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.
Many thanks in advance.

Last edited: Jun 13, 2017
2. Jun 18, 2017

PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.