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I'm having a little trouble with correlation functions wick theorem and ordering in the context of OPE and CFT, for string theory.

(1) My first question, the propagator is: $$<X(z) X(w)> = \frac{\alpha}{2} \ln(z-w).$$

In the context of primary operators it's easy to see that $X$ it's not a good conformal field. But $\partial X$ yes, so I need to get: $$<\partial X(z) \partial X(w) >$$ which I can get from the propagator of $X$ by taking two derivatives, if I take the first one:

$$\partial < X(z) X(w) > = <\partial X(z) X(w) > + <X(z) \partial X(w)>$$

But this seem to get the wrong result. So I guess that the derivative is:

$$\partial <X(z) X(w) > = <\partial X(z) X(w) >$$

If I want to take the second derivative the result seems to be:

$$\partial <\partial X(z) X(w)> = <\partial X(z) \partial X(w). $$

But I don't understand why I should want that derivative and not:

$$\partial <\partial X(z) X(w)> = <\partial^2 X(z) X(w)>.$$

(2) Regarding normal ordering and Wick's theorem, I have the following definition of normal ordering: $$T = \frac{-1}{\alpha} :\partial X \partial X: = \frac{-1}{\alpha} \lim_{z \to w} (\partial X(z) \partial X(w) - <\partial X(z)\partial X(w)>)$$

And the condition: $$<T> = 0$$

But what happens if I want to compute this:

$$T(z) T(w) = \frac{1}{\alpha^2} : \partial X(z) \partial X(z) : :\partial X(w) \partial X(w): $$

What's the meaning of product of normal ordered operators?

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You want to take the derivative with respect to both z and w.

Take $${X^\mu }\left( z \right){X^\nu }\left( w \right) \sim - {1 \over 4}{\eta ^{\mu \nu }}\ln \left( {z - w} \right)$$

and use the following derivative $${{{\partial ^2}} \over {\partial w\partial z}}\left[ {{X^\mu }\left( z \right){X^\nu }\left( w \right)} \right] = {\partial \over {\partial w}}\left[ {{X^\nu }\left( w \right){\partial \over {\partial z}}{X^\mu }\left( z \right)} \right] = {\partial \over {\partial z}}{X^\mu }\left( z \right){\partial \over {\partial w}}{X^\nu }\left( w \right)$$

If we do the same thing to the OPE... $${{{\partial ^2}} \over {\partial w\partial z}}\left[ { - {1 \over 4}{\eta ^{\mu \nu }}\ln \left( {z - w} \right)} \right] = {\partial \over {\partial w}}\left[ { - {1 \over 4}{\eta ^{\mu \nu }}{1 \over {z - w}}} \right] = - {1 \over 4}{\eta ^{\mu \nu }}{1 \over {{{\left( {z - w} \right)}^2}}}$$

Which gives us the correct result $$\partial {X^\mu }\left( z \right)\partial {X^\nu }\left( w \right) \sim - {1 \over 4}{\eta ^{\mu \nu }}{1 \over {{{\left( {z - w} \right)}^2}}}$$

This result can be verified using the relevant mode expansions. See Ex. 3.1 BBS ST & MT. http://www.nucleares.unam.mx/~alberto/apuntes/bbs.pdf

As far as the product of the normal ordered operators goes... EDIT: See user2309840's answer, better than what I had written.

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There is nothing to add to the first part of Jake Lebovic's answer.

With regard to the second part of the question -- how to compute the OPE of two stress tensors -- one uses Wick's theorem. Normal ordering means one does not contract together the individual fields making up the normal ordered operator, in this case the two $\partial X$'s making up $T(z)$. Thus in the calculation below, one sees there are only two ways of contracting everything together, but four ways of contracting only two of the $\partial X$ operators: $$ T(z) T(w) = \frac{1}{\alpha'^2} {:} \partial X^\mu \partial X_\mu (z) {:}\, {:} \partial X^\nu \partial X_\nu(w) {:} $$ $$ \sim \frac{2}{\alpha'^2} ( - \partial_z \partial_w \eta^\mu_\nu \frac{\alpha'}{2} \log |z-w|^2)(-\partial_z \partial_w \eta^\nu_\mu \frac{\alpha'}{2} \log |z-w|^2 ) $$ $$ + \frac{4}{\alpha'} (- \partial_z \partial_w \eta^\mu_\nu \frac{\alpha'}{2} \log |z-w|^2) {:} \partial X^\nu (z) \partial X_\mu(w) {:} $$ $$ \sim \frac{\eta^\mu_\mu}{2} \frac{1}{(z-w)^4} - \frac{2}{\alpha'} \frac{1}{(z-w)^2} {:} \partial X^\mu(z) \partial X_\mu(w){:} + \ldots $$ $$ \sim \frac{D}{2} \frac{1}{(z-w)^4} + \frac{2}{(z-w)^2} T(w) + \frac{1}{(z-w)} \partial T(w) + \ldots $$ The last line here is the canonical form of the OPE of two stress tensors in a 2d conformal field theory. The $D$ in the first term is the central charge, often denoted $c$, but here equal to the number of $X$ fields. The two in the second term is the scaling dimension of the stress tensor.

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OP last question essentially reads (v1):

What's the meaning of the product of normal-ordered operators $$: \partial X(z) \partial X(z) : :\partial X(w) \partial X(w): ~?$$

Strictly speaking, it is a radially ordered product of normal-ordered operators $${\cal R} \left[ : \partial X(z) \partial X(z) : :\partial X(w) \partial X(w): \right].$$

However, the radial ordering ${\cal R}$ is usually implicitly implied and not written explicitly in CFT texts. Nevertheless it is important. Without radial ordering (or other kinds of ordering, such as, e.g., time ordering, normal ordering, etc.) an operator product is often ill-defined. The contractions (and double contractions) that's performed in this calculation (cf. e.g. the answer by user2309840) is dictated by the rearrangements from radial ordering ${\cal R}$ to normal ordering in accordance with a nested version of the Wick's theorem. This is explained further in this Phys.SE post.

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