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

Abstract

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 ${T}_{22}=T({W}_{a}+{W}_{b}\ensuremath{\rightarrow}{W}_{c}+{W}_{d})$ is reexpressed in terms of the absorptive parts of Feynman diagrams constructed using only "soft" propagators $\frac{\ensuremath{-}{g}_{\ensuremath{\mu}\ensuremath{\nu}}}{({k}^{2}\ensuremath{-}{m}^{2})}$ and $\frac{1}{({k}^{2}\ensuremath{-}{m}^{2})}$. 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.