(-5.2,-3) [${N}$]{}; at (-5.2,-5) [${P}$]{}; at (-0.8,-5) [${Q}$]{}; at (-0.8,-3) [${M}$]{}; at (-4,-4) [$\mathcal{A}^{(0)}$]{}; at (-2,-4) [$\mathcal{A}^{(0)}$]{};

To proceed, in each case we use (\[eqn:ampcons\]) and the two-momentum conservation at the vertex involving the momentum $p_1$ to integrate over $l_2$ && \^[(1)]{}\^[[P]{}[Q]{}]{}\_[[M]{}[N]{}]{}(p\_1,p\_2,p\_3,p\_4)|\_[s-cut]{}=12i\^+([l\_1]{}\^2-[1]{}) i\^+(([l\_1]{}-[p\_1]{}-[p\_2]{})\^2-[1]{})\
&&\^[(0)]{}\_[[M]{}[N]{}]{}\^[[R]{}[S]{}]{}([p\_1,p\_2,l\_1,-l\_1+p\_1+p\_2]{}) \^[(0)]{}\_[[S]{}[R]{}]{}\^[[P]{}[Q]{}]{}([-l\_1+p\_1+p\_2,l\_1,p\_3,p\_4]{}),\[2\_8\]\
&& \^[(1)]{}\^[[P]{}[Q]{}]{}\_[[M]{}[N]{}]{}(p\_1,p\_2,p\_3,p\_4)|\_[t-cut]{}=12i\^+([l\_1]{}\^2-[1]{})i\^+(([l\_1]{}+[p\_1]{}-[p\_3]{})\^2-[1]{})\
&& \^[(0)]{}\_[[M]{}[R]{}]{}\^[[S]{}[P]{}]{}([p\_1,l\_1,l\_1+p\_1-p\_3,p\_3]{}) \^[(0)]{}\_[[S]{}[N]{}]{}\^[[R]{}[Q]{}]{}([l\_1+p\_1-p\_3,p\_2,l\_1,p\_4]{}) , \[2\_9\]\
&& \^[(1)]{}\^[[P]{}[Q]{}]{}\_[[M]{}[N]{}]{}(p\_1,p\_2,p\_3,p\_4)|\_[u-cut]{}= 12i\^+([l\_1]{}\^2-[1]{}) i\^+(([l\_1]{}+[p\_1]{}-[p\_4]{})\^2-[1]{})\
&& \^[(0)]{}\_[[M]{}[R]{}]{}\^[[S]{}[Q]{}]{}([p\_1,l\_1,l\_1+p\_1-p\_4,p\_4]{})\^[(0)]{}\_[[S]{}[N]{}]{}\^[[R]{}[P]{}]{}([l\_1+p\_1-p\_4,p\_2,l\_1,p\_3]{}).\[2\_10\] In each of these integrals the set of zeroes of the $\delta$-functions are discrete. This allows us to pull out the tree-level amplitudes with the loop-momenta evaluated at those zeroes, leaving scalar bubbles [^4]. Following standard unitarity computations [@Bern:1994zx], we apply the following replacement in the imaginary part of the amplitude – to the internal on-shell propagators: $i\pi \delta^+(l^2-1) \longrightarrow \tfrac{1}{l^2-1}$. This allows us to rebuild, from its imaginary part, the cut-constructible piece of the amplitude \^[(1)]{}\^[[P]{}[Q]{}]{}\_[[M]{}[N]{}]{}(p\_1,p\_2,p\_3,p\_4)&=&\
& +& \^[(0)]{}\_[[M]{}[R]{}]{}\^[[S]{}[P]{}]{}([p\_1,p\_3,p\_1,p\_3]{}) \^[(0)]{}\_[[S]{}[N]{}]{}\^[[R]{}[Q]{}]{}([p\_1,p\_2,p\_3,p\_4]{})\
& + & \^[(0)]{}\_[[M]{}[R]{}]{}\^[[S]{}[Q]{}]{}([p\_1,p\_4,p\_1,p\_4]{}) \^[(0)]{}\_[[S]{}[N]{}]{}\^[[R]{}[P]{}]{}([p\_1,p\_2,p\_4,p\_3]{})\[2\_11\]where we have introduced the bubble integral I(p)=   The structure of shows the difference between the $s$-channel, for which there are two solutions of the $\delta$-function constraints in (for positive energies), and the $t$- and $u$-channels, for which there is only one.

Choosing $p_3=p_1$, $p_4=p_2$, which corresponds to considering the amplitudes associated to the first product of 