equation}
\label{eq:neutmassmat}
{\bold M}_N=\left[  
\begin{array}{cc}  
{\cal M}_{\chi^0}& m^T \cr
m & 0 \cr
\end{array}
\right]
\end{equation}
%
with ${\cal M}_{\chi^0}$  given in Eq.~(\ref{eq:NeuM4x4}), and
%
\begin{equation}
  \label{eq:1}
m=\left(a_1, a_2,a_3\right)^T   
\end{equation}
%
where $a_i=(-\frac 12g^{\prime }v_i, \frac 12gv_i, 0,\epsilon_i)$. 
This neutralino/neutrino
mass matrix is diagonalized by a $7\times 7$ rotation matrix $\bold N$
such that

% 
\begin{equation} 
{\bold N}^*{\bold M}_N{\bold N}^{-1}={\rm
diag}(m_{\chi^0_1},m_{\chi^0_2},
m_{\chi^0_3},m_{\chi^0_4},m_{\nu_{e}},m_{\nu_{\mu}},m_{\nu_{\tau}})
\label{eq:NeuMdiag} 
\end{equation} 
% 
For future reference we note that
%
\begin{equation} \psi_j^0=\bold{N^*}_{kj}\, F_k^0 \end{equation}
%
and the four component Majorana neutral fermions are obtained from the
two component via the relation
%
\begin{equation} \chi_i^0=\left(\matrix{ F_i^0\cr \vb{18} \overline{F_i^0}\cr}
\right) \end{equation}

\ni
The neutralino masses coming from \Eq{eq:NeuMdiag} can have a negative
sign. We fix that by defining a sign parameter
%
\begin{equation}
\eta^0_i=\frac{m_{\chi^0_i}}{|m_{\chi^0_i}|}
\end{equation}
%
This corresponds to the following substitution
%
\begin{equation}
\label{epsilonsign}
\chi^0_i\rightarrow \gamma_5 \chi_i^0
\end{equation}
%
and therefore
%
\begin{equation}
\chi^0_i \rightarrow \left( \eta^0_i P_L + P_R \right) \chi^0_i
\quad ; \quad
\overline{\chi^0}_i \rightarrow \overline{\chi^0}_i 
\left(P_L +\eta^0_i  P_R \right) 
\end{equation}
 

\subsection{Quark Mass Matrices}


The BRpV model does not affect the quark sector, therefore the quark
mass matrices are exactly like in the SM or in the MSSM as discussed in
section~\ref{sec:QuarkMassMatrices}.

\section{Couplings in the BRpV Model}

\subsection{Charged Current Couplings}

The relevant part of the Lagrangian was already given in
Eq.~(\ref{cc1}). The difference with respect to the MSSM is that in
the BRpV model the neutral and charged fermions mix, respectively,
with the neutralinos and the charginos. 
To obtain the couplings in four component notation we need to express
the fields in terms of the mass eigenstates. Now, in addition to
Eq.~(\ref{eq:chareigMSSM}), we also have
%
\begin{equation}
\label{eq:chareigBRpV}
\ba{rclcrcl}
\nu_j &=& \bold{N^*}_{i,4+j} F_i^0& \hskip 1cm &
\ovl{\nu_j} &=& \bold{N}_{i,4+j} \ovl{F_i^0}\\[+1mm]
\ell^-_j&=& \bold{U^*}_{i,2+j} F_i^- & \hskip 1cm& 
\ovl{\ell^-}_j &=& \bold{U}_{i,2+j} \ovl{F_i^-}\\[+1mm]
\ell^c_j&=& \bold{V^*}_{i,2+j} F_i^+ & \hskip 1cm& 
\ovl{\ell^c}_j &=& \bold{V}_{i,2+j} \ovl{F_i^+}
\ea
\end{equation}
%
\ni
We obtain then
%
\begin{eqnarray}
\hskip -15pt
{\cal L}&\hskip -7pt=\hskip -7pt& 
g\, W_{\mu}^- \left[ \hskip 0.2cm
\ovl{F_i^0}\, \ovl{\sigma}^{\mu}\, F_j^+ 
\left( \bold{N}_{i2} \bold{V^*}_{j1} -\frac{1}{\sqrt{2}}\,
\bold{N}_{i4} \bold{V^*}_{j2} \right) \right. \cr
\vb{20}
&&\left.\hskip 1.2cm 
+\ovl{F_j^-}\, \ovl{\sigma}^{\mu}\, F_i^0 
\left( - \bold{N^*}_{i2} \bold{U}_{j1} -\frac{1}{\sqrt{2}}\,
\left( \bold{N^*}_{i3} \bold{U}_{j2} + \sum_{k=1}^3
\bold{N^*}_{i,4+k} \bold{U}_{j,2+k} \right) \right) \right]\cr
\vb{24}
&\hskip -7pt+\hskip -7pt& 
g\, W_{\mu}^+ \left[ \hskip 0.2cm
\ovl{F_j^+}\, \ovl{\sigma}^{\mu}\, F_i^0 
\left( \bold{N^*}_{i2} \bold{V}_{j1} -\frac{1}{\sqrt{2}}\,
\bold{N^*}_{i4} \bold{V}_{j2} \right) \right. \cr
\vb{20}
&&\left. \hskip 1.2cm 
+\ovl{F_i^0}\, \ovl{\sigma}^{\mu}\, F_j^- 
 \left( - \bold{N}_{i2} \bold{U^*}_{j1} -\frac{1}{\sqrt{2}}\,
\left( \bold{N}_{i3} \bold{U^*}_{j2} + \sum_{k=1}^3
\bold{N}_{i,4+k} \bold{U^*}_{j,2+k} \right) \right) \right]
\label{cc2-c7}
\end{eqnarray}  

Finally using Eq.~(\ref{eq:13}) we get
%
\begin{equation}
{\cal L}=
 g W_{\mu}^- \left[ \hskip 0.2cm
\ovl{\chi_j^-}\, \gamma^{\mu}\,  \left ( O^L_{ji} P_L +
O^R_{ji} P_R \right) \chi_i^0 \right]
+ g W_{\mu}^+ \left[ \hskip 0.2cm
\ovl{\chi_i^0}\, \gamma^{\mu}\,  \left ( (O^L_{ji})^* P_L +
(O^R_{ji})^* P_R\right) \chi_j^- \right] 
\label{cc3-c7}
\end{equation}  
%
where
%
\begin{eqnarray}
\label{eq:ChargedCurrent}
O^L_{ji}&=& \left[- \bold{N^*}_{i2} \bold{U}_{j1} -\frac{1}{\sqrt{2}}\,
\left( \bold{N^*}_{i3} \bold{U}_{j2} + \sum_{k=1}^3
\bold{N^*}_{i,4+k} \bold{U}_{j,2+k} \right) \right]\cr
\vb{24}
O^R_{ji}&=& \left(- \bold{N}_{i2} \bold{V^*}_{j1} +\frac{1}{\sqrt{2}}\,
\bold{N}_{i4} \bold{V^*}_{j2} \right) 
\end{eqnarray}



