are valid owing to the construction of event projections: [\_1]{}[x\_[1+]{}\^[(s)]{}]{}, [\_2]{}[x\_[2+]{}\^[(s)]{}]{}, in eq. (\[sigmaMCp\]), and [\_1]{}[x\_[1-]{}\^[(s)]{}]{}, [\_2]{}[x\_[2-]{}\^[(s)]{}]{}, in eq. (\[sigmaMCm\]). We can therefore perform a change of integration variables: d\^[()]{}[|\_[[MC]{}]{}]{}=\_[ab]{}dz\_1 dz\_2 d\_[ab]{}\^[()]{}[|\_[[MC]{}]{}]{}, \[sigmaMCt\] with d\_[ab]{}\^[(+)]{}[|\_[[MC]{}]{}]{}&=& f\_a\^[[(H\_1)]{}]{}([x\_[1+]{}\^[(s)]{}]{}/z)f\_b\^[[(H\_2)]{}]{}([x\_[2+]{}\^[(s)]{}]{})d\_[ab]{}\^[(+)]{}[|\_[[MC]{}]{}]{}\
d\_[ab]{}\^[(-)]{}[|\_[[MC]{}]{}]{}&=& f\_a\^[[(H\_1)]{}]{}([x\_[1-]{}\^[(s)]{}]{})f\_b\^[[(H\_2)]{}]{}([x\_[2-]{}\^[(s)]{}]{}/z)d\_[ab]{}\^[(-)]{}[|\_[[MC]{}]{}]{}\[bSigmaMCdef\] and [[|\_[[MC]{}]{}]{}]{}&=& \_c P\_[ca]{}\^[(0)]{}(z)[[M]{}\^[(b)]{}]{}\_[cb]{}([x\_[1+]{}\^[(s)]{}]{}P\_1,[x\_[2+]{}\^[(s)]{}]{}P\_2)\_+, \[dsigMCp\]\
[[|\_[[MC]{}]{}]{}]{}&=& \_c P\_[cb]{}\^[(0)]{}(z)[[M]{}\^[(b)]{}]{}\_[ac]{}([x\_[1-]{}\^[(s)]{}]{}P\_1,[x\_[2-]{}\^[(s)]{}]{}P\_2)\_-. \[dsigMCm\] Here, we have introduced the rescaled virtualities \_i=-t\_i/[m\_V]{}\^2. Furthermore, we have explicitly included the factors ${\Theta}_\pm$, which are related to the choice of the maximum virtuality allowed during shower evolution. We have e.g. [m\_V]{}\^2 \_=(y-), \[maxvirt\] with $y$ being the relevant angular variable associated with the FKS parton in the FKS subtraction (here the cosine of the angle between $\vec{k}_2$ and $\vec{p}_i$ for emissions from leg $i$), and $1-x$ the (normalized) energy of the FKS parton. In the case of , at variance with , $x\equiv z$.

MC@NLO short-distance cross sections\[sec:xsec\]
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At this point, we have all the ingredients needed to generate hard-subprocess events according to the MC@NLO formalism, which will be subsequently showered by . Recalling that we denote by ${\overline{\Sigma}}^{(\alpha)}$ the contributions $\Sigma^{(\alpha)}$ to the NLO parton-level cross sections after event projection, event generation will be performed following the procedure outlined in sect. 4.5 of ref. [@Frixione:2002ik] (see also sect. 