Abstract
Some time ago the general tree-level scattering amplitudes of N=4 Super Yang–Mills theory were expressed as certain Graßmannian contour integrals. These remarkable formulas allow to clearly expose the super-conformal, dual super-conformal, and Yangian symmetries of the amplitudes. Using ideas from integrability it was recently shown that the building blocks of the amplitudes permit a natural multi-parameter deformation. However, this approach had been criticized by the observation that it seemed impossible to reassemble the building blocks into Yangian-invariant deformed non-MHV amplitudes. In this note we demonstrate that the deformations may be succinctly summarized by a simple modification of the measure of the Graßmannian integrals, leading to a Yangian-invariant deformation of the general tree-level amplitudes. Interestingly, the deformed building blocks appear as residues of poles in the spectral parameter planes. Given that the contour integrals also contain information on the amplitudes at loop-level, we expect the deformations to be useful there as well. In particular, applying meromorphicity arguments, they may be expected to regulate all notorious infrared divergences. We also point out relations to Gelfand hypergeometric functions and the quantum Knizhnik–Zamolodchikov equations.
Highlights
We present some details on the derivation of the deformed Graßmannian formula (13) and prove that it is invariant under the action of the level-zero and the level-one Yangian generators (10)
In the previous section we have encountered the deformation of the A6,3 amplitude in terms of Lauricella hypergeometric functions
General hypergeometric functions make their appearance as solutions to the Knizhnik-Zamolodchikov equation. We suggest how the latter may be related to Yangian invariants
Summary
Nature prefers Yang-Mills theory in exactly 1+3 dimensions. There has been much recent interest in a mathematically exceedingly rich four-dimensional Yang-Mills model, the nearly unique N = 4 supersymmetric theory [1]. One is led to look for a more natural regulator, where natural means it should a) respect the fixed space-time dimensionality four and b) respect the Yangian symmetry, i.e. integrability. Such a regularization scheme was proposed in [8, 9]. Let us define shifted spectral parameters [10] Note that it is not really the Graßmannian space Gr(k, n) as such that is deformed, but the integration measure on this space. Where the subscripts indicate which matrix we consider when evaluating the minors, one proves that (8) deforms into Note that both the MHV-prefactor and the contour integral are deformed. From (14), we see that the total number of deformation parameters is k-independent and equals n−1, since (13),(16) depends only on differences of the {vj±}
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