Abstract
We show how conformal partial waves (or conformal blocks) of spinor/tensor correlators can be related to each other by means of differential operators in four dimensional conformal field theories. We explicitly construct such differential operators for all possible conformal partial waves associated to four-point functions of arbitrary traceless symmetric operators. Our method allows any conformal partial wave to be extracted from a few “seed” correlators, simplifying dramatically the computation needed to bootstrap tensor correlators.
Highlights
Apply to any CFT, with or without a Lagrangian description
For each exchanged primary operator, it is convenient not to talk of individual conformal blocks but of Conformal Partial Waves (CPW), namely the entire contribution given by several conformal blocks, one for each tensor structure
General three-point functions in 4D CFTs involving bosonic or fermionic operators in irreducible representations of the Lorentz group have recently been classified and computed in ref. [21] using the 6D embedding formalism [52,53,54,55] formulated in terms of twistors in an index-free notation [11]
Summary
General three-point functions in 4D CFTs involving bosonic or fermionic operators in irreducible representations of the Lorentz group have recently been classified and computed in ref. [21] (see refs. [44, 45] for important early works on tensor correlators and refs. [8, 9, 19, 46,47,48,49,50,51] for other recent studies) using the 6D embedding formalism [52,53,54,55] formulated in terms of twistors in an index-free notation [11] (see e.g. refs. [56,57,58,59,60,61] for applications mostly in the context of supersymmetric CFTs). General three-point functions in 4D CFTs involving bosonic or fermionic operators in irreducible representations of the Lorentz group have recently been classified and computed in ref. Any two fields O and O = O + XV or O = O + XW , for some multi twistors V and W , are equivalent uplifts of O. O1O2O3 which must be a sum of SU(2, 2) invariant quantities constructed out of the Xi, Si and Si, with the correct homogeneity properties under rescaling. As can be seen from eq (2.10), any operator Ol, ̄l has a non-vanishing two-point function with a conjugate operator Ol,l only. The most general three-point function O1O2O3 can be written as. In eq (2.12), K3 is a kinematic factor that depends on the scaling dimension and spin of the external fields, K3
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