Field content and the one-tree observable

The metric is

with $\hat{h}$ the free gauge-fixed graviton, $\hat{\varphi }$ the free scalar, and $h_{\mathrm{cl}},k_{\mathrm{cl}}$ classical and off shell. Write $h=h_{\mathrm{cl}}+\hat{h}$. The bare and completed coefficient functionals of X-B/X-C are polynomial in the metric perturbation with distributional coefficients; substituting $\hat{h}$ and Wick ordering gives elements of $\mathfrak{W}$, admitted at the fixed $\epsilon$ by X-B's joint admission of the multiplet.

Let ${\Theta }^{\mu \nu }:=:T^{\mu \nu }[\hat{\varphi }]:$ be the Wick stress tensor. By (eq:T-divergence), ${\partial }_{\mu }{\Theta }^{\mu \nu }=:(P_m\hat{\varphi }){\partial }^{\nu }\hat{\varphi }:\in \mathfrak{J}$.