Trying to derive Polchinski's equation $(2.2.11)$

I'm having trouble to derive Polchinski's equation $$(2.2.11)$$ which is an example of product of normal ordered product of $$\partial X^\mu$$. Precisely, Polchinski defines the product of normal ordered operators $$:\mathcal{F}:$$ and $$:\mathcal{G}:$$ (which are functionals of the fields $$X$$) in the following way,

$$:\mathcal{F}: :\mathcal{G}: \ =\ :\mathcal{F}\mathcal{G}: + \ \text{cross contractions}, \tag1$$

cf. eq. $$(2.2.9)$$, or as the formula given by $$(2.2.10)$$

$$:\mathcal{F}::\mathcal{G}: = \exp \left( -\frac{\alpha'}{2} \int d^2z_1 d^2z_2 \ln |z_1 - z_2|^2 \frac{\delta}{\delta X^\mu_F(z_1)} \frac{\delta}{\delta X_{G\mu}(z_2)} \right):\mathcal{F}\mathcal{G}: \tag2$$

Polchinski's first example is with $$:\partial X_\mu \partial X^\mu(z)::\partial' X_\nu \partial' X^\nu(z'):$$. If I use the "imprecise" first definition, I get the following:

We have that $$\mathcal{F} = \partial X^0 \partial X_0 (z) + \cdots + \partial X^{D-1} \partial X_{D-1}(z)$$ and that $$\mathcal{G} = \partial' X^0 \partial' X_0 (z') + \cdots + \partial' X^{D-1} \partial' X_{D-1}(z')$$

We can only make contractions between terms with same greek index. For each index, there are $$4$$ ways to perform a pair and $$2$$ ways to perform two pairs such that the sum of cross-contractions is given by

$$-4 \cdot \dfrac{\alpha'}{2} \partial \partial' \ln |z-z'|^2 \partial X^\mu(z) \partial' X_\mu (z') + 2 \eta^\mu_\mu \cdot \left( \dfrac{\alpha'}{2} \partial \partial' \ln |z-z'|^2 \right)^2. \tag4$$

The problem is that the first term right above differs from the RHS of $$(2.2.11)$$ by a normal ordering, and I can't figure out where is my mistake. Even if I use the equation

$$:\partial X^\mu(z)\partial' X_\mu(z'): =\partial X^\mu(z)\partial' X_\mu(z') - \dfrac{\alpha'}{2} \eta^\mu _\mu \partial \partial' \ln|z-z'|^2 \tag5$$

the terms do not match.

Besides that, I cannot find a way to get the equation by formula $$(2)$$, because I get divergent terms of the type $$\ln |z-z|^2$$. I will appreciate any help with this problem

Edit: here equation $$(5)$$ was obtained using the definition or normal ordering given in $$(2.2.7)$$ by Polchinski. It does not make sense to me that the terms multiplying the single pairs of $$ln|z|^2$$ have to be normal ordered since his formula tells you exactly the opposite.

• Possible duplicate: Virasoro TT OPE (2.2.11) in Polchinski's book Commented Sep 4, 2022 at 7:50
• In that question, OP already knows why the first equality $(2.2.11)$ has that specific form. Commented Sep 4, 2022 at 10:23