# Do descendant fields always die off quicker than the slowest primary?

Is the following true in a conformal field theory (CFT)?

Suppose I have some observable $\mathcal O$ which I know nothing about (e.g. I don't know if it is a primary), except that I know the leading term in its two-point correlation function: $$\langle \mathcal O(x) \mathcal O(0) \rangle \sim 1/x^{2\alpha} \; (+\textrm{subleading} \cdots)$$ Is it then true that there must exist a primary field $\phi$ with scaling dimension $\Delta \leq \alpha$ ?

Equivalently, but worded differently: is it true that all descendant fields ''fall off quicker'' than the ''slowest decaying'' primary field?

By translation and dilatation invariance, the two-point function of two scalar fields with conformal dimensions $\Delta_1$ and $\Delta_2$ is $$\langle \mathcal O_1(x_1) \mathcal O_2(x_2) \rangle \propto |x_1-x_2|^{-\Delta_1-\Delta_2}$$ This holds whenever the two fields are eigenvalues of the dilatation operator, which is true in particular for descendents and primaries. (If the fields are primary you have additional conformal Ward identities that say that the two-point function vanishes unless $\Delta_1=\Delta_2$.)