In (\[eqn:4ptampl\]) $\mathbb{S}$ is the scattering operator, the fields $\Phi$ carry flavor indices to account for different kinds of particles in the model and $p_i$ are their on-shell momenta. In this paper we will restrict to the case where all the particles have equal non-vanishing mass, which we set to unity. In the two-dimensional case the set of initial momenta is preserved under collision. This translates into the following identity for the energy-momentum conservation $\delta$-function of (\[eqn:ampcons\]) \[delta2d\] \^[(2)]{}(p\_1+p\_2-p\_3-p\_4)=J(p\_1,p\_2)((\_1 - \_3)(\_2 - \_4) +(\_1 - \_4)(\_2 - \_3) )  . Above, $\rmp$ is the spatial momentum and the Jacobian $J(p_1,p_2)=1/(\partial \e_{\rmp_1}/\partial \rmp_1-\partial\e_{\rmp_2}/\partial \rmp_2)$ depends on the dispersion relation $\e_\rmp$ (the on-shell energy associated to $\rmp$) for the theory at hand. Spatial momenta are assumed to be ordered $\rmp_1>\rmp_2$. Substituting (\[delta2d\]) in we can consider the amplitudes associated to the first product of $\delta$-functions $\delta(\text{p}_1 - \text{p}_3)\delta(\text{p}_2 - \text{p}_4)$ without loss of generality. The S-matrix elements relevant for the description of the $2\to2$ scattering in the two-dimensional case are then defined as [@Eden:1966] \[AandS\] S\_[[M]{}[N]{}]{}\^[[P]{}[Q]{}]{}(p\_1,p\_2) \_[[M]{}[N]{}]{}\^[[P]{}[Q]{}]{}(p\_1,p\_2,p\_1,p\_2)  , where the denominator is required to make contact with the standard definition of the S-matrix in two dimensions. In applying the unitarity method to 