\subsection{Neutral Current Couplings}

The relevant part of the Lagrangian was already given in
Eq.~(\ref{nc1}). Using Eqs.~(\ref{eq:chareigMSSM}),
(\ref{eq:AandZ}) and (\ref{eq:chareigBRpV}) to express the fields in
terms of the mass eigenstates we obtain,
%
\begin{eqnarray}
{\cal L}&\hskip -7pt=\hskip -7pt&
-e A_{\mu} \left[\
\sum_{k=1}^5 \bold{V}_{ik} \bold{V^*}_{jk} \ovl{F_i^+}\,
\ovl{\sigma}^{\mu}\, F_j^+ 
- \sum_{k=1}^5 \bold{U}_{ik} \bold{U^*}_{jk} \ovl{F_i^-}\,
\ovl{\sigma}^{\mu}\, F_j^- \right]\nn  \\[+2mm]
&\hskip -7pt+\hskip -7pt&
\frac{g}{\cos \theta_w} Z_{\mu} 
\left[\vb{20} \
\half \left(
\bold{N}_{i4} \bold{N^*}_{j4} -\bold{N}_{i3} \bold{N^*}_{j3}
-\sum_{k=1}^3\bold{N}_{i,4+k} \bold{N^*}_{j,4+k}
\right)\, \ovl{F_i^0} \ovl{\sigma}^{\mu}\, F_j^0 \right. \nn \\[+2mm]
&&\hskip 1.0cm
\left(
\half \bold{U}_{i2} \bold{U^*}_{j2} +\bold{U}_{i1} \bold{U^*}_{j1}
+ \half \sum_{k=1}^3 \bold{U}_{i,2+k} \bold{U^*}_{j,2+k} 
-\sin^2 \theta_w \sum_{k=1}^5 \bold{U}_{ik} \bold{U^*}_{jk}
\right)\, \ovl{F_i^-}\, \ovl{\sigma}^{\mu}\, F_j^- \nn \\[+2mm]
&&\hskip 1.0cm
\left. \left(
- \half \bold{V}_{i2} \bold{V^*}_{j2} - \bold{V}_{i1} \bold{V^*}_{j1}
+\sin^2 \theta_w \sum_{k=1}^5
\bold{V}_{ik} \bold{V^*}_{jk}
\right)\, \ovl{F_i^+}\, \ovl{\sigma}^{\mu}\, F_j^+ \right]
\label{nc2-c7}
\end{eqnarray}  
%
Now using the same techniques we can write in four component notation
%
\begin{eqnarray}
{\cal L}&\hskip -7pt=\hskip -7pt&
e A_{\mu} \
\sum_{i=1}^5 \ovl{\chi_i^-}\,
\gamma^{\mu}\, \chi_i^- 
+\frac{g}{\cos \theta_w} Z_{\mu} 
\left[\vb{18} \
\sum_{i,j=1}^5 \half  \ovl{\chi_i^0} \gamma^{\mu}\,
\left( O^L_{ij}{}'' P_L + O^R_{ij}{}'' P_R \right) \chi_j^0
\right. \nn \\[+2mm]
&&\left. \vb{18}\hskip 50mm
+ \sum_{i,j=1}^5 \ovl{\chi_i^-}\, \gamma^{\mu}\,
\left( O^L_{ij}{}' P_L + O^R_{ij}{}' P_R \right) \right] 
\label{nc4-c7}
\end{eqnarray}  
%
where
%
\begin{eqnarray}
O^L_{ij}{}''&=& \half \left(
\bold{N}_{i4} \bold{N^*}_{j4} -\bold{N}_{i3} \bold{N^*}_{j3}
-\sum_{k=1}^3\bold{N}_{i,4+k} \bold{N^*}_{j,4+k} \right)\cr
\vb{22}
O^R_{ij}{}''&=&-\left(O^L_{ji}{}''\right)^* \cr
\vb{22}
O^L_{ij}{}'&=&
\half \bold{U}_{i2} \bold{U^*}_{j2} +\bold{U}_{i1} \bold{U^*}_{j1}
+ \half \sum_{k=1}^3\bold{U}_{i,2+k} \bold{U^*}_{j,2+k} 
-\sin^2 \theta_w \delta_{ij}\cr
\vb{22}
O^R_{ij}{}'&=&
 \half \bold{V}_{j2} \bold{V^*}_{i2} + \bold{V}_{j1} \bold{V^*}_{i1}
-\sin^2 \theta_w \delta_{ij}
\end{eqnarray}


\subsection{Charged scalars couplings to fermions}

The Lagrangian containing the couplings of charged scalars to fermions
was already given in Eqs.~(\ref{eq:csgauge}) and (\ref{eq:csyukawa})
in terms of the weak states. In order to give simpler expressions we
give here the results in terms of the weak basis charged states 
$S'^+_i=(H_d^+,H_u^+,\widetilde{\ell_j}_L^+,\widetilde{\ell_j}_R^+)$. The
mass eigenstates $S_i^{\pm}$ are obtained from $S'^{\pm}_i$ through the 
rotation matrix $\bold{R_{S^{\pm}}}$. Then the final Lagrangian reads
%
\begin{eqnarray}
\hskip -15pt
{\cal L}&\hskip -7pt=\hskip -7pt& 
S'^+_i\, \ovl{\chi_A^0}\left[ C^{L(S+0-)}_{iAB}\ P_L
+C^{R(S+0-)}_{iAB}\ P_R \right] \chi_B^- 
+S'^+_i\, \ovl{u_j}\left[ C^{L(\ovl{u}d)}_{ijk}\ P_L
+C^{R((\ovl{u}d)}_{ijk}\ P_R \right] d_k \nn \\[+2mm]
&&+
\widetilde{u}_X\ \ovl{d}_i \left[ \vb{14}
C_{iAX}^{L(d)} P_L + C_{iAX}^{R(d)} P_R \right] \chi_A^- 
+\widetilde{d}_X\ \ovl{u}_i \left[ \vb{14}
C_{iAX}^{L(u)} P_L + C_{iAX}^{R(u)} P_R \right] \chi_A^c \nn \\[+2mm]
&&+
\widetilde{u}_X\ \ovl{u}_i \left[ \vb{14}
N_{iAX}^{L(u)} P_L + N_{iAX}^{R(u)} P_R \right] \chi_A^0 
+\widetilde{d}_X\ \ovl{d}_i \left[ \vb{14}
N_{iAX}^{L(d)} P_L + N_{iAX}^{R(d)} P_R \right] \chi_A^0 \nn \\[+2mm]
&&+
\hbox{h.c.}
\label{csff2}
\end{eqnarray}
%
where the coupling matrices have the following expressions $(i,j=1,2,3)$,
%
\begin{eqnarray}
C^{L(S+0-)}_{1AB}&=& 
\frac{g}{\sqrt{2}} \bold{N^*}_{A2} \bold{U^*}_{B2}
+\frac{g'}{\sqrt{2}} \bold{N^*}_{A1} \bold{U^*}_{B2}
-g \bold{N^*}_{A3} \bold{U^*}_{B1} 
\cr
\vb{18}
C^{L(S+0-)}_{2AB}&=& 0
\cr
\vb{18}
C^{L(S+0-)}_{2+i,AB}&=& 
\frac{g}{\sqrt{2}} \bold{N^*}_{A2} \bold{U^*}_{B,2+i}
+\frac{g'}{\sqrt{2}} \bold{N^*}_{A1} \bold{U^*}_{B,2+i}
-g \bold{N^*}_{A,4+i} \bold{U^*}_{B1} 
\cr
\vb{18}
C^{L(+0-)}_{5+j,AB}&=& 
(h_{E})_{ij} \left(
\bold{N^*}_{A,4+i} \bold{U^*}_{B2}-\bold{N^*}_{A3} \bold{U^*}_{B,2+i}
\right)
\end{eqnarray}
%
and
%
\begin{eqnarray}
C^{R(S+0-)}_{1AB}&=& 
(h^*_E)_{ij}  \bold{N}_{A,4+i} \bold{V}_{B,2+j}
\cr
\vb{18}
C^{R(S+0-)}_{2AB}&=&
-\frac{g}{\sqrt{2}} \bold{N}_{A2} \bold{V}_{B2}
-\frac{g'}{\sqrt{2}} \bold{N}_{A1} \bold{V}_{B2}
-g \bold{N}_{A4} \bold{V}_{B1}
\cr
\vb{18}
C^{R(S+0-)}_{2+i,AB}&=& 
- (h^*_E)_{ij}  \bold{N}_{A3} \bold{V}_{B,2+j}\cr
\vb{18}
C^{R(S+0-)}_{5+i,AB}&=& 
- g' \sqrt{2}\, \bold{N}_{A1} \bold{V}_{B,2+i}
\end{eqnarray}
%
\begin{eqnarray}
C^{L(\ovl{u}d)}_{ijk}&=& 
(h_U)_{kj}\ \delta_{i2} \cr
\vb{18}
C^{R(\ovl{u}d)}_{ijk}&=& 
(h^*_D)_{jk}\ \delta_{i1}
\end{eqnarray}

\ni
For the squarks, the couplings $C_{iAX}^{L,R(d,u)}$ and
$N_{iAX}^{L,R(d,u)}$ are formally the same as in the MSSM and were
given in Eqs.~(\ref{eq:sqdcn}), (\ref{eq:squcn}), (\ref{eq:squun}) and
(\ref{eq:sqddn}).  The differences with respect to the MSSM are in the
rotation matrices that now depend on the $\epsilon_i$.


