New Techniques for Worldline Integration

TL;DR

This paper reviews recent advances in worldline integration techniques, highlighting their mathematical connections and potential to unify Feynman diagram calculations in quantum field theory.

Contribution

It summarizes the development and current state of worldline integration methods, emphasizing new mathematical insights and connections to number theory.

Findings

Connections to Bernoulli numbers and polynomials

Relations to multiple zeta values

Overview of analytical integration challenges

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.