# Wick contraction with exponential operator in 2D CFT

Consider a 2d CFT in radial quantization. Let $$A(z)$$ be some primary field and $$Q$$ a charge that can be written as $$Q = \oint dz \, z^{(h-1)}J(z)\tag{1}$$ for some holomorphic current $$J(z)$$. I will denote Wick contraction by an overline, as I don't know how to typeset it in mathjax. The question is: is it true that inside expectation values (which implicitly include radial ordering) $$$$\langle A(z) \,:e^Q: \rangle \stackrel{?}{=} \langle \overline{A(z) Q}\, :e^Q:\rangle\tag{2}$$$$ where $$\overline{A(z)\, Q} = \oint dw\, w^{(h-1)}\overline{ A(z) J(w)},\tag{3}$$ and $$\overline{ A(z) J(w)}$$ is the singular part of the OPE between $$A$$ and $$J$$.

Here are some preliminary thoughts:

Suppose $$h=1$$ for simplicity. Expand the (normal-ordered) exponential $$:e^Q:\, \equiv \sum_{n=0}^\infty \frac{1}{n!} \oint_{\gamma_1} dz_1 \cdots \oint_{\gamma_n} dz_1 :J(z_1)\cdots J(z_n):\tag{4}$$ The contours $$\gamma_i$$ are non-intersecting and radially ordered. Define $$A_\varepsilon = \oint_\Gamma dz \,\varepsilon(z) A(z)\tag{5}$$ with $$\Gamma$$ larger than all $$\gamma_i$$. Then do the Wick contractions $$\langle A_\varepsilon :e^Q: \rangle = \sum_{n=0}^\infty \frac{1}{n!} \sum_{i=1}^n \Big\langle \oint_{\gamma_1} dz_1 J(z_1) \cdots \oint_{\gamma_i} dz_i \oint_{z_i} dz\, \varepsilon(z) \overline{A(z) J(z_i)} \cdots \oint_{\gamma_n} dz_n J(z_n)\Big\rangle\tag{6}$$

Suppose $$\langle A(z)\rangle =0$$, so the sum starts at $$n=1$$. The singular part of the OPE in general is written as $$\overline{A(z) J(z_i)} = \sum_k \frac{(AJ)_k(z_i)}{(z-z_i)^k}\tag{7}$$ Plugging this into the above formula, we see that the $$z$$-integral only has a pole in $$z = z_i$$. Does that imply we can deform the $$z$$-contour back to $$\Gamma$$? If so, can we also take out the contraction $$\oint_{\Gamma} dz\,\varepsilon(z)\oint_{\gamma_i} dz_i \overline{A(z) J(z_i)}$$ as a common factor? My confusion lies in whether radial ordering puts $$\oint_{\gamma_i} dz_i \overline{A(z) J(z_i)} \equiv B(z)$$ to the left of all other $$J(z_k)$$ insertions or not.

Eq. (2) holds if the contraction (3) commutes with $$Q$$. See e.g. this related Phys.SE post.