Implications for our measurements

Finally, we discuss the implication of resonant scattering in our atomic beam experiments.

Regarding the validity of the effective model, we used detailed Monte Carlo simulations to compare the fusion yields in our standard time-of-flight target arrangement for both model (a), the effective model with the renormalized rates, and model (b), with explicit resonant scattering. Assuming that resonant scattering removes $\mu t$ from the resonance region, which is well justified from the discussion above, we observed that model (a) overestimates the fusion yield in the DS layer by nearly 50%. The use of model (b) thus resolves the inconsistency of our data with our earlier analysis using model (a), as reported in Ref. [78].

As for our sensitivity to the scattered $\mu t$ energy, we have performed Monte Carlo calculations with different assumptions of the back-decayed $\mu t$ energies. As mentioned, the exact evaluation of our expressions given in this appendix for the back-decayed $\mu t$ energy distribution cannot be performed yet, since it requires the transition matrix elements which are not currently available to us. Our Monte Carlo calculations in the time-of-flight arrangement, with varied between 1 meV to 0.3 eV, suggest some 7% difference in the DS fusion yield. It should be noted that these variations depend also on the assumptions for $\mu t$interactions in a solid at low energies. Nonetheless, our present lack of precise knowledge of the back-decayed $\mu t$ energy gives, a significant, yet not overwhelming contribution to the total uncertainty in our measurements of the resonant molecular formation rate, hence significant improvement in the accuracy of the latter would require, among others, the exact numerical evaluation of our expression.

In summary, we have pointed out in this appendix the importance of the resonant scattering via back decay, a process which has not previously been well-considered, and (a) showed that the use of renormalized effective formation rates leads to an overestimate of fusion yield and (b) gave detailed expressions for the energy distribution of back-decayed $\mu t$, which is considerably different from the previously assumed thermal distribution. From the full Monte Carlo calculations, the effect of (a) is estimated to be about 50% and (b) less than about 10%, the latter depending on the possible solid state effects in $\mu t$ thermalization and molecular formation.