Conformal properties of four-gluon planar amplitudes and Wilson loops (0707.0243v3)

Abstract: We present further evidence for a dual conformal symmetry in the four-gluon planar scattering amplitude in N=4 SYM. We show that all the momentum integrals appearing in the perturbative on-shell calculations up to five loops are dual to true conformal integrals, well defined off shell. Assuming that the complete off-shell amplitude has this dual conformal symmetry and using the basic properties of factorization of infrared divergences, we derive the special form of the finite remainder previously found at weak coupling and recently reproduced at strong coupling by AdS/CFT. We show that the same finite term appears in a weak coupling calculation of a Wilson loop whose contour consists of four light-like segments associated with the gluon momenta. We also demonstrate that, due to the special form of the finite remainder, the asymptotic Regge limit of the four-gluon amplitude coincides with the exact expression evaluated for arbitrary values of the Mandelstam variables.

• The paper demonstrates that dual conformal symmetry underpins four-gluon amplitudes, simplifying complex scattering calculations in N=4 SYM.
• It reveals a striking correspondence between gluon amplitudes and light-like Wilson loops, linking infrared divergences with ultraviolet cusp anomalies.
• A detailed perturbative analysis up to four loops shows that conformal invariant integrals capture the dynamics, suggesting deeper integrability in the theory.

Overview of Dual Conformal Symmetry in Four-Gluon Planar Amplitudes

The paper "Conformal properties of four-gluon planar amplitudes and Wilson loops" presents a comprehensive analysis of the dual conformal symmetry observed in the four-gluon scattering amplitudes within the framework of maximally supersymmetric Yang-Mills theory ($\mathcal{N}=4$ SYM). The authors, Drummond, Korchemsky, and Sokatchev, propose that this symmetry could be instrumental in simplifying the computation of these scattering amplitudes, which play a crucial role in both theoretical and practical aspects of quantum field theory.

Key Contributions

1. Dual Conformal Symmetry: The paper posits that the four-gluon planar amplitudes possess a dual conformal symmetry—a hidden symmetry that emerges when amplitudes are expressed in terms of dual coordinates related to the momenta of the scattered particles. This symmetry is not immediately apparent in the on-shell calculations but becomes evident when analyzing the integrals off-shell.
2. Wilson Loops Correspondence: An intriguing aspect of this work is the connection between gluon scattering amplitudes and Wilson loops with light-like polygonal contours in the dual coordinate space. Specifically, the authors demonstrate that at both weak and strong coupling, the finite part of the four-gluon amplitude has a dual form corresponding to a Wilson loop with cusps.
3. Infrared (IR) and Ultraviolet (UV) Divergences: The paper elaborates on how the IR divergences of the gluon amplitudes in momentum space parallel the UV divergences found in configuration space Wilson loops. This is particularly elucidated through an analysis of the cusp anomaly and its implications for the conformal symmetry.
4. Perturbation Analysis: The authors provide a detailed loop-by-loop analysis up to four loops, discussing how dual conformal invariant integrals dominate the perturbative expansion of four-gluon amplitudes. They conjecture that these integrals, identifiable through specific conformal transformations, capture the essence of the dynamics to all loop orders.
5. Regge Limit and Gluon Trajectories: In the context of high-energy scattering, the paper addresses the Regge limit, illustrating that the results are Regge exact. The gluon Regge trajectory computed indeed straightforwardly connects with the dual Wilson loop observations, indicating a deeper integrability structure within the $\mathcal{N}=4$ SYM.

Implications and Future Directions

The confirmation of dual conformal symmetry in four-gluon amplitudes is significant because it offers avenues for the simplification of complex calculations within quantum field theory. The symmetry could represent a fundamental aspect of the $\mathcal{N}=4$ SYM theory, reflective of potential integrability in strongly interacting gauge theories.

The established connection between amplitudes and Wilson loops suggests that novel computational techniques—potentially exploiting dual conformal symmetry—could emerge, enhancing our ability to evaluate scattering processes in non-trivial backgrounds, such as those encountered in AdS/CFT correspondence scenarios.

For future developments, investigating these conformal properties in multi-gluon processes and extending the validity of these symmetries to different gauge theories beyond $\mathcal{N}=4$ SYM would be valuable. Understanding the profound theoretical underpinnings of such symmetries might unveil further connections between quantum field theories and their dual string theoretic counterparts.