\subsection{Neutral scalars couplings to fermions}

The Lagrangian containing the couplings of charged scalars to fermions
was already given in Eqs.~(\ref{eq:nscfg}) and (\ref{eq:nscfy}) in
terms of the weak states. In order to give simpler expressions, we
give here the results in terms of the weak basis charged states
$S'^0_i=(\sigma_d^0,\sigma_u^0,\widetilde{\nu}_j^R)$ and
$P'^0_i=(\varphi_d^0,\varphi_u^0,\widetilde{\nu}_j^I)$.  The mass
eigenstates are obtained using the rotation matrices $\bold{R_{S^0}}$
and $\bold{R_{P^0}}$ as in Eqs.~(\ref{eq:weakeigenrotS0}) and
(\ref{eq:weakeigenrotP0}). Then the final Lagrangian reads,
%
\begin{eqnarray}
\label{nscfermions}
{\cal L}&\hskip -7pt=\hskip -7pt& \half
S'^0_i\, \ovl{\chi_A^0}\left[ C^{L(S000)}_{iAB}\ P_L
+C^{R(S000)}_{iAB}\ P_R \right] \chi_B^0 
+
S'^0_i\, \ovl{\chi_A^-}\left[ C^{L(S0+-)}_{iAB}\ P_L
+C^{R(S0+-)}_{iAB}\ P_R \right] \chi_B^- \nn \\[+2mm] 
&\hskip -7pt+\hskip -7pt& \half
P'^0_i\, \ovl{\chi_A^0}\left[ C^{L(P000)}_{iAB}\ P_L
+C^{R(P000)}_{iAB}\ P_R \right] \chi_B^0 
+
P'^0_i\, \ovl{\chi_A^-}\left[ C^{L(P0+-)}_{iAB}\ P_L
+C^{R(P0+-)}_{iAB}\ P_R \right] \chi_B^- \nn \\
&\hskip -7pt+\hskip -7pt&
S'^0_i\, \ovl{u_j}\left[ C^{L(S\ovl{u}u)}_{ijk}\ P_L
+C^{R(S(\ovl{u}u)}_{ijk}\ P_R \right] u_k
+P'^0_i\, \ovl{u_j}\left[ C^{L(P\ovl{u}u)}_{ijk}\ P_L
+C^{R(P\ovl{u}u)}_{ijk}\ P_R \right] u_k
\end{eqnarray}
%
We get for the couplings with neutral fermions,
%
\begin{eqnarray}
C^{L(S000)}_{1AB}&=& 
-\frac{g}{2} \bold{N^*}_{A2} \bold{N^*}_{B3}
+\frac{g'}{2} \bold{N^*}_{A1} \bold{N^*}_{B3} + (A \leftrightarrow B)
\cr
\vb{18}
C^{L(S000)}_{2AB}&=& 
\frac{g}{2} \bold{N^*}_{A2} \bold{N^*}_{B4}
-\frac{g'}{2} \bold{N^*}_{A1} \bold{N^*}_{B4} + (A \leftrightarrow B)
\cr
\vb{18}
C^{L(S000)}_{2+i,AB}&=& 
-\frac{g}{2} \bold{N^*}_{A2} \bold{N^*}_{B,4+i}
+\frac{g'}{2} \bold{N^*}_{A1} \bold{N^*}_{B,4+i} + (A \leftrightarrow B)
\end{eqnarray}
%
%
\begin{eqnarray}
C^{L(P000)}_{iAB}&=& -i C^{L(S000)}_{iAB}
\cr
\vb{18}
C^{R(S000)}_{iAB}&=&  \left(C^{L(S000)}_{iAB}\right)^*
\cr
\vb{18}
C^{R(P000)}_{iAB}&=&  \left(C^{L(P000)}_{iAB}\right)^*
\end{eqnarray}


\ni
The couplings with the charged fermions are
%
\begin{eqnarray}
C^{L(S0+-)}_{1AB}&=& 
-\frac{g}{\sqrt{2}} \bold{V^*}_{A1} \bold{U^*}_{B2}
-\frac{(h_E)_{ij}}{\sqrt{2}}  \bold{V^*}_{A,2+j} \bold{U^*}_{B,2+i}
\cr
\vb{18}
C^{L(S0+-)}_{2AB}&=& 
-\frac{g}{\sqrt{2}} \bold{V^*}_{A2} \bold{U^*}_{B1}
\cr
\vb{18}
C^{L(S0+-)}_{2+i,AB}&=& 
-\frac{g}{\sqrt{2}} \bold{V^*}_{A1} \bold{U^*}_{B,2+i}
+\frac{(h_E)_{ij}}{\sqrt{2}}  \bold{V^*}_{A,2+j} \bold{U^*}_{B2}
\end{eqnarray}
%
%
\begin{eqnarray}
C^{L(P0+-)}_{1AB}&=& -i \left(
-\frac{g}{\sqrt{2}} \bold{V^*}_{A1} \bold{U^*}_{B2}
+\frac{(h_E)_{ij}}{\sqrt{2}}  \bold{V^*}_{A,2+j} \bold{U^*}_{k,2+i}
\right)
\cr
\vb{18}
C^{L(P0+-)}_{2AB}&=& 
i \frac{g}{\sqrt{2}} \bold{V^*}_{A2} \bold{U^*}_{B1}
\cr
\vb{18}
C^{L(P0+-)}_{2+i,AB}&=& -i \left(
-\frac{g}{\sqrt{2}} \bold{V^*}_{A1} \bold{U^*}_{B,2+i}
-\frac{(h_E)_{ij}}{\sqrt{2}}  \bold{V^*}_{A,2+j} \bold{U^*}_{B2}
\right)
\end{eqnarray}
%
and
%
\begin{eqnarray}
C^{R(S0+-)}_{iAB}&=&  \left(C^{L(S0+-)}_{iBA}\right)^*
\cr
\vb{18}
C^{R(P0+-)}_{iAB}&=&  \left(C^{L(P0+-)}_{iBA}\right)^*
\end{eqnarray}
%
Finally the couplings with the quarks are,
\begin{eqnarray}
C^{L(S\ovl{u}u)}_{ijk}&=& 
-(h_U)_{kj}\ \delta_{i2} \cr
\vb{18}
C^{L(S\ovl{d}d)}_{ijk}&=& 
-(h_D)_{kj}\ \delta_{i1} \cr
\vb{18}
C^{L(P\ovl{q}q)}_{ijk}&=& 
i\, C^{L(S\ovl{q}q)}_{ijk}\quad, \quad (q=u,d) \cr
\vb{18}
C^{R(Y^0\ovl{q}q)}_{ijk}&=& 
\left(C^{L(Y^0\ovl{q}q)}_{ikj} \right)^*
\quad, \quad (Y=S,P\ ;\ q=u,d)
\end{eqnarray}


