all the interaction matrix elements of the actual problem vanish at degeneracy due to the trigonometric factors in (\[Vcoef\]). We assume the same property in the toy model so that all energy integrals which appear when (\[tint\]) is inserted in (\[trip\]) converge despite the singular denominators.

We apply the $i\e$ prescription of the last line of (\[wno\]) in each of the six terms, multiplying and dividing by the energy denominators which are missing in the last four terms. It is easy to see that all contributions cancel pairwise in third and fourth and in the fifth and sixth terms. In the first and second terms we obtain \[trip1\] a\^n(0)\_[triple]{} &=& -\_2\_3’\_2 V\^n\_[\_2]{}V\^[\_2]{}\_[\_3]{} V\^[\_3]{}\_[’\_2]{}V\^[’\_2]{}\_n\
&&\
We now note that the term with $\d(E_{n\a_2})$ vanishes because $V^n_{\a_2}=0$ at degeneracy. Because of this vanishing one can combine the two principal value terms without ambiguity. The triple-trace contribution to the decay amplitude can then be written as \[tricon\] a\^n(0)\_[triple]{} = -d\_3 U\^n\_[\_3]{} \[(E\_[\_3 n]{})  + i()\] U\^[\_3]{}\_n where we have introduced the effective composite interactions which couple single- and triple-traces, namely U\^n\_[\_3]{} &=& d\_2 \[mel1\]\
U\^[\_3]{}\_n &=& d\_2 \[mel2\].

The single-trace contribution (\[sing\]) can be treated similarly. One makes the short-time replacement $a^m(t''') \to
