processes are simply obtained from eq. (\[Rproc\]) by removing $c(k_2)$ from the r.h.s.. Only soft and initial-state collinear singularities are present in the processes of eq. (\[Rproc\]). Hence, the ${{\cal S}}$ functions required for a separate treatment of the collinear singularities are two and shall be denoted by [[[S]{}]{}\_[+]{}]{},[[[S]{}]{}\_[-]{}]{}, with the following properties: [[[S]{}]{}\_[+]{}]{}+[[[S]{}]{}\_[-]{}]{}&=&1, \[Sfunnorm\]\
\_[\_2\_1]{}[[[S]{}]{}\_[+]{}]{}&=&1, \[Sfuncolp\]\
\_[\_2\_2]{}[[[S]{}]{}\_[-]{}]{}&=&1, \[Sfuncolm\]\
\_[k\_2\^00]{}[[[S]{}]{}\_]{}&& 0. \[Sfunsoft\]\
The expectation value for any observable $O$ will then be written as =\_+ + \_-, where (see eqs. (4.14)–(4.16) of ref. [@Frixione:2002ik]) \_&=&\_[ab]{}dx\_1 dx\_2 d\_2. \[nlosubtint\] As can be seen in ref. [@Frixione:2002ik], the $\Sigma$ terms in equation above are defined as the partonic short-distance cross sections, times the luminosity factors.

In writing eq. (\[nlosubtint\]), we have split (using the ${{\cal S}}$ functions) the Born ($\Sigma_{ab}^{(b)}$) and soft-virtual ($\Sigma_{ab}^{(sv)}$) contributions into two terms, and have associated them with the corresponding real-emission contributions (see refs. [@Frixione:2005vw; @Frederix:2009yq]). We have also used the collinear limits of eqs. (\[Sfuncolp\]) and (\[Sfuncolm\]), that imply that the ${{\cal S}}$ functions associated with the collinear counterterms are trivial. Finally, note that the collinear remainders $\Sigma_{ab}^{(c\pm)}$ need not be multiplied by the ${{\cal S}}$ functions.

Equation (\[nlosubtint\]) must be further manipulated in order to use 