
   The Table {\it Vertices} contains interaction vertices. 
The first four fields
$A1$, $A2$, $A3$, $A4$ include the names of the interacting particles.  
These fields
must contain particle names in \CompHEP~ notation.  $A4$ may be
empty.  The last two fields {\it 'Factor'} and {\it 'LorentzPart'} define a
vertex itself.  Let $S$ be the action, then a functional derivative of 
$S$ over fields is represented as  
\begin{eqnarray}
\label{eq1}
\frac{\delta S}{\delta A1_{[m1]}(p1)\, \delta A2_{[m2]}(p2)\, 
\delta A3_{[m3]}(p3)\,
 [\delta A4_{[m4]}(p4)]}
&  =  \nonumber   \\
(2 \pi)^4\delta(p1+p2+p3\, [+p4]) \, [\gamma_0] \, ColorStructure \cdot 
 Factor \cdot LorentzPart\;. &
\label{funcDeriv}
\end{eqnarray}   

  Here $p$ and $m$ denote 4-momenta and  Lorentz indices.  The brackets
\mbox{[\ ]} are used to mark the optional parts of  expression. Thus, 
 $A4$, $p4$, and  $m4$ appear only in the case of four particle 
vertex. In the case of anti-commuting fields the  right-side 
derivatives are assumed. The Fourier transformation  is defined by
\begin{equation}
       A(x) =  \int \! \exp(-i k x)  A(k) \,d^4k\;.
\label{Fourier}
\end{equation}
{\it 'Factor'} must be a rational  monomial constructed of the model
identifiers, integer numbers and imaginary unity.

   {\it 'LorentzPart'} must be  a tensor or  Dirac $\gamma$-matrix
expression. Coefficients of this expression are polynomials of the model
identifiers and scalar products of momenta.  The division '/' operator 
is forbidden in  {\it 'LorentzPart'}. It must be transferred to the {\it 'Factor'} field.

Similar to the \REDUCE~ notation, in order to  construct scalar products 
of momenta, momentum components, and metric tensors we use the  dot  symbol, for example, 
\begin{eqnarray*}
 p1.p2 & is & g_{\mu \nu}  p_1^\mu p_2^\nu\;; \\
 p1.m2 & is & p_1^{m_2}\;;\\
 m1.m2 & is & g_{m_1 m_2}\;.  
\end{eqnarray*}

  To  implement  the Dirac $\gamma$-matrix  with index $'m'$ we 
use a symbol $G(m)$, whereas  $G(p)$ denotes  $p_{\mu}\gamma^{\mu}$. Anti-commutation 
relations for $\gamma$  matrices should be written as
\begin{equation}
    G(v1)\, G(v2) + G(v2)\, G(v1) = 2 \;  v1.v2,
\label{anticommutation}
\end{equation}
where  $v1, v2$ are  momenta or  indices.

  The $\gamma_5$ matrix  is denoted by $G5$. It is defined by equation
   $$\gamma_5 = i\; \gamma_0\gamma_1\gamma_2\gamma_3$$

The number of fermion fields in one vertex must be two or zero. If you would like to 
implement a four-fermion interaction, use an auxiliary unphysical  field 
which may be constructed by means of the '*' symbol in the 'Aux' column 
of the particle table (see Section \ref{particles}).

 \CompHEP~ interprets the  anti-particle spinor  field as the Hermitian
conjugated particle field, rather than the Dirac conjugated one. Also it is
assumed that all spinor fields  are written in the
Majorana basis and  the matrix of C-conjugation is equal to
$(-\gamma_0)$ \footnote{ Above the Majorana spinor was defined as a
Hermitian self-conjugated one: $\psi=\psi^+$. In the same time 
it must be $C$-self-conjugated too:  $\psi=C \bar{\psi}^T$. Following 
these requests we get the phase of $C$ which is different from one 
chosen in the textbook \cite{BD} }. After substitution $C \to -\gamma_0$ all possible vertices could be
written as
$$  A1(p1) \, \gamma_0 \, G(v1) \, G(v2)  \ldots  G(vn) \, A2(p2) $$
where $A1$ and  $A2$ are  some spinor fields, each of them  corresponding to  particle 
or anti-particle.  This form of Lagrangian is assumed in the vertex
expression (\ref{eq1}).  $\gamma_0$ 
is substituted by \CompHEP~ automatically  in the case of a spinor 
particle vertex and is not expected in  {\it LorentzPart} of (\ref{funcDeriv}).

   Note that structures like $m1.m2$ and
$p1.m2$ are forbidden  for the vertex with fermions. In order to
implement these structures
use the equation (\ref{anticommutation}). 

   Let us note that by definition (\ref{funcDeriv}) the  {\it LorentzPart} 
has the  corresponding symmetry property in the case when  
identical particles appear in one vertex. This symmetry is not checked by
\CompHEP, but its absence will lead to  wrong results. The following 
equation may be used to check the symmetry in the case of fermion vertex:
\begin{eqnarray}
  A1(p1) \, \gamma_0 \, G(v1) \, G(v2)  \ldots [G5] \ldots  G(vn) \, A2(p2) = \nonumber \\
 A2(p2)\, \gamma_0  \, (-G(vn)) \ldots [G5] \ldots  (-G(v2)) \, (-G(v1)) \,
A1(p1) \;. & 
\label{reverse}
\end{eqnarray}
 It may be useful also to check the  Lagrangian self-conjugation property.
Note that the anticommutation  of {\it A1} and {\it A2}  is already taken into
account in (\ref{reverse}).
   

  {\it ColorStructure} is substituted by \CompHEP~ automatically. For a
colorless particle vertex it is equal to 1.  For $(3 \times \bar{3})$ and for
$(8 \times 8)$ vertices the unity tensor is substituted. If \CompHEP~ meets
a vertex
with three particles in the adjoint representation $(8 \times 8 \times 8)$, 
it substitutes
  $$ -\,i\, f(a1,a2,a3),$$ 
where $f^{\alpha_1}_{\alpha_2 \alpha_3}$  are the  structure constants of SU(3).
Color indices  $a1, a2, a3$  are taken in the same order as 
they appear in the  particle columns. 
For  the  $(3 \times \bar{3} \times 8)$  vertex  \CompHEP~ substitutes
        $$\frac{1}{2}\lambda(\bar{i},i,a),$$
where $\lambda(\bar{i},i,a)$ are the Gell-Mann  matrices.  
   More complicated color structures are not implemented yet, but it is
possible to  construct
them by means of  unphysical particles ({\it Aux}='*') or tensor ghosts (Section 
\ref{ghostFields}). In the case of tensor auxiliary 
field use the capital  {\it 'M'} for designation of the second Lorentz 
index of this field as it is shown in equation (\ref{tensor_propagator}).

