\subsection{Gauge theories}
\paragraph{Group.} Group is defined by its structure constants 
$ f^{\alpha}_{\beta \gamma} $ which are anti-symmetric 
$( f^{\alpha}_{\beta \gamma} = -f^{\alpha}_{\gamma \beta})$
and obey the Jacoby  identity:
$$  f^{\delta}_{\alpha \epsilon}f^{\epsilon}_{\beta \gamma} +
f^{\delta}_{\gamma \epsilon}f^{\epsilon}_{\alpha \beta } +
f^{\delta}_{\beta \epsilon}f^{\epsilon}_{\gamma \alpha} = 0\;. $$

Group generators are Hermitian matrices   $\hat\tau_{\alpha}$ which 
satisfy the commutation relations:
$$\hat\tau_{\alpha}\hat\tau_{\beta} - \hat\tau_{\beta}\hat\tau_{\alpha}=
 i\,f^{\gamma}_{\alpha \beta} \hat\tau_{\gamma}\;. $$ 
In particular the generators in the   adjoint representation are
$$ (\hat\tau_{\alpha})^i_j = -i\,f^i_{j\alpha}\;. $$  
Group transformation may be represented with the help of group 
generators as
   $$ \hat g(w)=exp(i\hat\tau_{\alpha}\omega^{\alpha})\;. $$
We   assume  that the Killing  metric is orthonormal: 
$$ -\, \frac{1}{2} f^{\epsilon}_{\gamma \alpha} f^{\gamma}_{\epsilon
\beta} = \delta_{\alpha \beta}\;. $$
This metric allows one to raise and lower the group indices. 
In the case of  orthonormal Killing metric the structure constants
are fully antisymmetric under interchange of any pair of indices.

   

\paragraph{Local gauge invariance.} It is an invariance of Lagrangian
under group     transformations $g(x)$ which depend on a point in the space-time 
manifold. Fields in such theories are divided into two classes: 
matter fields and gauge  fields. The gauge fields $A^{\alpha}_{\mu}(x)$
are  vector ones. The number of such fields is equal 
to the number of group generators. The matter fields $\psi^i(x)$  can  have 
an arbitrary 
Lorentz  structure. Their internal components are transformed
according to some representation of the group. Let $D(\omega)$ be an 
operator  which performs an  infinitesimal transformation of fields under 
the local gauge transformation. For the matter fields we have
$$ \hat D(\omega)\psi(x)= i\,\omega^{\alpha}(x){\hat \tau}_
{\alpha}\psi(x)\;. $$
For  gauge fields the local gauge transformations  are defined by 
$$ (\hat D(\omega) A_{\mu})^{\alpha}(x)= 
f^{\alpha}_{\beta \gamma} A^{\beta}_{\mu}(x) \omega^{\gamma}(x) 
 + \partial_{\mu} \omega^{\alpha}(x)\;.$$
The following expressions, namely a  covariant derivative and a gauge field 
tension, are used to construct a local invariant Lagrangian:
$$  \nabla_{\mu}\psi(x) = \partial_{\mu} \psi(x) - \hat D(A_{\mu})\psi(x) =
 \partial_{\mu} \psi(x) - iA^{\alpha}_{\mu}(x){\hat \tau}_{\alpha}\psi(x)\;; $$
$$ F^{\alpha}_{\mu \nu}(x) = \partial_{\mu}A^{\alpha}_{\nu}(x) 
-\partial_{\nu}A^{\alpha}_{\mu}(x) + 
f^{\alpha}_{\beta\gamma}A^{\beta}_{\mu}(x)A^{\gamma}_{\nu}(x)\;. $$
It can be proved that 
$$ \hat D(\omega)[\nabla_{\mu}\psi(x)]= \hat \tau_{\alpha}\omega^{\alpha}(x)
\nabla_{\mu}\psi(x)\;; $$
$$ (\hat D(\omega) F_{\mu \nu})^{\alpha}(x)= f^{\alpha}_{\beta \gamma}
F^{\beta}_{\mu \nu}(x)\omega^{\gamma}(x)\;.$$
In these terms the Lagrangian of gauge theory is defined by the following 
expression \cite{BD}
\begin{equation} 
 L = -\, \frac{1}{4g^2} {F^{\alpha}}_{\mu \nu} {F_{\alpha}}^{\mu \nu} 
+ L_m( \nabla_{\mu} \psi, \psi)
\label{physLag}
\end{equation}
where $g$ is the coupling constant and $L_m(\partial_{\mu}\psi,\psi)$ is some 
Lagrangian of the matter fields which is invariant under the global gauge transformations.

\paragraph{Gauge fixing terms.}
In order to quantize the gauge theory  one must add  to 
 (\ref{physLag})   gauge fixing term  and the corresponding Faddeev-Popov 
term. The first term breaks the gauge symmetry and in this way removes the 
divergence of the functional  integral. The second term improves the integration 
measure to provide  correct predictions for gauge invariant
observables.


