  We discuss shallow resonances in the nonrelativistic scattering of two
particles using an effective field theory (EFT) that includes an auxiliary
field with the quantum numbers of the resonance. We construct the manifestly
renormalized scattering amplitude up to next-to-leading order in a systematic
expansion. For a narrow resonance, the amplitude is perturbative except in the
immediate vicinity of the resonance poles. It naturally has a zero in the
low-energy region, analogous to the Ramsauer-Townsend effect. For a broad
resonance, the leading-order amplitude is nonperturbative almost everywhere in
the regime of validity of the EFT. We regain the results of an EFT without the
auxiliary field, which is equivalent to the effective-range expansion with
large scattering length and effective range. We also consider an additional
fine tuning leading to a low-energy amplitude zero even for a broad resonance.
We show that in all cases the requirement of renormalizability when the
auxiliary field is not a ghost ensures the resonance poles are in the lower
half of the complex momentum plane, as expected by other arguments. The
systematic character of the EFT expansion is exemplified with a toy model
serving as underlying theory.