\section{Neutrino Masses in the BRpV Model}


\subsection{Tree Level Neutrino Masses and Mixings}

The $7\times7$ neutralino/neutrino mass matrix was given in
Eq.~(\ref{eq:neutmassmat}). It can be shown exactly that it has two
zero eigenvalues. Therefore two of the neutrinos remain massless at
tree level. As we will see in the next section this degeneracy can be
lifted by including the one loop corrections. The values of the BRpV
parameters $\epsilon_i$ needed to account for the neutrino masses and
mixings turn out to be small compared with the electroweak scale. It it
then useful to use this fact to find an approximate expression for the
tree level neutrino mass matrix. For this we define a matrix
$\xi_{ij}$ with dimensionless small entries,
%
\begin{equation}
\xi = m \cdot {\cal M}_{\chi^0}^{-1}\hskip 10mm
\Rightarrow\hskip 10mm \forall \xi_{ij} \ll 1
\end{equation}
%
With this small quantities we can find an approximate solution for the
mixing  matrix ${\cal N}$. In leading order in $\xi$ the mixing matrix
${\cal N}$ is given by~\cite{hirsch:2000ef,romao:1999up}, 
%
\begin{equation}
\label{mixing}
{\cal N}^*  =  \left(\begin{array}{cc}
N^* & 0\\
0& V_\nu^T \end{array}\right)
\left(\begin{array}{cc}
1 -{1 \over 2} \xi^{\dagger} \xi& \xi^{\dagger} \\
-\xi &  1 -{1 \over 2} \xi \xi^\dagger
\end{array}\right) 
\end{equation}
%
The second matrix above block-diagonalize 
${\bold M}_N$ approximately to the form 
diag($m_{\hbox{\scriptsize eff}},{\cal M}_{\chi^0}$)
%
\begin{equation}
m_{\hbox{\scriptsize eff}} = - m \cdot {\cal M}_{\chi^0}^{-1} m^T = 
\frac{M_1 g^2 + M_2 {g'}^2}{4\, \det({\cal M}_{\chi^0})} 
\left(\begin{array}{ccc}
\Lambda_e^2 & \Lambda_e \Lambda_\mu
& \Lambda_e \Lambda_\tau \\
\Lambda_e \Lambda_\mu & \Lambda_\mu^2
& \Lambda_\mu \Lambda_\tau \\
\Lambda_e \Lambda_\tau & \Lambda_\mu \Lambda_\tau & \Lambda_\tau^2
\end{array}\right).
\end{equation}
%
The submatrices $N$ and $V_{\nu}$ in \Eq{mixing}  diagonalize 
${\cal M}_{\chi^0}$ and $m_{\hbox{\scriptsize eff}}$ 
%
\begin{equation}
N^{*}{\cal M}_{\chi^0} N^{\dagger} = \hbox{diag}(m_{\chi^0_i})
\quad ; \quad
V_{\nu}^T m_{\hbox{\scriptsize eff}} V_{\nu} = \hbox{diag}(0,0,m_{\nu}),
\end{equation}
%
where 
%
\begin{equation}
\label{eq:mnutree}
m_{\nu} = \hbox{Tr}(m_{\hbox{\scriptsize eff}}) = 
\frac{M_1 g^2 + M_2 {g'}^2}{4\, \det({\cal M}_{\chi^0})} 
|{\vec \Lambda}|^2.
\end{equation}
%

\subsection{One Loop Neutrino Masses and Mixings}
\subsubsection{Definition}

The Self--Energy for the neutralino/neutrino is

\vspace{-5mm}
\begin{equation}
\hskip 4cm \equiv
i \left\{ \slash{p} \left[ P_L \Sigma^L_{ij} + P_R \Sigma^R_{ij} \right]
-\left[ P_L \Pi^L_{ij} + P_R \Pi^R_{ij} \right]\right\}
\end{equation}

\begin{picture}(0,0)
\put(0,0.75){\includegraphics[width=3cm]{self.eps}}
\end{picture}

\vspace{-3mm}

\ni
Then 
%
\begin{equation}
M^{\rm pole}_{ij}= M^{\rm \overline{DR}}_{ij}(\mu_R) + \Delta M_{ij}
\end{equation}
%
with
%
\begin{equation}
\Delta M_{ij}\! =\! \left[ \half 
\left(\Pi^V_{ij}(m_i^2)\! +\! \Pi^V_{ij}(m_j^2)\right) 
\!-\! \half 
\left( m_{\chi^0_i} \Sigma^V_{ij}(m_i^2) \! +\! 
m_{\chi^0_j} \Sigma^V_{ij}(m_j^2) \right) \right]_{\Delta=0}
\end{equation}
%
where
%
\begin{equation}
\Sigma^V=\half \left(\Sigma^L+\Sigma^R\right)
\quad ; \quad
\Pi^V=\half \left(\Pi^L+\Pi^R\right)
\end{equation}
%
and
%
\begin{equation}
\ds \Delta=\frac{2}{4-d} -\gamma_E + \ln 4\pi
\end{equation}


\subsubsection{Diagrams Contributing}

In a generic way the diagrams contributing are
%
\begin{center}
\begin{tabular}{cccc}
\includegraphics[width=30mm]{wself.eps}
&
\includegraphics[width=30mm]{sself.eps}
&
\includegraphics[width=30mm]{sself.eps}
&
\includegraphics[width=22mm]{tadgoldstone.eps}
\end{tabular}
\end{center}
%
These diagrams can be calculated in a straightforward way. For
instance the $W$ diagram in the $\xi=1$ gauge gives
%
\begin{eqnarray}
\Sigma^V_{ij}&\!=\!\!& -\frac{1}{16\pi^2}\, \sum_{k=1}^5
2 \left(O^{\rm ncw}_{L jk} O^{\rm cnw}_{L ki} +
O^{\rm ncw}_{R jk} O^{\rm cnw}_{R ki}\right) B_1(p^2,m^2_k,m^2_W)\cr
\vb{35}
\Pi^V_{ij}&\!=\!\!& -\frac{1}{16\pi^2}\, \sum_{k=1}^5
(-4) \left(O^{\rm ncw}_{L jk} O^{\rm cnw}_{R ki} +
O^{\rm ncw}_{R jk} O^{\rm cnw}_{L ki}\right) m_k\, B_0(p^2,m^2_k,m^2_W)
\nn
\end{eqnarray}
%
where $B_0$ and $B_1$ are the Passarino-Veltman functions, and $O^{\rm
cnw}$, $O^{\rm ncw}$ are coupling matrices, that can be read from
Eq.~(\ref{eq:ChargedCurrent}). Explicit expressions for all the
diagrams can be found in~\cite{romao:1999up,hirsch:2000ef}.



\subsubsection{Gauge Invariance}


When calculating the self--energies the question of gauge invariance
arises. In the $R_{\xi}$ gauge the following diagrams
depend on $\xi$



\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=30mm]{wself.eps}
&
\includegraphics[width=30mm]{sself.eps}
&
\includegraphics[width=22mm]{tadgoldstone.eps}
\end{tabular}

\vspace{0mm}
Set 1

\end{center}



\begin{center}
\begin{tabular}{cc}
\includegraphics[width=18mm]{tadzij.eps}
&
\includegraphics[width=22mm]{tadczij.eps}
\end{tabular}

\vspace{-0mm}
Set 2
\end{center}



\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=19mm]{tadw.eps}
&
\includegraphics[width=22mm]{tadcw+.eps}
&
\includegraphics[width=22mm]{tadcw-.eps}
\end{tabular}

\vspace{-0mm}
Set 3
\end{center}


\ni
Calculating in an arbitrary $R_{\xi}$ gauge we have
shown~\cite{romao:1999up,hirsch:2000ef} that the gauge dependence
cancels among the diagrams in each set.





