We describe new on-shell recursion relations for tree amplitudes in $\mathcal{N}=1$ and $\mathcal{N}=2$ gauge theories and use these to show that the structure of the one-loop S-matrix in pure (i.e. without any matter) $\mathcal{N}=1$ and $\mathcal{N}=2$ gauge theories resembles that of pure Yang-Mills theory. 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 nonsupersymmetric theories that are free of bubbles. These theories see simplifications in their S-matrices that cannot be deduced just from naive power-counting. In particular, our results indicate that one-loop scattering amplitudes in the $\mathcal{N}=2$, $SU(N)$ theory with a symmetric tensor hypermultiplet and an antisymmetric tensor hypermultiplet are simple like those in the $\mathcal{N}=4$ theory.

• Received 2 December 2009

DOI:https://doi.org/10.1103/PhysRevD.81.105002