Perturbative Four-Point Functions from the Analytic Conformal Bootstrap

TL;DR

This paper uses the analytic conformal bootstrap method with twist conformal blocks to derive one-loop four-point functions in four-dimensional conformal gauge theories, avoiding traditional Feynman diagram calculations.

Contribution

It introduces a symmetry-based approach to compute one-loop four-point functions without regularisation or Feynman diagrams, applicable to various operators in super Yang-Mills theories.

Findings

Derived explicit one-loop four-point functions for Konishi operators.

Obtained correlation functions for half-BPS operators in $ ext{N}=4$ super Yang-Mills.

Method bypasses regularisation and Feynman redundancies.

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.