# Loss of Conformality in Efimov Physics

## TL;DR

This paper investigates how conformal invariance is lost in Efimov physics through the merging and disappearance of fixed points in the renormalization group flow as the spatial dimension varies, revealing a transition to limit cycle behavior.

## Contribution

It identifies the critical dimensions where fixed points merge and vanish, explaining the transition from conformal invariance to limit cycle behavior in Efimov physics.

## Key findings

- Critical dimensions at d=2.30 and d=3.76 where conformality is lost.
- Fixed points merge and become complex at critical dimensions.
- For intermediate dimensions, the RG flow exhibits limit cycle behavior.

## Abstract

The loss of conformal invariance in Efimov physics is due to the merger and disappearance of an infrared and an ultraviolet fixed point of a three-body renormalization group flow as the spatial dimension $d$ is varied. In the case of identical bosons at unitarity, it is known that there are two critical dimensions $d_{\rm 1}=2.30$ and $d_{\rm 2}=3.76$ at which there is loss of conformality. For $d<d_{\rm 1}$ and $d>d_{\rm 2}$, the beta function of the three-body coupling has real roots which correspond to infrared and ultraviolet fixed points. The fixed points merge and disappear into the complex plane at the critical dimensions $d_1$ and $d_2$. For $d_{\rm 1}<d<d_{\rm 2}$, the beta function has complex roots and the renormalization group flow for the three-body coupling constant is a limit cycle.