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 ...


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