# OPE of normal ordered operators

In what follows I use $\mathcal{N}\{\ldots\}$ for normal ordering, $\langle\ldots\rangle$ for contraction and $\operatorname{Reg}\{\ldots\}$ for the complete sequence of regular terms which is obtained via the Taylor series expansion.

In my ambitious studies of CFT, I need to calculate the OPE of the operator $\mathcal{N}\{(\exp(\sqrt{2} \Phi_1(z)))(\partial \Phi_2(z))\}$ with itself. The particular form of the operator is not very important for now. The question is more about the overall logic. More generally, I would like to know how to calculate the OPE of the following expression: $$\mathcal{N}\{AB\}\mathcal{N}\{CD\} \quad.$$

I want to use two basic relations ((6.128) and (6.205) in di Francesco): \begin{alignat}{6} X(z) Y(w) &= \langle X(z)Y(w)\rangle + \operatorname{Reg}\{X(z)Y(w)\} \quad&&,\\ \langle X(z),\, \mathcal{N}\{Y(w)Z(w) \} \rangle &= \langle X(z), Y(w) \rangle \,\mathcal{N}\{Z(w) \} + \langle X(z), Z(w) \rangle \,\mathcal{N}\{Y(w) \} \quad&&. \end{alignat}

I am using the first of these two, and then the second, twice: $$\begin{gathered} \mathcal{N}\{AB\}\mathcal{N}\{CD\} = \langle \mathcal{N}\{AB\},\,\mathcal{N}\{CD\} \rangle + \operatorname{Reg}\{\mathcal{N}\{AB\},\,\mathcal{N}\{CD\}\} \\ =\langle \mathcal{N}\{AB\},\,\mathcal{N}\{C\} \rangle \mathcal{N}\{D\} +\langle \mathcal{N}\{AB\},\,\mathcal{N}\{D\} \rangle \mathcal{N}\{C\} + \operatorname{Reg}\{\mathcal{N}\{AB\},\,\mathcal{N}\{CD\}\} \\ = \langle \mathcal{N}\{A\},\,\mathcal{N}\{C\} \rangle \mathcal{N}\{B\} \mathcal{N}\{D\} + \langle \mathcal{N}\{B\},\,\mathcal{N}\{C\} \rangle \mathcal{N}\{A\} \mathcal{N}\{D\} \\ + \langle \mathcal{N}\{A\},\,\mathcal{N}\{D\} \rangle \mathcal{N}\{B\} \mathcal{N}\{C\} + \langle \mathcal{N}\{B\},\,\mathcal{N}\{D\} \rangle \mathcal{N}\{A\} \mathcal{N}\{C\} + \operatorname{Reg}\{\mathcal{N}\{AB\},\,\mathcal{N}\{CD\}\} \end{gathered}$$

Questions:

1. Is everything correct by now? I'm surprised that I don't have terms like $\langle \mathcal{N}\{A\},\,\mathcal{N}\{B\} \rangle \langle \mathcal{N}\{C\},\,\mathcal{N}\{D\} \rangle$ or $\langle \mathcal{N}\{A\},\,\mathcal{N}\{C\} \rangle \langle \mathcal{N}\{B\},\,\mathcal{N}\{D\} \rangle$ .

2. Do I simply replace $\mathcal{N}\{A(z)\}$ with its Taylor expansion around $z=w$?

3. What should I do with the term $\operatorname{Reg}\{\mathcal{N}\{AB\},\,\mathcal{N}\{CD\}\}$?