The general form of the gauge fixing term is
\begin{equation}
 L_{FG}(x)= -\, \frac{1}{2}\sum_{\alpha} (\Phi^{\alpha}(x))^2\;. 
\label{gfLag}
\end{equation}
The corresponding Faddeev-Popov term is
\begin{equation}
 L_{FP}= - \bar{c}_{\alpha}(x) (D(c)[\Phi^{\alpha}])(x) \;,
\label{fpLag}
\end{equation}
where $c^{\alpha}(x)$ and $ \bar{c}_{\alpha}(x)$ are the auxiliary  
anti-commutative fields. They are called the Faddeev-Popov ghosts.
Note that we may multiply  (\ref{fpLag}) by an  arbitrary factor which 
can be hidden in the definition of ghost fields. As a rule
it is chosen in such a way to provide a convenient form of the  ghost propagator.

The well-known   choice of the gauge fixing terms, the Feynman-like gauge,  is 
$$ L_{GF}= -\,\frac{1}{2g^2}\sum_{\alpha} 
(\partial^{\mu} A^{\alpha}_{\mu} +...)^2\;.$$   
In this case the quadratic  part of gauge field  Lagrangian takes  
the simplest form
\begin{equation}
  -\,\frac{1}{2g^2} \partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}\;. 
\label{FeynmQ} 
\end{equation} 
The corresponding Faddeev-Popov Lagrangian is
%f^{\alpha}_{\beta \gamma} A^{\beta}_{\mu}(x) \omega^{\gamma}(x)
% + \partial_{\mu} \omega^{\alpha}(x)\;.$$

$$ L_{FP}= - \bar{c}_{\alpha}(x) (\Box c^{\alpha}(x) +
 \partial^{\mu}( f^{\alpha}_{\beta \gamma} A^{\beta}_{\mu}(x)
c^{\gamma}(x)) + ...)\;. $$
%
The normalization of $L_{FP}$ is chosen  to have
the Faddeev-Popov ghost propagator equal to the propagator of scalar
particle:
$$T(\bar{c}_{\alpha}(p),c^{\beta}(q))= \delta_{\alpha}^{\beta} \delta(p+q)
\frac{1}{i(2\pi)^4}\;\frac{1}{-p^2}\;. $$

\paragraph{Normalization.}
For the purposes of applying the  perturbation theory  
the  gauge field is rescaled by substitution  
$$ A  \rightarrow A/g\;. $$
In this way the coupling constant $g$ leaves the quadratic part of 
the Lagrangian (\ref{physLag}) and appears in the  interaction terms. 

\subsection{QCD Lagrangian}
QCD is a gauge theory based on the  $SU(3)$  group. The corresponding 
 gauge field  $G^{\alpha}_{\mu}(x)$ is called a {\it gluon}.  
The matter fields $q_k(x)$ are {\it quarks}.
They  are triplets and are transformed  according to the fundamental 
representation. Index $k$  enumerates a sort of quarks. 

QCD Lagrangian in the Feynman  gauge is written down 
following  the rules (\ref{physLag}), (\ref{gfLag}), (\ref{fpLag}). 


\begin{equation} 
 L_{phys} = -\, \frac{1}{4} {F^{\alpha}}_{\mu \nu}(x) {F_{\alpha}}^{\mu
\nu}(x) +  \sum_k \frac{i}{2}( \bar{q}_k(x)  \gamma^{\mu}\nabla_{\mu}q_k(x) -
\nabla_{\mu}\bar{q}_k(x)\gamma^{\mu}q_k(x) )\;.
\label{physLagQCD}
\end{equation}
where 
$$  \nabla_{\mu}q(x) = \partial_{\mu} q(x) - i\,g\, G^{\alpha}_{\mu}(x) 
\hat{t}_{\alpha}q(x)\;;  $$
$$ F^{\alpha}_{\mu \nu}(x) = \partial_{\mu}G^{\alpha}_{\nu}(x)
-\partial_{\nu}G^{\alpha}_{\mu}(x) +
g\,f^{\alpha}_{\beta\gamma}G^{\beta}_{\mu}(x)G^{\gamma}_{\nu}(x)\;; $$

\begin{equation}
L_{GF} = -\,\frac{1}{2} \sum_{\alpha}(\partial^{\mu} G^{\alpha}_{\mu})^2 \;;
\label{gfLagQCD}
\end{equation}


\begin{equation}
 L_{FP}= - \bar{c}_{\alpha}(x) (\Box c^{\alpha}(x) + g\,
 \partial^{\mu}( f^{\alpha}_{\beta \gamma} G^{\beta}_{\mu}(x)
c^{\gamma}(x)))\;,
\label{fpLagQCD}
\end{equation}
$ c^{\alpha}(x)$ are Faddeev-Popov ghosts,  $g$ is coupling constant, 
$ f^{\alpha}_{\beta \gamma} $ are the  $SU(3)$ structure constants,
$t_{\alpha}$ are the   generators in the fundamental representation\footnote{
$ t_{\alpha}$ are equal to   $ \lambda_{\alpha}/2$ where  $
\lambda_{\alpha}$ are the Gell-Mann matrices.}.  


