matrix element $V^i_j$ of the potential by that of the adjoint $(V^\dagger)^i_j=V^j_i$. Thus the results obtained in Section 3 in the gauge theory basis exactly describe the decay amplitude in string basis!

Computation of the decay rate
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In Section 3 we derived a formal expression for the decay rate that we reproduce here, \[decay\] \_n = 2\_3 U\^n\_[\_3]{} U\_n\^[\_3]{} (E\_[\_3]{}-E\_n). We explain the evaluation of this expression in some detail. Let us begin with the computation of the effective composite interaction matrices $U^n_{\a_3}$ and $U^{\a_3}_n$. Triple-trace state is labelled as $\a_3=\{m,s_0,s_1\}$. For convenience, let us also define $s_2=1-s_0-s_1$. Because of the $\d$-function in (\[decay\]), we need this matrix element only at degeneracy, $E_{\a_3}=E_n$, where the computation simplifies and has already appeared in [@umut2] and [@golm1]. Using the perturbation coefficients (\[Vcoef\]) in (\[mel2\]) one finds, \[U1\] U\_n\^[ms\_0s\_1]{}&=& \_0\^1s\_0’ (\_[s\_0’,s\_0+s\_1]{}+ \_[s\_0’,s\_0+s\_2]{})\^2(n s\_0’)\
&&\_[p=-]{}\^ . The evaluation of the sum in the second line is explained in Appendix C of [@umut2]. Finally performing the integral over $s_0'$ one arrives at the result, \[U13\] U\_n\^[\_3]{}= -\^2(ns\_1). A similar computation using (\[Vcoef2\]) in (\[mel1\]) gives the result[^6] $U_{\a_3}^n = 2 U_n^{\a_3}$ when $E_{\a_3}=E_n$. Inserting (\[U13\]) into (\[decay\]) one obtains, \[decay2\] \_n= \_[m=-]{}\^J\_0\^1s\_0 J\_0\^[1-s\_0]{}s\_1 s\_1s\_2 