Scalar shell integral in six dimensions ($I_r(\Lambda ,{\Lambda }_0;m)$)

For $m>0$ and $0<\Lambda <{\Lambda }_0$, defineThe 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 $r\ge 4$ at fixed $\Lambda ,m$. This follows directly by comparison with $q^{5-2r}$.