Example: A cyclic raw product need not converge

Two parallel delta lines between two vertices lead formally to $\delta (r{)}^2$. Take a nonzero real $\rho \in {C_c}^{\infty }(\mathbb{R})$ and ${\rho }_{\tau }(r)={\tau }^{-1}\rho (r/\tau )$. For $\varphi (0)>0$,

The raw regularization diverges. This does not exclude a separately authorized renormalized extension. It shows why the forest argument alone supplies no loop product.