
\subsubsection{Implementation of QCD Lagrangian}
\paragraph{ 3-gluon vertex.}
  Lagrangian (\ref{physLagQCD}) contains  the following  3-gluon vertex:
%
$$ S_{3G} =  -\frac{g}{2}\int(\partial_{\mu} G_{\nu}^{\alpha} - \partial_{\nu}
G_{\mu}^{\alpha})f^{\alpha}_{\beta \gamma} G^{\beta}_{\mu}
G^{\gamma}_{\nu}d^4x = 
-g\int \partial^{\mu_2} G_{\mu_1}^{\alpha_1} g^{\mu_1 \mu_3} 
 f_{\alpha_1 \alpha_2 \alpha_3} 
G^{\alpha_2}_{\mu_2} G^{\alpha_3}_{\mu_3} d^4x\;,    $$
%
where $G^{\alpha}_{\mu}$ is the gluon field, $g$ is the strong coupling
constant.  Applying the Fourier transformation (\ref{Fourier}) 
we get
 
$$ S_{3G} = (2\pi)^{4} g \int \delta(p_1+p_2+p_3)\;
i \; f_{\alpha_1 \alpha_2 \alpha_3} p^{\mu_2}_1  g^{\mu_1 \mu_3}
G_{\mu_1}^{\alpha_1}(p_1) G_{\mu_2}^{\alpha_2}(p_2)
G_{\mu_3}^{\alpha_3}(p_3)     d^4p_1 d^4p_2 d^4p_3\;. $$

This vertex contains three identical fields, so the  calculation of 
functional derivatives gives us six terms:
%
\begin{eqnarray*}
\lefteqn{\frac{\delta S_{3g}}{\delta G_{\mu_1}^{\alpha_1}(p_1)
\delta G_{\mu_2}^{\alpha_2}(p_2) \delta G_{\mu_3}^{\alpha_3}(p_3)}= 
 (2\pi)^4 \delta(p_1+p_2+p_3)\; i\; f_{\alpha_1 \alpha_2 \alpha_3}}\\
&&g 
(p_1^{\mu_2}g^{\mu_1 \mu_3} - p_3^{\mu_2}g^{\mu_1 \mu_3} 
+p_2^{\mu_3}g^{\mu_1 \mu_2} - p_1^{\mu_3}g^{\mu_1 \mu_2} 
+ p_3^{\mu_1}g^{\mu_2 \mu_3} -  p_2^{\mu_1}g^{\mu_2 \mu_3})\;.   
\end{eqnarray*}
%
Comparing this expression with the \CompHEP~  vertex  representation 
(\ref{funcDeriv}) where  {\it ColorFactor} is  $(-\,i\, f_{\alpha_1 \alpha_2
\alpha_3})$  we get  
%
$$Factor\cdot LorentzPart= -\,g \,
 \Bigl( (p_1^{\mu_2} - p_3^{\mu_2})g^{\mu_1 \mu_3}
+  (p_2^{\mu_3} - p_1^{\mu_3})g^{\mu_1 \mu_2}
+  (p_3^{\mu_1} -  p_2^{\mu_1})g^{\mu_2 \mu_3} \Bigr)\;.   $$
%

\CompHEP~ uses the notation $GG$ for the strong coupling constant $g$.
So for the 3-gluon vertex in the \CompHEP~ format we finally get
\vspace*{0.5cm}\\
{\footnotesize
\begin{tabular}{|l|l|l|l|l|l|}
\hline
A1&A2&A3&A4&Factor&Lorentz part\\
\hline
G&G&G& & GG & m1.m2*(p1-p2).m3+m2.m3*(p2-p3).m1+m3.m1*(p3-p1).m2 \\
\hline
\end{tabular}
}

\paragraph{ Quark-gluon interaction.}
 The interaction of a gluon with a quark is described by the 
following term of Lagrangian (\ref{physLagQCD}):
$$ 
S_{QqG}=  g \int G^{\alpha}_{\mu}(x) \bar{q}(x)\gamma^{\mu} 
\hat{t}_{\alpha}  q(x) \,  d^4x\;.
$$

