= One-loop shell quartic renormalization
{title2=$\delta u=-\frac32u^2I_2$}
For centered fast <Fourier modes> and $V=u\int\phi^4/4!$, the two-fast-field vertex is $(u/4)\int\varphi^2\eta^2$. Its connected second <cumulant> uses $\langle\eta^2(x)\eta^2(y)\rangle_c=2G_>(x-y)^2$, producing $-u^2\int\varphi^2(x)\varphi^2(y)G_>(x-y)^2/16$. Extracting the local zero-external-momentum quartic coefficient gives $u'=b^{4-D}(u-3u^2I_2/2)+\cdots$, where $I_2=\int_{\rm shell}d^Dp\,(2\pi)^{-D}K(p)^{-2}>0$. 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.
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