Notes on BPS Wilson Loops and the Cusp Anomalous Dimension in ABJM theory

Abstract: We introduce a new purely bosonic, $\frac{1}{6}$ BPS Wilson loop for ABJM theory on $S^3$ that couples scalar fields to a latitude at an angle $\theta$ on $S^2\in\mathbb{C}P^3$. Through localization of this operator, we relate the expansion of the cusp anomalous dimension at small cusp angles to the logarithmic derivative of the ABJM Wilson loop. This defines, non-perturbatively in the 't Hooft coupling, the bremsstrahlung function $B(\lambda)$ describing in three dimensions the soft radiation of a $W$-boson undergoing a sudden change in trajectory. We compare our results for $B(\lambda)$ to the known weak/strong coupling expansions of the function $h(\lambda)$ that enters integrability. At weak coupling we precisely match the previously known two-loop result. At strong coupling we find agreement at leading order in $\sqrt{\lambda}$, but a mismatch of the constant coefficient. We comment on the striking similarity that we observe between these two, in principle, unrelated functions.