Conformality Lost in Efimov Physics

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.