On collinear factorization of Wilson loops and MHV amplitudes in N=4 SYM

TL;DR

The paper investigates collinear factorization of Wilson loops in the planar N=4 SYM setting, linking MHV amplitudes to light-like Wilson loops across weak and strong coupling. It shows that at one loop the two-gluon Splitting function can be extracted from Wilson-loop diagrams with universal structure and explicit cancellations, aligning with the BDS framework; at strong coupling, it proves an anomalous Ward identity that constrains finite parts and supports BDS-like behavior for low-point functions. It further generalizes to a multi-splitting scenario with periodic collinear gluons, deriving a z-dependent finite function that does not scale with the number of collinear gluons, and discusses edge effects and the prospects for proving factorization to all orders. The work outlines routes to a rigorous all-orders proof, including potential gauge choices and numerical evidence from higher-loop studies, highlighting the ongoing interplay between perturbative and AdS/CFT perspectives on Wilson-loop factorization.

Abstract

We consider the (multi) Splitting function of Wilson loops and MHV gluon scattering S matrix elements in N=4 SYM. At strong coupling, one can utilize the methods of Alday and Maldacena and at weak coupling (one loop) the correspondence to light like Wilson loops is used. In both cases, the (multi) Splitting function corresponds to flattened cusps in the light like polygon, allowing for a clean disentanglement from the other gluons. We compute it in some cases and estimate some terms in other cases. We also prove the anomalous Ward identity of Drummond et al. in the strong coupling regime. Lastly, we briefly comment on a possible strategy for a proof of collinear factorization of Wilson loops at higher orders of perturbation theory.

Figures (7)

• Figure 1: The general, conjectured, correspondence between gluon scattering amplitudes and Wilson loops associates edges of the Wilson loop to momenta of gluons. It was shown to hold at one loop for any number of external legs Brandhuber:2007yx.
• Figure 2: The two contributions at one loop to the difference of finite parts in the collinear limit.
• Figure 3: The type of Wilson loop we consider. $v_1$ and $v_2$ are in the collinear limit and the lines $l_1$ and $l_2$ are the adjacent lines to the sequence of collinear gluons.
• Figure 4: The definitions of $p,q,P,Q,s,t$ applied for a one loop contribution to a hexagon Wilson loop. Solid lines are part of the contour defining the Wilson loop and the dashed lines just denote some effective momenta.
• Figure 5: We have to look for a minimal surface ending (on the boundary of $AdS_5$) on such a periodic piecewise linear function.