Vacuum functional ($Z[0]$)

The zero-source generating functional is the vacuum state persistence amplitude, conventionally written $Z[0]=\int \mathcal{D}\varphi e^{iS[\varphi ]}$ with a suitable Feynman i-epsilon prescription. Dividing by $Z[0]$ removes disconnected Vacuum Feynman diagrams from normalized Green functions. Adding a constant vacuum energy density to the Lagrangian density changes $Z[0]$ by a spacetime-volume phase but leaves these normalized Green functions unchanged.