Superconformal Partial Waves for Stress-tensor Multiplet Correlator in $4D$ $\mathcal{N}=2$ SCFTs

Abstract

We compute the superconformal partial waves of the four-point correlator $\langle JJJJ\rangle$, in which the external operator $J$ is the superconformal primary of the $4D$ $\mathcal{N}=2$ stress-tensor multiplet $\mathcal{J}$. We develop the superembedding formalism for the superconformal field theories (SCFTs) with extended supersymmetry. In $\mathcal{N}=2$ SCFTs, the three-point functions $\langle \mathcal{J}\mathcal{J}\mathcal{O}\rangle$ with general multiplet $\mathcal{O}$ contain two independent nilpotent superconformal invariants and new superconformal tensor structures, which can be nicely constructed from variables in superembedding space, and the three-point functions can be solved in compact forms. We compute the superconformal partial waves corresponding to the exchange of long multiplets using supershadow approach. The results are consistent with the non-trivial constraints by decomposing the $\mathcal{N}=2$ superconformal blocks into $\mathcal{N}=1$ superconformal blocks. Our results provide the necessary ingredient to study the fascinating $4D$ $\mathcal{N}=2$ SCFTs using conformal bootstrap.