Infrared asymptotics of the critical-mass subtraction ($J_D(R)=K_D{\int }_0^{\Lambda }{p}^{D-3}(p^2+R{)}^{-1}dp$)

For the scalar quartic one-loop critical-mass subtraction, the subtracted tadpole is $I_D(0)-I_D(R)=R{J}_D(R)$. Above four dimensions $J_D(0)=K_D{\Lambda }^{D-4}/(D-4)$ is finite. At four dimensions $J_4=(K_4/2)\log [({\Lambda }^2+R)/R]$. For $2<D<4$, $J_D\sim K_D{R}^{(D-4)/2}{\int }_0^{\infty }{q}^{D-3}(1+q^2{)}^{-1}dq$, which diverges as $R\to 0$. Hence a smooth linear thermal mass survives this test only above the ordinary upper critical dimension four. The massless subtraction is infrared divergent for $D\le 2$ and cannot prove absence of an interacting transition there.

For $R>0$, a spherical ultraviolet cutoff gives the following exact examples of the one-loop critical-mass subtraction integral: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 ${\int }_0^{\Lambda }dp/(p^2+R)$, ${\int }_0^{\Lambda }pdp/(p^2+R)$ and ${\int }_0^{\Lambda }{p}^{2}dp/(p^2+R)$ respectively. Elementary integration proves the formulas. As $R\to 0$, 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.