On planar gluon amplitudes/Wilson loops duality (0709.2368v2)

Abstract: There is growing evidence that on-shell gluon scattering amplitudes in planar N=4 SYM theory are equivalent to Wilson loops evaluated over contours consisting of straight, light-like segments defined by the momenta of the external gluons. This equivalence was first suggested at strong coupling using the AdS/CFT correspondence and has since been verified at weak coupling to one loop in perturbation theory. Here we perform an explicit two-loop calculation of the Wilson loop dual to the four-gluon scattering amplitude and demonstrate that the relation holds beyond one loop. We also propose an anomalous conformal Ward identity which uniquely fixes the form of the finite part (up to an additive constant) of the Wilson loop dual to four- and five-gluon amplitudes, in complete agreement with the BDS conjecture for the multi-gluon MHV amplitudes.

• The paper presents an explicit two-loop calculation of Wilson loops that validates their duality with four-gluon scattering amplitudes in planar N=4 SYM.
• The paper introduces an anomalous conformal Ward identity that constrains the finite part of the Wilson loop and aligns with the BDS conjecture for MHV amplitudes.
• The paper computes the cusp and collinear anomalous dimensions to two loops, offering deeper insights into the structural dynamics of supersymmetric gauge theories.

On Planar Gluon Amplitudes/Wilson Loops Duality: An Overview

This paper elucidates the growing body of evidence supporting the duality between planar gluon scattering amplitudes and Wilson loops in planar $\mathcal{N}=4$ SYM theory. The authors undertake a methodical investigation of this duality, beginning with a review of its initial hypothesis via the AdS/CFT correspondence and its subsequent verification at weak coupling to one loop in perturbation theory. Central to the paper is a detailed, explicit two-loop calculation of a Wilson loop which corresponds to the four-gluon scattering amplitude, confirming the duality's validity beyond one loop and thereby extending the previous findings.

Key Contributions

1. Two-Loop Calculation: The authors develop an explicit two-loop calculation of the Wilson loop, demonstrating its consistency with the dual four-gluon amplitude. This calculation showcases the duality's robustness and the intricate relationships underpinning these seemingly distinct theoretical constructs.
2. Conformal Ward Identity: The paper proposes an anomalous conformal Ward identity, which is significant as it constrains the form of the Wilson loop's finite part. This identity agrees with the Beisert-Dixon-Smirnov (BDS) conjecture for multi-gluon maximally helicity violating (MHV) amplitudes, which is an essential aspect of understanding the behavior of these amplitudes.
3. Implications for Anomalous Dimensions: Drawing on the conformal properties highlighted in their analysis, the authors calculate the cusp anomalous dimension and the collinear anomalous dimension to two loops. This provides further insight into the structural dynamics of $N=4$ SYM, contributing to the broader understanding of quantum field theories with extended symmetries.
4. Conformal Symmetry and its Consequences: The discussion extends into the field of how conformal symmetry at the level of Wilson loops influences known results. This symmetry imposes form constraints on amplitude calculations, revealing a potential pathway to further analytical understanding.

Implications and Speculations

This work strengthens the duality's foundational aspects, offering a connection between disparate regions of theoretical physics: Wilson loops, which are central in gauge theories, and scattering amplitudes, a cornerstone in perturbative quantum field theory. The implications extend toward a better grasp of the AdS/CFT correspondence—a pivotal conjecture unifying aspects of string theory and quantum field theories. Particularly, the authors' treatment suggests that many intricate properties of four-dimensional gauge theories could be encoded in geometric and combinatorial forms more typically associated with string theory paradigms.

Future Directions

Going forward, the authors' framework could guide calculations of Wilson loops at higher loops; this might reveal further substantial characteristics about the correspondence between gauge theories and string theories under the regime of dual conformal symmetry. Moreover, studying the ramifications within non-abelian gauge theories and exploring potential implications for integrability in $\mathcal{N}=4$ SYM might yield novel insights into exact result derivations, enabling new discoveries in quantum chromodynamics (QCD).

In conclusion, this paper consolidates and expands our understanding of the duality between planar gluon amplitudes and Wilson loops, underlining the theoretical elegance and physical relevance of these constructs in the ambit of supersymmetric gauge theories. As our grasp of these dualities improves, so too will our ability to tackle more complex phenomena within the universe using the powerful frameworks laid out in quantum field theory contexts.