Shadow Celestial Operator Product Expansions

TL;DR

This paper explores the operator product expansion (OPE) for shadow operators in celestial conformal field theories, providing tools to analyze their properties and relations to collinear limits in scattering amplitudes.

Contribution

It introduces a method to compute shadow OPE coefficients and studies their transformation under shadow, advancing understanding of non-local celestial operators.

Findings

Derived transformation rules for shadow OPE coefficients.

Established connections between celestial OPEs and collinear amplitude limits.

Provided a framework for calculating three-point functions involving shadow and celestial primaries.

Abstract

The linearized massless wave equation in four-dimensional asymptotically flat spacetimes is known to admit two families of solutions that transform in highest-weight representations of the Lorentz group ${\rm SL}(2, \mathbb{C})$. The two families are related to each other by a two-dimensional shadow transformation. The scattering states of one family are constructed from standard momentum eigenstates by a Mellin transformation with respect to energy. Their operator product expansion (OPE) is directly related to collinear limits of momentum space amplitudes. The scattering states of the other family are a priori non-local on the celestial sphere and lack a standard notion of OPE. Such states appear naturally in the context of asymptotic symmetries, but their properties as operators remain largely unexplored. Here we initiate a study, to be continued in a forthcoming companion paper, of a definition of an OPE for shadow operators. We present a useful technical ingredient: the transformation of the OPE coefficients associated to collinear limits under a shadow. Our results can be used to find the coefficients of all three-point functions involving any combination of celestial and shadow primaries. An OPE block is used to account for the contribution from a primary together with its global conformal descendants, all of which contribute when deriving the shadowed OPE coefficients. Applications involving $U(1)$ currents and stress tensors as well as a chiral current algebra of soft gluons are discussed.