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\setcounter{chapter}{4}
\chapter{Supersymmetry}

\section{Introduction and Motivation}


As we saw in the last chapter, it has been established in recent years
that the standard model (SM) describes, with great
precision~\cite{hagiwara:2002fs}, all the presently known phenomena,
with the important exception of neutrino masses and mixings.

Despite all its successes, the SM still has many unanswered
questions. Besides the question of the neutrinos, we saw that the SM
has too many free unrelated parameters, charge quantization is
introduced by hand and the mass spectrum is not explained. This seems
to suggest that it should be a low energy effective theory that should
embedded in some other theory valid up to much higher
energies. However the attempts at such embedding in Grand Unified
Theories seem to bring more problems, as we saw.
Among the various candidates to physics beyond the standard
model, supersymmetric theories play a special role. Although 
there is not yet direct experimental evidence for supersymmetry (SUSY),  
there are many theoretical arguments indicating that SUSY might be of
relevance for physics below the 1 TeV scale. 

\renewcommand{\theenumi}{\it \roman{enumi}}


The most commonly invoked theoretical arguments for SUSY are:
%
\begin{enumerate}
\item
Interrelates matter fields (leptons and quarks) with force fields
(gauge and/or Higgs bosons).
\item
As local SUSY implies gravity (supergravity)it could provide a way 
to unify gravity with the other interactions.
\item
As SUSY and supergravity have fewer divergences than conventional
field theories, the hope is that it could provide a consistent
(finite) quantum gravity theory.
\item
SUSY can help to understand the mass problem, in particular solve the
naturalness and hierarchy problems, 
if SUSY particles have masses $\le {\cal O} (1 \hbox{TeV})$.
\end{enumerate}
%
\renewcommand{\theenumi}{\bf \Alph{enumi}}
%
As it is the last argument that makes SUSY particularly attractive for
the experiments being done or proposed for the next decade, let us
explain the idea in more detail. As the SM is not asymptotically free,
at some energy scale $\Lambda$, the interactions must become strong
indicating the existence of new physics. Candidates for this scale
are, for instance, $M_X \simeq {\cal O}(10^{16}\ \hbox{GeV})$ in GUT's
or more fundamentally the Planck scale $M_P \simeq {\cal O}(10^{19}
\hbox{GeV})$.
This alone does not indicate that the new physics should be related to
SUSY, but the so--called mass problem does. The only consistent way to
give masses to the gauge bosons and fermions is through the Higgs
mechanism involving at least one spin zero Higgs boson. Although the
Higgs boson mass is not fixed by the theory, a value much bigger than
$< H^0>\simeq \left(\sqrt{2}G_F\right)^{-1/2}\simeq 250\ \hbox{GeV}$
would imply that the Higgs sector would be strongly coupled making it
difficult to understand why we are seeing an apparently successful
perturbation theory at low energies. We saw in
section~\ref{sec:hierarchy} that the one loop radiative
corrections to the Higgs boson mass would give
%
\begin{equation}
\delta m_H^2=\frac{1}{16\pi^2} \left[\vb{16} \left( -2 N_f \lambda_f^2 +
    \lambda_S N_S\right)
\Lambda^2  + \cdots \right]
\end{equation}
%

\ni
where $N_f$ and $N_S$ stand for the number of fermion and scalar
fields, respectively. This correction would be too large if $\Lambda$
is identified with $\Lambda_{GUT}$ or $\Lambda_{Planck}$. SUSY cures
this problem in the following way. If SUSY were exact, then for each
Dirac fermion we would have two complex scalar fields and the
couplings would be related in such a way that $\lambda_f^2=\lambda_S$.
Therefore dangerous terms in the radiative corrections to the scalar
masses squared would be absent because the contribution of fermion
loops would exactly cancel against the boson loops. This can be easily
understood if we anticipate, as we shall see in
section~\ref{sec:SUSYalgebra}, that exact supersymmetry requires the
degeneracy of the masses of scalar and fermion particles belonging to the same
supermultiplet. As it is well known that radiative corrections to
fermion masses diverge at most logarithmically, the same should also
happen to the scalar masses to all orders of perturbation theory.
However we know that SUSY is not exact, as we have not found in Nature
the scalar partners of the known particles with degenerate
masses. SUSY should therefore be broken as we will discuss later in
this chapter.

However we do not want the supersymmetry breaking to spoil the
solution to the hierarchy problem. In particular we do not want that
relations of the type $\lambda_f^2=\lambda_S$ would cease to hold
after the breaking of supersymmetry. It can be shown that for this to
happen the supersymmetry breaking lagrangian has to have a particular
form. We say that the breaking has to be ``soft''. The effective
lagrangian is then


\begin{equation}
  \label{eq:16}
  {\cal L}={\cal L}_{\hbox{\scriptsize SUSY}}+{\cal L}_{
  \hbox{\scriptsize soft}}
\end{equation}

\ni
where ${\cal L}_{\hbox{\scriptsize SUSY}}$ has the exact supersymmetry
and ${\cal L}_{\hbox{\scriptsize soft}}$, that breaks supersymmetry,
has only mass terms and couplings with positive mass dimension, to
ensure that relations of the type $\lambda_f^2=\lambda_S$ remain
unbroken. If we denote the largest mass scale in ${\cal
L}_{\hbox{\scriptsize soft}}$ by $m_{\hbox{\scriptsize soft}}$ then the
non supersymmetric corrections to the Higgs boson mass should vanish
in the $m_{\hbox{\scriptsize soft}}\rightarrow 0$ limit. In fact it
can be shown that


\begin{equation}
\delta m_H^2=m^2_{\hbox{\scriptsize soft}}\, \frac{\lambda}{16\pi^2}\, \ln
\frac{\Lambda}{m_{\hbox{\scriptsize soft}}} + \cdots
\end{equation}

\ni
for a generic dimensionless $\lambda$ coupling. For this solution to
the hierarchy problem to make sense, $m_{\hbox{\scriptsize soft}}$
cannot be too large, at most $\leq{\cal O}(1\ \hbox{TeV})$. As the
splitting between the known particles and their superpartners should
be of the order of $m_{\hbox{\scriptsize soft}}$ we conclude that SUSY
provides a solution for the the naturalness problem if the masses of
the superpartners are below ${\cal O}(1\ \hbox{TeV})$. This is the
main reason behind all the phenomenological interest in SUSY.

In the following we will give a brief review of the main aspects of
the SUSY extension of the SM, the so--called Minimal Supersymmetric
Standard Model (MSSM). Almost all the material is covered in many
excellent reviews that exist in the literature~\cite{nilles:1984ge,
haber:1985rc,martin:1997ns}.


\section{SUSY Algebra, Representations and Particle Content}

\subsection{SUSY Algebra}
\label{sec:SUSYalgebra}

The SUSY generators obey the following algebra
%
\begin{eqnarray}
\left\{Q_{\alpha},Q_{\beta} \right\} &=&0 \cr
\vb{18}
\left\{\ovl{Q}_{\dot \alpha},\ovl{Q}_{\dot \beta} \right\} &=&0 \cr
\vb{18}
\left\{Q_{\alpha},\ovl{Q}_{\dot \beta} \right\} &=& 
2 \left( \sigma^{\mu} \right)_{\alpha \dot \beta} \, P_{\mu}
\label{susyalgebra}
\end{eqnarray}
%
where 
%
\begin{equation}
\sigma^{\mu}\equiv (1, \sigma^i) \quad ; \quad \ovl{\sigma}^{\mu} \equiv (1,
-\sigma^i)
\label{sigma}
\end{equation}
%
and $\alpha,\beta,\dot \alpha, \dot \beta=1,2$ (Weyl 2--component
spinor notation) and we follow the conventions of
Ref.~\cite{haber:1985rc}.  The commutation relations with the
generators of the Poincar\'e group are
%
\begin{eqnarray}
\left[P^{\mu},Q_{\alpha} \right] &=&0 \cr
\vb{18}
\left[M^{\mu \nu},{Q}_{\alpha} \right] &=& -i \left( \sigma^{\mu \nu}
\right)_{\alpha}{}^{\beta}\, Q_{\beta}
 \nn
\end{eqnarray}
%
From these relations one can easily derive that the two invariants of
the Poincar\'e group,
%
\begin{equation}
\ba{l}
P^2=P_{\alpha} P^{\alpha} \\
W^2=W_{\alpha} W^{\alpha} 
\ea
\end{equation}
%
where $W^{\mu}$ is the Pauli--Lubanski vector operator
%
\begin{equation}
W_{\mu}=-\frac{i}{2} \epsilon_{\mu \nu \rho \sigma} M^{\nu \rho} P^{\sigma}
\end{equation}
%
are no longer invariants of the Super Poincar\'e group. In fact
%
\begin{eqnarray}
[Q_{\alpha},P^2] &=& 0 \cr
[Q_{\alpha},W^2] & \neq & 0
\end{eqnarray}
%
showing that the irreducible multiplets will have particles of the
same mass but with different spin.

\subsection{Simple Results from the Algebra}

From the supersymmetric algebra one can derive two important results:

\begin{enumerate}

\item
{\bf Number of  Bosons = Number of  Fermions}

We have
%
\begin{eqnarray}
Q_{\alpha} |B> =|F> \quad &;& \quad (-1)^{N_F} |B> =|B> \cr
\vb{18}
Q_{\alpha} |F> =|B> \quad &;& \quad (-1)^{N_F} |F> =- |F> 
\end{eqnarray}
%
where $(-1)^{N_F}$ is the fermion number of a given state. Then we obtain
%
\begin{equation}
Q_{\alpha} (-1)^{N_F}=-(-1)^{N_F} Q_{\alpha} 
\end{equation}
%
Using this relation we can show that
%
\begin{eqnarray}
Tr \left[(-1)^{N_F} \left\{Q_{\alpha},\ovl{Q}_{\dot \alpha} \right\}
\right] 
&\hskip -1mm=\hskip -1mm& 
Tr \left[(-1)^{N_F} Q_{\alpha} \ovl{Q}_{\dot \alpha} +
(-1)^{N_F} \ovl{Q}_{\dot \alpha} Q_{\alpha} \right] \cr
\vb{18}
&\hskip -1mm=\hskip -1mm& 
Tr \left[- Q_{\alpha} (-1)^{N_F} \ovl{Q}_{\dot \alpha} +
Q_{\alpha} (-1)^{N_F} \ovl{Q}_{\dot \alpha} \right] \cr
\vb{18}
&\hskip -1mm=\hskip -1mm& 0 \nn
\end{eqnarray}
%
But using \Eq{susyalgebra} we also have
%
\begin{equation}
Tr \left[(-1)^{N_F} \left\{Q_{\alpha},\ovl{Q}_{\dot \alpha} \right\}
\right] =Tr \left[(-1)^{N_F} 2 \sigma^{\mu}_{\alpha \dot \alpha} P_{\mu}
\right]
\end{equation}
%
This in turn implies
\ms

\begin{equation}
  \label{eq:12}
  Tr(-1)_{N_F}=\# \hbox{Bosons} -\# \hbox{Fermions}=0  
\end{equation}




showing that in a given representation the number of degrees of
freedom of the bosons equals those of the fermions.


\item
{\bf $\vev{0|H|0} \ge 0$}


From the algebra we get
%
\begin{eqnarray}
\left\{Q_1,\ovl{Q}_{\dot 1} \right\}+
\left\{Q_2,\ovl{Q}_{\dot 2} \right\} &=& 2 Tr \left( \sigma^{\mu}\right)
P_{\mu} \cr
\vb{18}
&=& 4 H
\end{eqnarray}
%
Then
%
\begin{equation}
H=\frac{1}{4}\, \left( Q_1 \ovl{Q}_1  + Q_2 \ovl{Q}_{\dot 2 }
+ \ovl{Q}_{\dot 1} Q_1+ \ovl{Q}_{\dot 2} Q_2 \right)
\end{equation}
%
and
%
\begin{eqnarray}
\vev{0|H|0}&\hskip -2mm=\hskip -2mm& 
\left( ||Q_1 \ket{0} ||^2 +||Q_1 \ket{0} ||^2 +
||\ovl{Q}_{\dot 1} \ket{0} ||^2 +||\ovl{Q}_{\dot 2} \ket{0} ||^2 \right)\cr
\vb{18}
&\hskip -2mm\ge\hskip -2mm&0
\end{eqnarray}
%
showing that the energy of the vacuum state is always positive definite.

\end{enumerate}


\subsection{SUSY Representations}


We consider separately the massive and the massless case.

\begin{enumerate}
\item
{\bf Massive case}


In the rest frame
%
\begin{equation}
\left\{Q_{\alpha},\ovl{Q}_{\dot \alpha} \right\}= 2\, m\ \delta_{\alpha \dot
\alpha} 
\end{equation}
%
This algebra is similar to the algebra of the spin 1/2 creation and
annihilation operators. 
Choose $\ket{\Omega}$ such that
%
\begin{equation}
Q_1 \ket{\Omega}=Q_2 \ket{\Omega}=0
\end{equation}
%
Then we have 4 states
%
\begin{equation}
\ket{\Omega} \ ; \ \ovl{Q}_1 \ket{\Omega} \ ; \ 
\ovl{Q}_2 \ket{\Omega} \ ; \ 
\ovl{Q}_1 \ovl{Q}_2 \ket{\Omega} 
\end{equation}
%
If $ J_3 \ket{\Omega} =j_3 \ket{\Omega} $ we  show in 
Table~\ref{massive} the values of $J_3$ for the 4 states.
%
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|}\hline
State& $J_3$ Eigenvalue \cr \hline\hline
$\ket{\Omega}$&$j_3$\cr
$\ovl{Q}_1 \ket{\Omega}$&$j_3 +\frac{1}{2}$\cr
$\ovl{Q}_2 \ket{\Omega}$&$j_3 -\frac{1}{2}$\cr
$\ovl{Q}_1 \ovl{Q}_2 \ket{\Omega}$&$j_3 $\cr\hline
\end{tabular}
\end{center}
\caption{Massive states}
\label{massive}
\end{table}
%
We notice that there two bosons and two fermions and that the states are
separated by one half unit of spin.


\item 
{\bf Massless case}

If $m=0$ then we can choose $P^{\mu}=(E,0,0,E)$. In this frame
%
\begin{equation}
\left\{Q_{\alpha},\ovl{Q}_{\dot \alpha}\right\} = M_{\alpha \dot \alpha} 
\end{equation}
%
where the matrix $M$ takes the form
%
\begin{equation}
M=\left(\matrix{0&0\cr
0&4E\cr}
\right)
\end{equation}
%
Then
%
\begin{equation}
\left\{Q_2,\ovl{Q}_2 \right\} = 4 E
\end{equation}
%
all others vanish. We have then just {\bf two} states
%
\begin{equation}
\ket{\Omega} \ ; \ \ovl{Q}_2 \ket{\Omega} 
\end{equation}
%
If $J_3 \ket{\Omega} =\lambda \ket{\Omega} $ we have the states shown
in Table~\ref{massless},

\bs
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|}\hline
State& $J_3$ Eigenvalue \cr \hline\hline
$\ket{\Omega}$&$\lambda$\cr
$\ovl{Q}_2 \ket{\Omega}$&$\lambda -\frac{1}{2}$\cr\hline
\end{tabular}
\end{center}
\caption{Massless states}
\label{massless}
\end{table}

\end{enumerate}


\section{How to Build a SUSY Model}

To construct supersymmetric Lagrangians one normally uses superfield
methods (see for instance~\cite{wess:xxxx,ross:xxxx,srivastava:xxxx,
freund:xxxx,west:xxxx}). Here we do not go into the details of that
construction.  We will take a more pragmatic view and give the results
in the form of a {\it recipe}~\cite{haber:1985rc,martin:1997ns}. To
simplify matters even further we just consider one gauge group
$G$. Then the gauge bosons $W_{\mu}^a$ are in the adjoint
representation of $G$ and are described by the massless gauge
supermultiplet
%
\begin{equation}
V^a\equiv(\lambda^a,W_{\mu}^a)
\end{equation}
%
where $\lambda^a$ are the superpartners of the gauge bosons, the
so--called {\it gauginos}. We also consider only one matter chiral
superfield 
%
\begin{equation}
\Phi_i\equiv(A_i,\psi_i) \qquad ; \qquad (i=1,\ldots,N)
\end{equation}
%
belonging to some $N$ dimensional representation of $G$. We will give
the rules for the different parts of the Lagrangian for these
superfields. The generalization to the case where we have more
complicated gauge groups and more matter
supermultiplets, like in the MSSM,  is straightforward.

