Celestial Conformal Primaries in Effective Field Theories

TL;DR

This paper explores the properties of conformal primary operators in effective field theories, establishing conditions on their scaling dimensions for completeness and consistency, and highlighting their potential in holographic descriptions of scattering amplitudes.

Contribution

It demonstrates that conformal primaries with real scaling dimensions can be constructed within effective field theories and identifies specific dimension restrictions under analyticity assumptions.

Findings

Conformal primaries with eltaorm a basis for massless particles.

Restrictions on eltansure completeness and CPT consistency.

Effective field theories can define primaries without UV knowledge.

Abstract

Scattering amplitudes in $d+2$ dimensions can be recast as correlators of conformal primary operators in a putative holographic CFT$_d$ by working in a basis of boost eigenstates instead of momentum eigenstates. It has been shown previously that conformal primary operators with $\Delta \in \frac{d}{2} + i {\mathbb R}$ form a basis for massless one-particle representations. In this paper, we consider more general conformal primary operators with $\Delta \in {\mathbb C}$ and show that completeness, normalizability, and consistency with CPT implies that we must restrict the scaling dimensions to either $\Delta \in \frac{d}{2} + i {\mathbb R}$ or $\Delta \in {\mathbb R}$. Unlike those with $\Delta \in \frac{d}{2} + i {\mathbb R}$, the conformal primaries with $\Delta \in {\mathbb R}$ can be constructed without knowledge of the UV and can therefore be defined in effective field theories. With additional analyticity assumptions, we can restrict $\Delta \in 2 - {\mathbb Z}_{\geq0}$ or $\Delta \in \frac{1}{2}-{\mathbb Z}_{\geq0}$ for bosonic or fermionic operators, respectively.