OPE of descendant fields How do I calculate OPE of two descendant fields $\chi_1$ and $\chi_2$, if I know OPE of $\phi_1$ and $\chi_2$. The field $\chi_1$ is a descendant of the primary field $\phi_1$.
 A: In $d>2$, all descendents are derivatives so all you have to do is to take derivatives of the OPE of primaries. 
In $d=2$, a generic descendent can always be described by a contour integrals of the OPE with the primary and several stress tensors. The OPE between two descendents then becomes the contour integral of the operator product of the two primaries with a bunch of stress-tensors. 
For e.g. let $O_1$ and $O_2$ be two primaries and consider the descendent $L_{-3} O_2$. This is
$$
L_{-3} O_2(0) = \oint \frac{dz}{2\pi i} z^{-2} T_{zz} (z) O_2(0)
$$
Thus,
$$
O_1(z) L_{-3} O_2(0) = \oint_C \frac{dw}{2\pi i} w^{-2} O_1(z)   T_{ww} (w) O_2(0)
$$
The contour $C$ surrounds the origin and $z$ is outside the contour. Then, I can push the contour outside to include $z$ at the cost of adding the poles at $z=w$. The residue at this pole is given entirely by the $TO_1$ OPE which is fixed since $O_1$ is primary. Finally, we take $z \to 0$ and the singularities one gets is controlled by the $O_1 O_2$ OPE which, I assume, is known.
A: It's pretty much trivial. I assume you're talking about global conformal symmetry i.e. $SL(2,C)$ for $d=2$ and $SO(d,2)$ in higher $d$. Say the OPE is given by
$$\phi_1(x)\phi_2(y) = \frac{C}{|x-y|^{a}} \left[ O(y) + b_1 (x-y)^\mu \frac{\partial}{\partial y^\mu} O(y) + b_2 (x-y)^2 \Box_y O(y) \ldots \right] + \text{other multiplets}
$$
for some other primary $O(y)$. Here $C$ is the OPE coefficient, and $b_1, b_2, \ldots$ are coefficients that are fixed by conformal symmetry. They depend on $d$ and the scaling dimensions and spins of $\phi_1$, $\phi_2$ and $O$. 
Now if $\chi_1$ is a descendant of $\phi_1$, it can be written as
$$\chi_1^{\mu_1 \dotsm \mu_J}(x) = \partial^{\mu_1} \dotsm \partial^{\mu_J} \Box^n \phi_1(x)$$
and likewise for $\chi_2$. Schematically, there exist some different operators $D_{1,2}$ such that
$$\chi_1(x) = D_1(\partial_x) \phi_1(x)$$ and similarly for $\chi_2$. But then we get the OPE $\chi_1 \times \chi_2$ just by applying these operators:
$$\chi_1(x) \chi_2(y) = D_1(\partial_x) D_2(\partial_y) \left[ \phi_1(x) \phi_2(y) \right] = D_1(\partial_x) D_2(\partial_y) \left[ \frac{C}{|x-y|^{2a}} O(y) + \ldots \right].$$
End of story!
