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Notes on Conformal Soft Theorems and Recursion Relations in Gravity

Alfredo Guevara

TL;DR

This work develops a Mellin-space (celestial) version of the BCFW recursion for gravitational amplitudes, revealing that the soft-energy piece exponentiates into conformal soft theorems governed by discrete poles in the $\Delta$-plane. The authors show that, in Mellin space, the soft sector acts as a set of Lorentz deformations on the celestial coordinates, enabling a celestial Hodges-like recursion in the MHV sector and yielding an infinite tower of conformal soft residues at $Δ=1-\mathbb{Z}_+$. They verify the framework with explicit 4-point examples, demonstrating consistency with known celestial soft theorems and delta-function support structures. The approach provides a unified, operator-based view of soft factorization on the celestial sphere and opens avenues for loop-level, gauge-theory extensions, and connections to conformal partial waves.

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 $Δ=1-\mathbb{Z}_+$. As a byproduct, in the case of the MHV sector we provide a direct celestial analogue of Hodges' recursion formula at all multiplicities.

Notes on Conformal Soft Theorems and Recursion Relations in Gravity

TL;DR

This work develops a Mellin-space (celestial) version of the BCFW recursion for gravitational amplitudes, revealing that the soft-energy piece exponentiates into conformal soft theorems governed by discrete poles in the -plane. The authors show that, in Mellin space, the soft sector acts as a set of Lorentz deformations on the celestial coordinates, enabling a celestial Hodges-like recursion in the MHV sector and yielding an infinite tower of conformal soft residues at . They verify the framework with explicit 4-point examples, demonstrating consistency with known celestial soft theorems and delta-function support structures. The approach provides a unified, operator-based view of soft factorization on the celestial sphere and opens avenues for loop-level, gauge-theory extensions, and connections to conformal partial waves.

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 . As a byproduct, in the case of the MHV sector we provide a direct celestial analogue of Hodges' recursion formula at all multiplicities.

Paper Structure

This paper contains 10 sections, 2 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction: Soft Expansion in Mellin Space
  2. Soft Theorems in Momentum Space
  3. Construction in Mellin Space
  4. Preliminaries
  5. Soft Theorems and Hodges' Recursion Relation
  6. Examples
  7. Recursion Formula
  8. Soft Limit and Tower of Singularities
  9. Discussion
  10. Elementary Properties of The Mellin Transform

Key Result

Theorem 1

Let $\tilde{f}(\Delta)$ defined above be integrable in the imaginary lime $c+i\mathbb{R}$, where $a<c<b$, then

Figures (1)

  • Figure 1: Soft and UV singularities in the $\Delta$-plane of a single particle. The strip $(1,b)$ provides the convergence region of the celestial amplitude. The existence of the strip, $b>1$, requires the amplitude to decay faster than $\omega^{-1}$ as $\omega \to \infty$. The red line, $\Delta=1+i\mathbb{R}$, denotes the principal continuous series of representations, whereas the left crosses, $\Delta=1-\mathbb{Z_+}$, correspond to the discrete series.

Theorems & Definitions (3)

  • Theorem 1: Inversion
  • Definition A.1
  • Theorem 2: Direct Mapping