Completing the square in the <Gaussian functional integral> and choosing
$$
\mathcal N^{-1}=\int\mathcal D\phi\,
\exp\left(i\int d^4x\,\mathcal L_b\right)
$$
normalizes $Z_0[0]=1$. The result is
$$
Z_0[J]=\exp\left[-\frac12\int d^4x,d^4y\,J(x)D_F(x-y)J(y)\right],
$$
where the <Feynman propagator> in the convention of the question is
$$
D_F(x-y)=\int\frac{d^4p}{(2\pi)^4}\frac{i,e^{-ip\cdot(x-y)}}{p^2-M^2+i\epsilon}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 2 / a / Solution

{wiki=Gaussian_integral#Functional_form}

A Gaussian functional integral has an exponent quadratic in a bosonic field. Completing the square expresses its source dependence through the inverse quadratic operator, the propagator.


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