# A How to prove 3.6.24 of Polchinski's big book

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1. Jun 13, 2017

### J.Hong

Hi
I'd like to show the equation 3.6.24 of Polchinski's big book(string theory volume 1). I think contents of page 35 to 36 is the key for the calculation, but I don't know how to carry out specific calculation. I think I need to know the form of $$\left \langle X(z,\bar{z})X(z',\bar{z'}) \right \rangle$$.
Using 2.1.18, I can guess $$\left \langle X(z,\bar{z})X(z',\bar{z'}) \right \rangle$$ is related with $$\eta ^{\mu \nu }\delta ^{2}\left ( z-z',\bar{z} -\bar{z'} \right )$$, but I don't know specific form because the world sheet is curved. If I naively yield $$\left \langle X(z,\bar{z})X(z',\bar{z'}) \right \rangle$$, I need to calculate $$\int_{-\infty }^{\infty } \frac{e^{ik\cdot (z-z')}}{k^{2}+i\varepsilon }d^{2}k$$, and I don't know how to deal with it by dimensional regularization even if the naive calculation is right.

Thus, my questions are
1. If my naive calculation is right, how can I carry out the dimensional regularization of $$\int_{-\infty }^{\infty } \frac{e^{ik\cdot (z-z')}}{k^{2}+i\varepsilon }d^{2}k$$ .
2. If wrong, I'd like to know how to get $$\left \langle X(z,\bar{z})X(z',\bar{z'}) \right \rangle$$ by dimensional regularization.