  The inherent complexity of system-bath interactions often requires making
critical approximations, which we here show to have a radical influence on the
renormalization group flow and the resulting phase diagram. Specifically, for
the Caldeira-Leggett model Schmid and Bulgadaev (SB) predicted a phase
transition, whose experimental verification in resistive superconducting
circuits is currently hotly debated. For normal metal and Josephson junction
array resistors, we show that the mapping to Caldeira-Leggett is only exact
when artificially decompactifying the superconducting phase. Retaining phase
compactness immediately leads to a phase diagram depending on four instead of
two parameters. While we still find an SB-like transition in the transmon
regime, the critical parameter is controlled exclusively by the capacitive
coupling. In contrast, the Cooper pair box maps to the anisotropic Kondo model,
where a pseudo-ferromagnetic phase is not allowed for regular electrostatic
interactions.
