Three-Body Scattering Hypervolume of Particles with Unequal Masses

TL;DR

This paper generalizes the three-body scattering hypervolume for particles with arbitrary mass ratios at zero energy, deriving asymptotic wave functions and analyzing implications for dilute Bose gases.

Contribution

It extends the concept of the three-body scattering hypervolume to particles with unequal masses and explores its effects on Bose gas properties.

Findings

Derived asymptotic expansions of the three-body wave function.

Connected the hypervolume parameter to ground state energy in large volumes.

Analyzed stability and decay rates of two-component Bose gases.

Abstract

We analyze the collision of three particles with arbitrary mass ratio at zero collision energy, assuming arbitrary short-range potentials, and generalize the three-body scattering hypervolume $D$ first defined for identical bosons in 2008. We solve the three-body Schr\"{o}dinger equation asymptotically when the three particles are far apart or one pair and a third particle are far apart, deriving two asymptotic expansions of the wave function, and the parameter $D$ appears at the order $1/B^4$, where $B$ is the overall size of the triangle formed by the particles. We then analyze the ground state energy of three such particles with vanishing or negligible two-body scattering lengths in a large periodic volume of side length $L$, where the three-body parameter contributes a term of the order $D/L^6$. From this result we derive some properties of a two-component Bose gas with negligible two-body scattering lengths: its energy density at zero temperature, the corresponding generalized Gross-Pitaevskii equation, the conditions for the stability of the two-component mixture against collapse or phase separation, and the decay rates of particle densities due to three-body recombination.