a loop amplitude into two pieces [^3]) and working in a *fixed* number of dimensions $d=2$, we propose a formula for constructing the one-loop $2\to 2$ scattering amplitude directly from the corresponding on-shell tree-level amplitudes.

As might be expected, the two-dimensional case is much simpler than its four-dimensional counterpart. Our proposals, equations and below, take a remarkably compact form. This is a consequence of the massive 2-d kinematics, which imply that the cut loop momenta are frozen to specific values. Therefore, the integral degenerates to a sum over discrete solutions of the on-shell conditions. This is reminiscent of the framework of generalized unitarity in the four-dimensional case when quadruple cuts (maximal cuts [@Britto:2004nc]) are used. There, the quadruple-cut integral is completely localized by the four delta-functions of the cut propagators, and it reduces to a product of four tree-level amplitudes.

It is important to note that our procedure is inherently finite. All the theories we consider are either UV-finite or renormalizable. In the latter case, our procedure implicitly chooses a particular regularization scheme. Hence the result we will compare to is the renormalized four-point amplitude, up to scheme ambiguities. A related point is that the expressions and , obtained via 