propagators can be written in the form $$\begin{aligned}
      \frac{1}{(r_0^\alpha)^2+E_\alpha^2}, \quad E_\alpha\equiv\sqrt{r_\alpha^2+m_\alpha^2}\, ,
     \end{aligned}$$ of which exactly $N$ can be chosen to correspond to the loop momenta $Q_i$, $i=1,...,N$, that are integrated over. The remaining $M-N$ $R_\alpha$ are then linear combinations of the loop momenta and the external momenta $P_k$, $k=1,...,n$, as dictated by momentum conservation at the vertices (see appendix \[sec:mom\] for a discussion of this issue).

    Our claim is now that the integral $$\begin{aligned}
     I&\equiv& \int_{-\infty}^\infty \prod_{i=1}^{N} \frac{dq_0^i}{2\pi}\, \prod_{\alpha=1}^M \frac{1}{(r_0^\alpha)^2+E_{\alpha}^2}, \label{Ires}
     \end{aligned}$$ where we have made some arbitrary choice for the loop momenta, can be written in a simple form that is explicitly independent of this choice. To write down this result, we introduce the following notation: denoting the set of all propagators by $P\equiv\{1,2,...,M\}$, we define $S$ to be the set of all possible choices of loop momenta, such that each element $S_r\in S$ corresponds to some subset of $N$ indices from $P$. The sets $S_r$ are limited only by momentum conservation, and several examples of them are given in appendix \[sec:mom\].

    With the above notation, our proposed result for $I$ reads $$\begin{aligned}
    I&=&\sum_{S_r\in S}\;\prod_{i\in S_r} \frac{1}{2E_i} \prod_{\alpha\in P\setminus S_r} \frac{1}{(r_0^\alpha(S_r))^2+E_\alpha^2(S_r)}\Big{|}_{\{q_0^i=iE_i\}}, \label{Iresmu0}\end{aligned}$$ where $P\setminus S_r$ denotes the propagators 