E_j}{(E_i +
   E_j)^2} k_i \cdot k_j \right]^b \Longrightarrow [z (1 - z) k_i \cdot k_j]^b$$ where $\lim_{i j}$ denotes the limit for particles $i$ and $j$ becoming collinear and $z$ is the energy fraction. The last expression is obviously Lorentz invariant in the collinear limit. Thanks to this property, it will turn out that the sum of all $R_{\alpha_r}$ associated with the same underlying Born full flavour structure factorizes in the collinear limit[^4] $$\lim_{i j} \sum_{\alpha_r {}{}}^{f_b (\alpha_r) = f_b}
  R_{\alpha_r} \propto B_{f_b} \times P_{i j} (z) .$$ This follows from the fact that all (and only) the $\alpha_r$ associated with particles $i, j$ becoming collinear dominate in this limit, and, being all equal, they simplify out in the numerator and denominator of eq. (\[eq:realpartition\]). We emphasize, however, that the $d_{ij}$ terms are not frame independent in the soft limit. This is quite clear from eq. (\[eq:dijdef\]), that in the $E_i\to 0$ limit becomes $$\label{eq:dijsoftlim}
d_{ij}\approx \left[\frac{E_i}{E_j} k_i\cdot k_j \right]^b,$$ that is clearly frame dependent.

As in the Born case, the scheme discussed here is not the only alternative for the partition of the singular regions and of the resonance structure. Using weights equal to the square of individual sub-amplitudes 