Finite Time Vacuum Survival Amplitude and Vacuum Energy Decay

The problem of the vacuum energy decay is studied for both signs of the cosmological constant, through the analysis of the vacuum survival amplitude, defined in terms of the {\em conformal time}, $z$, by ${\mathcal A}(z,z^\prime)\equiv <$\text{vac}$,z|$\text{vac}$,z^\prime>$. Transition amplitudes are computed for finite time-span, $Z\equiv z^{\mathrm{\prime }}-z$, and their {\em late time} behavior (directly related to the putative decay width of the state) as well as the transients are discussed up to first order in the coupling constant, $\lambda$.