Definition: Partial-diagonal ambiguity class; transcribed

The class ${\mathfrak{C}}_{\mathrm{part}}$ consists of finite sums of graphwise contact changes at order ${\hslash }^2$ in the time-ordered product, hence at order $\lambda \hslash$ in the selected retarded expansion. Before pushforward they have the form

Here:

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, of 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 are members. The bound six is a conservative frozen bound; an established scaling degree can impose a smaller bound on a particular locus. The total-diagonal subclass is ${\mathfrak{C}}_{\mathrm{tot}}\subseteq {\mathfrak{C}}_{\mathrm{part}}$.