Bulk graphs

D04: mechanism. The D04 bubble at $p\ne 0$ is not scaleless, so its zero needs an explicit reason.

Take the vertex $V_{ab}(p,q)=p_a{q}_b+p_b{q}_a-{\eta }_{ab}p\cdot q$ with $q=l-p$, and the graviton numerator $P=\frac{1}{2}(\eta \eta +\eta \eta )-c\eta \eta$, where the vertex-index trace is $\eta \cdot \eta =n$. Then

The $(p\cdot q{)}^2$ term is the coupling to the trace mode. It cancels exactly when $c=1/(n-2)$, which holds for the de Donder propagator in DRED ($n=4$, $c=\frac{1}{2}$, the declared prescription) and in CDR ($n=d$, $c=1/(d-2)$). What remains is $2p^2(l-p{)}^2$. This cancels the scalar propagator and leaves $p^2\int 1/l^2$, which is scaleless with $\omega =2$ and no logarithmic member, so D04 $=0$ exactly in $d$.

D04 $=0$ is therefore a statement about de Donder gauge, minimal coupling and $m=0$, off shell. It is not a general property of the bubble, and no free-EOM equivalence is used.

D04: routes.

• Engine route 1: Feynman parameters, with no scaleless rule.

• Engine route 2: an invariant fit identical to $\frac{1}{2}{p}^2(l-p{)}^2$, followed by IBP.

• Route 4: fully independent of the engine. It uses its own covariant algebra and its own Feynman-parameter integration with symbolic $D$ for arbitrary propagator powers.

In route 4, both the DRED and CDR cases give $0$. The mutations, as ratios to the master, are:

• mixed scheme ($n=4$, $c=1/(D-2)$): $\frac{D-4}{2(D-2)}{p}^4$, i.e.\ $O(ϵ)\times$pole;

• vertex trace normalisation $\frac{1}{2}$: $p^4/4$;

• non-de Donder gauge pieces of Fröb's propagator: $T^{(4)}$ gives $(D-1)p^4$ and $T^{(5)}$ gives $\frac{(D-2)(D-1)}{8}{p}^4$;

• non-minimal coupling $\xi$: $-\xi p^4$, which is linear in $\xi$;

• engine mutation dropping the trace term of the propagator: $-i/8$.

The gauge piece $T^{(3)}$ gives $0$, and this is expected rather than a failure: $l^a{V}_{ab}=p_b{q}^2+q_b{p}^2$, so every $T^{(3)}$ term carries $q^2$ and is scaleless. It is recorded as a Ward-identity zero, not counted as a detector.