In this paper we study a wide class of planar single-trace four point correlators in the chiral conformal field theory ($chi$CFT$_4$) arising as a double scaling limit of the $gamma$-deformed $mathcal{N}=4$ SYM theory. In the planar (tHooft) limit, each of such correlators is described by a single Feynman integral having the bulk topology of a square lattice fishnet and/or of an honeycomb lattice of Yukawa vertices. The computation of this class of Feynmann integrals at any loop is achieved by means of an exactly-solvable spin chain magnet with $SO(1,5)$ symmetry. In this paper we explain in detail the solution of the magnet model as presented in our recent letter and we obtain a general formula for the representation of the Feynman integrals over the spectrum of the separated variables of the magnet, for any number of scalar and fermionic fields in the corresponding correlator. For the particular choice of scalar fields only, our formula reproduces the conjecture of B. Basso and L. Dixon for the fishnet integrals.
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