Superconformal blocks for stress-tensor and chiral operators for 4D $\mathcal{N}=2$ superconformal field theories

We have computed superconformal partial wave for the mixed correlators involving $J$, $\phi$ and $\phi^\dagger$, where $J$ is the superconformal primary of 4D ${\mathcal N} = 2$ stress-tensor multiplet, $\phi$ and $\phi^\dagger$ are chiral and anti-chiral scalars respectively in the channel $\langle J J \phi \phi^\dagger \rangle$. We have used the superembedding formalism and computed conformal partial wave corresponding to the exchange of long multiplet using the supershadow approach. This study provides part of the ingredients necessary for the study of conformal bootstrap for mixed correlator in 4D ${\mathcal N} = 2$ superconformal field theories.