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A Two-Loop Octagon Wilson Loop in N = 4 SYM

Vittorio Del Duca, Claude Duhr, Vladimir A. Smirnov

TL;DR

This work delivers the first analytic computation of the two-loop remainder for an eight-edged Wilson loop in planar N=4 SYM, expressed in χ-kinematics as a constant plus a product of four logarithms. By employing Mellin–Barnes representations and exploiting Regge-exactness in the quasi-multi-Regge kinematics, the authors obtain a compact, weight-four remainder function R_{8,WL}^{(2)}(χ^+,χ^-) with manifest symmetries. They compare the result to the strong-coupling octagon remainder and find that, contrary to a proposed universality, the weak- and strong-coupling functions do not coincide, though their rescaled forms are only approximately aligned. The findings sharpen our understanding of the coupling dependence of polygonal Wilson loop remainders and motivate further connections to approaches such as the OPE for Wilson loops.

Abstract

In the planar N = 4 supersymmetric Yang-Mills theory at weak coupling, we perform the first analytic computation of a two-loop eight-edged Wilson loop embedded into the boundary of AdS3. Its remainder function is given as a function of uniform transcendental weight four in terms of a constant plus a product of four logarithms. We compare to the strong-coupling result, and test a conjecture on the universality of the remainder function proposed in the literature.

A Two-Loop Octagon Wilson Loop in N = 4 SYM

TL;DR

This work delivers the first analytic computation of the two-loop remainder for an eight-edged Wilson loop in planar N=4 SYM, expressed in χ-kinematics as a constant plus a product of four logarithms. By employing Mellin–Barnes representations and exploiting Regge-exactness in the quasi-multi-Regge kinematics, the authors obtain a compact, weight-four remainder function R_{8,WL}^{(2)}(χ^+,χ^-) with manifest symmetries. They compare the result to the strong-coupling octagon remainder and find that, contrary to a proposed universality, the weak- and strong-coupling functions do not coincide, though their rescaled forms are only approximately aligned. The findings sharpen our understanding of the coupling dependence of polygonal Wilson loop remainders and motivate further connections to approaches such as the OPE for Wilson loops.

Abstract

In the planar N = 4 supersymmetric Yang-Mills theory at weak coupling, we perform the first analytic computation of a two-loop eight-edged Wilson loop embedded into the boundary of AdS3. Its remainder function is given as a function of uniform transcendental weight four in terms of a constant plus a product of four logarithms. We compare to the strong-coupling result, and test a conjecture on the universality of the remainder function proposed in the literature.

Paper Structure

This paper contains 9 sections, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The two-loop Wilson loop
  3. Definitions
  4. The two-loop eight-edged Wilson loop
  5. The quasi-multi-Regge kinematics of four-of-a-kind along the ladder
  6. The octagon remainder function in $\chi$ kinematics
  7. Comparison to the strong coupling result
  8. Conclusion
  9. Eight-point kinematics

Figures (2)

  • Figure 1: Plot for $\overline{R}_{8}^{(2)}$ (red) and $\overline{R}_{8}^\textrm{strong}$ (blue) as a function of $|m|$ for $\phi = 0$ (left) and $\phi = \pi/4$ (right). Note that the two curves basically overlap and that the numerical difference between them is too small to be appreciated by eye.
  • Figure 2: Plot for $\overline{R}_{8}^{(2)}$ (red) and $\overline{R}_{8}^\textrm{strong}$ (blue) as a function of $\phi$ for $|m| = 0.2$ (left) and $|m| = 0.45$ (right).