# Path description of coordinate-space amplitudes

## TL;DR

This paper introduces a coordinate-space version of light-cone-ordered perturbation theory, linking path lengths to denominators and enabling new insights into scalar QED and Wilson lines, with extensions to massive fields.

## Contribution

It develops a novel coordinate-space formalism for light-cone perturbation theory, connecting path lengths to energy deficits and deriving identities for coordinate diagrams.

## Key findings

- Path lengths in coordinate space relate to denominators in perturbation theory.
- Eikonal approximation naturally emerges near the light cone.
- Applications demonstrated for scalar QED and Wilson line products.

## Abstract

We develop a coordinate version of light-cone-ordered perturbation theory, for general time-ordered products of fields, by carrying out integrals over one light-cone coordinate for each interaction vertex. The resulting expressions depend on the lengths of paths, measured in the same light-cone coordinate. Each path is associated with a denominator equal to a "light-cone deficit", analogous to the "energy deficits" of momentum-space time- or light-cone-ordered perturbation theory. In effect, the role played by intermediate states in momentum space is played by paths between external fields in coordinate space. We derive a class of identities satisfied by coordinate diagrams, from which their imaginary parts can be derived. Using scalar QED as an example, we show how the eikonal approximation arises naturally when the external points in a Green function approach the light cone, and we give applications to products of Wilson lines. Although much of our discussion is directed at massless fields in four dimensions, we extend the formalism to massive fields and dimensional regularization.