is convenient to express the amplitude $M$ in terms of the “square” propagator $$\label{D_0}
D(A)=\frac{1}{\hat{\Pi}^2-m^2+i\epsilon}$$ rather than in terms of $G(A)$. The details of the procedure to carry this out are reported in the Appendix A. Here, we only provide a summary of this procedure in terms of substitution rules. The amplitude $M$, in fact, turns out to be expressed as $$\label{M_s}
M=\frac{1}{2}\sum_{i=1}^{[N/2]+1}(M^{(i)}+\{1\ldots N\to N\ldots 1\}),$$ where $M^{(i)}$ are partial amplitudes, with $[N/2]$ indicating the integer part of $N/2$. The quantity $\{1\ldots N\to N\ldots 1\}$ refers to the fact that each partial amplitude $M^{(i)}$ will have $N$ indexes corresponding to the $N$ ordered operators $G_j(A)$ in Eq. (\[M\_op\]), and it indicates that the same partial amplitude $M^{(i)}$ has to be added, but with the indexes $1,\ldots,N$ appearing in the opposite order $N,\ldots,1$. In turn, each partial amplitude $M^{(i)}$ is expressed as a sum $\sum_{J=1}^{J_i} M_J^{(i)}$ of terms $M_J^{(i)}$ and the number $J_i$ of terms in each partial amplitude depends on the partial amplitude itself. Each term $M_J^{(i)}$ has the form $-e^N\int d^4x\,\text{Tr}\,\langle x\vert O_J^{(i)}\vert x\rangle$, with the operator $O_J^{(i)}$ being obtained from the original operator product $G_1(A)\cdots G_N(A)$ by means of the following substitution rules:

1.  Partial amplitude $M^{(1)}$: substitute 