Validity

Obviously, in transport calculations, neglecting one of the scattering channels leads to inaccurate results, if the contribution of that channel is significant. Therefore the back decay process has to be looked at carefully. On the other hand, in the description of fusion yield, the effective formation approximation has been used in many analyses (see for reviews Refs. [1,2,3,4]). Let us investigate the validity of this approximation in the fusion yield description.

We point out that in order for the effective formation approximation to be justified in terms of describing the fusion yield, at least one of the following criteria must be met: (a) trivial conditions that the back decay probability , (b) a negligible change in $\mu t$ energy before and after back decay in the laboratory frame, (c) fast (compared to the formation rate) ``re-thermalization'' of back-decayed $\mu t$ ($\mu d$) in a thermal equilibrium condition.

For example, at low temperatures the condition (a) is satisfied for $d\mu t$ formation, while the condition (c) applies for $d\mu d$ formation at least at high densities (one may need to be careful about this at very low density). In case of $d\mu t$ formation at epithermal energies, however, none of these conditions apply, hence the back decay process cannot be neglected. In the following section, we shall consider the implications of the resonant scattering.