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earth2

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I'm reading through chapter 7 of Green-Schwarz-Witten and I have a problem with the derivation of the M-tachyon correlation function. Basically I'm trying to get 7.A.17 from 7.A.12 and eq 7.A.22 in the appendix of the first volume.

Basically I want to prove:

$$\langle\frac{V(k_1,y_1)...V(k_M,y_M)}{y_1...y_M}\rangle=\prod_{i < j}(y_i-y_j)^{k_ik_j}$$

with $$V(k_i,y_i)=e^{ik_iX(k_i)}$$

If I try to get this by just plugging everything into $$\langle:e^{A_1}:...:e^{A_M}:\rangle=e^{\sum_{i<j}<A_iA_j>}$$ using $$\langle X(y_i)^{\mu}X(y_j)^\nu\rangle=-\eta^{\mu\nu}log(y_i-y_j)$$ I somehow still have the $$\frac{1}{y_1...y_M}$$ factor in the formula above existing on the right hand side of the first equation. How is that canceled in this approach? I don't see it.

More specifically I have

$$\langle\frac{V_1...V_M}{y_1...y_M}\rangle=\frac{1}{y_1...y_M}\langle V(k_1,y_1)...V(k_M,y_M)\rangle=\frac{1}{y_1...y_M}e^{(\sum_{i<j}k_i^\mu\langle X_iX_j\rangle k_j^\nu)}$$ $$=\frac{1}{y_1...y_M}e^{(\sum_{i<j}k_ik_jlog(y_i-y_j))}=\frac{1}{y_1...y_M}\prod_{i<j}(y_i-y_j)^{k_ik_j}$$

which is the result I want to have up to that nasty prefactor that I can't make sense of.

Edit: I also posted this on stackexchange but didnt get an answer :(

Btw, this is not a homework question. I'm reading it on my own.

Basically I want to prove:

$$\langle\frac{V(k_1,y_1)...V(k_M,y_M)}{y_1...y_M}\rangle=\prod_{i < j}(y_i-y_j)^{k_ik_j}$$

with $$V(k_i,y_i)=e^{ik_iX(k_i)}$$

If I try to get this by just plugging everything into $$\langle:e^{A_1}:...:e^{A_M}:\rangle=e^{\sum_{i<j}<A_iA_j>}$$ using $$\langle X(y_i)^{\mu}X(y_j)^\nu\rangle=-\eta^{\mu\nu}log(y_i-y_j)$$ I somehow still have the $$\frac{1}{y_1...y_M}$$ factor in the formula above existing on the right hand side of the first equation. How is that canceled in this approach? I don't see it.

More specifically I have

$$\langle\frac{V_1...V_M}{y_1...y_M}\rangle=\frac{1}{y_1...y_M}\langle V(k_1,y_1)...V(k_M,y_M)\rangle=\frac{1}{y_1...y_M}e^{(\sum_{i<j}k_i^\mu\langle X_iX_j\rangle k_j^\nu)}$$ $$=\frac{1}{y_1...y_M}e^{(\sum_{i<j}k_ik_jlog(y_i-y_j))}=\frac{1}{y_1...y_M}\prod_{i<j}(y_i-y_j)^{k_ik_j}$$

which is the result I want to have up to that nasty prefactor that I can't make sense of.

Edit: I also posted this on stackexchange but didnt get an answer :(

Btw, this is not a homework question. I'm reading it on my own.

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