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\setcounter{chapter}{3}
\chapter{The Standard Model}
\label{standard}
\section{Introduction}


The Standard Model of strong and electroweak interactions (SM) has
provided the cornerstone of elementary particle physics for nearly
three decades. Except for the neutrino sector, it has been well
established at the various high-energy collider experiments. In this
chapter we review of the boson and fermion structures in the SM in
order to set the basis for the subsequent discussion of of additional
structure. The basic principle of of the \sm is gauge invariance which
puts together in the same basic framework the matter particles, their
interactions and the gauge vector bosons which mediate them. It is
based on the gauge group $SU(3) \times SU(2) \times U(1)$. The theory
consists of three sectors: Quantum Chromodynamics (QCD) which deals
with the strong interaction, Quantum Electrodynamics (QED) responsible
for the electromagnetic force, and the weak interaction sector.  The
latter two get mixed up in the process of symmetry breaking which is
required in order to account for the short range nature of the weak
interaction.  QED and the weak forces are combined in what is called
the Electroweak standard model.  Apart from the recent important
confirmation of neutrino anomalies and the theoretical unaesthetical
aspects, the SM has been remarkably successful in account for all
aspects over the past three decades. 

\section{Standard Electroweak Model} 
\label{SEM}


The Standard Electroweak model is a gauge theory of the
electromagnetic and weak interactions and it was first formulated by
Glashow, Salam and Weinberg~\cite{weinberg:1967tq, Salam:1968rm,
  glashow:1970gm}.     
It is based on the gauge group
$SU(2)_L\times U(1)_Y $ and therefore contains four gauge bosons
corresponding to the four generators of the gauge group. The fermions
are assigned to the fundamental representations of the gauge group and 
the form of the Lagrangian is uniquely determined from the gauge
invariance principle with the exception of the symmetry breaking
sector (Higgs sector) as we will see below.
%
We now write the Lagrangian for the different sectors of the theory.

\subsection{Gauge Bosons}

As we mentioned before there are four gauge bosons, three
$W_\mu^i$ ($i=1,2,3$,  one for each generator $T^i$) transforming as the
adjoint representation of $SU(2)_L$, and one $B_\mu$ for $U(1)_Y$.
The corresponding field tensors are:
\begin{equation}
\begin{array}{l}
W^a_{\mu\nu}=\partial_\mu W^a_\nu-\partial_\nu W^a_\mu - g\,
\epsilon_{abc} W^b_\mu W^c_\nu\\ B_{\mu\nu}=\partial_\mu
B_\nu-\partial_\nu B_\mu\\
\end{array}
\end{equation}
where we call $g'$ and $g$ the coupling constants of the $U(1)_Y$ and
$SU(2)_L$ groups respectively.
%
The kinetic Lagrangian for the bosons is given by
\begin{equation}
{\cal L}_G=-\frac{1}{4}W^{a}_{\mu\nu}W_a^{\mu\nu}-\frac{1}{4}
B_{\mu\nu}B^{\mu\nu}
\end{equation}
%
and it is invariant under the (separate) local gauge transformations of the
$SU(2)_L$ and $U(1)_Y$ groups. The general form of these gauge
transformations for finite gauge parameters is,
%
\begin{eqnarray}
\label{eq:GTgaugegeral}
W_{\mu}^a \frac{\sigma^a}{2} &\rightarrow& W_{\mu}'^a
\frac{\sigma^a}{2}= {\cal U}_L\,  W_{\mu}^a \frac{\sigma^a}{2}\, {\cal
U}_L^{-1} +\frac{i}{g}\, \partial_{\mu} {\cal U}_L\, {\cal U}_L^{-1}\cr
\vb{16}
B_{\mu}&\rightarrow& B_{\mu}'=B_{\mu}+\frac{i}{g'}\, \partial_{\mu}
{\cal U}_Y\, {\cal U}_Y^{-1} 
\end{eqnarray}
%
where
%
\begin{eqnarray}
\label{eq:GT}
{\cal U}_L&=& e^{\displaystyle i \alpha^a \frac{\sigma^a}{2}}\cr
\vb{16}
{\cal U}_Y&=& e^{i \alpha_Y }
\end{eqnarray}
%
For infinitesimal transformations these reduce to
%
\begin{eqnarray}
\label{eq:gaugetransfomations}
\delta W_{\mu}^a&=&-\epsilon^{abc}\, \alpha^b\, W_{\mu}^c +
\frac{1}{g}\, \partial_{\mu} \alpha^a \cr
\vb{16}
\delta B_{\mu}&=&  \frac{1}{g'}\, \partial_{\mu} \alpha_Y
\end{eqnarray}
%
The most important feature of the gauge  Lagrangian is that it contains
gauge boson self couplings, as expected from the non-abelian nature of
the $SU(2)_L$ group.
%

\subsection{Matter Fields}

The matter fields of the SM are all the known fermions which are
classified in three generations. One can project each fermion in two
helicity states, left and right
\begin{equation}
\begin{array}{l}
\psi_L=\frac{1}{2}(1-\gamma_5) \psi\\
\psi_R=\frac{1}{2}(1+\gamma_5) \psi\\
\end{array}
\end{equation}
which transforms differently under the $SU(2)_L$ group. Left handed
components are assigned to doublet representation while right handed
ones transform as singlets, that is,
%
\begin{equation}
  L_L=\left(\matrix{\nu_e \cr e^-}\right)_L \, , \,
  Q_L=\left(\matrix{u \cr d}\right)_L \, , \,
  e^-_R, u_R, d_R,
\end{equation}
%
where we have only shown the particles in the first generation. The
other two generations are just copies of the first. The quantum
numbers with respect to the gauge group $SU(2)_L \times U(1)_Y $,
are given in Table \ref{tab:FermionQuantumNumbers},
where the electric charge is given by
%
\begin{equation}
  \label{eq:2}
  Q=T_3 + Y
\end{equation}
%
and $T_3=\half \sigma_3$ ($\sigma_a$ are the Pauli matrices).

