Named and not constructed

Two further terms live at ${\kappa }^2$ and are not constructed.

The gravitational self-interaction tree. By Lemma lem:quadratic, $\delta S_3/\delta h_{\mu \nu }=\frac{1}{2}{{\mathcal{G}}_2}^{\mu \nu }[h,h]$. The complete first-order retarded response of the metric is ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}(-\delta (V_{1,g}+S_3)/\delta h)$, and the second summand inserted in the affine slot gives

a Wick monomial of degree $(1,2)$ in $(\hat{\varphi },\hat{h})$. It exists at ${\kappa }^2{\hslash }^0$; the leg count in earlier planning that excluded it was wrong. It is not constructed because its kernel is a retarded propagator composed with a bilinear differential operator at coincident points, Wick ordered with ${\mathsf{H}}_{\mathrm{gr}}$, and X-B's admission lemmas are two-slot statements. The one-tree observable (eq:one-tree) omits it by declaration.

The mixed loop. Since ${\hat{A}}_1[\hat{h}]$ is bilinear in $(\hat{h},\hat{\varphi })$ and $V_{1,g}[\hat{h},\hat{\varphi }]$ is trilinear, a second cross-contraction exists along the scalar line. It is linear in $\hat{\varphi }$, of order $\hslash {\kappa }^2$, with kernel $f(x)g(y)\int \Lambda (x,z){{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}(z,y)\mathrm{d}z\partial \partial {\Delta }_m(x,y)$, whose extension to $y=x$ is a genuine renormalization problem. In X-B and X-C tree-exactness was a theorem because the insertion was linear in the quantum field. Here it is not. “Tree” in this manuscript means the ${\hslash }^0$ coefficient by definition.