Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 032, 56 pages      arXiv:2009.00426
Contribution to the Special Issue on Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday

An Introduction to Motivic Feynman Integrals

Claudia Rella
Section de Mathématiques, Université de Genève, Genève, CH-1211 Switzerland

Received August 30, 2020, in final form March 03, 2021; Published online March 26, 2021

This article gives a short step-by-step introduction to the representation of parametric Feynman integrals in scalar perturbative quantum field theory as periods of motives. The application of motivic Galois theory to the algebro-geometric and categorical structures underlying Feynman graphs is reviewed up to the current state of research. The example of primitive log-divergent Feynman graphs in scalar massless $\phi^4$ quantum field theory is analysed in detail.

Key words: scattering amplitudes; Feynman diagrams; multiple zeta values; Hodge structures; periods of motives; Galois theory; Tannakian categories.

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