Two-point functions of conformal primary operators in $\mathcal{N}=1$ superconformal theories

TL;DR

This paper derives explicit formulas for two-point functions of superconformal primary operators in four-dimensional $ =1$ superconformal theories, encompassing all representations, scaling dimensions, and R-charges, and introduces a computational tool for handling Grassmann expansions.

Contribution

It provides the complete determination of two-point function coefficients for any superconformal multiplet in $ =1$ theories, including non-unitary cases, and develops a Mathematica package for Grassmann variable expansions.

Findings

Recovered known unitarity bounds.

Identified all shortening conditions, even for non-unitary theories.

Developed a Mathematica package for Grassmann expansions.

Abstract

In $\mathcal{N}=1$ superconformal theories in four dimensions the two-point function of superconformal multiplets is known up to an overall constant. A superconformal multiplet contains several conformal primary operators, whose two-point function coefficients can be determined in terms of the multiplet's quantum numbers. In this paper we work out these coefficients in full generality, i.e. for superconformal multiplets that belong to any irreducible representation of the Lorentz group with arbitrary scaling dimension and R-charge. From our results we recover the known unitarity bounds, and also find all shortening conditions, even for non-unitary theories. For the purposes of our computations we have developed a Mathematica package for the efficient handling of expansions in Grassmann variables.