Regular Wilson loops and MHV amplitudes at weak and strong coupling

TL;DR

This paper introduces a new, regularisation-independent way to analyze Wilson loops and MHV amplitudes in N=4 SYM, connecting weak and strong coupling regimes through ratios of Wilson loops and their relation to integrable systems.

Contribution

It proposes a novel approach based on Wilson loop ratios that are finite, conformally invariant, and regularisation independent, bridging weak and strong coupling analyses.

Findings

Computed Wilson loop ratios for polygons with 6, 8, and 10 points at one and two loops.

Compared weak coupling results with recent strong coupling expressions.

Established a direct relation between these ratios and the free energy of an integrable system.

Abstract

Traditionally, the duality between Wilson loops and amplitudes beyond one loop in N=4 SYM is characterised by the remainder function. Because of the perturbative origins of the BDS expression, the remainder function is more natural at weak than at strong coupling. We advocate instead a more direct approach, based on considering ratios of Wilson loops. This allows us to define a manifestly finite, regularisation independent, conformally invariant quantity. It does not make a direct reference to the BDS expression and the definition is regularisation independent. It is a natural object at weak and at strong coupling, and in the latter case is directly related to the free energy of an auxiliary integrable system. We then compute these ratios for continuous families of regular polygons for 6,8 and 10 points at one and two-loops. These results are compared to expressions derived recently at strong coupling.