\subsection{The One--Loop Mass Matrix}


The one--loop corrected mass matrix is
%
\begin{equation}
M^{1L}= M^{0L}_{diag} + \Delta M^{1L}
\end{equation}
%
where
%
\begin{equation}
M^{0L}_{diag}= {\cal N} M_N {\cal N}^T
\end{equation}
%
Now we diagonalize the 1--loop mass matrix 
%
\begin{equation}
M^{1L}_{diag}={\cal N'} M^{1L} {\cal N'}^T
\end{equation}
%
Then the mass eigenstates are related to the weak basis states by
%
\begin{equation}
\chi_0^{mass}= {\cal N}^{1L}_{i \alpha}\, \chi_0^{weak}
\end{equation}
%
with
%
\begin{equation}
{\cal N}^{1L} = {\cal N'}\ {\cal N}
\end{equation}



\ni
The usual convention in neutrino physics
%
\begin{equation}
\nu_{\alpha} = U_{\alpha k}\,  \nu_k
\end{equation}
%
is recovered in our notation as
%
\begin{equation}
U_{\alpha k}= {\cal N}^{1L}_{4+k, 4+\alpha} 
\end{equation}
%
From this relation we can obtain the usual definiton of the neutrino
mixing angles.


\section{Approximate Formul{\ae} for the Neutrino Masses and Mixings}

In the last section we have sketched the full one loop calculation
that lifts the tree level degeneracy of the two massless neutrino
states. As we will shown in Chapter~\ref{BRpVphenomenology} this leads
to a pattern of neutrino masses and mixings compatible with the
experimental data. However, as the whole approach is numerical, it is
difficult to have a qualitative understanding.

In order to have a better understanding it has been identified that
the bottom-sbottom and the charged scalar loops are the most important
ones, at least in supersymmetric models with mSUGRA-like boundary
conditions.  Taking into account only these loops, explicit formulas
were given in Ref. \cite{diaz:2003as} were a complete discussion of
their validity and of the accuracy with which they describe solar
neutrino mass and mixing parameters.
The method is based on a systematic perturbative expansion of R-parity
violating vertices to leading order.  


It has been shown~\cite{diaz:2003as} that for the case of the large
mixing MSW solution our formulas yield good agreement with the full
numerical calculation, but are much simpler to implement than the full
numerical one-loop calculation.  The only solar neutrino ``solutions''
for which the analytical approximation is less accurate are those that
are now ruled out by the reactor neutrino data from KamLAND.
Therefore in this section we will just give the simplified formulas.


\subsection{Approximate Rotation Matrices}
\label{sec:rotation-matrices}


If the effective RpV parameters are smaller than the weak scale, we
can work in a perturbative expansion defined by $\xi \ll 1$, where
$\xi$ denotes a $3\times 4$ matrix given as~\cite{hirsch:1998kc} 

\begin{eqnarray}
\xi_{i1} &=& \frac{g' M_2 \mu}{2\Delta_0}\Lambda_i \cr
\vb{20}
\xi_{i2} &=& -\frac{g M_1 \mu}{2\Delta_0}\Lambda_i \cr
\vb{20}
\xi_{i3} &=& - \frac{\epsilon_i}{\mu} + 
          \frac{M_{\tilde\gamma} v_u}{4\Delta_0}\Lambda_i \cr
\vb{20}
\xi_{i4} &=& - \frac{M_{\tilde\gamma} v_d}{4\Delta_0}\Lambda_i
\label{xielementos}
\end{eqnarray}
%
where $\Delta_0$ is the determinant of the $4\times 4$ neutralino 
mass matrix, $M_{\tilde\gamma}=g^2M_1+g'^2M_2$ and
%

\begin{equation}
\Lambda_i = \mu v_i + v_d \epsilon_i 
\label{eq:lambdai}
\end{equation}

The neutralino/neutrino mass matrix is
diagonalized by a $7\times7$ rotation matrix ${\cal N}$ according to
%
\begin{equation}
{\cal N}^* {\bf M}_{F^0} {\cal N}^{-1}={\bf M}_{F^0}^{\mathrm diag}
\end{equation}
%
and the eigenvectors are given by
%
\begin{equation}
F^0_i={\cal N}_{ij}\psi_j
\end{equation}
%
using the basis $\psi=(-i\lambda',-i\lambda^3,\widetilde{H}_d^1,
\widetilde{H}_u^2, \nu_{e},\nu_{\mu}, \nu_{\tau} )$. In this 
approximation, the rotation matrix can be written as 
%
\begin{equation}
{\cal N}^*\approx\left(\matrix{
N^* & N^*\xi^{\dagger} \cr
-V_{\nu}^T\xi & V_{\nu}^T 
}\right)
\end{equation}
%
Here, $N$ is the rotation matrix that diagonalizes the $4\times4$ MSSM
neutralino mass matrix, $V_{\nu}$ is the rotation matrix that
diagonalizes the tree level neutrino $3\times3$ mass matrix, and
$\xi_{ij}\ll 1$ are the expansion parameters
\cite{hirsch:1998kc,nowakowski:1996dx}.
The terms we need are
%
\begin{equation}
V_{\nu}^T\xi=\left(\matrix{
0 & 0 & b\tilde\epsilon_1 & 0 \cr
0 & 0 & b\tilde\epsilon_2 & 0 \cr
a_1|\vec\Lambda| & a_2|\vec\Lambda| & a_3|\vec\Lambda|+b\tilde\epsilon_3 
& a_4|\vec\Lambda|
}\right)
\label{Rnuxi}
\end{equation}
%
where $b=-1/\mu$,
%
\begin{eqnarray}
a_1={{g'M_2\mu}\over{2\Delta_0}}\,,\quad
a_2=-{{gM_1\mu}\over{2\Delta_0}}\,,\quad
a_3={{M_{\tilde\gamma}v_u}\over{4\Delta_0}}\,,\quad
a_4=-{{M_{\tilde\gamma}v_d}\over{4\Delta_0}}\,,
\end{eqnarray}
%
The $\tilde\epsilon$ parameters in eq.~(\ref{Rnuxi}) are defined as
$\tilde\epsilon_i=\left(V_{\nu}^T\right)^{ij}\epsilon_j$, and are
given by
%
\begin{eqnarray}
\label{eq:TildeEpsilon}
\tilde\epsilon_1&=&{{
\epsilon_e(\Lambda_{\mu}^2+\Lambda_{\tau}^2)-\Lambda_e
(\Lambda_{\mu}\epsilon_{\mu}+\Lambda_{\tau}\epsilon_{\tau})
}\over{
\sqrt{\Lambda_{\mu}^2+\Lambda_{\tau}^2}
\sqrt{\Lambda_e^2+\Lambda_{\mu}^2+\Lambda_{\tau}^2}
}}
\nonumber\\
\tilde\epsilon_2&=&{{
\Lambda_{\tau}\epsilon_{\mu}-\Lambda_{\mu}\epsilon_{\tau}
}\over{
\sqrt{\Lambda_{\mu}^2+\Lambda_{\tau}^2}
}}
\\
\tilde\epsilon_3&=&{{
\vec\Lambda\cdot\vec\epsilon
}\over{
\sqrt{\Lambda_e^2+\Lambda_{\mu}^2+\Lambda_{\tau}^2}
}}\nonumber
\end{eqnarray}
%

