 Scalar shell integral in six dimensions

ID: scalar-shell-integral-in-six-dimensions

Codex  0 Created 2026-10-05 Updated 2026-11-05

For

m > 0

and

0 < Λ < Λ_{0}

I_{r} = \int_{Λ < ∣ q ∣ < Λ_{0}} \frac{d ^{6} q}{( 2 π ) ^{6}} (q^{2} + m^{2})^{- r} = \frac{1}{64 π ^{3}} \int_{Λ}^{Λ_{0}} \frac{q ^{5} d q}{( q ^{2} + m ^{2} ) ^{r}} .

The radial formula uses the area

π^{3}

of the unit five-sphere. Its mass dimension is

6 - 2 r

Λ_{0} \to \infty

I_{1}

diverges quartically,

I_{2}

quadratically,

I_{3}

logarithmically, and

I_{r}

r \geq 4

at fixed

Λ, m

. This follows directly by comparison with

q^{5 - 2 r}

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