# Universal Three-Body Parameter of Heavy-Heavy-Light systems with negative intraspecies scattering length

## TL;DR

This paper reveals that the three-body parameter in heavy-heavy-light systems with negative intraspecies scattering length exhibits universal behavior governed by van der Waals interactions and mass ratio, with an analytic formula provided.

## Contribution

It introduces a universal formula for the three-body parameter in HHL systems, linking it to van der Waals length and mass ratio, extending understanding beyond homonuclear cases.

## Key findings

- $a^{(1)}_{-}$ follows a universal behavior influenced by vdW interaction and mass ratio.
- An analytic expression for $a^{(1)}_{-}$ as a function of $a_{HH}$ is derived.
- In resonance conditions, $a^{(1)}_{-}$ is approximately a constant, about -6.3 times the vdW length.

## Abstract

The Three-Body Parameter(3BP) $a^{\scriptscriptstyle(1)}_{\scriptscriptstyle-}$ is crucial to understanding Efimov physics, and a universal 3BP has been shown in experiments and theory in ultracold homonuclear gases. The 3BP of heteronuclear systems was predicted to possess much richer properties than the homonuclear counterparts for the large parameter space. In this work, we investigate the universal properties of $a^{\scriptscriptstyle(1)}_{\scriptscriptstyle-}$ for the Heavy-Heavy-Light(HHL) system with negative intraspecies scattering length $a_{\scriptscriptstyle HH}$. We find that $a^{\scriptscriptstyle(1)}_{\scriptscriptstyle-}$ follows a universal behavior determined by the van der Waals(vdW) interaction and the mass ratio. An analytic formula of $a^{\scriptscriptstyle(1)}_{\scriptscriptstyle-}$ is given as a function of $a_{\scriptscriptstyle HH}$, which allows an intuitive understanding of how does $a^{\scriptscriptstyle(1)}_{\scriptscriptstyle-}$ depend on the mass ratio and the vdW length $r_{\scriptscriptstyle vdW}$. In a special case, when the two heavy atoms are in resonance, $a^{\scriptscriptstyle(1)}_{\scriptscriptstyle-}$ is approximately a constant: $a^{\scriptscriptstyle(1)}_{\scriptscriptstyle-} = -(6.3\pm0.6)\, r_{\scriptscriptstyle vdW,HL}$.