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The generalized Wilson loop operator interpolating between the supersymmetricand the ordinary Wilson loop in ${\cal N}=4$ SYM theory provides an interestingexample of renormalization group flow on a line defect: the scalar couplingparameter $\zeta$ has a non-trivial beta function and may be viewed as arunning coupling constant in a 1d defect QFT. In this paper we continue thestudy of this operator, generalizing previous results for the beta function andWilson loop expectation value to the case of an arbitrary representation of thegauge group and beyond the planar limit. Focusing on the scalar ladder limitwhere the generalized Wilson loop reduces to a purely scalar line operator in afree adjoint theory, and specializing to the case of the rank $k$ symmetricrepresentation of $SU(N)$, we also consider a certain semiclassical limit where$k$ is taken to infinity with the product $k\, \zeta^2$ fixed. This limit canbe conveniently studied using a 1d defect QFT representation in terms of $N$commuting bosons. Using this representation, we compute the beta function andthe circular loop expectation value in the large $k$ limit, and use it toderive constraints on the structure of the beta function for generalrepresentation. We discuss the corresponding 1d RG flow and comment on theconsistency of the results with the 1d defect version of the F-theorem.

Wilson loop in general representation and RG flow in 1d defect QFT