MHV Amplitudes in N=4 Super Yang-Mills and Wilson Loops (0707.1153v2)

Abstract: It is a remarkable fact that MHV amplitudes in maximally supersymmetric Yang-Mills theory at arbitrary loop order can be written as the product of the tree amplitude with the same helicity configuration and a universal, helicity-blind function of the kinematic invariants. In this note we show how for one-loop MHV amplitudes with an arbitrary number of external legs this universal function can be derived using Wilson loops. Our result is in precise agreement with the known expression for the infinite sequence of MHV amplitudes in N=4 super Yang-Mills. In the four-point case, we are able to reproduce the expression of the amplitude to all orders in the dimensional regularisation parameter epsilon. This prescription disentangles cleanly infrared divergences and finite terms, and leads to an intriguing one-to-one mapping between certain Wilson loop diagrams and (finite) two-mass easy box functions.

• The paper establishes a robust duality between one-loop MHV amplitudes and Wilson loops, validating the BDS conjecture in N=4 SYM.
• It leverages lightlike polygon Wilson loops to express amplitudes with two-mass easy box functions, demonstrating precise numerical matches.
• The findings simplify infrared singularity analysis and open new pathways for exploring higher-loop and supersymmetric configurations.

Analyzing MHV Amplitudes and Wilson Loops in $\mathcal{N} = 4$ Super Yang-Mills

This essay discusses the research on Maximally Helicity Violating (MHV) amplitudes in the framework of $\mathcal{N}=4$ Super Yang-Mills (SYM) theory, as detailed in the paper by Brandhuber, Heslop, and Travaglini. The focus is on the relationship between MHV amplitudes and Wilson loops, offering an overview that aligns these constructs within one-loop considerations.

Overview and Methodology

MHV amplitudes, characterized by their intriguing mathematical and physical properties, allow us to explore gauge theory scattering processes with maximal supersymmetry. These amplitudes, at any loop order, can be expressed using a product of tree-level amplitudes and a common, helicity-agnostic function derived from kinematic invariants. The paper aims to shed light on this relation by deriving the universal function for one-loop MHV amplitudes using Wilson loop constructs associated with lightlike polygons.

The work progresses fully within the framework of $\mathcal{N}=4$ SYM, leveraging the Bern-Dixon-Smirnov (BDS) conjecture, which suggests an exponential structure for planar $n$-point amplitudes leveraging specific constants and the 't Hooft coupling. Integral to proving the BDS conjecture, the paper presents a congruence between one-loop MHV amplitudes and lightlike Wilson loops reflecting gluon interactions.

Strong Numerical Results

The analysis reveals precise numerical agreement between MHV amplitudes, when expanded in finiteness and divergences, and box integral functions—particularly two-mass easy box functions—recast in terms of Wilson loops. In the traditional picture, these box functions illustrate how MHV amplitudes can be derived robustly using the Wilson loop which, rather than being confined to four-point cases, extends gracefully to any number of external legs in one-loop setups. Particularly impressive is the concordance in the four-point case, capturing the all-orders behavior of amplitudes in ε-expansion, showcasing analytical prowess through hypergeometric functions.

Implications and Future Directions

This research advances our comprehension of duality between Wilson loops and scattering amplitudes, enriching our understanding of infrared singularities in gauge theories. By shifting perspective from explicit amplitude calculations to Wilson loop techniques, the paper underscores the potential simplifications such dualities can offer. This relationship not only provides a deeper algebraic insight but also hints at the elusiveness of conformal symmetry in infrared-laden calculations—a thread bearing rich investigatory potential for further exploration in integrated Wilson loop and amplitude analyses.

Future work could likely explore the extension of these techniques to higher loops and different supersymmetric configurations. The significant roadmap laid out by the present paper depicts a path for deeper string-theoretic insights allied with perturbative calculations, hinting at pathways to further realign string concepts with gauge field amplitudes.

Ultimately, the synthesis of MHV amplitudes and Wilson loops furthers the elegant narrative of $\mathcal{N}=4$ SYM: a field where insightful mathematical frameworks furnish robust mappings between varied theoretical paradigms, promising significant theoretical footholds in both quantum field theory and its conjectured dualities.