
\subsubsection{Parameterization via  decay scheme}

The element of phase space volume  for a $n$-particle  state
 is equal to   \cite{ParticlesFields}
\begin{equation}
d\Gamma_n(q) = (2\pi)^4
\delta^4(q-p_1-p_2-p_3-...-p_n)\prod_{i=1}^{n}\frac{\delta(p_i^2-m_i^2)}
{(2\pi)^3}d^4p_i\;.
\label{phaseSpace}
\end{equation}
The same expression is valid for both  the decay of unstable particle with
momentum $q$ and the interaction of two particles with momenta $q_1$ and 
$q_2$  such that $q_1+q_2=q$.
For further discussion we need a designation for a 
phase space volume of some subset $S$ of the full $n$-particle set.
According to (\ref{phaseSpace}) 
\begin{equation}
d\Gamma(q,S)= (2\pi)^4
\delta^4(q-  \sum_{i \in S}p_i)\prod_{i \in S}\frac{\delta(p_i^2-m_i^2)}
{(2\pi)^3}d^4p_i\;.
\end{equation}


Let $S_1$ and $S_2$ be two disjoint particle subsets, then 

\begin{eqnarray}
\lefteqn{d\Gamma(q,S_1\cup S_2) =} \nonumber\\ &&\int ds_1ds_2\Bigl((2\pi)^4
\delta^4(q- q_1 -q_2 )  \frac{\delta(q_1^2-s_1)}{(2\pi)^3}d^4q_1
\frac{\delta(q_2^2-s_2)}{(2\pi)^3}d^4q_2\Bigr)  \nonumber \\ &&
\times \frac{d\Gamma(q,S_1)}{2\pi}\times\frac{ d\Gamma(q,S_2)}{2\pi}\;.
\label{recursion}
\end{eqnarray}

The above formula  expresses a   multi-particle volume 
in terms of  two-particle one,  the volumes $d\Gamma(q_1,S_1)$ and
$d\Gamma(q_2,S_2)$  with a  reduced 
number of particles, and the virtual squared masses  $s_1,\;s_2$ of clusters 
$ S_1,\;S_2$. 

Recursive application of this formula
allows one to   express the  multi-particle phase space in terms 
of two-particle phase space. In its turn the 
two-particle  phase space is explicitly  described by 
spherical angle $\Omega$ of motion  of the first decaying particle in the rest 
frame of initial state \cite{ParticlesFields}. 
\begin{equation}
\frac{d\Gamma(q,[1,2])}{2\pi}=\frac{k d\Omega}{4(2\pi)^3 \sqrt{q^2}}\;,
\label{1_2}
\end{equation}
where  $k$ is the absolute  value of three-dimensional 
momentum of outgoing particles in the rest frame.
Thus,  applying  recursively (\ref{recursion}) and (\ref{1_2})  to
(\ref{phaseSpace}) we obtain an explicit expression for the phase space
volume  in terms of the  squared  masses  $s_j$ of virtual  clusters and
the two-dimensional
spherical angles  $\Omega_j$, where $j$ is an ordinal number of decay:
\begin{equation}
d\Gamma _n(q) =   
\frac{k_1 d^2\Omega_1 }{4(2\pi)^2 \sqrt{q^2}}
\prod_{j=2}^{n-1} \frac{k_j d^2\Omega_j  }{4(2\pi)^3 \sqrt{s_j}}\;.
\prod_{j=2}^{n-1} ds_j 
\label{phaseSpace2}
\end{equation}
Here  $k_j$ is a  momentum of  outgoing clusters 
produced by decay of the $j^{th}$ cluster in its 
 center-of-mass.

The expression (\ref{phaseSpace2}) means some sequential \verb|1->2| decay 
scheme which starts from   incoming state  and finishes with outgoing 
particles of the process. For example, the integration domain for 
 $s_j$ parameters  depends on this scheme. Below we present 
two such schemes for a process with four outgoing particles: 

\begin{picture}(120,120)
\put(0,60){q} 
\put(5,60){\line(1,0){10}}
\put(22,55){$\Omega_1,k_1$}
\put(15,60){\line(1,1){30}} 
\put(18,75){$s_2$}  
\put(52,85){$\Omega_2,k_2$}
\put(45,90){\line(1,1){15}} \put(63,103){$p_1$}
\put(45,90){\line(1,-1){15}} \put(63,73){$p_2$}

