the production of the final state $F_b$, including all possible resonance histories allowed for the process. We separate the Born contribution in the following way: $$B_{F_b} = \sum_{f_b \in T (F_b)} B_{f_b}, \hspace{2em} B_{f_b} = \Pi_{f_b}
  B_{F_b},$$ where $T (F)$ is the set of all trees having the same bare flavour structure $F$. The factors $\Pi_{f_b}$ have the property $$\sum_{f_b \in T (F_b)} \Pi_{f_b} = 1 . \label{eq:pifbsum}$$ Furthermore, they must be such that $\Pi_{f_b} B_{F_b}$ must have resonance peaks compatible with the resonance history of $f_b$. One possible definition for the $\Pi_{f_b}$ is the following. With each resonance $i$ in the resonance history, we associate the factor $$\frac{M_i^4}{(s_i - M_i^2)^2 + \Gamma_i^2 M_i^2},$$ and define $$P^{f_b} = \prod_{i \in {\ensuremath{\operatorname{Nd}}} (f_b) }  \frac{M_i^4}{(s_i - M_i^2)^2 +
  \Gamma_i^2 M_i^2},$$ where $s_i$, $M_i$ and $\Gamma_i$ are respectively the invariant mass of the decay product system, the mass of the resonance and its width. By ${\ensuremath{\operatorname{Nd}}} (f_b)$ we denote the set of all nodes of the resonance tree for $f_b$ (excluding the root). We then define $$\Pi_{f_b} = \frac{P^{f_b}}{\sum_{f_b' \in T (F_b (f_b))} P^{f_b'}},$$ where we have introduced the notation $F_b (f_b)$ to denote the bare flavour structure associated to a given 