but can be applied to final states with arbitrary multiplicity. Thus, what done here is also valid for final states more complicated than single $V$.

The two procedures related to eq. (\[KonefKtwo\]) and (\[invmap\]) are equivalent, since the integrals are performed over the whole phase space (possibly subject to kinematic restrictions, which are however identical in the two cases). Equation (\[invmap\]) corresponds to the situation depicted in fig. \[fig:MCbran\], except for the seemingly different assignments of the momentum fractions of the incoming particles. These differences are however immaterial, since such fractions are simply integration variables which can be manipulated and renamed at will. We shall now proceed to perform such manipulations, starting from a change of variables in eq. (\[Itwodef\]) (z\_1,z\_2,\_2)([x\_[1]{}\^[(s)]{}]{},[x\_[2]{}\^[(s)]{}]{},\_2\^). \[vchange\] Furthermore, consistently with eq. (\[invmap\]), the integrands have to be computed with the suitable kinematic configurations, ${{\bf 2}}^+$ or ${{\bf 2}}^-$. We can take these two operations into account by the formal replacements \_[ab]{}\^[()]{}([[**2**]{}]{}(z\_i))\_[ab]{}\^[()]{}([[**2**]{}]{}\^(x\_[j]{}\^[(s)]{}))= \_[ab]{}\^[()]{}([[**2**]{}]{}(z\_i(x\_[j]{}\^[(s)]{}))). \[bStohS\] It is easy to realize that, in the case of the Born, soft-virtual, and soft counterterms contributions, we have by construction \_[ab]{}\^[()]{}=\_[ab]{}\^[()]{}, \[bStohSsoft\] since for these terms the jacobian appearing in eq. (\[bStohS\]) exactly compensates the one in the definition of the 