Considering the given information:
Given reaction is\({\mu ^ - } \to {e^ - } + {\nu _e} + {\nu _\mu }\)
We can check the following conservation laws to see if the decay\({\mu ^ - } \to {e^ - } + {\nu _e} + {\nu _\mu }\)is possible.
Decay:\({\mu ^ - } \to {e^ - } + {\nu _e} + {\nu _\mu }\)
Charge:\( - 1 \to - 1 + 0 + 0\)
\(\therefore \)the charge is conserved.
\({\rm{Baryon}}\;{\rm{number}}:0 \to 0 + 0 + 0\)
\(\therefore \)Baryon number is conserved
\({\rm{Electron}}\;{\rm{Lepton}}\;{\rm{number}}\;\left( {{L_e}} \right):0 \to + 1 + 1 + 0\)
\(\therefore \)electron lepton number is not conserved
\({\rm{Muon}}\;{\rm{Lepton}}\;{\rm{number}}\;\left( {{L_\mu }} \right): + 1 \to 0 + 0 + 1\)
\(\therefore \)muon lepton number is conserved
As a result, even though charge, muon lepton number\(\left( {{{\rm{L}}_{\rm{\mu }}}} \right)\), and baryon number are all conserved, the electron lepton number\(\left( {{{\rm{L}}_e}} \right)\)is not.
For the given decay,\(\left( {{{\rm{L}}_e}} \right)\)is not conserved. As a result, according to conservation laws, decay is not possible.
Therefore, the required decay\({\mu ^ - } \to {e^ - } + {\nu _e} + {\nu _\mu }\)is not possible as per the conservation laws.
