Operator product expansion in CFT

Question

I'm on Polchinski's p39.

1. Can someone please tell me the steps in the equivalence below? $$\exp\left[\frac{\alpha'}4\int d^2z_4 d^2z_5\ln|z_5-z_4|^2\frac{\delta}{\delta X^\mu(z_4,\bar z_4)}\frac{\delta}{\delta X_\mu(z_5,\bar z_5)}\right] $$ $$\times X^{\mu_1}(z_1,\bar z_1)X^{\mu_2}(z_2,\bar z_2)X^{\mu_3}(z_3,\bar z_3)$$ $$=X^{\mu_1}(z_1,\bar z_1)X^{\mu_2}(z_2,\bar z_2)X^{\mu_3}(z_3,\bar z_3)+\frac{\alpha'}{2}\eta^{\mu_1\mu_2}\ln|z_2-z_1|^2X^{\mu_3}(z_3,\bar z_3)+(2\text{ permutations})~?$$

2. Why do the variations in the exponential act on $X^{\mu_1}(z_1,\bar z_1)X^{\mu_2}(z_2,\bar z_2)X^{\mu_3}(z_3,\bar z_3)$?

3. Exactly how does the integration in the exponential go and how does the exponentiation give RHS?

4. The permutations look reasonable but the first term on RHS comes from when $\exp[...]=1$?

Answer 1

First of all, note that the radial operator ordering ${\cal R}$ is implicitly implied in many textbooks of CFT (e.g. Ref. 1). For instance, eq. (2.2.7) on p. 39 in Ref. 1 is discussing Wick's theorem between two operator ordering prescriptions. In this case between normal ordering $:~:$ and radial ordering ${\cal R}$. See also e.g. this Phys.SE post. The basic 2-point relation of Wick's theorem is

$$ \tag{1} : \hat{X}_i \hat{X}_j : ~=~ {\cal R}(\hat{X}_i \hat{X}_j)+ C_{ij}, $$

where the so-called contraction $C_{ij}$ is assumed to be a $c$-number. [More precisely: $C_{ij}$ is assumed to be a central element.] Here the indices $i,j,k,\ldots$ are a shorthand for all possible discrete and continuous labels of the operators $\hat{X}_i ,\hat{X}_j,\hat{X}_k, \ldots $, cf. DeWitt condensed notation.

OP's sought-for 3-point relation of Wick's theorem is$^1$

$$ \tag{2} : \hat{X}_i \hat{X}_j\hat{X}_k : ~=~ {\cal R}(\hat{X}_i \hat{X}_j\hat{X}_k) +C_{ij} {\cal R}(\hat{X}_k) + C_{ik}{\cal R}(\hat{X}_j) + {\cal R}(\hat{X}_i)C_{jk}. $$

Eqs. (1) and (2) can be formally generalized to eq. (2.2.7) of Ref. 1

$$\tag{3} :{\cal F}: ~=~ \exp \left(\frac{1}{2} \sum_{i,j} C_{ij}\frac{\partial}{\partial \hat{X}_i} \frac{\partial}{\partial \hat{X}_j} \right) {\cal R}({\cal F}), $$

where the operator ${\cal F}$ is a function of the operators $\hat{X}_i$. Note that the operators are treated as commutative objects under the two ordering symbols $:~:$ and ${\cal R}$. Eq. (3) is a convenient formal shorthand/mnemonic of the various $n$-point relations of Wick's theorem.

References:

1. J. Polchinski, String Theory, Vol. 1; p.39.

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$^1$ In this answer we have assumed for simplicity that all operators $\hat{X}_i$ are Grassmann-even. If some of the operators $\hat{X}_i$ are Grassmann-odd, there will be additional sign factors in eqs. (2) and (3).