\subsection{Kinetic Terms}

Like in any gauge theory we have
%
\begin{equation}
{\cal L}_{kin}= -\fourth F_{\mu \nu}^a F^{a \mu \nu} + \frac{i}{2}\
\ovl{\lambda^a} \gamma^{\mu} D_{\mu} \lambda^a 
+ \left(D_{\mu} A\right)^{\dagger} D^{\mu} A + i \ovl{\psi}
\gamma^{\mu} D_{\mu} P_L \psi
\label{Lkin}
\end{equation}
%
where the covariant derivative is
%
\begin{equation}
D_{\mu}=\partial_{\mu} + i g W_{\mu}^a T^a
\end{equation}
%
In \Eq{Lkin} one should note that $\psi$ is left handed and
that $\lambda$ is a Majorana spinor.


\subsection{Self Interactions of the Gauge Multiplet}

For a non Abelian gauge group $G$ we have the usual self--interactions
(cubic and quartic) of the gauge bosons with themselves. These are
well known and we do write them here again. 
But we have a new interaction of the gauge bosons with the
gauginos. In two component spinor notation it
reads~\cite{haber:1985rc},
%
\begin{equation}
{\cal L}_{\lambda \lambda W}=
i g f_{abc}\, \lambda^a \sigma^{\mu} \ovl{\lambda}^b\, W_{\mu}^c + \hbox{h.c.}
\end{equation}
%
where $f_{abc}$ are the structure constants of the gauge group $G$ and
the matrices $\sigma^{\mu}$ were introduced in \Eq{sigma}.



\subsection{Interactions of the Gauge and Matter Multiplets}

In the usual non Abelian gauge theories we have the interactions of
the gauge bosons with the fermions and scalars of the theory. In the
supersymmetric case we
also have interactions of the gauginos with the fermions and scalars
of the chiral matter multiplet. The general form, in two component
spinor notation, is~\cite{haber:1985rc},
%
\begin{eqnarray}
{\cal L}_{\Phi W}&=&-g T^a_{ij} W_{\mu}^a \left( \ovl{\psi}_i
\ovl{\sigma}^{\mu} \psi_j + i A_i^* \lrvpartial_{\mu} A_j  \right)
+ g^2 \left(T^a T^b\right)_{ij} W_{\mu}^a W^{\mu b}\, A_i^* A_j \cr
\vb{18}
&&+ ig \sqrt{2}\, T_{ij}^a \left(\lambda^a \psi_j A_i^*
-\ovl{\lambda}^a \ovl{\psi}_i A_j\right)
\end{eqnarray}
%
where the new interactions of the gauginos with the fermions and
scalars are given in the last term.


\subsection{Self Interactions of the Matter Multiplet}

These correspond in non supersymmetric gauge theories both to the
Yukawa interactions and to the scalar potential. In supersymmetric
gauge theories we have less freedom to construct these terms. The
first step is to construct the superpotential $W$. This must be a
gauge invariant polynomial function of the {\it scalar} components of
the chiral multiplet $\Phi_i$, that is $A_i$. It {\it does not} depend
on $A_i^*$. In order to have renormalizable theories the degree of the
polynomial must be at most three. This is in contrast with non
supersymmetric gauge theories where we can construct renormalizable
theories with a scalar potential being a polynomial up to the fourth
degree. The most general form is then\footnote{
We do not consider a term linear in $A$ because we are assuming that
$\Phi$, and hence $A$, are not gauge singlets.},
%
\begin{equation}
  \label{eq:21}
  W=\frac{1}{2}\, M_{ij} A_i A_j + \frac{1}{6}\, Y_{ijk} A_i A_j A_k
\end{equation}
%
where $M_{ij}$ and $Y_{ijk}$ should be such that $W$ is gauge invariant.
Once we have the superpotential $W$, then the theory is defined and 
the Yukawa interactions are
%
\begin{equation}
{\cal L}_{Yukawa}=-\half \left[
\frac{\partial^2 W}{\partial A_i \partial A_j}\ \psi_i \psi_j +
\left(\frac{\partial^2 W}{\partial A_i \partial A_j}\right)^*\ 
\ovl{\psi}_i \ovl{\psi}_j \right]
\label{eq:LYukawa}
\end{equation}
%
and the scalar potential is
%
\begin{equation}
V_{scalar}=\half D^a D^a + F_i F_i^*
\end{equation}
%
where
%
\begin{eqnarray}
F_i&=& \frac{\partial W}{\partial A_i}\cr
\vb{18}
D^a&=& g\ A_i^* T_{ij}^a A_j
\end{eqnarray}
%
We see easily from these equations that, if the polynomial degree of W
were higher than three, then the scalar potential would be a polynomial
of degree higher than four and hence non renormalizable.

\subsection{Supersymmetry Breaking Lagrangian}

As we have not discovered superpartners of the known particles with
the same mass, we conclude that
 SUSY has to be broken. How this done is the least
understood sector of the theory. In fact, as we shall see, the
majority of the unknown parameters come from this sector. As we do not
want to spoil the good features of SUSY, the form of these SUSY
breaking terms has to obey some restrictions. As we discussed before,
the added terms can only be mass terms, or have the same form
of the superpotential, with arbitrary coefficients. These are called
{\it soft terms}. Therefore, for the model that we are considering,
the general form would be,
%
\begin{equation}
 \label{eq:Lsoft}
{\cal L}_{\hbox{\scriptsize soft}}
=-(m^2)_{ij} A^*_i A_j - \frac{1}{2}\, M_{\lambda}  \left( \lambda^a
\lambda^a + \ovl{\lambda}^a \ovl{\lambda}^a \right)
-\left( 
\frac{1}{2}\, b_{ij} A_i A_j + \frac{1}{6}\, a_{ijk} A_i A_j A_k
+ \hbox{h.c} \right)
\end{equation}
%
where $a_{ijk}$ and $b_{ij}$ have dimensions of $mass$ and $(mass)^2$,
respectively, and are such that the resulting supersymmetry breaking
lagrangian, ${\cal L}_{\hbox{\scriptsize soft}}$, is gauge invariant.
Besides the requirement of gauge invariance these parameters are in
general arbitrary.  It is clear that ${\cal L}_{\hbox{\scriptsize
soft}}$ breaks supersymmetry as it gives masses to the gauginos and to
the scalar fields and not to their superpartners. It has been
shown~\cite{girardello:xxxx}, that a supersymmetry breaking of the form
of Eq.~(\ref{eq:Lsoft}) does not give rise to the dangerous quadratic
divergences that we discussed before.



\subsection{R--Parity}

As we see below, the requirements of gauge invariance for the
superpotential $W$ will in general not forbid terms that could lead to
baryon (B) and lepton (L) number violation. In particular it could be
possible to have models where the proton would decay. Although this is
not a fundamental principle, in many models these terms are forbidden
by requiring that the superpotential is invariant under a
multiplicatively quantum number, called {\it R--parity}. It is
defined as
%
\begin{equation}
R=(-1)^{2J + 3B +L}
\end{equation}
%
With this definition it has the value $+1$ for the known particles and
$-1$ for their superpartners. The MSSM it is a model where R--parity
is conserved. The conservation of R--parity has
three important consequences: {\it i)} SUSY particles are pair
produced, {\it ii)} SUSY particles decay into SUSY particles and {\it iii)}
The lightest SUSY particle is stable (LSP). 

As we said above, there is no fundamental principle that requires
R--parity to be conserved. In fact in the next chapters we will
introduce models where R--parity is not conserved. We will see that
these models are in perfect agreement with experiment, including the
bounds on lepton number violating processes, and can accommodate the
observed pattern of neutrino masses and mixing angles. However in this
chapter we will describe the MSSM and here we have R--parity
conserved.


\renewcommand{\theenumi}{\arabic{enumi}}


\section{The Minimal Supersymmetric Standard Model}


\subsection{The Gauge Group and Particle Content}

We want to describe the supersymmetric version of the SM. Therefore
the gauge group is considered to be that of the SM, that is
%
\begin{equation}
G=SU_c(3)\ot SU_L(2)\ot U_Y(1)
\end{equation}
%
We will now describe the minimal particle content. 

\begin{itemize}
\item
{\bf Gauge Fields}

We want to have gauge fields for the gauge group 
$G=SU_c(3)\ot SU_L(2)\ot U_Y(1)$. Therefore we will need three vector
superfields (or vector supermultiplets) $\widehat V_i$ with the following
components:

\begin{equation}
\ba{llll}
\widehat V_1\equiv(\lambda',W_1^{\mu})&\ra&U_Y(1)&\cr
\vb{18}
\widehat V_2\equiv(\lambda^a,W_2^{\mu a})&\ra&SU_L(2)\quad ,& a=1,2,3\cr
\vb{18}
\widehat V_3\equiv(\widetilde{g}^b,W_3^{\mu b})&\ra&SU_c(3)\quad ,&b=1,\ldots,8
\ea
\end{equation}
%
where $W_i^{\mu}$ are the gauge fields and $\lambda',\lambda$ and
$\widetilde{g}$ are the $U_Y(1)$ and $SU_L(2)$ gauginos and the
gluino, respectively.


\item
{\bf Leptons}

The leptons are described by chiral supermultiplets. As each chiral
multiplet only describes one helicity state, we will need two chiral
multiplets for each charged lepton\footnote{Here we will neglect the
mass of the neutrinos.}.
%
\begin{table}[ht]
%\vspace{-5mm}
\begin{center}
\begin{tabular}{|l|c|}\hline
Supermultiplet&$SU_c(3)\ot SU_L(2)\ot U_Y(1)$\cr
&Quantum Numbers\cr\hline\hline
\vb{14}
$\widehat L_i\equiv(\widetilde{L},L)_i$&$(1,2,-\half)$\cr
\vb{14}
$\widehat R_i\equiv(\widetilde{\ell}_R,\ell^c_L)_i$&$(1,1,1)$\cr\hline
\end{tabular}
\end{center}
\vspace{-5mm}
\caption{Lepton Supermultiplets}
\label{table:leptons}
\end{table}
%
The multiplets are given in Table~\ref{table:leptons}, where 
the $U_Y(1)$ hypercharge is defined through $Q=T_3+Y$. Notice that
each helicity state corresponds to a complex scalar and that $\hat
L_i$ is a doublet of $SU_L(2)$, that is
%
\begin{equation}
\widetilde{L}_i=\left(
\ba{l}\widetilde{\nu}_{Li}\cr
\vb{16}
\widetilde{\ell}_{Li}
\ea
\right)
\qquad ; \qquad
L_i=
\left(
\ba{l}
\nu_{Li}\cr
\ell_{Li}
\ea
\right)
\end{equation}


\item
{\bf Quarks}

The quark supermultiplets are given in Table~\ref{table:quarks}. The
supermultiplet $\widehat{Q}_i$ is also a  doublet of $SU_L(2)$, that
is
%
\begin{table}[ht]
%\vspace{-5mm}
\begin{center}
\begin{tabular}{|l|c|}\hline
Supermultiplet&$SU_c(3)\ot SU_L(2)\ot U_Y(1)$\cr
&Quantum Numbers\cr\hline\hline
\vb{14}
$\widehat Q_i\equiv(\widetilde{Q},Q)_i$&$(3,2,\frac{1}{6})$\cr
\vb{14}
$\widehat D_i\equiv(\widetilde{d}_R,d^c_L)_i$&$(3,1,\frac{1}{3})$\cr
\vb{14}
$\widehat U_i\equiv(\widetilde{u}_R,u^c_L)_i$&$(3,1,-\frac{2}{3})$\cr\hline
\end{tabular}
\end{center}
\vspace{-5mm}
\caption{Quark Supermultiplets}
\label{table:quarks}
\end{table}
%
%
\begin{equation}
\widetilde{Q}_i=\left(
\ba{l}
\widetilde{u}_{Li}\cr
\vb{16}
\widetilde{d}_{Li}
\ea
\right)
\qquad ; \qquad
Q_i=
\left(
\ba{l}
u_{Li}\cr
d_{Li}
\ea
\right)
\end{equation}



\item
{\bf Higgs Bosons}

Finally the Higgs sector. In the MSSM we need at least two Higgs
doublets. This is in contrast with the SM where only one Higgs doublet
is enough to give masses to all the particles. The reason can be explained in
two ways. Either the need to cancel the anomalies, or the fact that,
due to the analyticity of the superpotential, we have to have two
Higgs doublets of opposite hypercharges to give masses to the up and
down type of quarks. The two supermultiplets, with their quantum
numbers, are given in Table~\ref{table:higgs}.
%
\begin{table}[ht]
%\vspace{-5mm}
\begin{center}
\begin{tabular}{|l|c|}\hline
Supermultiplet&$SU_c(3)\ot SU_L(2)\ot U_Y(1)$\cr
&Quantum Numbers\cr\hline\hline
\vb{14}
$\widehat H_d\equiv(H_d,\widetilde{H}_d)$&$(1,2,-\frac{1}{2})$\cr
\vb{14}
$\widehat H_u\equiv(H_u,\widetilde{H}_u)$&$(1,2,+\frac{1}{2})$\cr\hline
\end{tabular}
\end{center}
\vspace{-5mm}
\caption{Higgs Supermultiplets}
\label{table:higgs}
\end{table}
%

\end{itemize}



\subsection{The Superpotential and SUSY Breaking Lagrangian}

The MSSM Lagrangian is specified by the R--parity conserving 
superpotential $W$ 
%
\begin{eqnarray}
\label{eq:WMSSM}
 W^{\hbox{\scriptsize MSSM}}&=&\varepsilon_{ab}\left[ 
 h_U^{ij}\widehat Q_i^a\widehat U_j\widehat H_u^b 
+h_D^{ij}\widehat Q_i^b\widehat D_j\widehat H_d^a 
+h_E^{ij}\widehat L_i^b\widehat R_j\widehat H_d^a 
-\mu\widehat H_d^a\widehat H_u^b \right] \
\end{eqnarray}
%
where $i,j=1,2,3$ are generation indices, $a,b=1,2$ are $SU(2)$
indices, and $\varepsilon$ is a completely antisymmetric $2\times2$
matrix, with $\varepsilon_{12}=1$. The symbol ``hat'' over each letter
indicates a superfield, with $\widehat Q_i$, $\widehat L_i$, $\widehat
H_1$, and $\widehat H_2$ being $SU(2)$ doublets with hypercharges
$\textstyle{1\over6}$, $-\frac{1}{2}$, $-\frac{1}{2}$, and
$\frac{1}{2}$ respectively, and $\widehat U$, $\widehat D$, and
$\widehat R$ being $SU(2)$ singlets with hypercharges
$-{\textstyle{2\over 3}}$, ${\textstyle{1\over 3}}$, and $1$
respectively. The coupling matrices $h_U,h_D$ and $h_E$ will give rise
to the usual Yukawa interactions needed to give masses to the leptons
and quarks, and $\mu$ is a parameter with units of mass.

