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I'm reading Tong's Lectures on String Theory chapter 4 on conformal field theory. The PDF can be found here. I'm trying to understand his proof of claim 2 in section 4.3.3, but I can't seem to grasp what happens from the first line to the second line in equation 4.26.

If I start doing the derivation myself, the initial expression is $$\partial X(z) :e^{ikX(w)} : = \sum_{n = 0}^\infty \frac{(ik)^n}{n!} \partial X(z) : X(w)^n:$$ It is implicit that $\partial = \partial_z$. Furthermore I know that $$\partial (X(z)X(w)) = \partial X(z) X(w) + X(z) \partial X(w) = \partial X(z) X(w)$$ and that $$\partial (X(z)X(w)) = \partial \left( - \frac{\alpha'}{2} \ln (z - w) + \dots \right) = -\frac{\alpha'}{2} \frac{1}{z - w} + \dots $$ This looks useful but I cannot figure out how to combine them into something useful.

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Hints:

  1. Tong is in eq. (4.26) calculating the OPE $${\cal R}[\partial X(z) :e^{ikX(w)}:]~=~\ldots\tag{4.26}$$ Note that the radial order ${\cal R}$ is implicitly written in Tong's text.

  2. He is using a nested Wick's theorem between radial order ${\cal R}$ and normal order $: :$, cf. e.g. my Phys.SE answer here.

  3. The contraction is $$ C(z,w)~=~{\cal R}[ X(z) X(w)]~-~ :X(z) X(w):~=~-\frac{\alpha^{\prime}}{2}\ln(z-w)\mathbb{1}.$$

  4. The singular terms on the RHS of eq. (4.26) come from all possible single-contractions between $\partial X(z)$ and the vertex operator $:e^{ikX(w)}:$.

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  • $\begingroup$ This cleared it up, thank you very much. $\endgroup$ Oct 5, 2020 at 15:19

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