[Paper Review] The SAGEX Review on Scattering Amplitudes, Chapter 11: Soft Theorems and Celestial Amplitudes

Tristan McLoughlin, Andrea Puhm+1 more|arXiv (Cornell University)|Mar 24, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This review establishes a comprehensive framework linking soft theorems, asymptotic symmetries, and celestial amplitudes in four-dimensional flat spacetime. By reformulating scattering amplitudes in a conformal primary basis, it demonstrates that celestial amplitudes exhibit conformal field theory (CFT) properties, providing strong evidence for a holographic duality between quantum gravity in asymptotically flat spacetimes and a two-dimensional celestial CFT (CCFT).

ABSTRACT

The soft limits of scattering amplitudes have been extensively studied due to their essential role in the computation of physical observables in collider physics. The universal factorisation that occurs in these kinematic limits has been shown to be related to conservation laws associated with asymptotic, or large, gauge symmetries. This connection has led to a deeper understanding of the symmetries of gauge and gravitational theories and to a reformulation of scattering amplitudes in a basis of boost eigenstates which makes manifest the two-dimensional global conformal symmetry of the celestial sphere. The recast, or celestial, amplitudes possess many of the properties of conformal field theory correlation functions which has suggested a path towards a holographic description of asymptotically flat spacetimes. In this review we consider these interconnected developments in our understanding of soft theorems, asymptotic symmetries and conformal field theory with a focus on the structure and symmetries of the celestial amplitudes and their holographic interpretation.

Motivation & Objective

• To unify the understanding of soft theorems, asymptotic symmetries, and the emerging structure of celestial amplitudes in four-dimensional Minkowski spacetime.
• To establish the connection between the Ward identities of asymptotic symmetries—such as supertranslations and superrotations—and universal soft theorems in QED and gravity.
• To demonstrate that scattering amplitudes expressed in conformal primary wavefunctions transform into correlation functions of a 2D celestial CFT (CCFT), revealing hidden conformal symmetry.
• To explore the algebraic structure of conformally soft symmetries and their realization via operator product expansions (OPEs) in the celestial CFT framework.
• To identify open challenges in formulating a consistent holographic dictionary for CCFT, including UV/IR mixing, OPE associativity, and the role of massive states.

Proposed method

• Reformulate scattering amplitudes using boost eigenstates (conformal primary wavefunctions), which diagonalize the Lorentz group and make the 2D conformal symmetry of the celestial sphere manifest.
• Apply the holographic map to transform standard momentum-space amplitudes into celestial amplitudes, where external states are labeled by scaling dimension and spin.
• Derive Ward identities for asymptotic symmetries (e.g., Virasoro symmetry from superrotations) and show their equivalence to soft theorems in gravity and gauge theory.
• Analyze the analytic structure of celestial amplitudes, including poles and branch cuts, and relate them to OPEs in the celestial CFT via collinear limits.
• Use the double copy construction to relate celestial amplitudes in gauge theory and gravity, extending the double copy to the celestial basis.
• Investigate conformally soft operators and their associated symmetry algebras, including the emergence of global conformal multiplets and conformal dressings.

Experimental results

Research questions

• RQ1 How do soft theorems in gauge and gravity theories arise as Ward identities of asymptotic symmetries, and what is their holographic interpretation?
• RQ2 To what extent do celestial amplitudes in the conformal primary basis realize the properties of 2D CFT correlation functions, and what are the implications for holography?
• RQ3 What is the algebraic structure of conformally soft symmetries, and how do they generate OPEs in the celestial CFT?
• RQ4 How do UV/IR entanglement and singularities in massless celestial amplitudes affect the applicability of standard CFT techniques?
• RQ5 What is the role of massive states in celestial holography, and can a consistent CCFT description be constructed for non-perturbative flat-space backgrounds?

Key findings

• Soft theorems in QED and gravity are shown to be Ward identities for asymptotic symmetries, including supertranslations and superrotations, with the Virasoro symmetry arising from superrotations in gravity.
• Celestial amplitudes expressed in conformal primary wavefunctions exhibit conformal covariance and can be interpreted as correlation functions in a 2D celestial CFT (CCFT), with external states labeled by scaling dimension and spin.
• The collinear limit of celestial amplitudes reproduces OPEs in the CCFT, providing a direct link between kinematic singularities in the bulk and operator product structures in the boundary theory.
• A tower of tree-level symmetry currents—beyond the sub-leading soft theorems—emerges from the conformal primary basis, suggesting a rich algebraic structure underlying scattering amplitudes.
• The double copy construction extends to the celestial basis, mapping celestial amplitudes in gauge theory to those in gravity, preserving conformal covariance and symmetry structures.
• Despite progress, challenges remain: massless celestial amplitudes inherit momentum conservation singularities absent in standard CFTs, and massive states appear non-local on the celestial sphere, complicating the holographic description.