Abstract
We continue the study of four-point correlation functions by the hexagon tessellation approach initiated in [38] and [39]. We consider planar tree-level correlation functions in mathcal{N}=4 supersymmetric Yang-Mills theory involving two non-protected operators. We find that, in order to reproduce the field theory result, it is necessary to include SU(N) colour factors in the hexagon formalism; moreover, we find that the hexagon approach as it stands is naturally tailored to the single-trace part of correlation functions, and does not account for multi-trace admixtures. We discuss how to compute correlators involving double-trace operators, as well as more general 1/N effects; in particular we compute the whole next-to-leading order in the large-N expansion of tree-level BMN two-point functions by tessellating a torus with punctures. Finally, we turn to the issue of “wrapping”, Lüscher-like corrections. We show that SU(N) colour-dressing reproduces an earlier empirical rule for incorporating single-magnon wrapping, and we provide a direct interpretation of such wrapping processes in terms of mathcal{N}=2 supersymmetric Feynman diagrams.
Highlights
The first objects to be studied in the framework of the AdS/CFT correspondence [1,2,3] were correlation functions of BPS operators in N = 4 supersymmetric Yang-Mills theory (N = 4 SYM), i.e. gauge-invariant composite operators without anomalous dimensions
In order to reproduce the field theory result, it is necessary to include SU(N ) colour factors in the hexagon formalism; we find that the hexagon approach as it stands is naturally tailored to the single-trace part of correlation functions, and does not account for multi-trace admixtures
The paper is organised as follows: in section 2, we review the computation of four-point functions at tree level in field theory for the case at hand; we introduce the DrukkerPlefka restricted kinematics [46, 47], which is natural for the hexagon formalism [19] and which we will employ for several computations throughout the paper
Summary
We propose that the correct way to account for this is to include SU(N ) colour factors in the hexagon formalism This prescription allows us to non-trivially reproduce several field-theory results, but it automatically incorporates the empirical rules for wrapping at one-loop proposed in ref. We see that the hexagon formalism does not capture multi-trace admixtures, even when those give leading effects in the 1/N expansion in field theory This is not entirely surprising, given that the whole integrability approach is naturally tailored to single-trace operators.
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