How to interpret "smooth momentum space slicing" in renormalization group analysis?

Question

Ref: [John B. Kogut, Rev. Mod. Phys. 51, 659 (1979), An introduction to lattice gauge theory and spin systems]. More precisely, please refer to Page 703 within the section of renormalization group analysis of Gaussian planar model in two dimensions.

When integrating the high momentum slicing for the Green function \begin{equation} G_{h}(x)=\int_{\Lambda-d\Lambda<p<\Lambda}\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{\text{i}\mathbf{px}}}{p^{2}}=\int_{\Lambda-d\Lambda<p<\Lambda}\frac{pdpd\theta}{(2\pi)^{2}}\frac{e^{\text{i}{px}\cos\theta}}{p^{2}}=\frac{1}{(2\pi)^{2}}\int\frac{dp}{p}\int{e}^{\text{i}px\cos\theta}d\theta=\frac{1}{2\pi}\frac{d\Lambda}{\Lambda}J_{0}(\Lambda{x}), \end{equation} where in the last step the Bessel function is used to expand it. Note that the Bessel function does not fall rapidly as its argument increases. Then the author argued that through the so called smooth momentum space slicing procedure, $G_{h}(x)$ becomes a really short-ranged function. What does it mean and how to do it explicitly?

Answer 1

I have learned something about this very question which was asked about one month ago by myself. I think I should post it here.

The point is that the Bessel function does not fall rapidly as its argument increases. Thus we cannot use the fact of non-negligible assumption with $|\mathbf{x}-\mathbf{y}|<\Lambda^{-1}$. The approximation fails here. This flaw can be solved by considering the smooth momentum space slicing, which is a "soft" cutoff in the momentum space instead. It is done by modifying the integration \begin{equation} \int_{0}^{\Lambda}d^{2}q \rightarrow \int_{0}^{\infty}d^{2}q\frac{\Lambda^{2}}{q^{2}+\Lambda^{2}}. \end{equation} Thereby we could make the Bessel function shorted ranged and the RG process is restored valid. This failure teaches us that there is nothing automatic in RG and new problems always require flexible techniques guided by physical intuition.

BTW, I also find a nice book related: A Modern Approach to Critical Phenomena, Igor Herbut, Cambridge University Press (2007).