  In a renormalizable theory the survival probability of an unstable quantum
state features divergences as a consequence of the rapid growth of the density
of states with energy. Introducing a high energy cutoff $\Lambda$, the
transient dynamics during a time scale $\simeq 1/\Lambda$ describes the
renormalization of the bare into a `quasiparticle' state which decays on longer
time scales. During this early transient the decay law features Zeno behavior
$e^{-(t/t_Z)^2}$ with the Zeno time-scale $t_Z \propto 1/\Lambda^{3/2}$. We
introduce a dynamical renormalization framework that allows to separate
consistently the dynamics of formation of the quasiparticle state and its decay
on longer time scales by introducing a renormalization time scale along with
alternative `schemes'. The survival probability obeys a renormalization group
equation with respect to this scale. We find a transient \emph{suppression} of
the decay law for large Lorentz factor as a consequence of the narrowing of the
phase space for decay different from the usual time dilation. In presence of
higher mass thresholds, the energy uncertainty associated with transient
dynamics leads to an \emph{anti Zeno enhancement} with a transient acceleration
of the decay into \emph{heavier particles}. There remains memory of the
transient effects in the survival probability even at long time. We discuss
possible consequences of these effects in cosmology.
