Still Another Conservation Law — Question 5

Question 5

A cascade removes strangeness in separate weak steps

Consider $\Xi^-\to\Lambda+\pi^-$ followed by $\Lambda\to p+\pi^-$. Assign S=-2,-1,0 to Xi, Lambda and proton, and S=0 to pions. Proper waiting times in the two rest frames are independent exponential variables with means 1.60 and 2.60 units of $10^{-10}$ s.

1. Write the net final state and the strangeness change in each step.

2. For the sum of the two proper waiting-time variables, find mean and standard deviation. Explain why it is not automatically a laboratory transit time.

Question 5 - Solution

A cascade removes strangeness in separate weak steps

(a) Method and calculation

$$\boxed{\Xi^-\to p+2\pi^-,\quad\Delta S_1=+1,\quad\Delta S_2=+1}.$$$$Q/e:\ -1=1-2,\quad B:\ 1=1,\quad\boxed{\Delta S_{\mathrm{net}}=+2}.$$

(b) Interpretation and further calculation

$$\boxed{\langle T_1+T_2\rangle=(1.60+2.60)10^{-10}=4.20\times10^{-10}\,\mathrm s}.$$$$\boxed{\sigma_{T_1+T_2}=\sqrt{1.60^2+2.60^2}\,10^{-10}=3.05287\times10^{-10}\,\mathrm s}.$$

Physical check

The two proper times refer to different moving particles. Laboratory durations require their separate Lorentz factors.