Seed Conformal Blocks in 4D CFT

TL;DR

This work derives closed analytic seed conformal blocks for the exchange of arbitrary mixed-symmetry spinor/tensor operators in 4D CFTs, parameterized by p=|\ell-\bar{\ell}|, via a compact system of Casimir equations and cross-checked with shadow-formalism integrals in a 6D twistor embedding. The seed blocks G_e^{(p)}(z,\bar z) are expressed as a finite sum over generalized hypergeometric eigenfunctions with coefficients c_{m,n}^e living on an octagon-shaped lattice; these coefficients are computable recursively and exhaust the p+1 independent tensor structures for each p. The results provide a complete analytic foundation for bootstrapping 4-point functions involving arbitrary spinor/tensor exchanges in 4D, and reveal a near-nearest-neighbour coupling structure in the Casimir system together with a rich geometric organization of block data akin to scalar blocks in even dimensions. The methods enable systematic study of tensor correlators, potential AdS/CFT holographic interpretations, and deeper analytic bootstrap directions, with explicit data and cross-checks for low-p cases and downloadable results for higher p.

Abstract

We compute in closed analytical form the minimal set of "seed" conformal blocks associated to the exchange of generic mixed symmetry spinor/tensor operators in an arbitrary representation (l,\bar l) of the Lorentz group in four dimensional conformal field theories. These blocks arise from 4-point functions involving two scalars, one (0,|l-\bar l|) and one (|l-\bar l|,0) spinors or tensors. We directly solve the set of Casimir equations, that can elegantly be written in a compact form for any (l,\bar l), by using an educated ansatz and reducing the problem to an algebraic linear system. Various details on the form of the ansatz have been deduced by using the so called shadow formalism. The complexity of the conformal blocks depends on the value of p=|l-\bar l | and grows with p, in analogy to what happens to scalar conformal blocks in d even space-time dimensions as d increases. These results open the way to bootstrap 4-point functions involving arbitrary spinor/tensor operators in four dimensional conformal field theories.

Figures (4)

• Figure 1: Set of points in the $(r,t)$ plane forming the regions ${\cal R}_0$ (13 points), ${\cal R}_+$ (12 points) and ${\cal R}_-$ (12 points) defined in eqs.(\ref{['eq:properties_group1']})-(\ref{['eq:properties_group3']}).
• Figure 2: The dimensions of the generic octagon enclosing the lattice of non-vanishing coefficients $c_{m,n}^e$ entering the ansatz for mixed tensor CBs in eq.(\ref{['eq:Ansatz']}).
• Figure 3: Set of non-vanishing coefficients $c_{m,n}^e$ (represented as black dots) entering the ansatz for mixed tensor CBs in eq.(\ref{['eq:Ansatz']}) for $p=3$ and $e=0,1,2,3$. For $e=0$ and $e=p$ the octagons collapse to hexagons.
• Figure 4: The dimensions of the generic slanted square enclosing the lattice of non-vanishing coefficients $x_{m,n}$ entering the ansatz for scalar symmetric CBs in eq.(\ref{['eq:object']}).