Particles, Particles, Particles — Question 4

Question 4

A decay-length distribution reveals the particle proper lifetime

A particle beam has $pc=3.00$ GeV and $mc^2=1.00$ GeV. The mean measured decay length is 9.00 cm. Use c=$3.00\times10^8$ m/s and an exponential proper-time law.

1. Infer the proper mean lifetime.

2. Find the fraction decaying between 3.00 cm and 12.0 cm from production.

Question 4 - Solution

A decay-length distribution reveals the particle proper lifetime

(a) Method and calculation

$$\bar L=\beta\gamma c\tau=\frac{pc}{mc^2}c\tau=3c\tau.$$$$\boxed{\tau=.0900/(3c)=1.00\times10^{-10}\,\mathrm s}.$$

(b) Interpretation and further calculation

$$P(a<L<b)=e^{-a/\bar L}-e^{-b/\bar L}.$$$$\boxed{P=e^{-1/3}-e^{-4/3}=.452934}.$$

Physical check

The mean flight distance depends on momentum. A broad momentum spectrum must be handled event by event rather than assigned a single boost.