Numbers and functions in quantum field theory

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 ${\varphi }^4$ theory up to eight loops and the renormalization functions $\beta$, $\gamma$, ${\gamma }_m$ of dimensionally regularized ${\varphi }^4$ theory in the minimal subtraction scheme up to seven loops.

Figure 1

Vertex connectivity 3 leads to a product of periods.

Figure 2

The completed graph $P_{7,11}$.

Figure 3

A four dimensional graphical function with three edges and its completion. The wiggly lines refer to edge weights $-1$.

Figure 4

Appending an edge to the vertex $z$ in $G$ gives $G_1$.

Figure 5

Completed ($\overline{Z_{•}}$) and uncompleted ($Z_{•}$) zig-zag graphs with five and six loops.

Figure 6

The inductive construction of GSVHs by a commutative hexagon.

Figure 7

The asymptotic expansion of graphical functions at $z=0$. The bold lines stand for sets of edges.