Applying  the Fourier transformation and substituting $ \bar{q} =
q^+\gamma^0$   we get 
$$
S_{QqG}=  g  (2\pi)^4 \int \delta(p_1+p_2+p_3)  G^{\alpha}_{\mu}(p_3)
q^+(p_1)\gamma^0\gamma^{\mu} \hat{t}_{\alpha}  q(p_2) \,  d^4p_1  d^4p_2
d^4p_3\;;
$$

$$
\frac{\delta S_{QqG}}
{\delta q^+_i(p_1)\delta q^j(p_2) \delta G_{\mu}^{\alpha}(p_3)}=
 g  (2\pi)^4 \delta(p_1+p_2+p_3) (\hat{t}_{\alpha})^i_j
\gamma^0\gamma^{\mu_3}\;.
$$

The factor $(2\pi)^4 \delta(p_1+p_2+p_3)
(\hat{t}_{\alpha})^i_j \gamma^0 $ is substituted by \CompHEP~ 
automatically.  Thus, the quark-gluon  interaction is implemented in the 
\CompHEP~ {\it Vertex} table as the following record:
\vspace*{0.5cm}\\
\begin{tabular}{|l|l|l|l|l|l|}
\hline
A1&A2&A3&A4&Factor&Lorentz part\\   
\hline
Q&q&G& &  GG & G(m3) \\
\hline
\end{tabular}

\bigskip

\noindent
where $q$ and $Q$  are  designations for a quark  and 
a corresponding antiquark. 
\paragraph{ Interaction of ghosts with gluon.}
This interaction is described by the non-linear term of (\ref{fpLagQCD}),
namely 
$$
S_{\bar{c}cG}=  -g \int \bar{c}_{\alpha}(x) 
 \partial^{\mu}( f^{\alpha}_{\beta \gamma} G^{\beta}_{\mu}(x)
c^{\gamma}(x)) d^4x\;;
$$

Fourier transformation and    subsequent evaluation of functional
derivatives  gives us 
$$
S_{\bar{c}cG}= -g (2\pi)^4 \int \delta(p_1+p_2+p_3)  \bar{c}_{\alpha}(p_1)
(-i\,p_2 -i\,p_3)^{\mu_3}
 f^{\alpha}_{\beta \gamma} G^{\beta}_{\mu_3}(p_3)
c^{\gamma}(p_2) d^4p_1  d^4p_2  d^4p_3   
$$
$$
\frac{\delta S_{\bar{c}cG}}
{\delta \bar{c}_{\alpha_1}(p_1)
\delta c^{\alpha_2}(p_2)
 \delta G_{\mu_3}^{\alpha_3}(p_3)}=
 g  (2\pi)^4 \delta(p_1+p_2+p_3)
p_1^{\mu_3}i\, f^{\alpha_1}_{\alpha_2 \alpha_3} \;.
$$

The factor $(2\pi)^4 \delta(p_1+p_2+p_3) (-\,i\,
f_{ \alpha_1  \alpha_2  \alpha_3})$ is substituted by \CompHEP.
 Thus, this interaction may be implemented  as the  
 following record in the {\it Vertex} table.
\vspace*{0.5cm}\\
\begin{tabular}{|l|l|l|l|l|l|}
\hline
A1&A2&A3&A4&Factor&Lorentz part\\
\hline
G.C&G.c&G& & -GG &p1.m3 \\
\hline
\end{tabular}
\vspace*{0.5cm}\\
where $G.C$ and $G.c$ are the \CompHEP~ notations for 
the Faddeev-Popov ghosts $\bar{c}$ and $c$ respectively.

\paragraph{4-gluon interaction.}

Besides  the  3-gluon interaction term the Lagrangian (\ref{physLagQCD})
contains also the term of 4-gluon interaction:
$$
S_{4G}= -\,\frac{g^2}{4} 
g^{\mu\mu^\prime} g^{\nu\nu^\prime} \delta_{\alpha \alpha^\prime}  \int
f^{\alpha}_{\beta\gamma}G^{\beta}_{\mu}(x)G^{\gamma}_{\nu}(x)
f^{\alpha^\prime}_{\beta^\prime\gamma^\prime}
   G^{\beta^\prime}_{\mu^\prime}(x)
   G^{\gamma^\prime}_{\nu^\prime}(x) d^4x\;.
$$ 

