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Physical Review C

Gaussian characterization of the unitary window for N=3 : Bound, scattering, and virtual states

Arnoldas Deltuva, M. Gattobigio, Alejandro Kievsky, M. Viviani

Published Dec 7, 2020• 16 citations
Publisher

Abstract

The three-body system inside the unitary window is studied for three equal bosons and three equal fermions having $1/2$ spin-isospin symmetry. We perform a Gaussian characterization of the window using a Gaussian potential to define trajectories for low-energy quantities as binding energies and phase shifts. On top of this trajectories experimental values are placed or, when not available, quantities calculated using realistic potentials that are known to reproduce experimental values. The intention is to show that the Gaussian characterization of the window, thought as a contact interaction plus range corrections, captures the main low-energy properties of real systems as for example three helium atoms or three nucleons. The mapping of real systems on the Gaussian trajectories is taken as indication of universal behavior. The trajectories continuously link the physical points to the unitary limit allowing for the explanation of strong correlations between observables appearing in real systems and which are known to exist in that limit. In the present study we focus on low-energy bound, scattering, and virtual states.

DOI
10.1103/physrevc.102.064001

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The three-body system inside the unitary window is studied for three equal bosons and three equal fermions having 1/21/21/2 spin-isospin symmetry. We perform a Gaussian characterization of the window using a Gaussian potential to define trajectories for low-energy quantities as binding energies and phase shifts. On top of this trajectories experimental values are placed or, when not available, quantities calculated using realistic potentials that are known to reproduce experimental values. The intention is to show that the Gaussian characterization of the window, thought as a contact interaction plus range corrections, captures the main low-energy properties of real systems as for example three helium atoms or three nucleons. The mapping of real systems on the Gaussian trajectories is taken as indication of universal behavior. The trajectories continuously link the physical points to the unitary limit allowing for the explanation of strong correlations between observables appearing in real systems and which are known to exist in that limit. In the present study we focus on low-energy bound, scattering, and virtual states.
Gaussian characterization of the unitary window for N=3N=3N=3: bound, scattering and virtual states
The three-body system inside the unitary window is studied for three equal bosons and three equal fermions having 1/21/21/2 spin-isospin symmetry. We perform a gaussian characterization of the window using a gaussian potential to define trajectories for low-energy quantities as binding energies and phase shifts. On top of this trajectories experimental values are placed or, when not available, quantities calculated using realistic potentials that are known to reproduce experimental values. The intention is to show that the gaussian characterization of the window, thought as a contact interaction plus range corrections, captures the main low-energy properties of real systems as for example three helium atoms or three nucleons. The mapping of real systems on the gaussian trajectories is taken as indication of universal behavior. The trajectories continuously link the physical points to the unitary limit allowing for the explanation of strong correlations between observables appearing in real systems and which are known to exist in that limit. In the present study we focus on low-energy bound, scattering and virtual states.
The equal mass three-fermion system having 1/2 spin-isospin symmetry is study around the unitary limit. The two body system has two different scattering lengths, a 0 and a 1 , corresponding to the spin singlet state S = 0 and the spin triplet state S = 1, respectively. The unitary limit is defined when the two quantities a 0 , a 1 → ∞ . The three-nucleon system is located very close to this limit, the singlet and triplet n − p scattering lengths are large with respect to the range of the nuclear interaction. The ratio of the two is about a 0 / a 1 ≈ −4.31. This value defines a plane in which a 0 and a 1 can be varied. Using a nucleon-nucleon spin dependent potential with variable strength it is possible to study the behavior of the three-nucleon binding energy along that plane. This analysis can be considered an extension of the Efimov plot for three bosons to the case of three 1/2-spin-isospin fermions.
Universal behavior in few-boson systems close to the unitary limit, where two bosons become unbound, has been intensively investigated in recent years both experimentally and theoretically. In this particular region, called the unitary window, details of the interparticle interactions are not important and observables, such as binding energies, can be characterized by a few parameters. With an increasing number of particles, the short-range repulsion, present in all atomic, molecular, and nuclear interactions, gradually induces deviations from the universal behavior. In the present paper, we discuss a simple way of incorporating nonuniversal behavior through one specific parameter which controls the smooth transition of the system from a universal to nonuniversal regime. Using a system of NNN helium atoms as an example, we calculate their ground-state energies as trajectories within the unitary window and also show that the control parameters can be used to determine the energy per particle in homogeneous systems when $N\ensuremath{\rightarrow}\ensuremath{\infty}$.
Two-, three-, and four-boson systems are studied close to the unitary limit using potential models constructed to reproduce the minimal information given by the two-body scattering length aaa and the two-body binding energy or virtual-state energy E2{E}_{2}E2​. The particular path used to reach the unitary limit is given by varying the potential strength. In this way the energy spectrum in the three- and four-boson systems is computed. The lowest-energy states show finite-range effects absorbed in the construction of level functions that can be used to study real systems. Higher-energy levels are free from finite-range effects, therefore the corresponding level functions tend to the zero-range universal function. Using this property, a zero-range equation for the four-boson system is proposed and the four-boson universal function is computed.
The universal character of non-relativistic large three-body bosonic systems is addressed, with focus on highly interesting situations that occur in low-energy nuclear and atomic physics. Investigations on the trajectory of the first excited Efimov state, in a renormalized zero-range three-body model for a system with two bound and one virtual two-body subsystems, are reported. The approach is applied to n-n-18C, where the n - n virtual energy and the three-body ground state are kept fixed. It is shown that the real part of the elastic s-wave phase-shift ([Formula: see text]) presents a zero, or a pole in [Formula: see text], when the system has an Efimov excited or virtual state. A brief discussion is given on the relevance of the approach for ultracold atom physics with tunable scattering lengths.
This work analyzes the three and four equal-mass fermionic systems near and at the sss- and ppp-wave unitary limits using hyperspherical methods. The unitary regime addressed here is where the two-body dimer energy is at zero energy. For fermionic systems near the sss-wave unitary limit, the hyperradial potentials in the NNN-body continuum exhibit a universal long-range ${R}^{\ensuremath{-}3}$ behavior governed by the sss-wave scattering length alone. The implications of this behavior on the low-energy phase shift are discussed. At the ppp-wave unitary limit, the four-body system is studied through a qualitative look at the structure of the hyperradial potentials at unitarity for the ${L}^{\ensuremath{\pi}}={0}^{+}$ symmetry. A quantitative analysis shows that there are tetramer states in the lowest hyperradial potentials for these systems. Correlations are made between these tetramers and the corresponding trimers in the two-body fragmentation channels. Universal properties related to the four-body recombination process $A+A+A+A\ensuremath{\leftrightarrow}{A}_{3}+A$ are discussed.