Recall the normal ordering of bosonic operators in QFT is defined by a re-arrangement of operators to put creation operators to the left of annihilation operators in the product. This is designed to avoid accidentally annihilating $|0\rangle$ when looking at an expectation value in relation to the vacuum state.
$$ : \hat{b}^\dagger\hat{b} : \: =\: \hat{b}^\dagger\hat{b} \\ : \hat{b}\hat{b}^\dagger: \: = \: \hat{b}^\dagger\hat{b} $$
In CFTs, I've seen defined the normal ordering of operators as the zeroth basis field of the Laurent expansion of the radial ordering product.
$$\mathcal{R}(a(z)b(w)) = \sum_{n = -n_0}^\infty (z-w)^n P_n(w),$$
and select
$$P_0(w) = \: : a(w)b(w) : $$
Is there an equivalence between these two definitions? What is the CFT analog of not annihilating the vaccuum/ how do we show this definition has that property?