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I) Recall first the $\phi\phi$-Operator Product Expansion (OPE): $$\tag{A} {\cal R}\left\{\phi(z,\bar{z})\phi(w,\bar{w})\right\} ~-~: \phi(z,\bar{z})\phi(w,\bar{w}): ~=~C(z,\bar{z};w,\bar{w}) ~{\bf 1}, $$ where the contraction is assumed to be a $c$-number: $$\tag{B} C(z,\bar{z};w,\bar{w})~=~ \langle 0 | {\cal ...


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Here $\begin{equation} \langle ik\phi(x)ik\phi(x)\rangle = \frac{\alpha'k^2}{\pi} \text{ln}(a/2R), \end{equation}$ where $a$ is an UV cutoff. Now we can write (as all the $\phi$'s are located at $x$ i.e. Radial ordered $\{\phi^n(x)\}=\phi^{n}(x)~$) $\begin{align} \{ik\phi\}^{n}(x) ~&=~ :\{ik\phi\}^n(x): +\sum_{\text{all contractions}} \\ ...


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There is a way to understand Wick's Theorem as a particular instance of a very general connection between differentiation and combinatorics. At the bottom of it is just Leibniz's product rule. Given variables $x_1,\ldots,x_d$, the basic identity is: $$ \frac{\partial}{\partial x_{i_1}}\ldots\frac{\partial}{\partial x_{i_n}}\ x_{j_1}\ldots x_{j_n}= ...



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