The Leptons — Question 9

Question 9

A supplied weak-decay scaling predicts a steep mass dependence

For two hypothetical charged leptons with the same weak coupling and negligible daughter masses, suppose $\Gamma=Cm^5$. Their masses satisfy $m_2=3m_1$. The lighter lepton mean life is $2.00\,\mu\mathrm s$.

1. Predict the heavier lepton mean life.

2. If an additional channel doubles the heavier particle total width, find its lifetime and the branching fraction of the original channel.

Question 9 - Solution

A supplied weak-decay scaling predicts a steep mass dependence

(a) Method and calculation

$$\Gamma_2/\Gamma_1=(m_2/m_1)^5=3^5=243.$$$$\boxed{\tau_2=\tau_1/243=8.23045\,\mathrm{ns}}.$$

(b) Interpretation and further calculation

$$\Gamma_{2,\mathrm{total}}=2\Gamma_2\Rightarrow\boxed{\tau_{2,\mathrm{total}}=4.11523\,\mathrm{ns}}.$$$$\boxed{b_{\mathrm{original}}=\Gamma_2/(2\Gamma_2)=1/2}.$$

Physical check

The scaling is a phase-space approximation. Real charged leptons have daughter-mass corrections, radiative corrections and additional channels.