the one-loop four point amplitude (\[eqn:ampcons\]) one follows the standard route of considering two-particle cuts, obtained by putting two intermediate lines on-shell.

In general there will be contributions from tadpole and bubble graphs to the one-loop four point amplitude. The former have no physical two-particle cuts and do not contribute an imaginary part to the amplitude. For this reason they should not be considered in our procedure, which is based on standard unitarity rules (derived from the optical theorem) [@Bern:1994zx; @Bern:1997nh]. Therefore the one-loop result following from unitarity techniques will receive contributions from the $s$- $t$- and $u$- channel cuts illustrated in Fig. \[stu\]. Explicitly, the imaginary part of the amplitude is given by the sum of the following three contributions \^[(1)]{}\^[[P]{}[Q]{}]{}\_[[M]{}[N]{}]{}(p\_1,p\_2,p\_3,p\_4)|\_[s-cut]{}=12 i\^+([l\_1]{}\^2-[1]{}) i\^+(l\_2\^2-[1]{})\
\[eqn:sch1\]\^[(0)]{}\_[[M]{}[N]{}]{}\^[[R]{}[S]{}]{}([p\_1,p\_2,l\_1,l\_2]{})\^[(0)]{}\_[[S]{}[R]{}]{}\^[[P]{}[Q]{}]{} ([l\_2,l\_1,p\_3,p\_4]{})\
\^[(1)]{}\^[[P]{}[Q]{}]{}\_[[M]{}[N]{}]{}(p\_1,p\_2,p\_3,p\_4)|\_[t-cut]{}=12 i\^+([l\_1]{}\^2-[1]{}) i\^+([l\_2]{}\^2-[1]{})\
\[eqn:tch1\]\^[(0)]{}\_[[M]{}[R]{}]{}\^[[S]{}[P]{}]{}([p\_1,l\_1,l\_2,p\_3]{})\^[(0)]{}\_[[S]{}[N]{}]{}\^[[R]{}[Q]{}]{}([l\_2,p\_2,l\_1,p\_4]{})\
\^[(1)]{}\^[[P]{}[Q]{}]{}\_[[M]{}[N]{}]{}(p\_1,p\_2,p\_3,p\_4)|\_[u-cut]{}=12 i\^+([l\_1]{}\^2-[1]{}) i\^+([l\_2]{}\^2-[1]{})\
\[eqn:uch1\]\^[(0)]{}\_[[M]{}[R]{}]{}\^[[S]{}[Q]{}]{}([p\_1,l\_1,l\_2,p\_4]{})\^[(0)]{}\_[[S]{}[N]{}]{}\^[[R]{}[P]{}]{}([l\_2,p\_2,l\_1,p\_3]{}) where $\mathcal{A}^{(0)}$ are tree-level amplitudes and a sum over the complete set of intermediate states ${R},{S}$ (all allowed particles for the cut lines) is understood. The on-shell propagator is given in terms of $\delta^+(k^2 - 1) = \theta(k^0) \delta(k^2-1)$ and we have included a symmetry factor of $\tfrac12$.

(-5,-3) – (-4,-4); (-5,-5) – (-4,-4); (-2,-4) – (-1,-3); (-2,-4) – (-1,-5); (-3,-4) circle (1cm); (-3,-2.5) – (-3,-5.5); 