If it were not for the need to break SUSY, the
number of parameters involved would be less than in the SM. This can
be seen in Table~\ref{SMvsMSSM}. However, as we have seen, we have to
break supersymmetry. The most general SUSY soft breaking is, 
%
\begin{eqnarray} 
V^{\hbox{\scriptsize MSSM}}_{\hbox{\scriptsize SB}}
&\hskip-3mm=\hskip-3mm& 
M_Q^{ij2}\widetilde Q^{a*}_i\widetilde Q^a_j+M_U^{ij2} 
\widetilde U_i\widetilde U^*_j+M_D^{ij2}\widetilde D_i 
\widetilde D^*_j
+M_L^{ij2}\widetilde L^{a*}_i\widetilde L^a_j 
+M_R^{ij2}\widetilde R_i\widetilde R^*_j\cr
\vb{18}
&\hskip-6mm& 
+m_{H_d}^2 H^{a*}_d H^a_d +m_{H_u}^2 H^{a*}_u H^a_u 
- \left[\half M_3\lambda_3\lambda_3+\half M_2\lambda\lambda 
+\half M_1\lambda'\lambda'+h.c.\right]\phantom{\hskip 5mm}\cr
\vb{18}
&\hskip-6mm& 
+\varepsilon_{ab}\left[ 
A_U^{ij}h_U^{ij}\widetilde Q_i^a\widetilde U_j H_u^b 
+A_D^{ij}h_D^{ij}\widetilde Q_i^b\widetilde D_j H_d^a 
+A_E^{ij}h_E^{ij}\widetilde L_i^b\widetilde R_j H_d^a 
-B\mu H_d^a H_u^b \right] 
\end{eqnarray} 
%
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
Theory&Gauge&Fermion& Higgs\cr
&Sector&Sector& Sector\cr\hline\hline
SM&$e,g,\alpha_s$&$h_U,h_D,h_E$&$\mu^2,\lambda$\cr\hline
MSSM&$e,g,\alpha_s$&$h_U,h_D,h_E$&$\mu$\cr\hline
Broken
MSSM&$e,g,\alpha_s$&$h_U,h_D,h_E$&$\mu,M_1,M_2,M_3,A_U,A_D,A_E,B$\cr
&&&$m_{H_u}^2,m_{H_d}^2,m_Q^2,m_U^2,m_D^2,m_L^2,m_R^2$\cr\hline
\end{tabular}
\end{center}
\caption{Comparative counting of parameters}
\label{SMvsMSSM}
\end{table}

\subsection{Symmetry Breaking}


The electroweak symmetry is broken when the two Higgs doublets  
$H_d$ and $H_u$ acquire VEVs
%
\begin{equation}
H_d={{{1\over{\sqrt{2}}}[\sigma^0_d+v_d+i\varphi^0_d]}\choose{ 
H^-_d}}\,,\qquad 
H_u={{H^+_u}\choose{{1\over{\sqrt{2}}}[\sigma^0_u+v_u+ 
i\varphi^0_u]}} 
\end{equation}
%
with $m_W^2=\quarter g^2v^2$ and $v^2\equiv v_d^2+v_u^2=(246 \;
\rm{GeV})^2$. The full scalar potential at tree level is 
%
\begin{equation}
V_{total}  = \sum_i \left| { \partial W \over \partial z_i} \right|^2 
        + V_D + V_{soft} 
\end{equation}
%
The scalar potential contains linear terms 
%
\begin{equation}
V_{linear}=t_d^0\sigma^0_d+t_u^0\sigma^0_u
\end{equation}
%
where
%
\begin{eqnarray} 
t_d&=&(m_{H_d}^2+\mu^2)v_d-B\mu v_u+
\eighth(g^2+g'^2)v_d(v_d^2-v_u^2)\,, 
\cr
\vb{18}
t_u&=&(m_{H_u}^2+\mu^2)v_u-B\mu v_d- 
\eighth(g^2+g'^2)v_u(v_d^2-v_u^2) 
\label{eq:tadpoles}
\end{eqnarray} 
%

One can determine in the tree-level approximation the minimum of the 
scalar potential by imposing the condition of vanishing tadpoles in 
\Eq{eq:tadpoles}. One--loop corrections change these equations to 
\begin{equation} 
t_i=t_i^0-\delta t_i+T_i(Q) 
\label{eq:onelooptad} 
\end{equation} 
%
where $t_i$, with $i=d,u$, are the renormalized tadpoles, $t_i^0$ 
are the tree level tadpoles given in \Eq{eq:tadpoles}, $\delta t_i$ 
are the tadpole counter-terms, and $T_i(Q)$ are the sum of all 
one--loop contributions to the corresponding one--point functions with 
zero external momentum. The contribution from quarks and squarks to 
these tadpoles can be found in Ref.~\cite{diaz:1998xc}.  In an 
on shell scheme we identify the tree level tadpoles with the 
renormalized ones. Therefore, to find the correct minima we use 
\Eq{eq:tadpoles} unchanged, where now all the parameters are 
understood to be renormalized quantities. Eqs.~(\ref{eq:tadpoles}) can be
solved for $B$ and $\mu$ up to a sign. We have
%
\begin{eqnarray}
&&\mu^2=-\half m_Z^2 + \frac{m_{H_u}^2 \sin^2 \beta -
m_{H_d}^2 \cos^2 \beta}{\cos 2 \beta}\cr
\vb{18}
&&B \mu = \half \sin 2 \beta \left(m_{H_d}^2 +m_{H_u}^2 + 2 \mu^2
\right)
\label{eq:minimum}
\end{eqnarray} 
%
It can be shown the necessary condition for the existence of a
stable minimum is
%
\begin{equation}
\label{eq:MinimumCondition}
(B\mu)^2 \gt (m_{H_d}^2 +\mu^2) (m_{H_u}^2+\mu^2) 
\end{equation}
%
Notice that \Eq{eq:minimum} only makes sense if $\mu^2\gt0$
and, as we shall see in Section~\ref{sec:NeutralHiggs}, also 
$B \mu \gt 0$ because is related with the mass squared, $m_A^2$,
of the physical CP--odd state.


\subsection{UFB directions and CCB minima}

In the last section we discussed the minimization of the potential
assuming that the conditions were such that only the neutral Higgs
bosons acquired vevs. Let us go back and discuss this in more
detail. The potential for the Higgs bosons fields is,
%
\begin{eqnarray}
  \label{eq:HiggsPotential}
  V_{\hbox{\scriptsize Higgs}}&=& \frac{1}{8}\left( g^2 + g'^2\right)
  \left( |H_u^0|^2 + |H_u^+|^2 - |H_d^0|^2 - |H_d^-|^2 \right)^2
+ \frac{1}{2}g^2 \left|H_u^+ H_d^{0*} + H_u^0 H_d^{-*} \right|^2 \nn
  \\[+2mm]
&&+\left( \mu^2 +m_{H_u}^2 \right) \left(|H_u^0|^2 + |H_u^+|^2\right)
\left( \mu^2 +m_{H_d}^2 \right)\left(|H_d^0|^2 + |H_d^-|^2\right)
\nn \\[+2mm]
&&+ B\mu \left[ \left( H_u^+ H_d^- - H_u^0 H_d^0 \right) + \hbox{h.c.} 
\right]
\end{eqnarray}
%
We can use the $SU(2)$ gauge transformations to choose one of the
charged Higgs bosons to be zero. So without loss of generality we can
take $H_u^+=0$ at the minimum. Now from $\frac{\partial V}{\partial
 H_u^+}=0$ we also get $H_d^-=0$. So after setting the charged Higgs
boson fields to zero we are left with the neutral scalar potential,
%
\begin{eqnarray}
  \label{eq:NeutralHiggsPotential}
  V_{\hbox{\scriptsize Higgs}}&=& \frac{1}{8}\left( g^2 + g'^2\right)
  \left( |H_u^0|^2 - |H_d^0|^2  \right)^2 \nn
  \\[+2mm]
&&+\left( \mu^2 +m_{H_u}^2 \right) |H_u^0|^2 
\left( \mu^2 +m_{H_d}^2 \right) |H_d^0|^2 
- B\mu \left(  H_u^0 H_d^0  + \hbox{h.c.} \right)
\end{eqnarray}

\ni
It was the potential in Eq.~(\ref{eq:NeutralHiggsPotential}) that we
minimized in the last section. However it is not enough that we found
a non trivial minimum that breaks $SU_L(2)\times U_Y(1)$ into
$U_{\hbox{\scriptsize EM}}(1)$. We have to make sure that the
potential is bounded from below for arbitrary large values of the
scalar fields. The quartic terms in $V_{\hbox{\scriptsize Higgs}}$
will stabilize the potential. Therefore the dangerous situation
happens when the D terms, responsable for those quartic terms,
vanish. From Eq.~(\ref{eq:NeutralHiggsPotential}) one can see that
this occurs when $|H_d^0|=|H_u^0|$. Then the potential is bounded from
below if the coefficient of the quadratic term is positive. This gives
the requirement,
%
\begin{equation}
  \label{eq:UFB}
  m_{H_d}^2 + m_{H_u}^2 + 2 \mu^2 \gt 2 | B \mu |  
\end{equation}
%
One can check that the requirements in Eqs.~(\ref{eq:MinimumCondition})
and (\ref{eq:UFB}) are indeed compatible with the minimization
conditions, Eq.~(\ref{eq:tadpoles}). 

We have shown above that for the MSSM the minimization of the Higgs
potential implies that the vevs of the charged Higgs have to
vanish. However one has to worry about other situations when the
squarks or sleptons take nonvanishing vevs. These can lead to other
UFB directions and to charge and color breaking (CCB) minima. The most
famous of these is the traditional bound of Frere
etal~\cite{abel:1997eb,alvarez-gaume:1983gj}.  In this case it is
assumed that only $H_u$ and $\widetilde{Q}$ take vevs. Then if
$|H_u|=|\widetilde{Q_u}|$ the D-term vanishes. Then it is shown that a
very deep CCB minimum appears unless
%
\begin{equation}
  \label{eq:CCBminimum}
  |A_u|^2 \leq 3 \left( m^2_{Q_u} + m_u^2 + M^2_{H_u} + \mu^2 \right)
\end{equation}
%
The other famous case is an UFB direction realized when [KOMATSU]
%
\begin{eqnarray}
  \label{eq:UFBKomatsu}
  |L_i|^2&=&|H_u|^2+|Q_j|^2 \nn \\[+2mm]
  h_{D_j} Q_j d_{R_j} &=&-\mu H_u \nn \\[+2mm]
  |Q_j|&=&|d_{R_j}|
\end{eqnarray}
%
This direction will be UFB unless
%
\begin{equation}
  \label{eq:UFBKomatsu-2}
  m^2_{H_u} + m^2_{L_i} \geq 0
\end{equation}
%
Other UFB and CCB cases can be found if we allow the squark and
sleptons to take non vanishing vevs. A complete classification of all
the dangerous cases can be found in Ref.\cite{casas:1996pd}.


\subsection{The Constrained Minimal Supersymmetric Standard Model}

We have seen in the previous section that the parameters of the MSSM
can be considered arbitrary at the weak scale. This is completely
consistent. However the number of independent parameters in
Table~\ref{SMvsMSSM} can be
reduced if we impose some further constraints. That is usually done by
embedding the MSSM in a grand unified scenario. Different schemes are
possible but in all of them some kind of unification is imposed at the
GUT scale. Then we run the Renormalization Group (RG) equations down
to the weak scale to get the values of the parameters at that
scale. This is sometimes called the constrained MSSM model. 


Among the possible scenarios, the most popular is the MSSM coupled to
$N=1$ Supergravity (SUGRA). 
Here at $M_{GUT}$ one usually takes the conditions:
%
\bea
&&A_t = A_b = A_{\tau} \equiv A \:, B=A-1 \:, \cr
&&\vb{18}
m_{H_d}^2 = m_{H_u}^2 = M_{L}^2 = M_{R}^2 = m_0^2 \:,
M_{Q}^2 =M_{U}^2 = M_{D}^2 = m_0^2 \:, \cr
&&\vb{18}
M_3 = M_2 = M_1 = M_{1/2}
\eea

The counting of free parameters\footnote{For one family and without counting
the gauge couplings.} is done in Table~\ref{table:SUGRA} where we
introduced the usual notation 
%
\begin{equation}
\tan \beta= \frac{v_u}{v_d}
\end{equation}

\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|}\hline
Parameters & Conditions & Free Parameters \\[+3pt] \hline\hline
$h_t$, $h_b$, $h_{\tau}$, $v_d$, $v_u$
&$m_W$, $m_t$, $m_b$, $m_{\tau}$ & $\tan
\beta$ \\[+3pt] \hline
$A$, $m_0$, $M_{1/2}$, $\mu$&$t_i=0$, $i=1,2$& 2 Extra free
parameters\\[+3pt] \hline
Total = 9&Total = 6 &Total = 3\\[+3pt] \hline
\end{tabular}
\end{center}
\vspace{-5mm}
\caption{\small
Counting of free parameters in the MSSM coupled to N=1 SUGRA
}
\label{table:SUGRA}
\end{table}

\ni
It is remarkable that with so few parameters we can get the correct
values for the parameters, in particular $m_{H_u}^2 <0$. This is known
as \textit{radiative electroweak symmetry breaking}. For this to
happen the top Yukawa coupling has to be large which we know to be
true. This provides additional motivation for this constrained
model. The complete RG equations up to two loops can be found in
Ref.\cite{barger:1993ac}. A very clear discussion of the approximate
solutions of the one loop RG equations is given in
Ref.\cite{martin:1997ns}.


\section{Mass Matrices in the MSSM}



\subsection{The Chargino Mass Matrices}


The charged gauginos mix with the charged higgsinos giving the
so--called charginos. In a basis where  
$\psi^{+T}=(-i\lambda^+,\widetilde H_u^+)$ 
and $\psi^{-T}=(-i\lambda^-,\widetilde H_d^-)$, the chargino 
mass terms in the Lagrangian are 
% 
\begin{equation} 
{\cal L}_m=-{1\over 2}(\psi^{+T},\psi^{-T}) 
\left(\matrix{{\bold 0} & \bold {\bold M}_{\chi^{\pm}}^T \cr 
{\bold M}_{\chi^{\pm}} &  
{\bold 0} }\right) 
\left(\matrix{\psi^+ \cr \psi^-}\right) + \hbox{h.c.} 
\label{eq:chFmtermMSSM} 
\end{equation} 
% 
where the chargino mass matrix is given by 
% 
\begin{equation} 
{\bold M}_{\chi^{\pm}}=
\left[\matrix{ 
M_2 & {\textstyle{1\over{\sqrt{2}}}}gv_u \cr 
\vb{18}
{\textstyle{1\over{\sqrt{2}}}}gv_d & \mu }
\right]
\label{eq:ChaM2x2} 
\end{equation} 
% 
and $M_2$ is the $SU_L(2)$ gaugino soft mass. It is sometimes more
useful to write this mass matrix in terms of the physical masses, that
is
%
\begin{equation} 
{\bold M}_{\chi^{\pm}}
=\left[\matrix{ 
M_2 & \sqrt{2} m_W \sin \beta \cr 
\vb{18}
\sqrt{2} m_W \cos \beta & \mu }
\right]
\label{eq:ChaM2x2b} 
\end{equation} 
%
The chargino mass matrix is diagonalized by two rotation matrices 
$\bold U$ and $\bold V$ defined by
%
\begin{equation}
F_i^-=\bold{U}_{ij}\, \psi_j^- \quad ; \quad F_i^+=\bold{V}_{ij}\, \psi_j^+
\end{equation}
%
Then 
%
\begin{equation}
\bold{U}^* {\bold M}_{\chi^{\pm}} \bold{V}^{-1} = 
{\bold M}_{\chi^{\pm}}^{\hbox{\scriptsize diag}}
\end{equation}
%
where ${\bold M}_{\chi^{\pm}}^{\hbox{\scriptsize diag}}$ is the
diagonal chargino mass matrix.  To determine $\bold{U}$ and $\bold{V}$
we note that
%
\begin{equation}
\left({\bold M}_{\chi^{\pm}}^{\hbox{\scriptsize diag}}\right)^2
=\bold{V} {\bold M}_{\chi^{\pm}}^{\dagger} 
{\bold M}_{\chi^{\pm}} \bold{V}^{-1} =
\bold{U^*} {\bold M}_{\chi^{\pm}} 
{\bold M}_{\chi^{\pm}}^{\dagger}
(\bold{U^*})^{-1}
\end{equation}
%
implying that $\bold{V}$ diagonalizes ${\bold M}_{\chi^{\pm}}^{\dagger} 
{\bold M}_{\chi^{\pm}}$ and 
$\bold{U^*}$ diagonalizes ${\bold M}_{\chi^{\pm}} 
{\bold M}_{\chi^{\pm}}^{\dagger}$. 
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 $\chi^{-}$
to be the particle and not the anti--particle.},
%
\begin{equation}
\chi_i^-=\left(\matrix{
F_i^- \cr
\vb{16}
\overline{F_i^+}\cr}
\right)
\end{equation}

Because the matrix in Eq.~(\ref{eq:ChaM2x2b}) is quite simple it is
possible to write analytical expressions for the chargino
masses. These are,
%
\begin{equation}
  \label{eq:charginoeigenvalues}
  m^2_{\chi_{1,2}}=
\frac{1}{2} \left[ \left( M_2^2 + \mu^2 + 2 m_W^2 \right)
\mp \sqrt{\left( M_2^2 + \mu^2 + 2 m_W^2 \right)^2 -4 \left(\mu M2
    -m_W^2 \sin 2 \beta \right)^2} \right]
\end{equation}
%
The present lower limit on the chargino mass~\cite{hagiwara:2002fs},
$m_{\chi_{1}} > 103$ GeV, translates into an exclusion in the
$(\mu,M_2)$ plane, as is shown in Fig.~\ref{fig:mum2} for
$\tan\beta=2$ and $\tan\beta=30$. 
%
\begin{figure}[htbp]
  \centering
  \begin{tabular}{cc}
    \includegraphics[clip,width=0.45\textwidth]{mum2tb2.eps}&
    \includegraphics[clip,width=0.45\textwidth]{mum2tb30.eps}
  \end{tabular}
  \caption{Region excluded in the $(\mu,M_2)$ plane by the bound
    $m_{\chi_{1}} > 103$ GeV} 
  \label{fig:mum2}
\end{figure}


