Nonlocal Symmetry Operations, the Cabibbo Angle, and the Leptonic Decays of Vector Mesons

Symmetries which are not exact but which leave the vacuum invariant cannot be implemented locally. In this note we consider simple nonlocal symmetries. For the in- and out-fields, the nonlocality involved can be simply some trivial scale transformations that accompany the internal-symmetry index transformations. In particular, with proper relative normalizations, a set of in-fields ${\hat{\phi }}_{\alpha }^{\mathrm{in}}\mathrm{}\left(0\right)\mathrm{}$ will transform irreducibly under the nonlocal group. If one assumes that the vector currents $V_{\alpha }^{\mu }\mathrm{}\left(0\right)\mathrm{}$ behave like vector-meson fields $\square {\hat{\phi }}_{\alpha ,\mathrm{in}}^{\mu }\mathrm{}\left(0\right)\mathrm{}$, as far as the vacuum to one-vector-meson matrix elements are concerned, one finds $tan{\theta }_Y=tan{\theta }_B{\left(\frac{m_{\phi }}{m_{\omega }}\right)}^2 \mathrm{and} \frac{1}{3}{m}_{\rho }\Gamma (\rho \to ee)=m_{\omega }\Gamma (\omega \to ee)+m_{\phi }\Gamma (\phi \to ee)$, both previously derived from Weinberg's first sum rule. If one assumes that the weak axial-vector currents behave like the pseudoscalar meson fields ${\partial }_{\mu }{\hat{\phi }}_{\alpha }^{\mathrm{in}}\mathrm{}\left(0\right)\mathrm{}$, as far as the vacuum to one-pseudoscalar-meson matrix elements are concerned, one finds $\frac{m_{\pi }}{m_K}=\left(\frac{f_K}{f_{\pi }}\right)tan{\theta }_A$.