 Solution

Write $\rho =m^2{\Lambda }^{-2}$ for the PDF's signed mass coordinate $u^2$, and $K_D={\Omega }_D/(2\pi {)}^D$. The superscript on $u^2$ is notation here, not a restriction to positive mass. In a momentum-shell renormalization group step, remove momenta $\Lambda e^{-d\ell }<|p|<\Lambda$ and set $\ell =\log ({\Lambda }_0/\Lambda )$, the PDF's $b$. Split the scalar field into a slow field $\phi$ and a fast field $\eta$, whose Gaussian shell covariance is $(p^2+m^2{)}^{-1}$. The shell must have a positive quadratic kernel; $1+\rho >0$ near the fixed point meets this requirement.

Expand the coarse action by the cumulant expansion of a coarse-grained free energy. In the first cumulant of $V=g\int (\phi +\eta {)}^4/4!$, the term with two fast fields is $(g/4)\int {\phi }^2⟨{\eta }^2⟩$. Thus the one-loop shell mass renormalization in scalar quartic theory isFor the quartic correction, the two-fast-field vertex is $(g/4)\int {\phi }^2{\eta }^2$. Its connected second cumulant has a minus sign and uses $⟨{\eta }^2(x){\eta }^2(y){⟩}_c=2G_{>}(x-y{)}^2$. It givesTaking the local zero-external-momentum quartic part produces $\delta g=-3g^2{I}_2/2$, where $I_2={\int }_{\mathrm{shell}}{d}^{D}p(2\pi {)}^{-D}(p^2+m^2{)}^{-2}$. This derives the combinatorial factors in the one-loop shell quartic renormalization. The mass tadpole is independent of external momentum; kinetic normalization changes only at higher order for the leading epsilon calculation.

For an infinitesimal spherical shell, radial integration givesLowering the cutoff adds the engineering terms $2\rho$ and $ϵ\lambda$ to the running dimensionless couplings. Since $g=\lambda {\Lambda }^{ϵ}$ and $D=4-ϵ$, the remaining cutoff powers cancel exactly. ThereforeThese are the requested flows, to the stated one-loop order. They implement cutoff independence of long-wavelength observables after changing the couplings and retaining the generated constant and other effective operators to the relevant approximation order. They are not an exact closure of the complete effective action.