\subsection{Lagrangian of electroweak interactions}

\subsubsection{Vector bosons}
Gauge theory of electroweak interactions is based on the $SU(2)\times U(1)$ group.
So, we have a triplet of $SU(2)$ vector fields $W^{\alpha}_\mu(x)$ and a single 
vector field $B_\mu(x)$. The $SU(2)$ structure constants are
presented by the absolutely antisymmetric tensor
$\epsilon^{\alpha\beta\gamma}$. The  Lagrangian of gauge fields is 
written down    according to (\ref{physLag}):
\begin{equation}
\label{SU2U1}
 L_{GF} = -\, \frac{1}{4} {FW^{\alpha}}_{\mu \nu}(x) {FW_{\alpha}}^{\mu \nu}(x)
     -\, \frac{1}{4} FB_{\mu \nu}(x) FB^{\mu \nu}(x)\;,
\end{equation}
where  
\begin{eqnarray*}
FW^{\alpha}_{\mu \nu}(x)& =& \partial_{\mu}W^{\alpha}_{\nu}(x)
-\partial_{\nu}W^{\alpha}_{\mu}(x) + g_2
\epsilon^{\alpha\beta\gamma}W^{\beta}_{\mu}(x)W^{\gamma}_{\nu}(x)
\;;\\
 FB_{\mu \nu}(x)& =& \partial_{\mu}B_{\nu}(x) - \partial_{\nu}B_{\mu}(x)
\;;
\end{eqnarray*}
$g_2$ is a coupling constant for the $SU(2)$ gauge interaction.



 Infinitesimal local gauge transformations are defined as follows
\begin{eqnarray}
(\hat D(w,b) W_{\mu})^{\alpha}(x) &=& 
g_2 \epsilon^{\alpha\beta \gamma} W^{\beta}_{\mu}(x) w^{\gamma}(x) 
 + \partial_{\mu} w^{\alpha}(x) \;;\nonumber \\
\label{SU2U1tranf}
  D(w,b) B_{\mu}(x)&=&\partial_{\mu} b(x) \;.
\end{eqnarray}

Let us express $W^1_\mu$ and $W^2_\mu$ 
in terms of a mutually conjugated couple  $W^+_\mu$ and $W^-_\mu$ 
\begin{eqnarray*}
 W^1_\mu&=&(W^-_\mu+\,W^+_\mu)/\sqrt{2}\;;  \\
 W^2_\mu&=& (W^-_\mu-\,W^+_\mu)/(i \sqrt{2})\;.
\end{eqnarray*}
Thus, the Lagrangian of  self-interaction for the  $SU(2)$ gauge fields  
in term of $W^+$ and $W^-$  has the form:
\begin{eqnarray}
i\,g_2 \left(\partial_\mu W^3_\nu(W^+_\mu W^-_\nu - W^-_\mu W^+_\nu)
              +\partial_\mu W^+_\nu(W^-_\mu W^3_\nu - W^3_\mu W^-_\nu) \right.
&&\nonumber \\
    \left.     -\partial_\mu W^-_\nu(W^+_\mu W^3_\nu - W^3_\mu W^+_\nu) \right)
&&\nonumber \\
    +g_2^2 \left( W^3_\mu W^+_\nu(W^-_\mu W^3_\nu - W^3_\mu W^-_\nu) 
    +\frac{1}{2}  W^+_\mu W^-_\nu  (W^+_\mu W^-_\nu - W^-_\mu W^+_\nu) \right)\;.
&&\label{3V4V}
\end{eqnarray}



All matter fields in the electroweak theory are either  $SU(2)$
 invariant singlets or  belong  to its fundamental 
representation. In the latter case  they form  doublets. 
Generators for these doublets   are expressed via the Pauli $\sigma$-matrices
$$
 \hat \tau_{\alpha} = \hat \sigma_{\alpha}/2 \;.          
$$
 Thus the infinitesimal local gauge transformations for doublets take a
form:

$$
    D(w,b) \psi(x)= \frac{i\,g_2}{2}w^{\alpha}(x)\hat \sigma_{\alpha}\psi(x) + 
    \frac{i\, g_1}{2} Y\, b(x) \psi(x) \;.
$$   
Here $g_1$ is the coupling constant of $U(1)$ gauge interaction.
The constant $Y$ depends on a type of the doublet. It is called a {\it hypercharge}.

In the gauge theory of electroweak interaction the gauge fields  
interact  with a  scalar ({\it Higgs})  doublet  which  has a nonzero 
vacuum state. 
Without loss of generality one can put  $Y=1$ for the Higgs doublet.
By means of the gauge transformation  the vacuum state of this field 
may be presented in the form:
$$
   \Phi_{\Omega}= \left( \begin{array}{c} 0 \\ \phi_0/\sqrt{2} 
 \end{array} \right)\;,
$$     
where $\phi_0$ is a  real constant.

As a result of spontaneous symmetry breaking the $ W^{\alpha}_{\mu}$ and 
$B_{\mu}$ fields do not correspond to physical particles. Physical 
particles in this model are the photon ($A_\mu$), W-bosons ($W^+_{\mu}\,,
W^-_{\mu}$) and Z-boson ($Z_{\mu}$).
 The photon field $A_{\mu}$ is a combination of  gauge fields 
responsible for the local gauge transformations which save  
the Higgs vacuum $\Phi_{\Omega}$: 
$$
     A_{\mu} = B_{\mu} \cos{\Theta_w} + W^3_{\mu} \sin{\Theta_w}\;,
$$
where the mixing angle $\Theta_w = \arctan(g_2/g_1)$. To complete $W^+$,  $W^-$,
 and $A$ up to the orthonormal  basis of gauge fields we introduce   
$$  Z_{\mu} = -B_{\mu} \sin{\Theta_w} + W^3_{\mu} \cos{\Theta_w}\;. $$

