The Leptons — Question 4

Question 4

Pion decay produces a nearly monoenergetic muon neutrino at rest

A positive pion at rest decays as $\pi^+\to\mu^++\nu_\mu$. Take $m_\pi c^2=140$ MeV, $m_\mu c^2=106$ MeV and zero neutrino mass.

1. Find the neutrino energy and muon total energy.

2. Find the muon kinetic energy and compare the two kinetic-energy shares.

Question 4 - Solution

Pion decay produces a nearly monoenergetic muon neutrino at rest

(a) Method and calculation

$$E_\mu+E_\nu=m_\pi c^2,\quad p_\mu c=E_\nu,\quad E_\mu^2=m_\mu^2c^4+E_\nu^2.$$$$\boxed{E_\nu=\frac{140^2-106^2}{2(140)}=29.87143\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{E_\mu=110.12857\,\mathrm{MeV},\quad K_\mu=4.12857\,\mathrm{MeV}}.$$$$\boxed{K_\mu+E_\nu=34.0\,\mathrm{MeV}=(m_\pi-m_\mu)c^2}.$$

Physical check

The neutrino receives most of the available kinetic energy; equal momenta do not imply equal kinetic energies. The pion is a hadron but the decay products are leptons.