\begin{table}[hbt]
\begin{center}
\begin{tabular}{|c|ccccccc|}\hline
\vb{13}
Particle&$\nu_{e L}$&$e_L$&$u_L$&$d_L$&$e_R$&$u_R$&$d_R$\\ \hline \hline
\vb{13}
$T_3$&$\half$&$-\half$&$\half$&$-\half$&$0$&$0$&$0$ \\ \hline
\vb{13}
$Y$&$-\half$&$-\half$&$\smallfrac{1}{6}$&$\smallfrac{1}{6}$&$-1$&
$\smallfrac{2}{3}$ & $-\smallfrac{1}{3}$ \\ \hline
\vb{13}
$Q$&$0$&$-1$&$\smallfrac{2}{3}$&$-\smallfrac{1}{3}$&$-1$&$\smallfrac{2}{3}$&
$-\smallfrac{1}{3}$
\vb{13}
\\ \hline
\end{tabular}
\end{center}
\caption{Quantum numbers of the particles of the first generation 
with respect to the gauge group $SU(2)_L \times U(1)_Y $.}
\label{tab:FermionQuantumNumbers}
\end{table}
%
Under finite local gauge transformations the $\Psi_L$ and $\psi_R$
fields transform as follows
%
\begin{eqnarray}
  \label{eq:GTfermions}
  \Psi_L\rightarrow \Psi'_L&=& e^{i\alpha^a \frac{\sigma^a}{2}}\, 
  e^{i \alpha_Y Y}\, \Psi_L\cr
\vb{16}
  \psi_R\rightarrow \psi'_R&=&   e^{i \alpha_Y Y}\, \psi_R
\end{eqnarray}
%
The principle of gauge invariance establishes that the piece of the
Lagrangian describing the gauge interactions of the fermions is
obtained from kinetic energy part of the Lagrangian, after
substituting the derivative by the covariant derivative,
%
\begin{eqnarray}
\label{eq:3}
\partial_\mu\Psi_L\rightarrow {\cal D}_\mu \Psi_L&=&
\left(\partial_\mu +i g \frac{\sigma_a}{2} W^a_\mu +i g'\, Y B_\mu\right)
\Psi_L \cr
\vb{18}
\partial_\mu\psi_R\rightarrow {\cal D}_\mu \psi_R&=&
\left(\partial_\mu +i g'\, Y B_\mu\right) \psi_R
\end{eqnarray}
%
Using Eq.{~(\ref{eq:GTgaugegeral})} and Eq.~(\ref{eq:GTfermions}) one
can easily verify that the covariant derivatives have the 
appropriate transformations properties (that is, they transform in the
same way as the fields),
%
\begin{eqnarray}
  \label{eq:GTcovder}
  {\cal D}_\mu \Psi_L\rightarrow {\cal D}_\mu \Psi'_L&=& e^{i\alpha^a
  \frac{\sigma^a}{2}}\, 
  e^{i \alpha_Y Y}\, {\cal D}_\mu \Psi_L\cr
\vb{16}
  {\cal D}_\mu \psi_R\rightarrow {\cal D}_\mu\psi'_R&=&   e^{i
  \alpha_Y Y}\, {\cal D}_\mu \psi_R
\end{eqnarray}
%
After symmetry breaking (see below) we have
%
\begin{eqnarray}
W_{\mu}^3&=&\sin \theta_W A_{\mu} +\cos \theta_W Z_{\mu}\cr
B_{\mu}&=&\cos \theta_W A_{\mu} -\sin \theta_W Z_{\mu}
\label{rotWQ}
\end{eqnarray}
%
with
%
\begin{equation}
e=g\ \sin\theta_W=g'\cos\theta_W;\hskip 5mm ; \hskip 5mm
\frac{g'}{g}=\tan \theta_W
\label{tree1}
\end{equation}
%
and
%
\begin{equation}
  W_{\mu}^{\pm}=\frac{W_{\mu}^1 \mp i\, W_{\mu}^2}{\sqrt{2}}
\end{equation}
%
We can then write the covariant derivative in the more useful form
%
\begin{eqnarray}
D_{\mu} \Psi_L&=&\partial_{\mu} + i\, \left[\vb{20} 
\frac{g}{\sqrt{2}}\,  \left(\matrix{0&W_{\mu}^+\cr0&0}\right)
+\frac{g}{\sqrt{2}}\,  \left(\matrix{0&0\cr W_{\mu}^-&0}\right)\right.\cr
&&
\hskip 15mm 
\left.\vb{20}
+\frac{g}{\cos \theta_W}\left(T_3 -\sin^2\theta_W Q\right)\, Z_{\mu} + 
e\, Q\, A_{\mu} \right] \Psi_L\cr
\vb{18}
D_{\mu} \psi_R&=&\partial_{\mu} + i\, \left[
-\frac{g}{\cos \theta_W} \sin^2\theta_W Q\, Z_{\mu} + 
e\, Q\, A_{\mu} \right] \psi_R
\end{eqnarray}
%
This way we get for the Lagrangian of the fermion fields, invariant
under local gauge transformations,
%
\begin{eqnarray}
  \label{eq:LagSU2U1}
  {\cal L}^{\hbox{\scriptsize kinetic}}_F&=& 
\sum_{doublets} i\, \overline{\Psi}_L
  \gamma^{\mu}\, D_{\mu} \Psi_L +
  \sum_{singlets} i\, \overline{\psi}_R
  \gamma^{\mu}\, D_{\mu} \psi_R \cr
\vb{18}
  &=& \sum_f i\, \overline{\psi}_f \gamma^{\mu}
  \partial_{\mu} \psi_f \cr
\vb{18}
&&-e\, \sum_f Q^f \overline{\psi}_f \gamma^{\mu} \psi_f \ A_{\mu} 
-\frac{g}{\cos\theta_W}\, \sum_f \overline{\psi}_f \gamma^{\mu}
  \left(g_V^f -g_A^f\, \gamma_5\right)\, Z_{\mu}\cr
\vb{18}
&&
-\frac{g}{\sqrt{2}}\, \sum_{doublets} \overline{\psi_u} \gamma^{\mu}
  \frac{1-\gamma_5}{2} \psi_d W_{\mu}^+ 
-\frac{g}{\sqrt{2}}\, \sum_{doublets} \overline{\psi_d} \gamma^{\mu}
  \frac{1-\gamma_5}{2} \psi_u W^-_{\mu}
\end{eqnarray}
%
where the last sum is over all the doublets of the theory
%
\begin{equation}
  \Psi_L=\left(\matrix{\psi_u\cr \psi_d}\right)_L=
\left(\matrix{\nu_e\cr e}\right)_L, 
\left(\matrix{u\cr d}\right)_L, \cdots 
\end{equation}
and
\begin{equation}
\begin{array}{lr}
g_v^f=\frac{1}{2} T^f_{3L} -\sin^2\theta_W Q^f &
 \;\;\;\;\;\;\;\;\;\;\; g_a^f=\frac{1}{2} T^f_{3L}\\
\end{array}
\end{equation} 

\section{Spontaneous Symmetry Breaking: Mass Generation}

The Lagrangian given in Eq.~(\ref{eq:LagSU2U1}) is invariant under the
gauge group $SU(2)_L \times U(1)_Y $. Because the left and right
components of the fermion fields transform differently under the gauge
group we can not write a mass term for them. Also there is no mass
term for the gauge bosons compatible with the symmetry. So at this
point all the fermions and gauge bosons have zero mass. However in
Nature only the photon is massless. If we were to add fermion and
gauge boson masses by hand we would break the gauge invariance and
therefore the renormalizability of the theory would be spoiled.  In
order to prevent this from happening, it is necessary to introduce the
masses by a mechanism that preserves the gauge invariance of the
Lagrangian. This is achieved by the spontaneous symmetry breaking
mechanism, the so-called {\it Higgs mechanism}~\cite{Higgs:1966ev,
Guralnik:1964eu}.
 
