Conformal Partial Wave Expansion of Celestial Correlators

TL;DR

This paper demonstrates that celestial correlators in Minkowski space can be expanded into conformal partial waves with meromorphic spectral density, providing a new perspective on holographic correlation functions in flat spacetime.

Contribution

It introduces a conformal partial wave expansion for celestial correlators in Minkowski space, extending harmonic analysis techniques from AdS to flat spacetime.

Findings

Celestial correlators admit a conformal partial wave expansion with meromorphic spectral density.

Four-point contact and exchange diagrams can be expanded into conformal blocks, with simple forms for massless scalars.

Non-perturbative spectral density positivity relates to Lorentz unitarity.

Abstract

A novel definition of holographic correlation functions on the celestial sphere of Minkowski space was recently introduced in arXiv:2301.01810 as the extrapolation of bulk time-ordered correlation functions to the celestial sphere. In this work, focusing on theories of scalar fields in $(d+2)$-dimensional Minkowski space, we show that in perturbation theory such celestial correlation functions admit a conformal partial wave expansion with meromorphic spectral density, and hence also an expansion into conformal blocks. This is achieved in the hyperbolic slicing of Minkowski space by extending the harmonic function (``spectral") decomposition of AdS bulk-to-bulk propagators to the Minkowski Feynman propagator. We study the conformal partial wave expansion of celestial correlators for four-point contact and tree-level exchange diagrams, and extract the contributions to their conformal block expansions in the direct channel. When all scalar fields are massless, the tree-level exchange diagram takes a remarkably simple form and is given by a finite sum of conformal blocks (and, for $d=2$, their derivatives as well). We also discuss the conformal partial wave expansion at the non-perturbative level, where Lorentz unitarity manifests as positivity of the spectral density.