\subsection{The Neutralino Mass Matrices}


In the basis $\psi^{0T}= 
(-i\lambda',-i\lambda^3,\widetilde{H}_d^1,\widetilde{H}_u^2)$ 
the neutral fermions mass terms in the Lagrangian are given by 
% 
\begin{equation} 
{\cal L}_m=-\frac 12(\psi^0)^T{\bold M}_{\chi^0}\psi^0+h.c.   
\label{eq:NeuMLagMSSM} 
\end{equation} 
% 
where the neutralino mass matrix is 
 
\begin{equation} 
{\bold M}_{\chi^0}=\left[  
\begin{array}{cccc}  
M_1 & 0 & -\frac 12g^{\prime }v_d & \frac 12g^{\prime }v_u \\   
0 & M_2 & \frac 12gv_d & -\frac 12gv_u \\   
-\frac 12g^{\prime }v_d & \frac 12gv_d & 0 & -\mu  \\   
\frac 12g^{\prime }v_u & -\frac 12gv_u & -\mu  & 0 \\   
\end{array}  
\right] 
\label{eq:NeuM4x4} 
\end{equation} 

\ni 
and $M_1$ and $M_2$ are the $U_Y(1)$ and $SU_L(2)$ gaugino soft
mass. The neutralino mass matrix can also be written in terms of the
physical masses.


\begin{equation} 
{\bold M}_{\chi^0}=\left[  
\begin{array}{cccc}  
M_1 & 0 & -m_Z \sin \theta_W \cos \beta & m_Z \sin \theta_W \sin \beta \\   
0 & M_2 & m_Z \cos \theta_W \cos \beta & - m_Z \cos \theta_W \sin \beta \\   
-m_Z \sin \theta_W \cos \beta & m_Z \cos \theta_W \cos \beta & 0 & -\mu  \\   
m_Z \sin \theta_W \sin \beta  & -m_Z \cos \theta_W \sin \beta & -\mu  & 0 \\   
\end{array}  
\right] 
\label{eq:NeuM4x4b} 
\end{equation} 

\ni
This neutralino mass matrix is diagonalized by a $4\times 4$ rotation
matrix $\bold N$ such that


\begin{equation} 
{\bold N}^*{\bold M}_{\chi^0}{\bold N}^{-1}={\rm diag}(m_{F^0_1},m_{F^0_2}, 
m_{F^0_3},m_{F^0_4}) 
\label{eq:NeuMdiagMSSM} 
\end{equation} 
% 
and
%
\begin{equation}
F^0_k=\bold{N}_{kj}\, \psi^0_j
\end{equation}
%
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{16}
\overline{F_i^0}\cr}
\right)
\end{equation}

\ni
Because of the complicated form of Eq.~(\ref{eq:NeuM4x4b}) it is in
general necessary to perform the diagonalization numerically.


\subsection{Neutral Higgs Mass Matrices} 
\label{sec:NeutralHiggs}

The quadratic scalar potential includes 
% 
\begin{equation} 
V_{quadratic}=\half[\varphi'^0_d,\varphi'^0_u]\ 
{\bold M^2_{P^0}}\left[\matrix{ 
\varphi'^0_d \cr \varphi'^0_u \cr 
}\right] 
+\half[\sigma'^0_d,\sigma'^0_u] \
{\bold M^2_{S^0}}\left[\matrix{ 
\sigma'^0_d \cr \sigma'^0_u \cr 
}\right]+ \cdots
\label{eq:NeutScalLagMSSM} 
\end{equation} 
% 
where the CP-odd neutral scalar mass matrix is 
% 
\begin{equation} 
{\bold M^2_{P^0}}=\left[\matrix{ 
B\mu \tan \beta+{{t_d}\over{v_d}} 
& B\mu \cr
\vb{18} 
B\mu & B\mu \cot \beta +{{t_u}\over{v_u}} 
}\right] 
\label{eq:neuscaM} 
\end{equation} 
% 
This matrix has a vanishing determinant after the tadpoles are set to
zero (the condition for the minimum of the potential), and the zero
eigenvalue corresponds to the mass of the neutral Goldstone boson. The
mass of the CP-odd state, usually called $A^0$, is
%
\begin{equation}
m_{A^0}^2=B 
\mu \left(\tan\beta + \cot\beta\right) = \frac{2 B \mu}{\sin 2 \beta}
\end{equation}

In the following sections we will give the mass matrices for all the
fields of the MSSM. So it is useful to define an uniform notation for
all those cases. We will always denote the weak states with a prime
and for the mass eigenstates we will use the same letter without the
prime. For any weak field $\lambda'$ the relation with the mass
eigenstate $\lambda$ is,
%
\begin{equation}
  \label{eq:defRotation}
  \lambda_i=\bold{R^{\lambda}}_{ij}\, \lambda_i
\end{equation}
%
and
%
\begin{equation}
  \label{eq:defMassMatrixDiag}
  \bold{R^{\lambda}} \bold{M^{\lambda}} \bold{R^{\lambda}}^{\dagger} =
\bold{M^{\lambda}}_{\hbox{\scriptsize diag}}
\end{equation}
 
\ni 
For the CP-odd scalars we have $P'^0=(\varphi'^0_d,\varphi'^0_u)$ and
$P^0=(G^0,A^0)$ with

\begin{equation}
  \label{eq:RotCP-odd}
  \bold{R^{P^0}}=\left[
  \begin{array}{cc}
    \cos\beta & -\sin\beta \\
    \sin\beta & \cos\beta 
  \end{array}
  \right]
\end{equation}

\ni
The neutral CP-even scalar sector mass matrix in \Eq{eq:NeutScalLagMSSM} 
is given by 
%
\begin{equation}
{\bold M_{S^0}^2}= 
\left[\matrix{ 
B\mu\tan\beta+m_Z^2 \sin^2\beta +{{t_d}\over{v_d}}  
& -B\mu-m_Z^2\sin\beta\cos\beta  \cr 
\vb{18}
-B\mu-m_Z^2\sin\beta\cos\beta  
& B\mu\cot\beta+m_Z^2\cos^2\beta
+ {{t_u}\over{v_u}}  \cr 
}\right]
\label{eq:PseScalM}
\end{equation}

\ni 
The mass eigenstates that are usually denoted $S^0=(h^0,H^0)$ are
related to the weak states $S'^0=(\sigma'^0_d,\sigma'^0_d)$ by the
rotation matrix

\begin{equation}
  \label{eq:RotCP-even}
  \bold{R^{P^0}}=\left[
  \begin{array}{cc}
    \sin\alpha & -\cos\alpha \\
    \cos\alpha & \sin\alpha 
  \end{array}
  \right]
\end{equation}
%
which defines the mixing angle.
\ni
The masses of the neutral CP-even states are 
%
\begin{equation}
  \label{eq:CP-evenmasses}
  m^2_{h^0,H^0}=\frac{1}{2}\, \left( m^2_A + m^2_Z \mp
\sqrt{(m^2_A+m^2_Z)^2 -4 m^2_Z m^2_A \cos^2 2\beta} \right)
\end{equation}
%
obeying the sum rule,
%
\begin{equation}
  \label{eq:CPevenMassesRelation}
  m^2_{h^0}+m^2_{H^0}=m^2_{A^0} + m^2_Z
\end{equation}
%
At tree level the following relations between the angles $\alpha$ and
$\beta$ hold,
%
\begin{equation}
  \label{eq:alphabetaRelations}
  \frac{\sin 2\alpha}{\sin 2\beta} = -
  \frac{m^2_{A^0}+m^2_Z}{m^2_{H^0}-m^2_{h^0}},\hskip 10mm
  \frac{\cos 2\alpha}{\cos 2\beta} = -
  \frac{m^2_{A^0}-m^2_Z}{m^2_{H^0}-m^2_{h^0}}.
\end{equation}
%
The masses of the $A^0$ and $H^0$ can be arbitrarily large as they grow
with $\frac{2 B \mu}{\sin 2\beta}$. In contrast the mass of the
lightest neutral CP-even Higgs boson, $h^0$ is bounded from above. It
can be shown from Eq.~(\ref{eq:CP-evenmasses}) that, at tree level, we
have,
\begin{equation}
  \label{eq:h0masslimit}
  m_{h^0} < | \cos 2 \beta |\, m_Z
\end{equation}
%
that is, the $h^0$ would be lighter than the $Z^0$ boson. If this
relation were not modified by higher order corrections the MSSM would
by now probably be excluded taking in account the results from
LEP2~\cite{hagiwara:2002fs}. However the supersymmetry breaking
introduces a large splitting between the particles and their
superpartners (large compared with the mass of the particles, of
course). This allows the one loop contributions for the Higgs boson
self energies for the quarks and squarks not to cancel, as would be
the case if supersymmetry were exact. As these self energies are
proportional to the fourth power of the quark mass, the contribution
from the top--stop loop can be quite large. After taking this in
account the bound is weaker~\cite{haber:1991aw,okada:1991gg,ellis:1991nz},
%
\begin{equation}
  \label{eq:mh0upperlimit}
  m_{h^0} \leq 130 GeV
\end{equation}
%
but still very interesting for the coming experiments at the LHC.


\subsection{Charged Higgs Mass Matrices}

The mass matrix of the charged Higgs bosons follows from the 
quadratic terms in the scalar potential 
% 
\begin{equation} 
V_{quadratic}=[H_d^-,H_u^-,]\ 
\bold{M_{H^{\pm}}^2}
\left[
\matrix{H_d^+ \cr H_u^+}
\right] 
\label{eq:VquadraticMSSM} 
\end{equation} 
%
where the charged Higgs mass matrix is
% 


\begin{equation}
{\bold M_{H^{\pm}}^2}= 
\left[\matrix{ 
B\mu\tan\beta+m_W^2 \sin^2\beta +{{t_d}\over{v_d}} 
& B\mu+m_W^2 \sin\beta\cos\beta \cr 
\vb{18}
B\mu+ m_W^2 \sin\beta\cos\beta 
& B\mu\cot\beta+m_W^2 \cos^2\beta +{{t_u}\over{v_u}} 
}\right] 
\label{eq:chargedHiggs} 
\end{equation}
% 

\ni
This mass matrix has one zero eigenvalue corresponding to the charged
Goldstone boson. The mass of the remaining physical state, the charged
Higgs boson $H^\pm$ is,
\begin{equation}
  \label{eq:chargedHiggsMass}
  m^2_{H^\pm}=m^2_{A^0} + m^2_W
\end{equation}
%
The mass eigenstates $S^+=(G^+,H^+)$ are obtained from the weak
states $S'^+=(H^+_d,H^+_u)$ with the same rotation matrix as in
Eq.~(\ref{eq:RotCP-odd}), that is, $\bold{R^{S^+}}=\bold{R^{P^0}}$.


\subsection{Lepton Mass Matrices}

The lepton and quark sectors of the MSSM do not differ from the
SM. However as the MSSM lagrangian is more naturally written in terms
of two component spinor notation, we go through all the details to
define our conventions.
In the superpotential in \Eq{eq:WMSSM} the Yukawa couplings are
matrices in generation space. For the case of the leptons it is
possible to start with the matrix $h_E$ already diagonalized. 
However, for some applications, could be useful to have a general
matrix. Here we consider an arbitrary matrix. In 2--component spinor
notation the relevant mass terms in the Lagrangian are
%
\begin{equation}
{\cal L}_M= -\frac{v_d}{\sqrt{2}}\, 
\left(h_E\right)_{ij}\, \ell'_{Li}\, \ell'^c_{Lj}
-\frac{v_d}{\sqrt{2}}\, \left(h_E^*\right)_{ij}\, 
\ovl{\ell'_L}_i\, \ovl{\ell'^c_L}_j
\end{equation}
%
where $\ell'$ are the interaction eigenstates. The 4--component
spinors are 
%
\begin{equation}
\ell'=
\left(\matrix{
\ell'_L\cr
\vb{18}
\ovl{\ell'^c_L}}
\right)
\end{equation}
%
and therefore we have 
%
\begin{eqnarray}
{\cal L}_M&=&
-\ovl{\ell'}\, \bold{M_E} P_R \ell'
-\ovl{\ell'}\, \bold{M_E}^{\dagger} P_L \ell'\cr
\vb{18}
&=&
-\ovl{\ell'_L}\, \bold{M_E} \ell'_R
-\ovl{\ell'_R}\, \bold{M_E}^{\dagger} \ell'_L
\end{eqnarray}
%
where
%
\begin{equation}
\label{eq:ElectronMassMatrix}
\left(\bold{M_E}\right)_{ij}= \frac{v_d}{\sqrt{2}}\, \left(h^*_E\right)_{ij}
\end{equation}
%
This equation is equivalent to Eq.~(\ref{eq:10}) if we make the
correspondence $Y^l_{ij}\rightarrow \left(h^*_E\right)_{ij}$.
To diagonalize the mass matrix $\bold{M_E}$ we need different
rotations for the left handed and right handed components. We
introduce
%
\begin{equation}
\ell_R=\bold{R^{\ell}_R}\, \ell'_R \qquad ; \qquad
\ovl{\ell_L}=\ovl{\ell'_L}\,\bold{R^{\ell}_L} 
\end{equation}
%
Then
%
\begin{equation}
\bold{R^{\ell}_L{}^{\dagger}}\, \bold{M_E} \bold{R^{\ell}_R{}^{\dagger}}
= \bold{M_E^{diag}} \qquad \hbox{or}\qquad
\bold{R^{\ell}_L}\, \bold{M_E^{diag}} \bold{R^{\ell}_R}
= \bold{M_E} 
\end{equation}
%
where $\bold{M_E^{diag}}$ is a diagonal matrix. The rotation matrices
are obtained by noticing that
%
\begin{equation}
\bold{M_E} \bold{M_E}^{\dagger}= \bold{R^{\ell}_L}\,
\left(\bold{M_E^{diag}}\right)^2 \bold{R^{\ell}_L}^{\dagger}
\qquad \hbox{and} \qquad
\bold{M_E}^{\dagger} \bold{M_E}= \bold{R^{\ell}_R}^{\dagger}\,
\left(\bold{M_E^{diag}}\right)^2 \bold{R^{\ell}_R}
\end{equation}
%
that tell us that $\bold{R^{\ell}_L}$ diagonalizes 
$\bold{M_E} \bold{M_E}^{\dagger}$ and $ \bold{R^{\ell}_R}$
diagonalizes $\bold{M_E}^{\dagger} \bold{M_E}$. For future reference
we write the relations between the mass and the interaction
eigenstates. We have 
%
\begin{eqnarray}
&&\ell'_{R i}= \left( \bold{R^{\ell}_R} \right)^*_{ji}\, \ell_{Rj}
\qquad ; \qquad
\ovl{\ell'_R}_i= \ovl{\ell_R}_j\, \left( \bold{R^{\ell}_R} \right)_{ji} 
\cr
\vb{18}
&&\ell'_{L i}= \left( \bold{R^{\ell}_L} \right)_{ij}\, \ell_{Lj}
\qquad ; \qquad
\ovl{\ell'_L}_i= \ovl{\ell_L}_j\, \left( \bold{R^{\ell}_L} \right)^*_{ij} 
\end{eqnarray}


