Permutation relations of generalized Yangian Invariants, unitarity cuts, and scattering amplitudes

We find a permutation relation among the generalized Yangian Invariants-two Yangian Invariants with adjacent external lines exchanged are related by a simple kinematic factor-which is shown to be equivalent to U(1) decoupling and Bern-Carrasco-Johansson (BCJ) relation at the level of maximal helicity violating (MHV) amplitudes. We propose using unitarity cuts to study nonplanar amplitudes and to systematically reconstruct the integrands of nonplanar MHV amplitudes, up to a rational function which vanishes under all possible unitarity cuts. This is made possible with the newly found permutation relations by converting nonplanar on-shell diagrams into planar ones. As explicit examples the construction of one-loop double-trace MHV amplitudes of 4- and 5-point interactions are presented using on-shell diagrams. The kinematic factors and the resultant planar diagrams are carefully dealt with using the unitarity cut conditions. The first next-to-MHV amplitudes are addressed using generalized unitarity cuts. Their leading singularities can be identified as residues of the Grassmanian integrals. These examples also serve to demonstrate the power of the newly found relation of the generalized Yangian Invariants.