Universal Angular Probability Distribution of Three Particles near Zero Energy Threshold

TL;DR

This paper proves that near the zero energy threshold, the angular probability distribution of a three-particle system approaches a universal form, independent of specific pair interactions, with implications for Efimov physics and halo nuclei.

Contribution

The paper establishes the universal angular distribution of three particles near zero energy threshold, regardless of pair interaction specifics.

Findings

Angular distribution approaches a universal form.

Result applies to systems with zero energy resonance.

Implications for Efimov physics and halo nuclei.

Abstract

We study bound states of a 3--particle system in $\mathbb{R}^3$ described by the Hamiltonian $H(\lambda_n) = H_0 + v_{12} + \lambda_n (v_{13} + v_{23})$, where the particle pair $\{1,2\}$ has a zero energy resonance and no bound states, while other particle pairs have neither bound states nor zero energy resonances. It is assumed that for a converging sequence of coupling constants $\lambda_n \to \lambda_{cr}$ the Hamiltonian $H(\lambda_n)$ has a sequence of levels with negative energies $E_n$ and wave functions $\psi_n$, where the sequence $\psi_n$ totally spreads in the sense that $\lim_{n \to \infty}\int_{|\zeta| \leq R} |\psi_n (\zeta)|^2 d\zeta = 0$ for all $R>0$. We prove that for large $n$ the angular probability distribution of three particles determined by $\psi_n$ approaches the universal analytical expression, which does not depend on pair--interactions. The result has applications in Efimov physics and in the physics of halo nuclei.