\put(15,60){\line(1,-1){30}}
\put(20,40){$s_3$}
\put(52,25){$\Omega_3,k_3$}
\put(45,30){\line(1,-1){15}}  \put(63,43){$p_3$}
\put(45,30){\line(1,1){15}}    \put(63,13){$p_4$}

\end{picture}
~~~~~~~~
\begin{picture}(120,120)
\put(0,90){q} 
\put(5,90){\line(1,0){10}}  \put(20,87){$\Omega_1,k_1$}

\put(15,90){\line(1,1){15}}  \put(33,103){$p_1$}

\put(15,90){\line(1,-2){15}} \put(35,57){$\Omega_2,k_2$}

\put(10,75){$s_2$} 
\put(30,60){\line(1,1){15}}  \put(48,73){$p_2$}


\put(30,60){\line(1,-2){15}} \put(50,27){$\Omega_3,k_3$}

\put(25,45){$s_3$} 
\put(45,30){\line(1,1){15}}   \put(63,43){$p_3$}

\put(45,30){\line(1,-1){15}}   \put(63,13){$p_4$}

\end{picture}

  In the case of \CompHEP~ project  such  decay scheme
is defined by the user via the `Kinematics' menu (see Section \ref{Kinematics}).

  

\subsubsection{Polar vectors}

To complete  phase space parameterization we must 
fix a polar coordinate system  choosing the polar and
the azimuthal  angles for each of decays
\begin{equation}
 d^2\Omega_j=d\cos{\Theta_j}d\Phi_j
 \label{polarCoordinate} 
\end{equation}

We have  an ambiguity in the choice of  polar coordinate.  Let us remind 
that  our goal is not only  parameterization of phase space but also 
 regularization of the squared matrix element in  the 
phase space manifold.  The main idea of such regularization is 
a  cancellation of integrand sharp peaks  by the  phase space measure.
Originally the  phase space measure (\ref{phaseSpace2})  has no 
 cancellation factors, but we can create them by  means of a Jacobian of
transformed variables. To get an appropriate Jacobian  we need to have
the initial phase space variables  related  to   poles  of the
 squared matrix element. 

   In their  turn the poles of  squared matrix element are
caused by virtual particle propagators and  generally  have  
one of the   forms (\ref{pole_1}), (\ref{pole_2}) or (\ref{pole_w})
 (Section \ref{regul_menu})
 depending on  a  squared sum of momenta. Variables $s_j$ in
(\ref{phaseSpace2}) are also equal  to  squared sums of momenta. 
So, the parameterization (\ref{phaseSpace2}) allows us to 
 smooth some peaks of the matrix element.

It  appears to be that the  polar coordinates can be chosen in such a way 
that all  $cos{\Theta_j}$  have  simple linear 
relations to the  squared sums of momenta \cite{kinemat, AIHENP5}.
 The polar angle $\Theta_j$ can  be  unambiguously fixed by  the
{\it polar vector} $Pole_j$  whose space components 
in the rest frame of decay  correspond to the 
$\Theta_j=0$ direction. 
Let $q_{j1}$  and $q_{j2}$ be the   momenta
of the first and the second clusters produced by the $j^{th}$ decay. 
Then 
\begin{eqnarray*}
(Pole_j+q_{j1})^2= (Pole_j^0+q_{j1}^0)^2 - \mid \overline{Pole}_j\mid ^2 -
  \mid\overline{q}_{j1}\mid ^2 - 2cos{\Theta_j} \mid
\overline{Pole}_j  
\mid \mid \overline{q}_{j1}  \mid  \nonumber \\   
(Pole_j+q_{j2})^2= (Pole_j^0+q_{j2}^0)^2 -  \mid \overline{Pole}_j\mid ^2 -
  \mid\overline{q}_{j2}\mid ^2 + 2cos{\Theta_j} \mid
\overline{Pole}_j 
\mid \mid \overline{q}_{j2}  \mid 
\end{eqnarray*}

Thus, in order to get    $cos{\Theta_j}$
 related to  a squared sum of some particle momenta we may 
construct the polar vector as a sum of particle   momenta
\cite{kinemat,AIHENP5}.

 
For the  non-contradictory construction  
we need to  set  the decays  in some order with a natural requirement 
that the sub-decays of clusters produced by the $j^{th}$ decay  have the ordinal 
numbers  larger than $j$. In giving such ordering we can 
  construct a polar vector  for each decay 
based on the incoming momenta and on those of particles produced by 
decays possessing smaller ordinal numbers.

