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

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.

Efimov Physics and Connections to Nuclear Physics

“…Figure 3b also shows the doublet neutron-deuteron scattering length, 2 a nd , calculated with the Gaussian interaction, as a violet curve (in units of the energy length a B ). It corresponds to a fit of the numerical results using the form given by Equation 26(see 79). Two branches are shown in Figure 3b; the dashed black vertical line, the asymptote at r 0 /a B = 0.101, indicates the position at which 2 a nd diverges, and the first excited state disappears into the 1 + 2 continuum.…”

“…The axis value of r 0 /a B = 0.061, which corresponds to the ground state, can be used to determine the characteristic Gaussian range r (0) 0 = 11.15 a 0 with which a Gaussian potential reproduces the dimer and ground state trimer energies. From that value, the three-body parameters of the ground and excited states of the helium trimer can be estimated (79):…”

Efimov Physics and Connections to Nuclear Physics

“…( 26), see Ref. (77). Two branches are shown, with the dashed vertical line, the asymptote at r0/aB = 0.101, indicating the position at which 2 a nd diverges and the first excited state disappears into the 1+2 continuum.…”

“…Following Ref. (77), we show in Fig. 3 (left panel) two branches of the function aAD/aB (violet solid line) using γ The two-body energy is represented by the solid red line and the two asymptotes (dashed lines), extracted from Eq.…”

“…Following Refs. (77,86) the S-matrix energy pole, EP = −3h 2 k 2 p /4m is determined from the s-wave low energy phase-shifts calculated using the Gaussian potential of Eq. ( 27).…”