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

Abstract

In N = 1 superconformal theories in four dimensions the form of two-point functions 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 Rcharge. From our results we recover the known unitarity bounds, and also find all shortening conditions, even in non-unitary theories. For the purposes of our computations we have developed a Mathematica package for the efficient handling of expansions in Grassmann variables.

Implications of ANEC for SCFTs in four dimensions

“…The coefficient c O is usually set to 1 by normalizing the operator in the two-point function, but here we do not do this rescaling of operator because in the supersymmetric case the relative normalizations of the operators in the same superconformal multiplet are fixed. We will assume that the superconformal primary is normalized to c O = 1 and use the results of [21] to fix the normalization of its superdescendants.…”

“…Finally one could also consider O → QO; the result is obtained by a simple rescaling of the coefficients in Table 5 and a replacement j → j ± 1. For the reader's convenience we report here the relative normalizations for the operators in the O multiplet as derived in [21]:…”

“…Every order that contains at least a θ and a θ at the same point will mix with conformal descendants due to {Q α , Qα} = 2P αα . The results of [21] can be used to subtract these contributions. We will only perform this expansion to first order in θ i , θ i and not for all possible combinations but only the ones of interest.…”

Polology of Superconformal Blocks

“…One remark worth mentioning is that our criterion is different from the Kac's irreducibility criterion [40], which is often invoked in the literature of superconformal field theories. The criterion of [27] reduces to Kac's criterion [40] when the Levi subalgebra is a Borel subalgebra, however for applications to the physics of SCFTs the Levi algebra should be given by (15), and is not a Borel subalgebra.…”

“…This is equivalent with the condition that λ + ρ is on the boundary of the Weyl chambers. 15 If this condition is not satisfied, then the module is irreducible thanks to Corollary 1 of [27]; otherwise we proceed to the last step.…”