  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.
