  Competing interactions are often responsible for intriguing phase diagrams in
correlated electron systems. Here we analyze the competition of instantaneous
short range Coulomb interaction $U$ with the retarded electron-electron
interaction induced by an electron-phonon coupling $g$ as described by the
Hubbard-Holstein model. The ground state phase diagram of this model in the
limit of large dimensions at half filling is established. The study is based on
dynamical mean field theory combined with the numerical renormalization group.
Depending on $U$, $g$, and the phonon frequency $\omega_0$, the ground state is
antiferromagnetically (AFM) or charge ordered (CO). We find quantum phase
transitions from the AFM to CO state to occur when $U-\lambda\simeq 0$, where
$\lambda$ characterizes the phonon induced effective attraction. The transition
is continuous for small couplings and large phonon frequencies $\omega_0$ and
becomes discontinuous for large couplings and small values of $\omega_0$. We
comment on the possible relevance of this work for
Ba${}_{1-x}$K${}_x$BiO${}_3$.