Let  $w^+(x)$, $w^-(x)$, $a(x)$, and $z(x)$ be parameters of the local
gauge transformation corresponding to the fields $W^+_\mu$,  $W^-_\mu$,
$A_\mu$, and  $Z_\mu$. Then  for a matter  doublet  with a hypercharge $Y$
the gauge transformation is given  by the following expression:
\begin{eqnarray}
\lefteqn{\hat D(w^+,w^-,a,z) \Psi=}\nonumber \\ & &\frac{i\,g_2}{2} \left[\sqrt{2} 
\left( \begin{array}{cc} 0 & w^+ \\ w^-&0 \end{array} \right) +
\sin\Theta_w\,a\, \left( \begin{array}{cc} 1+Y & 0 \\ 0&Y-1 \end{array} \right) \right.
\nonumber \\
\label{DPsi}
&+&  \cos\Theta_w  \,z\,
\left.\left( 
\begin{array}{cc} 1 -Y\tan^2\Theta_w &  0 \\ 0& -1  -Y\tan^2\Theta_w \end{array}
\right)   \right] \Psi\;.
\end{eqnarray}



\subsubsection{Lagrangian of Higgs field}

In the framework of renormalizable field theory  the general 
expression for the  gauge invariant  Higgs  Lagrangian is:  
$$
 L_{Higgs}= (\nabla_\mu \Phi)^+(\nabla^\mu \Phi) -\, \frac{\lambda}{2}
(\Phi^+ \Phi - \frac{1}{2} \phi_0^2)^2 \;, 
$$
where $\lambda$ is a new coupling constant. 


 Some perturbations of the Higgs vacuum  can be realized by 
means of the local gauge transformation which do not correspond to physical
degrees of freedom. To separate physical and gauge degrees of freedom for a
small perturbation of the Higgs vacuum we present $\Phi$ in the
following form: 
\begin{equation}
 \Phi= \left(\begin{array}{c} i\,W^+_f \\(\phi_0 + H -i\,Z_f)/\sqrt{2} \end{array}
\right)\;.
\label{HiigsField}
\end{equation}
Here $W^+_f$ and $Z_f$ are the unphysical {\it Goldstone} fields
corresponding
to variation of the Higgs vacuum caused by the  $D(w^+,0,0,0)$ and 
$D(0,0,0,z)$ gauge transformations. The real field  $H$ corresponds to 
a physical degree of freedom which is assoshated with the Higgs patricle.  

The  term $(-\, \frac{\lambda}{2}( \Phi^+ \Phi - \frac{1}{2} \phi_0^2)^2)$ 
 contains  the mass term for the $H$ field:
$$
     -\, \frac{\lambda}{2} (\phi_0)^2  \, H^2 \;,
$$
and the following terms of self-interaction for  $H$ and 
the Goldstone fields:
\begin{equation}
-\, \frac{\lambda}{2} \left( (W^+_f W^-_f + ( H^2 + Z_f^2)/2)^2 
   + 2\phi_0 H(W^+_f W^-_f + ( H^2 + Z_f^2)/2) \right)\;.
\label{HiggsSelfInt}
\end{equation}
Usually   $\lambda$ is expressed via the  mass of $H$-boson $M_H$:
$$
   \lambda=\left(\frac{M_H}{\phi_0}\right)^2\;.
$$
Let us remind that 
$$\nabla_\mu \Phi = \partial_\mu \Phi - \hat D(W+_\mu,W-_\mu,A_\mu,Z_\mu)
\Phi \;\;,$$
where due to (\ref{DPsi})
\begin{eqnarray}
\lefteqn {\hat D(W+_\mu,W-_\mu,A_\mu,Z_\mu) \Phi=}&&  \nonumber \\
&& \frac{i\,g_2}{2} 
\left( \begin{array}{c}  
W^+_\mu (\phi_0 + H -i\,Z_f) + i\,(2 \sin{\Theta_w} A_\mu + \cos{\Theta_w}
(1-\tan^2{\Theta_w}) Z_\mu) \,W^+_f
\\
 ( 2 i W^-_\mu   W^+_f    - Z_\mu(\phi_0 + H -i\,Z_f)/\cos{\Theta_w} )/\sqrt{2}
\end{array} \right)\;. \nonumber \\
\label{DPhi}
\end{eqnarray}

The term $(\hat D_\mu \Phi)^+(\hat D^\mu \Phi)$  gives us the
mass terms for the $W$ and $Z$-bosons: 
\begin{equation}
\left( \frac{g_2 \phi_0}{2} \right)^2 W^+_\mu {W^-}^\mu + \frac{1}{2}
\left( \frac{g_2 \phi_0}{2 \cos{\Theta_w} } \right)^2 Z_\mu Z^\mu \;,
\label{WZ-mass-terms}
\end{equation}
and the following terms  describing the interaction of a couple of 
vector bosons with Higgs and Goldstones:
\begin{eqnarray}
\frac{g_2^2}{4}(
&& W^+_\mu {W^-}^\mu ( H^2 +Z_f^2 +2\phi_0 H +2 W^+_f W^-_f) \nonumber \\
&& + 4 A_\mu A^\mu \sin^2{\Theta_w} W^+_f W^-_f \nonumber \\
&& + \frac{Z_\mu Z^\mu}{\cos^2{\Theta_w}}(  (H^2 + Z_f^2)/2 + \phi_0 H + (1-2
\sin^2{\Theta_w})^2  W^+_f W^-_f) \nonumber  \\ 
&& + 4  A_\mu  Z^\mu \tan{\Theta_w}(1 -2 \sin^2{\Theta_w}) W^+_f W^-_f \nonumber\\
&& -2i\tan{\Theta_w} W^+_\mu (\cos{\Theta_w} A^\mu - \sin{\Theta_w} Z^\mu) 
           (\phi_0 + H -i\,Z_f) W^-_f\nonumber \\
&& + 2i\tan{\Theta_w} W^-_\mu (\cos{\Theta_w} A^\mu - \sin{\Theta_w} Z^\mu)
           (\phi_0 + H +i\,Z_f) W^+_f\nonumber\\
 )\;.&&  \label{VVH}
\end{eqnarray} 