\subsection{Quark Mass Matrices}
\label{sec:QuarkMassMatrices}

In 2--component spinor notation the relevant terms in the Lagrangian are 

%
\begin{equation}
{\cal L}_M= -\frac{v_d}{\sqrt{2}}\, 
\left(h_D\right)_{ij}\, d'_{Li}\, d'^c_{Lj}
-\frac{v_u}{\sqrt{2}}\, \left(h_U^*\right)_{ij}\, 
\left(h_U\right)_{ij}\, u'_{Li}\, u'^c_{Lj} + h.c.
\end{equation}
%
where the primed states are again the interaction eigenstates. In
4--component spinor notation with the definitions
%
\begin{equation}
d'=
\left(\matrix{
d'_L\cr
\vb{18}
\ovl{d'^c_L}}
\right) 
\qquad ; \qquad
u'=
\left(\matrix{
u'_L\cr
\vb{18}
\ovl{u'^c_L}}
\right) 
\end{equation}
%
we get
%
\begin{equation}
{\cal L}_M=
-\ovl{d'_L}\, \bold{M_D} d'_R
-\ovl{u'_L}\, \bold{M_U} u'_R + h.c.
\end{equation}
%
where
%
\begin{equation}
\left(\bold{M_D}\right)_{ij}= \frac{v_d}{\sqrt{2}}\, \left(h^*_D\right)_{ij}
\qquad ; \qquad
\left(\bold{M_U}\right)_{ij}= \frac{v_u}{\sqrt{2}}\, \left(h^*_U\right)_{ij}
\end{equation}
%
To obtain the eigenstates of the mass we rotate the quark fields through
%
\begin{equation}
d_R=\bold{R^d_R}\, d'_R 
\qquad ; \qquad
d_L=\bold{R^d_L}\, d'_L
\qquad ; \qquad
u_R=\bold{R^u_R}\, u'_R 
\qquad ; \qquad
u_L=\bold{R^u_L}\, u'_L
\end{equation}
%
For future reference we write the relations between the mass and the 
interaction eigenstates. We have 
%
\begin{eqnarray}
&&q'_{R i}= \left( \bold{R^q_R} \right)^*_{ji}\, q_{Rj}
\qquad ; \qquad
\ovl{q'_R}_i= \ovl{q_R}_j\, \left( \bold{R^q_R} \right)_{ji} 
\cr
&&\hskip 9cm q=(d,u)\cr
&&q'_{L i}= \left( \bold{R^q_L} \right)^*_{ji}\, q_{Lj}
\qquad ; \qquad
\ovl{q'_L}_i= \ovl{q_L}_j\, \left( \bold{R^q_L} \right)_{ji} 
\end{eqnarray}
%
Then
%
\begin{equation}
\bold{R^d_L}\, \bold{M_D} \bold{R^d_R{}^{\dagger}}
= \bold{M_D^{diag}} \qquad \hbox{and}\qquad
\bold{R^u_L}\, \bold{M_U} \bold{R^u_R{}^{\dagger}}
= \bold{M_U^{diag}}
\end{equation}
%
where $\bold{M_{D(U)}^{diag}}$ are a diagonal matrices. These are
diagonalized by noticing that
%
\begin{eqnarray}
&&\bold{R^d_L}\, \bold{M_D} \bold{M_D}^{\dagger}\, \bold{R^d_L}^{\dagger}= 
\left(\bold{M_D^{diag}}\right)^2 
\qquad ; \qquad
\bold{R^d_R}\, \bold{M_D}^{\dagger} \bold{M_D}\, \bold{R^d_R}^{\dagger} = 
\left(\bold{M_D^{diag}}\right)^2 \cr
\vb{20}
&&\bold{R^u_L}\, \bold{M_U} \bold{M_U}^{\dagger}\, \bold{R^u_L}^{\dagger}= 
\left(\bold{M_U^{diag}}\right)^2 
\qquad ; \qquad
\bold{R^u_R}\, \bold{M_U}^{\dagger} \bold{M_U}\, \bold{R^u_R}^{\dagger}=
\left(\bold{M_U^{diag}}\right)^2 
\end{eqnarray}
%
Before we close this section let us write down the
Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix with our conventions. The
couplings of the $W^{\pm}$ with the quarks are
%
\begin{equation}
{\cal L}_{CC}=
- \frac{g}{\sqrt{2}}\, W_{\mu}^-  
\ovl{d'_{Li}}\, \gamma^{\mu}\, u'_{Li} 
- \frac{g}{\sqrt{2}}\, W_{\mu}^+  
\ovl{u'_{Li}}\, \gamma^{\mu}\, d'_{Li} 
\end{equation}

Then in terms of the mass eigenstates the charged current Lagrangian
reads
%
\begin{equation}
{\cal L}_{CC}=
- \frac{g}{\sqrt{2}}\,\VCKM{ij}^*\,   W_{\mu}^-\,
\ovl{d_L}_j\, \gamma^{\mu}\, u_{Li} 
- \frac{g}{\sqrt{2}}\,\VCKM{ij}\,   W_{\mu}^+\,   
\ovl{u_L}_i\, \gamma^{\mu}\, d_{Lj} 
\end{equation}
%
where the CKM matrix $\VCKM{}$ is defined through
%
\begin{equation}
\VCKM{}=\bold{R^u_L}\, \bold{R^d_L}^{\dagger} 
\label{eq:CKM}
\end{equation}
%
in agreement with Eq.~(\ref{eq:VCKM-SM}).




\subsection{Slepton Mass Matrices}

In the unrotated basis $\widetilde{\ell'}_i=(\widetilde{e}_L,
\widetilde{\mu}_L,\widetilde{\tau}_L,\widetilde{e}^*_R,
\widetilde{\mu}^*_R,\widetilde{\tau}^*_R)$ we get
%
\begin{equation}
{\cal L}_M=- \widetilde{\ell'}^{\dagger}\, \bold{M_{\widetilde{\ell}}}^2\,
\widetilde{\ell'}
\end{equation}
%
where
%
\begin{equation}
\bold{M_{\widetilde{\ell}}}^2=
\left(
\matrix{
\bold{M^2_{LL}}&\bold{M^2_{LR}}\cr
\vb{18}
\bold{M^2_{RL}}&\bold{M^2_{RR}}
}
\right)
\label{eq:MassSleptons}
\end{equation}
%
and
%
\begin{eqnarray}
\bold{M^2_{LL}}&=&\smallfrac{1}{2}\, v_d^2\, h_E^* h_E^T + M^2_L
-\smallfrac{1}{2}\, (2 m_W^2 -m_Z^2) \cos 2 \beta \cr
\vb{18}
\bold{M^2_{RR}}&=& \smallfrac{1}{2}\, v_d^2\, h_E^T h_E^* + M^2_R
-( m_Z^2 -m_W^2) \cos 2 \beta \cr
\vb{18}
\bold{M^2_{LR}}&=&\frac{v_d}{\sqrt{2}}\, A_E^* -\mu
\frac{v_u}{\sqrt{2}}\, h_E^* \cr
\vb{18}
\bold{M^2_{RL}}&=&\bold{M^2_{LR}}^{\dagger}
\end{eqnarray}
%
We define the mass eigenstates
%
\begin{equation}
\widetilde{\ell}=\bold{R^{\widetilde{\ell}}}\, \widetilde{\ell'}
\end{equation}
%
which implies
%
\begin{equation}
\widetilde{\ell'}_i=\bold{R^{\widetilde{\ell}}}_{ji}^{\ *} \, \widetilde{\ell}_j
\end{equation}
%
The rotation matrices are obtained from
%
\begin{equation}
\bold{R^{\widetilde{\ell}}}^{\ \dagger}
\left(\bold{M_{\widetilde{\ell}}^{diag}}\right)^2
\bold{R^{\widetilde{\ell}}}= \bold{M_{\widetilde{\ell}}}^2
\end{equation}
%
In most applications the matrices in \Eq{eq:MassSleptons} are real and
therefore the rotation matrices $\bold{R^{\widetilde{\ell}}}$ are
orthogonal matrices.



\subsection{Sneutrino Mass Matrices}

In the unrotated basis $\widetilde{\nu}'_i=\widetilde{\nu}_{iL}$
we have
%
\begin{equation}
{\cal L}_M=-\half\, \widetilde{\nu}'^{\dagger}\, \bold{M_{\widetilde{\nu}}}^2\,
\widetilde{\nu}'
\end{equation}
%
where
%
\begin{equation}
\bold{M_{\widetilde{\nu}}}^2
=M^2_L
+\smallfrac{1}{2}\, m_Z^2 \cos 2 \beta 
\label{eq:MassSneutrinos}
\end{equation}
%
We define the mass eigenstates
%
\begin{equation}
\widetilde{\nu}=\bold{R^{\widetilde{\nu}}}\, \widetilde{\nu}'
\end{equation}
%
which implies
%
\begin{equation}
\widetilde{\nu}'_i=\bold{R^{\widetilde{\nu}}}_{ji}^{\ *} \, 
\widetilde{\nu}_j
\end{equation}
%
The rotation matrices are obtained from
%
\begin{equation}
\bold{R^{\widetilde{\nu}}}^{\ \dagger}
\left(\bold{M_{\widetilde{\nu}}^{diag}}\right)^2
\bold{R^{\widetilde{\nu}}}= \bold{M_{\widetilde{\nu}}}^2
\end{equation}
%
In most applications the matrix in \Eq{eq:MassSneutrinos} is real and
therefore the rotation matrices $\bold{R^{\widetilde{\ell}}}$ are
orthogonal matrices.




\subsection{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}
{\cal L}_M=-\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:MassSquarksMSSM}
\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) \cos 2 \beta \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) \cos 2 \beta \cr
\vb{18}
\bold{M^2_{\widetilde{u}LR}}&=&\frac{v_u}{\sqrt{2}}\, A_U^* -\mu
\frac{v_d}{\sqrt{2}}\, h_U^* \cr
\vb{18}
\bold{M^2_{\widetilde{u}RL}}&=&\bold{M^2_{\widetilde{u}LR}}^{\dagger}
\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) \cos 2 \beta \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) \cos 2 \beta \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}
\end{eqnarray}
%
We define the mass eigenstates
%
\begin{equation}
\widetilde{q}=\bold{R^{\widetilde{q}}}\, \widetilde{q}'
\end{equation}
%
which implies
%
\begin{equation}
\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}
%
In many applications the matrices in \Eq{eq:MassSquarksMSSM} are real and
therefore the rotation matrices $\bold{R^{\widetilde{q}}}$ are
orthogonal matrices.




\section{Couplings in the MSSM}

\subsection{Charged Current Couplings}

Using two component spinors and following the notation of
ref.~\cite{haber:1985rc} the relevant part of the Lagrangian can be written
as
%
\begin{eqnarray}
\hskip -15pt
{\cal L}&=&
g\, W_{\mu}^- \left[ \left(\ovl{\lambda^3}\,
\ovl{\sigma}^{\mu}\, \lambda^+ - \ovl{\lambda^-}\,
\ovl{\sigma}^{\mu}\, \lambda^3 \right) 
-{1 \over \sqrt{2}} \left( \ovl{\widetilde{H}_u^0}\,\ovl{\sigma}^{\mu}\,
\widetilde{H}_u^+ + \ovl{\widetilde{H}_d^-}\,\ovl{\sigma}^{\mu}\,
\widetilde{H}_d^0 \right) \right. \cr
\vb{22}
&&\left. \hskip 1.5cm
-{1 \over \sqrt{2}} \left( \ovl{\ell'_L}_i\, 
\ovl{\sigma}^{\mu}\, \nu'_{Li} 
+ \ovl{d'_L}_i\, \ovl{\sigma}^{\mu}\, u'_{Li} \right) \right]\cr
\vb{24}
&+&
g\, W_{\mu}^+ \left[ \left(\ovl{\lambda^+}\,
\ovl{\sigma}^{\mu}\, \lambda^3 - \ovl{\lambda^3}\,
\ovl{\sigma}^{\mu}\, \lambda^- \right) 
-{1 \over \sqrt{2}} \left( \ovl{\widetilde{H}_u^+}\,\ovl{\sigma}^{\mu}\,
\widetilde{H}_u^0 + \ovl{\widetilde{H}_d^0}\,\ovl{\sigma}^{\mu}\,
\widetilde{H}_d^- \right) \right.\cr
\vb{22}
&&\left. \hskip 1.5cm
-{1 \over \sqrt{2}} \left( \ovl{\nu'_L}_i\, 
\ovl{\sigma}^{\mu}\, \ell'_{Li} 
+ \ovl{u'_L}_i\, \ovl{\sigma}^{\mu}\, d'_{Li} \right) \right]
\label{cc1}
\end{eqnarray}  
%
To obtain the couplings in four component notation we first write
\Eq{cc1} in terms of the mass eigenstates in two component
notation, $F_i^{\pm}$ and $F_i^0$. In order to do that we recall that
for the neutralinos
%
\begin{equation}
\ba{rclcrcl}
-i \lambda'& = & \bold{N^*}_{i1} F_i^0 &\hskip 1cm &
i \ovl{\lambda'}& = & \bold{N}_{i1} \ovl{F_i^0}\cr
\vb{18}
-i \lambda^3& =& \bold{N^*}_{i2} F_i^0 & \hskip 1cm & 
i \ovl{\lambda^3} &=& \bold{N}_{i2} \ovl{F_i^0}\cr
\vb{18}
\widetilde{H}_d^0 &=& \bold{N^*}_{i3} F_i^0 & \hskip 1cm & 
\ovl{\widetilde{H}_d^0}& = &\bold{N}_{i3} \ovl{F_i^0}\cr
\vb{18}
\widetilde{H}_u^0 &=& \bold{N^*}_{i4} F_i^0 & \hskip 1cm & 
\ovl{\widetilde{H}_u^0} &=& \bold{N}_{i4} \ovl{F_i^0}
\ea
\end{equation}
%
while for the charginos
%
\begin{equation}
\label{eq:chareigMSSM}
\ba{rclcrcl}
-i \lambda^- &=& \bold{U^*}_{j1} F_j^- &\hskip 1cm& 
i \ovl{\lambda^-} &=& \bold{U}_{j1} \ovl{F_j^-}\cr
\vb{18}
 \widetilde{H}_d^-&=& \bold{U^*}_{j2} F_j^- & \hskip 1cm& 
\ovl{\widetilde{H}_d^-} &=& \bold{U}_{j2} \ovl{F_j^-}\cr
\vb{18}
-i \lambda^+ &=& \bold{V^*}_{j1} F_j^+ & \hskip 1cm& 
i \ovl{\lambda^+} &=& \bold{V}_{j1} \ovl{F_j^+}\cr
\vb{18}
\widetilde{H}_u^+&=& \bold{V^*}_{j2} F_j^+ & \hskip 1cm& 
\ovl{\widetilde{H}_u^+} &=& \bold{V}_{j2} \ovl{F_j^+}
\ea
\end{equation}
%
We obtain then
%
\begin{eqnarray}
{\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) 
+\ovl{F_j^-}\, \ovl{\sigma}^{\mu}\, F_i^0 
\left( - \bold{N^*}_{i2} \bold{U}_{j1} -\frac{1}{\sqrt{2}}\,
 \bold{N^*}_{i3} \bold{U}_{j2}  \right.
\right) \cr
\vb{20}
&&\hskip 1.2cm 
\left.
-\frac{1}{\sqrt{2}}\left(
\ovl{\ell'_L}_i\, \ovl{\sigma}^{\mu}\, \nu_{Li}
+ \ovl{d'_L}_i\, \ovl{\sigma}^{\mu}\, u'_{Li} \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) 
+\ovl{F_i^0}\, \ovl{\sigma}^{\mu}\, F_j^- 
 \left( - \bold{N}_{i2} \bold{U^*}_{j1} -\frac{1}{\sqrt{2}}\,
\bold{N}_{i3} \bold{U^*}_{j2} \right) \right. \cr
\vb{20}
&&\hskip 1.2cm 
\left.
-\frac{1}{\sqrt{2}}\left(
\ovl{\ell'_L}_i\, \ovl{\sigma}^{\mu}\, \nu_{Li}
+ \ovl{d'_L}_i\, \ovl{\sigma}^{\mu}\, u'_{Li} \right)
\right] 
\label{cc2}
\end{eqnarray}  
%
Finally using 
%
\begin{equation}
  \label{eq:13}
  \begin{array}{rclcrcl}
\ovl{F_i^0}\, \ovl{\sigma}^{\mu}\, F_j^+ &=& -\ovl{\chi_j^-}
\gamma^{\mu} P_R \chi_i^0 &\hskip 1cm& 
\ovl{F_j^-}\, \ovl{\sigma}^{\mu}\, F_i^0 &=& \ovl{\chi_j^-}
\gamma^{\mu} P_L \chi_i^0 \\[+3mm]
\ovl{F_i^0}\, \ovl{\sigma}^{\mu}\, F_j^- &=& \ovl{\chi_i^0}
\gamma^{\mu} P_L \chi_j^- &\hskip 1cm& 
\ovl{F_j^+}\, \ovl{\sigma}^{\mu}\, F_i^0 &=& - \ovl{\chi_i^0}
\gamma^{\mu} P_R \chi_j^- \\[+3mm]
\ovl{\ell'_L}_i\, \ovl{\sigma}^{\mu}\, \nu'_{Li} &=& 
\ovl{\ell'_L}_i \gamma^{\mu} P_L \nu'_{Li} &\hskip 1cm&
 \ovl{\nu'_L}_i\, \ovl{\sigma}^{\mu}\, \ell'_{Li} &=& 
\ovl{\nu_L}_i
\gamma^{\mu} P_L \ell_{Li} 
  \end{array}
\end{equation}
%
we get
%
\begin{eqnarray}
{\cal L}\hskip -7pt&=&\hskip -7pt 
 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 
-\frac{1}{\sqrt{2}}\left( 
\ovl{\ell'_L}_i \gamma^{\mu} P_L \nu'_{Li} +
\ovl{d'_L}_i \gamma^{\mu} P_L u'_{Li} \right)
\right] \phantom{\hskip 18mm}\cr
\vb{20}
\hskip -7pt&+&\hskip -7pt 
 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^-  
-\frac{1}{\sqrt{2}}\left(
\ovl{\nu'_L}_i \gamma^{\mu} P_L \ell'_{Li} +
\ovl{u'_L}_i \gamma^{\mu} P_L d'_{Li} \right)
\right] 
\label{cc3}
\end{eqnarray}  
%
where
%
\begin{eqnarray}
O^L_{ji}&=& \left(- \bold{N^*}_{i2} \bold{U}_{j1} -\frac{1}{\sqrt{2}}\,
 \bold{N^*}_{i3} \bold{U}_{j2} \right)\cr
\vb{20}
O^R_{ji}&=& \left(- \bold{N}_{i2} \bold{V^*}_{j1} +\frac{1}{\sqrt{2}}\,
\bold{N}_{i4} \bold{V^*}_{j2} \right) 
\end{eqnarray}
%
Finally we rotate the leptons and quarks to the mass eigenstates using
the relations\footnote{Notice that as the neutrinos 
are massless we can rotate them
by the same matrix as the leptons. Then the charged current for
neutrinos and leptons will remain diagonal.} 
%
\begin{equation}
\ba{rclcrcl}
u'_{Li} &=& \bold{R^u_L}_{ij} u_{Lj} &\hskip 1cm& 
\ovl{u'_L}_i &=& \ovl{u_L}_j \left(\bold{R^u_L}\right)^*_{ij}\cr
\vb{18}
d'_{Li} &=& \bold{R^d_L}_{ij} d_{Lj} &\hskip 1cm& 
\ovl{d'_L}_i &=& \ovl{d_L}_j \left(\bold{R^d_L}\right)^*_{ij}\cr
\ea
\end{equation}
%
and
%
\begin{equation}
\ba{rclcrcl}
\ell'_{Li} &=& \bold{R^{\ell}_L}_{ij} \ell_{Lj} &\hskip 1cm& 
\ovl{\ell'_L}_i &=& \ovl{\ell_L}_j \left(\bold{R^{\ell}_L}\right)^*_{ij}\cr
\vb{18}
\nu'_{Li} &=& \bold{R^{\ell}_L}_{ij} \nu_{Lj} &\hskip 1cm& 
\ovl{\nu'_L}_i &=& \ovl{\nu_L}_j \left(\bold{R^{\ell}_L}\right)^*_{ij}\cr
\ea
\end{equation}
%
to get
%
\begin{eqnarray}
\hskip -30pt
{\cal L}\hskip -7pt&=&\hskip -7pt 
 g W_{\mu}^- \! \left[ 
\ovl{\chi_j^-}\, \gamma^{\mu}\!  \left ( O^L_{ji} P_L +
O^R_{ji} P_R \right) \chi_i^0 
-\frac{1}{\sqrt{2}}\left( 
\ovl{\ell_L}_i \gamma^{\mu} P_L \nu_{Li} +
\VCKM{ij}^*\, \ovl{d_L}_j \gamma^{\mu} P_L u_{Li} \right)
\right] \cr
\vb{24}
\hskip -7pt&+&\hskip -7pt 
 g W_{\mu}^+ \!\! \left[ 
\ovl{\chi_i^0}\, \gamma^{\mu}\!  \left ( (O^L_{ji})^* P_L \!+\!
(O^R_{ji})^* P_R\right)\! \chi_j^-  
\!-\! \frac{1}{\sqrt{2}}\! \left(
\ovl{\nu_L}_i \gamma^{\mu} P_L \ell_{Li} \!+\!
\VCKM{ij}\, \ovl{u_L}_i \gamma^{\mu} P_L d_{Lj} \right)\!
\right] \hbox{\hskip 5mm}
\label{cc4}
\end{eqnarray}  
%
where the CKM matrix $\VCKM{}$ was defined in Eqs.~(\ref{eq:VCKM-SM})
and (\ref{eq:CKM}). One can easily verify that the charged current
couplings of the fermions $\ell,\nu,u,d$ are in agreement with
Eq.~(\ref{eq:LagSU2U1}). 


