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Celestial IR divergences and the effective action of supertranslation modes

Kevin Nguyen, Jakob Salzer

TL;DR

The paper demonstrates that the celestial two-point function of the supertranslation Goldstone mode arises from an infrared effective action derived from the Compere–Dehouck action near spatial infinity. By isolating the IR-divergent sector of flat-space gravity, it shows that the on-shell IR action is S_IR = - (log Lambda)/(32 pi G) ∫ d^2x (□ C_plane)^2, whose Green's function reproduces the celestial correlator ⟨C plane(x) C plane(y)⟩ ∝ |x−y|^2 log|x−y|^2. It then reformulates the soft graviton factor as a celestial CFT path integral with this action, yielding the known A_soft factor through a Gaussian integral over vertex operators. The work connects IR aspects of gravity to a two-dimensional celestial CFT framework, clarifying how supertranslation memory, soft theorems, and boundary symmetries interrelate, and discusses implications for dS/CFT-like interpretations and future extensions to additional asymptotic symmetries. Overall, it provides a concrete, action-based route from IR GR near i0 to celestial CFT correlators and soft factors, highlighting the central role of the infrared counterterm in shaping the celestial structure of gravity.

Abstract

Infrared divergences in perturbative gravitational scattering amplitudes have been recently argued to be governed by the two-point function of the supertranslation Goldstone mode on the celestial sphere. We show that the form of this celestial two-point function simply derives from an effective action that also controls infrared divergences in the symplectic structure of General Relativity with asymptotically flat boundary conditions. This effective action finds its natural place in a path integral formulation of a celestial conformal field theory, as we illustrate by re-deriving the infrared soft factors in terms of celestial correlators. Our analysis relies on a well-posed action principle close to spatial infinity introduced by Compère and Dehouck.

Celestial IR divergences and the effective action of supertranslation modes

TL;DR

The paper demonstrates that the celestial two-point function of the supertranslation Goldstone mode arises from an infrared effective action derived from the Compere–Dehouck action near spatial infinity. By isolating the IR-divergent sector of flat-space gravity, it shows that the on-shell IR action is S_IR = - (log Lambda)/(32 pi G) ∫ d^2x (□ C_plane)^2, whose Green's function reproduces the celestial correlator ⟨C plane(x) C plane(y)⟩ ∝ |x−y|^2 log|x−y|^2. It then reformulates the soft graviton factor as a celestial CFT path integral with this action, yielding the known A_soft factor through a Gaussian integral over vertex operators. The work connects IR aspects of gravity to a two-dimensional celestial CFT framework, clarifying how supertranslation memory, soft theorems, and boundary symmetries interrelate, and discusses implications for dS/CFT-like interpretations and future extensions to additional asymptotic symmetries. Overall, it provides a concrete, action-based route from IR GR near i0 to celestial CFT correlators and soft factors, highlighting the central role of the infrared counterterm in shaping the celestial structure of gravity.

Abstract

Infrared divergences in perturbative gravitational scattering amplitudes have been recently argued to be governed by the two-point function of the supertranslation Goldstone mode on the celestial sphere. We show that the form of this celestial two-point function simply derives from an effective action that also controls infrared divergences in the symplectic structure of General Relativity with asymptotically flat boundary conditions. This effective action finds its natural place in a path integral formulation of a celestial conformal field theory, as we illustrate by re-deriving the infrared soft factors in terms of celestial correlators. Our analysis relies on a well-posed action principle close to spatial infinity introduced by Compère and Dehouck.

Paper Structure

This paper contains 11 sections, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conventions.
  3. Gravity at spatial infinity
  4. Action principle.
  5. Supertranslations and Goldstone modes.
  6. Matching to null infinity
  7. Infrared effective action on the celestial sphere
  8. Path integral representation of the soft factor
  9. Discussion
  10. Variational principle and counterterms
  11. Evaluation of the on-shell action

Figures (1)

  • Figure 1: Schematic representation of the geometric setup required to define the action principle near spatial infinity $i^0$. The de Sitter hyperboloids $\mathcal{H}_\pm$ are surfaces of constant $\rho=\Lambda_\pm$. The action principle is defined for the yellow region of spacetime bounded by $\mathcal{H}_\pm$ and two Cauchy surfaces $\Sigma_\pm$. The intersections of $\mathcal{H}_+$ with $\Sigma_\pm$ are denoted $\mathcal{S}_\pm$ and possess the geometry of a two-sphere. In the limit $\Lambda_+ \to \infty$, these two-spheres are identified with $\mathscr{I}^+_-$ and $\mathscr{I}^-_+$, respectively.