Universal spectrum of isolated three-body resonances

TL;DR

This work develops a universal framework for isolated three-body resonances (ITBR) near two-body unitarity, formulating the problem with a zero-range contact model and solving the STM equation to obtain universal functions that describe the ITBR spectrum. It shows that, for a broad class of configurations (2FI, 2BI, 3B) and detuning regimes, the spectrum is governed by a two-parameter boundary condition at short hyperradius, often with $\upsilon'=0$, and it provides analytic results for small detuning when $s \gtrsim 1.5$, as well as a numerical path for other cases through the universal functions and the log-derivative condition. The paper further validates universality by analyzing a finite-range, separable two-body model, demonstrating that the reference model reproduces the contact-model spectrum and ITBR thresholds, and discusses practical implications for ultracold-atom experiments (e.g., Yb-Cs systems) where ITBRs may be observable via rf association. These results deepen understanding of three-body universal physics beyond the Efimov limit and offer a concrete route to extract three-body parameters from measured spectra. The approach bridges exact STM-based theory with finite-range realizations, enabling quantitative predictions across mass ratios, statistics, and partial waves relevant to experiments.

Abstract

The exact wavefunction of an isolated three-body resonance at finite scattering length is obtained for two identical particles interacting with another one via a pairwise zero-range potential. The corresponding universal spectrum is studied as a function of the scattering length. The universality of the results is illustrated by considering a model with finite-range interactions.

Figures (11)

• Figure 1: (a) Spectrum of a 2FI system with ${\ell=1}$, ${s=0.25}$, ${s-\upsilon=-5.960\dots\times10^{-2}}$ and ${\upsilon'=0}$. Black solid line: ITBR; red solid line: deepest KM state; dotted line: dimer spectrum; green solid line: example of ITBR of the separable model of Eqs. (\ref{['eq:V_separable']},\ref{['eq:form-factor']}) for ${\alpha=0}$ (see sec.\ref{['sec:example_ITBR_0']}). (b) Plot of the balance coefficient ${r_{\ell,s,\tau}}$ defined in the series of Eq. \ref{['eq:expansion_A']}.
• Figure 2: Plot of the universal spectrum for ${\upsilon=s+10^{-3}}$, ${\upsilon'=0}$ and for increasing values of ${s}$ (Solid line). Black dashed line: asymptotic law in Eq. \ref{['eq:E_s>1']} for the same values of ${s}$.
• Figure 3: a) Critical mass ratio at which the ITBR occurs at unitarity in the p-wave sector ${\ell=1}$ for the separable model of Eqs. (\ref{['eq:V_separable']},\ref{['eq:form-factor']}), as a function of the parameter ${\alpha}$; b) same but for the value of the scaling exponent ${s}$.
• Figure 4: (a) Plot of ${\langle k | \mathcal{A}_1^{\rm ref} \rangle}$ for three values of the binding wavenumber associated with the spectrum of Fig. \ref{['fig:1']} where ${s=0.25}$. The functions ${\langle k | \mathcal{A}_1 \rangle}$ for ${q=q_{\rm u}}$ and ${q=2q_u}$ coincide with the black solid line; (b) Plot of the ratio ${\langle k | \mathcal{A}_1^{\rm ref} \rangle/\langle k|\mathcal{A}_1 \rangle}$ for the same values of the binding wavenumber.
• Figure 5: Plot of the ITBR spectrum (red line) of the reference model in Eqs.(\ref{['eq:form-factor']},\ref{['eq:V_separable']}) near the ITBR threshold for ${s = 1.25}$; Black line: universal spectrum ; Dash line: approximate spectrum of Eq. \ref{['eq:approx_spectrum_s>1']}; Dotted line: dimer spectrum. The numerical values of the parameters are given in the text of sec. \ref{['sec:s>1']}