\subsection{Neutral Current Couplings}

Using two component spinors and following the notation of
ref.~\cite{haber:1985rc} the relevant part of the Lagrangian can be written
as

\begin{eqnarray}
{\cal \hskip -5mm L}\hskip -7pt&=&\hskip -7pt
-\smallfrac{1}{2}\, \left(g W^3_{\mu} +g'B_{\mu} \right)\
\left(\ovl{\widetilde{H}_u^+}\, \ovl{\sigma}^{\mu}\, \widetilde{H}_u^+
- \ovl{\widetilde{H}_d^0}\, \ovl{\sigma}^{\mu}\, \widetilde{H}_d^-
\right)
-g W^3_{\mu} \left(\ovl{\lambda^+}\, \ovl{\sigma}^{\mu}\,
\lambda^+
-\ovl{\lambda^-}\, \ovl{\sigma}^{\mu}\,
\lambda^- \right)\cr
\vb{24}
&&\hskip -7pt
-\smallfrac{1}{2}\, \left(-g W^3_{\mu} +g'B_{\mu} \right)\
\left(\ovl{\widetilde{H}_u^0}\, \ovl{\sigma}^{\mu}\, \widetilde{H}_u^0
- \ovl{\widetilde{H}_d^0}\, \ovl{\sigma}^{\mu}\, \widetilde{H}_d^0
\right)\cr
\vb{24}
&&\hskip -7pt
+\smallfrac{1}{2}\,  \left(-g W^3_{\mu} +g'B_{\mu} \right)\
\ovl{\nu'_L}_i\, \ovl{\sigma}^{\mu}\, \nu'_{Li} 
+\smallfrac{1}{2}\, \left(g W^3_{\mu} +g'B_{\mu} \right)\,
\ovl{\ell'_L}_i\, \ovl{\sigma}^{\mu}\, \ell'_{Li}
-g' B_{\mu}\,
\ovl{\ell'^c_L}_i\, \ovl{\sigma}^{\mu}\, \ell'^c_{Li}\cr
\vb{24}
&&\hskip -7pt
+\smallfrac{1}{2}\,  \left(-g W^3_{\mu} -\smallfrac{1}{3}g'B_{\mu} \right)\
\ovl{u'_L}_i\, \ovl{\sigma}^{\mu}\, u'_{Li} 
+\smallfrac{1}{2}\, \left(g W^3_{\mu} -\smallfrac{1}{3}g'B_{\mu} \right)\,
\ovl{d'_L}_i\, \ovl{\sigma}^{\mu}\, d'_{Li}\cr
\vb{24}
&&\hskip -7pt
+\smallfrac{2}{3}\, g' B_{\mu}\,
\ovl{u'^c_L}_i\, \ovl{\sigma}^{\mu}\, u'^c_{Li}
-\smallfrac{1}{3}\, g' B_{\mu}\,
\ovl{d'^c_L}_i\, \ovl{\sigma}^{\mu}\, d'^c_{Li}
\label{nc1}
\end{eqnarray}  
%
To obtain the couplings in four component notation we first write
\Eq{nc1} in terms of the mass eigenstates in two component
notation, $F_i^{\pm}$ and $F_i^0$. We also use Eq.~(\ref{rotWQ}) to write,
%
\begin{eqnarray}
\label{eq:AandZ}
g W_{\mu}^3&=&e A_{\mu} + \frac{g}{\cos \theta_W} \left(1 -\sin^2
\theta_W\right) Z_{\mu} \cr
\vb{20}
g' B_{\mu}&=&e A_{\mu} - \frac{g}{\cos \theta_W} \sin^2 \theta_W  Z_{\mu} 
\end{eqnarray}
%
We then get in two component notation,
%
\begin{eqnarray}
\hskip -2cm
{\cal L}&\hskip -7pt=\hskip -7pt&
-e A_{\mu} \left[\vb{18}
\bold{V}_{ik} \bold{V^*}_{jk} \ovl{F_i^+}\,
\ovl{\sigma}^{\mu}\, F_j^+ 
- \bold{U}_{ik} \bold{U^*}_{jk} \ovl{F_i^-}\,
\ovl{\sigma}^{\mu}\, F_j^- 
-\ovl{\ell'_L}_i\, \ovl{\sigma}^{\mu}\, \ell'_{Li} 
+\ovl{\ell^c_L}_i\, \ovl{\sigma}^{\mu}\, \ell'^c_{Li} \right. \cr
\vb{20}
&&
\left. \hskip 1.5cm
+\smallfrac{2}{3}\, 
\ovl{u'_L}_i\, \ovl{\sigma}^{\mu}\, u'_{Li}
-\smallfrac{1}{3}\, 
\ovl{d'_L}_i\, \ovl{\sigma}^{\mu}\, d'_{Li}
-\smallfrac{2}{3}\, 
\ovl{u'^c_R}_i\, \ovl{\sigma}^{\mu}\, u'^c_{Ri}
+\smallfrac{1}{3}\, 
\ovl{d'^c_R}_i\, \ovl{\sigma}^{\mu}\, d'^c_{Ri}
\right]\cr
\vb{24}
&\hskip -7pt+\hskip -7pt&
\frac{g}{\cos \theta_W} Z_{\mu} 
\left[\vb{18} \
\half \left(
\bold{N}_{i4} \bold{N^*}_{j4} -\bold{N}_{i3} \bold{N^*}_{j3}
\right)\, \ovl{F_i^0} \ovl{\sigma}^{\mu}\, F_j^0 \right.\cr
\vb{22}
&&\hskip 2cm
+\left(
\half \bold{U}_{i2} \bold{U^*}_{j2} +\bold{U}_{i1} \bold{U^*}_{j1}
-\sin^2 \theta_W 
\bold{U}_{ik} \bold{U^*}_{jk}
\right)\, \ovl{F_i^-}\, \ovl{\sigma}^{\mu}\, F_j^- \cr
\vb{22}
&&\hskip 2cm
+\left(
- \half \bold{V}_{i2} \bold{V^*}_{j2} - \bold{V}_{i1} \bold{V^*}_{j1}
+\sin^2 \theta_W 
\bold{V}_{ik} \bold{V^*}_{jk}
\right)\, \ovl{F_i^+}\, \ovl{\sigma}^{\mu}\, F_j^+ \cr
\vb{22}
&&\hskip 2cm
-\half \ovl{\nu'_L}_i\, \ovl{\sigma}^{\mu}\, \nu'_{Li}
+(\half -\sin^2 \theta_W )\, \ovl{\ell'_L}_i\, \ovl{\sigma}^{\mu}\, \ell'_{Li}
+ \sin^2 \theta_W\, \ovl{\ell^c_L}_i\, \ovl{\sigma}^{\mu}\,
\ell'^c_{Li}\cr
\vb{22}
&&\hskip 2cm
+(-\half +\smallfrac{2}{3}\sin^2\theta_W )
\ovl{u'_L}_i\, \ovl{\sigma}^{\mu}\, u'_{Li}
+(\half -\smallfrac{1}{3}\sin^2 \theta_W )\, 
\ovl{d'_L}_i\, \ovl{\sigma}^{\mu}\, d'_{Li} \cr
\vb{22}
&&\hskip 2cm\left. \vb{18}
+\smallfrac{2}{3} \sin^2 \theta_W\, \ovl{u'^c_L}_i\, 
\ovl{\sigma}^{\mu}\, u'^c_{Li}
-\smallfrac{1}{3} \sin^2 \theta_W\, \ovl{d'^c_L}_i\, 
\ovl{\sigma}^{\mu}\, d'^c_{Li} \right]
\label{nc2}
\end{eqnarray}  
%
Now using the unitarity of the diagonalization matrices we get, still
in two component spinor notation,
%
\begin{eqnarray}
\hskip -2cm
{\cal L}&\hskip -7pt=\hskip -7pt&
e A_{\mu} \left[\, \vb{18}
\left( \ovl{F_i^-}\,
\ovl{\sigma}^{\mu}\, F_i^-
-\ovl{F_i^+}\,
\ovl{\sigma}^{\mu}\, F_i^+ \right) 
+\left( 
\ovl{\ell'_L}_i\, \ovl{\sigma}^{\mu}\, \ell'_{Li} 
- \ovl{\ell'^c_L}_i\, \ovl{\sigma}^{\mu}\, \ell'^c_{Li} \right)
\right. \cr
\vb{22}
&&\hskip 1.5cm
\left. \vb{18}
-\smallfrac{2}{3}
\left( 
\ovl{u'_L}_i\, \ovl{\sigma}^{\mu}\, u'_{Li} 
- \ovl{u'^c_L}_i\, \ovl{\sigma}^{\mu}\, u'^c_{Li} \right)
+\smallfrac{1}{3}
\left( 
\ovl{d'_L}_i\, \ovl{\sigma}^{\mu}\, d'_{Li} 
- \ovl{d'^c_L}_i\, \ovl{\sigma}^{\mu}\, d'^c_{Li} \right)
\right]\cr
\vb{24}
&\hskip -7pt+\hskip -7pt&
\frac{g}{\cos \theta_W} Z_{\mu} 
\left[\vb{18} \
\half \left(
\bold{N}_{i4} \bold{N^*}_{j4} -\bold{N}_{i3} \bold{N^*}_{j3}
\right)\, 
\ovl{F_i^0} \ovl{\sigma}^{\mu}\, F_j^0 \right.\cr
&&\hskip 2cm
\vb{22}
+\left(
\half \bold{U}_{i2} \bold{U^*}_{j2} +\bold{U}_{i1} \bold{U^*}_{j1}
-\sin^2 \theta_W\, \delta_{ij}
\right)\, \ovl{F_i^-}\, \ovl{\sigma}^{\mu}\, F_j^- \cr
&&\hskip 2cm
\vb{22}
+\left(
- \half \bold{V}_{i2} \bold{V^*}_{j2} - \bold{V}_{i1} \bold{V^*}_{j1}
+\sin^2 \theta_W\, \delta_{ij}
\right)\, \ovl{F_i^+}\, \ovl{\sigma}^{\mu}\, F_j^+ \cr
\vb{22}
&&\hskip 2cm
-\half \ovl{\nu'_L}_i\, \ovl{\sigma}^{\mu}\, \nu'_{Li}
+(\half -\sin^2 \theta_W )\, \ovl{\ell'_L}_i\, \ovl{\sigma}^{\mu}\, \ell'_{Li}
+ \sin^2 \theta_W\, \ovl{\ell^c_L}_i\, \ovl{\sigma}^{\mu}\,
\ell'^c_{Li}\cr
\vb{22}
&&\hskip 2cm
+(-\half +\smallfrac{2}{3}\sin^2\theta_W )
\ovl{u'_L}_i\, \ovl{\sigma}^{\mu}\, u'_{Li}
+(\half -\smallfrac{1}{3}\sin^2 \theta_W )\, 
\ovl{d'_L}_i\, \ovl{\sigma}^{\mu}\, d'_{Li} \cr
\vb{22}
&&\hskip 2cm\left. \vb{18}
+\smallfrac{2}{3} \sin^2 \theta_W\, \ovl{u'^c_L}_i\, 
\ovl{\sigma}^{\mu}\, u'^c_{Li}
-\smallfrac{1}{3} \sin^2 \theta_W\, \ovl{d'^c_L}_i\, 
\ovl{\sigma}^{\mu}\, d'^c_{Li} \right]
\label{nc3}
\end{eqnarray}  
%
Finally we get in four component notation\footnote{We can see that the
neutral current interaction 
of the leptons and quarks is diagonal in generation
space after we perform the rotation into the mass eigenstates.} 
%
\begin{eqnarray}
\hskip -2cm
{\cal L}&\hskip -7pt=\hskip -7pt&
e A_{\mu} \left[\ \vb{18}
\ovl{\chi_i^-}\,
\gamma^{\mu}\, \chi_i^-
+\ovl{\ell}_i\, \gamma^{\mu}\, \ell_i
-\smallfrac{2}{3}\, \ovl{u}_i\, \gamma^{\mu}\, u_i
+\smallfrac{1}{3}\, \ovl{d}_i\, \gamma^{\mu}\, d_i
\right]\cr
\vb{24}
&\hskip -7pt+\hskip -7pt&
\frac{g}{\cos \theta_W} Z_{\mu} 
\left[\vb{18} \
\half  \ovl{\chi_i^0} \gamma^{\mu}\,
\left( O^L_{ij}{}'' P_L + O^R_{ij}{}'' P_R \right) \chi_j^0 
+ \ovl{\chi_i^-}\, \gamma^{\mu}\,
\left( O^L_{ij}{}' P_L + O^R_{ij}{}' P_R \right) \right.
\chi_j^- \cr
\vb{22}
&&\hskip 2cm
+\half \ovl{\nu}_i\, \gamma^{\mu}\, \left( -\half P_L +\half P_R
\right) \nu_i \\
\vb{22}
&&\hskip 2cm\left. \vb{18}
+\sum_{f=\ell,u,d}
\ovl{f}_i\, \gamma^{\mu}\, \left[\vb{16} \left(-T_3^f +Q^f \sin^2 \theta_W 
\right)\, P_L +Q^f \sin^2 \theta_W\, P_R \right] f_i
\right] \nn
\label{nc4}
\end{eqnarray}  
%
where
%
\begin{eqnarray}
O^L_{ij}{}''&=& \half \left(
\bold{N}_{i4} \bold{N^*}_{j4} -\bold{N}_{i3} \bold{N^*}_{j3}
\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}
-\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}
%
and $T_3^f$ and $Q^f$ are, respectively, the weak isospin and the
charge of fermion $f=\ell,u,d$ as given in
Table~\ref{tab:FermionQuantumNumbers}. We can verify that the neutral
current couplings of the fermions $\ell,u,d$ are in agreement with
Eq.~(\ref{eq:LagSU2U1}). 


