\theta_i)]^b, \\  \label{eq:dipmdef}
  d_i^{\pm} & = & [E_i^2 2 (1 \pm \cos \theta_i)]^b, \\ \label{eq:dijdef}
  d_{i j} & = & \left[ \frac{E_i^2 E^2_j}{(E_i + E_j)^2}  (1 - \cos \theta_{i
  j}) \right]^b, \end{aligned}$$ where $b$ is a positive constant parameter. Eq. (\[eq:didef\]) is used for the collinear region characterized by parton $i$, with energy $E_i$ and angle $\theta_i$ (relative to the beam) in the partonic CM, becoming collinear to either incoming hadrons. Eq. (\[eq:dipmdef\]) is again for initial state collinear regions, but distinguishes among the two collinear directions. Eq. (\[eq:dijdef\]) is used for the region characterised by final state partons $i$ and $j$ becoming collinear. They are commonly evaluated in the partonic rest frame. In the present case, however, in case of final state singularities associated with the decay products of a resonance, it seems more natural to compute them in the resonance rest frame. They thus become dependent upon the full flavour structure $f_r$ of the real contribution. It is however important for the following developments that the $d_{i j}$ factors do not depend upon the resonance structure in the collinear limit. This is in fact the case with our definition, since $$\lim_{i j} d_{i j} = \lim_{{\ensuremath{\operatorname{coll}}}} \left[ \frac{E_i 