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

SIGMA 17 (2021), 065, 19 pages      arXiv:2106.12071      https://doi.org/10.3842/SIGMA.2021.065
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

New Techniques for Worldline Integration

James P. Edwards a, C. Moctezuma Mata a, Uwe Müller b and Christian Schubert a
a) Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacan, Mexico
b) Brandenburg an der Havel, Brandenburg, Germany

Received March 01, 2021, in final form June 23, 2021; Published online July 03, 2021

Abstract
The worldline formalism provides an alternative to Feynman diagrams in the construction of amplitudes and effective actions that shares some of the superior properties of the organization of amplitudes in string theory. In particular, it allows one to write down integral representations combining the contributions of large classes of Feynman diagrams of different topologies. However, calculating these integrals analytically without splitting them into sectors corresponding to individual diagrams poses a formidable mathematical challenge. We summarize the history and state of the art of this problem, including some natural connections to the theory of Bernoulli numbers and polynomials and multiple zeta values.

Key words: worldline formalism; Bernoulli numbers; Bernoulli polynomials; Feynman diagram.

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