Worldline Proof of Eikonal Exponentiation

TL;DR

This paper proves that the eikonal phase for scalar particle scattering in electromagnetism and gravity exponentiates at all orders using the worldline formalism, providing a first quantized approach to this quantum field theory result.

Contribution

It introduces a worldline formalism proof for the all-order exponentiation of the 2-body eikonal phase in scalar scattering, offering a novel perspective beyond standard QFT methods.

Findings

Demonstrates factorization of the scattering amplitude in impact parameter space at each loop order.

Shows the natural emergence of exponentiation in the worldline formalism.

Confirms previous conjectures about eikonal exponentiation in a new formalism.

Abstract

In this paper, working in the eikonal approximation, we present a proof for the exponentiation of the 2-body eikonal phase to {\it all orders in the eikonal expansion}, for scalar particles interacting electromagnetically or gravitationally. The proof is based on the worldline formalism, which is an alternative, first quantized method to the standard QFT calculation of the scattering amplitude. We show that in the worldline formalism the 2-body scattering amplitude written in impact parameter space naturally factorizes at each loop order. This factorization is responsible for the exponentiation of the eikonal phase, a result which was anticipated in the work of Mogull, Plefka, and Steinhoff [2010.02865 [hep-th]].