On the vanishing of certain cuts or residues of loop integrals with higher powers of the propagators

Starting from two-loops, there are Feynman integrals with higher powers of the propagators. They arise from self-energy insertions on internal lines. Within the loop-tree duality approach or within methods based on numerical unitarity one needs (among other things) the residue when a raised propagator goes on-shell. We show that for renormalised quantities in the on-shell scheme these residues can be made to vanish already at the integrand level.