A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries

TL;DR

This paper introduces the concept of rigidity for Feynman integrals, linking their complexity to Calabi-Yau geometries, and provides bounds and examples of integrals saturating these bounds in four-dimensional massless theories.

Contribution

It establishes a bound on the rigidity of massless Feynman integrals in four dimensions and connects marginal integrals to Calabi-Yau geometries, with explicit examples.

Findings

Rigidity of Feynman integrals is bounded by 2(L-1) in four dimensions.

Marginal integrals involve Calabi-Yau geometries.

Examples of finite four-dimensional integrals saturate the rigidity bound.

Abstract

We define the rigidity of a Feynman integral to be the smallest dimension over which it is non-polylogarithmic. We argue that massless Feynman integrals in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show that this bound may be saturated for integrals that we call marginal: those with (L+1)D/2 propagators in (even) D dimensions. We show that marginal Feynman integrals in D dimensions generically involve Calabi-Yau geometries, and we give examples of finite four-dimensional Feynman integrals in massless $\phi^4$ theory that saturate our predicted bound in rigidity at all loop orders.