dimers can be taken into account by analytically continuing the cutoff $\Lambda$ into the complex plane [@Braaten:2002sr] $$\Lambda\to \Lambda e^{i \eta_*/s_0},
\label{cutoff}$$ where $\eta_*$ is a width parameter associated with the effects of the deep dimers. $\Lambda$ and $\eta_*$ are determined from experimental measurements of three-body recombination in the $^6$Li system [@Williams:2009zzd] and their numerical values are $\Lambda=456~a_0^{-1}$ and $\eta_*=0.016$ [@Braaten:2009ey] where $a_0$ is the Bohr radius.

In order to obtain the T-matrix element, the amplitude ${\cal A}_{ij}$ must be multiplied with the dimer-wavefunction renormalization factors $\sqrt{Z_i Z_j}$, where $Z_i=8\pi/a_i$, $${\cal T}^{\rm AD}_{ij}(k,p;E) 
=\frac{8\pi}{\sqrt{a_i a_j}}{\cal A}_{ij}(k,p;E).
\label{Tmatrix}$$ By solving the STM equation and using the relation in Eq. , we can calculate the relaxation rates in Eqs.  and .

Zero Temperature Results {#sec:results}
========================

In this section we present our numerical results for the dimer relaxation rate constants $\beta_i$ at zero temperature and compare them with recent measurements in Refs. [@Lompe:2010; @Nakajima:2010]. We solve the STM equation in Eq.  numerically with 5 input parameters: the three pair scattering lengths $a_i$, $i=1,2,3$, the cutoff $\Lambda$ and the width parameter $\eta_*$. We use the values $\Lambda=456~a_0^{-1}$ and $\eta_*=0.016$ that have been determined from measurements of the three-body recombination rate in Ref. [@Williams:2009zzd] 