Defects in scalar field theories, RG flows and Dimensional Disentangling

We consider defect operators in scalar field theories in dimensions $d=4-\epsilon $ and $d=6-\epsilon$ with self-interactions given by a general marginal potential. In a double scaling limit, where the bulk couplings go to zero and the defect couplings go to infinity, the bulk theory becomes classical and the quantum defect theory can be solved order by order in perturbation theory. We compute the defect $\beta $ functions to two loops and study the Renormalization Group flows. The defect fixed points can move and merge, leading to fixed point annihilation; and they exhibit a remarkable factorization property where the $\epsilon$-dependence gets disentangled from the coupling dependence.