Abstract
We classify and compute, by means of the six-dimensional embedding formalism in twistor space, all possible three-point functions in four dimensional conformal field theories involving bosonic or fermionic operators in irreducible representations of the Lorentz group. We show how to impose in this formalism constraints due to conservation of bosonic or fermionic currents. The number of independent tensor structures appearing in any three-point function is obtained by a simple counting. Using the Operator Product Expansion (OPE), we can then determine the number of structures appearing in 4-point functions with arbitrary operators. This procedure is independent of the way we take the OPE between pairs of operators, namely it is consistent with crossing symmetry, as it should be. An analytic formula for the number of tensor structures for three-point correlators with two symmetric and an arbitrary bosonic (non-conserved) operators is found, which in turn allows to analytically determine the number of structures in 4-point functions of symmetric traceless tensors.
Highlights
To correlators involving traceless symmetric conserved operators,see e.g. refs. [1,2,3,4,5,6,7,8]
We will see, generalizing the results found in ref. [9] for traceless symmetric operators, that 4D current conservation conditions can be covariantly lifted to 6D only if the conserved operator saturates the unitarity bound
We have computed in this paper the most general three point function occurring in a 4D CFT between bosonic and fermionic primary fields in arbitrary representations of the Lorentz group
Summary
The embedding formalism idea dates back to Dirac [15]. It is based on the simple observation that the 4D conformal group is isomorphic to SO(4, 2), that is the Lorentz group of a 6D flat space with signature (− − + + ++). Using the local isomorphism between SO(4, 2) and SU(2, 2), the embedding formalism can be reformulated in twistor space. In this form it has sporadically been used in the literature, mainly in the context of super conformal field theories [13] to study correlation functions in 4D CFTs. For completeness, we briefly review here the embedding formalism in twistor space, essentially following the analysis made in section 5 of ref. Given the relation (B.6) between symmetric traceless tensors written in vector and spinor notation, the spinors sα and sαappearing in eq (2.19) are exactly the ones defined in eq (2.12). There is not a simple relation between the 6D coordinates ZA and the 6D twistors Sa and Sa
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