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?