= Solution
{c}
The <scalar field path integral> integrates over real field histories with vacuum <boundary conditions>. With $S_0[\phi]=\int d^4x\,\mathcal L_0$, the <partition function> is
$$
\boxed{Z=\mathcal N\int\mathcal D\phi\,
\exp\left\{i\int d^4x\left[\frac12\partial_\mu\phi\partial^\mu\phi-\frac12(m^2-i0)\phi^2-\frac\lambda{4!}\phi^4\right]\right\}.}
$$
The constant $\mathcal N$ fixes the desired vacuum normalization. A <generating functional> adds $\int d^4x\,J\phi$ to the <action>; the required <partition function> is its zero-source value. Equivalently, a <Wick rotation> gives $Z_E=\mathcal N_E\int\mathcal D\phi\,e^{-S_E[\phi]}$, with the positive-sign kinetic and potential terms in the <Euclidean action> when $m^2\geq0$ and $\lambda>0$.
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