 One-loop shell quartic renormalization

For centered fast Fourier modes and $V=u\int {\varphi }^4/4!$, the two-fast-field vertex is $(u/4)\int {\phi }^2{\eta }^2$. Its connected second cumulant uses $⟨{\eta }^2(x){\eta }^2(y){⟩}_c=2G_{>}(x-y{)}^2$, producing $-u^2\int {\phi }^2(x){\phi }^2(y)G_{>}(x-y{)}^2/16$. Extracting the local zero-external-momentum quartic coefficient gives $u^{\prime }=b^{4-D}(u-3u^2{I}_2/2)+\cdots$, where $I_2={\int }_{\mathrm{shell}}{d}^{D}p(2\pi {)}^{-D}K(p{)}^{-2}>0$. At $D=4$ its sign makes weak positive quartic coupling marginally irrelevant for increasing length scale, after tuning the critical mass. This local truncation omits generated derivative and higher-order operators; it is not the exact complete finite-shell effective action.