Tree-level Graviton Scattering in the Worldline Formalism

TL;DR

This paper employs the worldline formalism to analyze tree-level graviton scattering, revealing symmetry-breaking requirements for coupling to gravity and uncovering double-copy relations in MHV amplitudes.

Contribution

It demonstrates that coupling to background gravity requires breaking $O(4)$ symmetry to $O(2) imes O(2)$ and identifies a new value for the counterterm coefficient, advancing understanding of higher spin interactions.

Findings

Reproduces Einstein's 3-point graviton vertex using worldline formalism.

Discovers symmetry-breaking from $O(4)$ to $O(2) imes O(2)$ in graviton coupling.

Identifies squaring relations leading to double-copy structures in MHV amplitudes.

Abstract

We use the worldline formalism to study tree-level scattering processes involving gravitons. A massless spin 2 particle is described by an $N=4$ supersymmetric worldline action which is also $O(4)$ symmetric. More generally, $N=2S$ supersymmetric worldline actions exhibiting $O(N)$ symmetry describe free spin $S$ particles. Recently a BRST approach was used to construct the on-shell background graviton emission vertex from a graviton worldline. Nonetheless, an action describing the coupling of higher spin ($S\geq 2$) particles with generic background gravity is unknown. In this paper, we found that in order to reproduce Einstein's general relativity 3-point graviton vertex, interpreted as the emission of an off-shell graviton from the worldline, the coupling to background gravity must break the $O(4)$ symmetry to $O(2) \times O(2)$. In addition to this symmetry-breaking feature, we also found that the coefficient $\beta$ of the worldline action counterterm $\beta R$ differs from previous results in the literature. By comparing the linearized graviton and photon emission vertex operators from different worldlines, we noticed that they obey a squaring relation. For MHV (Maximal Helicity Violating) amplitudes, these squaring relations among the linearized vertex operators directly result in double-copy-like relations between the scattering amplitudes.