Three-point functions in $\mathcal{N}=4$ SYM: the hexagon proposal at three loops

Basso, Komatsu and Vieira recently proposed an all-loop framework for the computation of three-point functions of single-trace operators of $\mathcal{N}=4$ super-Yang-Mills, the “hexagon program”. This proposal results in several remarkable predictions, including the three-point function of two protected operators with an unprotected one in the $SU(2)$ and $SL(2)$ sectors. Such predictions consist of an “asymptotic” part—similar in spirit to the asymptotic Bethe Ansatz of Beisert and Staudacher for two-point functions—as well as additional finite-size “wrapping” L"uscher-like corrections. The focus of this paper is on such wrapping corrections, which we compute at three-loops in the $SL(2)$ sector. The resulting structure constants perfectly match the ones obtained in the literature from four-point correlators of protected operators.