Worldline geometries for scattering amplitudes

TL;DR

This paper develops a worldline path integral framework for scalar scattering amplitudes, automatically implementing LSZ reduction and offering a first-quantized approach that could extend to gauge theories and aid in understanding color-kinematics duality.

Contribution

It introduces a novel worldline representation of scattering amplitudes with generalized vertex operators and recursive correlation functions, avoiding reliance on field-theoretic inputs.

Findings

Constructed the path integral for scalar worldlines with vertex operators at infinity.

Demonstrated recursive relations for correlation functions on semi-infinite lines.

Proposed potential extensions to gauge theories and insights into color-kinematics duality.

Abstract

In this paper, we construct the path integral for infinite and semi-infinite scalar worldlines. We show that, at the asymptotic endpoints, on-shell physical states can be generated by inserting vertex operators at infinity. This procedure implements automatically the LSZ reduction, thus leading to a direct worldline representation of scattering amplitudes. To obtain it, we introduce generalized vertex operators, to be viewed as the gluing of entire tree subdiagrams to a given worldline. We demonstrate that the subdiagrams themselves are given, via a recursive relation, by correlation functions on the semi-infinite line. In this sense, the approach we take is fully first-quantized, in that it does not need any field theoretic quantity as input. We envisage that, when suitably extended to gauge theories, it could provide useful insights in addressing current research issues, such as color-kinematics duality.