  We consider a special double scaling limit, recently introduced by two of the
authors, combining weak coupling and large imaginary twist, for the
$\gamma$-twisted $\mathcal{N}=4$ SYM theory. We also establish the analogous
limit for ABJM theory. The resulting non-gauge chiral 4D and 3D theories of
interacting scalars and fermions are integrable in the planar limit. In spite
of the breakdown of conformality by double-trace interactions, most of the
correlators for local operators of these theories are conformal, with
non-trivial anomalous dimensions defined by specific, integrable Feynman
diagrams. We discuss the details of this diagrammatics. We construct the
doubly-scaled asymptotic Bethe ansatz (ABA) equations for multi-magnon states
in these theories. Each entry of the mixing matrix of local conformal operators
in the simplest of these theories - the bi-scalar model in 4D and tri-scalar
model in 3D - is given by a single Feynman diagram at any given loop order. The
related diagrams are in principle computable, up to a few scheme dependent
constants, by integrability methods (quantum spectral curve or ABA). These
constants should be fixed from direct computations of a few simplest graphs.
This integrability-based method is advocated to be able to provide information
about some high loop order graphs which are hardly computable by other known
methods. We exemplify our approach with specific five-loop graphs.
