$m= \pm 2, ~s_0 =2/3$ and when $m=\pm 1,~ s_0=1/3$, and so forth. One must thus question the validity of non-degenerate perturbation theory, and this was discussed in [@mit2]. One signal for breakdown of perturbation theory is a divergence due to a vanishing energy denominator in the summation formulas for shifts of eigenvalues. It was pointed out in [@mit2] that perturbation theory would remain valid if the matrix elements in the numerator happened to vanish at degeneracy, and that the matrix element of the effective interaction between single- and double-trace states does indeed so vanish. This is a necessary condition for the validity of the order $g_2^2$ calculation leading to (\[eig1\]), but it is not sufficient. The single/triple mixing matrix element is of order $g_2^2$. If it does not vanish at degeneracy, it would also require [@mit2] the use of degenerate perturbation theory with possible modification of the result (\[eig1\]) even though a divergence would not appear until order $g_2^4$.

An effective quantum mechanical formulation for the gauge theory, which simplifies previous computations was developed in [@golm1]. The single/triple matrix element was computed in this formalism, and it [**does not**]{} vanish at degeneracy. It is this fact that motivates the 