for atom-dimer scattering which is the sum of elastic and inelastic rates. However, in the low energy limit $k\to 0$, the elastic rate scales as $k$ because ${\cal T}^{\rm AD}_{ii}(k,k;E)$ is constant and the two-body phase space gives one power of $k$ [@Braaten:2004rn]. The elastic rate therefore vanishes at zero energy and the optical theorem gives the total relaxation rate $\beta_i$ $$\beta_i =\frac{2\hbar}{m}\,
{\rm Im} {\cal T}^{\rm AD}_{ii}(0,0,-1/(m a_i^2)).
\label{beta:opt}$$ The rate into the shallow dimer $\beta^{\rm sh}_{i\to j}$ is determined by the square of the T-matrix element multiplied by the two-body phase space: $$\beta^{\rm sh}_{i\to j} =
\frac{2p\hbar}{3\pi m} 
\left| {\cal T}^{\rm AD}_{ij}(0,p,-1/(m a_i^2))
\right|^2
\,\theta(a_i-a_j),
\label{beta:sh}$$ where the wave number $p=(2/\sqrt{3})\sqrt{a_j^{-2}-a_i^{-2}}$ and the $\theta$-function is inserted because the relaxation is allowed only when the initial dimer binding energy is smaller than the final one. The prefactor $2p/(3\pi)$ comes from the two-body phase space integral. From Eq. (\[beta:sh\]), one can deduce that $\beta^{\rm sh}_{i\to j}$ vanishes as $\sqrt{a_i-a_j}$ near the crossing of the $ik$- and $jk$-dimers where $a_i$ approaches $a_j$.

The relaxation rate into deep dimers $\beta^{\rm deep}_i$ could also be calculated by using the T-matrix in a similar way to Eq.  if the theory described deep dimers explicitly. 