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

The three-body system inside the unitary window is studied for three equal bosons and three equal fermions having $1/2$ 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.

Figure 1

Binding momentum as a function of the inverse scattering length for a Gaussian potential, both multiplied by the effective range $r_e$. Experimental data of four real systems are shown by the filled circles.

Figure 2

Binding momentum as a function of the inverse scattering length for a Gaussian potential, both multiplied by the Gaussian range $r_0$. Data of real systems are located in the plot through the ratio $a/a_B$.

Figure 3

Three-body binding momentum as a function of the two-body binding momentum for a Gaussian potential, both multiplied by the Gaussian range $r_0$ to build dimensionless quantities. Notable points are indicated as solid circles as explained in the text.

Figure 4

The atom-dimer scattering length (in units of $a_B$) as a function of the dimer binding momentum ${\kappa }_2=1/a_B$ times $r_0$. The blue solid points are the calculations using a Gaussian potential and the blue solid line is the parametrization of Eq. (11). The binding momentum (times $r_0$) of the ground state (green solid line) and the first excited state (black solid line) are shown too. The solid red diamonds indicate the positions of the first excited state of the helium trimer and the atom-dimer scattering length. The two dashed lines indicate the position in which the second and third excited states cross the atom-dimer threshold.

Figure 5

The doublet neutron-deuteron scattering length ${}^{2}a_{\text{nd}}$ in units of the triplet two-body energy length $a_B$ as a function of the inverse of $a_B$ in units of the Gaussian range $r_0$ (blue solid points). The blue line is the parametrization of Eq. (11). In the upper panel the dimensionless binding momenta ${\kappa }_3^{\left(n\right)}{r}_0$ of the three-nucleon ground state (orange curve), $n=0$, and excited state (green curve), $n=1$, are shown together with the binding momentum ${\kappa }_2{r}_0$ of the two-body bound state (red curve). The excited state almost overlaps with the two-body state. The red diamonds indicate the physical point of the ground state curve and the corresponding value on the ${}^{2}a_{\text{nd}}$ curve. The lower panel shows ${}^{2}a_{\text{nd}}/a_B$ in a larger region. The vertical dashed line is the asymptote at which ${}^{2}a_{\text{nd}}$ diverges.

Figure 6

The effective range functions $S_k^B$ (upper panel) and $S_k^F$ (lower panel) as functions of the energy momentum $k$, multiplied by energy length $a_B$. The different curves correspond to different dimer binding energies.

Figure 7

The virtual state pole momentum for three bosons and three fermions for different dimer energies. The realistic cases for fermions are on top of the Gaussian characteristic curve.

Figure 8

The effective range function for the three realistic cases discussed.