= One-loop shell mass renormalization in scalar quartic theory
{title2=$\delta r=\frac u2G_>(0)$}
For interaction $V=u\int\phi^4/4!$, split $\phi=\varphi+\eta$ into slow and fast <Fourier modes>, with a centered fast <Gaussian measure>. The first <cumulant expansion> term is $\langle V\rangle=(u/24)\int[\varphi^4+6G_>(0)\varphi^2+3G_>(0)^2]$ by <Wick theorem>. The constant shifts the <free energy>, the quadratic term gives $\delta r=uG_>(0)/2$, and there is no <gradient> correction at this order. Rescaling gives $r'=b^2[r+uG_>(0)/2]+O(u^2)$ and $u'=b^{4-D}u+O(u^2)$. The mass shift mixes thermal and quartic perturbations; the interacting critical tuning need not have zero bare mass.
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