Contemporary Mathematics Renormalization, Hopf algebras and Mellin

Abstract. This article aims to give a short introduction into Hopf-algebraic aspects of renormalization, enjoying growing attention for more than a decade by now. As most available literature is concerned with the minimal subtraction scheme, we like to point out properties of the kinematic subtraction scheme which is also widely used in physics (under the names of MOM or BPHZ). In particular we relate renormalized Feynman rules φR in this scheme to the universal property of the Hopf algebra HR of rooted trees, exhibiting a refined renormalization group equation which is equivalent to φR: HR → K[x] being a morphism of Hopf algebras to the polynomials in one indeterminate. Upon introduction of analytic regularization this results in efficient com-binatorial recursions to calculate φR in terms of the Mellin transform. We find that different Feynman rules are related by a distinguished class of Hopf alge-bra automorphisms of HR that arise naturally from Hochschild cohomology. Also we recall the known results for the minimal subtraction scheme and shed light on the interrelationship of both schemes. Finally we incorporate combinatorial Dyson-Schwinger equations to study the effects of renormalization on the physical meaningful correlation functions. This yields a precise formulation of the equivalence of the two different renor-malization prescriptions mentioned before and allows for non-perturbative def-initions of quantum field theories in special cases. Motivation: The renormalization problem Suppose we want to assign a value to the logarithmically divergent integral φs ( ):= 0 dx x+s, which we associate to the tree. Observing the (absolutely) integrable difference (0.1