Numbers and Functions in Quantum Field Theory
Oliver Schnetz
TL;DR
The paper surveys how number-theoretic structures such as graphical functions and generalized single-valued hyperlogarithms encode the analytic content of quantum field theory, showing that not all periods arise from multiple zeta values. It develops a conformal-completion framework and existence criteria for Feynman periods, and establishes a product identity that organizes periods under graph operations. Leveraging these tools, it enables epsilon-expansions and high-loop computations in the $\\phi^4$ theory, including the seven-loop renormalization functions $\beta$, $\gamma$, and $\gamma_m$, via Maple-based methods and connections to the Galois theory of algebraic integrals. This work strengthens the bridge between number theory and QFT, yielding practical computational techniques for multiloop renormalization and deeper insights into the algebraic structure of Feynman integrals.
Abstract
We review recent results in the theory of numbers and single-valued functions on the complex plane which arise in quantum field theory. We use the results to calculate the renormalization functions $β$, $γ$, $γ_m$ of dimensionally regularized $φ^4$ theory in the minimal subtraction scheme up to seven loops.