An off-shell Wilson loop

It is well-known that on-shell maximally helicity-violating gluon scattering amplitudes in planar maximally supersymmetric Yang-Mills theory are dual to a bosonic Wilson loop on a null-polygonal contour. The light-like nature of the intervals is a reflection of the mass-shell condition for massless gluons involved in scattering. Presently, we introduce a Wilson loop prototype on a piece-wise curvilinear contour that can be interpreted in the T-dual language to correspond to nonvanishing gluon off-shellness. We analyze it first for four sites at one loop and demonstrate that it coincides with the four-gluon amplitude on the Coulomb branch. Encouraged by this fact, we move on to the two-loop order. To simplify our considerations, we only focus on the Sudakov asymptotics of the Wilson loop, when the off-shellness goes to zero. The latter serves as a regulator of short-distance divergences around the perimeter of the loop, i.e., divergences when gluons are integrated over a small vicinity of the Wilson loop cusps. It does not however regulate conventional ultraviolet divergences of interior closed loops. This unavoidably introduces a renormalization scale dependence and thus scheme dependence into the problem. With a choice of the scale setting and a finite renormalization, we observe exponentiation of the double logarithmic scaling of the Wilson loop with the accompanying exponent being given by the so-called hexagon anomalous dimension, which recently made its debut in the origin limit of six-leg gluon amplitudes. This is contrary to the expectation for the octagon anomalous dimension to rather emerge from our analysis suggesting that the current object encodes physics different from the Coulomb branch scattering amplitudes.