The Fourier transformation and functional differentiation  lead us to the 
expression which contains three different $SU(3)$ color structures:

\begin{eqnarray}
\nonumber
&&\frac{\delta S_{4G}}{\delta G_{\mu_1}^{\alpha_1}(p_1)   
\delta G_{\mu_2}^{\alpha_2}(p_2) \delta G_{\mu_3}^{\alpha_3}(p_3)
 \delta G_{\mu_4}^{\alpha_4}(p_4)} =
 - g^2 (2\pi)^4 \delta(p_1+p_2+p_3+p_4) \delta_{\epsilon \epsilon^\prime
}\times\\ \nonumber
&&\Bigl( f^{\epsilon}_{\alpha_1 \alpha_2}f^{\epsilon^\prime}_{\alpha_3\alpha_4}
(g^{\mu_1\mu_3}g^{\mu_2\mu_4} - g^{\mu_1\mu_4}g^{\mu_2\mu_3})
+ f^{\epsilon}_{\alpha_1 \alpha_3}f^{\epsilon^\prime}_{\alpha_2\alpha_4}
(g^{\mu_1\mu_2}g^{\mu_3\mu_4} - g^{\mu_1\mu_4}g^{\mu_2\mu_3}) \\
&&+ 
f^{\epsilon}_{\alpha_1 \alpha_4}f^{\epsilon^\prime}_{\alpha_2\alpha_3}
(g^{\mu_1\mu_2}g^{\mu_3\mu_4} - g^{\mu_1\mu_3}g^{\mu_2\mu_4}) \Bigr)\;.
\label{4Gvertex}
\end{eqnarray}
The complicated color structure of 
this vertex  cannot be 
directly  written  down in the \CompHEP~ format. To implement this vertex 
we use the following trick. We introduce an auxiliary  tensor  field  
$t^{\alpha}_{\mu \nu}(x) $
and the Lagrangian of its interaction with
the gluon field:   
$$
   S_{aux}=  \int \left( \frac{i\,g}{\sqrt{2}} f^{\alpha}_{\beta\gamma} {t_{\alpha}}^{\mu \nu}(x)
 G^{\beta}_{\mu}(x)G^{\gamma}_{\nu}(x)
-\,\frac{1}{2}
  {t^{\alpha}}_{\mu \nu}(x) {t_{\alpha}}^{\mu \nu}(x)\right)d^4x\;.
$$

It is easy to notice that the  functional integration over the auxiliary field
  $t^{\alpha}_{\mu \nu}(x) $ reproduces  the term of 4-gluon
interaction in the partition function:

$$
 e^{i\,S_{4G}(G)}= \int e^{i\,S_{aux}(G,t)} \prod_{x,\alpha,\mu,\nu}
d\,t^{\alpha}_{\mu\nu}(x) \;.
$$  

For each colored  vector particle  \CompHEP~ 
adds     a tensor field with the same 
color to the  internal list of quantum fields.
The propagator of this field (\ref{tensor_propagator})  corresponds to the 
Lagrangian 
$( -\,\frac{1}{2} {t^{\alpha}}_{\mu \nu}(x) {t_{\alpha}}^{\mu\nu}(x))$. 
Consequently, in order to realize the 4-gluon
interaction  we must  introduce a vertex for the  interaction of the gluon  with 
this tensor field 
\begin{eqnarray*}
 S_{tGG}&=&\frac{i\, g}{\sqrt{2}}\int  f^{\alpha}_{\beta\gamma} {t_{\alpha}}^{\mu \nu}(x)
 G^{\beta}_{\mu}(x)G^{\gamma}_{\nu}(x) d^4x\;; \\
\frac{\delta S_{tGG}} { 
G^{\alpha_1}_{\mu_1}(p1)
G^{\alpha_2}_{\mu_2}(p2)
\delta t^{\alpha_3}_{\mu_3 {\mu_3}\prime}(p_3)   
} 
&=& (2\pi)^4\delta(p_1+p_2+p_3) (-\,i\, f_{\alpha_1 \alpha_2 \alpha_3})
\\&\times& \frac{g}{\sqrt{2}}
 (g^{\mu_2 \mu_3} g^{\mu_1{\mu_3}\prime} 
  - g^{\mu_1 \mu_3} g^{\mu_2{\mu_3}\prime} 
 )\;.
\end{eqnarray*}
 
