Angular momentum of zero-frequency gravitons

TL;DR

This paper calculates the angular momentum contribution of zero-frequency gravitons in scattering processes, revealing a cutoff-independent result localized at zero frequency, based on Weinberg's soft theorem.

Contribution

It provides a novel calculation of zero-frequency graviton angular momentum contributions, extending soft theorem analysis to angular momentum in arbitrary collisions.

Findings

Angular momentum from zero-frequency gravitons is cutoff-independent.

The result depends only on the kinematics of the hard particles.

Application to 2-to-2 scattering in GR and supergravity demonstrates the formula's utility.

Abstract

By following closely Weinberg's soft theorem, which captures the $1/\omega$ pole contribution to the amplitude for soft graviton emissions ($\omega<\Lambda$) on top of an arbitrary background hard process, we calculate the expectation value of the graviton's angular momentum operator for arbitrary collisions dressed with soft radiation. We find that the result becomes independent of the cutoff $\Lambda$ on the graviton's frequency, effectively localizing at $\omega=0$. In this way, our result captures the contribution to the angular momentum that comes from the zero-frequency modes. Like the soft theorem, our formula has an exact dependence on the kinematics of the hard particles and is only a function of their momenta. As an example, we discuss in some detail the case of the $2 \to 2$ scattering of spinless particles in General Relativity and ${\cal N}=8$ supergravity.