Two-dimensional S-matrices from unitarity cuts

TL;DR

The paper develops a unitarity-cut framework for massive two-dimensional theories to compute the one-loop $2\to 2$ S-matrix from tree-level data in fixed dimension $d=2$, producing a finite, regulator-sensitive cut-constructible piece. It shows that in supersymmetric or integrable 2D theories this piece often equals the full one-loop S-matrix, while non-SUSY integrable models exhibit missing rational terms that can be absorbed by a finite coupling shift. In the AdS$_5\times S^5$ worldsheet context, the method reproduces the near-BMN one-loop S-matrix known from integrability up to an overall linear-in-momentum phase, supporting the link between unitarity and integrability in 2D. The results offer a finite, perturbative handle on testing exact quantum S-matrices and hint at regularization-scheme interpretations of discrepancies, with potential extensions to higher loops and other string backgrounds.

Abstract

Using unitarity methods, we compute, for several massive two-dimensional models, the cut-constructible part of the one-loop 2->2 scattering S-matrices from the tree-level amplitudes. We apply our method to various integrable theories, finding evidence that for supersymmetric models the one-loop S-matrix is cut-constructible, while for models without supersymmetry (but with integrability) the missing rational terms are proportional to the tree-level S-matrix and therefore can be interpreted as a shift in the coupling. Finally, applying our procedure to the world-sheet theory for the light-cone gauge-fixed AdS_5 x S^5 superstring we reproduce, at one-loop in the near-BMN expansion, the S-matrix known from integrability techniques.

Figures (1)

• Figure 1: Diagrams representing s-, t- and u-channel cuts contributing to the four-point one-loop amplitude.