An spontaneously broken symmetry is preserved by the Lagrangian but it
is not a symmetry of the ground state of the system, the vacuum state.
In order to implement this idea in the SM a $SU(2)_L$ scalar doublet
$\Phi$ is introduced in the theory
%
\begin{equation}
  \Phi= \left(\matrix{\phi^+\cr \phi^0}\right)
\end{equation}
%
with the quantum numbers given in Table \ref{HiggsQuantumNumbers}.
%
\begin{table}[hbt]
\begin{center}
\begin{tabular}{|c|cc|}\hline
\vb{13}
Particle&$\phi^+$&$\phi^0$\\ \hline \hline
\vb{13}
$T_3$&$\half$&$-\half$\\ \hline
\vb{13}
$Y$&$\half$&$\half$ \\ \hline
\vb{13}
$Q$&$1$&$0$
\vb{13}
\\ \hline
\end{tabular}
\end{center}
\caption{Quantum numbers of the Higgs doublet 
with respect to the gauge group $SU(2)_L \times U(1)_Y $.}
\label{HiggsQuantumNumbers}
\end{table}
%
We can write the following $SU(2)_L \times U(1)_Y $ invariant 
Lagrangian
\begin{equation}
\label{eq:HiggsLag}
\begin{array}{l}
{\cal L}_H=({\cal D}_\mu \Phi)^\dagger ({\cal D}^\mu \Phi)
-\mu^2\, \Phi^\dagger \Phi  -\lambda 
\left(\Phi^\dagger \Phi \right)^2 \cr 
\vb{18}
{\cal L}_{Yuk}=
-{\displaystyle \sum_{ij}} \left[Y^l_{ij}\ \overline {l'}_{iL}\,
\Phi\,
 l'_{jR} + Y^u_{ij}\ \overline {u'}_{iL}\, \tilde{\Phi}\, u'_{jR}
+Y^d_{ij}\ \overline {d'_{iL}}\, \Phi\, d'_{jR} + h.c. \right]
\end{array}
\end{equation}
where $\tilde{\Phi}=i\sigma_2\, {\Phi}^*$ is a doublet with
$Y=-1/2$ as needed to make ${\cal L}_{Yuk}$ invariant under
the gauge group. We have used the notation $l',u',d'$ to distinguish
these weak states from the mass eigenstates that we will discuss below.

In Fig. \ref{Higgspotential} we sketch the potential part in ${\cal
L}_H$, $V=\mu^2|\Phi|^2 +\lambda |\Phi|^4$ as a function of
$|\Phi|=\sqrt{\Phi^\dagger \Phi}$. 
For $\mu^2>0$ $V$ has a unique minimum at $|\Phi|=0$. 
However when $\mu^2<0$ the classical ground state occurs at
$|\Phi|^2=-\frac{1}{2}\mu^2/\lambda$. In the quantized theory this
is equivalent to the appearance of a non-zero vacuum expectation value
of $\Phi$
\begin{equation}
\vev {|\Phi|}=\frac{v}{\sqrt{2}}= \sqrt{-\frac{\mu^2}{2\lambda}}
\label{eq:SMvev}
\end{equation}

\begin{figure}
\begin{center}
\includegraphics[height=5cm]{HiggsPotmuPos.ps}\hskip 2cm
\includegraphics[height=5cm]{HiggsPotmuNeg.ps}
\caption{Classical potential $V$ of the scalar field for different signs of 
$\mu^2$.}
\label{Higgspotential}
\end{center}
\end{figure}

If we now perform perturbation theory around the true vacuum it is
convenient to parameterize the scalar field as
\begin{equation}
\label{eq:PolarParametrization}
\Phi=e^{i\frac{\theta^a(x)
\sigma_a}{ v}}
\left(\begin{array}{c} 0\\ 
\displaystyle
\frac{v+H(x)}{\sqrt{2}}\end{array}\right)
\end{equation}
%
where the fields $\theta^a$ and $H$ are real and have zero vacuum
expectation value. If the $SU(2)_L$ symmetry was a {\it global}
symmetry of the Lagrangian the three $\theta$ fields would correspond
to physical fields with zero mass since the potential is flat in those
directions, as stated by the Goldstone Theorem\cite{Goldstone:1961eq,
Goldstone:1962es}. In
fact if we substitute Eq.~(\ref{eq:PolarParametrization}) in the Higgs
Lagrangian we get
%
\begin{eqnarray}
  \label{eq:4}
  {\cal L}_H&=& \frac{1}{2}\, \partial_{\mu} H \partial^{\mu} H 
+ \mu^2\, H^2 +
\frac{1}{2}\, \partial_{\mu} \theta^a \partial^{\mu} \theta^a +
\frac{1}{4}\, g^2\, v^2\, W^+_{\mu} W^{- \mu} +
\frac{1}{8}\, g^2\, v^2\, Z_{\mu} Z^{\mu} \cr
\vb{18}
&&+
\frac{gv}{2\sqrt{2}}\, W_{\mu}^- \partial^{\mu} \theta^+ +
\frac{gv}{2\sqrt{2}}\, W_{\mu}^+ \partial^{\mu} \theta^- +
\frac{gv}{2\cos\theta_W}\, Z_{\mu} \partial^{\mu} \theta^3
+ \cdots
\end{eqnarray}
%
where the dots stand for cubic and quartic terms and we have used
Eq.~(\ref{eq:SMvev}). Looking at Eq.~(\ref{eq:4}) one would think that
the three $\theta^a$ fields are massless and the remaining $H$ field
has a mass squared $m_H^2=-2\mu^2 >0$. Notice however, that there is a
mixing between the gauge fields and the $\theta^a$ fields, the {\it
would be} Goldstone bosons. One has therefore to be more careful in
the analysis of the spectrum. The best way to do it is to realize that
the three $\theta^a$ fields can be gauged away by a finite
transformation under the local $SU(2)_L$ group. This can be
immediately seen if we compare Eq.~(\ref{eq:PolarParametrization})
with Eq.~(\ref{eq:GTfermions}). We see that a local $SU(2)_L$
transformation with parameter
%
\begin{equation}
  \label{eq:5}
\alpha^a=-\frac{2 \theta^a}{v}  
\end{equation}
%
will rotate away the $\theta^a$ fields. We then have
%
\begin{eqnarray}
  \label{eq:UnitaryGauge}
  \Phi(x) &\rightarrow&\Phi'(x)=
e^{ -i \frac{2 \theta^a(x) }{v} \frac{\sigma^a}{2}}\ 
\Phi= 
\left(\begin{array}{c} 0\\ \displaystyle
\frac{v+H(x)}{\sqrt{2}}
\end{array}
\right) \cr
\vb{16}
W_{\mu}^a &\rightarrow&  W_{\mu}'^a \cr
\vb{16}
B_{\mu}& \rightarrow & B'_{\mu}=B_{\mu}
\end{eqnarray}
%
This particular choice of gauge is called
the {\it unitary gauge}. In this gauge there is only one physical
scalar field, the Higgs boson $H$, and the $\theta^a$ degrees of
freedom become the longitudinal components of the 3 gauge bosons of
$SU(2)_W$ which are now massive.

