Loss of Conformality in Efimov Physics

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.