  Resonances are of particular importance to the scattering of composite
particles in quantum mechanics. We build an effective field theory for two-body
scattering which includes a low-energy $S$-wave resonance. Our starting point
is the most general Lagrangian with short-range interactions. We demonstrate
that these interactions can be organized into various orders so as to generate
a systematic expansion for an $S$ matrix with two low-energy poles. The pole
positions are restricted by renormalization at leading order, where the common
feature is a non-positive effective range. We carry out the expansion
explicitly to next-to-leading order and illustrate how it systematically
accounts for the results of a toy model -- a spherical well with a delta shell
at its border.
