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While calculating OPE of $T(z)\partial_w\phi(w)$ in Francesco CFT book, I can't understand how Wick theorem is used. The calculation is like following: $$T(z)\partial_w\phi(w)=-2\pi g:\partial_z\phi(z)\partial_z\phi(z):\partial_w\phi(w)$$ $$\color{red}{\sim \frac{\partial_z\phi(z)}{(z-w)^2}}$$ where while doing the contraction we have used $<\partial_z\phi(z)\partial_w\phi(w)>=-\frac{1}{4\pi g}\frac{1}{(z-w)^2}$, $\phi$ is a masless scalar field.

My doubt is how Wick theorem is used to get the red colored equation. Wick theorem relates time ordered product with normal ordered product and the contracted correlators i.e. $$\mathcal{T\{\phi_1\phi_2...\}}=:\phi_1\phi_2...:+:(...):<(...)>$$ Authors do tell that in OPE it will be implicitly assumed the fields appear as correlator $<>$ so the LHS is $<T(z)\partial_w\phi(w)>$. Still I can't understand how Wick theorem is used to simplify the mentioned realtion. There is also a similiar question but there the issue is about modulo regular function.

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    $\begingroup$ Hi @aitfel: Which page? Which eq? Is it eq. (5.81)? $\endgroup$
    – Qmechanic
    Commented Jun 17, 2021 at 19:19

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There are implicitly written radial ordering symbols ${\cal R}$ on the first line of OP's first equation. De Francesco et. al. is using a Wick theorem that translates between between radial ordering ${\cal R}$ and normal ordering $:~:$ $$\begin{align} {\cal R}[\partial_z\phi(z)\partial_w\phi(w)] ~=~&-\frac{1}{4\pi g}\frac{1}{(z-w)^2} \cr ~+~& :\partial_z\phi(z)\partial_w\phi(w):, \end{align}\tag{5.77}$$ cf. my Phys.SE answer here. OP's red expression is caused by 2 single-contractions.

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  • $\begingroup$ After reading ch-6 the issue of radial ordering is clear but it's against the pedagogy that they used radial ordering without mentioning it beforehand in ch-5. $\endgroup$
    – aitfel
    Commented Jun 22, 2021 at 16:40

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