Three-body limit cycle: Universal form for general regulators

The Efimov effect, a remarkable realization of discrete scale invariance, emerges in the three-body problem with short-range interactions and is understood as a renormalization group (RG) limit cycle within Short-Range Effective Field Theory (SREFT). While the analytic form of the three-body renormalization relation has been established for a sharp cutoff regulator, its universality for other regulators remains underexplored. In this work, we derive the universal functional form of the three-body renormalization relation for general separable regulators through a detailed analysis of the Skorniakov-Ter-Martirosian and Faddeev equations. We find that the relation follows from a real Möbius transformation characterized by three parameters. This universality is verified numerically for various regulators. Although the functional form remains the same, the parameters characterizing the limit cycle exhibit regulator dependence. These findings broaden the class of RG limit cycles in SREFT and offer a more complete understanding of three-body renormalization.