Introducing Eq.~(\ref{eq:UnitaryGauge}) into the Higgs Lagrangian,
Eq~(\ref{eq:HiggsLag}), and dropping the prime in $W_{\mu}'^a$, we get
after rotating the gauge bosons according to Eq.~(\ref{rotWQ}),
%
\begin{eqnarray}
{\cal L}_H&\hskip -2mm=\hskip -2mm& 
\frac{1}{2}(\partial_\mu H)^2-\frac{1}{2}m_H^2 H^2
-\frac{1}{4}\lambda\, H^4 -\lambda\, v\, H^3 
+ \frac{1}{2}\, v g^2\, W^+_{\mu} W^{- \mu} H  
+ \frac{1}{4}\, v \frac{g^2}{\cos\theta_W}\, Z_{\mu} Z^{\mu} H  \cr
\vb{18}
&&\hskip -3mm
+\frac{1}{4} g^2\, W^+_{\mu} W^{- \mu} H^2  
+\frac{1}{8} \frac{g^2}{\cos\theta_W}\, Z_{\mu} Z^{\mu} H^2  
+\frac{1}{2} m_Z^2\, Z_\mu Z^\mu 
+ m_W^2\, W^+_{\mu} W^{- \mu} +\lambda \frac{v^4}{4}
\end{eqnarray}
%
where the masses $m_W$, $m_Z$, and $m_H$ are given by
\begin{equation}
\begin{array}{lcr}
m_W=\frac{1}{2} g v \;\;\;\; &
m_Z=\frac{1}{2}g'v=\displaystyle \frac{m_W}{\cos\theta_W}
\;\;\;\;\; & m_H=\sqrt{-2\mu^2}
\label{tree2}
\end{array}
\end{equation}
We see now the physical meaning of the rotation (\ref{rotWQ}). After
the spontaneous symmetry breaking of the electroweak symmetry the
fields $Z$ and $A$ are the physical mass eigenstates. Since
electromagnetism remains unbroken the photon field stays massless. In
other words the assignment of hypercharges for the Higgs field has
been chosen in such a way that the field acquiring a vacuum
expectation value was neutral and the charge conservation holds
exactly even after the spontaneous symmetry breaking.

We now turn to the Yukawa Lagrangian ${\cal L}_{Yuk}$. In the unitary
gauge we have
%
\begin{equation}
  \label{eq:9}
 \Phi= 
\left(\begin{array}{c} 0\\ \displaystyle
\frac{v+H(x)}{\sqrt{2}}
\end{array}
\right)
\quad ; \quad
\tilde{\Phi}= 
\left(\begin{array}{c} \displaystyle
\frac{v+H(x)}{\sqrt{2}}\\
0
\end{array}
\right)
\end{equation}
%
and we get
%
\begin{eqnarray}
{\cal L}_{Yuk}&= & -{
\sum_{ij}} \left[\vb{16}\
\overline{l'}_{iL}\, M^{l}_{ij} l'_{jR} 
+ \overline {u'}_{iL}\, M^u_{ij} u'_{jR} + 
\overline {d'}_{iL}\, M^d_{ij} d'_{jR}\right. \cr
\vb{18}
 && \left. +
\frac{H}{\sqrt{2}}\, \overline{l'}_{iL}\, Y^{l}_{ij} l'_{jR} 
+ \frac{H}{\sqrt{2}}\, \overline {u'}_{iL}\, Y^u_{ij} u'_{jR} + 
\frac{H}{\sqrt{2}}\, \overline {d'}_{iL}\, Y^d_{ij} d'_{jR} + h.c.
\right]
\end{eqnarray}
%
where 
%
\begin{equation}
  \label{eq:10}
  M^{l}_{ij}=Y^{l}_{ij}\, \frac{v}{\sqrt{2}}\quad , \quad
  M^{u}_{ij}=Y^{u}_{ij}\, \frac{v}{\sqrt{2}}\quad , \quad
  M^{d}_{ij}=Y^{d}_{ij}\, \frac{v}{\sqrt{2}}
\end{equation}
%
Let us denote by $l,u$ and $d$ the mass eigenstates obtained via the rotations
%
\begin{equation}
\begin{array}{lcr}
l_{iL}=\bold{R^l_L}_{ij} l'_{jL} \;\;\;\;\;\; & 
u_{iL}=\bold{R^u_L}_{ij} u'_{jL} \;\;\;\;\;\; & 
d_{iL}=\bold{R^d_L}_{ij} d'_{jL} \\ 
l_{iR}=\bold{R^l_R}_{ij} l'_{jR} \;\;\;\;\;\; & 
u_{iR}=\bold{R^u_R}_{ij} u'_{jR} \;\;\;\;\;\; & 
d_{iR}=\bold{R^d_R}_{ij} d'_{jR} \\ 
\end{array}
\end{equation}
where the $\bold{R^f}$'s are unitary matrices.  The neutrinos are massless
because the model does not contain right-handed neutrinos. We can
choose any arbitrary vector as the mass eigenstates. For convenience we
choose $\nu_{iL}=\bold{R^l_L}_{ij} \nu'_{jL}$. With this rotation we get
%
\begin{equation}
{\cal L}_{Yuk}=  -{
\sum_{i}} \left[\vb{16}\ m_i^l\
\overline{l_{i}}\, l_{i} 
+ m_i^u\ \overline {u_{i}}\,  u_{i} + 
m_i^d\ \overline {d_{i}}\,  d_{i} \right]
+ \cdots
\end{equation}
%
where $m_i^f$ are the physical fermion masses.


We can now write the Lagrangian ${\cal L}^{\hbox{\scriptsize
    kinetic}}_F$ 
in term of the physical
mass eigenstates. The derivative and the neutral current coupling
remain diagonal since it always involves $\bold{R^f}^\dagger \bold{R^f}=I$. The
diagonal form of the neutral current coupling implies that there are
not flavour changing neutral currents. This is the Glashow, Iliopoulos
and Miani (GIM) mechanism\cite{glashow:1970gm}. It is due to the fact that 
neutral current only connect fermions with the same electroweak charges.

