Mixed five-point correlators in the 3d Ising model

We study five-point correlators of the $\sigma$, $\epsilon$, and ${\epsilon }^{\mathrm{\prime }}$ operators in the critical 3d Ising model. We consider the $\sigma \times \sigma$ and $\sigma \times \epsilon$ operator product expansions (OPEs) and truncate them by including a finite set of exchanged operators. We approximate the truncated operators by the corresponding contributions in appropriate disconnected five-point correlators. We compute a number of OPE coefficients that were previously unknown and show that these are consistent with predictions obtained using the fuzzy sphere regularization of the critical 3d Ising model.