A dictionary between R-operators, on-shell graphs and Yangian algebras

Abstract

We translate between different formulations of Yangian invariants relevant for the computation of tree-level scattering amplitudes in N=4 super-Yang--Mills theory. While the R-operator formulation allows to relate scattering amplitudes to structures well known from integrability, it can equally well be connected to the permutations encoded by on-shell graphs.Comment: 44 pages; replaced with published versio

Deformed one-loop amplitudes in N = 4 $$ \mathcal{N}=4 $$ super-Yang-Mills theory

“…The relations between these two formulations and permutations, which can be used to uniquely label tree-level Yangian invariants, have been thoroughly investigated in refs. [4,5]. Here we will be rather brief and refer the reader to these references for further details.…”

“…On-shell diagrams representing invariants for Y[u(2, 2|4)] are composed by gluing two types of deformed three-point vertices [1,2] relations between evaluation parameters and central charges. Generally, solutions to this system of equations are of the form [3][4][5] …”

“…In a series of recent papers, deformations of Yangian invariants in the context of N = 4 super-Yang-Mills (sYM) theory have been investigated [1][2][3][4][5]. As opposed to the undeformed situation, a deformed Yangian invariant allows for nonzero expectation values of the central charge operator for each external leg -which amounts to shifting the helicities of the external particles.…”

“…[4,5]. At tree level, the permutation labels a Yangian invariant unambiguously and can be translated into on-shell graphs [6] and R-operators [7].…”

“…[3][4][5]. All tree-level amplitudes in the maximally-helicityviolating (MHV) sector are represented by a single Yangian invariant, and can thus be deformed.…”