Abstract
Planar N=4 Super Yang-Mills theory appears to be a quantum integrable four-dimensional conformal theory. This has been used to find equations believed to describe its exact spectrum of anomalous dimensions. Integrability seemingly also extends to the planar space-time scattering amplitudes of the N=4 model, which show strong signs of Yangian invariance. However, in contradistinction to the spectral problem, this has not yet led to equations determining the exact amplitudes. We propose that the missing element is the spectral parameter, ubiquitous in integrable models. We show that it may indeed be included into recent on-shell approaches to scattering amplitude integrands, providing a natural deformation of the latter. Under some constraints, Yangian symmetry is preserved. Finally we speculate that the spectral parameter might also be the regulator of choice for controlling the infrared divergences appearing when integrating the integrands in exactly four dimensions.
Highlights
Introduction and overviewDespite its seeming complexity on the Lagrangian level the maximally supersymmetric Yang-Mills theory (N = 4 SYM) [1, 2] might be the simplest interacting four-dimensional quantum field theory known
We provide initial evidence that the amplitudes suitably deformed by a spectral parameter regulate the theory while preserving the full superconformal symmetry
This seemingly enables us to derive an arbitrary deformed non-maximally helicity violating (MHV) amplitude, but there is a caveat: since we are currently lacking a deformed version of the BCFW recursion relation, we are not able to recombine these deformed on-shell diagrams into what should be called deformed non-MHV scattering amplitudes
Summary
In the present paper we aim at unifying these developments with the observation of Zwiebel [37] who connected the tree-level four-point MHV scattering amplitude to the oneloop dilatation operator of N = 4 SYM The latter being the Hamiltonian of an integrable spin-chain is generated by an R-matrix satisfying the Yang-Baxter equation, the cornerstone of the Quantum Inverse Scattering Method, see e.g. Our result reproduces the harmonic action form of the Hamiltonian, up to an overall minus sign
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