Renormalization as a functor on bialgebras

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra BB and renormalization endows T(T(B)+)T(T(B)+), the double tensor algebra of BB, with the structure of a noncommutative bialgebra. When the bialgebra BB is commutative, renormalization turns S(S(B)+)S(S(B)+), the double symmetric algebra of BB, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S1(B)S1(B) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When BB is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)+)S(S(B)+) and the Faà di Bruno bialgebra of composition of series. The relation with the Connes–Moscovici Hopf algebra is given. Finally, the bialgebra S(S(B)+)S(S(B)+) is shown to give the same results as the standard renormalization procedure for the scalar field.