Computing OPEs of linear dilaton CFT

[snypehype46]

Homework Statement

Determine the transformation of $X$ under conformal transformations

Relevant Equations

None

I'm trying to do the following question from David Tong's problem sheets on string theory:

> A theory of a free scalar field has OPE $$\partial X(z)\partial X(w) = \frac{\alpha'}{2}\frac{1}{(z-w)^2}+...$$. Consider the putative candidate for the stress energy tensor $$T(z) = \frac{1}{\alpha '}: \partial X (z) \partial X (z) : -Q \partial^2 X(z)$$. Use $TX$ OPE to determine the transformation of $X$ under conformal transformations $\delta z = \epsilon(z)$

Now to determine $T(z)X(w)$, I thought I would contract the normally ordered field with $X(w)$. So to get:

$$2\langle \partial X (z) \partial X (w) \rangle -Q^2 \partial^3 X(z)$$

Is that the correct way of proceeding? I'm not sure if I need to contract the not-normally ordered fields $\partial^2 X(z)$ with $X(w)$ as well?

Also, I don't quite understand how would I continue with this question after I worked out the OPE of $TX$.

 

[BigJDubs]

How do I use this to determine the transformation of $X$ under conformal transformations?Thanks in advance!