Renormalization and Hopf algebraic structure of the five-dimensional quartic tensor field theory

This paper gives a complete selfcontained proof of our result announced in hep-th/9909126 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra \(\Hc\) which is commutative as an algebra. It is the dual Hopf algebra of the envelopping algebra of a Lie algebra \(\ud G\) whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group \(G\) is the group of characters of \(\Hc\). We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop \[ \g (z) \in G \qquad z \in C \] where \(C\) is a small circle of complex dimensions around the integer dimension \(D\) of space-time. Our main result is that the renormalized theory is just the evaluation at \(z = D\) of the holomorphic part \(\g_+\) of the Birkhoff decomposition of \(\g\). We begin to analyse the group \(G\) and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.

We prove that the real four-dimensional Euclidean noncommutative \phi^4-model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative R^4. The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials.