Bananas: multi-edge graphs and their Feynman integrals

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 loop integrals and their automatic computer aided evaluation

Multi-loop Feynman Integrals and Conformal Quantum Mechanics

New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts and star-triangle relation methods, can be drastically simplified by using this algebraic approach. To demonstrate the advantages of the algebraic method of ...

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...