Scattering of gauge bosons in sixth order and nonrenormalizability of massive Yang-Mills theories

We consider the unitarity relation in order $g^6$ for the canonically quantized massive Yang-Mills theory. The three-particle intermediate-state contribution to the absorptive part of $T_{22}=T(W_a+W_b\to W_c+W_d)$ is reexpressed in terms of the absorptive parts of Feynman diagrams constructed using only "soft" propagators $\frac{-g_{\mu \nu }}{(k^2-m^2)}$ and $\frac{1}{(k^2-m^2)}$. The vertices which appear in this decomposition of Abs $T_{22}$ are the usual ones which occur in order $g^4$, together with two new vertices of dimension five which couple a $W$ to three ghost particles; these vertices are similar to those found by Veltman in his study of radiative corrections to the $W$ propagator. It is shown that, as a consequence, the Feynman rules proposed recently by Hsu and Sudarshan for a quantized massive Yang-Mills theory do not yield a unitary $S$ matrix. Our result is in harmony with Boulware's formal argument that such theories are not renormalizable.