Narrow resonances and short-range interactions

Abstract

Narrow resonances in systems with short-range interactions are discussed in an effective field theory (EFT) framework. An effective Lagrangian is formulated in the form of a combined expansion in powers of a momentum $Q\ensuremath{\ll}\ensuremath{\Lambda}$---a short-distance scale---and an energy difference $\ensuremath{\delta}\ensuremath{\epsilon}=|E\ensuremath{-}{\ensuremath{\epsilon}}_{0}|\ensuremath{\ll}{\ensuremath{\epsilon}}_{0}$---a resonance peak energy. At leading order in the combined expansion, a two-body scattering amplitude is the sum of a smooth background term of order ${Q}^{0}$ and a Breit-Wigner term of order ${Q}^{2}(\ensuremath{\delta}\ensuremath{\epsilon}){}^{\ensuremath{-}1}$ which becomes dominant for $\ensuremath{\delta}\ensuremath{\epsilon}\ensuremath{\lesssim}{Q}^{3}$. Such an EFT is applicable to systems in which short-distance dynamics generates a low-lying quasistationary state. The EFT is generalized to describe a narrow low-lying resonance in a system of charged particles. It is shown that in the case of Coulomb repulsion, a two-body scattering amplitude at leading order in a combined expansion is the sum of a Coulomb-modified background term and a Breit-Wigner amplitude with parameters renormalized by Coulomb interactions.