1} (\alpha_r')} R_{F_r (\alpha_r)}, \label{eq:realpartition}$$ where $F_r (\alpha_r)$ stands for the bare flavour structure associated with $\alpha_r$. We require that the real weights $P^{f_r (\alpha_r)}$ are compatible with the Born weights, in the sense that, in the soft or collinear limit, the $P^{f_r (\alpha_r)}$ must approach smoothly a $P^{f_b}$ factor of the corresponding underlying Born. This is certainly the case if they are defined as in the Born case.

We notice that in the standard `POWHEG` scheme, the real contribution to a given region is enhanced if the collinear pair has a smaller transverse momentum than all other possible collinear pairings. In the present scheme, the relative transverse momentum of the pair is no longer the only element that decides about the partition of singular regions. As an example, consider three final state partons $i$, $j$ and $k$. The cross section is parted among the $i,j$ and $i,k$ singular regions, depending upon how small are the relative transverse momenta in the two cases, and how far from the resonance peaks are the resonances containing respectively the $i,j$ and $i,k$ partons.

The $d^{- 1}$ factors used in the have the form $$\begin{aligned}
  \label{eq:didef}
  d_i & = & [E_i^2 (1 - \cos^2 