On the other hand the chargino/charged slepton mass matrix is
diagonalized with two different $5\times5$ mass matrices,
%
\begin{equation}
{\cal U}^*\, {\bf M}_{F^+} {\cal V}^{-1}={\bf M}_{F^+}^{\mathrm diag}
\end{equation}
%
with the eigenvectors satisfying
%
\begin{equation}
F^+_{Ri}={\cal V}_{ij}\psi^+_j\,,\qquad 
F^-_{Li}={\cal U}_{ij}\psi^-_j
\end{equation}
%
in the basis 
$\psi^+=(-i\lambda^+,\widetilde H_2^1,e_R^+,\mu_R^+,\tau_R^+)$ and
$\psi^-=(-i\lambda^-,\widetilde H_1^2,e_L^-,\mu_L^-,\tau_L^-)$, and with
the Dirac fermions being
%
\begin{equation}
F_i^+=\left(\matrix{F^+_{Ri} \cr \cr \overline{F^-_{Li}}}\right)
\end{equation}
%
To first order in the R-Parity violating parameters we have
%
\begin{equation}
{\cal V}\approx\left(\matrix{
V & V\xi^{T}_R \cr
-V^{\ell}_R\xi^{*}_R & V^{\ell}_R
}\right)\,,\qquad
{\cal U}\approx\left(\matrix{
U & U\xi^{\dagger}_L \cr
-V^{{\ell}*}_L\xi_L & V^{{\ell}*}_L
}\right)
\end{equation}
%
where $V^{{\ell}*}_L$ and $V^{\ell}_R$ diagonalize the charged lepton
mass matrix according to $V^{{\ell}*}_L{\bf
  M}^{\ell}V^{\ell\dagger}_R={\bf M}^{\ell}_{\mathrm diag}$.  For the
purposes of our approximate formula, it is sufficient to take 
$\xi_R={\bf 0}_{2\times3}$, because the mixing between right-handed 
leptons and the charginos is supressed with respect to $\xi_L$ by 
a factor of $m_l/M_{SUSY}$ \cite{hirsch:1998kc,nowakowski:1996dx}.
Note, that we can choose $V^{{\ell}*}_L = 
V^{\ell\dagger}_R = {\bf 1}_{3\times 3}$. We then have
%
\begin{equation}
\xi_L^{i1}=a^L_1\Lambda_i\,,\qquad \xi_L^{i2}=a^L_2\Lambda_i+b\epsilon_i
\end{equation}
%
and
%
\begin{equation}
a_1^L={g\over{\sqrt{2}\Delta_+}}\,,\qquad 
a_2^L=-{{g^2v_u}\over{2\mu\Delta_+}}
\end{equation}
%
where $\Delta_+$ is the determinant of the $2\times2$ chargino mass matrix.

In the BRpV model the charged Higgs fields mix with the charged
sleptons forming an $8\times8$ mass matrix \cite{hirsch:2000ef}, which
is diagonalized by a rotation matrix ${\bf R}_{S^{\pm}}$. The 
construction of ${\bf R}_{S^{\pm}}$ to first order in small (RpV) 
parameters is quite straightforward but lengthy. The interested 
reader can find the details in Ref.~\cite{diaz:2003as}.




\subsection{Approximate Couplings}
\label{sec:appr-coupl}


The relevant Feynman rules for the bottom-sbottom loops are, in the
case of left sbottoms:
%
\begin{center}
\includegraphics{vert-1.eps}
\end{center}
%
with
%
\begin{eqnarray}
O^{bn\tilde b}_{Lij}&=&-R^{\tilde b}_{j1}h_b{\cal N}^*_{i3}
-R^{\tilde b}_{j2}{{2g}\over{3\sqrt{2}}}t_W{\cal N}^*_{i1}
\nonumber\\
O^{bn\tilde b}_{Rij}&=&R^{\tilde b}_{j1}{g\over{\sqrt{2}}}
\left({\cal N}_{i2}-{\textstyle{1\over3}}t_W{\cal N}_{i1}\right)
-R^{\tilde b}_{j2}h_b{\cal N}^*_{i3}
\label{Obnsb}
\end{eqnarray}
%
where $t_W=\tan\theta_W$. After approximating the rotation matrix 
${\cal N}$ we find that expressions similar to eq.~(\ref{Obnsb}) with
the replacement ${\cal N}\rightarrow{\mathrm N}$ are valid when the
neutral fermion is a neutralino. When the neutral fermion $F^0$ is a 
neutrino, the following expressions hold
%
\begin{eqnarray}
O^{bn\tilde b}_{Lij} &\approx& R^{\tilde b}_{j1}h_b\left(
a_3|\vec\Lambda|\delta_{i'3}+b\tilde\epsilon_{i'}\right)
+R^{\tilde b}_{j2}{{2g}\over{3\sqrt{2}}}t_Wa_1|\vec\Lambda|\delta_{i'3}
\nonumber\\
O^{bn\tilde b}_{Rij} &\approx& R^{\tilde b}_{j1}{g\over{\sqrt{2}}}
\left({\textstyle{1\over3}}t_Wa_1-a_2\right)|\vec\Lambda|\delta_{i'3}
+R^{\tilde b}_{j2}h_b\left(a_3|\vec\Lambda|\delta_{i'3}+
b\tilde\epsilon_{i'}\right)
\end{eqnarray}
%
where $i'=i-4$ label one of the neutrinos. $R^{\tilde b}_{jk}$ are the 
rotation matrices connecting weak and mass eigenstate basis for the 
scalar bottom quarks. In case of no intergenerational mixing in the 
squark sector $R^{\tilde b}_{jk}$ can be parameterized by just one 
diagonalizing angle $\theta_{\tilde b}$.


\subsection{Relevant Topologies}
\label{sec:relevant-topologies}

We now give the structure of the mass matrices relevant for the
determination of solar neutrino masses and mixings. While in the BRpV
model the atmospheric anomaly is explained at the tree-level, the
solar neutrino masses and mixings are both generated
radiatively. In particular, the ``solar angle'' has no meaning at the
tree level due to the degeneracy of the two lightest neutrinos in this
limit.

Diagonalizing the tree-level neutrino mass matrix first and adding 
then the 1-loop corrections before re-diagonalization the resulting 
neutrino/neutralino mass matrix has non-zero entries in the 
neutrino/neutrino, the neutrino/neutralino and in the neutralino/neutralino 
sectors. We have found that the most important part of the 1-loop 
neutrino masses derives from the neutrino/neutrino sector and that 
the 1-loop induced neutrino/neutralino mixing is usually subdominant. 


The relevant topologies for the one loop calculation of neutrino
masses are then illustrated in Fig.\ref{fig:topologies}.
%
\begin{figure}[ht]
  \begin{center}
  \begin{tabular}{ccc}
    \includegraphics[width=0.28\linewidth]{topo-1.eps}&
    \includegraphics[width=0.28\linewidth]{topo-2.eps}&
    \includegraphics[width=0.28\linewidth]{topo-5.eps}
  \end{tabular}
  \begin{tabular}{ccc}
    \includegraphics[width=0.28\linewidth]{topo-4.eps}&
     & \includegraphics[bb= 115 628 260 678,height=15mm]{topo-7.eps}
  \end{tabular}
\caption{Topologies for neutrino self-energies in the BRpV 
supersymmetric model}
\label{fig:topologies}
\end{center}
\end{figure}
%
Here our conventions are as follows: open circles with a cross inside
indicate genuine mass insertions which flip chirality.  On the other
hand open circles without a cross correspond to small R-Parity
violating projections, indicating how much of an Rp-even/odd mass
eigenstate is present in a given Rp-odd/even weak eigenstate.
Strictly speaking these projections are really coupling matrices
attached to the vertices, and this is what appears in the numerical
code.  However, given the smallness of Rp-violating effects, the
``insertion-method'' proves to be a rather useful tool to develop an
analytical perturbative expansion and to acquire some simple
understanding of the results.


\begin{equation}
  \label{eq:PV}
  \Delta B_0^{\tilde b_1\tilde b_2}=
  B_0(0,m_b^2,m_{\tilde b_1}^2)-B_0(0,m_b^2,m_{\tilde b_2}^2)
\end{equation}
as follows
%
\begin{equation}
\label{eq:bsb}
\Delta\widetilde\Pi_{ij}=-{{N_c m_b}\over{16\pi^2}}
2s_{\tilde b}c_{\tilde b}h_b^2
\Delta B_0^{\tilde b_1\tilde b_2}
\left[
 {{\tilde\epsilon_i\tilde\epsilon_j}\over{\mu^2}}
  + a_3 b \left(\tilde\epsilon_i\delta_{j3}+
 \tilde\epsilon_j\delta_{i3}\right)|\vec\Lambda|
 + \left( a_3^2 + \frac{ a_L a_R}{h^2_b}\right)
 \delta_{i3}\delta_{j3} 
 |\vec\Lambda|^2 \right] 
 \end{equation}
%
where we have defined
%
\begin{equation}
a_R={g\over{\sqrt{2}}}\left({1\over3}t_Wa_1-a_2\right)
\,,\qquad
a_L={g\over{\sqrt{2}}}\,{2\over3}t_Wa_1
\end{equation}
%

