Nonrelativistic effective field theory with a resonance field

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