Nesting and Dressing

TL;DR

The paper computes all-loop anomalous dimensions for field-strength operators Tr F^L in planar N=4 SYM using an asymptotic nested Bethe ansatz. It derives a single effective integral equation in the thermodynamic limit and shows that nesting and dressing contribute similar kernel structures, suggesting a common origin. The authors hypothesize that the dressing phase arises from hidden infinite auxiliary Bethe roots describing a nontrivial BPS vacuum, and show this mechanism via an all-loop analysis of a related so(6) singlet state. They provide exact one-loop solutions for arbitrary L, derive the all-loop energy expansion with transcendental terms tied to dressing, and illustrate a generic nesting-dressing mechanism with stack dynamics, linking to Hubbard-like long-range models and AdS/CFT integrability.

Abstract

We compute the anomalous dimensions of field strength operators Tr F^L in N=4 SYM from an asymptotic nested Bethe ansatz to all-loop order. Starting from the exact solution of the one-loop problem at arbitrary L, we derive a single effective integral equation for the thermodynamic limit of these dimensions. We also include the recently proposed phase factor for the S-matrix of the planar AdS/CFT system. The terms in the effective equation corresponding to, respectively, the nesting and the dressing are structurally very similar. This hints at the physical origin of the dressing phase, which we conjecture to arise from the hidden presence of infinitely many auxiliary Bethe roots describing a non-trivial "filled" structure of the theory's BPS vacuum. We finally show that the mechanism for creating effective nesting/dressing kernels is quite generic by also deriving the integral equation for the all-loop dimension of a certain one-loop so(6) singlet state.

Figures (2)

• Figure 1: Oscillator realization and fundamental magnons. There are four adjoint complex scalars $\mathcal{X},\bar{\mathcal{X}},\mathcal{Y},\bar{\mathcal{Y}}$, four light-cone covariant derivatives, $\mathcal{D}, \bar{\mathcal{D}}, \dot \mathcal{D}, \dot{\bar{\mathcal{D}}}$, and eight adjoint fermions $\mathcal{U},\mathcal{V},\dot \mathcal{U},\dot \mathcal{V},\bar{\mathcal{U}},\bar{\mathcal{V}}, \dot{\bar{\mathcal{U}}},\dot{\bar{\mathcal{V}}}.$ The nodes of the Dynkin diagram are labeled from 1 to 7, starting at the south-west end and going clockwise until the north-east end is reached.
• Figure 2: Schematic scattering notation for the dressed asymptotic Bethe ansatz equations \ref{['MAIN']} - \ref{['FULL SET4']}. The horizontal branch represents the Dynkin diagram of $\mathfrak{psu}(2,2|4)$ and the scattering of elementary magnons. The vertical branch (dotted lines) indicates the emulation of the dressing phase by the scattering of the dressing roots among themself and the momentum carrying main node. Note that the auxiliary "dressing" roots do not scatter with any of the auxiliary "nesting" roots.