In  \CompHEP~ notations this vertex looks like the following
\vspace*{0.5cm}\\
\begin{tabular}{|l|l|l|l|l|l|}
\hline
A1&A2&A3&A4&Factor&Lorentz part\\
\hline
G&G&G.t& & GG/Sqrt2 & m2.m3*m1.M3 -m1.m3*m2.M3   \\
\hline
\end{tabular}
\vspace*{0.5cm}\\
where $G.t$ is a \CompHEP~ notation for the auxiliary  tensor field 
$t^{\alpha}_{\mu\nu}$ associated
with the vector field $G$. Capital $M$  denotes the  second Lorentz index of
the tensor field. 

From the viewpoint of the  Feynman diagram technique such
realization of the 4-gluon interaction means that
instead of one 4-gluon vertex we substitute
three sub-diagrams presented  in Fig.\ref{4Gsplitt}.
Contribution of each of these diagrams corresponds to
one of the terms of expression (\ref{4Gvertex}).

\subsubsection{Neutrino as a Majorana fermion}

It is well known that  in the  Standard Model  only the left component 
of massless neutrino takes part in  interactions.
So one can describe neutrino by a Majorana  field which has the 
same number of degrees of freedom as a left Dirac one. To realize such a
particle  in the framework of \CompHEP one should add the  
following record to the table of particles:
\vspace*{0.5cm}\\
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
 Full  name    &A &A+ &2*spin &  mass    &width                  & color&aux \\
\hline
neutrino &MN&MN& 1      & 0 & 0 & 1 &  \\
\hline
\end{tabular}
\vspace*{0.5cm}\\

In terms of  Dirac  field  $\Psi_\nu$ a neutrino appears in the Standard Model
 Lagrangian  in the following way  (\ref{V-fermion})\footnote{where $Y=-1$,
$\Psi_1=\Psi_\nu$, $\Psi_2=\Psi_e$, $g_2=e/\sin{\Theta_w}$,
$g_1=e/\cos{\Theta_w}$ }:
\begin{eqnarray}
L_\nu &=& \frac{i}{2}(\bar{\Psi_\nu} \gamma_\mu \partial^\mu \Psi_\nu -
(\partial^\mu \bar{\Psi_\nu})\gamma_\mu \Psi_\nu) 
+ \frac{e}{4 \sin{\Theta_w} \cos{\Theta_w} } Z_\mu 
\bar{\Psi}_\nu \gamma^\mu ( 1 - \gamma^5) \Psi_\nu  \nonumber \\
&& + \frac{e}{2\sqrt{2}\sin{\Theta_w} }
\left(  W^-_\mu  \bar{\Psi}_e  \gamma^\mu (1 - \gamma^5)\Psi_\nu
+   W^+_\mu  \bar{\Psi}_\nu  \gamma^\mu (1 - \gamma^5)\Psi_e \right)\;, 
\label{neutrinoLag}
\end{eqnarray}
where  $\Psi_e$ is the  electron field. To rewrite it in terms of a Majorana
fermion let us perform the  substitution
$$ \Psi_\nu = \frac{1}{2}(1-\gamma^5) \psi_l + \frac{1}{2}(1+\gamma^5)
\psi_r\;,
$$
 where $\psi_l$ and  $\psi_r$ are   Majorana fermions.
Omitting  Lagrangian for  $\psi_r$  and applying the following identities for Majorana
fermions
\begin{eqnarray*}
\frac{i}{4}(\bar{\psi} \gamma_\mu \gamma^5 \partial^\mu \psi -
  (\partial^\mu \bar{\psi})\gamma_\mu \gamma^5 \psi) = \bar{\psi} \gamma^\mu  \psi &=&0
\end{eqnarray*}
which can be obtained  by means of (\ref{reverse}), we get  

