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