The charged current Lagrangian for quarks becomes
%
\begin{equation}
\label{eq:VCKM-SM}
{\cal L}=\frac{g}{2\sqrt{2}} \overline{u_i} \gamma^\mu (1-\gamma^5)
\VCKM{ij} d_j
\end{equation}
%
where $\VCKM{}=\bold{R^u_L} \bold{R^d_L}^\dagger$ is the
Cabbibo-Kobayashi-Maskawa
matrix\cite{Cabibbo:1963yz,Kobayashi:1973fv}.  It contains 4 free
parameters, 3 angles and a phase which leads to CP violating
terms. Like the value of the masses, the values of the angles in the
CKM matrix have no explanation in the SM.
 
The charged currents for leptons remain diagonal because we have chosen
the neutrino states to be rotated by the same matrix as the charge
leptons. As mentioned before this is only possible because in the SM 
the neutrinos are massless. In extensions of the SM where the neutrinos
have a mass the charge current Lagrangian for the leptons is in
general not diagonal and it can contain also sources of CP
violation. Also for right-handed neutrinos it is possible to write
another type of mass term which is gauge invariant, a Majorana mass
term $\psi^T \psi$. This mass term breaks any $U(1)$ symmetry and
therefore it can only be written for neutral particles. In future
chapters we will study interesting phenomenological implications of
the existence of these type of mass terms in models beyond the SM.

\section{Renormalization}
\label{renor}
In previous sections we have written the tree level Lagrangian of the
minimal standard model as a gauge theory. The model is well defined
and one can perform calculations at the tree level and obtain finite
answers to compare with the experimental results. However when trying
to calculate in higher order in perturbation theory one finds
divergences which have to be removed in a consistent way.  The virtue
of gauge invariant theories relies on their {\it renormalizability} as
proved by `t Hooft\cite{thooft:1971fh,thooft:1971rn,thooft:1972fi}.
In a renormalizable theory the divergences appearing in any order of
perturbation theory can be absorbed in the definition of the fields
and couplings of the original {\it bare} theory. The redefined
parameters are referred to as {\it renormalized} parameters. When
expressed in terms of these renormalized parameters all physical
quantities are finite. Although this is easily stated the actual
proof\cite{thooft:1971fh,thooft:1971rn,thooft:1972fi,baulieu:1985tg}, 
even after more than thirty years, is quite complicated and we
will not attempt to show it here.

The strategy to follow is to split the bare Lagrangian into a
renormalized piece and a counter term Lagrangian which compensates for
the infinities and leads to the final finite results. This can be done
for example using the multiplicative renormalization procedure where
each parameter of the Lagrangian becomes
\begin{equation}
g^{0}_i\rightarrow Z_i g^{ren}_i
\end{equation}
and the fields
\begin{equation}
\phi^{0}_j=\sqrt{Z_j} \phi^{ren}
\end{equation}
%
We have denoted with the index $0$ the bare quantities.  The
renormalization constants $Z$'s are infinite but are written formally
as $Z_i=1+\delta Z_i$. These constants are not all independent. They
are related by the Slavnov-Taylor identities which express the gauge
invariance of the theory. The masses of the particles are also
renormalized
%
\begin{equation}
m^2_{i, ren}=m^2_{i,0}+\delta m_i^2
\end{equation}
%
The procedure to obtain finite Green functions is first to compute
them in some regularization procedure where the divergences are
parameterized in a well-defined way and then they are absorbed in the
definition of the renormalization constants $Z$'s. It is important to
realize that this procedure is not unique. There are infinite ways of
splitting the Lagrangian in a renormalized piece and a counterterm
depending on the point $\mu$ at which we define the renormalized
parameters. Different subtraction points $\mu$ define different
renormalization schemes and therefore the renormalized couplings and
masses depend formally of the scale $\mu$.

Let's call $Z(\mu)$'s the renormalization constants in certain
renormalization scheme that we characterize by the subtraction point
$\mu$. If we call $\Gamma_0$ a bare quantity and $\Gamma (\mu)$ 
the corresponding renormalized quantity in that renormalization scheme
and $\Gamma(\mu')$ in a different renormalization scheme, then
%
\begin{equation}
\begin{array}{l}
\Gamma (\mu)=Z(\mu)\Gamma_0 \\
\Gamma (\mu')=Z(\mu') \Gamma_0\equiv Z(\mu',\mu) \Gamma(\mu)
\end{array}
\end{equation}
%
where
%
\begin{equation}
Z(\mu',\mu)\equiv Z(\mu')/Z(\mu)
\end{equation}
Therefore the $Z(\mu,\mu')$ satisfy a group multiplication law
\begin{equation}
Z(\mu'',\mu') Z(\mu',\mu) =Z(\mu'',\mu)
\end{equation}
%
and $Z(\mu,\mu)=1$. This structure is the {\it renormalization group}.

Interesting physics consequences can be derived from the simple fact
that the physical quantities cannot depend on the renormalization
scheme, or in other words on the subtraction point $\mu$. The
formalization of this condition leads to the renormalization group
equations.

Let's take a renormalized one particle irreducible (OPI) Green's
function.  In a mass independent regularization scheme it can be
written formally as
\begin{equation}\
\Gamma_R(p_i,g_i,m_i,\mu)=Z_\Gamma \Gamma_0 (p_i,g_{0,i},m_{0,i})
\end{equation}
where $Z_\Gamma$ is the product of renormalization constants for the
external particles in the Green's function.

The bare Green's function $\Gamma_0$ must be independent of the scale
$\mu$ and therefore
%
\begin{eqnarray}
  \label{eq:18}
  0&=& \mu \frac{d}{d\mu} \Gamma_0=
\mu\frac{\partial Z_\Gamma^{-1}}{\partial \mu} \Gamma_0+ Z_\Gamma^{-1}  \mu
\frac{d}{d\mu}\Gamma_R \nn \\
&=&
Z_\Gamma^{-1} \left(- \mu  \frac{1}{Z_\Gamma} 
\frac{\partial Z_\Gamma}{\partial \mu} + \mu \frac{d}{d\mu} \right)\Gamma_R 
\end{eqnarray}
%
We now define
%
\begin{eqnarray}
  \label{eq:beta}
  \beta_i(g_j,m_k)&\equiv& \mu\frac{\partial g_i}{\partial \mu}\nn \\
\gamma_{\Gamma}(g_j,m_k)&\equiv& \mu \frac{\partial \ln Z_\Gamma}
{\partial \mu}\nn \\ 
\gamma_{m_i}(g_j,m_k)&\equiv& \mu\frac{\partial
\ln m_i}{\partial \mu}
\end{eqnarray}
%
to finally obtain
%
\begin{equation}
\left( \mu\frac{\partial}{\partial \mu} +\beta_i
\frac{\partial}{\partial g_i} + m_i \gamma_{m_i}
\frac{\partial}{\partial m_i} - \gamma_\Gamma \right)\Gamma_R=0
\end{equation}
%
These are the renormalization group equations. They are in general a
set of coupled differential equations for coupling constants and
masses (or Yukawa couplings). In a general renormalization scheme they
are not easy to solve because the functions in Eq.~(\ref{eq:beta})
depend \textit{both} on the mass and on the couplings. There is
however one scheme, the (modified) Minimal Subtraction 
($\overline{\hbox{MS}}$) where those function depend only on the
coupling constants. For this scheme the renormalization group
equations are easy to solve. It is usual to introduce the 
variable $t=\frac{1}{2} \ln(\frac{Q^2}{\mu^2})$ and
define the {\it running couplings} $g_i(t)$ through the equation
%
\begin{equation}
\frac{d g_i(t)}{dt}=\beta_i(g_j(t),m_k(t))
\end{equation}
%
We will come back to the running coupling constants in more detail
in section~\ref{sec:gaugeunification}.

