Two-vertex scalar logarithm and its local ambiguity

The graph in the Example has two distinct vertices and two parallel internal edges, so its numerical relative kernel is the product of the two Euclidean edge kernels, $C_0(z{)}^2=u(z)$. By [F2], this punctured distribution has scaling degree $4$; the causal extension theorem [F1] supplies the separately stated Lorentzian graph construction for admissible scalar free data at the same formal order. The Euclidean calculation uses [F2] and makes no identification of the Euclidean kernel with that Lorentzian graph.

Here the relative dimension is $4$ and the scaling degree is also $4$. Thus $\lfloor 4-4\rfloor =0$, so [F4] says every two same-degree extensions differ by $c{\delta }_0$ for a scalar $c\in \mathbb{C}$; no derivative of ${\delta }_0$ is allowed. This proves the local ambiguity claim without identifying the Euclidean kernel with $H_F$.

Applying the exact Euclidean calculation in [F2] to the chosen scale parameters gives $T_{\mu }-T_{\nu }=(8{\pi }^2{)}^{-1}\log (\mu \mathrm{/}\nu ){\delta }_0$. Since the difference is supported at $z=0$ and is proportional to ${\delta }_0$, it is local. The calculation concerns only the stated Euclidean extension family, so it supplies no Lorentzian amplitude or measured probability.