Non-Relativistic Effective Field Theory of Narrow S-Wave Resonances with No Auxiliary Field

We present a potential in the context of effective field theory (EFT) that allows sharp resonances to form without the use of dimer fields. We show that our theory leads to the same results previously proved for the on-shell T-matrix of such states and demonstrate that this potential leads to a scattering amplitude which up to next-to-leading order is renormalizable on and off mass shell. We show that this potential leads to the corresponding T-matrix for the bound state and is also renormalizable for the bound state on and off mass shell up to next-to-leading order.