\section{Experimental Tests of the Standard Model}

At tree level the standard electroweak model contains 3 parameters in
the gauge sector which can be chosen to be the coupling constants $g$,
$g'$ and the vacuum expectation value $v$. The gauge boson masses and
the Weinberg angle are given in term of these parameters by
Eq.~(\ref{tree1}) and Eq.~(\ref{tree2}).
 After renormalization these relations are modified by
higher order corrections and we can use different set of parameters as
inputs to compare the SM predictions with the experimental data and in
this way to constraint the model and to search for new physics. 

The general strategy is to choose a set of input parameters which are
well measured quantities such as the electromagnetic
coupling constant at zero energy $\alpha(0)=1/137.03599976(50)$, the
$Z$ mass $M_Z=91.1876\pm 0.0021$ GeV and the
Fermi constant determined from the muon lifetime $G_F=1.16637(1)\times
10^{-5}$ GeV$^{-2}$. With
these inputs $\sin^2\theta_W$ and the $W$ boson mass, $m_W$, can be
calculated for any given value of $m_t$ and $m_H$. At present, $m_H$
is not known and can be constrained by $\sin^2\theta_W$ and $m_W$.
The value of $\sin^2\theta_W$ is extracted from $Z$-pole observables
and neutral current processes and depends on the renormalization
prescription.  The tree level relation
%
\begin{equation}
\label{eq:sw2onshell}
\sin^2\theta_W=1-\frac{m_W^2}{m_Z^2}=\frac{{g'}^2}{g^2+{g'}^2}
\end{equation}
can be rewritten in terms of the electromagnetic coupling
constant $\alpha$ and the Fermi constant $G_F$ as
%
\begin{equation}
m_W=\frac{(\pi \alpha/\sqrt{2} G_F)^{\frac{1}{2}}}
{\sin\theta_W}=\frac{A_0}{\sin\theta_W}
\end{equation}
%
and it is modified by radiative corrections. The precise form that these take
depends on the renormalization scheme. Here we just present the two
most popular schemes, a more complete discussion can be found in
\cite{hagiwara:2002fs}. 

\begin{itemize}
\item 
The on-shell scheme promotes the tree-level formula in
Eq.~(\ref{eq:sw2onshell}) to a definition of the renormalized
$\sin^2\theta_W \equiv s_W^2$ to all orders in perturbation theory. From this
it follows
%
\begin{eqnarray}
  m_W&=&\frac{A_0}{s_W (1- \Delta r)^{1/2}}\nn \\[+1mm]
  m_Z&=&\frac{m_W}{c_W}
\end{eqnarray}
%
where $c_W=\cos\theta_W$, $A_0=37.2805(2) \hbox{GeV}$ and $\Delta r=
0.0355\mp 0.0019\pm 0.0002$ are the radiative corrections.

\item
The modified minimal subtraction ($\overline{\hbox{MS}}$) scheme
introduces the quantity
%
\begin{equation}
  \label{eq:20}
  \sin^2\hat{\theta}_W \equiv
  \frac{\hat{g}'^2(\mu)}{\hat{g}^2(\mu)+\hat{g}'^2(\mu) }
\end{equation}
%
where the couplings $\hat{g}$ and $\hat{g}'$ are defined by modified minimal
subtraction and the scale $\mu$ is chosen to be $m_Z$. Then
%
\begin{eqnarray}
  m_W&=&\frac{A_0}{\hat{s}_Z (1- \Delta \hat{r}_W)^{1/2}}\nn \\[+1mm]
  m_Z&=&\frac{m_W}{\hat{\rho}^{1/2} c_Z}
\end{eqnarray}
%
where $\Delta \hat{r}_W=0.0695 \pm 0.001 \pm 0.0002$ and
$\hat{\rho}^{1/2}=1.0106\pm0.0006$. 


\end{itemize}


There are many experimental results which have been used to test to
high precision the standard model. At low energies a precise
determination of the on-shell $\sin\theta_W$ is obtained from deep
inelastic neutrino scattering from isoscalar targets. $e D$ scattering
asymmetry, atomic parity violation, $\nu e$ and $\nu p$ scattering
experiments have also provided data.

At higher energies LEP and SLC have measured different observables on
the $Z$ pole, the $Z$ mass and widths, forward-backward asymmetries,
left-right asymmetries, tau polarization and more. At the Tevatron
besides the measurement of the top mass also
measurements of the $W$ and $Z$ masses and decay widths have
been performed.  In table \ref{smdata} we summarized the present
status of these measurements\cite{hagiwara:2002fs}.
%


