Definition: Frozen total-diagonal class

Use the finite rooted graph presentations of $\mathcal{O}$ supplied by XII-D and XII-E, together with the local vertex (eq:vertex). The total graph diagonal is the locus where all spacetime vertices in a graph coincide.

The class ${\mathfrak{C}}_{\mathrm{tot}}$ permits changes only at order ${\hslash }^2$ in ${{\mathsf{T}}_2}^{\mathrm{ren}}$, hence at order $\lambda \hslash$ in (eq:order). After pushforward these changes are finite sums

where:

1. $A,B\in \{\gamma ,\varphi \}$, with constant fiber contractions;

2. the total field-derivative order is at most eight;

3. each $P$ is a polynomial of total degree at most eight in the listed external jets, with fixed complex coefficients depending on the frozen mass;

4. no higher derivative of $g$ or ${\sigma }_{\epsilon }$ is admitted as an independent coefficient;

5. the unintegrated change is supported on the total graph diagonal and preserves the degree-two Wick expansion;

6. the tree inverse and off-diagonal first-completion root normalization are unchanged.