Three-boson scattering hypervolume for a nonzero orbital angular momentum

TL;DR

This work extends three-body scattering theory to the $L=2$ sector for three identical bosons with short-range interactions by introducing the three-body scattering hypervolume $D$ with dimension $L^{8}$. It develops two asymptotic wave-function expansions, the 111- and 21-expansions, in which $D$ appears at leading subleading orders and connects $D$ to energy shifts in finite and thermodynamic systems, as well as to the low-energy three-body T-matrix and recombination rates. A Born-approximation formula for $D$ in the weak-coupling limit is provided, and an effective Hamiltonian is constructed to incorporate three-body effects in a dilute Bose gas. The results quantify how $D$ and its $L=0$ counterpart $D^{(0)}$ influence energy shifts and recombination, with particular attention to the absence of Efimov physics for $L=2$ and possible resonance-enhanced effects when $D$ is large.

Abstract

We analyze the zero energy collision of three identical bosons in the same internal state with total orbital angular momentum $L=2$, assuming short range interactions. By solving the Schrödinger equation asymptotically, we derive two expansions of the wave function when three bosons are far apart or a pair of bosons and the third boson are far apart. The scattering hypervolume $D$ is defined for this collision. Unlike the scattering hypervolume defined by one of us in 2008, whose dimension is length to the fourth power, the dimension of $D$ studied in the present paper is length to the eighth power. We then derive the expression of $D$ when the interaction potentials are weak, using the Born's expansion. We also calculate the energy shift of such three bosons with three different momenta $\hbar \mathbf{k_{1}}$, $\hbar\mathbf{k_{2}}$ and $\hbar\mathbf{k_{3}}$ in a large periodic box. The obtained energy shift depends on $D^{(0)}/Ω^{2}$ and $D/Ω^{2}$, where $D^{(0)}$ is the three-body scattering hypervolume defined for the three-body $L=0$ collision and $Ω$ is the volume of the periodic box. We also calculate the contribution of $D$ to the three-body T-matrix element for low-energy collisions. We then calculate the shift of the energy and the three-body recombination rate due to $D^{(0)}$ and $D$ in the dilute homogeneous Bose gas. The contribution to the three-body recombination rate constant from $D$ is proportional to $T^2$ if the temperature $T$ is much larger than the quantum degeneracy temperature but still much lower than the temperature scale at which the thermal de Broglie wave length becomes comparable to the physical range of interaction.

Figures (2)

• Figure 1: The energy shift of the dilute Bose gas caused by the adiabatic introduction of the three-body $L=0$ scattering hypervolume $D^{(0)}$ vs. the reduced temperature $T/T_d$. The red line shows the energy shift above the critical temperature, and the blue curve shows the energy shift below the critical temperature.
• Figure 2: The energy shift of the dilute Bose gas by the adiabatic introduction of the three-body $L=2$ scattering hypervolume $D$ vs. the reduced temperature $T/T_d$. The red curve shows the energy shift above the critical temperature, and the blue curve shows the energy shift below the critical temperature.