$\mathcal{N}=1$ Superconformal Blocks for General Scalar Operators

TL;DR

$Using a covariant ${\cal N}=1$ supershadow formalism in the superembedding space, the paper derives explicit expressions for four-point blocks with general scalar external operators and $R$-neutral exchange. External operators are lifted to multi-twistor representations and blocks are computed as monodromy-projected superconformal integrals that decompose into linear combinations of bosonic conformal blocks $g_{\Delta,\ell}$. The formalism reproduces known blocks for chiral-antichiral and current correlators as special cases, and provides nontrivial consistency checks by decomposing ${\cal N}=2$ blocks into ${\cal N}=1$ blocks with explicit coefficients. The results supply new atomic ingredients for the ${\cal N}=1$ conformal bootstrap and set the stage for extensions to ${\cal N}>1$. $

Abstract

We use supershadow methods to derive new expressions for superconformal blocks in 4d $\mathcal{N}=1$ superconformal field theories. We analyze the four-point function $\langle\mathcal{A}_1 \mathcal{A}_2^\dagger \mathcal{B}_1 \mathcal{B}_2^\dagger\rangle$, where $\mathcal{A}_i$ and $\mathcal{B}_i$ are scalar superconformal primary operators with arbitrary dimension and $R$-charge and the exchanged operator is neutral under $R$-symmetry. Previously studied superconformal blocks for chiral operators and conserved currents are special cases of our general results.