# Tag Info

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 ... 0 Here $$\langle ik\phi(x)ik\phi(x)\rangle = \frac{\alpha'k^2}{\pi} \text{ln}(a/2R),$$ 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}} \\ ... 0 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}= ...