Abstract
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.
Highlights
Arkani-Hamed et al [35] proposed using positive Grassmannian to study N = 4 super Yang-Mills along with the constructions of the bipartite ribbon on-shell diagrams [36] — in which all internal legs are on shell — for planar Yang-Mills interactions
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-CarrascoJohansson (BCJ) relation at the level of maximal helicity violating (MHV) amplitudes
For planar bipartite on-shell diagrams, we can use the criterion (2.8) to justify if any two adjacent legs fall into a box
Summary
The set of rules governing the permutations of external legs for the bipartite on-shell diagrams have been introduced in [35]. For MHV on-shell diagrams, it is always possible to connect a box directly to the pair of the permuting legs [35], evident from the expression Yn(2) = Y4(2) Y3(1) . Y3(1) together n−4 with the cyclic symmetry of the external legs This box is nothing but the 4-point on-shell amplitude. A BCFW-Bridge decomposition to a Box. Starting with a given permutation σ and picking two consecutive legs i and i+1, if σ(i) = i mod n and σ(i+1) = i+1 mod n and σ for the other legs is not identical to the identity modulus n (a “dressed” identity2), one can decompose σ as (j1j2) ◦ σ , where 1 j1 < j2 n and σ(j1) < σ(j2), with j1 = {i, i + 1}, and j2 = {i, i + 1}. When permuting the pair of legs n − 1 and n, as shown in figure 3
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