I'm trying to work out in detail how the Wick theorem is used for constructing OPEs in CFT. One of the first things which bothers me is the difference in definitions of normal ordered product and contraction in usual QFT and in CFT. Namely, wiki and Peskin tell us that: \begin{equation} X(z) Y(w) = \langle X(z) Y(w) \rangle+ \mathcal{N}\{X(z) Y(w)\} \quad. \end{equation} On the contrary, di Francesco tells us: \begin{equation}\begin{gathered} X(z) Y(w) = \langle X(z) Y(w) \rangle + \operatorname{Reg}\{X(z) Y(w) \} \\ = \underbrace{\sum \limits_{n=1}^{N} \dfrac{\{XY\}_n(w)}{(z-w)^n}}_{\text{contraction}} + \underbrace{\sum \limits_{n=-\infty}^0 \dfrac{\{XY\}_n(w)}{(z-w)^n}}_{\text{regular terms}} \quad. \end{gathered}\end{equation}

The Wick theorem for standard definitions reads as: \begin{equation}\begin{alignedat}{9} ABCD\ldots = \mathcal{N}\{ABCD\ldots\} &+ \sum (\text{single contraction})\times (\text{normal ordered terms})\\ &+\sum (\text{double contraction})\times (\text{normal ordered terms})\\ &+\ldots\\ & +\sum (\text{products of all possible contractions}) \end{alignedat}\end{equation}

I would like to adopt Wick's theorem for the CFT definitions of contraction and normal ordering. Will it be correct to simply replace everywhere normal ordering with the 'regular terms' (which are obtained via the Taylor series expansion)? Then, the OPE will take form: \begin{equation}\begin{alignedat}{9} ABCD\ldots = \operatorname{Reg}\{ABCD\ldots\} &+ \sum (\text{single contraction})\times (\text{regular terms})\\ &+\sum (\text{double contraction})\times (\text{regular terms})\\ &+\ldots\\ & +\sum (\text{products of all possible contractions}) \end{alignedat}\end{equation}

The whole point of this effort is to understand how to compute the non-singular terms in the OPEs (I believe, for the most singular terms one can simply use the usual Wick theorem).



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