 Solution

The scalar field path integral integrates over real field histories with vacuum boundary conditions. With $S_0[\varphi ]=\int d^{4}x{\mathcal{L}}_0$, the partition function isThe constant $\mathcal{N}$ fixes the desired vacuum normalization. A generating functional adds $\int d^{4}xJ\varphi$ to the action; the required partition function is its zero-source value. Equivalently, a Wick rotation gives $Z_E={\mathcal{N}}_E\int \mathcal{D}\varphi e^{-S_E[\varphi ]}$, with the positive-sign kinetic and potential terms in the Euclidean action when $m^2\ge 0$ and $\lambda >0$.