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Дата изменения: Wed Aug 9 20:40:48 2000
Дата индексирования: Mon Oct 1 22:53:07 2012
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Summation of ghost diagrams in CompHEP

CompHEP  uses the squared diagram technique with summation over polarizations. Basically one squared diagram corresponds to the

$$\sum_{i \in S_{all}} A_i^k A_i^{*l}$$ (12)

contribution in a squared matrix element, where $A^k$ and $A^l$ are the amplitudes related to some Feynman diagrams.

If we follow an idea of the previous section and take into account the ghost incoming particles, a number of squared diagrams increases significantly. For example, in the simplest case of $e^-,\gamma \rightarrow n_e, W^-$ process one physical squared diagram (the diagram (a) on the figure below) is accompanied by three ghost diagrams (b,c,d) with the similar topology.

   e1    n1   !  n1    e1            e1    n1   !  n1    e1
 ==>==@==>====!==>===@==>==        ==>==@==>====!==>===@==>=
      |       !      |                  |       !      |
    W+|       !    W+|                W+|       !    W+|
      |       !      |                  |       !      |
 -----@--<----!--<---@-----        -----@--<----!--<---@-----
 A       W-      W-       A        A      W-.f    W-.f      A
             a)                                 b)

   e1    n1   !  n1    e1            e1    n1   !  n1    e1
 ==>==@==>====!==>===@==>==        ==>==@==>====!==>===@==>==
      |       !      |                  |       !      |
    W+|       !    W+|                W+|       !    W+|
      |       !      |                  |       !      |
 --<--@--<----!--<---@--<--        -->--@--<----!--<---@-->--
A.C     W-.C    W-.C     A.C       A.c     W-.c    W-.c     A.c
              c)                                d)

Let us note that all these diagrams of have the same denominators. The numerators of these diagrams are polynomials of scalar products of momenta. The powers of these polynomials are the same for all diagrams, so one might expect that the symbolic sum of all these diagrams has approximately the same size as one term.

Following this note CompHEP  calculates the symbolic sum of all sets of diagrams which become identical after replacing the ghost particles by their parents.