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 image and renormalization endows image, the double tensor algebra of image, with the structure of a noncommutative bialgebra. When the bialgebra image is commutative, renormalization turns image, the double symmetric algebra of image, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of image are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When image is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra image and the Faà di Bruno bialgebra of composition of series. The relation with the Connes–Moscovici Hopf algebra is given. Finally, the bialgebra image is shown to give the same results as the standard renormalization procedure for the scalar field.