Goldilocks Modes and the Three Scattering Bases

TL;DR

This work develops a unified celestial holography framework for massless scattering by carefully analyzing three bases—position (u), momentum (ω), and celestial (Δ)—and clarifying how soft theorems, memory effects, and asymptotic symmetries reorganize within the celestial CFT. It introduces Goldilocks (subleading) conformally soft modes and their memory pairings, resolves order-of-limits tensions, and extends the structure to the infinite w_{1+∞} tower, revealing a natural multipole interpretation of infrared data. The paper demonstrates how to renormalize symplectic forms to define finite charges for overleading modes, links subleading soft theorems to celestial currents, and shows that the Δ-basis is optimally suited to identify and classify infrared symmetries and their associated memory effects. Overall, it provides a coherent, transform-consistent picture where celestial CFT organizes infrared physics of gauge theory and gravity, including subleading and sub-subleading sectors, through a tower of conformally soft generators and their memories. These insights have implications for understanding the symmetry structure of quantum gravity in asymptotically flat spacetimes and for practical computations of soft charges in CCFT.

Abstract

We consider massless scattering from the point of view of the position, momentum, and celestial bases. In these three languages different properties of physical processes become manifest or obscured. Within the soft sector, they highlight distinct aspects of the infrared triangle: quantum field theory soft theorems arise in the limit of vanishing energy $ω$, memory effects are described via shifts of fields at the boundary along the null time coordinate $u$, and celestial symmetry algebras are realized via currents that appear at special values of the conformal dimension $Δ$. We focus on the subleading soft theorems at $Δ=1-s$ for gauge theory $(s=1)$ and gravity $(s=2)$ and explore how to translate the infrared triangle to the celestial basis. We resolve an existing tension between proposed overleading gauge transformations as examined in the position basis and the `Goldstone-like' modes where we expect celestial symmetry generators to appear. In the process we elucidate various order-of-limits issues implicit in the celestial formalism. We then generalize our construction to the tower of $w_{1+\infty}$ generators in celestial CFT, which probe further subleading-in-$ω$ soft behavior and are related to subleading-in-$r$ vacuum transitions that measure higher multipole moments of scatterers. In the end we see that the celestial basis is `just right' for identifying the symmetry structure.

Figures (5)

• Figure 1: When the saddle-point approximation is valid, the boundary operators correspond to operators inserted on $\mathscr{I}^\pm$. For the momentum and celestial bases they are smeared along the null generators, as illustrated by the antipodally placed blue (past) and red (future) contours. The celestial basis views scattering in terms of constant-Rindler time cuts Pasterski:2022jzc (dotted), in contrast to the constant-$u$ cuts considered in Laddha:2020kvp (dashed).
• Figure 2: Celestial diamond illustrating the nested submodule structure that appears at special values of the conformal dimension such that $h= \frac{1-k}{2}, \bar{h}=\frac{1-\bar{k}}{2}$ for $k,\bar{k}\in\mathbb{Z}_>$.
• Figure 3: Mapping between contact and non-contact terms under descendants (blue) and shadows (green). Both operations commute with the large-$r$ expansion. The dashed maps between terms of the same type will be less relevant for what follows.
• Figure 4: Celestial 'Diamonds' for the Subleading Soft Photon.
• Figure 5: Celestial 'Diamonds' for the Sub-subleading Soft Graviton.