\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Quantity & Experimental Value & Standard Model Value \\[+3pt] \hline 
$m_t$ [GeV] &$174.3 \pm 5.1$ & $175.3\pm 4.4$ \\[+3pt] 
$m_W$ [GeV] & $80.451 \pm 0.0061$ & $80.391\pm0.0019$ \\[+3pt] 
$m_Z$ [GeV] & $91.1876 \pm 0.0021$ & $91.1874\pm0.0021$ \\[+3pt] 
$\Gamma_Z$ [GeV] & $2.4952 \pm 0.0023$ & $2.4966\pm0.0016 $ \\[+3pt] 
$\Gamma(\hbox{had})$ [GeV] & $1.7444 \pm 0.0020$ & $1.7429\pm0.0015 $ \\[+3pt] 
$\Gamma(\hbox{inv})$ [MeV] & $ 499.0\pm 1.5$ & $501.76\pm0.14 $ \\[+3pt] 
$\Gamma(\ell^+\ell^-)$ [MeV] & $83.984 \pm 0.086$ 
& $84.019\pm0.027 $ \\[+3pt] 
$\sigma_{\rm had}^0 $ [nb] & $41.541 \pm 0.037$ & $41.477\pm 0.014 $ \\[+3pt] 
$R_e = \Gamma({\rm had})/\Gamma(e^+ e^-)$ & $20.804 \pm 0.050$ &
$20.744 \pm 0.014$ \\[+3pt] 
$R_{\mu} 
= \Gamma({\rm had})/\Gamma(\mu^+ \mu^-)$ & $20.785 \pm 0.033$ &
$20.744 \pm 0.018$ \\[+3pt] 
$R_{\tau} 
= \Gamma({\rm had})/\Gamma(\tau^+ \tau^-)$ & $20.764 \pm 0.045$ &
$20.790\pm 0.018$ \\[+3pt] 
$R_b = \Gamma(b \bar{b})/ \Gamma({\rm had})$ &$0.21664 \pm 0.00068$ 
& $0.21569\pm 0.00016$ \\[+3pt] 
$R_c = \Gamma(c\bar{c})/\Gamma({\rm had})$ & $0.1729 \pm 0.0032$ 
& $0.1723\pm0.00007$ \\[+3pt] 
$A_{FB}^{0e}$ & $0.0145 \pm 0.0025$ & $0.01637\pm0.00026$ \\[+3pt] 
$A_{FB}^{0\mu}$ & $0.0169 \pm 0.0013$ & $0.01637\pm0.00026$ \\[+3pt] 
$A_{FB}^{0\tau}$ & $0.0188 \pm 0.0017$ & $0.01637\pm0.00026$ \\[+3pt] 
$A_{FB}^{0b}$ & $0.0982 \pm 0.0017$ & $0.1036\pm0.0008$ \\[+3pt] 
$A_{FB}^{0c}$ & $0.0689 \pm 0.0035$ & $0.0740\pm0.0006$ \\[+3pt] 
\hline
\end{tabular}
\caption{Summary of measurements from LEP, SLC and from $p\bar p$ colliders
and $\nu N$ scattering.}
\label{smdata}
\end{center}
\end{table}


The extraordinary agreement between the SM and the experimental data
is given in the table \ref{smdata} where we compare the experimental
results with the standard model prediction for a large set of
observables. For the standard model results a best fit was performed
with the results
%
\begin{eqnarray}
  \label{eq:smvaluesatMZ}
  m_H&=&99^{+51}_{-35}\hbox{GeV} \nn \\[+1mm]
  m_t&=&175\pm4.4 \hbox{GeV} \nn \\[+1mm]
  \hat{s}^2_Z&=&0.23113\pm0.00015 \nn \\[+1mm]
\alpha^{-1}(m_Z)&=&127.922\pm 0.027\nn \\[+1mm]
  \alpha_s(m_Z)&=&0.1200\pm0.0028
\end{eqnarray}
%
We see that the agreement between theory and experiment is always better than
2-$\sigma$. We should also notice that we also get a prediction for the
Higgs mass. This prediction is not very accurate because the
dependence of the observables on the Higgs boson mass is only
logarithmic. This is known as the screening theorem of
Veltman\cite{veltman:1968ki}. It is nevertheless remarkable that the
prediction for the Higgs boson mass is relatively light. This is also
shown in Fig.~\ref{fig:mhmt} taken from the Particle Data
Group\cite{hagiwara:2002fs}. 


\begin{figure}[htbp]
  \centering
  \includegraphics[clip,height=80mm]{mhmt.eps}
  \caption{ }
  \label{fig:mhmt}
\end{figure}




\section{Unification: Problems}
\label{Problems}
Despite the standard model is in an impressive agreement with the
experimental data it does not seem to be the complete theory of
particle interactions for several reasons. It contains too many free
parameters that remain unexplained, charged quantization is introduced
by hand and the mass spectrum is not explained, among other
limitations.

All this seems to suggest that the standard model is only a low energy
effective theory which should break down at some scale. Which is the
physics beyond this scale is an open subject. There are in principle
two different principles to apply in the extension of the standard
model. The first is to continue in the direction of searching for more
fundamental level of structure in the fields of the standard model
which would not be elementary but composite. The second possibility is
to extend the symmetries of the fields and the interactions.  Grand
Unified Theories belong to this second group.

Grand Unified Theories (GUT) try to unify the strong and electroweak
interactions in a single gauge group such as $SU(5)$. All the matter
fields belong to a unique representation of the gauge group. According
to GUT theory these unification must take place at some large scale
$M_X$ of about $10^{15}$ GeV.  The first problem which appear in GUT
theories is the instability of the proton. Since now quarks and
leptons are belong to the same multiplet there can be decays of quarks
into leptons and vice versa which would force the proton to decay much
faster than what is experimentally allowed. There are however some GUT
theories which can accommodate longer proton lifetimes.

The most striking problems with GUT's however are related with the
postulation of a {\it desert} which extends for more than twelve
orders of magnitude from the grand unification scale till the
electroweak scale. We are going to discuss now two of these problems,
the hierarchy problem and the lack of coupling constant
unification. As we will see in the next chapter supersymmetry will
solve these inconsistencies in an elegant manner.
 
\subsection{The Hierarchy Problem}
\label{sec:hierarchy}

In any GUT theory there should be superheavy gauge bosons with masses
of the order of the grand unification scale $M_X$. Between them and
the electroweak scale there should be a desert. The scalar Higgs
responsible for the electroweak symmetry breaking must have a mass of
the order of the electroweak scale $m_W$ if we want the standard model
to satisfy unitarity . On the other hand the corresponding Higgs
bosons associated with the breaking of the GUT symmetry must have
masses of the order of the grand unification scale. Therefore the
scalar potential must be such that can give rise to this hierarchy of
vacuum expectation values.

To achieve this hierarchy one can think of adjusting by hand the
parameters of the Higgs potential. However this does not only imply an
unnatural fine tune of the parameters but it is also unstable under
radiative corrections.  In principle to keep the two scales separated
we should avoid mixing terms in the scalar potential. However at one
loop level there are corrections which lead to interactions connecting
the light and heavy scalars. In order to keep this mixing small one
has to adjust the parameters in the potential to one part in
$10^{24}$. What is worse this adjustment would be ruined by the two
loop corrections and so on. This is the {\it hierarchy problem}. 

