Null Wilson loops with a self-crossing and the Wilson loop/amplitude conjecture

TL;DR

This paper investigates the singularities and analytic structure of self-crossing null Wilson loops and their relation to scattering amplitudes, revealing divergences and branch cuts that inform the Wilson loop/amplitude conjecture.

Contribution

It provides a detailed analysis of the singular behavior and analytic continuation of Wilson loops with self-crossings and their connection to amplitude deviations from the BDS conjecture.

Findings

Self-crossing Wilson loops develop additional 1/ε singularities.

The remainder function R diverges as iπ log^3(1-u_2) near the self-crossing limit.

R exhibits a branch cut in the negative u_2 axis, indicating complex analytic structure.

Abstract

The present study illuminates the relation between null cusped Wilson loops and their corresponding amplitudes. We find that, compared to the case with no self-crossing, the one loop expectation value of a self-intersecting Wilson loop develops an additional 1/\epsilon singularity associated to the intersection. Interestingly, the same 1/\epsilon pole exists in the finite part of the one loop amplitude, appearing in the BDS conjecture, at the corresponding kinematic limit. At two loops, we explore the behaviour of the remainder function R, encoding the deviation of the amplitude from the BDS conjecture. By analysing the renormalisation group equations for the Wilson loop with a simple self-crossing, we argue that, when approaching the configuration with a self-crossing (u_2 \to 1, u_1\approx u_3), R diverges in the imaginary direction like R ~ i \pi \log^3(1-u_2). This behaviour can be attributed to the non-trivial analytic continuation needed when passing from the Euclidean to the physical region and suggests that R has a branch cut in the negative u_2 axis when the two other cross ratios are approximately equal (u_1 \approx u_3).