full flavour structure $f_b$. This definition clearly satisfies the property (\[eq:pifbsum\]). Thus $B_{f_b}$ exhibits resonance peaks only in correspondence with resonances in its own resonance history. In fact the $P^{f'_b}$ factors for all alternative resonance histories in the denominator of $\Pi_{f_b}$ cancel the resonance peaks due to alternative resonance histories in $B_{F_b}$. Only the peaks compatible with the $f_b$ resonance structure, that have a corresponding enhancement factor in the numerator, will remain.

It is worth pointing out that our definition of the $\Pi$ factor is certainly not unique. In particular, there is an alternative possibility that is easily implemented if one has access to the individual sub-amplitudes contributing to the total amplitude characterized by $F_b$: $$  B_{F_b} = \Bigl|^{} \sum_i \mathcal{A}_i  \Bigr|^2 .$$ The structure of each sub-amplitude represents in this case a resonance history, so that we can create a correspondence $i \leftrightarrow f_b$, and define $$P^{f_b} = | \mathcal{A}_{f_b} |^2 .$$ This possibility may prove convenient with current numerical matrix elements programs, where the numerical calculation of the individual amplitude is a necessary step for the computation of the full matrix element. Since this procedure is gauge dependent, care should be taken in the choice of an appropriate 