Notes on the scattering amplitude / Wilson loop duality (1010.1167v1)

Abstract: We consider the duality between the four-dimensional S-matrix of planar maximally supersymmetric Yang-Mills theory and the expectation value of polygonal shaped Wilson loops in the same theory. We extend the duality to amplitudes with arbitrary helicity states by introducing a suitable supersymmetric extension of the Wilson loop. We show that this object is determined by a host of recursion relations, which are valid at tree level and at loop level for a certain "loop integrand" defined within the Lagrangian insertion procedure. These recursion relations reproduce the BCFW ones obeyed by tree-level scattering amplitudes, as well as their extension to loop integrands which appeared recently in the literature, establishing the duality to all orders in perturbation theory. Finally, we propose that a certain set of finite correlation functions can be used to compute all first derivatives of the logarithm of MHV amplitudes.

• The paper extends the duality to all multiplicities and helicities by introducing a supersymmetric Wilson loop framework.
• It employs tree-level BCFW recursion and loop-level Lagrangian insertion to confirm the duality across perturbative orders.
• Results refine MHV amplitude computations by enabling the calculation of all first derivatives of the logarithm of amplitudes, guiding future research.

Analyzing the Duality Between Scattering Amplitudes and Wilson Loops in N=4 SYM Theory

The paper by S. Caron-Huot revisits the intriguing duality between scattering amplitudes in N=4 super Yang-Mills (SYM) theory and polygonal Wilson loops. The paper explores this duality within four-dimensional space-time for planar maximally supersymmetric Yang-Mills theory, proposing extensions and confirmations of the duality that broaden our understanding of both theoretical frameworks.

Supersymmetric Extension and Recursion Relations

Caron-Huot extends the duality to include all multiplicities and helicities by introducing a supersymmetric extension of the Wilson loop. This extension recognizes the recursion relations at tree and loop levels. Specifically, at tree level, the BCFW recursion relations that are widely known in the context of scattering amplitudes emerge naturally within this framework. Moreover, these recursion relations extend to loop integrands by employing a Lagrangian insertion procedure, thus confirming the duality to all orders in perturbation theory.

Results on MHV Amplitudes

The paper further refines the understanding of the maximally helicity violating (MHV) amplitudes. By proposing finite correlation functions, it establishes a method to compute all first derivatives of the logarithm of MHV amplitudes. This insight not only aids in the computation of amplitudes but also offers potential pathways for future research exploration.

Implications and Future Directions

The work has profound implications for the understanding of the integrability of N=4 SYM theories and their dualities. The connection established between Wilson loops and scattering amplitude anticpiates significant effects on practical computations within this theory. Moreover, the development of the supersymmetric extension offers new tools that can facilitate calculations for more complex, higher-order amplitudes.

Anticipating further development, this research encourages future investigations to explore other supersymmetric theories where similar dualities may exist. Furthermore, the integration of these findings within the broader landscape of string theory and the AdS/CFT correspondence could unravel being a deeper, more unified understanding of quantum field theories.

This research contributes a comprehensive framework that stands to influence the landscape of theoretical physics, providing significant insights which could prove instrumental in addressing complex problems within the domain of quantum field theories. Researchers engaged in this field may find these developments helpful in extending the reach and applicability of the SYM theories and their intrinsic dualities.