Current-Density—Current-Density Commutators and Vector-Meson Decays

Abstract

The techniques developed in a previous paper for investigating equal-time current-density-current-density commutators by making use of the Dyson representation, and the Ward-identity techniques of Schnitzer and Weinberg are applied to a study of the three-point function relevant to the decays of the $\omega$, $\varphi$, and ${\pi }^o$ mesons. It is found that the algebra-of-fields commutation relations do not permit these mesons to decay, and that the pole model of Gell-Mann, Sharp, and Wagner is incorrect, in that the $\omega -\rho -\pi$ vertex is not a constant in the region appropriate to the decays but rather depends linearly on the squared momenta. The commutation relations of $U\left(12\right)\mathrm{}$ are then assumed, and the Kawarabayashi-Suzuki relation $g_{\rho }=m_{\rho }{f}_{\pi }$ is obtained as a consistency condition. The unknown constants arising from the equal-time commutators of the $\pi$ field with the vector currents, and from $\varphi -\omega$ mixing, are fixed by requiring, as a first approximation, the vanishing of the $\varphi -\rho -\pi$ vertex, and by feeding the measured value of the width of the decay ${\pi }^o\to 2\gamma$. The results then obtained for the widths $\Gamma (\omega \to {\pi }^o+\gamma )$ and $\Gamma (\omega \to 3\pi )$ are in excellent agreement with experiment. This agreement is maintained when provision is made to allow for the decays $\varphi \to \rho +\pi$ and $\varphi \to {\pi }^o+\gamma$.