The Leptons — Question 3

Question 3

The muon-decay endpoint follows from the smallest missing invariant mass

A muon at rest decays to an electron and two effectively massless neutrinos. Use $m_\mu c^2=105.66$ MeV and $m_ec^2=.511$ MeV. Treat the neutrino pair as a composite system of invariant mass $M_{\nu\nu}$.

1. Derive electron total energy in terms of $M_{\nu\nu}$ and find its maximum.

2. Find the maximum electron kinetic energy and explain why the electron generally has a continuous energy spectrum.

Question 3 - Solution

The muon-decay endpoint follows from the smallest missing invariant mass

(a) Method and calculation

$$M_{\nu\nu}^2c^4=(m_\mu c^2-E_e)^2-p_e^2c^2=m_\mu^2c^4+m_e^2c^4-2m_\mu c^2E_e.$$$$\boxed{E_{e,\max}=\frac{m_\mu^2+m_e^2}{2m_\mu}c^2=52.83124\,\mathrm{MeV}}\quad(M_{\nu\nu}=0).$$

(b) Interpretation and further calculation

$$\boxed{K_{e,\max}=E_{e,\max}-m_ec^2=52.32024\,\mathrm{MeV}}.$$$$M_{\nu\nu}\text{ varies}\Rightarrow E_e\text{ varies continuously between allowed limits}.$$

Physical check

The endpoint is a limiting collinear-neutrino configuration. A two-body decay would instead fix the electron energy in the parent rest frame.