The Three-Body Limit Cycle: Universal Form for General Regulators

TL;DR

This work shows that the three-body renormalization relation in Short-Range EFT exhibits a universal Möbius transformation form when general separable regulators are used, extending the familiar sharp-cutoff result. By analyzing the Skorniakov–Ter-Martirosian and Faddeev equations in the low-energy limit, the authors demonstrate that the running of the three-body coupling $H_0$ is governed by a real Möbius map of the phase and that regulator-dependent parameters $\delta_0$, $h_0$, and $b_0$ encode the details of the regulator. Numerical solutions for multiple regulators validate the universal form and reveal regulator-specific values of the Möbius parameters. The limit-cycle structure is recast on a unit circle, clarifying how poles and zeros determine the Efimov spectrum and its winding as the cutoff changes. These results provide a rigorous Hamiltonian-based foundation for three-body renormalization in a broad class of calculations and illuminate how regulator choices shape the three-body sector while preserving universal scaling behavior.

Abstract

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.

Figures (4)

• Figure 1: (a) Self-energy diagram of the dimer field $d$. Solid and dashed lines represent the propagators of the particle $\psi$ and the dimer $d$, respectively. (b) Scattering process between a dimer $d$ and a particle $\psi$. The double-dashed lines indicate the renormalized propagator of the dimer field $d$, which can depend on the two-body regulator $g_2$. Only the momenta of the fields are labeled, since they correspond to the arguments of the two-body and three-body regulators, $g_2$ and $g_3$.
• Figure 2: The three-body low-energy constant $H_0$ at the unitarity limit as a function of the momentum cutoff $\Lambda$ (in units of $\kappa_{*}$) for the sharp cutoff (black diamonds), Gaussian ($n=1$, blue circles), quartic super-Gaussian ($n=2$, red squares), and sextic super-Gaussian ($n=3$, green triangles). Results obtained from solving the STM and Faddeev equations are shown in the left and right panels, respectively. The black dashed line is obtained with parameter values from the literature for the sharp cutoff (see Table \ref{['H0paramvalues']}). The lines for (super-)Gaussians are obtained by fitting the data points of the same color with Eq. \ref{['H0_0_expr']}.
• Figure 3: Dimensionless parameters $\delta_0$ (panel a), $h_0$ (panel b), and $b_0$ (panel c) appearing in Eqs. \ref{['H0_0_expr']} and \ref{['eq:b0']} as functions of $n$, the index of a super-Gaussian regulator, Eq. \ref{['eq:SuperGaussians']}. Blue circles are obtained by numerically solving the STM equation \ref{['eq:STM_general_reg']} and fitting to Eq. \ref{['H0_0_expr']}, green dashed lines represent the approximate values from Eqs. \ref{['eq:delta0approx']} and \ref{['eq:h0approx']}, and black stars denote the exact sharp-cutoff values from Ref. Chen:2025rti.
• Figure 4: The unit-circle representation of the three-body limit cycle. The arrow indicates the direction of the RG flow as $\Lambda$ increases. The red and black points mark the poles and the fixed point $\tilde{H}_0$ of the limit cycle, respectively. The blue (green) star, square, and triangle on the circle denote the positions of the zeros and of the flow at cutoff values $\Lambda' = 15 \kappa_\ast$ and $\Lambda" = 800 \kappa_\ast$ on the limit cycle obtained by solving the Faddeev equation with the $n = 1$ ($n = 3$) (super-)Gaussian regulator.