Charge commutation relations, asymptotic SU(2)_{L} symmetry, and the mass of the second Z boson in electroweak gauge theories

Abstract

Electroweak gauge theories are discussed, using charge commutation relations and asymptotic symmetry. The mass of the second $Z$ boson (${m}_{2}$) in the $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}_{1}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}_{2}$ models is predicted using the $W$- and first-$Z$-boson masses (${m}_{W}$ and ${m}_{1}$) to be ${{m}_{2}}^{2}={3\ensuremath{-}r+{[{(3\ensuremath{-}r)}^{2}+\frac{4(r\ensuremath{-}1)(2\ensuremath{-}r)}{(1\ensuremath{-}{{c}_{\ensuremath{\theta}}}^{2}r)}]}^{\frac{1}{2}}}\frac{{{m}_{W}}^{2}}{2}$, where $r={(\frac{{m}_{1}}{{m}_{W}})}^{2}$ with $1\ensuremath{\le}r\ensuremath{\le}2$ and ${{c}_{\ensuremath{\theta}}}^{2}=1\ensuremath{-}{{s}_{\ensuremath{\theta}}}^{2}$ with ${{s}_{\ensuremath{\theta}}}^{2}={(\frac{37.2 \mathrm{GeV}}{{m}_{W}})}^{2}$ defined at zero-four-momentum-transfer-squared limit. At present, ${m}_{2}$ is bounded as ${m}_{2}\ensuremath{\gtrsim}(1.6\ensuremath{-}1.7){m}_{W}$ by the recent data on ${s}_{\ensuremath{\theta}}$ and $r$ from the $\overline{p}p$ collider experiments.