  We consider the $\gamma$-deformed $\mathcal{N}=4$ SYM in the double scaling
limit of large imaginary twists and small coupling, which discards the gauge
fields and retains only certain Yukawa and scalar interactions with three
arbitrary effective couplings. In the 't~Hooft limit, these 4D theories are
integrable, with most of the correlators being conformal such that the whole
arsenal of AdS/CFT integrability remains applicable. In particular, for one
non-zero effective coupling, we obtain a QFT of two complex scalars with a
chiral, quartic interaction. The BMN vacuum anomalous dimension is dominated in
each non-zero loop order by a single "wheel" graph, in principle computable by
integrability. Thus we also provide an explicit conjecture for the periods of
double-wheel graphs with an arbitrary number of spokes.
