= Radial critical-mass subtraction in dimensions three to five
{title2=$J_D(R)=\int_{|p|<\Lambda}\frac{d^Dp}{(2\pi)^D p^2(p^2+R)}$}
For $R>0$, a spherical <ultraviolet cutoff> gives the following exact examples of the <one-loop critical-mass subtraction> integral:
$$
J_3(R)=\frac{1}{2\pi^2\sqrt R}\arctan\frac\Lambda{\sqrt R},\qquad
J_4(R)=\frac{1}{16\pi^2}\log\frac{\Lambda^2+R}{R},\qquad
J_5(R)=\frac{1}{12\pi^3}\left(\Lambda-\sqrt R\arctan\frac\Lambda{\sqrt R}\right).
$$
The areas of the unit spheres in dimensions three, four and five are $4\pi$, $2\pi^2$ and $8\pi^2/3$. Dividing by $(2\pi)^D$ reduces the integrals to constants times $\int_0^\Lambda dp/(p^2+R)$, $\int_0^\Lambda p\,dp/(p^2+R)$ and $\int_0^\Lambda p^2dp/(p^2+R)$ respectively. Elementary integration proves the formulas. As $R\to0$, the first has a power divergence, the second a logarithmic divergence, and the third a finite limit. These examples make the ordinary <upper critical dimension> four visible without treating a self-consistent one-loop equation as an exact interacting <critical exponent> prediction.
Back to article page
