  We develop a functional renormalization group approach which describes the
low-energy single-particle properties of the Anderson impurity model up to
intermediate on-site interactions $U \lesssim 15 \Delta$, where $\Delta$ is the
hybridization in the wide-band limit. Our method is based on a generalization
of a method proposed by Sch\"{u}tz, Bartosch and Kopietz [Phys. Rev. B 72,
035107 (2005)], using two independent Hubbard-Stratonovich fields associated
with transverse and longitudinal spin fluctuations. Although we do not
reproduce the exponentially small Kondo scale in the limit $U \to \infty$, the
spin fluctuations included in our approach remove the unphysical Stoner
instability predicted by mean-field theory for $U > \pi \Delta$. We discuss
different decoupling schemes and show that a decoupling which manifestly
respects the spin-rotational invariance of the problem gives rise to the lowest
quasiparticle weight. To obtain a closed flow equation for the fermionic
self-energy we also propose a new truncation scheme of the functional
renormalization group flow equations using Dyson-Schwinger equations to express
bosonic vertex functions in terms of fermionic ones.
