Abstract
In this work we discuss an analytic bootstrap approach [1, 2] in the context of spinning 4D conformal blocks [3, 4]. As an example we study the simplest spinning case, the scalar-fermion correlator leftlangle phi psi phi overline{psi}rightrangle . We find that to every pair of primary scalar ϕ and fermion ψ correspond two infinite towers of fermionic large spin primary operators. We compute their twists and products of OPE coefficients using both s-t and u-t bootstrap equations to the leading and sub-leading orders. We find that the leading order is represented by the scalar-fermion generalized free theory and the sub-leading order is governed by the minimal twist bosonic (light scalars, currents and the energy-momentum tensor) and fermionic (light fermions and the suppersymmetric current) operators present in the spectrum.
Highlights
The obtained CFT result can be interpreted in AdS as a system of scalar particles with large relative angular momentum
We find that to every pair of primary scalar φ and fermion ψ correspond two infinite towers of fermionic large spin primary operators
We start by discussing the main ingredients of general bootstrap equations in section 2.1 preparing the ground for section 2.2, where we provide an analytic bootstrap recipe in a schematic form
Summary
We work in 4D Minkowski space where the conformal group is SO(2, 4) SU(2, 2). The local primary operators transform in a finite-dimensional representation of the U(1)×SU(2)× SU(2) sub-group and are labeled by the U(1) charge (the scaling dimension) ∆ and the pair of integers ( , ̄) which describes spin. In what follows it will be more convenient to use instead of the scaling dimension another quantity called the twist defined as. A generic 4-point function of local operators has the following form when expanded in the s-channel. The parameters a and b depend on the twists of external operators, parameter c on the spin difference p defined in (2.8). After decomposing it into a set of independent equations by stripping of independent tensor structures TI , all the equations get the following schematic form (zz)A. Equation (2.17) leads to the following set of bootstrap equations We will study this equation in the z, z ∈ [1, +∞] region. One can consider a different correlation function (with positions of operators re-ordered) and to study an s-t channel. In this case the standard s-t techniques straightforwardly apply and no analytic continuation is needed
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