How to do the first OPE in Polchinski?

Question

I can't get the first OPE in Polchinski's String Theory book. It is

$$:\partial X^\mu(z)\partial X_\mu(z)::\partial'X^\nu(z')\partial'X_\nu(z'):=:\partial X^\mu(z)\partial X_\mu(z)\partial'X^\nu(z')\partial'X_\nu(z'):$$ $$-4\cdot \frac{\alpha'}{2}(\partial\partial'\ln|z-z'|^2):\partial X^\mu(z)\partial' X_\mu(z'):+2\cdot\eta^\mu_{\;\mu}\left(-\frac{\alpha'}{2}\partial\partial'\ln|z-z'|^2\right)^2$$ $$\sim\frac{D\alpha'^2}{2(z-z')^4}-\frac{2\alpha'}{(z-z')^2}:\partial' X^\mu(z')\partial' X_\mu(z'):-\frac{2\alpha'}{z-z'}:\partial'^2 X^\mu(z')\partial'X_\mu(z'):$$ I can see how the first equality comes about from Eq. (2.2.9) and Polchinski's hint. However, I don't know how to get part of the asymptotic. Thanks to Prahar I got the first term. I definitely don't know how to get the other terms. I'm not really sure how to make use of his hint to Taylor expand. So if you want to just give a hint, that's fine, just please don't tell me to Taylor expand ;)

Any help would be greatly appreciated.

This post imported from StackExchange Physics at 2015-02-01 13:24 (UTC), posted by SE-user 0celo7

Comments

@Prahar: I figured it out while typing up my work. The term $-4\cdot \frac{\alpha'}{2}(\partial\partial'\ln|z-z'|^2):\partial X^\mu(z)\partial' X_\mu(z'):$ actually contributes two terms to the asymptotic. Thank you for all your help. Post one of your hints below if you want the check.

This post imported from StackExchange Physics at 2015-02-01 13:24 (UTC), posted by SE-user 0celo7

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@Prahar: Sorry for pulling the high school card. I just get frustrated when P.S.E. people won't answer questions because they think I am a lazy college student or whatever. When I ask a question, then I'm really stuck. Its not like I can ask anyone I know.

This post imported from StackExchange Physics at 2015-02-01 13:24 (UTC), posted by SE-user 0celo7

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@0celo7 - For future reference, no one is going to help you unless you actually show what you have done and ask specific questions about where you are stuck. You did not do that at all in this question. For HW help, you are supposed to elaborate extensively on EVERYTHING that you have tried.

This post imported from StackExchange Physics at 2015-02-01 13:24 (UTC), posted by SE-user Prahar

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@Prahar: I see what you mean in this question. If I had gone through the Taylor expansion fully in the OP, I probably could have figured it out right then.

This post imported from StackExchange Physics at 2015-02-01 13:24 (UTC), posted by SE-user 0celo7

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@0celo7 - Exactly! That is why we require that it be written up. More often than not, this clears it up for the OP.

This post imported from StackExchange Physics at 2015-02-01 13:24 (UTC), posted by SE-user Prahar

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@Prahar: I'm in high school. I'm afraid I need a little more help than that! I don't have an expression of the form $\frac{1}{(z-w)^2}f(z)f(w)$, or if I do, then I don't see it anywhere.

This post imported from StackExchange Physics at 2015-02-01 13:24 (UTC), posted by SE-user 0celo7

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$f(z) = \partial X^\mu \partial X_\mu$

This post imported from StackExchange Physics at 2015-02-01 13:24 (UTC), posted by SE-user Prahar