each block $G_j(A)$ by $D_j(A)\equiv D(A)\exp[-i(k_jx)][2(\Pi e_j)+\hat{k}_j\hat{e}_j]$ (this partial amplitude contains one term).

2.  Partial amplitude $M^{(2)}$: combine two successive blocks $G_j(A) G_{j+1}(A)$ (for $j=1,\ldots, N$) and substitute this quantity with the “contraction”\
    $-C_{j,j+1}(A)=-D(A)\hat{e}_j\exp[-i(k_jx)] \hat{e}_{j+1}\exp[-i(k_{j+1}x)]$, then substitute the remaining blocks as in 1.; it is understood that $G_{N+1}(A)\equiv G_1(A)$ and that $C_{N,N+1}(A)\equiv C_{N,1}(A)$; this partial amplitude contains $N$ terms.

3.  Partial amplitude $M^{(3)}$: combine twice two successive blocks $G_j(A)G_{j+1}(A)$ and $G_{j'}(A)G_{j'+1}(A)$ (for $j=1,\ldots,N-2$, and for $j'=3,\ldots,N-1$ (if $j=1$) or for $j'=j+2,\ldots,N$ (if $j>1$)), and substitute these quantities with the contractions $-C_{j,j+1}(A)$ and $-C_{j',j'+1}(A)$, respectively; then substitute the remaining blocks as in 1.; it is understood that $G_{N+1}(A)\equiv G_1(A)$ and that $C_{N,N+1}(A)\equiv C_{N,1}(A)$; this partial amplitude has to be considered only if $N\ge 4$ and it contains $N(N-3)/2$ terms.

4.  The above procedure continues by increasing by one the number of combinations of successive blocks. The last partial amplitude $M^{([N/2]+1)}$ contains the two terms $(-1)^{N/2}C_{1,2}(A)C_{3,4}(A)\cdots C_{N-1,N}(A)$ and\
    $(-1)^{N/2}C_{N,1}(A)C_{2,3}(A)\cdots C_{N-2,N-1}(A)$ with $N/2$ contractions if $N$ is even, or the $N$ terms $(-1)^{(N-1)/2}D_1(A)C_{2,3}(A)\cdots C_{N-1,N}(A)$,\
    $(-1)^{(N-1)/2}C_{N,1}(A)D_2(A)C_{3,4}(A)\cdots C_{N-2,N-1}(A)$,...,\
    $(-1)^{(N-1)/2}C_{1,2}(A)C_{3,4}(A)\cdots C_{N-2,N-1}(A)D_N(A)$ with $(N-1)/2$ contractions if $N$ is odd.

Now, a useful exponential representation of the square propagator $D(A)$ has been found in [@Baier_1976_a; @Baier_1976_b] (see also 