Quantum Spectral Curve for a cusped Wilson line in N = 4 $$ mathcal{N}=4 $$ SYM

We show that the quantum spectral curve (qsc) formalism, initially formulated for the spectrum of anomalous dimensions of all local single trace operators in $$ mathcal{n}=4 $$ sym, can be extended to the generalized cusp anomalous dimension for all values of the parameters. we find that the large spectral parameter asymptotics and some analyticity properties have to be modified, but the functional relations are unchanged. as a demonstration, we find an all-loop analytic expression for the first two nontrivial terms in the small |ϕ ± θ| expansion. we also present nonperturbative numerical results at generic angles which match perfectly 4-loop perturbation theory and the classical string prediction. the reformulation of the problem in terms of the qsc opens the possibility to explore many open questions. we attach to this paper several mathematica notebooks which should facilitate future studies.