The constant $\phi_0$ now can be expressed in terms of the $W$-boson mass
$M_W$:
$$
 \phi_0= 2\,M_W/g_2\;.
$$
The  $Z$-boson mass is related to  the $W$-boson mass by means of
constraint:
$$ M_Z=M_W/\cos{\Theta_w}\;.$$
 
The term $( - (\hat D_\mu \Phi)\partial^\mu \Phi^+  - (\hat D_\mu
\Phi)^+ \partial^\mu \Phi) $  gives us  off-diagonal quadratic terms:
\begin{equation} \label{offDiagonal}
- g_2 \phi_0 \left( \frac{1}{2}( W^+_\mu \partial^\mu W^-_f + 
W^-_\mu \partial^\mu W^+_f )  
+\frac{1}{\cos{\Theta_w}}Z_\mu \partial^\mu Z_f \right)  
\end{equation}
and the following  terms of interaction:
\begin{eqnarray}
\frac{g_2}{2}\left(\right. 
&&-H ( W^-_\mu \partial^\mu W^+_f  + W^+_\mu \partial^\mu W^-_f ) 
+  (\partial^\mu H)(W^-_\mu W^+_f +W^+_\mu W^-_f) \nonumber \\
&&-i(Z_f (  W^-_\mu \partial^\mu W^+_f  -W^+_\mu \partial^\mu W^-_f ) 
+ (\partial^\mu Z_f)  (W^+_\mu W^-_f - W^-_\mu W^+_f)) \nonumber \\
&&+i( 2\sin{\Theta_w} A_\mu + \cos{\Theta_w}(1-\tan^2{\Theta_w}) Z_\mu)
( W^-_f \partial^\mu W^+_f  -  W^+_f \partial^\mu W^-_f )\nonumber \\
&&+  (\partial^\mu H) Z_\mu Z_f/\cos{\Theta_w} 
 - (\partial^\mu Z_f) Z_\mu H/cos{\Theta_w} \nonumber \\
\left. \right)\;. \label{VHH}
\end{eqnarray}

The off-diagonal quadratic terms are canceled by the gauge fixing terms.
See below.

\subsubsection{Gauge fixing and ghost terms for the t'Hooft-Feynman gauge}
In the case of t'Hooft-Feynman gauge the   gauge fixing terms are
$$
   -\frac{1}{2} (\partial^\mu A_\mu)^2  
-\frac{1}{2} (\partial^\mu Z_\mu  + M_Z\, Z_f)^2
-| \partial^\mu W^+_\mu + M_W\, W^+_f|^2\;.
$$
The squared  divergences  of fields  transform the  quadratic part of Lagrangian 
for vector 
bosons to a diagonal form like (\ref{FeynmQ}). The squared Goldstone field 
terms  give a mass to the
Goldstone particle   equal to the mass of the corresponding 
vector boson field. The off-diagonal quadratic terms, which follow from
the gauge fixing Lagrangian, cancel the off-diagonal terms (\ref{offDiagonal})
 up to  complete  divergency terms. 

