The most general 4 D $$ \mathcal{D} $$ N $$ \mathcal{N} $$ = 1 superconformal blocks for scalar operators

Abstract

We compute the most general superconformal blocks for scalar operators in 4D N = 1 superconformal field theories. Specifically we employ the supershadow formalism to study the four-point correlator Φ 1 Φ 2 Φ 3 Φ 4 , in which the four scalars, Φ i , have arbitrary scaling dimensions and R-charges. The only constraint on the R-charges is from R-symmetry invariance of the four-point correlator and the exchanged operators can have arbitrary R-charges. Our results extend previous studies on 4D N = 1 superconformal blocks to the most general case, which are the essential ingredient for bootstrapping mixed correlators of scalars with independent scaling dimensions and R-charges.

Long multiplet bootstrap

“…In this work we initiated, in two dimensions, the bootstrap of long multiplets, using the whole superfield as the external operator. While long multiplets have been considered in the past, from the point of view of kinematics [15,26,34,36,39,102], and recently through a numerical analysis of dynamical information [14], all previous work has been restricted to considering only the superconformal primary of long multiplets. Unlike the case of external chiral operators (or BPS operators in general) where the four-point function depends only on the supersymmetrization of the regular bosonic conformal and R-symmetry invariants, for more general external fields one starts finding nilpotent superconformal invariants.…”

“…Such four-point functions can depend on nilpotent superconformal invariants, and information is lost when restricting the external operators to the superconformal primaries. For the case of four generic long multiplets this might mean, as was the case in [14,36,39] for four-dimensional N = 1 long multiplets, that correlation functions of superprimaries can (only) be decomposed into bosonic conformal blocks with independent coefficients. While supersymmetry relates the various operators in the exchanged multi- 1 While the presence of a "kink" is not enough to guarantee the existence of a fully consistent SCFT, it provides hints it might correspond to a new N = 1 SCFT.…”

Superconformal blocks: general theory

“…Also the bootstrap programme, both in its numerical and analytical incarnation, has been applied to higher dimensional superconformal field theories, see [22][23][24]. Significant progress has been made by studying four-point blocks of half-BPS operators [25][26][27][28][29][30][31][32][33][34][35], or superprimary components of correlators for operators that are not half-BPS [36][37][38][39]. For correlation functions of BPS operators, the conformal blocks are similar to those of the ordinary bosonic conformal symmetry and hence they are well known.…”

Conformal manifolds: ODEs from OPEs

“…This is not the case for long multiplets with nilpotent superconformal invariants. In these superconformal blocks, the coefficients of bosonic blocks that appear are theory dependent [41][42][43]. Even before these exercises are done, it is clear that (2.7) will not allow us to see individual cancellation within a given block.…”