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Null Zig-Zag Wilson Loops in N=4 SYM

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.