do not have the resonance as an ancestor.

In the general context when finite width effects are to be included, more than one resonance cascade chain (from now on “resonance history”) may be present, and interference between amplitudes with different resonance histories must also be included. We thus need to perform a separation of the cross section into a sum of contributions, each one of them dominating only for a single resonance history. For each of these contributions we should apply the resonance aware subtraction method of ref. [[@Campbell:2014kua]]{}.

In the following we will describe in great detail the procedure adopted for the separation of the cross section contributions into terms with a definite resonance structure. We will discuss the procedure for the terms that have the Born kinematics (i.e. the Born, Virtual and Collinear Remnant terms), and for real terms. For the latter, subtraction terms having the Born kinematics are also present. We will require that in the collinear and soft limits the separation of the contributions associated with given resonance histories in the real term smoothly matches the corresponding separation in the Born kinematics.

The Born resonance histories
----------------------------

We need to single out contributions from the Born 