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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

$$d^2\Omega_j=d\cos{\Theta_j}d\Phi_j$$ (6)

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 (5) 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 (.2), (.3) or (.4) (Section ) depending on a squared sum of momenta. Variables $s_j$ in (5) are also equal to squared sums of momenta. So, the parameterization (5) 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 [Ilyin-1996,Kovalenko-1997]. The polar angle $\Theta_j$ can be unambiguously fixed by the 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

$$\begin{eqnarray*} (Pole_j+q_{j1})^2= (Pole_j^0+q_{j1}^0)^2 - \mid \overline{Pole... ...eta_j} \mid \overline{Pole}_j \mid \mid \overline{q}_{j2} \mid \end{eqnarray*}$$

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 [Ilyin-1996,Kovalenko-1997].

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. 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 (.2), (.3), (.4) 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 [Ilyin-1996,Kovalenko-1997].

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.