In the (anti-)muon decay process from the weak interaction, specifically

$\mu^+ \rightarrow e^+ + \nu_e + \bar{\nu}_\mu$

how should I address the angular momentum conservation in the muon rest frame? The following is what I thought reasonable. The orbital angular momentum of the three daughter particles in the muon rest frame are zero, so only the spin angular momentum should be conserved before/after the decay.

$\mathbf{s}_{\mu^+} = \mathbf{s}_{e^+} + \mathbf{s}_{\nu_e} + \mathbf{s}_{\bar{\nu}_\mu}$

Also, the neutrino and anti-neutrino are assumed to be left-handed and right-handed respectfully. Decay positron is also preferred to be right-handed. That means,

$\mathbf{s}_{\mu^+} = \frac{\hbar}{2} \left( \hat{p}_{e^+} - \hat{p}_{\nu_e} + \hat{p}_{\bar{\nu}_\mu} \right)$

holds, where $\hat{p}$ is a momentum unit vector.

Hence, together with the momentum conservation

$0 = \mathbf{p}_{e^+} + \mathbf{p}_{\nu_e} + \mathbf{p}_{\bar{\nu}_\mu}$,

and the energy conservation,

$m_{\mu^+} = E_{e^+} + E_{\nu_e} + E_{\bar{\nu}_\mu}$,

Could we solve for the relationships between the momentum of the decay positron in terms of the momenta of the neutrinos? Or prove the Michel parameters, something?

My question is that, is this whole argument correct?


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