 According to the general rule (\ref{fpLag})  the  Faddeev-Popov Lagrangian is
\begin{eqnarray*}
 &&  -A_{\bar{c}} D(W^+c,W^-_c,A_c,Z_c)(\partial^\mu A_\mu) \\  
 &&  -Z_{\bar{c}} D(W^+c,W^-_c,A_c,Z_c) (\partial^\mu Z_\mu  + M_Z\, Z_f)\\
 && -W^-_{\bar{c}} D(W^+c,W^-_c,A_c Z_c)(\partial^\mu W^+_\mu +
M_W\,W^+_f)\\ 
 && -W^+_{\bar{c}} D(W^+c,W^-_c,A_c,Z_c)(\partial^\mu W^-_\mu + 
M_W\,W^-_f)\;.
\end{eqnarray*}
Note that due to (\ref{SU2U1tranf}) 
\begin{eqnarray*}
\hat D W^+_{\mu}& =&  i\,g_2\, ( W^+_\mu( \sin{\Theta_w} A_c+ \cos{\Theta_w} Z_c)
-W^+_c(\sin{\Theta_w}A_\mu+\cos{\Theta_w} Z_\mu) ) +\partial_\mu W^+_c   \;;\\
%
\hat D A_{\mu}& =& -i\,g_2 \sin{\Theta_w}(W^+_\mu W^-_c - W^-_\mu W^+_c) +\partial_\mu
A_c \;;\\
%
\hat D Z_{\mu }& =& -i\,g_2 \cos{\Theta_w} (W^+_\mu W^-_c - W^-_\mu W^+_c)
+\partial_\mu Z_c\;,
\end{eqnarray*}
and according to (\ref{DPhi})
\begin{eqnarray*}
 D W^+_f &=& W^+_c M_W +  \frac{g_2}{2} 
( W^+_c ( H -i\,Z_f) + i\,(2 \sin{\Theta_w} A_c + \cos{\Theta_w}
(1-\tan^2{\Theta_w}) Z_c) \,W^+_f)\;;
\\
 D Z_f & =& M_Z Z_c +\frac{g_2}{2}( \frac{Z_c H} 
{\cos{\Theta_w}} - i(W^-_c W^+_f - W^+_c W^-_f) 
)\;.
\end{eqnarray*}
After substitution of these derivatives to the Faddeev-Popov Lagrangian 
  we see  that it contains the quadratic part:
$$ -A_{\bar{c}}\Box A_c -Z_{\bar{c}}\Box Z_c - W^-_{\bar{c}}(\Box
W^+_c +M_W W^+_c) - W^+_{\bar{c}}(\Box W^-_c +M_W W^-_c)\;,
$$
and the following  vertices of interaction:
\begin{eqnarray}
-g_2(&&  i \,\sin{\Theta_w} (\partial^\mu A_{\bar{c}})  (W^+_\mu W^-_c - W^-_\mu W^+_c)
\nonumber \\
%
&&+ i\,\cos{\Theta_w}  (\partial^\mu Z_{\bar{c}}) (W^+_\mu W^-_c - W^-_\mu W^+_c)
\nonumber\\ 
&&+  \frac{M_Z Z_{\bar{c}}}{2}\left(\frac{Z_c H}{\cos{\Theta_w}} - 
i(W^-_c W^+_f - W^+_c W^-_f) \right) \nonumber\\
%
&&- i(\partial^\mu  W^-_{\bar{c}}) ( W^+_\mu( \sin{\Theta_w} A_c+ \cos{\Theta_w} Z_c)
-W^+_c(\sin{\Theta_w}A_\mu+\cos{\Theta_w} Z_\mu)) \nonumber \\
&&+ \frac{M_W W^-_{\bar{c}}}{2} 
( W^+_c ( H -i\,Z_f) + i\,(2 \sin{\Theta_w} A_c + \cos{\Theta_w}
(1-\tan^2{\Theta_w}) Z_c) \,W^+_f)  )  )\nonumber  \\
%
&&+ i(\partial^\mu  W^+_{\bar{c}}) ( W^-_\mu( \sin{\Theta_w} A_c+ \cos{\Theta_w} Z_c)
-W^-_c(\sin{\Theta_w}A_\mu+\cos{\Theta_w} Z_\mu)) \nonumber \\
&&+ \frac{M_W W^+_{\bar{c}}}{2} 
( W^-_c ( H + i\,Z_f) - i\,(2 \sin{\Theta_w} A_c + \cos{\Theta_w}
(1-\tan^2{\Theta_w}) Z_c) \,W^-_f)  )  ) \nonumber \\
)\;.&& \label{FP-interaction}
\end{eqnarray}

\subsubsection{Unitary gauge}

The {\it unitary} gauge  may be considered as a limit of $\xi \rightarrow
\infty$ of the following  gauge fixing Lagrangian:
$$
 -\frac{1}{2} (\partial^\mu A_\mu)^2  -\frac{M_Z^2}{2\xi} (Z_f)^2
-\frac{M_W^2}{\xi} (W^+_f W^-_f)\;. 
$$
In this limit one gets $Z_f=0$ and $W_f=0$, what decreases a number
of vertices. Under these constraints the Faddeev-Popov Lagrangian
(\ref{fpLag}) takes the form:  

\begin{eqnarray*}
  -A_{\bar{c}} \partial^\mu (-i\,g_2 \sin{\Theta_w}(W^+_\mu W^-_c - W^-_\mu W^+_c) +\partial_\mu
A_c ) 
  -Z_{\bar{c}}  M_Z\ ( M_Z  + \frac{g_2 H} {2\cos{\Theta_w}} ) Z_c &&  \\
 -W^-_{\bar{c}}  M_W ( M_W +  \frac{g_2}{2}   H )  W^+_c 
 -W^+_{\bar{c}} M_W ( M_W +  \frac{g_2}{2}   H  )  W^-_c \;. &&
\end{eqnarray*}
Integration over $A_c$ and $A_{\bar{c}}$ can be performed explicitly 
and gives a result which does not depend on other fields. So  $A_c$ and
$A_{\bar{c}}$ ghosts may be omitted.  

In the unitary gauge only  physical polarization states of the 
incoming and outgoing $W^\pm$ and $Z$ bosons are considered. So 
$W^\pm_c$ and $Z_c$ are not needed in the external lines of Feynman diagrams
and may be omitted also in tree level calculations.
Consequently, in the unitary gauge all ghost and Goldstone fields may be omitted. 

