Rescattering corrections and self-consistent metric in Planckian scattering

TL;DR

This work extends the ACV reduced-action framework for transplanckian scattering by embedding the leading eikonal in a 3D setting and incorporating rescattering corrections to achieve a self-consistent description of particle motion and the gravitational metric up to order $R^2/b^2$. It derives shifted, rotated shock-wave fields from an improved eikonal representation, and develops a RAM-based semiclassical field theory that includes H-diagram and rescattering contributions, yielding corrected scattering angles and time delays that are consistent with the action. The analysis shows that rescattering is essential for aligning metric shifts with the action at subleading order, and it introduces a self-consistent, shifted metric with a redistributed support for the transverse field; it also discusses irregular UV-sensitive solutions that may carry information across the interaction region, pointing to intriguing links with string-scale dynamics. Overall, the paper provides a more complete, self-consistent picture of Planckian scattering, clarifying how trajectory shifts, shock-wave shifts, and rescattering corrections interplay to shape high-energy gravitational interactions.

Abstract

Starting from the ACV approach to transplanckian scattering, we present a development of the reduced-action model in which the (improved) eikonal representation is able to describe particles' motion at large scattering angle and, furthermore, UV-safe (regular) rescattering solutions are found and incorporated in the metric. The resulting particles' shock-waves undergo calculable trajectory shifts and time delays during the scattering process --- which turns out to be consistently described by both action and metric, up to relative order $R^2/b^2$ in the gravitational radius over impact parameter expansion. Some suggestions about the role and the (re)scattering properties of irregular solutions --- not fully investigated here --- are also presented.

Figures (13)

• Figure 1: One- and two-rung effective ladder diagrams determining the elastic $S$-matrix in the eikonal approximation. Solid lines: on-shell external particles; dashed lines: eikonal gravitational exchanges.
• Figure 2: Kinematics of the wave packets' relative coordinates.
• Figure 3: Space-time diagram of the double-shift picture of a test particle (solid blue) by the shock-wave of particle 2 (not shown) and then by the shifted shock wave of particle 1 (thick pink line). In the upper part, the motion in the longitudinal plane is shown. In the lower part, the coordinate $\xi$ denote the transverse direction starting from the center of mass (CM) of the two colliding particles. At the point $A$ ($B$) the test particle leaves the first (second) wave front.
• Figure 4: Simplest eikonal diagrams with one insertion of the $h_{--}$ field (red cross). The parametrization of momenta is choosen so as to simplify the the calculations.
• Figure 5: Higher order eikonal diagrams with one insertion of the field $h_{--}$ (red cross). The blobs represent ladders of eikonal exchanges.