Short-range interactions and narrow resonances in effective field theory

We consider the effective field theory (EFT) treatment of two-body systems with narrow resonances. Within this approach, an $s$-wave scattering amplitude can be expanded in powers of a typical momentum scale of a system $Q\ll \Lambda$, where $\Lambda$ represents a hard scale of a scattering system and an energy difference $\delta \epsilon=|E-\epsilon_0|\ll \epsilon_0$, where $\epsilon_0$ is a resonance peak energy. It is shown that at leading order in the double expansion a universal form of a two-body scattering amplitude is a sum of a Breit-Wigner term of order $Q^{-1}$, a smooth background term of order $Q^0$, and an interference term of order $Q^0$. The techniques developed in this paper can be used to investigate the properties of narrow resonances that are produced by short-distance dynamics.