MHV Amplitudes in N=4 Super Yang-Mills and Wilson Loops

TL;DR

The paper shows that one-loop MHV amplitudes in N=4 SYM for any number of external legs can be computed from a lightlike Wilson loop, with the amplitude factorizing into the tree amplitude times a universal helicity-blind function of kinematics. The finite part of this function is captured by a sum of finite two-mass-easy box functions, while IR divergences arise from cusp-related adjacent segments; for four points, the authors reproduce the all-orders in epsilon box result. This Wilson-loop approach cleanly separates IR and finite pieces and reveals a direct mapping between Wilson-loop diagrams and box functions, supporting the BDS exponentiation and the AdS/CFT-inspired duality between scattering amplitudes and Wilson loops. The work offers both a practical calculation strategy and deeper structural insight into planar N=4 SYM amplitudes, suggesting avenues for extending the correspondence to higher loops.

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.

Figures (3)

• Figure 1: A two-mass easy box function. The momenta $p$ and $q$ are null, whereas, in general, $P^2\neq 0$ and $Q^2\neq 0$. The cases when either $P^2$ or $Q^2$, or both, are also null, correspond to the one-mass and zero-mass boxes, obtained as smooth limits from the expression \ref{['2mebst']} of the two-mass box function.
• Figure 2: A one-loop correction to the Wilson loop, where the gluon stretches between two lightlike momenta meeting at a cusp. Diagrams in this class provide the infrared-divergent terms in the $n$-point scattering amplitudes, given in \ref{['MIR']}.
• Figure 3: Diagrams in this class -- where a gluon connects two non-adjacent segments -- are finite, and give a contribution equal to the finite part of a two-mass easy box function $F^{\rm 2me} (p, q, P, Q)$, second line of \ref{['2mebst']}. $p$ and $q$ are the massless legs of the two-mass easy box, and correspond to the segments which are connected by the gluon. The diagram depends on the other gluon momenta only through the combinations $P$ and $Q$.