\subsubsection{Summary of vertices for the boson sector}
In the case of t'Hooft-Feynman gauge the full set of vertices is 
described by  expressions (\ref{3V4V}), (\ref{HiggsSelfInt}), (\ref{VVH}),
(\ref{VHH}), and (\ref{FP-interaction}), where 
\begin{eqnarray*}
 W^3_\mu &=& \sin{\Theta_w}  A_\mu + \cos{\Theta_w} Z_\mu \;; \\
\lambda&=&\left(\frac{g_2 M_H}{2 M_w}\right)^2 \;.
\end{eqnarray*}
The coupling constants $g_2$ and $g_1$ may be expressed in terms of the electromagnetic
coupling constant: $g_2 = e \sin{\Theta_w}$ and  $g_1 = e \cos{\Theta_w}$ .

In the case of unitary gauge the interaction is defined by a
 subset of vertices which appears after removing the Faddeev-Popov ghosts and the 
Goldstone  fields.

\subsubsection{Interaction of vector bosons  with fermions}
Experiments in particle physics show that the  $W$-bosons 
interact with left-handed components of fermions
$$ \Psi^L = \frac{1 - \gamma^5}{2} \Psi\;. $$
Thus, the $SU(2)$ group  must transform only the left-handed fermion components.
The initial Lagrangian of  fermion field has the corresponding  
global  symmetry only if all fermions are massless. Indeed, the
Lagrangian of free massless fermion field  splits into two independent
parts: 
\begin{eqnarray*}
 L &=& \frac{i}{2}(\bar{\Psi} \gamma_\mu \partial^\mu \Psi -
(\partial^\mu \bar{\Psi})\gamma_\mu \Psi)\\ 
&=&  \frac{i}{2}(\bar{\Psi}^L \gamma_\mu \partial^\mu \Psi^L -
(\partial^\mu \bar{\Psi}^L)\gamma_\mu \Psi^L) +  
  \frac{i}{2}(\bar{\Psi}^R \gamma_\mu \partial^\mu \Psi -
(\partial^\mu \bar{\Psi}^R)\gamma_\mu \Psi^R)\;,
\end{eqnarray*} 
which allows to  apply  the $SU(2)$ gauge transformations to the left-handed 
components of fermion doublets. In the same time the mass term contains a 
product of left-handed and right-handed  fermion components:
$$ m \bar{\Psi} \Psi = m ( \bar{\Psi}^L \Psi^R + \bar{\Psi}^R \Psi^L)\;,$$
what forbids an appearance of such terms in the  invariant Lagrangian.
Later on we shall show how such fermion particles will  acquire masses in  
result  of the gauge invariant 
interaction  of formerly massless fermion fields with the Higgs doublet.

Whereas $SU(2)$ transforms only the left-handed components of doublets,
$U(1)$ interacts with both. Left-handed  and right-handed components of fermions 
must have the same electric charge. It allows to find the hypercharge 
of right-handed components if a hypercharge of left-handed doublet is known.
The  $U(1)$ gauge field $B_\mu$ is equal to $ \cos{\Theta_w} A_\mu
- \sin{\Theta_w} Z_\mu$. So, the electromagnetic  coupling constant for 
right-handed fermion with a hypercharge $Y^R$ is $(g_1 Y^R \cos{\Theta_w})/2$.
Comparing it with the expression (\ref{DPsi}) we see that the hypercharges of
right-handed components  of doublet are 
\begin{eqnarray*}
Y^R_1&=&1+Y\;, \\
Y^R_2&=&Y-1\;.
\end{eqnarray*}
Thus, we may unambiguously write down the vertices of interactions 
associated with the covariant derivative:
\begin{eqnarray*}
&&\frac{g_2}{2} \left( \begin{array}{cc} \bar{\Psi}_1 & \bar{\Psi}_2 \end{array} \right)
\left[    \sqrt{2} 
\left( \begin{array}{cc} 0 & W^+_\mu \\ W^-_\mu&0 \end{array} \right) +
\sin\Theta_w\,A_\mu\, \left( \begin{array}{cc} 1+Y & 0 \\ 0&Y-1 \end{array} \right) \right.\\
&+&  \cos\Theta_w  \,Z_\mu
\left.\left( 
\begin{array}{cc} 1 -Y\tan^2\Theta_w &  0 \\ 0& -1  -Y\tan^2\Theta_w \end{array}
\right)        \right]
\left( \begin{array}{c} \gamma^\mu \Psi^L_1 \\ \gamma^\mu \Psi^L_2 \end{array} \right) \\
&+& \frac{g_1}{2} (\cos{\Theta_w} A_\mu- \sin{\Theta_w} Z_\mu )  
\left( \begin{array}{cc} \bar{\Psi}_1 & \bar{\Psi}_2 \end{array} \right)
 \left( \begin{array}{cc} Y+1 &0\\0&Y-1 \end{array}  \right)  
\left( \begin{array}{c}
 \gamma^\mu \Psi^R_1 \\  \gamma^\mu  \Psi^R_2 \end{array} \right)\;. 
\end{eqnarray*}