\subsection{Scalar couplings to fermions}


\subsubsection{Charged scalars couplings to fermions}

Using two component spinors and following the notation of
ref.~\cite{haber:1985rc} the relevant part of the Lagrangian can be written
as
%
\begin{equation}
{\cal L}={\cal L}_{gauge} +{\cal L}_{Yukawa} 
\end{equation}
%
where the gauge part is determined by the gauge group and quantum
numbers of the matter multiplets and is given by
%
\begin{eqnarray}
{\cal L}_{gauge}&\hskip -7pt=\hskip -7pt& 
H_d^+\, \left[
\frac{g}{\sqrt{2}}\, (-i\lambda^3)\, \widetilde{H}_d^-
+\frac{g'}{\sqrt{2}}\, (-i\lambda')\, \widetilde{H}_d^-
-g\, (-i\lambda^-)\, \widetilde{H}_d^0
\right]\cr
\vb{22}
&\hskip -7pt + \hskip -7pt&
H_u^+\, \left[
-\frac{g}{\sqrt{2}}\, (i \ovl{\lambda^3)}\ \ovl{\widetilde{H}_u^+}
-\frac{g'}{\sqrt{2}}\, (i\ovl{\lambda'})\ \ovl{\widetilde{H}_u^+}
-g\, (i\ovl{\lambda^+})\ \ovl{\widetilde{H}_u^0}
\right]\cr
\vb{22}
&\hskip -7pt + \hskip -7pt&
\widetilde{\ell}^*_{Li}\, \left[
\frac{g}{\sqrt{2}}\, (-i\lambda^3)\, \ell'_{Li}
+\frac{g'}{\sqrt{2}}\, (-i\lambda')\, \ell'_{Li}
-g\, (-i\lambda^-)\, \nu'_{Li}\ 
\right]
+ \widetilde{\ell}_{Ri}\, \left[
- g' \sqrt{2}\, (i \ovl{\lambda'})\ \ovl{\ell^c_L}_i
\right]\cr
\vb{22}
&\hskip -7pt + \hskip -7pt&
\widetilde{u}^*_{Li}\, \left[
-\frac{g}{\sqrt{2}}\, (-i\lambda^3)\, u'_{Li}
-\frac{g'}{3 \sqrt{2}}\, (-i\lambda')\, u'_{Li}
-g\, (-i\lambda^+)\, d'_{Li}\ 
\right]
+ \widetilde{u}_{Ri}\, \left[
\frac{4g'}{3\sqrt{2}}\, (i \ovl{\lambda'})\ \ovl{u^c_L}_i
\right]\cr
\vb{22}
&\hskip -7pt + \hskip -7pt&
\widetilde{d}^*_{Li}\, \left[
+\frac{g}{\sqrt{2}}\, (-i\lambda^3)\, d'_{Li}
-\frac{g'}{3 \sqrt{2}}\, (-i\lambda')\, d'_{Li}
-g\, (-i\lambda^-)\, u'_{Li}\ 
\right]
+ \widetilde{d}_{Ri}\, \left[
-\frac{2g'}{3\sqrt{2}}\, (i \ovl{\lambda'})\ \ovl{d^c_L}_i
\right]\cr
\vb{22}
&\hskip -7pt + \hskip -7pt&
\hbox{h.c.}
\label{eq:csgauge}
\end{eqnarray}
%
The Yukawa part is derived from the superpotential $W$ by use of 
\Eq{eq:LYukawa}. In order to use this equation it is helpful to write
down explicitly the superpotential. We have
%
\begin{eqnarray}
W&=&-\left(h_E\right)_{ij} \widehat L_i^1 \widehat R_j \widehat H_d^2
-\left(h_D\right)_{ij} \widehat Q_i^1 \widehat D_j \widehat H_d^2
-\left(h_U\right)_{ij} \widehat Q_i^2 \widehat U_j \widehat H_u^1\cr
\vb{18}
&&
+\left(h_E\right)_{ij} \widehat L_i^2 \widehat R_j \widehat H_d^1
+\left(h_D\right)_{ij} \widehat Q_i^2 \widehat D_j \widehat H_d^1
+\left(h_U\right)_{ij} \widehat Q_i^1 \widehat U_j \widehat H_u^2
\label{eq:Wexplicit}
\end{eqnarray}
%
Then in two component spinor notation the Yukawa part of the couplings
of charged scalar to fermions is
%
\begin{eqnarray}
\label{eq:csyukawa}
{\cal L}_{Yukawa}&\hskip -5pt=\hskip -5pt&
-\left(h_E\right)_{ij} \widetilde{\ell}_{Li} \ell'^c_{Lj} \widetilde{H}_d^0
-\left(h_E\right)_{ij} \widetilde{\ell}_{Rj} \ell'_{Li} \widetilde{H}_d^0
+\left(h_E\right)_{ij} \widetilde{\ell}_{Rj} \nu'_{Li} \widetilde{H}_d^-
+\left(h_E\right)_{ij} H_d^- \nu'_{Li} \ell'^c_{Lj}\cr
\vb{22}
&\hskip -10pt&
-\left(h_D\right)_{ij} \widetilde{d}_{Li} d'^c_{Lj} \widetilde{H}_d^0
-\left(h_D\right)_{ij} \widetilde{d}_{Rj} d'_{Li} \widetilde{H}_d^0
+\left(h_D\right)_{ij} \widetilde{d}_{Rj} u'_{Li} \widetilde{H}_d^-
+\left(h_D\right)_{ij} H_d^- u'_{Li} d'^c_{Lj}\cr
\vb{22}
&\hskip -10pt&
-\left(h_U\right)_{ij} \widetilde{u}_{Li} u'^c_{Lj} \widetilde{H}_u^0
-\left(h_U\right)_{ij} \widetilde{u}_{Rj} u_{Li} \widetilde{H}_u^0
+\left(h_U\right)_{ij} \widetilde{u}_{Rj} d_{Li} \widetilde{H}_u^+
+\left(h_U\right)_{ij} H_u^+ d'_{Li} u'^c_{Lj}\cr
\vb{22}
&\hskip -10pt&
+\left(h_D\right)_{ij} \widetilde{u}_{Li} d'^c_{Lj} \widetilde{H}_d^-
+\left(h_U\right)_{ij} \widetilde{d}_{Li} u'^c_{Lj} \widetilde{H}_u^+\cr
\vb{22}
&\hskip -10pt&\hskip 2pt
+\hbox{h.c.}
\end{eqnarray}
%
Now this can be written in four component spinor notation in terms of
the mass eigenstates. We will do this separately for ${\cal L}_{gauge}$ and
${\cal L}_{Yukawa}$. We obtain\footnote{We still keep the unrotated
fields for the leptons and quarks. The final rotation will be done in
the end of this section.}
%
\begin{eqnarray}
{\cal L}_{gauge}&=&
H_d^+\left[\left(
\frac{g}{\sqrt{2}}\, \bold{U}^*_{B2}
\left(\bold{N}^*_{A2}+\tan\theta_W \bold{N}^*_{A1}\right)\
-g\, \bold{U}^*_{B1}\bold{N}^*_{A3} \right)\
\ovl{\chi^0}_A P_L \chi^-_B \right]
\cr
\vb{24}
&\hskip -5pt+\hskip -5pt&
H_u^+\left[\left(
- \frac{g}{\sqrt{2}}\, \bold{V}_{B2}
\left(\bold{N}_{A2}+\tan\theta_W \bold{N}_{A1}\right)\
-g\, \bold{V}_{B1}\bold{N}_{A4} \right)\
\ovl{\chi^0}_A P_L \chi^-_B \right]
\cr
\vb{24}
&\hskip -5pt+\hskip -5pt&
\bold{R^{\widetilde{\ell}}}_{X,i}\ \widetilde{\ell}^*_X
\left[
\frac{g}{\sqrt{2}}\,
\left(\bold{N}^*_{A2}+\tan\theta_W \bold{N}^*_{A1}\right)\
\ovl{\chi^0}_A P_L \ell'_i 
-g\, \bold{U}^*_{B1}\
\ovl{\nu'_i} P_L \chi^-_B \right]
\cr
\vb{24}
&\hskip -5pt+\hskip -5pt&
\bold{R^{\widetilde{\ell}}}_{X,i+3}\ \widetilde{\ell}^*_X
\left[
- g \sqrt{2}\tan\theta_W\, \bold{N}_{A1}\
\ovl{\chi^0}_A P_R \ell'_i \right]
\cr
\vb{24}
&\hskip -5pt+\hskip -5pt&
\bold{R^{\widetilde{u}}}_{X,i}\ \widetilde{u}^*_X
\left[
-\frac{g}{\sqrt{2}}\,
\left(\bold{N}^*_{A2}+\smallfrac{1}{3}\tan\theta_W \bold{N}^*_{A1}\right)\
\ovl{\chi^0}_A P_L u'_i 
-g\, \bold{V}^*_{B1}\
\ovl{\chi^-_B} P_L d'_i  \right]
\cr
\vb{24}
&\hskip -5pt+\hskip -5pt&
\bold{R^{\widetilde{u}}}_{X,i+3}\ \widetilde{u}^*_X
\left[
\frac{4}{3} \frac{g}{\sqrt{2}}\tan\theta_W\, \bold{N}_{A1}\
\ovl{\chi^0}_A P_R u'_i \right]
\cr
\vb{24}
&\hskip -5pt+\hskip -5pt&
\bold{R^{\widetilde{d}}}_{X,i}\ \widetilde{d}^*_X
\left[
\frac{g}{\sqrt{2}}\,
\left(\bold{N}^*_{A2}-\smallfrac{1}{3}\tan\theta_W \bold{N}^*_{A1}\right)\
\ovl{\chi^0}_A P_L d'_i 
-g\, \bold{U}^*_{B1}\
\ovl{\chi^c}_B P_L u'_i \right]
\cr
\vb{24}
&\hskip -5pt+\hskip -5pt&
\bold{R^{\widetilde{d}}}_{X,i+3}\ \widetilde{d}^*_X
\left[
- \frac{2}{3}\frac{g}{\sqrt{2}}\tan\theta_W\, \bold{N}_{A1}\
\ovl{\chi^0}_A P_R d'_i \right]
\cr
\vb{24}
&\hskip -5pt+\hskip -5pt&
\hbox{h.c.}
\label{eq:csffg4component}
\end{eqnarray}
%
and
%
\begin{eqnarray}
{\cal L}_{Yukawa}&=&
-\left(h_E\right)_{ij} \bold{N}^*_{A3}\left(
\bold{R^{\widetilde{\ell}}}^{\; *}_{X,i}\, \widetilde{\ell}_X\
\ovl{\ell'}_j P_L \chi^0_A 
+\bold{R^{\widetilde{\ell}}}_{X,j+3}\, \widetilde{\ell}^*_X\
\ovl{\chi^0}_A P_L \ell'_i \right)
\cr
\vb{24}
&\hskip -5pt\hskip -5pt&
+\left(h_E\right)_{ij} \left(\bold{U}^*_{A2}\,
\bold{R^{\widetilde{\ell}}}_{X,j+3}\, \widetilde{\ell}^*_X\
\ovl{\nu'}_i P_L \chi^-_A 
+H_d^-\, \ovl{\ell'}_j P_L \nu'_i \right)
\cr
\vb{24}
&\hskip -5pt\hskip -5pt&
-\left(h_D\right)_{ij} \bold{N}^*_{A3}\left(
\bold{R^{\widetilde{d}}}^{\; *}_{X,i}\, \widetilde{d}_X\
\ovl{d'}_j P_L \chi^0_A 
+\bold{R^{\widetilde{d}}}_{X,j+3}\, \widetilde{d}^*_X\
\ovl{\chi^0}_A P_L d'_i \right)
\cr
\vb{24}
&\hskip -5pt\hskip -5pt&
+\left(h_D\right)_{ij} \left(\bold{U}^*_{A2}\,
\bold{R^{\widetilde{d}}}_{X,j+3}\, \widetilde{d}^*_X\
\ovl{\chi^c}_A P_L u'_i
+H_d^-\, \ovl{d'}_j P_L u'_i \right)
\cr
\vb{24}
&\hskip -5pt\hskip -5pt&
-\left(h_U\right)_{ij} \bold{N}^*_{A4}\left(
\bold{R^{\widetilde{u}}}^{\; *}_{X,i}\, \widetilde{u}_X\
\ovl{u'}_j P_L \chi^0_A 
+\bold{R^{\widetilde{u}}}_{X,j+3}\, \widetilde{u}^*_X\
\ovl{\chi^0}_A P_L u'_i \right)
\cr
\vb{24}
&\hskip -5pt\hskip -5pt&
+\left(h_U\right)_{ij} \left(\bold{V}^*_{A2}\,
\bold{R^{\widetilde{u}}}_{X,j+3}\, \widetilde{u}^*_X\
\ovl{\chi^-}_A P_L d'_i
+H_u^+\, \ovl{u'}_j P_L d'_i \right)
\cr
\vb{24}
&\hskip -5pt\hskip -5pt&
+\left(h_D\right)_{ij}\ \bold{U}^*_{A2}\,
\bold{R^{\widetilde{u}}}^{\; *}_{X,i}\, \widetilde{u}_X\
\ovl{d'}_j P_L \chi_A^-
+\left(h_U\right)_{ij}\ \bold{V}^*_{A2}\,
\bold{R^{\widetilde{d}}}^{\; *}_{X,i}\, \widetilde{d}_X\
\ovl{u'}_j P_L \chi^c_A
\cr
\vb{24}
&\hskip -5pt\hskip -5pt&
+\hbox{ h.c.}
\label{eq:csffy4component}
\end{eqnarray}
%
In Section~\ref{sffFinal} we will give the final formulas.