To illustrate the problem we use an argument borrowed from
Ref.~\cite{martin:1997ns}. Consider the one loop corrections to the
Higgs boson mass coming from a fermion $f$ and from a scalar $S$. Let
us assume that these fields couple to  the Higgs boson through the
following lagrangian
%
\begin{equation}
  \label{eq:22}
  {\cal L}= -\lambda_f H \overline{f}f - \lambda_S |H|^2 |S|^2 + \cdots
\end{equation}
%
The one loop contributions to the Higgs boson self energy are shown in
Fig.~\ref{fig:HiggsSelf}. The contribution from 
Fig~\ref{fig:HiggsSelf}a reads,


\begin{figure}[htbp]
  \centering
  \includegraphics{bookc4HiggsSelf.eps}
  \vspace{-5mm}
  \caption{One loop correction the Higgs boson mass}
  \label{fig:HiggsSelf}
\end{figure}

\begin{equation}
  \label{eq:23}
  \delta m^2_H= \frac{\lambda^2_f}{16\pi^2} \left( -2 \Lambda^2 + 6
  m^2_f \ln \frac{\Lambda}{m_f} + \cdots \right)
\end{equation}
%
where $\Lambda$ is the ultraviolet momentum cutoff used to regulate the
divergence of the integral. It should be interpreted as the energy scale
where new physics will enter. In GUT's this should be at least of the
order of $M_X$, hence many orders of magnitude above $m_H$. If we now
compute the diagram in Fig~\ref{fig:HiggsSelf}b we get
%
\begin{equation}
  \label{eq:24}
  \delta m^2_H= \frac{\lambda_S}{16\pi^2} \left(  \Lambda^2 -2
  m^2_S \ln \frac{\Lambda}{m_S} + \cdots \right)
\end{equation}
%
and the same problem arises. Notice however that the coefficient of
the $\Lambda^2$ term have opposite signs in Eqs.~(\ref{eq:23}) and
(\ref{eq:24}). This difference originates from the fact that the
particle in loop is a fermion in Fig~\ref{fig:HiggsSelf}a and a boson
in Fig~\ref{fig:HiggsSelf}b. Therefore the problem can be evaded by
advocating to a boson-fermion symmetry, which would imply some
relation between $\lambda_f$ and $\lambda_S$ to ensure that the
corrections to the dangerous terms in the scalar potential should
vanish. As we will see in the next chapter such a symmetry, called
supersymmetry, does indeed exist. In supersymmetry for each fermion we
have two scalars and $\lambda_f^2=\lambda_S$ and therefore the terms
proportional to $\Lambda^2$ in Eqs.~(\ref{eq:23}) and
(\ref{eq:24}) exactly cancel. Supersymmetry also ensures that this
success persists in higher order in perturbation theory.



\subsection{Coupling Constant Unification}
\label{sec:gaugeunification}

We have seen in section \ref{renor} that the renormalization group
equations imply that the gauge couplings as well as the masses {\it
run} with the scale $Q$. At a certain scale only the particles with
masses lighter than the scale contribute to the running. In Grand
Unifies Theories there is a scale $M_X$ above which the coupling
constants should be the same. Below $M_X$ the coupling constants
evolve according to the $\beta$ functions of the gauge group.

If we assume that the SM is valid all the way till the scale $M_X$ we
can compute the $\beta$ functions in perturbation theory and we can
verify if the coupling constants do indeed meet.

In the two loop approximation the RGE for the gauge couplings are
given by:
%
\begin{equation}
\label{eq:gaugecoupevolution}
\frac{d g_i}{ dt}= \frac{g^3_i}{16 \pi^2}\left[ b_i+\frac{1}{16\pi^2}
{\displaystyle\sum_{j=1}^3} b_{ij}g_i^2g_j^2\right]
\end{equation}
%
where we have neglected the effect of the running masses. We have
denoted by $g_1=g'\sqrt{5/3}$, $g_2=g$, and $g_3$ the coupling constant of the
strong interaction and, as before,$t=\frac{1}{2}
\ln(\frac{Q^2}{\mu^2})$.  In the SM the $b$'s functions
are~\cite{barger:1993ac},
%
\begin{equation}
b_i= (\frac{41}{10},-\frac{19}{6},-7) \quad , \quad
b_{ij}=\left(\begin{array}{lll} \frac{199}{50} & \frac{27}{10} &
\frac{44}{5}
\\ \frac{9}{10} &\frac{35}{6} & 12 \\ \frac{11}{10} & \frac{9}{2} & -26
\end{array}\right)
\end{equation}
%
To solve these equations it is convenient to introduce the quantities 
%
\begin{equation}
  \label{eq:11}
  \alpha_i=\frac{g_i^2}{4\pi}
\end{equation}
%
and to define a new variable,
%
\begin{equation}
  \label{eq:14}
  x=x(Q)=\frac{1}{2\pi} \left[\vb{12} t(Q)-t(m_Z) \right] = \frac{1}{2\pi} \ln
  \frac{Q}{m_Z}   
\end{equation}
%
that satisfies $x(m_Z)=0$. 
With these definitions the differential equations read
%
\begin{equation}
  \label{eq:15}
  \frac{d \alpha_i^{-1}}{d x}= - b_i - \sum_{j=1}^3 b_{ij} \alpha_i
  \alpha_j   
\end{equation}
%
and have the solutions
\begin{equation}
  \label{eq:gcevolution}
  \alpha_i^{-1}(x)=\alpha_i^{-1}(0) - b_i x - b_{ii} 
  \left[\vb{12} f_i(x) -f_i(0)\right] - \sum_{j\not=i} b_{ij}
  \left[\vb{12} g_{ij}(x)-g_{ij}(0) \right]
\end{equation}
%
where
%
\begin{eqnarray}
  \label{eq:17}
  f_i(x)&=&\frac{1}{b_i} \frac{1}{\alpha_i^{-1}(0)-b_i x}\\[+1mm]
  g_{ij}(x)&=&\frac{1}{\alpha_i^{-1}(0) b_j -\alpha_j^{-1}(0) b_i}\,
\ln\left[\frac{\alpha_i^{-1}(0)-b_i x}{\alpha_j^{-1}(0)-b_j x}\right]
\end{eqnarray}
%
Starting from the experimental values at $Q=m_Z$ given in
Eq.~(\ref{eq:smvaluesatMZ}) we can extract 
%
\begin{eqnarray}
\label{eq:alfmz_exp}
\alpha_1^{-1}(0)&=&\frac{3}{5}\, \alpha^{-1} \left(1
  -\sin^2\theta_W\right) = 59.013\pm 0.017 \nn\\
\alpha_2^{-1}(0)&=&\alpha^{-1} \sin^2\theta_W = 29.567\pm 0.020 \nn \\
\alpha_3^{-1}(0)&=&\alpha_s^{-1}=8.33\pm 0.20 
\end{eqnarray}
%


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

\ni
One can then integrate the equations and obtain the couplings at any scale
$Q$. In Fig.\ref{smcouplings} we see the evolution of the coupling
constants with the scale $Q$. They do not intersect in a common
point. This seems to indicate the existence of new physics at some
intermediate scale between the electroweak scale and the Grand
Unification scale. As we will see in the next chapter Supersymmetry is
a perfect candidate for new physics which solves this problem.



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