Angular zeros of Brown, Mikaelian, Sahdev, and Samuel and the factorization of tree amplitudes in gauge theories

Abstract

The zero in the angular distribution of the process $q\overline{q}\ensuremath{\rightarrow}\ensuremath{\gamma}W$ (discovered by Brown, Mikaelian, Sahdev, and Samuel) when the magnetic moment of the $W$ has the Yang-Mills value, is shown to be a consequence of a factorizability of the amplitude into one factor which contains the dependence on the charge or other internal-symmetry indices, and another which contains the dependence on the spin or polarization indices. In gauge theories generally, this factorization is found to hold for any four-particle tree-approximation amplitude, when one or more of the four particles is a gauge-field quantum. The factorization hinges on a "spatial generalized Jacobi identity" obeyed by the polarization-dependent factors of the vertices, in analogy to the generalized Jacobi identity obeyed by the charge-index-dependent factors. We emphasize that observation of the process $q\overline{q}\ensuremath{\rightarrow}\ensuremath{\gamma}W$ in $p\overline{p}$ collisions or the decay $W\ensuremath{\rightarrow}q\overline{q}\ensuremath{\gamma}$ provides a direct test of the prediction of gauge (Yang-Mills) theories for vector-vector-vector couplings, just as much as would ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}Z\ensuremath{\rightarrow}{W}^{+}{W}^{\ensuremath{-}}$.