# Bounded Collection of Feynman Integral Calabi-Yau Geometries

Bourjaily, Jacob L. (Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark) ; McLeod, Andrew J. (Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark) ; von Hippel, Matt (Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark) ; Wilhelm, Matthias (Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark)

25 January 2019

Abstract: We define the rigidity of a Feynman integral to be the smallest dimension over which it is nonpolylogarithmic. We prove that massless Feynman integrals in four dimensions have a rigidity bounded by $2\left(L-1\right)$ at $L$ loops provided they are in the class that we call marginal: those with $\left(L+1\right)D/2$ 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.

Published in: Physical Review Letters 122 (2019)