%
The different contributions can be understood as coming from the
graphs corresponding to the first topology of
Fig.~\ref{fig:topologies}. They have been depicted in more detail in
Fig.~\ref{fig:BottomSbottomLoop},
%
\begin{figure}[ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{diag-1.eps}
&\includegraphics[width=0.45\linewidth]{diag-3.eps}
\end{tabular}
\end{center}
\vspace{-3mm}
\caption{Bottom--Sbottom diagrams for solar neutrino mass in the BRpV model}
\label{fig:BottomSbottomLoop}
\end{figure}
%
where we have adopted the following conventions: a) as before, open
circles correspond to small R-parity violating projections, indicating
how much of a weak eigenstate is present in a given mass eigenstate,
(b) full circles correspond to R-parity conserving projections and (c)
open circles with a cross inside indicate genuine mass insertions
which flip chirality.

The open and full circles should really appear at the vertices since
the particles propagating in the loop are the mass eigenstates. We
have however separated them to better identify the origin of the
various terms.
%
There is another set of graphs analogous to the previous ones which
corresponds to the heavy sbottom. They are obtained from the previous
graphs making the replacement $\tilde b_1\rightarrow\tilde b_2$,
$s_{\tilde b}\rightarrow c_{\tilde b}$ and $c_{\tilde b}\rightarrow
-s_{\tilde b}$.  Note that for all contributions to the $2\times2$
submatrix corresponding to the light neutrinos the divergence from
$B_0(0,m_b^2,m_{\tilde b_1}^2)$ is canceled by the divergence from
$B_0(0,m_b^2,m_{\tilde b_2}^2)$, making finite the contribution from
bottom-sbottom loops to this submatrix, as it should be, since the
mass is fully ``calculable''.


\subsection{The solar mass}
\label{sec:solar-mass}

First we note that for nearly all points in our random sets we find
that $m_{\nu_2} \ll m_{\nu_3}$.  In other words, bilinear R-parity
breaking favors a hierarchical neutrino spectrum. Moreover, we have found
numerically that the terms proportional to ${\tilde \epsilon_i}\times
{\tilde \epsilon_j}$ in the self energies in Eq.~(\ref{eq:bsb}) give
the most important contribution to $m_{\nu_2}$ in the bottom-sbottom 
loop calculation in most points of our sets. If these terms are dominant 
one can find a very simple approximation for the bottom-sbottom loop
contribution to $m_{\nu_2}$.  It is given by
\begin{equation}
\label{Simplest}
m_{\nu_2} \simeq \frac{3}{16 \pi^2} \sin(2\theta_{\tilde b}) 
m_b \Delta B_0^{\tilde\tau_2\tilde\tau_1}\ 
\frac{({\tilde \epsilon}_1^2 + {\tilde \epsilon}_2^2)}{\mu^2}
\end{equation}

We have checked numerically that Eq.~(\ref{Simplest}) reproduces 
the result of the full approximative formula to high 
accuracy if $m_{\nu_2} \le 0.3 m_{\nu_3}$ . Note also that 
Eq. (\ref{Simplest}) holds only if the 1-loop contributions to 
the neutrino mass matrix are smaller than the tree-level one. 
This condition requires that $|{\vec \epsilon}|^2/|\Lambda| \le 1$ 
approximately, i.e. the bilinear parameters $\epsilon_i$ must 
be suppressed with respect to $\mu$. Note that such a suppression 
could, in principle, be motivated by suitable flavour 
symmetries \cite{Mira:2000gg}.


Due to the more complicated structure of the charged scalar loop it is
not possible to give a simple equation for $m_{\nu_2}$ similar to
Eq.~(\ref{Simplest}) for the bottom-sbottom loop.  However, for
$m_{\nu_2}$ larger than (few) $\times 10^{-4}$ we have found that the
most important contributions to the charged scalar loop are the
terms proportional to $\Delta B_0^{\tilde\tau_2\tilde\tau_1}$, 
$\Delta B_0^{H^{\pm}\tilde\tau_1}$ and $\Delta B_0^{H^{\pm}\tilde\tau_2}$ 
in Eq.CHARGEDHIGGSLOOP. We note in passing that Eq.  (\ref{Simplest}),
with appropriate replacements, allows us to estimate the typical
contributions to the charged scalar loop within a factor of $\sim 3$.
However, such an estimate will be biased toward too small (large)
$m_{\nu_2}$ for scalar tau (neutralino) LSPs.

\begin{figure}[htbp] \centering
  \begin{tabular}{cc}
    \includegraphics[width=0.45\linewidth]{Nf2002Sim-v4.eps}
    &\includegraphics[width=0.45\linewidth]{StauSim-v4.eps}
  \end{tabular}
\caption[] {($m_{\nu_2}^{Appr}/m_{\nu_2}^{exact}$) versus 
  $\Dms$ [$eV^2$] for the set Ntrl (left) and Stau (right). Shown is
  the result of the simplified approximation formula in
  Eq.~(\ref{Simplest}) for the sbottom-bottom loop and taking into 
  account only coefficients
  $C_{H^{\pm}\tilde\tau_2}$, $C_{H^{\pm}\tilde\tau_1}$ and
  $C_{\tilde\tau_2\tilde\tau_1}$ in the charged scalar loop.}
\label{fig:Simp}
\end{figure}

In Fig. (\ref{fig:Simp}) we show a comparison of our simplified approximation 
formula, including the simple form of the sbottom-bottom loop and 
the three most important coefficients for the charged scalar loop, 
as discussed above, to the full numerical calculation including all 
loops. As one can see, even the simplified version of our formula 
works surprisingly well in the LMA-MSW regime, although the agreement 
with the full calculation is now less good for the LOW region, as 
could have been expected from the results discussed previously. 