\subsubsection{Neutral scalars couplings to fermions}
%
Using two component spinors and following the notation of
ref.~\cite{haber:1985rc} the relevant part of the Lagrangian can be written
as
%
\begin{equation}
{\cal L}={\cal L}_{gauge} +{\cal L}_{Yukawa} 
\end{equation}
%
where the gauge part is 
%
\begin{eqnarray}
\label{eq:nscfg}
\hskip -15pt
{\cal L}_{gauge}&\hskip -7pt=\hskip -7pt& 
\left(H_d^0\right)^*\, \left[
- \frac{g}{\sqrt{2}}\, (-i\lambda^3)\, \widetilde{H}_d^0
+\frac{g'}{\sqrt{2}}\, (-i\lambda')\, \widetilde{H}_d^0
-g\, (-i\lambda^+)\, \widetilde{H}_d^-
\right]\cr
\vb{22}
&\hskip -7pt+\hskip -7pt& 
\left(H_u^0\right)^*\, \left[
\frac{g}{\sqrt{2}}\, (-i \lambda^3)\ \widetilde{H}_u^0
-\frac{g'}{\sqrt{2}}\, (-i \lambda')\ \widetilde{H}_u^0
-g\, (-i \lambda^-)\ \widetilde{H}_u^+
\right]\cr
\vb{22}
&\hskip -7pt+\hskip -7pt& 
\widetilde{\nu}_i^*\, \left[
-\frac{g}{\sqrt{2}}\, (-i\lambda^3)\, \nu_i
+\frac{g'}{\sqrt{2}}\, (-i\lambda')\, \nu_i
-g\, (-i\lambda^+)\, \ell_i^- 
\right]\cr
\vb{22}
&\hskip -7pt+\hskip -7pt& 
\hbox{h.c.}
\end{eqnarray}
%
and the Yukawa part is 
%
\begin{eqnarray}
\label{eq:nscfy}
{\cal L}_{Yukawa}&\hskip -5pt=\hskip -5pt&
\left(h_E\right)_{ij} \widetilde{\nu'}_{Li} \ell'^c_{Lj} \widetilde{H}_d^- 
-\left(h_E\right)_{ij} H_d^0 \ell'_{Li} \ell'^c_{Lj} 
-\left(h_D\right)_{ij} H_d^0 d'_{Li} d'^c_{Lj} 
-\left(h_U\right)_{ij} H_u^0 u'_{Li} u'^c_{Lj} 
\cr
\vb{22}
&\hskip -10pt&\hskip 2pt
+\hbox{h.c.}
\end{eqnarray}
%
Now this can be written in four component spinor notation in terms of
the mass eigenstates. We will do this separately for ${\cal L}_{gauge}$ and
${\cal L}_{Yukawa}$. We obtain
%
\begin{eqnarray}
{\cal L}_{gauge}&\hskip -7pt=\hskip -7pt&
\left(H_d^0\right)^* \left[-\frac{g}{\sqrt{2}}\left(\vb{14}
\bold{N}^*_{A2} -\tan\theta_W \bold{N}^*_{A1} \right)
\bold{N}^*_{B3}\ 
\ovl{\chi^0}_A P_L \chi^0_B
-g \bold{V}^*_{A1} \bold{U}^*_{B2}\ \ovl{\chi^-}_A P_L \chi^-_B
\right]\cr
\vb{22}
&\hskip -14pt &
+\left(H_u^0\right)^* \left[\frac{g}{\sqrt{2}}\left(\vb{14}
\bold{N}^*_{A2} -\tan\theta_W \bold{N}^*_{A1}\right)
 \bold{N}^*_{B4}\
\ovl{\chi^0}_A P_L \chi^0_B
-g \bold{V}^*_{A2} \bold{U}^*_{B1}\ \ovl{\chi^-}_A P_L \chi^-_B
\right]\cr
\vb{22}
&\hskip -14pt &
+\bold{R^{\widetilde{\nu}}}{}_{X,i}\,
\widetilde{\nu}^*_X \left[
-\frac{g}{\sqrt{2}}\left(\vb{14}
\bold{N}^*_{A2} 
-\tan\theta_W \bold{N}^*_{A1} \right)\ 
\ovl{\chi^0}_A P_L \nu'_i
-g \bold{V}^*_{A1}\ \ovl{\chi^-}_A P_L \ell'_i
\right]\cr
\vb{22}
&\hskip -14pt &
+\hbox{h.c.}
\label{eq:nsffg4component}
\end{eqnarray}
%
and
%
\begin{eqnarray}
{\cal L}_{Yukawa}&=&
-\left(h_E\right)_{ij} H_d^0\, \ovl{\ell'}_j P_L \ell'_i
-\left(h_D\right)_{ij} H_d^0\, \ovl{d'}_j P_L d'_i
-\left(h_U\right)_{ij} H_u^0\, \ovl{u'}_j P_L u'_i\cr
\vb{20}
&&
+\left(h_E\right)_{ij} \bold{U}^*_{A2}\,
\bold{R^{\widetilde{\nu}}}{}_{X,i}^*\,
\widetilde{\nu}_X\, \ovl{\ell'}_j P_L \chi_A^-
+\hbox{h.c.}
\label{eq:nsffy4component}
\end{eqnarray}
%
In Section~\ref{sffFinal} we will give the final formulas.

\subsubsection{Final formulas for the scalar couplings to fermions}
\label{sffFinal}

In this Section we present the final formulas for the scalar couplings
to fermions. To be more precise we are going to only to write down
those for the couplings {\sl fermion--sfermion--chargino} and
{\sl fermion--sfermion--neutralino}. These leave out the couplings of
the neutral and charged Higgs bosons with the fermions. These can be
read from
Eqs.~(\ref{eq:csffg4component}),~(\ref{eq:csffy4component}), 
(\ref{eq:nsffg4component}) and~(\ref{eq:nsffy4component}) not
forgetting that we still have to perform the rotation into the Higgs
bosons mass eigenstates.

We interaction lagrangian can be written as
%
\begin{eqnarray}
{\cal L}&=&
\widetilde{\nu}_X\ \ovl{\ell}_i \left[ \vb{14}
C_{iAX}^{L(\ell)} P_L + C_{iAX}^{R(\ell)} P_R \right] \chi_A^- 
+\widetilde{\ell}^*_X\ \ovl{\nu}_i \left[ \vb{14}
C_{iAX}^{L(\nu)} P_L + C_{iAX}^{R(\nu)} P_R \right] \chi_A^- 
\cr
\vb{22}
&+&
\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 
\cr
\vb{22}
&+&
\widetilde{f}_X\ \ovl{f}_i \left[ \vb{14}
N_{iAX}^{L(f)} P_L + N_{iAX}^{R(f)} P_R \right] \chi_A^0 
\cr
\vb{22}
&+&
\hbox{   h.c.}
\end{eqnarray}
%
where the indices $i,A,X$ apply to generations, charginos (or
neutralinos) and sfermions, respectively. All repeated indices are
summed over and $f=\ell,\nu,u,d$. The coefficients $C$ and $N$ can be
read from 
Eqs.~(\ref{eq:csffg4component}),~(\ref{eq:csffy4component}), 
(\ref{eq:nsffg4component}) and~(\ref{eq:nsffy4component}). We list
them below.
%
\begin{equation}
\left\{
\ba{l}
C_{iAX}^{L(\ell)}=\left(h_E\right)_{kj}\, \bold{U}^*_{A2}\,
\bold{R^{\widetilde{\nu}}}^{\; *}_{X,k}\,
\bold{R^{\ell}_R}_{ij}\hbox{\hskip 4.85cm}
\cr
\vb{22}
C_{iAX}^{R(\ell)}=-g \bold{V}_{A1}\,
\bold{R^{\widetilde{\nu}}}^{\; *}_{X,k}\,
\bold{R^{\ell}_L}^*_{ki}
\ea
\right.
\end{equation}


\begin{equation}
\left\{
\ba{l}
C_{iAX}^{L(\nu)}=-g \bold{U}^*_{A1}\,
\bold{R^{\widetilde{\ell}}}_{X,k}\,
\bold{R^{\ell}_L}^*_{ki}
+\left(h_E\right)_{kj}\, \bold{U}^*_{A2}\,
\bold{R^{\widetilde{\ell}}}_{X,j+3}\,
\bold{R^{\ell}_L}^*_{ki} \hbox{\hskip 8mm}
\cr
\vb{22}
C_{iAX}^{R(\nu)}=0\hfill
\ea
\right.
\end{equation}


\begin{equation}
\label{eq:sqdcn}
\left\{
\ba{l}
C_{iAX}^{L(d)}=
\left(h_D\right)_{kj}\, \bold{U}^*_{A2}\,
\bold{R^{\widetilde{u}}}^{\; *}_{X,k}\,
\bold{R^{d}_R}_{ij}
\cr
\vb{22}
C_{iAX}^{R(d)}=
-g \bold{V}_{A1}\,
\bold{R^{\widetilde{u}}}^{\; *}_{X,k}\,
\bold{R^{d}_L}^*_{ki} + \left(h_U^*\right)_{kj}\,
\bold{V}_{A2}\, 
\bold{R^{\widetilde{u}}}^{\; *}_{X,j+3}\,
\bold{R^{d}_L}^*_{ki}\hbox{\hskip 6.5mm}
\ea
\right.
\end{equation}


\begin{equation}
\label{eq:squcn}
\left\{
\ba{l}
C_{iAX}^{L(u)}=
\left(h_U\right)_{kj}\, \bold{V}^*_{A2}\,
\bold{R^{\widetilde{d}}}^{\; *}_{X,k}\,
\bold{R^{u}_R}_{ij}
\cr
\vb{22}
C_{iAX}^{R(u)}=
-g \bold{U}_{A1}\,
\bold{R^{\widetilde{d}}}^{\; *}_{X,k}\,
\bold{R^{u}_L}^*_{ki} + \left(h_D^*\right)_{kj}\,
\bold{U}_{A2}\, 
\bold{R^{\widetilde{d}}}^{\; *}_{X,j+3}\,
\bold{R^{u}_L}^*_{ik}\hbox{\hskip 7mm}
\ea
\right.
\end{equation}

and for the $N's$

\begin{equation}
\left\{
\ba{l}
N_{iAX}^{L(\ell)}=
-g\, \sqrt{2} \tan\theta_W\ \bold{N}^*_{A1}\,
\bold{R^{\widetilde{\ell}}}^{\; *}_{X,k+3}\,
\bold{R^{\ell}_R}_{ik}\,
-\left(h_E\right)_{kj}\, \bold{N}^*_{A3}\,
\bold{R^{\widetilde{\ell}}}^{\; *}_{X,k}\,
\bold{R_R^{\ell}}_{ij}
\cr
\vb{22}
N_{iAX}^{R(\ell)}=
\ds \frac{g}{\sqrt{2}}\left(\vb{14} \bold{N}_{A2} +\tan\theta_W
\bold{N}_{A1}\right)
\bold{R^{\widetilde{\ell}}}^{\; *}_{X,k}\,
\bold{R^{\ell}_L}^{\; *}_{ki}\,
-\left(h^*_E\right)_{kj}\, \bold{N}_{A3}\,
\bold{R^{\widetilde{\ell}}}^{\; *}_{X,j+3}\,
\bold{R_L^{\ell}}^{\; *}_{ki}\hbox{\hskip 8mm}
\ea
\right.
\end{equation}

\begin{equation}
\left\{
\ba{l}
N_{iAX}^{L(\nu)}=0
\cr
\vb{22}
N_{iAX}^{R(\nu)}=
\ds -\frac{g}{\sqrt{2}}\left(\vb{14} \bold{N}_{A2} -\tan\theta_W
\bold{N}_{A1}\right)
\bold{R^{\widetilde{\nu}}}^{\; *}_{X,k}\,
\bold{R^{\ell}_L}^{\; *}_{ki}\hbox{\hskip 52.8mm}
\ea
\right.
\end{equation}

\begin{equation}
\label{eq:squun}
\left\{
\ba{l}
N_{iAX}^{L(u)}=
\ds \frac{4}{3}\, \frac{g}{\sqrt{2}}\, \tan\theta_W\,
\bold{N}^*_{A1}\, \bold{R^{\widetilde{u}}}^{\; *}_{X,k+3}\,
\bold{R^u_R}_{ik}
-\left(h_U\right)_{kj}\, \bold{N}^*_{A4}\,
\bold{R^{\widetilde{u}}}^{\; *}_{X,k}\,
\bold{R_R^u}_{ij}
\cr
\vb{22}
N_{iAX}^{R(u)}=
\ds -\frac{g}{\sqrt{2}}\left(\vb{14} \bold{N}_{A2} 
+\smallfrac{1}{3}\tan\theta_W \bold{N}_{A1}\right)
\bold{R^{\widetilde{u}}}^{\; *}_{X,k}\,
\bold{R^{u}_L}^{\; *}_{ki}\,
-\left(h^*_U\right)_{kj}\, \bold{N}_{A4}\,
\bold{R^{\widetilde{u}}}^{\; *}_{X,j+3}\,
\bold{R_L^{u}}^{\; *}_{ki}
\ea
\right.
\end{equation}

\begin{equation}
\label{eq:sqddn}
\left\{
\ba{l}
N_{iAX}^{L(d)}=
\ds -\frac{2}{3}\, \frac{g}{\sqrt{2}}\, \tan\theta_W\,
\bold{N}^*_{A1}\, \bold{R^{\widetilde{d}}}^{\; *}_{X,k+3}\,
\bold{R^d_R}_{ik}
-\left(h_D\right)_{kj}\, \bold{N}^*_{A3}\,
\bold{R^{\widetilde{d}}}^{\; *}_{X,k}\,
\bold{R_R^d}_{ij}
\cr
\vb{22}
N_{iAX}^{R(d)}=
\ds \frac{g}{\sqrt{2}}\left(\vb{14} \bold{N}_{A2} 
-\smallfrac{1}{3}\tan\theta_W \bold{N}_{A1}\right)
\bold{R^{\widetilde{d}}}^{\; *}_{X,k}\,
\bold{R^{d}_L}^{\; *}_{ki}\,
-\left(h^*_D\right)_{kj}\, \bold{N}_{A3}\,
\bold{R^{\widetilde{d}}}^{\; *}_{X,j+3}\,
\bold{R_L^{d}}^{\; *}_{ki}\hbox{\hskip 4mm}
\ea
\right.
\end{equation}

%\section{Supersymmetric Grand Unification} 


\section{Coupling Constant Unification} 
We have seen in section~\ref{sec:gaugeunification} that the SM fails
to unify the gauge couplings at the scale $M_X$ in Grand Unified
Theories. We repeat here the exercise for the MSSM. 
As in the SM case we assume that the MSSM is valid all the way till the scale
$M_X$. In the two loop approximation the RGE for the gauge couplings are
still given by Eq.~(\ref{eq:gaugecoupevolution}) where we have neglected the
effect of the running masses. However the $b$'s functions are now
%
\begin{equation}
b_i= (\frac{33}{5},1,-3) \quad , \quad
b_{ij}=\left(\begin{array}{lll} \frac{199}{25} & \frac{27}{5} &
\frac{88}{5}
\\ \frac{9}{5} &25 & 24 \\ \frac{11}{5} & 9 & 14
\end{array}\right)
\end{equation}
%
The solution of the gauge couplings evolution is again given by
Eq.~(\ref{eq:gcevolution}) with the appropriate coefficients $b_i$ and
$b_{ij}$. Starting from the experimental values of the couplings at
$Q=m_Z$, Eq.~(\ref{eq:alfmz_exp}) one can integrate the equations and obtain the couplings at any scale
$Q$. In Fig.\ref{mssmcouplings} we see the evolution of the coupling
constants with the scale $Q$. We see that now they intersect all at
the same point. This is a major point in favour of SUSY. If we look in
more detail around the unification point and take in account the
errors in the experimental values at $m_Z$ we still get a very good
agreement, as shown in Fig.\ref{mssmcouplings-zoom}. 
However we should be more carefull,
because we have neglected the running of the Yukawa couplings that, at
two loops, mix with the gauge couplings. Once this is taken in account
the crossing is not completely perfect. This is not a problem because
we have neglected threshold effects. Also extension of the MSSM as
those with breaking of R-parity provide small changes that go in the
right direction. For a complete discussion see Ref.~\cite{diaz:1999is}. 

\begin{figure}
\centering
\includegraphics[clip,height=80mm]{mssmcouplings.eps}
\caption{Evolution of the gauge coupling constants with mass scale $Q$ in the
MSSM}
\label{mssmcouplings}
\end{figure}



\begin{figure}
\centering
\includegraphics[clip,height=80mm]{mssmcouplings-zoom.eps}
\caption{A closer look at the intersection of the gauge couplings in
  the MSSM.}
\label{mssmcouplings-zoom}
\end{figure}



\section{ Neutrino Masses } %%V will do

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