Lemma: Vertex normalization

Equation (eq:vertex) is the full contraction of (eq:Hessian), with the remaining scalar derivative transferred to the output kernel. It includes the kinetic, trace-kinetic, and mass terms.

Proof. Symmetry of the metric tensor converts the symmetrized kinetic term into $h^{\mu \nu }{\partial }_{\nu }\varphi {\partial }_{\mu }\psi$. Contracting the crossed $\psi$ produces ${\partial }_{2,\mu }{C_I}^{\mu \nu }$. The trace kinetic term produces $-{\eta }_{\mu \lambda }{{\partial }_2}^{\nu }{C_I}^{\mu \lambda }/2$. These are the two contributions to ${Z_I}^{\nu }$.

The polarized mass term is $m^2{\eta }_{\mu \nu }\varphi \psi /2$. The leading minus sign of the Hessian gives the negative mass coefficient in (eq:vertex). Integration by parts in the remaining scalar slot gives the output divergence and (eq:vertex-test). The differentiation of the quadratic stress has already used the factor $1/2$ in the interaction, so there is no additional polarization factor.