The following statements can be proved.   
{\it
In the framework of any ordered  scheme of decays and for any sum $P$ 
of particle  momenta one can find the decay  number  $j$ such that 
either  $P^2=s_j$   or  $P$ might  be represented as  $Pole_j + q_j$, 
where   $q_j$ is the momentum of one of the clusters  in the  $j^{th}$ 
decay and   $Pole_j$ is a polar vector constructed according to the above
rule.
}
In other words, any of poles (\ref{pole_1}), (\ref{pole_2}), (\ref{pole_w}) 
can be expressed   either in terms  of  $s_j$ parameters or in terms some of  $cos{\Theta_j}$ for
an  appropriate choice of the polar vector \cite{kinemat,AIHENP5}. 



In  \CompHEP~  the  ordering is arranged automatically, so
that   all sub-decays of the first cluster 
have smaller numbers than those of the second cluster.
Polar vectors are also constructed automatically  according to the
list of peaks prepared by the user.

\subsubsection{Smoothing}

The general idea of the integrand smoothing is trivial.
Let us need to evaluate 
\begin{equation}
\int_a^b F(x)dx \;\;\;,
\label{integral} 
\end{equation}
and let $F(x)$ have a peak like  $f(x)$,  where $f(x)$    is a simple
symbolically integrable function  in contrast to 
 $F(x)$:
\begin{equation}
g(x)=\int_a^x f(x') dx'\;.
\end{equation} 

Now we may represent the integral (\ref{integral}) as 
\begin{equation}
\int_a^b F(x)dx = \int_0^{g(b)}dy\frac{F(g^{-1}(y))}{f(g^{-1}(y))},  
\end{equation}
where $g^{-1}(y)$ is the inverse  function for   $g(x)$. The 
integrand is a smooth function now.

We face very often  squared matrix elements which have several 
poles in one of variables. For example, the
$\gamma \rightarrow b,\bar{b}$, $Z \rightarrow b,\bar{b}$ and
$H \rightarrow b,\bar{b}$ virtual  subprocesses may contribute just to 
the same amplitude.
Although in this case we can evaluate  the  integral function  $g(x)$
symbolically,  the inverse  function $g^{-1}(y)$ can be computed
only as a  numerical solution of the corresponding equation.
To bypass   the calculation of inverse function  
\CompHEP~  uses the   multi-channel  Monte Carlo  
(branching) method to  smooth a sum of peaks.
   

   The idea of the branching method is the following. Let  
$F(x)$ have two peaks, one is similar to $f_1(x)$ and
another to $f_2(x)$.
 $f_1(x)$ and  $f_2(x)$  are singular but elementary functions.
Then, instead of 
one integration (\ref{integral}),  we could  perform two ones:
\begin{equation}
\int F(x)dx=
 \int \frac{F(x)f_1(x)}{f_1(x)+f_2(x)} dx + 
\int \frac{F(x)f_2(x)}{f_1(x)+f_2(x)} dx\;, 
\label{branching}
\end{equation}
 but now each integration  has only a single peak! It is easy to
extend this method for an arbitrary number of peaks. 

The branching method was used in \cite{excalibur} to separate  peaks
which came from various diagrams. In that paper there was also  proposed
to use the expression (\ref{branching}) where  $f_i(x)$ is replaced by 
$\alpha_i f_i(x)$  with a subsequent 
search for optimal coefficients $\alpha_i$. 
\CompHEP~ passes on this weight optimization to \VEGAS,
combining two integrals in one \VEGAS~ hypercube.


As was mentioned above, \CompHEP~ automatically  searches for a  polar 
vector for each angle integration in order to  reach a  linear relation
between   $cos\Theta$  and one of the squared sum of
momenta which is responsible for the peak. 
It could happen that various peaks  need different polar vectors 
for the same decay. In this case \CompHEP~ uses the branching method again,
but now for the whole two-dimension sphere integration.  In other words,
we use the branching equation (\ref{branching})  where  $x$ is the
two dimensional sphere angle \cite{kinemat,AIHENP5}.
 
