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 $Q\ll \mathrm{\Lambda }$, where $\mathrm{\Lambda }$ 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.

Figure 1

The Feynman graphs that contribute to ${\mathcal{A}}_{-1}$. This amplitude, which represents the full dimeron propagator (double solid lines), gets a contribution from the bare dimeron propagator (solid line) dressed with particle bubbles. The scattering particles are represented by solid lines with arrows.

Figure 2

The amplitude $A_0$ arises from (a) a tree graph with $C_0$ at the vertex and (b)–(c) two one-loop graphs that consist of the full dimeron propagator and have one $C_0$ at the vertices. The notation is the same as in Fig. 1.

Figure 3

The amplitude ${\mathcal{A}}_1$ receives contributions from all Feynman graphs with two $C_0$’s at the vertices. In this figure, (a) provides a correction to Fig. 2 and 2 provides a correction to Figs. 2 and 2. The notation is the same as in Fig. 1.