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This is probably a very basic question but I tried physics stack exchange already and I got no answers, so I'm asking the same question here.

I was reading this article and the author considers the following propagator: $$D_{\kappa}(x-y) = \frac{1}{(2\pi)^{2}}\int dp \frac{-i\not{p}+m^{2}}{p^{2}+m^{2}}\chi_{\kappa}(p)e^{ip(x-y)}$$ where $\chi_{\kappa}$ is a cutoff function. Of course, if we take: $$\chi_{\kappa}(p) = e^{-(p^{2}+m^{2})/\kappa^{2}} $$ then this cutoff is going to supress high momenta. But the author mentions the alternative form: $$\chi_{\kappa}(p) = e^{-(p^{2}+m^{2})/\kappa^{2}} - e^{-\ell^{2} (p^{2}+m^{2})/\kappa^{2}}$$ with $\ell > 1$, which he claims "selects a slice in momentum space". What does this mean? In which sense it selects a slice in momentum space and how can one see it?

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  • $\begingroup$ This interpretation would make more sense when $m^2=0$, then $\chi_\kappa(p) \to 0$ when $p^2\to \infty$ as well as when $p^2\to 0$. So the support is roughly concentrated in some shell of finite momenta. I think the general idea should be pretty clear. $\endgroup$
    – Igor Khavkine
    Commented Apr 7, 2022 at 7:53
  • $\begingroup$ @IgorKhavkine but even in the case $m^{2} = 0$, $\chi_{\kappa}$ is not really zero for $p^{2}$ large, right? I was expecting that the statement meant exactly what you wrote about the $\chi_{\kappa}$ having support in some shell of finite momenta, but it does never seem to be the case. $\endgroup$
    – JustWannaKnow
    Commented Apr 7, 2022 at 12:25
  • $\begingroup$ You could raise the same objection about the original "cutoff" function $e^{-(p^2+m^2)/\kappa^2}$ as well. Practically, it is sufficient that it goes to zero sufficiently fast, not that it is exactly zero for large $p^2$. $\endgroup$
    – Igor Khavkine
    Commented Apr 7, 2022 at 12:36

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