Subthreshold resonances

Rather constant behaviour of the formation rate $\lambda _{mf} (T)$ is observed by experiments, especially at low temperature [62,65], together with an unexpected density dependence of the formation rate [63]. This led theorists to consider extensions of the classical Vesman mechanism of resonant molecular formation which assumed an isolated two-body collision and the $\delta$function resonance profile, as adopted in Refs. [133,71,72]. Recall that the Vesman model was very successful in the $d\mu d$ case [12]. In the $d\mu t$ case, however, strong resonance levels are expected to exist just below the $\mu t+ d$ threshold, e.g., $K_i\rightarrow K_f = 0\rightarrow 1$ at -14.0meV, $0\rightarrow 2$ at -4.3 meV, and $1\rightarrow 2$ at -11.7 meV for F=0 [4]. These transitions have large matrix elements because of the strong overlap of the wave functions.

Two main mechanisms to access these subthreshold (i.e., negative energy) resonances are: (1) intrinsic resonance width due to the finite lifetime of the molecular complex (mainly the Auger decay width), and (2) three-body or many-body collisions in which the other body (or bodies) absorbs the excess energy. The mechanism (1) is density independent, but (2) depends on the surrounding environment.

There have been many attempts to treat the subthreshold resonances  [158,159,160,161,162,163,144], however, complete understanding has not yet been achieved. For example, the use of the Breit-Wigner profile adopted in Ref. [158] was criticized in Ref. [160,161] at least for the high density situation. Despite these criticisms, Petrov's calculation for the formation rate for $\lambda ^{F=0}_{d\mu t}$ at low temperature and low density [145,146] seems to agree with the value suggested by the PSI measurement ( $130 \pm 20 \; \mu$s) [22]. Armour recently proposed a new approach beyond the Born approximation [164,165], which also predicts non-zero width resonance profiles, but its application is still limited [121] and comparison with experiment is not yet possible.

We note, however, that for epithermal molecular formation, the resonant widths considered in these models are smaller than the Doppler broadening; thus the effects are negligible.