Matching the amplitude of a massive critical correlation tail ($G(r)\sim {\xi }^{-\eta }\frac{\xi }{(r\xi {)}^{(D-1)/2}}{e}^{-r/\xi }$)

A connected correlation function with critical anomalous dimension $\eta$ scales as $G(r)={\xi }^{-(D-2+\eta )}\mathcal{G}(r/\xi )$. If its large-distance shape is $s^{-(D-1)/2}{e}^{-s}$, substitution of $s=r/\xi$ produces the displayed amplitude. The extra ${\xi }^{-\eta }$ relative to the massive Gaussian field correlation tail is necessary for matching at $r$ of order $\xi$. Integrating over space gives magnetic susceptibility proportional to ${\xi }^{2-\eta }$; omitting that factor would instead make the long-distance contribution scale as ${\xi }^2$.