Quantum graviton scattering with definite helicities in the null surface formulation

TL;DR

This paper develops a second-order quantum gravity perturbation theory within the Null Surface Formulation, focusing on radiative data at null infinity, leading to finite, on-shell graviton scattering amplitudes and insights into gravitational memory and symmetries.

Contribution

It introduces a novel on-shell, finite perturbative framework for quantum gravity using null infinity data, avoiding bulk propagators and renormalization.

Findings

Derives helicity-resolved Bondi shear and out-operators for graviton processes.

Shows graviton scattering factorizes into tail vertices with on-shell intermediate states.

Reproduces Weinberg's tree-level amplitude in the Poincare limit and demonstrates perturbative finiteness.

Abstract

We develop the second-order quantum perturbation theory of gravity in the Null Surface Formulation (NSF) of asymptotically flat spacetimes. In this framework all dynamical degrees of freedom are radiative data defined at null infinity; no bulk fields or off-shell propagators enter the construction. Working directly at null infinity, we derive the helicity-resolved Bondi shear and the corresponding out-operators governing nonlinear graviton processes. The formalism naturally generates a gravitational tail amplitude requiring opposite incoming helicities, and a graviton scattering amplitude that factorizes into two tail vertices connected by an on-shell intermediate graviton. Imposing the Poincare limit reproduces the s-channel contribution of the Weinberg tree-level amplitude, while the crossed channels are shown to arise at higher perturbative order. The theory is perturbatively finite for two independent reasons: all intermediate gravitons are strictly on-shell, so no loop integrals over virtual bulk momenta are generated; and the perturbative regime requires Gaussian-smeared graviton states (small Bondi mass relative to the Planck scale), whose suppression propagates recursively through the hierarchy, rendering all energy integrals absolutely convergent at every order without renormalization or counterterms. This finiteness is structurally distinct from the ultraviolet problem of covariant perturbative gravity, where divergences originate in off-shell bulk propagators and asymptotic states are defined only indirectly via an i{\epsilon} bulk prescription. The natural observables of the NSF are spectral-angular distributions on the celestial sphere, which encode BMS supermomentum flux rather than ordinary Poincare momentum conservation. Gravitational memory, MHV helicity selection rules, and the coherent-state classical limit arise naturally within the same framework.