Non-supersymmetric Wilson loop in $N=4$ SYM and defect 1d CFT

Following Polchinski and Sully (https://arxiv.org/abs/1104.5077), we consider a generalized Wilson loop operator containing a constant parameter $\zeta$ in front of the scalar coupling term, so that $\zeta =0$ corresponds to the standard Wilson loop, while $\zeta =1$ to the locally supersymmetric one. We compute the expectation value of this operator for circular loop as a function of $\zeta$ to second order in the planar weak coupling expansion in $N=4$ SYM theory. We then explain the relation of the expansion near the two conformal points $\zeta =0$ and $\zeta =1$ to the correlators of scalar operators inserted on the loop. We also discuss the AdS${}_5\times S^5$ string 1-loop correction to the strong-coupling expansion of the standard circular Wilson loop, as well as its generalization to the case of mixed boundary conditions on the five-sphere coordinates, corresponding to general $\zeta$. From the point of view of the defect CFT${}_1$ defined on the Wilson line, the $\zeta$-dependent term can be seen as a perturbation driving a RG flow from the standard Wilson loop in the UV to the supersymmetric Wilson loop in the IR. Both at weak and strong coupling we find that the logarithm of the expectation value of the standard Wilson loop for the circular contour is larger than that of the supersymmetric one, which appears to be in agreement with the 1d analog of the F-theorem.