Higher order RG flow on the Wilson line in $\mathcal{N}=4$ SYM

Extending earlier work, we find the two-loop term in the beta-function for the scalar coupling $ζ$ in a generalized Wilson loop operator of the $\mathcal{N}=4$ SYM theory, working in the planar weak-coupling expansion. The beta-function for $ζ$ has fixed points at $ζ=\pm 1$ and $ζ=0$, corresponding respectively to the supersymmetric Wilson-Maldacena loop and to the standard Wilson loop without scalar coupling. As a consequence of our result for the beta-function, we obtain a prediction for the two-loop term in the anomalous dimension of the scalar field inserted on the standard Wilson loop. We also find a subset of higher-loop contributions (with highest powers of $ζ$ at each order in `t Hooft coupling) coming from the scalar ladder graphs determining the corresponding terms in the five-loop beta-function. We discuss the related structure of the circular Wilson loop expectation value commenting, in particular, on consistency with a 1d defect version of the F-theorem. We also compute (to two loops in the planar ladder approximation) the two-point correlators of scalars inserted on the Wilson line.

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