Rational Terms in Theories with Matter

TL;DR

The paper addresses rational remainders in gluon amplitudes within gauge theories coupled to matter in arbitrary representations and shows these terms depend only on low-order representation indices, specifically the second- and fourth-order indices $$I^{(2)}$ and $$I^{(4)}$. Using Badger's large-mass limit technique, the authors express rational terms in terms of these indices and derive linear Diophantine equations whose solutions identify an infinite class of non-supersymmetric theories with vanishing rational terms for gluon amplitudes. This class includes the next-to-simplest quantum field theories and reveals amplitudes that are not naively cut-constructible yet exhibit no rational contributions. The results extend the scope of tractable one-loop gluon amplitudes beyond supersymmetric theories and provide new cut-constructible examples despite apparent power-counting obstructions.

Abstract

We study rational remainders associated with gluon amplitudes in gauge theories coupled to matter in arbitrary representations. We find that these terms depend on only a small number of invariants of the matter-representation called indices. In particular, rational remainders can depend on the second and fourth order indices only. Using this, we find an infinite class of non-supersymmetric theories in which rational remainders vanish for gluon amplitudes. This class includes all the "next-to-simplest" quantum field theories of arXiv:0910.0930. This provides new examples of amplitudes in which rational remainders vanish even though naive power counting would suggest their presence.