Numbers and functions in quantum field theory

Abstract

We review recent results in the theory of numbers and single-valued functions on the complex plane which arise in quantum field theory. These results are the basis for a new approach to high-loop-order calculations. As concrete examples, we provide scheme-independent counterterms of primitive log-divergent graphs in φ4 theory up to eight loops and the renormalization functions β, γ, γm of dimensionally regularized φ4 theory in the minimal subtraction scheme up to seven loops.