Entanglement Entropy Distributions of a Muon Decay

Abstract

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, these regularizations are shown to give the
expected helicities of final particles. As an illustration, the density matrix is derived for the weak decay of a
polarized muon at rest, µ¯→ νµ(e¯ν¯_e), with Lorentz invariant density matrix entries and unitarity upheld
at tree level. The electron’s von Neumann entanglement entropy distributions are derived with respect
to both the electron’s emission angle and energy. The angular entropy distribution peaks for an electron
emitted backward with respect to the muon’s polarization given a minimum volume regularization larger
than the cube of the muon’s Compton wavelength. 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 in terms of either ratios of
areas or volumes.