Wilsonian renormalization group analysis of nonrelativistic three-body systems without introducing dimerons

TL;DR

This paper analyzes nonrelativistic three-body systems using Wilsonian RG without auxiliary dimeron fields, revealing limit cycle behavior and a novel fixed point, and accurately estimates the Efimov parameter.

Contribution

It derives correct RGEs for three-body systems without dimerons, demonstrating limit cycle behavior and identifying new fixed points in the RG flow.

Findings

Limit cycle behavior confirmed in three-body RGEs.

A new nontrivial fixed point found when two-body interactions are absent.

Efimov parameter $s_{0}$ estimated within a few percent accuracy.

Abstract

Low-energy effective field theory describing a nonrelativistic three-body system is analyzed in the Wilsonian renormalization group (RG) method. No effective auxiliary field (dimeron) that corresponds to two-body propagation is introduced. The Efimov effect is expected in the case of an infinite two-body scattering length, and is believed to be related to the limit cycle behavior in the three-body renormalization group equations (RGEs). If the one-loop property of the RGEs for the nonrelativistic system without the dimeron field, which is essential in deriving RGEs in the two-body sector, persists in the three-body sector, it appears to prevent the emergence of limit cycle behavior. We explain how the multi-loop diagrams contribute in the three-body sector without contradicting the one-loop property of the RGEs, and derive the correct RGEs, which lead to the limit cycle behavior. The Efimov parameter, $s_{0}$, is obtained within a few percent error in the leading orders. We also remark on the correct use of the dimeron formulation. We find rich RG-flow structure in the three-body sector. In particular, a novel nontrivial fixed point of the three-body couplings is found when the two-body interactions are absent. We also find, on the two-body nontrivial fixed point, the limit cycle is realized as a loop of finite size in the space of three-body coupling constants when terms with derivatives are included.