[Paper Review] Notes on Conformal Soft Theorems and Recursion Relations in Gravity

Alfredo Guevara|arXiv (Cornell University)|Jun 18, 2019

Black Holes and Theoretical Physics50 references50 citations

TL;DR

The paper develops a tree-level BCFW-inspired recursion in Mellin space for gravitational celestial amplitudes, revealing conformal soft theorems as residues at discrete Δ poles and deriving a celestial Hodges-type recursion for the MHV sector.

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.

Motivation & Objective

• Investigate how soft theorems in gravity appear in Mellin (celestial) space through a BCFW-based recursion.
• Show that the collinear part of the amplitude exponentiates in the soft energy and translates into conformal soft theorems.
• Derive a celestial analogue of Hodges’ recursion for MHV gravity amplitudes at all multiplicities.
• Clarify the role of discrete Δ-poles in the Δ-plane as residues encoding soft expansions.
• Connect soft factorization, Lorentz transformations, and SL(2,C) properties in celestial CFT language.

Proposed method

• Review BCFW factorization and Hodges recursion in momentum space for gravity.
• Translate the BCFW construction to Mellin space to obtain celestial amplitudes.
• Identify the soft piece of the amplitude as an exponential in the soft energy and study its Mellin transform.
• Characterize conformal soft theorems as residues at discrete Δ = 1 − Z+ in the Δ-plane.
• Derive a celestial Hodges-type recursion formula for the MHV sector from the Mellin-space soft piece.
• Use SL(2,C) generators to systematically expand in the soft energy and extract residues.

Experimental results

Research questions

• RQ1 How do soft theorems in gravity manifest when amplitudes are Mellin-transformed to the celestial sphere?
• RQ2 Can a BCFW/Hodges-type recursion be formulated directly in Mellin space for gravitational amplitudes?
• RQ3 What is the role of discrete Δ-poles (Δ = 1 − Z+) in encoding conformal soft theorems?
• RQ4 Does the celestial analogue of Hodges’ recursion hold for MHV gravity amplitudes at all multiplicities?
• RQ5 How are Lorentz transformations and SL(2,C) representations realized in celestial coordinates to produce soft factorization?

Key findings

• A recursion in Mellin space for gravitational celestial amplitudes is obtained via a tree-level BCFW construction.
• The collinear soft piece M_{n+1}^{c} exponentiates in the soft energy and maps to conformal soft theorems as residues in Δ-space.
• Discrete Δ-poles at Δ = 1 − Z+ capture the conformal soft expansion, with leading orders corresponding to universal soft graviton factorizations.
• In the MHV sector, M_{n+1}^{c} provides the full amplitude, yielding a celestial Hodges-type formula at all multiplicities.
• A direct celestial analogue of Hodges’ recursion is derived, relating deformed MHV n-point amplitudes to lower-point celestial amplitudes.
• The framework connects soft theorems, Lorentz transformations, and SL(2,C) structures in celestial CFT language.