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

Daliang Li, Andreas Stergiou

TL;DR

This work determines the full set of two-point function coefficients for all conformal primaries contained in an arbitrary $\mathcal{N}=1$ superconformal multiplet in four dimensions, expressed in terms of the lowest component’s coefficient and the multiplet quantum numbers $(j,\bar{j},q,\bar{q})$ with $\Delta=q+\bar{q}$ and $R=\tfrac{2}{3}(q-\bar{q})$. By expanding the general superconformal two-point function in Grassmann variables and matching to the component primaries, the authors derive explicit relations among the coefficients for sixteen primaries, recover the unitarity bounds, and catalog shortening conditions, including non-unitary cases. A dedicated Mathematica package is developed to automate $\theta$-expansions and the handling of Lorentz/spinor structures, enabling systematic decomposition of superfield correlators into conformal-primary correlators. The results are exemplified by the Ferrara–Zumino supercurrent multiplet, where the nonvanishing component two-point functions reproduce standard current and energy-momentum tensor correlators, illustrating the approach’s consistency and utility for constructing superconformal blocks and guiding three-point function analyses.

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.

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

TL;DR

This work determines the full set of two-point function coefficients for all conformal primaries contained in an arbitrary superconformal multiplet in four dimensions, expressed in terms of the lowest component’s coefficient and the multiplet quantum numbers with and . By expanding the general superconformal two-point function in Grassmann variables and matching to the component primaries, the authors derive explicit relations among the coefficients for sixteen primaries, recover the unitarity bounds, and catalog shortening conditions, including non-unitary cases. A dedicated Mathematica package is developed to automate -expansions and the handling of Lorentz/spinor structures, enabling systematic decomposition of superfield correlators into conformal-primary correlators. The results are exemplified by the Ferrara–Zumino supercurrent multiplet, where the nonvanishing component two-point functions reproduce standard current and energy-momentum tensor correlators, illustrating the approach’s consistency and utility for constructing superconformal blocks and guiding three-point function analyses.

Abstract

In 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.

Paper Structure

This paper contains 32 sections, 109 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Introduction
  3. N=1 superconformal primary operators and their descendants
  4. N=1 superconformal primary operators and their descendants
  5. Components of O
  6. Components of O
  7. Summary of results
  8. Summary of results
  9. Comments
  10. Comments
  11. Two-point functions
  12. Two-point functions
  13. Orders theta1 thetabar2 and thetabar1 theta2
  14. Orders theta1 thetabar2 and thetabar1 theta2
  15. Orders theta1sq thetabar2sq and thetabar1sq theta2sq
  16. ...and 17 more sections

Figures (1)

  • Figure 1: A superconformal multiplet consists of a finite number of conformal primary operators, related by supersymmetry. Some of these primaries may become null when the multiplet's quantum numbers $(j,{\bar{\jmath}}\space,q,{\bar{q}})$ satisfy shortening conditions.