Computing descendants of primary operator in 2D CFT

Question

I computed the four-point function of primaries $\partial\phi$ in the 2D free boson CFT to be $$\langle \partial\phi(z_1)\partial\phi(z_2)\partial\phi(z_3)\partial\phi(z_4) \rangle = \frac{1}{16 \pi^2} \left(\frac{1}{z_{12}^{2}z_{34}^{2}} + \frac{1}{z_{13}^{2}z_{24}^{2}} + \frac{1}{z_{14}^{2}z_{23}^{2}}\right).$$

I now want to use this result to obtain the level-two descendants of $\partial\phi$ (i.e. $[L_{-2}, \partial\phi]$ and $[L_{-1},[L_{-1}, \partial \phi]]$). I don't really know how I can adapt my previous result to obtain this. Moreover, when computing $[L_{-2}, \partial\phi]$, it doesn't seem like I can obtain a result from this. Since we can calculate $$[L_{-2}, \partial \phi(w)] = \frac{1}{2\pi i} \oint_{c_w} dz \; z^{-1} T(z) \partial \phi(w).$$ However, in the OPE expansion, we have $$T(z) \partial \phi(w)\sim \frac{\partial \phi(w)}{(z-w)^2} + \frac{\partial^2 \phi(w)}{(z-w)}.$$ In other words, will the $(z-w)^{-1}$ factor in the integrand of the contour integral make the regular terms in the OPE also contribute?

Answer 1

For any field $V$, the identity $\partial V = L_{-1}V$ allows you to compute correlation functions of $L_{-1}$-descendants. For correlation functions of $L_{-2}$-descendants or higher, use the local Ward identities, see https://arxiv.org/abs/1406.4290 eq. (2.2.15).

Answer 2

For the descendant obtained by hitting $\partial \phi(w)$ with $L_{-2}$, we obtain $$[L_{-2}, \partial \phi(w)] = \frac{1}{2\pi i} \oint_{c_w} dz \; z^{-1} T(z) \partial \phi(w) = - \frac{\partial \phi(w)}{w^2} + \frac{\partial^2 \phi(w)}{w},$$ provided we assume $c_w$ does not contain the origin.

Similarly, we can find the other level-two descendant as $$ [L_{-1},[L_{-1}, \partial\phi(w)]] = \partial^3 \phi(w).$$