Vector Space of Feynman Integrals and Multivariate Intersection Numbers

Feynman integrals and motives

This article gives an overview of recent results on the relation between quantum field theory and motives, with an emphasis on two different approaches: a “bottom-up” approach based on the algebraic geometry of varieties associated to Feynman graphs, and a “top-down” approach based on the comparison of the properties of associated categorical structures. This survey is mostly based on joint wor...

Periods and Feynman integrals

We consider multi-loop integrals in dimensional regularisation and the corresponding Laurent series. We study the integral in the Euclidean region and where all ratios of invariants and masses have rational values. We prove that in this case all coefficients of the Laurent series are periods.

Feynman path integrals on phase space and the metaplectic representation

Feynman parameter integrals

We often deal with products of many propagator factors in loop integrals. The trick is to combine many propagators into a single fraction so that the four-momentum integration can be done easily. This is done commonly using so-called Feynman parameters. We rewrite the product of propagators 1 (A 1 + ii)(A 2 + ii) · · · (A n + ii) , (1) where A i has the form of p 2 − m 2. The sign of A i is not...

Blowing up Feynman integrals

In this talk we discuss sector decomposition. This is a method to disentangle overlapping singularities through a sequence of blow-ups. We report on an open-source implementation of this algorithm to compute numerically the Laurent expansion of divergent multi-loop integrals. We also show how this method can be used to prove a theorem which relates the coefficients of the Laurent series of dime...