The real resonance histories and singular regions
-------------------------------------------------

In the case of real graphs, we have more resonance histories, because we have one more final state particle that can belong to resonances. In analogy with the Born case, we introduce $F_r$ and $f_r$ as before, labeling the bare and full flavour structure for a real graph. We will now introduce a label $\alpha_r$, that labels a singular region[^3] compatible (in a sense that we will specify in the following) with a given $f_r$ $$\alpha_r \in {\ensuremath{\operatorname{Sr}}} (f_r) .$$ Also here we will use the notation $f_r (\alpha_r)$ and $F_r (\alpha_r)$ to denote the full and bare flavour structure associated with a given singular region.

We only consider singular regions that are compatible with the given resonance history in the following sense: [*the particles that become collinear should be siblings, i.e. should arise directly from the decay of the same resonance or from the root (if they are directly produced in the hard reaction)*]{}.

We now perform the separation of the real cross section with a given bare flavour structure into singular region contributions: $$R_{\alpha_r} = \frac{P^{f_r (\alpha_r)} d^{- 1} (\alpha_r)}{\sum_{f_r'
  {}\in T (F_r (\alpha_r))} P^{f'_r}  \sum_{\alpha'_r \in {\ensuremath{\operatorname{Sr}}}
  (f'_r)} d^{- 