The Next-to-Simplest Quantum Field Theories

TL;DR

The paper probes which gauge theories beyond ${ mathcal{N}}=4$ can exhibit a simple one-loop S-matrix for gluon scattering by extending on-shell methods to ${ mathcal{N}}=1$ and ${ mathcal{N}}=2$ theories. It develops new tree-level recursion relations via an on-shell superspace formalism and analyzes how matter representations affect one-loop bubble and triangle coefficients through higher-order Indices, yielding Diophantine constraints that can force a no-bubble/no-triangle structure. Several explicit theories are constructed that display box-only one-loop amplitudes, including a finite N=2 SU(K) theory with symmetric and antisymmetric hypermultiplets, indicating N=4-like simplicity beyond the planar limit. The work thus broadens the class of “next-to-simplest” quantum field theories with highly constrained S-matrices and provides a practical framework for analyzing their perturbative structure.

Abstract

We describe new on-shell recursion relations for tree-amplitudes in N=1 and N=2 gauge theories and use these to show that the structure of the S-matrix in pure N=1 and N=2 gauge theories resembles that of pure Yang-Mills. We proceed to study gluon scattering in gauge theories coupled to matter in arbitrary representations. The contribution of matter to individual bubble and triangle coefficients can depend on the fourth and sixth order Indices of the matter representation respectively. So, the condition that one-loop amplitudes be free of bubbles and triangles can be written as a set of linear Diophantine equations involving these higher-order Indices. These equations simplify for supersymmetric theories. We present new examples of supersymmetric theories that have only boxes (and no triangles or bubbles at one-loop) and non-supersymmetric theories that are free of bubbles. In particular, our results indicate that one-loop scattering amplitudes in the N=2, SU(K) theory with a symmetric tensor hypermultiplet and an anti-symmetric tensor hypermultiplet are simple like those in the N=4 theory.