  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.
