Abstract
We consider a light-like Wilson loop in \( \mathcal{N}=4 \) SYM evaluated on a regular n-polygon contour. Sending the number of edges to infinity the polygon approximates a circle and the expectation value of the light-like WL is expected to tend to the localization result for the circular one. We show this explicitly at one loop, providing a prescription to deal with the divergences of the light-like WL and the large n limit. Taking this limit entails evaluating certain sums of dilogarithms which, for a regular polygon, evaluate to the same constant independently of n. We show that this occurs thanks to underlying dilogarithm identities, related to the so-called “polylogarithm ladders”, which appear in rather different contexts of physics and mathematics and enable us to perform the large n limit analytically.
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Bianchi, M.S., Leoni, M. Dilogarithm ladders from Wilson loops. J. High Energ. Phys. 2015, 180 (2015). https://doi.org/10.1007/JHEP02(2015)180
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DOI: https://doi.org/10.1007/JHEP02(2015)180