Consider a closed string scattering (worldsheet topology $S^2$ sphere). Given that the OPE:
$i \partial X^\mu(z) e^{ik X(w)} \sim \frac{k^\mu}{z - w} e^{ik X (w)} + ... \ \ , \ \ i \partial X^\mu (z) i \partial X^\nu (w) \sim \frac{\eta^{\mu\nu}}{z-w}+ ...$
How can I show the following relation (mainly, what happened with all the non-singular part?):
$\Bigg\langle \prod^A_{l=1} i \partial X^{\mu_l} (w_l) \prod^B_{m=1} e^{ik_m X(z_m)} \Bigg\rangle_{S^2} = \sum^A_{n=2} \frac{\eta^{\mu_1 \mu_n}}{(w_1 - w_n)^2} \Bigg\langle \prod^A_{l=2; l \neq n} i \partial X^{\mu_l} (w_l) \prod^B_{m=1} e^{ik_m X(z_m)} \Bigg\rangle_{S^2} + \sum^B_{n=1} \frac{k_n^{\mu_1}}{(w_1 - z_n)^2} \Bigg\langle \prod^A_{l=2} i \partial X^{\mu_l} (w_l) \prod^B_{m=1} e^{ik_m X(z_m)} \Bigg\rangle_{S^2}$
(Basically, replacing $i\partial X^{\mu_1}$ by all singularities it can produced via OPEs)
The prefactor might be sloppy ...
This post imported from StackExchange Physics at 2015-03-04 12:48 (UTC), posted by SE-user user109798