= Infrared asymptotics of the critical-mass subtraction
{title2=$J_D(R)=K_D\int_0^\Lambda p^{D-3}(p^2+R)^{-1}\,dp$}
For the scalar quartic <one-loop critical-mass subtraction>, the subtracted tadpole is $I_D(0)-I_D(R)=RJ_D(R)$. 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 $2<D<4$, $J_D\sim K_DR^{(D-4)/2}\int_0^\infty q^{D-3}(1+q^2)^{-1}dq$, which diverges as $R\to0$. Hence a smooth linear thermal mass survives this test only above the ordinary <upper critical dimension> four. The massless subtraction is infrared divergent for $D\leq2$ and cannot prove absence of an interacting transition there.
Back to article page
