Note that, having treated separately the two collinear singularities in the NLO cross section, the case $c-$ need not be considered when studying branching from leg 1). We find: && [x\_[1+]{}\^[(s)]{}]{}=z[z\_1]{}, [x\_[2+]{}\^[(s)]{}]{}=[z\_2]{}, \[xtilsp\]\
&& [x\_[1+]{}\^[(c+)]{}]{}=[x\_[1+]{}\^[(s)]{}]{}/z, [x\_[2+]{}\^[(c+)]{}]{}=[x\_[2+]{}\^[(s)]{}]{}. \[xtilcpp\] For an emission from leg 2, we find instead && [x\_[1-]{}\^[(s)]{}]{}=[z\_1]{}, [x\_[2-]{}\^[(s)]{}]{}=z[z\_2]{}, \[xtilsm\]\
&& [x\_[1-]{}\^[(c-)]{}]{}=[x\_[1-]{}\^[(s)]{}]{}, [x\_[2-]{}\^[(c-)]{}]{}=[x\_[2-]{}\^[(s)]{}]{}/z. \[xtilcmm\] At this point, following what was done in sect. 4.4 of ref. [@Frixione:2002ik], one uses eqs. (\[xtilsp\]) and (\[xtilcpp\]) to perform the event projection transformation of $\VEV{O}_+$, and eqs. (\[xtilsm\]) and (\[xtilcmm\]) to deal with $\VEV{O}_-$. Once event projection has been achieved, the MC@NLO short distance cross section can be defined by including the MC subtraction terms. We shall do this in the next subsection.

MC subtraction terms
--------------------

At the NLO in ${\alpha_{{\scriptscriptstyle}S}}$, the cross section resulting from  is d[|\_[[MC]{}]{}]{}=d\^[(+)]{}[|\_[[MC]{}]{}]{}+d\^[(-)]{}[|\_[[MC]{}]{}]{}, where the two terms on the r.h.s. account for emissions from leg 1 and 2 respectively. They read: d\^[(+)]{}[|\_[[MC]{}]{}]{}&=&\_[abc]{}d[\_1]{}d[\_2]{} d\_[ab]{}\^[(b)]{}([\_1]{}P\_1,[\_2]{}P\_2) P\^[(0)]{}\_[ac]{}(z)f\_c\^[[(H\_1)]{}]{}([\_1]{}/z)f\_b\^[[(H\_2)]{}]{}([\_2]{}), \[sigmaMCp\]\
d\^[(-)]{}[|\_[[MC]{}]{}]{}&=&\_[abc]{}d[\_1]{}d[\_2]{} d\_[ab]{}\^[(b)]{}([\_1]{}P\_1,[\_2]{}P\_2) P\^[(0)]{}\_[bc]{}(z)f\_a\^[[(H\_1)]{}]{}([\_1]{})f\_c\^[[(H\_2)]{}]{}([\_2]{}/z). \[sigmaMCm\] Following what was done in sect. A.5 of ref. [@Frixione:2002ik], we can manipulate the equations above to render them suitable for an integration together with the NLO parton-level cross section. First of all, one notes that the following identifications 