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Notes on Integrable Boundary Interactions of Open $SU(4)$ Alternating Spin Chains

In arXiv:1704.05807, it was shown that the planar flavored ABJM theory is integrable in the scalar sector at two-loop order using coordinate Bethe ansatz. A salient feature of this case is that the boundary reflection matrices are anti-diagonal with respect to the chosen basis. In this paper, we relax the coefficients of the boundary terms to be general constants in order to search for integrable systems among this class. We found that at each end of the spin chain, the only integrable boundary interaction, besides the one in arXiv:1704.05807, is the one with vanishing boundary interactions leading to diagonal reflection matrices. We also construct non-supersymmetric plannar flavored ABJM theory which leads to trivial boundary interaction at both ends of the open chain from two-loop anomalous dimension matrix in the scalar sector.