On Four-Point Functions of Half-BPS Operators in General Dimensions

TL;DR

This work develops a universal framework for four-point functions of half-BPS operators in all superconformal dimensions $d=3$–$6$ where such theories exist. By employing harmonic/analytic superspace, the authors derive universal superconformal Ward identities and solve them using Jack-polynomial expansions, revealing that the general solution is parameterized by a set of arbitrary two-variable functions; in $d=4$ there is an additional one-variable sector. A key methodological advance is the frame-fixing strategy that isolates conformal and R-symmetry invariants, allowing a clean separation of the dynamical two-variable data from the protected single-variable contributions. The paper also connects these Ward-identity solutions to conformal partial waves, distinguishing long multiplet OPE content from shortened multiplets and showing absorption of restricted sectors into the two-variable sector for $d\neq 4$. The results generalize known four-dimensional ${\cal N}=4$ structures and provide a dimension-spanning, basis-independent description that can aid CPWA analyses in arbitrary dimensions. Overall, the work clarifies how superconformal kinematics and representation theory shape four-point functions and their OPE data across diverse spacetime dimensions.

Abstract

We study four-point correlation functions of half-BPS operators of arbitrary weight for all dimensions d=3,4,5,6 where superconformal theories exist. Using harmonic superspace techniques, we derive the superconformal Ward identities for these correlators and present them in a universal form. We then solve these identities, employing Jack polynomial expansions. We show that the general solution is parameterized by a set of arbitrary two-variable functions, with the exception of the case d=4, where in addition functions of a single variable appear. We also discuss the operator product expansion using recent results on conformal partial wave amplitudes in arbitrary dimension.