The five-point bootstrap

We study five-point correlation functions of scalar operators in d-dimensional conformal field theories. We develop a new approach to computing the five-point conformal blocks for exchanged primary operators of arbitrary spin by introducing a generalization of radial coordinates, using an appropriate ansatz, and perturbatively solving two quadratic Casimir differential equations. We then study five-point correlators $⟨\sigma \sigma \epsilon \sigma \sigma ⟩$ in the critical 3d Ising model. We truncate the operator product expansions (OPEs) in the correlator by including a finite number of primary operators with conformal dimension below a cutoff $\mathrm{\Delta }⩽{\mathrm{\Delta }}_{\mathrm{c}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{f}\mathrm{f}}$. We then compute several OPE coefficients involving $\epsilon$ and two spinning operators by demanding that the truncated correlator approximately satisfies the crossing relation.