\begin{eqnarray*}
L_\nu &=& \frac{i}{4}(\bar{\psi}_l \gamma_\mu  \partial^\mu \psi_l - 
   (\partial^\mu \bar{\psi}_l)\gamma_\mu \psi_l) 
- \frac{e}{4 \sin{\Theta_w} \cos{\Theta_w} } Z_\mu\bar{\psi_l} \gamma^\mu
\gamma^5  \psi_l  \\    
&+& \frac{e}{2\sqrt{2}\sin{\Theta_w} }\left(  W^-_\mu  \bar{\Psi}_e
\gamma^\mu (1 - \gamma^5)\psi_l    
+   W^+_\mu  \bar{\psi}_l  \gamma^\mu (1 - \gamma^5)\Psi_e \right)\;. \nonumber\\
\end{eqnarray*}
Here the first term is the  free Lagrangian for a massless  Majorana fermion.  
Other terms define the interaction. Using the definition 
(\ref{funcDeriv}) we can rewrite them in the \CompHEP~ notations:
\vspace*{0.5cm}\\
\begin{tabular}{|l|l|l|l|l|l|}
\hline
A1&A2&A3&A4&Factor&Lorentz part\\
\hline
MN &MN&Z& &-EE/(2*SW*CW) & G(m3)*G5   \\
\hline
E1&MN&W-& & EE/(2*Sqrt2*SW)& G(m3)*(1-G5)\\
\hline
MN&e1&W+& & EE/(2*Sqrt2*SW)& G(m3)*(1-G5)\\
\hline
\end{tabular}
\vspace*{0.5cm}\\


Let us emphasize  that there are two identical neutrino fields 
in that  term of the Lagrangian which describes the interaction of neutrinos
with a $Z$-boson. It leads to the additional factor  $2$  and to the symmetry
property
of the  corresponding  vertex. One of the typical mistakes in
realization of such a vertex is an introduction 
of  the G(m3)*(1-G5)  term which breaks the symmetry property.
Correct evaluation of the  functional derivative (\ref{funcDeriv})  with the 
help of the identity  (\ref{reverse}) never produces such a term.

\subsubsection{Leptoquarks}

In this section we  present an example of Lagrangian which contains 
the matrix of  {\it C}-conjugation. Such  matrix appears in interactions 
violating  the fermion number conservation. Let 
$\Psi_e$ and $\Psi_u$  be the fermion fields of electron and  {\it u}-quark, 
respectively, interacting   with the scalar complex leptoquark field
{\it F}
$$ L = \lambda  \bar{\Psi}_u^c (1+\gamma^5) \Psi_e F + \lambda  \bar{\Psi}_e
(1+\gamma^5) \Psi_u^c F^+   $$
where $\bar{\Psi}_u^c=\Psi_u^T C$ and  $\Psi_e^c=C \bar{\Psi}_e^T$.

In the Majorana basis which is used in \CompHEP~ the charge conjugation
operator  $C=-\gamma^0$. So the
Lagrangian  can be presented in the form
\begin{eqnarray*}
 L &=& - \lambda  \Psi_u^T \gamma^0 (1+\gamma^5) \Psi_e F - \lambda
\Psi_e^+\gamma_0 (1+\gamma^5) \gamma^0 (\gamma^0)^T \Psi_u^+ F^+ \\
&=&- \lambda  \Psi_u^T \gamma^0 (1+\gamma^5) \Psi_e F + \lambda \Psi_u^+  
\gamma^0  (1+\gamma^5)  \Psi_e^+ F^+\;.
\end{eqnarray*}

 Direct  implementation of the  definition (\ref{funcDeriv}) gives us 
the {\it Vertex} table 
\vspace*{0.5cm}\\
\begin{tabular}{|l|l|l|l|l|l|}
\hline
A1&A2&A3&A4&Factor&Lorentz part\\
\hline
u &e1&F& &-lambda & (1+G5)   \\
\hline
E1&U&F+& & lambda& (1+G5)\\
\hline
\end{tabular}
\vspace*{0.5cm}\\
By means of  equation (\ref{reverse}) we can rewrite this table  in the
equivalent form: 
\vspace*{0.5cm}\\
\begin{tabular}{|l|l|l|l|l|l|}
\hline
A1&A2&A3&A4&Factor&Lorentz part\\
\hline
e1 &u&F& &-lambda & (1+G5)   \\
\hline
U&E1&F+& & lambda& (1+G5)\\
\hline
\end{tabular}
\vspace*{0.5cm}
