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Physical Review C

Efimov effect in the three-nucleon system

Sadhan K. Adhikari, Lauro Tomio

Published Jul 1, 1982• 35 citations
Publisher

Abstract

The Efimov effect in the $S$ wave spin doublet trinucleon system is studied and the virtual state of this system is found to be an Efimov virtual state. In the Efimov limit this state becomes a bound state. But as the strength of the spin triplet nucleon-nucleon interaction is increased, it first moves away from the lowest physical scattering threshold, then stops and turns around and eventually becomes a physical bound state. The movement of other Efimov states of this system were studied and once they become a virtual state they move into the left-hand cut with the increase of the strength of the triplet nucleon-nucleon potential and do not become resonances.NUCLEAR REACTIONS Efimov effect, trinucleon system, Efimov virtual states, Amado model.

DOI
10.1103/physrevc.26.83

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The Efimov effect in the SSS wave spin doublet trinucleon system is studied and the virtual state of this system is found to be an Efimov virtual state. In the Efimov limit this state becomes a bound state. But as the strength of the spin triplet nucleon-nucleon interaction is increased, it first moves away from the lowest physical scattering threshold, then stops and turns around and eventually becomes a physical bound state. The movement of other Efimov states of this system were studied and once they become a virtual state they move into the left-hand cut with the increase of the strength of the triplet nucleon-nucleon potential and do not become resonances.NUCLEAR REACTIONS Efimov effect, trinucleon system, Efimov virtual states, Amado model.
The emergence of Efimov levels in a three-body system is investigated near the unitarity limit characterized by a resonant two-body interaction. No direct evidence of Efimov levels is seen in the three-nucleon system since the triton is the only physical bound state. We provide a model-independent analysis of nucleon–deuteron scattering at low energy by formulating a consistent effective field theory. We show that virtual states evolve into shallow bound states, which emerge as excited triton levels as we drive the system towards unitarity. Even though we consider this specific system, our results for the emergence of the Efimov levels are universal.
The Efimov phenomenon manifests itself as an emergent discrete scaling symmetry in the quantum three-body problem. In the unitarity limit, it leads to an infinite tower of three-body bound states with energies forming a geometric sequence. In this work, we study the evolution of these so-called Efimov states using relativistic scattering theory. We identify them as poles of the three-particle SSS matrix in the complex energy plane, and we study how they transform from virtual states through bound states to resonances when we change the interaction strength. We dial the scattering parameters toward the unitarity limit and observe the emergence of the universal scaling of energies and couplings---a behavior known from the nonrelativistic case. We additionally find that Efimov resonances follow unusual, cyclic trajectories accumulating at the three-body threshold and then disappear at some values of the two-body scattering length. We propose a partial resolution to this ``missing states'' problem.
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.
Introducing a suitable cutoff parameter r0{r}_{0}r0​ for the effective potential in the three-body subsystem, we show that the number of four-body bound states is roughly $k\mathrm{ln}[\ensuremath{\epsilon}a({r}_{0})]$, where $\ensuremath{\epsilon}$ is a momentum cutoff and $\frac{2}{3\ensuremath{\pi}}\ensuremath{\le}k\ensuremath{\le}\frac{1}{3}$, if the scattering length a(r0)a({r}_{0})a(r0​) for a two-body bound state and a third particle tends to infinity. In a numerical calculation we found that the value ${r}_{0}\ensuremath{\approx}1.82$ fm, which provides a sufficiently resonant effective interaction in the three-body subsystem, is independent of the applied two-body potentials within the numerical accuracy.NUCLEAR REACTIONS Efimov effect.
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.
By studying the eigenvalue spectrum of the Faddeev kernel in a certain singular limit, we give an independent proof of an effect recently deduced by Efimov: When three identical particles interact via short-range pairwise potentials, the number of three-body bound states grows without limit when the pairwise scattering length aaa becomes large. [The number of bound states is then roughly $(\frac{1}{\ensuremath{\pi}})\mathrm{ln}(\ensuremath{\Lambda}|a|)$, where $\ensuremath{\Lambda}$ is a momentum cutoff]. We extend our proof to the case where only two particles are identical and show that Efimov's effect persists in the special limiting cases with two heavy and one light particle, and with two light and one heavy particle.