Nonrelativistic Effective Field Theory with a Resonance Field

TL;DR

This paper develops a systematic nonrelativistic effective field theory with an auxiliary resonance field to analyze shallow resonances, providing a renormalized amplitude up to next-to-leading order and exploring different resonance regimes.

Contribution

It introduces a new EFT framework with an auxiliary field for resonances, clarifies the perturbative and nonperturbative regimes, and ensures renormalizability and correct pole placement.

Findings

Perturbative amplitude near narrow resonance poles.

Nonperturbative leading-order amplitude for broad resonances.

Resonance poles are in the lower half-plane, ensuring physical consistency.

Abstract

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