Question Statement
Which of the following decay modes is FORBIDDEN by the conservation of individual lepton family numbers?
(A)
$\mu^- \to e^- + \gamma$
(B)
$\mu^- \to e^- + \bar{\nu}_e + \nu_\mu$
(C)
$\tau^- \to \mu^- + \bar{\nu}_\mu + \nu_\tau$
(D)
$n \to p + e^- + \bar{\nu}_e$
Step-by-Step KaTeX Derivation & Detailed Explanation
### Step-by-Step Derivation & Verification: 1. **Standard Model Lepton Family Numbers**: - Electron number: $L_e = +1$ for $(e^-, \nu_e)$, $-1$ for $(e^+, \bar{\nu}_e)$, $0$ for others. - Muon number: $L_\mu = +1$ for $(\mu^-, \nu_\mu)$, $-1$ for $(\mu^+, \bar{\nu}_\mu)$, $0$ for others. - Tau number: $L_\tau = +1$ for $(\tau^-, \nu_\tau)$, $-1$ for $(\tau^+, \bar{\nu}_\tau)$, $0$ for others. 2. **Inspection of Process $\mu^- \to e^- + \gamma$**: - Initial: $L_\mu = +1, \; L_e = 0$ - Final: $L_\mu = 0, \; L_e = +1$ - $\Delta L_\mu = -1 \neq 0$, $\Delta L_e = +1 \neq 0$ (VIOLATED) 3. **Inspection of Other Options**: - $\mu^- \to e^- \bar{\nu}_e \nu_\mu$: $L_\mu: 1 \to 0 + 0 + 1 = 1$; $L_e: 0 \to 1 - 1 + 0 = 0$ (CONSERVED) - $\tau^- \to \mu^- \bar{\nu}_\mu \nu_\tau$: $L_\tau: 1 \to 1$; $L_\mu: 0 \to 1 - 1 = 0$ (CONSERVED) - $n \to p e^- \bar{\nu}_e$: $L_e: 0 \to 1 - 1 = 0$ (CONSERVED) 4. **Conclusion**: $\mu^- \to e^- + \gamma$ is forbidden. Option (A) is correct.