It is worth noting that while eq. (\[Obs1\]) is the same as in , eq. (\[Obs2\]) does not coincide with either of the choices considered in ref. [@Frixione:2002ik]; the implication of this fact is that we obtain two different event projections, depending on whether it is leg 1 or 2 that emits. We stress that eqs. (\[Obs1\]) and (\[Obs2\]), and the resulting manipulations we are now going to describe, do not depend on the nature of the Born-level final state (a $V$ boson rather than – say – a three-jet configuration), owing to the fact that  adopts the so-called $s$-approach [@Sjostrand:2006za] when doing initial-state branchings for all hadroproduction processes. In the context of the $s$-approach, the invariant mass ($\sqrt{s_0}$) of the final-state system at the Born level is kept constant during the branching. This allows one to formally replace $V$ with the set of final-state particles at the Born level, and $k_1$ with the sum of their four momenta, which does not entail any changes to eqs. (\[spart\])–(\[ss0z\]), from which we ultimately derive eqs. (\[Obs1\]) and (\[Obs2\]). The procedure

described here is thus fully general, and is not restricted to $V$-boson production. We recall that event projection is a way 