Differential Cross Section for $e^{+}+e^{-}\to W^{+}+W^{-}\to e^{-}+{\overline{\nu }}_e+{\mu }^{+}+{\nu }_{\mu }$

The cross section $e^{+}+e^{-}\to W^{+}+W^{-}\to {\mu }^{+}+{\nu }_{\mu }+e^{-}+{\overline{\nu }}_e$ in which $e^{-}$ and ${\mu }^{+}$ are detected in coincidence in the colliding-beam experiment is computed with the mass, magnetic moment, and leptonic mode branching ratio of the $W$ boson as parameters. The kinematical correlations necessary for the identification and mass determination of the $W$ meson are discussed. Numerical examples show that the energy-angle correlations of the final $e$ and $\mu$ are very sensitive to the $W$ mass. The analytical expression for the cross section was obtained by an electronic computer. The characteristics of dynamical correlations were investigated by numerical examples of angular distributions of $e^{-}$ and ${\mu }^{+}$ for different values of magnetic moment of $W$. It was found that the rate of increase of cross section with respect to the relative angle between the final electron and muon is the most sensitive dynamical correlation needed for the determination of the $W$ magnetic moment. We ignore the possibility that $W$ may have form factors and an anomalous quadrupole moment. Symmetries in the differential cross section are discussed. Because of one-photon exchange, the differential cross section $e^{-}$ and ${\mu }^{+}$ must be symmetric with respect to the plane perpendicular to the incident beam. Because of time-reversal invariance, the differential cross section for ${\mu }^{+}$ must be symmetric with respect to the plane formed by the incident beam and the final electron. Similarly the differential cross section for $e^{-}$ must be symmetric with respect to the plane formed by the incident beam and the ${\mu }^{+}$. It is also shown that the charge-conjugate decay mode $e^{+}+e^{-}\to W^{+}+W^{-}\to {\mu }^{+}+{\overline{\nu }}_{\mu }+e^{+}+{\nu }_e$ can be obtained from our result by simply putting ${\mu }^{+}\to {\mu }^{-}$ and $e^{-}\to e^{+}$ in the final state if one considers only the lowest order process. It is pointed out that the techniques used in this paper can be employed to calculate many other processes in which two unstable particles are produced.