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.
Highlights
Four-point functions can be expressed as a sum in terms of conformal blocks, whose explicit form has been worked out in some even dimensions in [6, 7]
For many physical applications it is necessary to work out explicitly these component correlation functions and their relations as imposed by superconformal symmetry
From the superconformal two- and three-point functions one can extract the operator product expansion, and use it to explore the phenomenology of models of supersymmetry breaking with a superconformal hidden sector in the ultraviolet [22,23,24]
Summary
One can construct the full superconformal multiplet by applying raising operators Pμ, Qα, and Qαon O. If we apply Q2, the result Q2O is a superconformal descendant It is still a conformal primary with quantum numbers (j, ̄, q + 1, q) Since1 [Kμ, Qα] = −σαμα Sα , the two operators on the right-hand side are conformal primaries, characterized by quantum numbers They can be denoted unambiguously by (QO)j±1, ̄. Where, the dotted indices do not participate in the symmetrization of the undotted ones and vice-versa These four operators can be denoted by (QQO)j±1, ̄±1. The multiplet contains sixteen different conformal primary operators, but if the quantum numbers (j, ̄, q, q) obey special conditions, certain higher components may become null.
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