The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space (1009.2225v3)

Abstract: We propose that the complete planar S-matrix of N=4 super Yang-Mills - including all N^kMHV partial amplitudes to all loops - is equivalent to the correlation function of a supersymmetric Wilson loop in twistor space. Remarkably, the entire classical S-matrix arises from evaluating the correlation function in the self-dual sector, while the expansion of the correlation function in powers of the Yang-Mills coupling constant provides the loop expansion of the amplitudes. We support our proposal with explicit computations of the n particle NMHV and NNMHV trees, the integrands of the 1-loop MHV and NMHV amplitudes, and the n particle 2-loop MHV amplitude. These calculations are performed using the twistor action in axial gauge. In this gauge, the Feynman diagrams of the correlation function are the planar duals of the usual MHV diagrams for the scattering amplitude. The results are presented in the form of a sum of products of dual superconformal invariants in (momentum) twistor space, and agree with the expressions derived in arXiv:1009.1854 directly from the MHV rules. The twistor space Wilson loop is a natural supersymmetric generalization of the standard Wilson loop used to compute MHV amplitudes. We show how the Penrose-Ward transform can be used to determine a corresponding supersymmetrization on space-time.

• The paper proposes a novel framework representing the complete planar S-matrix of N=4 SYM theory as a supersymmetric Wilson loop correlation function in twistor space.
• The authors validate this conjecture with explicit computations using the twistor action, showing agreement with known tree and loop level amplitudes like NMHV and N2MHV.
• This work provides a geometrically intuitive dual picture for scattering amplitudes, potentially simplifying computations and offering new ways to handle divergences.

The Complete Planar S-Matrix of $N=4$ SYM as a Wilson Loop in Twistor Space

The paper by Lionel Mason and David Skinner develops a novel framework to express the complete planar S-matrix for $N=4$ super Yang-Mills (SYM) theory through the lens of a supersymmetric Wilson loop in twistor space. This approach is striking for its holistic nature, capturing all N$^k$MHV partial amplitudes across tree and loop levels.

The authors' main proposal is a bold conjecture that all $n$-particle planar amplitudes in $N=4$ SYM can be represented through a correlation function of a Wilson loop when translated into twistor space. This proposal is informed by the pre-existing hypothesis linking Wilson loops with MHV amplitudes in dual space-time and builds on foundational work suggesting that the area of minimal surfaces within AdS$_5$ correlates with planar MHV amplitudes at strong coupling.

This paper validates the conjecture with explicit computations, strategically utilizing the twistor action in an axial gauge. Employing the twistor action facilitates aligning Feynman diagrams of correlation functions with the planar duals of the conventional MHV diagrams for scattering amplitudes, simplifying the computational aspect of the proposal. The core of their results focuses on leveraging the structure of dual superconformal invariants wrought in momentum twistor space to express results matching those from the direct application of MHV rules.

Central to this paradigm shift is the introduction of a supersymmetric Wilson loop on twistor space, embodied fully in the $N=4$ twistor superfield $A(Z, \chi)$. This loop can be seen as a supersymmetric analogue of the classical twistor space Wilson loop, which allows spacetime Wilson loop frameworks to incorporate the full N$^k$MHV structure of scattering amplitudes into the dual space-time. The twistor space Wilson loop is analyzed using a superconnection derived from a novel form of off-shell Penrose-Ward transformation, which is reflected in the construction of a chiral superfield connection on space-time.

The paper does not shy away from computational rigor. It explicitly shows NMHV and N$^2$MHV tree amplitudes through detailed analysis of two- and four-point correlation functions, respectively. At loop levels, the MHV and NMHV amplitudes are deduced by integrating over auxiliary twistor lines, representing MHV vertices on space-time. The results convincingly resonate with known formulations in the literature, such as the conventional MHV rules in momentum space.

Furthermore, the results suggest that all N$^k$MHV amplitudes could be obtained through this dual picture, with loop integrals being inherently represented within twistor space — offering a promising avenue for future research to address the potential divergence problems with regularizations directly within this framework.

In essence, this work represents a significant advancement in the representation of planar scattering amplitudes, transforming highly intricate amplitude formulations into geometrically intuitive constructs within twistor space. It sets a robust precedent for translating between twistor space operators and field theories of scattering amplitudes. Future endeavors could further refine the spacetime superconnection framework, potentially advancing computational techniques in planar $N=4$ SYM and even exploring perturbative expansions beyond the planar limits. The implications extend beyond computational efficiency: they offer a conceptual consolidation of ideas bridging Wilson loops, MHV rules, and twistor space geometries, a unifying perspective resonant across contemporary research in quantum field theories.