Renormalization of vacuum expectation values in spontaneously broken gauge theories

We compute one-loop and two-loop beta-functions for vacuum expectation values (VEVs) in gauge theories. In R-xi gauge the VEVs renormalize differently from the respective scalar fields. We focus particularly on the origin and behaviour of this difference and show that it can be interpreted as the anomalous dimension of a certain scalar background field, leading to simple direct computation and qualitative understanding. The results are given for generic as well as supersymmetric gauge theories. These complement the set of well-known gamma- and beta-functions of Machacek/Vaughn. As an application, we compute the beta-functions for VEVs and tan beta in the MSSM, NMSSM, and E6SSM.