Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes

We find a permutation relation among Yangian Invariants -- two Yangian Invariants with adjacent external lines exchanged are related by a simple kinematic factor. This relation is shown to be equivalent to U(1) decoupling and Bern-Carrasco-Johansson (BCJ) relation at the level of maximal helicity violating (MHV) amplitudes. Together with the newly found permutation relation and using unitarity cuts to remove ambiguity in the definition of loop momenta of cut amplitudes due to the nonplanar legs, we propose a systematic way of reconstructing the the integrands of nonplanar MHV amplitudes up to a rational function which vanishes under all possible unitarity cuts. This method when applied to planar diagrams reproduces results from the single cut method. 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 condition. The first next-to-MHV amplitudes are addressed using generalized unitarity cuts. Their leading singularities can be identified as residues of the Grassmanian integral. This example also serve to demonstrate the power of the newly found relation of Yangian Invariants.