After  matrix multiplication we obtain:
\begin{eqnarray}
&& \frac{g_2}{2\sqrt{2}} 
\left(  W^-_\mu  \bar{\Psi}_2  \gamma^\mu (1 - \gamma^5)\Psi_1 
+   W^+_\mu  \bar{\Psi}_1  \gamma^\mu (1 - \gamma^5)\Psi_2 \right) \nonumber \\
&+& \frac{g_1}{4 \sin{\Theta_w}} Z_\mu \left(\;
\bar{\Psi}_1 \gamma^\mu ( 1 - \gamma^5 -2(Y+1) \sin^2{\Theta_w} ) \Psi_1 \right.
\nonumber \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left.
- \bar{\Psi}_2 \gamma^\mu (1-\gamma^5 +2(Y-1)\sin^2{\Theta_w} ) \Psi_2 \;\right) 
\nonumber \\
&+& \frac{g_1}{2}\cos{\Theta_w} A_\mu \left( (Y+1) \bar{\Psi}_1 \gamma^\mu
\Psi_1 + (Y-1) \bar{\Psi}_2 \gamma^\mu \Psi_2 \right)\;.
\label{V-fermion} 
\end{eqnarray}


\subsubsection{Interaction of the Higgs doublet with fermions and generation
of fermion masses}
The mass terms for fermions  in the electroweak theory are generated via their
$SU(2)\times U(1)$ invariant Yukawa interaction with Higgs doublet. 
Namely, the  Yukawa Lagrangian
$$
 - \frac{m_1 \sqrt{2}}{\phi_0} \left( \bar{\Psi}^L_i \epsilon^{ij} \stackrel{*}{\Phi^{j}} \Psi^R_1 -
\bar{\Psi}^R_1 \Phi^{i} \epsilon^{ij} \Psi^L_j \right)
-\,\frac{m_2 \sqrt{2}}{\phi_0} \left( \bar{\Psi}^L_i \Phi^i \Psi^R_2 +
\bar{\Psi}^R_2 \stackrel{*}{\Phi^{i}} \Psi^L_i \right) 
$$
produces the mass terms for doublet components
$$  -m_1 \bar{\Psi}_1 \Psi_1 -m_2 \bar{\Psi}_2 \Psi_2\;, $$ 
which are accompanied by  vertices of interactions of fermions 
with Goldstone fields: 
\begin{eqnarray}
&-&  \frac{ m_1\, g_2}{2 M_W} \left ( \frac{i}{\sqrt{2}}(W^-_f \bar{\Psi}_2 
(1 + \gamma^5) \Psi_1 - W^+_f \bar{\Psi}_1 (1-\gamma^5) \Psi_2) 
+  H \bar{\Psi}_1 \Psi_1 + i Z_f \bar{\Psi}_1 \gamma^5 \Psi_1 \right)
\nonumber\\
&-&  \frac{ m_2\, g_2}{2 M_W} \left ( \frac{i}{\sqrt{2}}(W^+_f \bar{\Psi}_1 
(1 + \gamma^5) \Psi_2 - W^-_f \bar{\Psi}_2 (1-\gamma^5) \Psi_1) 
+  H \bar{\Psi}_2 \Psi_2 -i Z_f \bar{\Psi}_2 \gamma^5 \Psi_2 \right)\;.
\nonumber\\ \label{H-fermion}
\end{eqnarray}

If there are several doublets with the same hypercharge, then 
a general form of Yukawa   Lagrangian  contains a  product of terms from
different doublets. Such terms form  two  mass matrices: one  for
upper   and another for lower fermions. Each of these matrices can be diagonalized 
by means of the unitary transformation of doublets, but this cannot be done for both 
of them at the
same time.  In this case the basis  of doublets is chosen in such a way 
to  present one of these matrices, for example,  for upper fermions,
in the diagonal form. 
Then   the   physical particles correspond to linear combinations of
lower doublet fields   realized by some unitary matrix which is called 
a {\it mixing matrix}.    

Generally the Lagrangian is written down in terms of fermion fields which 
directly correspond to particles.  Interactions of such  fields
with $A$, $Z$, $H$, and $Z_f$  are the same as defined by (\ref{V-fermion}) 
and (\ref{H-fermion}), whereas interactions with $W^\pm$ and $W^\pm_f$
contain elements of the  mixing matrix.
 

\subsubsection{Quarks and leptons}

The Standard Model contains three doublets of leptons with a hypercharge
$Y=-1$:  
$$
\left( \begin{array}{c} \nu_e \\ e^-  \end{array} \right)
\left( \begin{array}{c} \nu_\mu   \\ \mu^-  \end{array} \right)
\left( \begin{array}{c} \nu_\tau \\ \tau^-  \end{array} \right)\;,
$$ 
and three  doublets of quarks with a hypercharge $Y=\frac{1}{3}$:
$$
\left( \begin{array}{c} u \\d^\prime  \end{array} \right)
\left( \begin{array}{c} c \\s^\prime  \end{array} \right)
\left( \begin{array}{c} t \\b^\prime  \end{array} \right)\;.
$$ 

Neutrinos, the upper components of lepton doublets, are so far massless as 
a result of experimental measurements. So the mixing matrix for 
leptons is not needed. Meantime both the  upper and down components of  quark
doublets have nonzero   masses. The corresponding mixing matrix is
named the Cabibbo-Kobayashi-Maskawa matrix. It expresses the lower components of 
quark doublets in terms of the real quark fields:
$$
\left(\begin{array}{c} d^\prime \\ s^\prime \\ b^\prime \end{array}\right) =
\left(\begin{array}{c} Kobayashi \\ Moskawa \\ matrix \end{array}\right)
\left(\begin{array}{c} d \\ s \\ b \end{array}\right)\;.
$$
