The Leptons — Question 1

Question 1

Muon decay balances two separate flavor labels at the weak vertex

Use lepton-family numbers +1 for a charged lepton or its neutrino, -1 for their antiparticles, and zero for other families. Consider $\mu^-\to e^-+\bar\nu_e+\nu_\mu$. Ignore oscillation during this local decay.

1. Check electric charge and both electron- and muon-family numbers.

2. Replace the final electron antineutrino by an electron neutrino and identify the violated number.

Question 1 - Solution

Muon decay balances two separate flavor labels at the weak vertex

(a) Method and calculation

$$\boxed{Q_i=-e=Q_f,\quad L_{e,i}=0=1-1+0}.$$$$\boxed{L_{\mu,i}=1=0+0+1=L_{\mu,f}}.$$

(b) Interpretation and further calculation

$$e^-+\nu_e+\nu_\mu:\quad\boxed{L_{e,f}=1+1=2\ne0}.$$$$\boxed{L_{\mathrm{total,f}}=3\ne L_{\mathrm{total,i}}=1}.$$

Physical check

These flavor labels track the charged-current process. Neutrino propagation with mixing does not conserve each separate flavor population.