Goldilocks Modes and the Three Scattering Bases

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 ∆. We focus on the subleading soft theorems at ∆ = 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 orderof-limits issues implicit in the celestial formalism. We then generalize our construction to the tower of w 1+∞ 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.

Notes on Resonances and Unitarity from Celestial Amplitudes

“…In other words, the original Bose symmetry of the S-matrix implies relationships between the celestial amplitudes in their different physical regions, and those relations are necessary for the celestial amplitude to literally "be a CFT four point function of bosons" on the celestial circle. This is consistent with the extrapolate dictionary [46,47].…”

Celestial Geometry

“…We generalize this statement to n-point functions. This exchange symmetry of the CCFT data follows directly from the extrapolate dictionary [37,38], whereby the celestial operators can be expressed as limits of local bulk operators smeared along null generators of the conformal boundary.…”

“…To be more precise, if the former is invariant under exchanging all the labels of the j-th and k-th particle, then so is the latter. It is an exact statement, not relying on perturbation theory, that is manifest in the extrapolate dictionary [37,38], where it follows from the way the correlators are constructed from the same bulk field pushed to the conformal boundary. A simple example was given in (1.10).…”

A Comment on Loop Corrections to the Celestial Stress Tensor

“…These are observed to decouple in scattering amplitudes for the known soft theorems. For recent discussions on the tower of symmetry currents phrased in the celestial diamond language see [26,38].…”