Note on integrability of marginally deformed ABJ(M) theories

Abstract

We study the anomalous dimensions of operators in the scalar sector of [beta]-deformed ABJ(M) theories. We show that the anomalous dimension matrix at two-loop order gives an integrable Hamiltonian acting on an alternating SU(4) spin chain with the spins at odd lattice sides in the fundamental representation and the spins at even lattices in the anti-fundamental representation. We get a set of [beta]-deformed Bethe ansatz equations which give the eigenvalues of Hamiltonian of this deformed spin chain system. Based on our computations, we also extend our study to non-supersymmetric three-parameter γ-deformation of ABJ(M) theories and find that the corresponding Hamiltonian is the same as the one in [beta]-deformed case at two-loop level in the scalar sector.