We have
calculated the rates for a few cuts, including those
given in the literature.
For the case of unequal leptons, the results are given
in table 2.1 for the decay
. These include the cuts used in Refs.
[20] and [21],
and 
,
respectively.
Table 2.1:
Theoretical values for the branching ratios for the decay
for various cuts.
It can be seen that for this decay most of the branching ratio is generated at very low electron-positron invariant masses. As can be seen from the result for the cuts used in Ref. [21], the effect of the structure dependent terms is most visible at high invariant electron-positron invariant mass. Our calculation, including the effect of the form factors agrees well with their data. We disagree, however, with the numerical result obtained by Ref. [20] by about an order of magnitude.
For the decay 
, we obtain for the tree level
or IB contribution a branching ratio
and, including the form factors,
Here the structure
dependent terms are the leading-contribution since the inner
Bremsstrahlung contribution is helicity suppressed as can be seen
from the factor 
in 
.
For the decays with identical leptons we obtain for the muon case a branching ratio of
for the full phase space including the effects of the form factors. The inner Bremsstrahlung or the tree level branching ratio for this decay is
For the decay with two positrons and one electron the integration over full phase space for the tree level results is very sensitive to the behaviour for small pair masses. We have given the tree level and the full prediction, including form factor effects in table 2.2. The cuts are always on both invariant masses :
Table 2.2:
Theoretical values for the branching ratios for the decay
for various cuts.
The values for the masses used are those of 
and 
.
For 
and 
we used the values given in table 1 in Ref.
[2],
