Conformality Lost in Efimov Physics

TL;DR

This paper investigates how conformal invariance is lost in Efimov physics by analyzing the behavior of fixed points in the renormalization group flow of identical bosons as the spatial dimension varies, revealing a transition to limit cycles.

Contribution

It explicitly demonstrates the mechanism of conformality loss via fixed point merger and disappearance in Efimov physics across different spatial dimensions.

Findings

Fixed points merge and become complex at critical dimensions.

Efimov effect appears when fixed points are complex.

Renormalization group flow exhibits limit cycles between critical dimensions.

Abstract

A general mechanism for the loss of conformal invariance is the merger and disappearance of an infrared fixed point and an ultraviolet fixed point of a renormalization group flow. We show explicitly how this mechanism works in the case of identical bosons at unitarity as the spatial dimension $d$ is varied. For $d$ between the critical dimensions $d_{\rm 1}=2.30$ and $d_{\rm 2}=3.76$, there is loss of conformality as evidenced by the Efimov effect in the three-body sector. The beta function for an appropriate three-body coupling is a quadratic polynomial in that coupling. For $d<d_{\rm 1}$ and for $d>d_{\rm 2}$, the beta function has two real roots that correspond to infrared and ultraviolet fixed points. As $d$ approaches $d_{\rm 1}$ from below and as $d$ approaches $d_{\rm 2}$ from above, the fixed points merge and disappear into the complex plane. For $d_{\rm 1}<d<d_{\rm 2}$, the beta function has complex roots and the renormalization group flow for the three-body coupling is a limit cycle.