Entanglement Entropy Distributions of a Muon Decay

Divergences that occur in density matrices of decay and scattering processes are shown to be regularized by tracing and unitarity or the optical theorem. These divergences are regularized by the lifetime of the decaying particle or the total scattering cross section. Also, this regularization is shown to give the expected helicities of final particles. The density matrix is derived for the weak decay of a polarized muon at rest, $\mu^- \rightarrow \nu_{\mu} (e^- \bar \nu_e)$, with Lorentz invariant density matrix entries and unitarity upheld at tree level. The electron's von Neumann entanglement entropy distributions are calculated with respect to both the electron's emission angle and energy. The angular entropy distribution favors an electron emitted backwards with respect to the muon's polarization given a minimum volume regularization. The kinematic entropy distribution is maximal at half the muon's rest mass energy. These results are similar to the electron's angular and kinematic decay rate distributions. Both the density matrix and entanglement entropy can be cast either in terms of ratios of areas or volumes.