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

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: $a^{\scriptscriptstyle(1)}_{\scriptscriptstyle-} = -(6.3\pm0.6)\, r_{\scriptscriptstyle vdW,HL}$.