  We consider the operator spectrum of a three-dimensional ${\cal N} = 2$
superconformal field theory with moduli spaces of one complex dimension, such
as the fixed point theory with three chiral superfields $X,Y,Z$ and a
superpotential $W = XYZ$. By using the existence of an effective theory on each
branch of moduli space, we calculate the anomalous dimensions of certain
low-lying operators carrying large $R$-charge $J$. While the lowest primary
operator is a BPS scalar primary, the second-lowest scalar primary is in a
semi-short representation, with dimension exactly $J+1$, a fact that cannot be
seen directly from the $XYZ$ Lagrangian. The third-lowest scalar primary lies
in a long multiplet with dimension $J+2 - c_{-3} \, J^{-3} + O(J^{-4})$, where
$c_{-3}$ is an unknown positive coefficient. The coefficient $c_{-3}$ is
proportional to the leading superconformal interaction term in the effective
theory on moduli space. The positivity of $c_{-3}$ does not follow from
supersymmetry, but rather from unitarity of moduli scattering and the absence
of superluminal signal propagation in the effective dynamics of the complex
modulus. We also prove a general lemma, that scalar semi-short representations
form a module over the chiral ring in a natural way, by ordinary multiplication
of local operators. Combined with the existence of scalar semi-short states at
large $J$, this proves the existence of scalar semi-short states at all values
of $J$. Thus the combination of ${\cal N}=2$ superconformal symmetry with the
large-$J$ expansion is more powerful than the sum of its parts.
