  We study quantum phase transitions in the Bose-Fermi Kondo problem, where a
local spin is coupled to independent bosonic and fermionic degrees of freedom.
Applying a second order expansion in the anomalous dimension of the Bose field
we analyze the various non-trivial fixed points of this model. We show that
anisotropy in the couplings is relevant at the SU(2) invariant non Fermi liquid
fixed points studied earlier and thus the quantum phase transition is usually
governed by XY or Ising-type fixed points. We furthermore derive an exact
result that relates the anomalous exponent of the Bose field to that of the
susceptibility at any finite coupling fixed point. Implications on the
dynamical mean field approach to locally quantum critical phase transitions are
also discussed.
