Recombination rates from potential models close to the unitary limit

We investigate universal behavior in the recombination rate of three bosons close to threshold. Using the He-He system as a reference, we solve the three-body Schrödinger equation above the dimer threshold for different potentials having large values of the two-body scattering length $a$. To this aim, we use the hyperspherical adiabatic expansion and we extract the $S$ matrix through the integral relations recently derived. The results are compared to the universal form, $\alpha \approx 67.1{\sin }^2[s_0\ln \left({\kappa }_{*}a\right)+\gamma ]$, for different values of $a$ and selected values of the three-body parameter ${\kappa }_{*}$. A good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections, which have been recently proposed [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. Furthermore, we analyze the validity of the above formula in the description of a very different system: neutron-neutron-proton recombination. Our analysis confirms the universal character of the process in systems of very different scales having a large two-body scattering length.

Figure 1

Ratio between the energy of the ground (upper panel) and first excited (lower panel) state of the trimer and the dimer binding energy as a function of ${\kappa }_{*}a$. The dashed line is the universal prediction of the Efimov law given by Eq. (9) without shift (${\Gamma }_n=0$), while the solid line is the translated universal curve. The circles and squares are the calculations using the TBG and TBG+H3B potentials, respectively.

Figure 2

The effective-range function as a function of ${\left(ka\right)}^2$ for two different values of ${\kappa }_{*}a$; in the upper panel, ${\kappa }_{*}a=0.36$, and, in the lower panel, ${\kappa }_{*}a=0.56$. The dimer threshold corresponds to ${\left(ka\right)}^2\approx 4/3$. The (red) circles are the calculations below the dimer threshold of Ref. [8]. The (green) triangles are the present calculations. The solid line is Eq. (12), whereas the dashed line is an effective-range parametrization (see text).

Figure 3

Atom-dimer elastic (solid line) and breakup (dashed line) cross sections as a function of the atom incident energy for a three-helium atom system and the TBG+H3B potential. The breakup channel is open at $E_i=E_2=1.303$ mK. The zoom shows the behavior close to both thresholds, i.e., $E_i=0$ for the elastic case and $E_i=E_2$ for the breakup case. The circles correspond to actual calculations, while the curves are interpolations.

Figure 4

The recombination rate $K_3$ and the dissociation rate $D_3$ for three-helium atoms with the TBG+H3B potential ($\lambda =1$) as a function of the three-body energy $E$.

Figure 5

The recombination rate $K_3$ at threshold for different values of the product ${\kappa }_{*}a$. The red circles are the results using the TBG potential, whereas the green squares are the results using the TBG+H3B potential. The points have been fitted with Eq. (21), obtaining ${\Gamma }_{+}\simeq 6\times {10}^{-2}$.

Figure 6

The recombination rate $K_3$ at threshold as a function of $E_3^1/E_2$. The red circles are the results using the TBG potential, whereas the green squares are the results using the TBG+H3B potential. The dashed line corresponds to the zero-range theory, while the solid line is the translation of the zero-range curve.

Figure 7

The recombination rate $K_3$ and the dissociation rate $D_3$ for the $1/2^{+}$ state in the nucleon-deuteron system as a function of the energy.

Figure 8

The coefficients of the scattering wave function as a function of the hyperradius $\rho$. The picture show the Jacobi set of coordinates used, whereas the shadow box indicates the two coefficients surviving at long distances.