and have been used in Ref. [@Braaten:2009ey] to predict the atom-dimer relaxation rate.

![(Color online) The scattering lengths for the three lowest hyperfine states of $^6$Li in units of $10^3~a_0$ as a function of the magnetic field B from [@Julienne]: $a_{13} \equiv a_2$ (long-dashed), $a_{12} \equiv a_3$ (short-dashed), and $a_{23}  \equiv a_1$ (dashed-dotted). The vertical lines at 690 G, 811 G, and 834 G indicate the Feshbach resonances for $a_{13}$, $a_{23}$, and $a_{12}$, respectively.[]{data-label="fig:aij"}](aij.pdf){width="15cm"}

Fig. \[fig:aij\] shows the scattering lengths for the 3 lowest hyperfine state of $^6$Li as a function of the magnetic field. The atom-dimer relaxation rate $\beta_i$ is non-zero in the region of positive scattering length $a_i$. The upper limit of the region is set by the Feshbach resonances at 811 G, 690 G, and 834 G for $a_1$, $a_2$, and $a_3$, respectively. If the scattering length is much larger than the van der Waals length $\ell_{\rm vdW}\approx 65 a_0$, the universal theory is valid. We denote this region as the [ *universal region*]{}. Corrections due to the finite range of the interaction should be small in the universal region. In practice, we apply the universal zero-range theory when all scattering lengths are at least two times 