  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.
