loop momenta $S_r\in S$, we may clearly express the integral we wish to evaluate \[cf. eq. (\[Ires\])\] in the form $$\begin{aligned}
I(S_r)&\equiv& \int_{-\infty}^\infty \prod_{i \in S_r}\frac{dq_0^i}{2\pi}\, \prod_{\alpha=1}^M \frac{1}{(r_0^\alpha(S_r))^2+E_{\alpha}^2(S_r)}\,, \label{Iresx}\end{aligned}$$ where our notation highlights the fact that the $R_\alpha$ and corresponding $E_\alpha$ depend on $S_r$. Picking some $i \in S_r$ as the first integration to be performed gives a sum of residues of the form $\frac{1}{2E_{\alpha}}\left(\ldots\right)|_{q_0^i = iE_\alpha+...}$, where each $\mathbf{r}_\alpha$ depends linearly on the $\mathbf{q}_i$. Placing the momentum in question on shell shifts the poles of some of the remaining propagators but does not affect the corresponding residues. Upon subsequent integrations, complicated combinations of $\theta$-functions typically appear in the numerator of the result due to these shifts. What remains unchanged, however, is that each integration produces an additional factor of $1/(2E_{\alpha'})$, originating from the residue of one of the remaining propagators.

To illustrate the above reasoning, consider the following simple example, where we perform three $q_0$ integrations, always picking up the pole from the highlighted propagator: $$\begin{aligned}
&&\int_{q_0^1}\int_{q_0^2}\int_{q_0^3}\mathbf{\frac{1}{(q_0^1)^2+E_1^2}}\frac{1}{(q_0^2)^2+E_2^2}\frac{1}{(q_0^3)^2+E_3^2} \frac{1}{(q_0^1+q_0^2)^2+E_4^2}\frac{1}{(q_0^1+q_0^2-q_0^3)^2+E_5^2} \nonumber \\
&=& \int_{q_0^2}\int_{q_0^3}\frac{1}{2E_1}\frac{1}{(q_0^2)^2+E_2^2}\frac{1}{(q_0^3)^2+E_3^2}\mathbf{\frac{1}{(iE_1+q_0^2)^2+E_4^2}}\frac{1}{(iE_1+q_0^2-q_0^3)^2+E_5^2}+\cdots \nonumber \\
&=& \int_{q_0^3}\frac{1}{2E_1}\frac{\theta(E_4-E_1)}{2E_4}\frac{1}{(iE_4-iE_1)^2+E_2^2}\frac{1}{(q_0^3)^2+E_3^2}\mathbf{\frac{1}{(iE_4-q_0^3)^2+E_5^2}} +\cdots\nonumber \\
&=& \frac{1}{2E_1}\frac{\theta(E_4-E_1)}{2E_4}\frac{\theta(E_5-E_4)}{2E_5}\frac{1}{(iE_4-iE_1)^2+E_2^2}\frac{1}{(iE_4-iE_5)^2+E_3^2} +\cdots\, .\end{aligned}$$ Note that additional terms of the exact same form but with different $\theta$-function factors originate from taking 