Algebraic geometry informs perturbative quantum field theory

TL;DR

This paper explores how algebraic geometry helps identify the special functions, such as modular forms and multiple zeta values, that appear in high-loop Feynman diagram integrals in quantum field theory.

Contribution

It provides explicit examples of Feynman diagrams evaluated to modular forms and polylogarithms, revealing algebraic geometry's role in understanding these integrals.

Findings

Feynman integrals evaluate to Dirichlet L-series of modular forms.

At high loops, polylogarithms are insufficient for beta-function evaluations.

Modular forms act as obstructions to polylogarithmic evaluations.

Abstract

Single-scale Feynman diagrams yield integrals that are periods, namely projective integrals of rational functions of Schwinger parameters. Algebraic geometry may therefore inform us of the types of number to which these integrals evaluate. We give examples at 3, 4 and 6 loops of massive Feynman diagrams that evaluate to Dirichlet $L$-series of modular forms and examples at 6, 7 and 8 loops of counterterms that evaluate to multiple zeta values or polylogarithms of the sixth root of unity. At 8 loops and beyond, algebraic geometry informs us that polylogs are insufficient for the evaluation of terms in the beta-function of $\phi^4$ theory. Here, modular forms appear as obstructions to polylogarithmic evaluation.