\subsection{The solar mixing angle}
\label{sec:solar}

In the basis where the tree-level neutrino mass matrix is diagonal the
mass matrix at one--loop level can be written as 
%
\begin{eqnarray}
  \label{eq:2}
  \widetilde m_\nu= V_\nu^{(0)T} m_\nu V_\nu^{(0)} 
      =\left( \begin{array}{ccc}
      c_1 \widetilde \epsilon_1 \widetilde \epsilon_1 & c_1 \widetilde
      \epsilon_1 \widetilde \epsilon_2  
      & c_1 \widetilde \epsilon_1 \widetilde \epsilon_3 \\
      c_1 \widetilde \epsilon_2 \widetilde \epsilon_1 & c_1 \widetilde
      \epsilon_2 \widetilde \epsilon_2  
      & c_1 \widetilde \epsilon_2 \widetilde \epsilon_3 \\
      c_1 \widetilde \epsilon_3 \widetilde \epsilon_1 & c_1 \widetilde
      \epsilon_3 \widetilde \epsilon_2  
      & c_0 |\vec\Lambda|^2 + c_1 \widetilde \epsilon_3 \widetilde \epsilon_3 
     \end{array} \right) + \cdots
\end{eqnarray}
%
where the $\widetilde \epsilon_i$ were defined before in
Eq.~(\ref{eq:TildeEpsilon}). Coefficients $c_0$ and $c_1$ contain 
couplings and supersymmetric masses. Since they cancel in the final 
expression for the angle their exact definition is not necessary in 
the following. Dots stand for other terms which we will 
assume to be less important in the following, see the discussion at 
the end of this subsection.  
This matrix can be diagonalized approximately under the condition 
%
\begin{eqnarray}
  \label{eq:3}
  x\equiv\frac{c_1 |\vec{\widetilde{\epsilon}}|^2}{c_0 |\vec \Lambda|^2 }\ll 1
\end{eqnarray}
%
i.e. if the 1-loop contribution to the neutrino mass matrix is smaller 
than the tree-level contribution, as also discussed above for 
Eq. (\ref{Simplest}). Then
%
\begin{eqnarray}
  \label{eq:5}
  \widetilde m_\nu= c_0 |\vec\Lambda|^2 
      \left( \begin{array}{ccc} \displaystyle
      x\, \frac{\widetilde \epsilon_1 \widetilde \epsilon_1 }{
      |\vec{\widetilde \epsilon|^2} } 
      &\displaystyle x\, \frac{\widetilde \epsilon_1 \widetilde
      \epsilon_2 }{|\vec{\widetilde \epsilon|^2}} 
      & \displaystyle x\, \frac{\widetilde \epsilon_1 \widetilde
      \epsilon_3 }{|\vec{\widetilde \epsilon|^2}} \\[+4mm]
      \displaystyle
      x\, \frac{\widetilde \epsilon_2 \widetilde \epsilon_1 }{
      |\vec{\widetilde \epsilon|^2 }}
      &\displaystyle x\, \frac{\widetilde \epsilon_2 \widetilde
      \epsilon_2 }{|\vec{\widetilde \epsilon|^2}}
      & \displaystyle x\, \frac{\widetilde \epsilon_2 \widetilde
      \epsilon_3 }{|\vec{\widetilde \epsilon|^2}} \\[+4mm] 
      \displaystyle
      x\, \frac{\widetilde \epsilon_3 \widetilde \epsilon_1 }{
        |\vec{\widetilde \epsilon|^2 }}
      &\displaystyle x\, \frac{\widetilde \epsilon_3 \widetilde
      \epsilon_2 }{|\vec{\widetilde \epsilon|^2}}
      & \displaystyle 1 + x\, \frac{\widetilde \epsilon_3 \widetilde
      \epsilon_3 }{|\vec{\widetilde \epsilon|^2}}  
     \end{array} \right)
\end{eqnarray}
%
We now calculate the eigenvalues and eigenvectors of this matrix as
series expansions in the small $x$ parameter. For the eigenvalues we
get
%
\begin{eqnarray}
  \label{eq:6}
  m_1&=&0\nonumber  \\
  m_2&=& x\, c_0 \frac{|\vec\Lambda|^2}{|\vec{\widetilde \epsilon|^2}}
 + \mathcal{O}(x^2)  = 
  c_1 \left( \widetilde \epsilon_1^2+\widetilde \epsilon_2^2 \right)
  + \mathcal{O}(x^2)\\
  m_3&=& c_0 |\vec\Lambda|^2 + c_1 \widetilde \epsilon_3^2 +
  \mathcal{O}(x^2) \nonumber 
\end{eqnarray}
%
and for the first two eigenvalues (the third can also be easily
obtained but it will not be necessary for the discussion of the solar
mixing angle),
%
\begin{eqnarray}
  \label{eq:7}
  e_1&=&\left(-\frac{\widetilde \epsilon_2}{\widetilde \epsilon_1}\, 
  \sqrt{\frac{\widetilde \epsilon_1^2}{\widetilde \epsilon_1^2+
   \widetilde \epsilon_2^2}},\ \sqrt{\frac{\widetilde
  \epsilon_1^2}{\widetilde \epsilon_1^2+ \widetilde
  \epsilon_2^2}},\ 0\right)\nonumber \\[+2mm]
  e_2&=&\left( e_{2,1}, e_{2,2}, e_{2,3}\right)
\end{eqnarray}
%
where up to $\mathcal{O}(x^2)$ we have,
%
\begin{eqnarray}
  \label{eq:8}
  e_{2,1}&=& -\frac{\widetilde \epsilon_1 \widetilde \epsilon_3}
  {\sqrt{\widetilde \epsilon_3^2(\widetilde \epsilon_1^2 +\widetilde
  \epsilon_2^2 )}} +\frac{1}{2}\, \frac{\widetilde \epsilon_1
  \widetilde \epsilon_3 \sqrt{\widetilde \epsilon_3^2(\widetilde
  \epsilon_1^2 +\widetilde \epsilon_2^2 )}}{|\vec{\widetilde
  \epsilon|^4}}\ x^2 + \mathcal{O}(x^3)\nonumber
 \\[+2mm]
  e_{2,2}&=& -\frac{\widetilde \epsilon_2 \widetilde \epsilon_3}
  {\sqrt{\widetilde \epsilon_3^2(\widetilde \epsilon_1^2 +\widetilde
  \epsilon_2^2 )}}+\frac{1}{2}\, \frac{\widetilde \epsilon_2
  \widetilde \epsilon_3 \sqrt{\widetilde \epsilon_3^2(\widetilde
  \epsilon_1^2 +\widetilde \epsilon_2^2 )}}{|\vec{\widetilde
  \epsilon|^4}}\ x^2 + \mathcal{O}(x^3)
 \\[+2mm]
  e_{2,3}&=& \frac{\sqrt{\widetilde \epsilon_3^2(\widetilde
  \epsilon_1^2 +\widetilde  \epsilon_2^2 )}}{|\vec{\widetilde
  \epsilon|^2}}\ x + \frac{\left(\widetilde  \epsilon_1^2 +\widetilde
  \epsilon_2^2-\widetilde \epsilon_3^2 \right) \sqrt{\widetilde
  \epsilon_3^2(\widetilde \epsilon_1^2 +\widetilde
  \epsilon_2^2)}}{|\vec{\widetilde  \epsilon|^4}}\ x^2 +
  \mathcal{O}(x^3) \nonumber 
\end{eqnarray}
%
Knowing the eigenvectors we can write down the rotation matrix that
diagonalizes $\widetilde m_\nu$,
%
\begin{eqnarray}
  \label{eq:9}
  \widetilde V_\nu^T \widetilde m_\nu \widetilde V_\nu
  =\hbox{diag}(m_1,m_2,m_3) 
\end{eqnarray}
%
where
%
\begin{eqnarray}
  \label{eq:10}
   \widetilde V_\nu^T= \left( \begin{array}{ccc}
       e_{1,1} & e_{1,2}  & e_{1,3} \\
       e_{2,1} & e_{2,2}  & e_{2,3} \\
       e_{3,1} & e_{3,2}  & e_{3,3} 
       \end{array}\right)
\end{eqnarray}
%
The neutrino mixing matrix is then given by
%
\begin{eqnarray}
  \label{eq:11}
  U=\left(  V_\nu^T  \widetilde V_\nu^T\right)^T
\end{eqnarray}
%
Using the fact that $U_{e3}$ has to be small one can get the following
expression for the solar mixing angle:
\begin{eqnarray}
  \label{eq:sol1}
  \tan^2\theta_\Sol  = \frac{U_{e2}^2}{U_{e1}^2}
\end{eqnarray}
%
Now using the Eqs.~(\ref{eq:10}), (\ref{eq:8}) and substituting in
Eq.~(\ref{eq:11}) we obtain the very simple expression for the solar
mixing angle,
%
\begin{eqnarray}
  \label{eq:12}
   \tan^2\theta_\Sol  = \frac{\widetilde  \epsilon_1^2}{\widetilde
  \epsilon_2^2} 
\end{eqnarray}
%
This formula is a very good approximation if the one--loop matrix has 
the structure $\epsilon_i \times \epsilon_j$, as is the case of the
bottom-sbottom loop (and, to a lesser extend also for the charged scalar 
loop, which has one coefficient with the same index structure), and if 
$m_{\nu_3} \gg m_{\nu_2}$ . 





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