RG flows and stability in defect field theories

We investigate defects in scalar field theories in four and six dimensions in a double-scaling (semiclassical) limit, where bulk loops are suppressed and quantum effects come from the defect coupling. We compute fi-functions up to four loops and find that fixed points satisfy dimensional disentanglement - i.e. their dependence on the space dimension is factorized from the coupling dependence - and discuss some physical implications. We also give an alternative derivation of the fi functions by computing systematic logarithmic corrections to the Coulomb potential. In this natural scheme, fi functions turn out to be a gradient of a `Hamiltonian' function H. We also obtain closed formulas for the dimension of scalar operators and show that instabilities do not occur for potentials bounded from below. The same formulas are reproduced using Rigid Holography.