Summing Feynman diagrams in the worldline formalism

TL;DR

This paper reviews methods to sum Feynman diagrams using the worldline formalism, demonstrating its advantages in simplifying complex multiloop calculations and extending non-perturbative gauge transformations.

Contribution

It generalizes the Landau-Khalatnikov-Fradkin formula for fermion propagators and introduces a framework for analytical evaluation of worldline integrals.

Findings

Generalization of gauge transformation formulas in QED

Integration techniques for photon contributions in low-energy expansions

A framework linking worldline integrals to quantum mechanics on the circle

Abstract

The worldline formalism shares with string theory the property that it allows one to write down master integrals that effectively combine the contributions of many Feynman diagrams. While at the one-loop level these diagrams differ only by the position of the external legs along a fixed line or loop, at multiloop they generally involve different topologies. Here we summarize various efforts that have been made over the years to exploit this property in a computationally meaningful way. As a first example, we show how to generalize the Landau-Khalatnikov-Fradkin formula for the non-perturbative gauge transformation of the fermion propagator in QED to the general $2n$ - point case by pure manipulations at the path-integral level. At the parameter-integral level, we show how to integrate out individual photons in the low-energy expansion, and then sketch a recently introduced general framework for the analytical evaluation of such worldline integrals involving a reduction to quantum mechanics on the circle and the relation between inverse derivatives and Bernoulli polynomials.