Notes on Conformal Soft Theorems and Recursion Relations in Gravity

TL;DR

This paper develops a recursion relation for gravitational celestial amplitudes using BCFW techniques, revealing conformal soft theorems and providing a celestial analogue of Hodges' recursion for MHV amplitudes.

Contribution

It introduces a BCFW-based recursion for celestial amplitudes and explores conformal softness, connecting soft theorems with celestial correlators and extending Hodges' recursion.

Findings

Derived a recursion relation for celestial gravitational amplitudes.

Identified exponential form of conformal soft theorems.

Provided celestial Hodges' recursion for MHV amplitudes.

Abstract

Celestial amplitudes are flat-space amplitudes which are Mellin-transformed to correlators living on the celestial sphere. In this note we present a recursion relation, based on a tree-level BCFW recursion, for gravitational celestial amplitudes and use it to explore the notion of conformal softness. As the BCFW formula exponentiates in the soft energy, it leads directly to conformal soft theorems in an exponential form. These appear from a soft piece of the amplitude characterized by a discrete family of singularities with weights $\Delta=1-\mathbb{Z}_+$. As a byproduct, in the case of the MHV sector we provide a direct celestial analogue of Hodges' recursion formula at all multiplicities.