  A new family of S-matrix theories with resonance poles is constructed and
conjectured to correspond to the Homogeneous sine-Gordon theories associated
with simply laced compact Lie groups, where some of the resonance poles can be
traced to the presence of unstable particles in the spectrum. These theories
are unitary in the usual S S^\dagger =1 sense, they are not parity invariant,
and they exhibit continuous coupling constants that determine both the mass
spectrum of stable particles and the masses and the position of the resonance
poles.
