Hopf Algebraic Structures in Proving Perturbative Unitarity

The coproduct of a Feynman diagram is set up through identifying the perturbative unitarity of the S-matrix with the cutting equation from the cutting rules. On the one hand, it includes all partitions of the vertex set of the Feynman diagram and leads to the circling rules for the largest time equation. Its antipode is the conjugation of the Feynman diagram. On the other hand, it is regarded as the integration of incoming and outgoing particles over the on-shell momentum space. This causes the cutting rules for the cutting equation. Its antipode is an advanced function vanishing in retarded regions. Both types of coproduct are well-defined for a renormalized Feynman diagram since they are compatible with the Connes--Kreimer Hopf algebra.