Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory (0902.2987v3)

Abstract: Tree-level scattering amplitudes in N=4 super Yang-Mills theory have recently been shown to transform covariantly with respect to a 'dual' superconformal symmetry algebra, thus extending the conventional superconformal symmetry algebra psu(2,2|4) of the theory. In this paper we derive the action of the dual superconformal generators in on-shell superspace and extend the dual generators suitably to leave scattering amplitudes invariant. We then study the algebra of standard and dual symmetry generators and show that the inclusion of the dual superconformal generators lifts the psu(2,2|4) symmetry algebra to a Yangian. The non-local Yangian generators acting on amplitudes turn out to be cyclically invariant due to special properties of psu(2,2|4). The representation of the Yangian generators takes the same form as in the case of local operators, suggesting that the Yangian symmetry is an intrinsic property of planar N=4 super Yang-Mills, at least at tree level.

• The paper demonstrates that combining conventional and dual superconformal symmetries manifests Yangian symmetry in tree-level N=4 SYM scattering amplitudes.
• It outlines the use of on-shell superspace to derive non-local Yangian generators that parallel structures found in integrable spin chain models.
• The study speculates on extending the symmetry to loop-level amplitudes, suggesting that deformed Yangian structures could simplify complex quantum field computations.

Yangian Symmetry in Scattering Amplitudes within N = 4 Super Yang-Mills Theory

The paper "Yangian Symmetry of Scattering Amplitudes in N = 4 Super Yang-Mills Theory," authored by James Drummond, Johannes Henn, and Jan Plefka, explores the complex relationship between integrability and symmetry in the context of scattering amplitudes in N = 4 supersymmetric Yang-Mills (SYM) theory. This work is situated within the broader AdS/CFT framework, extending the understanding of symmetries in such highly symmetric field theories.

N = 4 SYM is known for its rich mathematical structure, characterized by maximal supersymmetry and conformal invariance. These properties are preserved even at the quantum level with no necessity for renormalization of the coupling constant, which is rather unusual in typical gauge theories. Within its planar limit, N = 4 SYM exhibits hidden integrability that has revolutionized the comprehension of its dynamics. In this regard, the paper in question explores how this integrability manifests in the context of scattering amplitudes through Yangian symmetry.

Dual Superconformal Symmetry and Scattering Amplitudes

A significant advancement discussed in the paper is the dual superconformal symmetry of scattering amplitudes in N = 4 SYM, initially found by examining the structure of Maximally Helicity Violating (MHV) amplitudes. This symmetry, along with conventional superconformal symmetry, suggests a profound algebraic structure governed by a Yangian symmetry, a type of quantum group symmetry first recognized in integrable systems in two dimensions.

The analysis begins with a review of dual superconformal symmetry. The authors derive the action of dual superconformal generators on amplitudes in on-shell superspace, ensuring these amplitudes remain invariant. They explore the interplay between conventional and dual superconformal symmetries and identify the emergence of a Yangian algebra through the commutation of these symmetry generators.

Yangian Algebra and its Implications

The paper demonstrates that the combination of conventional and dual superconformal symmetries leads to a Yangian structure at tree-level in N = 4 SYM. The generators of the Yangian algebra exhibit non-local characteristics, cyclically invariant under the action on scattering amplitudes. This intrinsic property is attributed to the special properties of the underlying psu(2,2|4) Lie superalgebra, prevalent in the theory’s formulation. The authors show that Yangian generators have similar formulations and implications as in other integrable spin chain systems, strengthening the identification of N = 4 SYM as an integrable system at planar limit.

The paper also speculates on higher-order implications, addressing the potential extension of Yangian symmetry to loop-level calculations. Loop corrections complicate direct symmetry applications due to infrared divergences; however, understanding these corrections through a deformed Yangian symmetry could illuminate new pathways in quantum field theories.

Numerical Results and Bold Claims

Several intricate proofs reinforce the authors' assertions. They present the closure of the dual superconformal symmetry algebra as a manifestation of the Yangian symmetry. Furthermore, the distinctive presentation of generators in super-twistor space highlights second-order differential operators, mapping out a clearer picture of integration within twistor frameworks. Importantly, the text aligns well with existing results from integrable spin chains and the AdS/CFT correspondence, highlighting theoretical plausibility and practical relevance.

Future Directions and Theoretical Speculation

Practical implications are also touched upon, including the potential simplification in computing higher-point and higher-loop amplitudes. The identification of dual conformal anomalies as a path to understand deviations at loop levels underscores future research directions: an exploration of novel symmetries may uncover new facets in quantum theories or contribute to resolving current computational challenges.

In conclusion, this paper successfully bridges the gap between symmetries in gauge theories and integrability, underscoring the emergence of a Yangian structure in N = 4 SYM. By augmenting the understanding of well-established symmetries and integrating them into a Yangian framework, it opens new perspectives on the structural underpinnings of field theories, potentially impacting computational techniques and theoretical approaches within the field of super Yang-Mills theories and beyond.