Null Zig-Zag Wilson Loops in N=4 SYM

TL;DR

This paper demonstrates that in planar N=4 SYM, null zig-zag Wilson loops' finite parts at one-loop match those of smooth spacelike Wilson loops, extending previous rectangular results to circular and disconnected shapes.

Contribution

It extends the equivalence of null zig-zag Wilson loops to smooth spacelike loops from rectangles to circles and disconnected segments at one-loop order.

Findings

Finite parts of null zig-zag and smooth spacelike Wilson loops are equal at one-loop.

Analytic proof for circular and disconnected zig-zag contours.

Comments on potential generalizations to arbitrary shapes.

Abstract

In planar ${\cal N}=4$ supersymmetric Yang-Mills theory we have studied supersymmetric Wilson loops composed of a large number of light-like segments, i.e., null zig-zags. These contours oscillate around smooth underlying spacelike paths. At one-loop in perturbation theory we have compared the finite part of the expectation value of null zig-zags to the finite part of the expectation value of non-scalar-coupled Wilson loops whose contours are the underlying smooth spacelike paths. In arXiv:0710.1060 [hep-th] it was argued that these quantities are equal for the case of a rectangular Wilson loop. Here we present a modest extension of this result to zig-zags of circular shape and zig-zags following non-parallel, disconnected line segments and show analytically that the one-loop finite part is indeed that given by the smooth spacelike Wilson loop without coupling to scalars which the zig-zag contour approximates. We make some comments regarding the generalization to arbitrary shapes.