Celestial Conformal Primaries in Effective Field Theories

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 $\mathrm{\Delta }\in \frac{d}{2}+i\mathbb{R}$ form a basis for massless one-particle representations. In this paper, we consider more general conformal primary operators with $\mathrm{\Delta }\in \mathbb{C}$ and show that completeness, normalizability, and consistency with CPT implies that we must restrict the scaling dimensions to either $\mathrm{\Delta }\in \frac{d}{2}+i\mathbb{R}$ or $\mathrm{\Delta }\in \mathbb{R}$. Unlike those with $\mathrm{\Delta }\in \frac{d}{2}+i\mathbb{R}$, the conformal primaries with $\mathrm{\Delta }\in \mathbb{R}$ can be constructed without knowledge of the UV and can therefore be defined in effective field theories. With additional analyticity assumptions, we can restrict $\mathrm{\Delta }\in 2-{\mathbb{Z}}_{\ge 0}$ or $\mathrm{\Delta }\in \frac{1}{2}-{\mathbb{Z}}_{\ge 0}$ for bosonic or fermionic operators, respectively.