= Scalar shell integral in six dimensions
{title2=$I_r(\Lambda,\Lambda_0;m)$}
For $m>0$ and $0<\Lambda<\Lambda_0$, define
$$
I_r=\int_{\Lambda<|q|<\Lambda_0}\frac{d^6q}{(2\pi)^6}(q^2+m^2)^{-r}=\frac1{64\pi^3}\int_\Lambda^{\Lambda_0}\frac{q^5\,dq}{(q^2+m^2)^r}.
$$
The radial formula uses the area $\pi^3$ of the unit five-sphere. Its <mass dimension> is $6-2r$. As $\Lambda_0\to\infty$, $I_1$ diverges quartically, $I_2$ quadratically, $I_3$ logarithmically, and $I_r$ is ultraviolet finite for $r\ge4$ at fixed $\Lambda,m$. This follows directly by comparison with $q^{5-2r}$.
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