# Clarification of derivation in Tong's lectures on String theory

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 $$\partial X(z) :e^{ikX(w)} : = \sum_{n = 0}^\infty \frac{(ik)^n}{n!} \partial X(z) : X(w)^n:\tag{4.26}$$ 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.

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 \begin{align}C(z,w)~=~&{\cal R}[ X(z) X(w)]~-~ :X(z) X(w):\cr ~=~&-\frac{\alpha^{\prime}}{2}\ln(z-w)\mathbb{1}.\end{align}\tag{4.22}

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 $$:e^{ikX(w)}:$$.

• This cleared it up, thank you very much. Commented Oct 5, 2020 at 15:19
• 1. On pg 69 Tong writes "We will write a lot of operator equations of the form 4.10 [OPE] and it's important to clarify exactly what they mean: they are always to be understood as statements which holds as operator insertions inside time-ordered correlation functions." Looking at hint 1, I think the R comes from the "time-ordered", but I don't see "correlation functions"? Should hint 1 be: $<R[\partial X (z): e^{ikX(w):}]>=<...>$? 2. Also, does this implicit convention apply only to OPEs or to all equations relating operators? (It's a bit unclear what "they" refers to in the sentence of Tong.) Commented Apr 24 at 18:26
• Tong's statement sounds more like a physical paradigm rather than a mathematical fact. Commented Apr 24 at 18:52