= Massive Gaussian field correlation tail
{title2=$G(r)\sim r^{-(D-1)/2}e^{-\sqrt\kappa mr}$}
For the positive quadratic kernel $\widetilde\Delta(p)=\kappa^{-1}p^2+m^2$, $\kappa,m>0$, the <connected correlation function> is its inverse. Fourier inversion gives $G(r)=\kappa(\sqrt\kappa m/r)^{D/2-1}K_{D/2-1}(\sqrt\kappa mr)/(2\pi)^{D/2}$. The <Modified Bessel function of the second kind> has an exponentially decaying large-argument tail, so the <correlation length> is $1/(\sqrt\kappa m)$ at fixed positive <gradient> coefficient. In three dimensions this becomes $\kappa e^{-\sqrt\kappa mr}/(4\pi r)$. The statement concerns the continuum long-distance theory; regulator-dependent contact terms do not define the physical <correlation length>.
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