Doubt in using Wick thoerem in using OPE

Question

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.

Answer 1

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.