Perturbative Four-Point Functions from the Analytic Conformal Bootstrap
Johan Henriksson, Tomasz Lukowski
TL;DR
This work develops an analytic bootstrap framework to compute one-loop four-point functions in four-dimensional weakly coupled CFTs. By formulating twist conformal blocks and their spin-dependent generalisations as H-functions, the authors resum infinite towers of high-spin operators and extract the large-spin CFT data necessary to reconstruct the full one-loop correlator, all without relying on Feynman diagrams. They obtain a general LT-resummed form and, using a small set of external CFT data, reproduce the known Konishi four-point function and, separately, the four-point function of four half-BPS operators in $\mathcal{N}=4$ SYM, thereby validating the method. The approach provides a symmetry-based, regularisation-free route to loop-level correlators, with clear pathways to higher-order and mixed correlators in conformal and superconformal theories, and offers deeper insight into the structure of perturbative CFT data at large spin.
Abstract
We apply the analytic conformal bootstrap method to study weakly coupled conformal gauge theories in four dimensions. We employ twist conformal blocks to find the most general form of the one-loop four-point correlation function of identical scalar operators, without any reference to Feynman calculations. The method relies only on symmetries of the model. In particular, it does not require introducing any regularisation and it is free from the redundancies usually associated with the Feynman approach. By supplementing the general solution with known data for a small number of operators, we recover explicit forms of one-loop correlation functions of four Konishi operators as well as of four half-BPS operators $\mathcal{O}_{20'}$ in $\mathcal{N}=4$ super Yang-Mills.