Definition: Partial-diagonal ambiguity class

The class ${\mathfrak{C}}_{\mathrm{part}}$ consists of finite sums of graphwise contact changes at order ${\hslash }^2$ in ${\mathsf{T}}_2$, hence order $\lambda \hslash$ in (eq:order). Before pushforward they have the form

where:

1. $r_I$ is a system of relative coordinates transverse to ${\Delta }_I$;

2. $S_{\Gamma /I}$ is an inherited spectator graph coefficient, with the two remaining field slots retained;

3. $w_{\Gamma }$ is built from $g,\partial g,{\sigma }_{\epsilon },{\sigma }_{\epsilon }^{\prime },{\sigma }_{\epsilon }^{\prime \prime }$ and the smooth external graph weights already present;

4. $D_{\Gamma }$ is the specified finite graph differential operator;

5. the change is even, has ghost number zero and field degree two, and is supported in the existing compact graph closure;

6. it leaves the tree datum and the inherited observable subgraph normalizations unchanged.

Only well-defined products and pushforwards of (eq:partial-counterterm) are members of the class. The bound six is a conservative frozen bound; an established scaling degree may impose a smaller bound on a particular locus. The subclass supported on the total graph diagonal is denoted ${\mathfrak{C}}_{\mathrm{tot}}\subseteq {\mathfrak{C}}_{\mathrm{part}}$.