= Solution
The <connected generating functional> is $W_0[J]=\log Z_0[J]$. Its second functional source derivative gives the <connected correlation function>; the source-independent determinant drops out. Translational invariance therefore gives
$$
\langle\widetilde\phi(p)\widetilde\phi(q)\rangle_c=(2\pi)^D\delta^{(D)}(p+q)\widetilde G_0(p),\qquad
\boxed{\widetilde G_0(p)=\widetilde\Delta(p)^{-1}=\frac{\alpha}{p^2+\mu^2},\quad \mu^2=\alpha m^2.}
$$
This covariance is independent of the source in a purely Gaussian theory; at zero source the mean field vanishes.
Fourier inversion can be evaluated with the <Schwinger parameterization>. Since $(p^2+\mu^2)^{-1}=\int_0^\infty ds\,e^{-s(p^2+\mu^2)}$, the elementary <Fourier transform of a Gaussian> gives
$$
G_0(r)=\alpha\int_0^\infty\frac{ds}{(4\pi s)^{D/2}}
\exp\left[-\mu^2s-\frac{r^2}{4s}\right]
=\frac{\alpha}{(2\pi)^{D/2}}\left(\frac\mu r\right)^{D/2-1}K_{D/2-1}(\mu r),
$$
where $K$ is the <Modified Bessel function of the second kind>. This integral formula also gives both asymptotic regimes directly.
For $\mu r\ll1$, set $u=r^2/(4s)$. The integral tends to the <gamma function> integral $\int_0^\infty u^{D/2-2}e^{-u}du=\Gamma(D/2-1)$; this is finite for $D>2$. Hence, at distances larger than the microscopic cutoff,
$$
\boxed{G_0(r)\sim\frac{\alpha\Gamma(D/2-1)}{4\pi^{D/2}}\frac1{r^{D-2}},\qquad \Lambda^{-1}\ll r\ll\xi.}
$$
For $\mu r\gg1$, the exponent $\mu^2s+r^2/(4s)$ has its minimum at $s=r/(2\mu)$, with value $\mu r$. Gaussian expansion about that saddle, or the <large-argument asymptotic expansion of a modified Bessel function>, gives $K_\rho(z)\sim\sqrt{\pi/(2z)}e^{-z}$. Therefore
$$
\boxed{G_0(r)\sim\frac{\alpha}{2(2\pi)^{(D-1)/2}}
\frac{\xi}{(r\xi)^{(D-1)/2}}e^{-r/\xi},\qquad r\gg\xi,\quad \xi=\frac1{\sqrt{\alpha m^2}}.}
$$
This is the <massive Gaussian field correlation tail> with its normalization displayed. For example $D=3$ gives exactly $G_0(r)=\alpha e^{-r/\xi}/(4\pi r)$, verifying both regimes and the exponential length scale.
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