Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 43 / 3 / 3 / 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 givesThis 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 giveswhere $K$ is the Modified Bessel function of the second kind. This integral formula also gives both asymptotic regimes directly.

For $\mu r\ll 1$, 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,For $\mu r\gg 1$, the exponent ${\mu }^{2}s+r^2/(4s)$ 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 $K_{\rho }(z)\sim \sqrt{\pi /(2z)}{e}^{-z}$. ThereforeThis 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.