A shortcut to general tree‐level scattering amplitudes in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} = 4$\end{document} SYM via integrability

Abstract

We combine recent applications of the two-dimensional quantum inverse scattering method to the scattering amplitude problem in four-dimensional N = 4 Super Yang-Mills theory. Integrability allows us to obtain a general, explicit method for the derivation of the Yangian invariants relevant for tree-level scattering amplitudes in the N = 4 model.

Yangian symmetry for the tree amplituhedron

“…This is indeed the case and Figure 3 seed cluster mutations maps a given on-shell diagram to a version of itself where the external legs have been cyclically relabelled. As a further remark, it was shown in [9] that it is also possible to construct deformed Grassmannian integrals following a similar procedure to that of (3.10). So far we have used solely the operators B ij (0): in order to obtain deformed Grassmannian integrals, we need to allow a non-trivial dependence on u-parameters.…”

“…We conclude this section with a simple example illustrating all concepts we have introduced so far. More examples (in the context of scattering amplitudes) can be found in [9]. Let us consider the case of Ω This procedure can be depicted as in Figure 4 where B ij is represented as the so-called BCFW bridge composed of one black and one white trivalent vertex.…”

“…As a result, we will prove that there exists a matrix of functions closely related to the amplituhedron volume function which is invariant under the Yangian of gl(m + k). To this purpose, we follow the steps of [9], where Yangian invariants relevant for tree-level amplitudes in N = 4 SYM have been obtained using the Quantum Inverse Scattering Method. Indeed, the infinite-dimensional symmetry algebra governing planar N = 4 SYM allows us to employ methods and techniques proper to (or inspired by) integrable theories.…”

Graßmannian integrals in Minkowski signature, amplitudes, and integrability

“…Here the exponents are defined by [28] can even be derived using tools rooted in the QISM [31]. These tools were introduced in [14,33] and studied more systematically in [29,34]. However, this method uses somewhat formal integral operators and does not yield a suitable contour for the resulting deformed Graßmannian integral (2.24).…”

A connection between R-invariants and Yang-Baxter R-operators in $$ \mathcal{N} $$ = 4 super-Yang-Mills theory

“…This approach exploits Yangian symmetry of the amplitudes, that has been studied e.g. in [2], [3] The framework of R-operators was developed in a number of papers [4], [5], [6], [7]. For example, in [6], a connection was established between the graded permutations encoding the on-shell graphs and chains of R-operators acting on a suitable basic state, as well as a connection to the top-cell graphs.…”